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1868 1866 y(CLN)23 b(relies)h(on)g(semi-automatic)f(ordering)h(of)f
1869 (initializations)j(of)e(static)h(and)e(global)i(v)-5
1870 b(ariables,)25 b(a)f(feature)g(whic)m(h)-30 1976 y(I)30
1871 b(could)h(implemen)m(t)e(for)h(GNU)h(g)p Fp(++)f Fr(only)-8
1872 b(.)-30 2196 y Fn(2.1.2)63 b(Mak)m(e)40 b(utilit)m(y)-30
1873 2387 y Fr(T)-8 b(o)31 b(build)e(CLN,)h(y)m(ou)h(also)g(need)g(to)g(ha)m
1874 (v)m(e)g(GNU)g Fp(make)e Fr(installed.)-30 2607 y Fn(2.1.3)63
1875 b(Sed)41 b(utilit)m(y)-30 2798 y Fr(T)-8 b(o)39 b(build)e(CLN)h(on)g
1876 (HP-UX,)h(y)m(ou)g(also)g(need)f(to)h(ha)m(v)m(e)g(GNU)g
1877 Fp(sed)e Fr(installed.)66 b(This)37 b(is)h(b)s(ecause)h(the)f(libto)s
1878 (ol)-30 2907 y(script,)29 b(whic)m(h)g(creates)h(the)e(CLN)h(library)-8
1879 b(,)29 b(relies)g(on)g Fp(sed)p Fr(,)f(and)g(the)h(v)m(endor's)g
1880 Fp(sed)f Fr(utilit)m(y)i(on)e(these)h(systems)f(is)-30
1881 3017 y(to)s(o)j(limited.)-30 3270 y Fs(2.2)68 b(Building)45
1882 b(the)g(library)-30 3461 y Fr(As)30 b(with)g(an)m(y)h(auto)s
1883 (con\014guring)f(GNU)h(soft)m(w)m(are,)h(installation)h(is)d(as)h(easy)
1884 g(as)f(this:)210 3589 y Fp($)47 b(./configure)210 3692
1885 y($)g(make)210 3796 y($)g(make)g(check)-30 3929 y Fr(If)30
1886 b(on)g(y)m(our)g(system,)g(`)p Fp(make)p Fr(')g(is)g(not)h(GNU)g
1887 Fp(make)p Fr(,)e(y)m(ou)i(ha)m(v)m(e)h(to)f(use)f(`)p
1888 Fp(gmake)p Fr(')f(instead)i(of)g(`)p Fp(make)p Fr(')e(ab)s(o)m(v)m(e.)
1889 -30 4062 y(The)39 b Fp(configure)d Fr(command)h(c)m(hec)m(ks)k(out)e
1890 (some)g(features)g(of)h(y)m(our)f(system)f(and)h(C)p
1891 Fp(++)f Fr(compiler)h(and)f(builds)-30 4172 y(the)i Fp(Makefile)p
1892 Fr(s.)66 b(The)39 b Fp(make)f Fr(command)g(builds)g(the)i(library)-8
1893 b(.)68 b(This)39 b(step)h(ma)m(y)f(tak)m(e)i(4)f(hours)e(on)i(an)f(a)m
1894 (v)m(er-)-30 4281 y(age)d(w)m(orkstation.)57 b(The)35
1895 b Fp(make)29 b(check)34 b Fr(runs)f(some)i(test)h(to)g(c)m(hec)m(k)g
1896 (that)g(no)f(imp)s(ortan)m(t)g(subroutine)f(has)h(b)s(een)-30
1897 4391 y(miscompiled.)-30 4524 y(The)30 b Fp(configure)e
1898 Fr(command)g(accepts)k(options.)41 b(T)-8 b(o)30 b(get)i(a)f(summary)c
1899 (of)j(them,)g(try)210 4651 y Fp($)47 b(./configure)e(--help)-30
1900 4784 y Fr(Some)29 b(of)i(the)g(options)f(are)h(explained)g(in)f(detail)
1901 h(in)f(the)h(`)p Fp(INSTALL.generic)p Fr(')26 b(\014le.)-30
1902 4917 y(Y)-8 b(ou)21 b(can)g(sp)s(ecify)f(the)g(C)g(compiler,)i(the)f(C)
1903 p Fp(++)e Fr(compiler)h(and)g(their)g(options)h(through)f(the)g(follo)m
1904 (wing)i(en)m(vironmen)m(t)-30 5027 y(v)-5 b(ariables)31
1905 b(when)e(running)g Fp(configure)p Fr(:)-30 5183 y Fp(CC)384
1906 b Fr(Sp)s(eci\014es)30 b(the)g(C)g(compiler.)-30 5340
1907 y Fp(CFLAGS)192 b Fr(Flags)32 b(to)f(b)s(e)f(giv)m(en)h(to)g(the)g(C)f
1908 (compiler)f(when)h(compiling)g(programs)f(\(not)i(when)e(linking\).)p
1911 4 5 bop -30 -116 a Fr(Chapter)30 b(2:)41 b(Installation)2999
1912 b(4)-30 299 y Fp(CXX)336 b Fr(Sp)s(eci\014es)30 b(the)g(C)p
1913 Fp(++)g Fr(compiler.)-30 452 y Fp(CXXFLAGS)96 b Fr(Flags)32
1914 b(to)f(b)s(e)f(giv)m(en)h(to)g(the)g(C)p Fp(++)e Fr(compiler)h(when)f
1915 (compiling)h(programs)f(\(not)i(when)f(linking\).)-30
1916 604 y(Examples:)210 730 y Fp($)47 b(CC="gcc")f(CFLAGS="-O")f(CXX="g++")
1917 g(CXXFLAGS="-O")f(./configure)210 834 y($)j(CC="gcc)f(-V)h
1918 (egcs-2.91.60")e(CFLAGS="-O)g(-g")i(\\)305 937 y(CXX="g++)f(-V)h
1919 (egcs-2.91.60")d(CXXFLAGS="-O)h(-g")i(./configure)210
1920 1041 y($)g(CC="gcc)f(-V)h(2.95.2")f(CFLAGS="-O2)f(-fno-exceptions")e
1921 (\\)305 1145 y(CXX="g++)j(-V)h(2.95.2")f(CFLAGS="-O2)f
1922 (-fno-exceptions")e(./configure)-30 1276 y Fr(Note)32
1923 b(that)g(for)e(these)i(en)m(vironmen)m(t)e(v)-5 b(ariables)32
1924 b(to)f(tak)m(e)i(e\013ect,)g(y)m(ou)e(ha)m(v)m(e)h(to)f(set)h(them)e
1925 (\(assuming)g(a)h(Bourne-)-30 1386 y(compatible)i(shell\))g(on)g(the)g
1926 (same)f(line)h(as)g(the)g Fp(configure)d Fr(command.)46
1927 b(If)32 b(y)m(ou)h(made)f(the)h(settings)h(in)e(earlier)-30
1928 1495 y(shell)g(commands,)e(y)m(ou)i(ha)m(v)m(e)h(to)g
1929 Fp(export)d Fr(the)i(en)m(vironmen)m(t)f(v)-5 b(ariables)33
1930 b(b)s(efore)e(calling)i Fp(configure)p Fr(.)43 b(In)31
1931 b(a)h Fp(csh)-30 1605 y Fr(shell,)f(y)m(ou)g(ha)m(v)m(e)g(to)g(use)f
1932 (the)h(`)p Fp(setenv)p Fr(')e(command)f(for)i(setting)i(eac)m(h)f(of)g
1933 (the)f(en)m(vironmen)m(t)g(v)-5 b(ariables.)-30 1736
1934 y(Curren)m(tly)25 b(CLN)g(w)m(orks)h(only)g(with)f(the)h(GNU)g
1935 Fp(g++)f Fr(compiler,)i(and)e(only)g(in)h(optimizing)g(mo)s(de.)37
1936 b(So)26 b(y)m(ou)g(should)-30 1846 y(sp)s(ecify)33 b(at)i(least)g
1937 Fp(-O)e Fr(in)h(the)g(CXXFLA)m(GS,)g(or)g(no)g(CXXFLA)m(GS)g(at)h(all.)
1938 52 b(\(If)34 b(CXXFLA)m(GS)g(is)g(not)g(set,)h(CLN)-30
1939 1955 y(will)c(use)f Fp(-O)p Fr(.\))-30 2086 y(If)44 b(y)m(ou)h(use)f
1940 Fp(g++)g Fr(v)m(ersion)h(2.8.x)h(or)e(egcs-2.91.x)k(\(a.k.a.)85
1941 b(egcs-1.1\))47 b(or)e(gcc-2.95.x,)52 b(I)44 b(recommend)e(adding)-30
1942 2196 y(`)p Fp(-fno-exceptions)p Fr(')27 b(to)k(the)f(CXXFLA)m(GS.)i
1943 (This)d(will)i(lik)m(ely)h(generate)g(b)s(etter)e(co)s(de.)-30
1944 2327 y(If)37 b(y)m(ou)g(use)g Fp(g++)g Fr(v)m(ersion)g(egcs-2.91.x)k
1945 (\(egcs-1.1\))f(or)e(gcc-2.95.x)i(on)d(Sparc,)i(add)e(either)g(`)p
1946 Fp(-O)p Fr(',)i(`)p Fp(-O1)p Fr(')e(or)h(`)p Fp(-O2)-30
1947 2437 y(-fno-schedule-insns)p Fr(')26 b(to)32 b(the)f(CXXFLA)m(GS.)h
1948 (With)g(full)f(`)p Fp(-O2)p Fr(',)g Fp(g++)f Fr(miscompiles)g(the)i
1949 (division)f(routines.)-30 2546 y(Also,)41 b(if)e(y)m(ou)g(ha)m(v)m(e)g
1950 Fp(g++)f Fr(v)m(ersion)h(egcs-1.1.1)j(or)c(older)h(on)f(Sparc,)j(y)m
1951 (ou)d(m)m(ust)g(sp)s(ecify)g(`)p Fp(--disable-shared)p
1952 Fr(')-30 2656 y(b)s(ecause)30 b Fp(g++)g Fr(w)m(ould)g(miscompile)f
1953 (parts)h(of)h(the)f(library)-8 b(.)-30 2787 y(By)32 b(default,)g(b)s
1954 (oth)f(a)h(shared)e(and)h(a)h(static)h(library)e(are)h(built.)44
1955 b(Y)-8 b(ou)32 b(can)g(build)f(CLN)g(as)g(a)h(static)h(\(or)f(shared\))
1956 -30 2897 y(library)k(only)-8 b(,)38 b(b)m(y)f(calling)g
1957 Fp(configure)d Fr(with)i(the)h(option)f(`)p Fp(--disable-shared)p
1958 Fr(')d(\(or)j(`)p Fp(--disable-static)p Fr('\).)-30 3006
1959 y(While)f(shared)e(libraries)g(are)i(usually)e(more)g(con)m(v)m(enien)m
1960 (t)j(to)e(use,)h(they)f(ma)m(y)f(not)h(w)m(ork)g(on)g(all)g(arc)m
1961 (hitectures.)-30 3116 y(T)-8 b(ry)32 b(disabling)g(them)g(if)g(y)m(ou)h
1962 (run)d(in)m(to)k(link)m(er)e(problems.)45 b(Also,)34
1963 b(they)f(are)f(generally)i(somewhat)e(slo)m(w)m(er)h(than)-30
1964 3225 y(static)f(libraries)e(so)h(run)m(time-critical)h(applications)f
1965 (should)e(b)s(e)h(link)m(ed)h(statically)-8 b(.)-30 3440
1966 y Fn(2.2.1)63 b(Using)42 b(the)f(GNU)g(MP)h(Library)-30
1967 3629 y Fr(Starting)32 b(with)f(v)m(ersion)g(1.1,)i(CLN)e(ma)m(y)g(b)s
1968 (e)f(con\014gured)h(to)h(mak)m(e)f(use)g(of)g(a)h(preinstalled)f
1969 Fp(gmp)g Fr(library)-8 b(.)43 b(Please)-30 3738 y(mak)m(e)d(sure)f
1970 (that)h(y)m(ou)g(ha)m(v)m(e)h(at)f(least)h Fp(gmp)e Fr(v)m(ersion)h
1971 (3.0)h(installed)f(since)g(earlier)h(v)m(ersions)f(are)g(unsupp)s
1972 (orted)-30 3848 y(and)30 b(lik)m(ely)i(not)f(to)h(w)m(ork.)41
1973 b(Enabling)31 b(this)f(feature)h(b)m(y)g(calling)h Fp(configure)c
1974 Fr(with)j(the)f(option)i(`)p Fp(--with-gmp)p Fr(')c(is)-30
1975 3958 y(kno)m(wn)i(to)h(b)s(e)f(quite)h(a)f(b)s(o)s(ost)g(for)g(CLN's)g
1976 (p)s(erformance.)-30 4089 y(If)35 b(y)m(ou)g(ha)m(v)m(e)i(installed)f
1977 (the)f Fp(gmp)f Fr(library)h(and)f(its)i(header)f(\014le)g(in)g(some)g
1978 (place)h(where)e(y)m(our)i(compiler)e(cannot)-30 4198
1979 y(\014nd)27 b(it)j(b)m(y)f(default,)h(y)m(ou)f(m)m(ust)f(help)h
1980 Fp(configure)d Fr(b)m(y)j(setting)h Fp(CPPFLAGS)d Fr(and)h
1981 Fp(LDFLAGS)p Fr(.)38 b(Here)30 b(is)f(an)g(example:)210
1982 4324 y Fp($)47 b(CC="gcc")f(CFLAGS="-O2")e(CXX="g++")i(CXXFLAGS="-O2)e
1983 (-fno-exceptions")f(\\)305 4428 y(CPPFLAGS="-I/opt/gmp/inclu)o(de")e
1984 (LDFLAGS="-L/opt/gmp/lib")g(./configure)k(--with-gmp)-30
1985 4675 y Fs(2.3)68 b(Installing)47 b(the)e(library)-30
1986 4864 y Fr(As)30 b(with)g(an)m(y)h(auto)s(con\014guring)f(GNU)h(soft)m
1987 (w)m(are,)h(installation)h(is)d(as)h(easy)g(as)f(this:)210
1988 4990 y Fp($)47 b(make)g(install)-30 5121 y Fr(The)66
1989 b(`)p Fp(make)30 b(install)p Fr(')65 b(command)f(installs)k(the)f
1990 (library)f(and)h(the)g(include)f(\014les)h(in)m(to)h(public)e(places)
1991 -30 5230 y(\(`)p Fp(/usr/local/lib/)p Fr(')41 b(and)j(`)p
1992 Fp(/usr/local/include/)p Fr(',)f(if)i(y)m(ou)g(ha)m(v)m(en't)g(sp)s
1993 (eci\014ed)f(a)h Fp(--prefix)d Fr(option)j(to)-30 5340
1994 y Fp(configure)p Fr(\).)39 b(This)29 b(step)i(ma)m(y)f(require)g(sup)s
1995 (eruser)e(privileges.)p eop
1997 5 6 bop -30 -116 a Fr(Chapter)30 b(2:)41 b(Installation)2999
1998 b(5)-30 299 y(If)42 b(y)m(ou)g(ha)m(v)m(e)i(already)f(built)f(the)g
1999 (library)g(and)g(wish)f(to)i(install)g(it,)j(but)c(didn't)f(sp)s(ecify)
2000 h Fp(--prefix=...)d Fr(at)-30 408 y(con\014gure)j(time,)k(just)c
2001 (re-run)f Fp(configure)p Fr(,)i(giving)h(it)f(the)g(same)f(options)g
2002 (as)h(the)g(\014rst)f(time,)j(plus)d(the)h Fp(--)-30
2003 518 y(prefix=...)28 b Fr(option.)-30 775 y Fs(2.4)68
2004 b(Cleaning)46 b(up)-30 968 y Fr(Y)-8 b(ou)31 b(can)g(remo)m(v)m(e)f
2005 (system-dep)s(enden)m(t)f(\014les)i(generated)g(b)m(y)f
2006 Fp(make)f Fr(through)210 1097 y Fp($)47 b(make)g(clean)-30
2007 1231 y Fr(Y)-8 b(ou)31 b(can)g(remo)m(v)m(e)f(all)i(\014les)e
2008 (generated)h(b)m(y)g Fp(make)p Fr(,)e(th)m(us)h(rev)m(erting)h(to)h(a)e
2009 (virgin)h(distribution)e(of)i(CLN,)f(through)210 1360
2010 y Fp($)47 b(make)g(distclean)p eop
2012 6 7 bop -30 -116 a Fr(Chapter)30 b(3:)41 b(Ordinary)29
2013 b(n)m(um)m(b)s(er)f(t)m(yp)s(es)2517 b(6)-30 299 y Fo(3)80
2014 b(Ordinary)55 b(n)l(um)l(b)t(er)c(t)l(yp)t(es)-30 539
2015 y Fr(CLN)30 b(implemen)m(ts)f(the)h(follo)m(wing)i(class)f(hierarc)m(h)
2016 m(y:)1355 670 y Fp(Number)1260 773 y(cl_number)1165 877
2017 y(<cln/number.h>)1451 981 y(|)1451 1085 y(|)1021 1189
2018 y(Real)47 b(or)g(complex)f(number)1355 1292 y(cl_N)1165
2019 1396 y(<cln/complex.h>)1451 1500 y(|)1451 1604 y(|)1212
2020 1707 y(Real)h(number)1355 1811 y(cl_R)1212 1915 y(<cln/real.h>)1451
2021 2019 y(|)496 2123 y(+-------------------+-----)o(---)o(----)o(----)o
2022 (---)o(+)496 2226 y(|)1861 b(|)210 2330 y(Rational)46
2023 b(number)1000 b(Floating-point)44 b(number)401 2434 y(cl_RA)1669
2024 b(cl_F)210 2538 y(<cln/rational.h>)1189 b(<cln/float.h>)496
2025 2641 y(|)1861 b(|)496 2745 y(|)764 b(+--------------+-------)o(----)o
2026 (---+)o(---)o(----)o(----)o(---)o(+)353 2849 y(Integer)619
2027 b(|)668 b(|)g(|)f(|)401 2953 y(cl_I)476 b(Short-Float)188
2028 b(Single-Float)140 b(Double-Float)188 b(Long-Float)210
2029 3057 y(<cln/integer.h>)282 b(cl_SF)476 b(cl_FF)g(cl_DF)g(cl_LF)1021
2030 3160 y(<cln/sfloat.h>)44 b(<cln/ffloat.h>)g(<cln/dfloat.h>)g
2031 (<cln/lfloat.h>)-30 3297 y Fr(The)28 b(base)h(class)h
2032 Fp(cl_number)c Fr(is)j(an)g(abstract)g(base)g(class.)41
2033 b(It)29 b(is)g(not)g(useful)f(to)i(declare)f(a)h(v)-5
2034 b(ariable)29 b(of)g(this)g(t)m(yp)s(e)-30 3406 y(except)g(if)f(y)m(ou)g
2035 (w)m(an)m(t)h(to)g(completely)f(disable)g(compile-time)g(t)m(yp)s(e)h
2036 (c)m(hec)m(king)g(and)f(use)f(run-time)g(t)m(yp)s(e)h(c)m(hec)m(king)
2037 -30 3516 y(instead.)-30 3653 y(The)h(class)i Fp(cl_N)d
2038 Fr(comprises)h(real)h(and)f(complex)g(n)m(um)m(b)s(ers.)39
2039 b(There)29 b(is)g(no)h(sp)s(ecial)g(class)h(for)e(complex)g(n)m(um)m(b)
2040 s(ers)-30 3762 y(since)i(complex)f(n)m(um)m(b)s(ers)e(with)i(imaginary)
2041 g(part)g Fp(0)g Fr(are)h(automatically)h(con)m(v)m(erted)g(to)f(real)g
2042 (n)m(um)m(b)s(ers.)-30 3899 y(The)f(class)h Fp(cl_R)e
2043 Fr(comprises)h(real)h(n)m(um)m(b)s(ers)d(of)i(di\013eren)m(t)h(kinds.)
2044 40 b(It)30 b(is)h(an)f(abstract)h(class.)-30 4035 y(The)37
2045 b(class)g Fp(cl_RA)f Fr(comprises)g(exact)j(real)f(n)m(um)m(b)s(ers:)51
2046 b(rational)39 b(n)m(um)m(b)s(ers,)d(including)h(in)m(tegers.)62
2047 b(There)37 b(is)g(no)-30 4145 y(sp)s(ecial)31 b(class)f(for)g(non-in)m
2048 (tegral)i(rational)f(n)m(um)m(b)s(ers)d(since)i(rational)h(n)m(um)m(b)s
2049 (ers)d(with)i(denominator)f Fp(1)h Fr(are)g(auto-)-30
2050 4254 y(matically)h(con)m(v)m(erted)h(to)f(in)m(tegers.)-30
2051 4391 y(The)f(class)h Fp(cl_F)e Fr(implemen)m(ts)g(\015oating-p)s(oin)m
2052 (t)j(appro)m(ximations)e(to)h(real)g(n)m(um)m(b)s(ers.)38
2053 b(It)31 b(is)f(an)g(abstract)i(class.)-30 4653 y Fs(3.1)68
2054 b(Exact)46 b(n)l(um)l(b)t(ers)-30 4848 y Fr(Some)24 b(n)m(um)m(b)s(ers)
2055 f(are)j(represen)m(ted)g(as)f(exact)i(n)m(um)m(b)s(ers:)36
2056 b(there)26 b(is)f(no)h(loss)f(of)h(information)f(when)f(suc)m(h)h(a)h
2057 (n)m(um)m(b)s(er)-30 4957 y(is)43 b(con)m(v)m(erted)i(from)d(its)h
2058 (mathematical)h(v)-5 b(alue)44 b(to)g(its)f(in)m(ternal)h(represen)m
2059 (tation.)80 b(On)43 b(exact)i(n)m(um)m(b)s(ers,)f(the)-30
2060 5067 y(elemen)m(tary)31 b(op)s(erations)g(\()p Fp(+)p
2061 Fr(,)f Fp(-)p Fr(,)g Fp(*)p Fr(,)h Fp(/)p Fr(,)f(comparisons,)35
2062 b(.)23 b(.)f(.)11 b(\))31 b(compute)e(the)i(completely)g(correct)h
2063 (result.)-30 5203 y(In)e(CLN,)g(the)g(exact)i(n)m(um)m(b)s(ers)c(are:)
2064 45 5340 y Fq(\017)60 b Fr(rational)32 b(n)m(um)m(b)s(ers)c(\(including)
2065 i(in)m(tegers\),)p eop
2067 7 8 bop -30 -116 a Fr(Chapter)30 b(3:)41 b(Ordinary)29
2068 b(n)m(um)m(b)s(er)f(t)m(yp)s(es)2517 b(7)45 299 y Fq(\017)60
2069 b Fr(complex)30 b(n)m(um)m(b)s(ers)e(whose)i(real)h(and)f(imaginary)g
2070 (parts)g(are)h(b)s(oth)e(rational)j(n)m(um)m(b)s(ers.)-30
2071 463 y(Rational)g(n)m(um)m(b)s(ers)27 b(are)k(alw)m(a)m(ys)h(normalized)
2072 e(to)g(the)h(form)e Fl(n)m(umerator)7 b Fp(/)p Fl(denominator)34
2073 b Fr(where)29 b(the)i(n)m(umerator)-30 573 y(and)c(denominator)h(are)g
2074 (coprime)f(in)m(tegers)j(and)d(the)h(denominator)g(is)g(p)s(ositiv)m
2075 (e.)41 b(If)27 b(the)i(resulting)f(denominator)-30 682
2076 y(is)i Fp(1)p Fr(,)h(the)f(rational)i(n)m(um)m(b)s(er)c(is)i(con)m(v)m
2077 (erted)i(to)f(an)g(in)m(teger.)-30 820 y(Small)g(in)m(tegers)i(\(t)m
2078 (ypically)h(in)d(the)h(range)g Fp(-2^30)6 b Fr(.)20 b(.)j(.)11
2079 b Fp(2^30-1)p Fr(,)30 b(for)i(32-bit)h(mac)m(hines\))e(are)h(esp)s
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2083 (\\\014xn)m(ums"\))d(and)g(heap)i(allo)s(cated)h(in)m(tegers)g
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2097 Fr(,)h Fp(-)p Fr(,)f Fp(*)p Fr(,)g Fp(/)p Fr(,)35 b(.)22
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2100 y(of)36 b(the)g(expression)f Fp(\(cl_F\))29 b(0.3)g(+)i(\(cl_F\))d(0.4)
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2114 Fp(0)g Fr(in)g(its)h(least)h(signi\014can)m(t)f(man)m(tissa)f(bit)h(is)
2115 f(c)m(hosen.)-30 2791 y(Similarly)-8 b(,)26 b(testing)h(\015oating)f(p)
2116 s(oin)m(t)f(n)m(um)m(b)s(ers)e(for)i(equalit)m(y)i(`)p
2117 Fp(x)j(==)g(y)p Fr(')25 b(is)g(gam)m(bling)h(with)f(random)e(errors.)38
2118 b(Better)-30 2901 y(c)m(hec)m(k)32 b(for)e(`)p Fp(abs\(x)f(-)h(y\))g(<)
2119 g(epsilon)p Fr(')f(for)h(some)g(w)m(ell-c)m(hosen)i Fp(epsilon)p
2120 Fr(.)-30 3039 y(Floating)g(p)s(oin)m(t)f(n)m(um)m(b)s(ers)d(come)i(in)g
2121 (four)g(\015a)m(v)m(ors:)45 3176 y Fq(\017)60 b Fr(Short)28
2122 b(\015oats,)i(t)m(yp)s(e)g Fp(cl_SF)p Fr(.)38 b(They)29
2123 b(ha)m(v)m(e)h(1)f(sign)g(bit,)h(8)f(exp)s(onen)m(t)g(bits)g
2124 (\(including)g(the)g(exp)s(onen)m(t's)g(sign\),)150 3286
2125 y(and)h(17)h(man)m(tissa)f(bits)h(\(including)f(the)g(\\hidden")g
2126 (bit\).)41 b(They)30 b(don't)h(consume)e(heap)h(allo)s(cation.)45
2127 3422 y Fq(\017)60 b Fr(Single)43 b(\015oats,)k(t)m(yp)s(e)c
2128 Fp(cl_FF)p Fr(.)78 b(They)42 b(ha)m(v)m(e)i(1)g(sign)f(bit,)j(8)e(exp)s
2129 (onen)m(t)f(bits)g(\(including)g(the)g(exp)s(onen)m(t's)150
2130 3532 y(sign\),)29 b(and)e(24)i(man)m(tissa)e(bits)h(\(including)g(the)g
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2133 (um)m(b)s(ers.)39 b(This)29 b(corresp)s(onds)g(closely)j(to)f(the)g
2134 (C/C)p Fp(++)e Fr(t)m(yp)s(e)i(`)p Fp(float)p Fr('.)45
2135 3777 y Fq(\017)60 b Fr(Double)38 b(\015oats,)i(t)m(yp)s(e)e
2136 Fp(cl_DF)p Fr(.)61 b(They)37 b(ha)m(v)m(e)i(1)f(sign)f(bit,)j(11)e(exp)
2137 s(onen)m(t)g(bits)g(\(including)f(the)h(exp)s(onen)m(t's)150
2138 3887 y(sign\),)29 b(and)e(53)i(man)m(tissa)e(bits)h(\(including)g(the)g
2139 (\\hidden")f(bit\).)40 b(In)27 b(CLN,)h(they)g(are)g(represen)m(ted)g
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2141 (um)m(b)s(ers.)38 b(This)30 b(corresp)s(onds)f(closely)j(to)f(the)g
2142 (C/C)p Fp(++)e Fr(t)m(yp)s(e)i(`)p Fp(double)p Fr('.)45
2143 4133 y Fq(\017)60 b Fr(Long)27 b(\015oats,)i(t)m(yp)s(e)e
2144 Fp(cl_LF)p Fr(.)38 b(They)27 b(ha)m(v)m(e)h(1)g(sign)f(bit,)h(32)g(exp)
2145 s(onen)m(t)f(bits)g(\(including)g(the)g(exp)s(onen)m(t's)g(sign\),)150
2146 4242 y(and)k(n)g(man)m(tissa)g(bits)h(\(including)f(the)h(\\hidden")f
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2148 (a)g(long)g(\015oat)150 4352 y(is)e(unlimited,)g(but)f(once)i(created,)
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2150 b(\(No)32 b(\\lazy)g(recomputation".\))-30 4516 y(Of)f(course,)i
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2154 (features)g(lik)m(e)i(NaNs,)j(denormalized)44 b(n)m(um)m(b)s(ers)f(and)
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2159 (with)g(larger)h(exp)s(onen)m(t)g(range.)-30 5121 y(As)37
2160 b(a)g(user)e(of)i(CLN,)g(y)m(ou)g(can)g(forget)g(ab)s(out)f(the)h
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2162 (t)i(t)m(yp)s(es)f(and)-30 5230 y(just)i(declare)h(all)g(y)m(our)g
2163 (\015oating-p)s(oin)m(t)g(v)-5 b(ariables)40 b(as)g(b)s(eing)f(of)g(t)m
2164 (yp)s(e)h Fp(cl_F)p Fr(.)66 b(This)39 b(has)g(the)g(adv)-5
2165 b(an)m(tage)41 b(that)-30 5340 y(when)30 b(y)m(ou)i(c)m(hange)h(the)f
2166 (precision)f(of)h(some)f(computation)g(\(sa)m(y)-8 b(,)34
2167 b(from)c Fp(cl_DF)g Fr(to)i Fp(cl_LF)p Fr(\),)f(y)m(ou)h(don't)f(ha)m
2170 8 9 bop -30 -116 a Fr(Chapter)30 b(3:)41 b(Ordinary)29
2171 b(n)m(um)m(b)s(er)f(t)m(yp)s(es)2517 b(8)-30 299 y(c)m(hange)34
2172 b(the)g(co)s(de,)g(only)f(the)g(precision)h(of)f(the)g(initial)i(v)-5
2173 b(alues.)49 b(Also,)35 b(man)m(y)d(transcenden)m(tal)i(functions)f(ha)m
2174 (v)m(e)-30 408 y(b)s(een)f(declared)i(as)f(returning)f(a)h
2175 Fp(cl_F)f Fr(when)g(the)h(argumen)m(t)f(is)h(a)g Fp(cl_F)p
2176 Fr(,)g(but)f(suc)m(h)g(declarations)j(are)e(missing)-30
2177 518 y(for)f(the)h(t)m(yp)s(es)g Fp(cl_SF)p Fr(,)f Fp(cl_FF)p
2178 Fr(,)g Fp(cl_DF)p Fr(,)g Fp(cl_LF)p Fr(.)45 b(\(Suc)m(h)33
2179 b(declarations)h(w)m(ould)e(b)s(e)g(wrong)g(if)h(the)f(\015oating)i(p)s
2180 (oin)m(t)-30 628 y(con)m(tagion)f(rule)d(happ)s(ened)e(to)j(c)m(hange)h
2181 (in)e(the)h(future.\))-30 941 y Fs(3.3)68 b(Complex)46
2182 b(n)l(um)l(b)t(ers)-30 1156 y Fr(Complex)29 b(n)m(um)m(b)s(ers,)f(as)j
2183 (implemen)m(ted)e(b)m(y)h(the)g(class)h Fp(cl_N)p Fr(,)e(ha)m(v)m(e)j
2184 (a)f(real)f(part)h(and)e(an)h(imaginary)g(part,)h(b)s(oth)-30
2185 1266 y(real)42 b(n)m(um)m(b)s(ers.)70 b(A)41 b(complex)g(n)m(um)m(b)s
2186 (er)e(whose)i(imaginary)f(part)h(is)h(the)f(exact)h(n)m(um)m(b)s(er)d
2187 Fp(0)i Fr(is)g(automatically)-30 1375 y(con)m(v)m(erted)32
2188 b(to)f(a)g(real)g(n)m(um)m(b)s(er.)-30 1532 y(Complex)36
2189 b(n)m(um)m(b)s(ers)f(can)i(arise)h(from)d(real)j(n)m(um)m(b)s(ers)d
2190 (alone,)40 b(for)d(example)g(through)f(application)j(of)e
2191 Fp(sqrt)f Fr(or)-30 1642 y(transcenden)m(tal)31 b(functions.)-30
2192 1955 y Fs(3.4)68 b(Con)l(v)l(ersions)-30 2170 y Fr(Con)m(v)m(ersions)46
2193 b(from)e(an)m(y)h(class)i(to)f(an)m(y)f(its)h(sup)s(erclasses)f
2194 (\(\\base)h(classes")h(in)e(C)p Fp(++)g Fr(terminology\))h(is)f(done)
2195 -30 2280 y(automatically)-8 b(.)-30 2437 y(Con)m(v)m(ersions)28
2196 b(from)f(the)h(C)f(built-in)h(t)m(yp)s(es)g(`)p Fp(long)p
2197 Fr(')f(and)h(`)p Fp(unsigned)g(long)p Fr(')f(are)i(pro)m(vided)e(for)h
2198 (the)g(classes)h Fp(cl_I)p Fr(,)-30 2546 y Fp(cl_RA)p
2199 Fr(,)g Fp(cl_R)p Fr(,)h Fp(cl_N)f Fr(and)h Fp(cl_number)p
2200 Fr(.)-30 2703 y(Con)m(v)m(ersions)35 b(from)e(the)i(C)g(built-in)f(t)m
2201 (yp)s(es)h(`)p Fp(int)p Fr(')f(and)h(`)p Fp(unsigned)28
2202 b(int)p Fr(')34 b(are)h(pro)m(vided)g(for)f(the)h(classes)h
2203 Fp(cl_I)p Fr(,)-30 2813 y Fp(cl_RA)p Fr(,)30 b Fp(cl_R)p
2204 Fr(,)h Fp(cl_N)e Fr(and)i Fp(cl_number)p Fr(.)40 b(Ho)m(w)m(ev)m(er,)34
2205 b(these)e(con)m(v)m(ersions)g(emphasize)f(e\016ciency)-8
2206 b(.)44 b(Their)30 b(range)i(is)-30 2922 y(therefore)f(limited:)19
2207 3079 y Fq(\000)60 b Fr(The)30 b(con)m(v)m(ersion)i(from)c(`)p
2208 Fp(int)p Fr(')i(w)m(orks)h(only)f(if)h(the)f(argumen)m(t)g(is)g
2209 Fp(<)g Fr(2)p Fp(^)p Fr(29)i(and)e Fp(>)g Fr(-2)p Fp(^)p
2210 Fr(29.)19 3225 y Fq(\000)60 b Fr(The)30 b(con)m(v)m(ersion)i(from)c(`)p
2211 Fp(unsigned)h(int)p Fr(')h(w)m(orks)g(only)g(if)h(the)f(argumen)m(t)g
2212 (is)h Fp(<)f Fr(2)p Fp(^)p Fr(29.)-30 3418 y(In)j(a)h(declaration)i
2213 (lik)m(e)f(`)p Fp(cl_I)29 b(x)h(=)g(10;)p Fr(')j(the)h(C)p
2214 Fp(++)f Fr(compiler)g(is)h(able)h(to)f(do)g(the)g(con)m(v)m(ersion)h
2215 (of)f Fp(10)f Fr(from)f(`)p Fp(int)p Fr(')-30 3528 y(to)f(`)p
2216 Fp(cl_I)p Fr(')f(at)i(compile)e(time)g(already)-8 b(.)43
2217 b(On)30 b(the)g(other)h(hand,)f(co)s(de)h(lik)m(e)h(`)p
2218 Fp(cl_I)d(x)h(=)g(1000000000;)p Fr(')e(is)j(in)f(error.)-30
2219 3637 y(So,)f(if)f(y)m(ou)g(w)m(an)m(t)h(to)g(b)s(e)e(sure)g(that)i(an)f
2220 (`)p Fp(int)p Fr(')f(whose)h(magnitude)f(is)h(not)h(guaran)m(teed)g(to)
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2222 3747 y(con)m(v)m(erted)45 b(to)f(a)g(`)p Fp(cl_I)p Fr(',)i(\014rst)c
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2224 b(Similarly)-8 b(,)47 b(if)c(a)h(large)g(`)p Fp(unsigned)28
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2226 b(to)f(a)g(`)p Fp(cl_I)p Fr(',)f(\014rst)f(con)m(v)m(ert)j(it)f(to)g
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2228 b(from)e(the)i(C)g(built-in)f(t)m(yp)s(e)h(`)p Fp(float)p
2229 Fr(')f(are)h(pro)m(vided)f(for)h(the)g(classes)g Fp(cl_FF)p
2230 Fr(,)g Fp(cl_F)p Fr(,)f Fp(cl_R)p Fr(,)h Fp(cl_N)-30
2231 4123 y Fr(and)c Fp(cl_number)p Fr(.)-30 4280 y(Con)m(v)m(ersions)h
2232 (from)e(the)i(C)f(built-in)h(t)m(yp)s(e)g(`)p Fp(double)p
2233 Fr(')e(are)i(pro)m(vided)g(for)f(the)h(classes)h Fp(cl_DF)p
2234 Fr(,)d Fp(cl_F)p Fr(,)h Fp(cl_R)p Fr(,)g Fp(cl_N)-30
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2236 b(from)e(`)p Fp(const)29 b(char)g(*)p Fr(')40 b(are)g(pro)m(vided)f
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2238 Fp(cl_SF)p Fr(,)f Fp(cl_FF)p Fr(,)h Fp(cl_DF)p Fr(,)-30
2239 4656 y Fp(cl_LF)p Fr(,)26 b Fp(cl_F)p Fr(,)h Fp(cl_R)p
2240 Fr(,)g Fp(cl_N)p Fr(.)38 b(The)27 b(easiest)h(w)m(a)m(y)g(to)g(sp)s
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2246 b(that)f(this)f(con)m(v)m(ersion)i(is)e(done)g(at)h(run)m(time,)f(not)g
2247 (at)h(compile-time.)-30 5230 y(Con)m(v)m(ersions)37 b(from)e
2248 Fp(cl_I)h Fr(to)h(the)g(C)f(built-in)h(t)m(yp)s(es)f(`)p
2249 Fp(int)p Fr(',)i(`)p Fp(unsigned)29 b(int)p Fr(',)38
2250 b(`)p Fp(long)p Fr(',)f(`)p Fp(unsigned)29 b(long)p Fr(')36
2251 b(are)-30 5340 y(pro)m(vided)30 b(through)g(the)g(functions)p
2254 9 10 bop -30 -116 a Fr(Chapter)30 b(3:)41 b(Ordinary)29
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2256 b(cl_I_to_int)f(\(const)g(cl_I&)h(x\))-30 408 y(unsigned)f(int)i
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2259 (\(const)i(cl_I&)g(x\))450 737 y Fr(Returns)d Fp(x)g
2260 Fr(as)h(elemen)m(t)h(of)f(the)g(C)f(t)m(yp)s(e)h Fl(ct)m(yp)s(e)p
2261 Fr(.)40 b(If)27 b Fp(x)f Fr(is)h(not)g(represen)m(table)g(in)g(the)g
2262 (range)g(of)g Fl(ct)m(yp)s(e)p Fr(,)450 847 y(a)k(run)m(time)e(error)h
2263 (o)s(ccurs.)-30 1006 y(Con)m(v)m(ersions)38 b(from)f(the)h(classes)h
2264 Fp(cl_I)p Fr(,)g Fp(cl_RA)p Fr(,)g Fp(cl_SF)p Fr(,)f
2265 Fp(cl_FF)p Fr(,)h Fp(cl_DF)p Fr(,)g Fp(cl_LF)p Fr(,)f
2266 Fp(cl_F)f Fr(and)h Fp(cl_R)e Fr(to)j(the)f(C)-30 1116
2267 y(built-in)30 b(t)m(yp)s(es)h(`)p Fp(float)p Fr(')e(and)h(`)p
2268 Fp(double)p Fr(')f(are)i(pro)m(vided)f(through)f(the)i(functions)-30
2269 1275 y Fp(float)e(float_approx)e(\(const)i Fl(t)m(yp)s(e)5
2270 b Fp(&)30 b(x\))-30 1385 y(double)f(double_approx)d(\(const)j
2271 Fl(t)m(yp)s(e)5 b Fp(&)30 b(x\))450 1494 y Fr(Returns)f(an)h(appro)m
2272 (ximation)f(of)h Fp(x)g Fr(of)g(C)f(t)m(yp)s(e)h Fl(ct)m(yp)s(e)p
2273 Fr(.)41 b(If)29 b Fp(abs\(x\))f Fr(is)i(to)s(o)h(close)g(to)f(0)g
2274 (\(under\015o)m(w\),)450 1604 y(0)h(is)f(returned.)40
2275 b(If)30 b Fp(abs\(x\))e Fr(is)j(to)s(o)g(large)g(\(o)m(v)m(er\015o)m
2276 (w\),)i(an)d(IEEE)g(in\014nit)m(y)g(is)h(returned.)-30
2277 1763 y(Con)m(v)m(ersions)k(from)e(an)m(y)i(class)g(to)g(an)m(y)g(of)g
2278 (its)g(sub)s(classes)f(\(\\deriv)m(ed)h(classes")h(in)e(C)p
2279 Fp(++)g Fr(terminology\))h(are)g(not)-30 1873 y(pro)m(vided.)40
2280 b(Instead,)29 b(y)m(ou)f(can)h(assert)g(and)e(c)m(hec)m(k)j(that)f(a)g
2281 (v)-5 b(alue)29 b(b)s(elongs)f(to)h(a)g(certain)g(sub)s(class,)f(and)g
2282 (return)f(it)-30 1983 y(as)f(elemen)m(t)h(of)f(that)h(class,)h(using)d
2283 (the)h(`)p Fp(As)p Fr(')g(and)g(`)p Fp(The)p Fr(')f(macros.)39
2284 b Fp(As\()p Fl(t)m(yp)s(e)5 b Fp(\)\()p Fl(v)-5 b(alue)5
2285 b Fp(\))25 b Fr(c)m(hec)m(ks)j(that)e Fl(v)-5 b(alue)32
2286 b Fr(b)s(elongs)-30 2092 y(to)d Fl(t)m(yp)s(e)k Fr(and)28
2287 b(returns)f(it)i(as)f(suc)m(h.)40 b Fp(The\()p Fl(t)m(yp)s(e)5
2288 b Fp(\)\()p Fl(v)-5 b(alue)5 b Fp(\))27 b Fr(assumes)g(that)i
2289 Fl(v)-5 b(alue)33 b Fr(b)s(elongs)28 b(to)h Fl(t)m(yp)s(e)34
2290 b Fr(and)27 b(returns)g(it)-30 2202 y(as)k(suc)m(h.)40
2291 b(It)31 b(is)f(y)m(our)g(resp)s(onsibilit)m(y)h(to)g(ensure)e(that)i
2292 (this)g(assumption)d(is)j(v)-5 b(alid.)41 b(Example:)353
2293 2330 y Fp(cl_I)47 b(x)g(=)h(...)o(;)353 2434 y(if)f(\(!\(x)g(>=)g
2294 (0\)\))g(abort\(\);)353 2538 y(cl_I)g(ten_x)f(=)i
2295 (The\(cl_I\)\(expt\(10,x\)\);)41 b(//)48 b(If)f(x)g(>=)g(0,)h(10^x)e
2296 (is)h(an)h(integer.)974 2642 y(//)f(In)g(general,)f(it)h(would)f(be)h
2297 (a)h(rational)d(number.)p eop
2299 10 11 bop -30 -116 a Fr(Chapter)30 b(4:)41 b(F)-8 b(unctions)31
2300 b(on)f(n)m(um)m(b)s(ers)2523 b(10)-30 299 y Fo(4)80 b(F)-13
2301 b(unctions)53 b(on)g(n)l(um)l(b)t(ers)-30 600 y Fr(Eac)m(h)36
2302 b(of)g(the)g(n)m(um)m(b)s(er)d(classes)k(declares)f(its)g(mathematical)
2303 g(op)s(erations)g(in)f(the)h(corresp)s(onding)e(include)i(\014le.)-30
2304 709 y(F)-8 b(or)27 b(example,)g(if)f(y)m(our)g(co)s(de)h(op)s(erates)g
2305 (with)f(ob)5 b(jects)27 b(of)f(t)m(yp)s(e)g Fp(cl_I)p
2306 Fr(,)h(it)f(should)g Fp(#include)i(<cln/integer.h>)p
2307 Fr(.)-30 1022 y Fs(4.1)68 b(Constructing)45 b(n)l(um)l(b)t(ers)-30
2308 1237 y Fr(Here)31 b(is)f(ho)m(w)h(to)g(create)h(n)m(um)m(b)s(er)c(ob)5
2309 b(jects)31 b(\\from)e(nothing".)-30 1505 y Fn(4.1.1)63
2310 b(Constructing)41 b(in)m(tegers)-30 1720 y Fp(cl_I)28
2311 b Fr(ob)5 b(jects)30 b(are)f(most)g(easily)h(constructed)f(from)f(C)g
2312 (in)m(tegers)i(and)f(from)f(strings.)40 b(See)29 b(Section)h(3.4)g
2313 ([Con)m(v)m(er-)-30 1830 y(sions],)h(page)g(8.)-30 2098
2314 y Fn(4.1.2)63 b(Constructing)41 b(rational)g(n)m(um)m(b)s(ers)-30
2315 2313 y Fp(cl_RA)e Fr(ob)5 b(jects)41 b(can)g(b)s(e)f(constructed)h
2316 (from)e(strings.)71 b(The)40 b(syn)m(tax)h(for)f(rational)i(n)m(um)m(b)
2317 s(ers)c(is)j(describ)s(ed)e(in)-30 2423 y(Section)44
2318 b(5.1)g([In)m(ternal)g(and)f(prin)m(ted)g(represen)m(tation],)48
2319 b(page)c(28.)80 b(Another)44 b(standard)e(w)m(a)m(y)i(to)g(pro)s(duce)f
2320 (a)-30 2532 y(rational)32 b(n)m(um)m(b)s(er)c(is)i(through)g
2321 (application)h(of)g(`)p Fp(operator)d(/)p Fr(')j(or)f(`)p
2322 Fp(recip)p Fr(')f(on)i(in)m(tegers.)-30 2801 y Fn(4.1.3)63
2323 b(Constructing)41 b(\015oating-p)s(oin)m(t)h(n)m(um)m(b)s(ers)-30
2324 3016 y Fp(cl_F)h Fr(ob)5 b(jects)45 b(with)f(lo)m(w)h(precision)f(are)g
2325 (most)g(easily)h(constructed)g(from)d(C)i(`)p Fp(float)p
2326 Fr(')f(and)h(`)p Fp(double)p Fr('.)81 b(See)-30 3125
2327 y(Section)31 b(3.4)h([Con)m(v)m(ersions],)f(page)g(8.)-30
2328 3282 y(T)-8 b(o)34 b(construct)h(a)f Fp(cl_F)e Fr(with)i(high)g
2329 (precision,)h(y)m(ou)f(can)g(use)g(the)g(con)m(v)m(ersion)h(from)d(`)p
2330 Fp(const)d(char)h(*)p Fr(',)k(but)g(y)m(ou)-30 3392 y(ha)m(v)m(e)e(to)f
2331 (sp)s(ecify)f(the)h(desired)f(precision)h(within)f(the)h(string.)41
2332 b(\(See)32 b(Section)f(5.1)h([In)m(ternal)f(and)f(prin)m(ted)g(repre-)
2333 -30 3501 y(sen)m(tation],)i(page)g(28.\))41 b(Example:)353
2334 3652 y Fp(cl_F)47 b(e)g(=)h("0.27182818284590452353)o(6028)o(7471)o
2335 (352)o(6624)o(9775)o(724)o(7093)o(6999)o(6e+)o(1_40)o(";)-30
2336 3809 y Fr(will)31 b(set)g(`)p Fp(e)p Fr(')f(to)h(the)g(giv)m(en)g(v)-5
2337 b(alue,)31 b(with)f(a)h(precision)g(of)f(40)h(decimal)g(digits.)-30
2338 3966 y(The)e(programmatic)g(w)m(a)m(y)h(to)h(construct)f(a)g
2339 Fp(cl_F)e Fr(with)i(high)f(precision)h(is)f(through)g(the)h
2340 Fp(cl_float)e Fr(con)m(v)m(ersion)-30 4075 y(function,)c(see)e(Section)
2341 h(4.11.1)h([Con)m(v)m(ersion)f(to)f(\015oating-p)s(oin)m(t)i(n)m(um)m
2342 (b)s(ers],)d(page)i(25.)39 b(F)-8 b(or)22 b(example,)i(to)f(compute)-30
2343 4185 y Fp(e)43 b Fr(to)h(40)g(decimal)f(places,)k(\014rst)c(construct)g
2344 (1.0)h(to)g(40)g(decimal)f(places)h(and)f(then)g(apply)g(the)g(exp)s
2345 (onen)m(tial)-30 4294 y(function:)353 4445 y Fp(cl_float_format_t)g
2346 (precision)j(=)h(cl_float_format\(40\);)353 4549 y(cl_F)g(e)g(=)h
2347 (exp\(cl_float\(1,precisio)o(n\)\);)-30 4818 y Fn(4.1.4)63
2348 b(Constructing)41 b(complex)f(n)m(um)m(b)s(ers)-30 5032
2349 y Fr(Non-real)31 b Fp(cl_N)f Fr(ob)5 b(jects)31 b(are)g(normally)e
2350 (constructed)i(through)e(the)i(function)353 5183 y Fp(cl_N)47
2351 b(complex)f(\(const)g(cl_R&)g(realpart,)f(const)i(cl_R&)f(imagpart\))
2352 -30 5340 y Fr(See)31 b(Section)g(4.4)g([Elemen)m(tary)g(complex)f
2353 (functions],)g(page)h(12.)p eop
2355 11 12 bop -30 -116 a Fr(Chapter)30 b(4:)41 b(F)-8 b(unctions)31
2356 b(on)f(n)m(um)m(b)s(ers)2523 b(11)-30 299 y Fs(4.2)68
2357 b(Elemen)l(tary)47 b(functions)-30 491 y Fr(Eac)m(h)27
2358 b(of)f(the)h(classes)g Fp(cl_N)p Fr(,)f Fp(cl_R)p Fr(,)g
2359 Fp(cl_RA)p Fr(,)g Fp(cl_I)p Fr(,)g Fp(cl_F)p Fr(,)g Fp(cl_SF)p
2360 Fr(,)g Fp(cl_FF)p Fr(,)g Fp(cl_DF)p Fr(,)g Fp(cl_LF)f
2361 Fr(de\014nes)g(the)h(follo)m(wing)-30 601 y(op)s(erations:)-30
2362 760 y Fl(t)m(yp)s(e)36 b Fp(operator)28 b(+)i(\(const)e
2363 Fl(t)m(yp)s(e)5 b Fp(&,)31 b(const)d Fl(t)m(yp)s(e)5
2364 b Fp(&\))450 870 y Fr(Addition.)-30 1029 y Fl(t)m(yp)s(e)36
2365 b Fp(operator)28 b(-)i(\(const)e Fl(t)m(yp)s(e)5 b Fp(&,)31
2366 b(const)d Fl(t)m(yp)s(e)5 b Fp(&\))450 1139 y Fr(Subtraction.)-30
2367 1298 y Fl(t)m(yp)s(e)36 b Fp(operator)28 b(-)i(\(const)e
2368 Fl(t)m(yp)s(e)5 b Fp(&\))450 1407 y Fr(Returns)29 b(the)i(negativ)m(e)i
2369 (of)d(the)h(argumen)m(t.)-30 1567 y Fl(t)m(yp)s(e)36
2370 b Fp(plus1)29 b(\(const)f Fl(t)m(yp)s(e)5 b Fp(&)31 b(x\))450
2371 1676 y Fr(Returns)e Fp(x)h(+)h(1)p Fr(.)-30 1835 y Fl(t)m(yp)s(e)36
2372 b Fp(minus1)28 b(\(const)h Fl(t)m(yp)s(e)5 b Fp(&)30
2373 b(x\))450 1945 y Fr(Returns)f Fp(x)h(-)h(1)p Fr(.)-30
2374 2104 y Fl(t)m(yp)s(e)36 b Fp(operator)28 b(*)i(\(const)e
2375 Fl(t)m(yp)s(e)5 b Fp(&,)31 b(const)d Fl(t)m(yp)s(e)5
2376 b Fp(&\))450 2214 y Fr(Multiplication.)-30 2373 y Fl(t)m(yp)s(e)36
2377 b Fp(square)28 b(\(const)h Fl(t)m(yp)s(e)5 b Fp(&)30
2378 b(x\))450 2483 y Fr(Returns)f Fp(x)h(*)h(x)p Fr(.)-30
2379 2642 y(Eac)m(h)43 b(of)g(the)f(classes)i Fp(cl_N)p Fr(,)h
2380 Fp(cl_R)p Fr(,)f Fp(cl_RA)p Fr(,)h Fp(cl_F)p Fr(,)f Fp(cl_SF)p
2381 Fr(,)g Fp(cl_FF)p Fr(,)h Fp(cl_DF)p Fr(,)f Fp(cl_LF)d
2382 Fr(de\014nes)h(the)g(follo)m(wing)-30 2751 y(op)s(erations:)-30
2383 2911 y Fl(t)m(yp)s(e)36 b Fp(operator)28 b(/)i(\(const)e
2384 Fl(t)m(yp)s(e)5 b Fp(&,)31 b(const)d Fl(t)m(yp)s(e)5
2385 b Fp(&\))450 3020 y Fr(Division.)-30 3179 y Fl(t)m(yp)s(e)36
2386 b Fp(recip)29 b(\(const)f Fl(t)m(yp)s(e)5 b Fp(&\))450
2387 3289 y Fr(Returns)29 b(the)i(recipro)s(cal)g(of)g(the)f(argumen)m(t.)
2388 -30 3448 y(The)h(class)h Fp(cl_I)d Fr(do)s(esn't)i(de\014ne)g(a)g(`)p
2389 Fp(/)p Fr(')h(op)s(eration)f(b)s(ecause)g(in)g(the)g(C/C)p
2390 Fp(++)g Fr(language)h(this)f(op)s(erator,)h(applied)-30
2391 3558 y(to)j(in)m(tegral)h(t)m(yp)s(es,)g(denotes)f(the)g(`)p
2392 Fp(floor)p Fr(')e(or)h(`)p Fp(truncate)p Fr(')f(op)s(eration)i(\(whic)m
2393 (h)g(one)f(of)h(these,)h(is)f(implemen)m(ta-)-30 3667
2394 y(tion)g(dep)s(enden)m(t\).)51 b(\(See)35 b(Section)g(4.6)g([Rounding)f
2395 (functions],)h(page)g(13.\))53 b(Instead,)36 b Fp(cl_I)d
2396 Fr(de\014nes)g(an)h(\\exact)-30 3777 y(quotien)m(t")e(function:)-30
2397 3936 y Fp(cl_I)d(exquo)g(\(const)g(cl_I&)g(x,)h(const)e(cl_I&)h(y\))450
2398 4046 y Fr(Chec)m(ks)i(that)g Fp(y)f Fr(divides)g Fp(x)p
2399 Fr(,)g(and)g(returns)f(the)h(quotien)m(t)i Fp(x)p Fr(/)p
2400 Fp(y)p Fr(.)-30 4205 y(The)e(follo)m(wing)i(exp)s(onen)m(tiation)f
2401 (functions)f(are)h(de\014ned:)-30 4364 y Fp(cl_I)e(expt_pos)f(\(const)h
2402 (cl_I&)g(x,)h(const)f(cl_I&)g(y\))-30 4474 y(cl_RA)g(expt_pos)f
2403 (\(const)h(cl_RA&)f(x,)i(const)f(cl_I&)g(y\))450 4583
2404 y(y)h Fr(m)m(ust)f(b)s(e)h Fp(>)g Fr(0.)41 b(Returns)30
2405 b Fp(x^y)p Fr(.)-30 4742 y Fp(cl_RA)f(expt)g(\(const)g(cl_RA&)f(x,)i
2406 (const)f(cl_I&)g(y\))-30 4852 y(cl_R)g(expt)g(\(const)g(cl_R&)g(x,)h
2407 (const)f(cl_I&)g(y\))-30 4962 y(cl_N)g(expt)g(\(const)g(cl_N&)g(x,)h
2408 (const)f(cl_I&)g(y\))450 5071 y Fr(Returns)g Fp(x^y)p
2409 Fr(.)-30 5230 y(Eac)m(h)43 b(of)g(the)f(classes)i Fp(cl_R)p
2410 Fr(,)h Fp(cl_RA)p Fr(,)f Fp(cl_I)p Fr(,)h Fp(cl_F)p Fr(,)f
2411 Fp(cl_SF)p Fr(,)g Fp(cl_FF)p Fr(,)h Fp(cl_DF)p Fr(,)f
2412 Fp(cl_LF)d Fr(de\014nes)h(the)g(follo)m(wing)-30 5340
2413 y(op)s(eration:)p eop
2415 12 13 bop -30 -116 a Fr(Chapter)30 b(4:)41 b(F)-8 b(unctions)31
2416 b(on)f(n)m(um)m(b)s(ers)2523 b(12)-30 299 y Fl(t)m(yp)s(e)36
2417 b Fp(abs)29 b(\(const)g Fl(t)m(yp)s(e)5 b Fp(&)30 b(x\))450
2418 408 y Fr(Returns)f(the)i(absolute)g(v)-5 b(alue)31 b(of)f
2419 Fp(x)p Fr(.)41 b(This)29 b(is)i Fp(x)f Fr(if)g Fp(x)g(>=)g(0)p
2420 Fr(,)g(and)g Fp(-x)g Fr(if)g Fp(x)g(<=)g(0)p Fr(.)-30
2421 588 y(The)g(class)h Fp(cl_N)e Fr(implemen)m(ts)g(this)h(as)h(follo)m
2422 (ws:)-30 767 y Fp(cl_R)e(abs)h(\(const)e(cl_N)i(x\))450
2423 877 y Fr(Returns)f(the)i(absolute)g(v)-5 b(alue)31 b(of)f
2424 Fp(x)p Fr(.)-30 1057 y(Eac)m(h)d(of)f(the)h(classes)g
2425 Fp(cl_N)p Fr(,)f Fp(cl_R)p Fr(,)g Fp(cl_RA)p Fr(,)g Fp(cl_I)p
2426 Fr(,)g Fp(cl_F)p Fr(,)g Fp(cl_SF)p Fr(,)g Fp(cl_FF)p
2427 Fr(,)g Fp(cl_DF)p Fr(,)g Fp(cl_LF)f Fr(de\014nes)g(the)h(follo)m(wing)
2428 -30 1166 y(op)s(eration:)-30 1346 y Fl(t)m(yp)s(e)36
2429 b Fp(signum)28 b(\(const)h Fl(t)m(yp)s(e)5 b Fp(&)30
2430 b(x\))450 1455 y Fr(Returns)f(the)h(sign)g(of)g Fp(x)p
2431 Fr(,)g(in)f(the)i(same)e(n)m(um)m(b)s(er)e(format)j(as)g
2432 Fp(x)p Fr(.)40 b(This)29 b(is)h(de\014ned)f(as)h Fp(x)g(/)g(abs\(x\))e
2433 Fr(if)450 1565 y Fp(x)i Fr(is)g(non-zero,)i(and)d Fp(x)h
2434 Fr(if)h Fp(x)f Fr(is)g(zero.)42 b(If)30 b Fp(x)g Fr(is)g(real,)h(the)g
2435 (v)-5 b(alue)31 b(is)f(either)h(0)g(or)f(1)h(or)f(-1.)-30
2436 1856 y Fs(4.3)68 b(Elemen)l(tary)47 b(rational)f(functions)-30
2437 2062 y Fr(Eac)m(h)31 b(of)g(the)f(classes)i Fp(cl_RA)p
2438 Fr(,)d Fp(cl_I)g Fr(de\014nes)g(the)i(follo)m(wing)h(op)s(erations:)-30
2439 2241 y Fp(cl_I)d(numerator)f(\(const)h Fl(t)m(yp)s(e)5
2440 b Fp(&)30 b(x\))450 2351 y Fr(Returns)f(the)i(n)m(umerator)e(of)i
2441 Fp(x)p Fr(.)-30 2524 y Fp(cl_I)e(denominator)f(\(const)g
2442 Fl(t)m(yp)s(e)5 b Fp(&)31 b(x\))450 2633 y Fr(Returns)e(the)i
2443 (denominator)f(of)g Fp(x)p Fr(.)-30 2813 y(The)g(n)m(umerator)g(and)g
2444 (denominator)g(of)h(a)g(rational)h(n)m(um)m(b)s(er)c(are)k(normalized)e
2445 (in)g(suc)m(h)h(a)g(w)m(a)m(y)g(that)h(they)f(ha)m(v)m(e)-30
2446 2922 y(no)f(factor)i(in)e(common)e(and)i(the)h(denominator)e(is)i(p)s
2447 (ositiv)m(e.)-30 3213 y Fs(4.4)68 b(Elemen)l(tary)47
2448 b(complex)e(functions)-30 3419 y Fr(The)30 b(class)h
2449 Fp(cl_N)e Fr(de\014nes)h(the)g(follo)m(wing)i(op)s(eration:)-30
2450 3599 y Fp(cl_N)d(complex)g(\(const)f(cl_R&)h(a,)h(const)f(cl_R&)g(b\))
2451 450 3708 y Fr(Returns)35 b(the)h(complex)g(n)m(um)m(b)s(er)e
2452 Fp(a+bi)p Fr(,)i(that)h(is,)g(the)f(complex)g(n)m(um)m(b)s(er)e(with)h
2453 (real)i(part)f Fp(a)g Fr(and)450 3818 y(imaginary)30
2454 b(part)g Fp(b)p Fr(.)-30 3997 y(Eac)m(h)h(of)g(the)f(classes)i
2455 Fp(cl_N)p Fr(,)d Fp(cl_R)g Fr(de\014nes)h(the)g(follo)m(wing)i(op)s
2456 (erations:)-30 4177 y Fp(cl_R)d(realpart)f(\(const)h
2457 Fl(t)m(yp)s(e)5 b Fp(&)30 b(x\))450 4286 y Fr(Returns)f(the)i(real)g
2458 (part)f(of)h Fp(x)p Fr(.)-30 4459 y Fp(cl_R)e(imagpart)f(\(const)h
2459 Fl(t)m(yp)s(e)5 b Fp(&)30 b(x\))450 4569 y Fr(Returns)f(the)i
2460 (imaginary)f(part)g(of)h Fp(x)p Fr(.)-30 4742 y Fl(t)m(yp)s(e)36
2461 b Fp(conjugate)27 b(\(const)i Fl(t)m(yp)s(e)5 b Fp(&)31
2462 b(x\))450 4851 y Fr(Returns)e(the)i(complex)f(conjugate)i(of)e
2463 Fp(x)p Fr(.)-30 5031 y(W)-8 b(e)32 b(ha)m(v)m(e)f(the)g(relations)150
2464 5179 y Fp(x)f(=)g(complex\(realpart\(x\),)25 b(imagpart\(x\)\))150
2465 5320 y(conjugate\(x\))i(=)j(complex\(realpart\(x\),)25
2466 b(-imagpart\(x\)\))p eop
2468 13 14 bop -30 -116 a Fr(Chapter)30 b(4:)41 b(F)-8 b(unctions)31
2469 b(on)f(n)m(um)m(b)s(ers)2523 b(13)-30 299 y Fs(4.5)68
2470 b(Comparisons)-30 489 y Fr(Eac)m(h)27 b(of)f(the)h(classes)g
2471 Fp(cl_N)p Fr(,)f Fp(cl_R)p Fr(,)g Fp(cl_RA)p Fr(,)g Fp(cl_I)p
2472 Fr(,)g Fp(cl_F)p Fr(,)g Fp(cl_SF)p Fr(,)g Fp(cl_FF)p
2473 Fr(,)g Fp(cl_DF)p Fr(,)g Fp(cl_LF)f Fr(de\014nes)g(the)h(follo)m(wing)
2474 -30 598 y(op)s(erations:)-30 752 y Fp(bool)j(operator)f(==)i(\(const)f
2475 Fl(t)m(yp)s(e)5 b Fp(&,)30 b(const)f Fl(t)m(yp)s(e)5
2476 b Fp(&\))-30 862 y(bool)29 b(operator)f(!=)i(\(const)f
2477 Fl(t)m(yp)s(e)5 b Fp(&,)30 b(const)f Fl(t)m(yp)s(e)5
2478 b Fp(&\))450 971 y Fr(Comparison,)29 b(as)i(in)f(C)g(and)f(C)p
2479 Fp(++)p Fr(.)-30 1125 y Fp(uint32)g(equal_hashcode)d(\(const)j
2480 Fl(t)m(yp)s(e)5 b Fp(&\))450 1235 y Fr(Returns)35 b(a)h(32-bit)h(hash)e
2481 (co)s(de)h(that)g(is)g(the)g(same)f(for)h(an)m(y)g(t)m(w)m(o)h(n)m(um)m
2482 (b)s(ers)c(whic)m(h)i(are)i(the)f(same)450 1345 y(according)24
2483 b(to)h Fp(==)p Fr(.)38 b(This)22 b(hash)h(co)s(de)h(dep)s(ends)e(on)h
2484 (the)h(n)m(um)m(b)s(er's)d(v)-5 b(alue,)26 b(not)d(its)h(t)m(yp)s(e)g
2485 (or)g(precision.)-30 1499 y Fp(cl_boolean)k(zerop)h(\(const)f
2486 Fl(t)m(yp)s(e)5 b Fp(&)31 b(x\))450 1608 y Fr(Compare)e(against)j
2487 (zero:)41 b Fp(x)30 b(==)g(0)-30 1762 y Fr(Eac)m(h)43
2488 b(of)g(the)f(classes)i Fp(cl_R)p Fr(,)h Fp(cl_RA)p Fr(,)f
2489 Fp(cl_I)p Fr(,)h Fp(cl_F)p Fr(,)f Fp(cl_SF)p Fr(,)g Fp(cl_FF)p
2490 Fr(,)h Fp(cl_DF)p Fr(,)f Fp(cl_LF)d Fr(de\014nes)h(the)g(follo)m(wing)
2491 -30 1872 y(op)s(erations:)-30 2026 y Fp(cl_signean)28
2492 b(compare)g(\(const)h Fl(t)m(yp)s(e)5 b Fp(&)30 b(x,)g(const)f
2493 Fl(t)m(yp)s(e)5 b Fp(&)30 b(y\))450 2135 y Fr(Compares)f
2494 Fp(x)h Fr(and)g Fp(y)p Fr(.)40 b(Returns)30 b Fp(+)p
2495 Fr(1)g(if)g Fp(x>y)p Fr(,)g(-1)h(if)g Fp(x<y)p Fr(,)e(0)i(if)f
2496 Fp(x)p Fr(=)p Fp(y)p Fr(.)-30 2289 y Fp(bool)f(operator)f(<=)i(\(const)
2497 f Fl(t)m(yp)s(e)5 b Fp(&,)30 b(const)f Fl(t)m(yp)s(e)5
2498 b Fp(&\))-30 2399 y(bool)29 b(operator)f(<)i(\(const)f
2499 Fl(t)m(yp)s(e)5 b Fp(&,)30 b(const)f Fl(t)m(yp)s(e)5
2500 b Fp(&\))-30 2508 y(bool)29 b(operator)f(>=)i(\(const)f
2501 Fl(t)m(yp)s(e)5 b Fp(&,)30 b(const)f Fl(t)m(yp)s(e)5
2502 b Fp(&\))-30 2618 y(bool)29 b(operator)f(>)i(\(const)f
2503 Fl(t)m(yp)s(e)5 b Fp(&,)30 b(const)f Fl(t)m(yp)s(e)5
2504 b Fp(&\))450 2727 y Fr(Comparison,)29 b(as)i(in)f(C)g(and)f(C)p
2505 Fp(++)p Fr(.)-30 2881 y Fp(cl_boolean)f(minusp)g(\(const)h
2506 Fl(t)m(yp)s(e)5 b Fp(&)30 b(x\))450 2991 y Fr(Compare)f(against)j
2507 (zero:)41 b Fp(x)30 b(<)g(0)-30 3145 y(cl_boolean)e(plusp)h(\(const)f
2508 Fl(t)m(yp)s(e)5 b Fp(&)31 b(x\))450 3255 y Fr(Compare)e(against)j
2509 (zero:)41 b Fp(x)30 b(>)g(0)-30 3408 y Fl(t)m(yp)s(e)36
2510 b Fp(max)29 b(\(const)g Fl(t)m(yp)s(e)5 b Fp(&)30 b(x,)g(const)f
2511 Fl(t)m(yp)s(e)5 b Fp(&)30 b(y\))450 3518 y Fr(Return)g(the)g(maxim)m
2512 (um)d(of)k Fp(x)f Fr(and)g Fp(y)p Fr(.)-30 3672 y Fl(t)m(yp)s(e)36
2513 b Fp(min)29 b(\(const)g Fl(t)m(yp)s(e)5 b Fp(&)30 b(x,)g(const)f
2514 Fl(t)m(yp)s(e)5 b Fp(&)30 b(y\))450 3782 y Fr(Return)g(the)g(minim)m
2515 (um)d(of)j Fp(x)g Fr(and)g Fp(y)p Fr(.)-30 3936 y(When)35
2516 b(a)g(\015oating)h(p)s(oin)m(t)f(n)m(um)m(b)s(er)d(and)i(a)i(rational)g
2517 (n)m(um)m(b)s(er)c(are)k(compared,)f(the)g(\015oat)h(is)f(\014rst)f
2518 (con)m(v)m(erted)i(to)-30 4045 y(a)f(rational)h(n)m(um)m(b)s(er)c
2519 (using)i(the)h(function)f Fp(rational)p Fr(.)51 b(Since)35
2520 b(a)g(\015oating)g(p)s(oin)m(t)g(n)m(um)m(b)s(er)d(actually)k(represen)
2521 m(ts)-30 4155 y(an)h(in)m(terv)-5 b(al)39 b(of)e(real)h(n)m(um)m(b)s
2522 (ers,)f(the)h(result)f(migh)m(t)g(b)s(e)g(surprising.)60
2523 b(F)-8 b(or)38 b(example,)h Fp(\(cl_F\)\(cl_R\)"1/3")26
2524 b(==)-30 4264 y(\(cl_R\)"1/3")h Fr(returns)i(false)i(b)s(ecause)g
2525 (there)f(is)h(no)f(\015oating)h(p)s(oin)m(t)g(n)m(um)m(b)s(er)d(whose)i
2526 (v)-5 b(alue)31 b(is)f(exactly)i Fp(1/3)p Fr(.)-30 4513
2527 y Fs(4.6)68 b(Rounding)45 b(functions)-30 4703 y Fr(When)40
2528 b(a)g(real)h(n)m(um)m(b)s(er)d(is)i(to)h(b)s(e)f(con)m(v)m(erted)h(to)g
2529 (an)f(in)m(teger,)45 b(there)40 b(is)g(no)g(\\b)s(est")h(rounding.)69
2530 b(The)39 b(desired)-30 4813 y(rounding)27 b(function)g(dep)s(ends)f(on)
2531 i(the)g(application.)41 b(The)28 b(Common)d(Lisp)i(and)g(ISO)g(Lisp)g
2532 (standards)g(o\013er)i(four)-30 4923 y(rounding)g(functions:)-30
2533 5076 y Fp(floor\(x\))96 b Fr(This)30 b(is)g(the)h(largest)g(in)m(teger)
2534 h Fp(<)p Fr(=)p Fp(x)p Fr(.)-30 5230 y Fp(ceiling\(x\))450
2535 5340 y Fr(This)e(is)g(the)h(smallest)f(in)m(teger)i Fp(>)p
2536 Fr(=)p Fp(x)p Fr(.)p eop
2538 14 15 bop -30 -116 a Fr(Chapter)30 b(4:)41 b(F)-8 b(unctions)31
2539 b(on)f(n)m(um)m(b)s(ers)2523 b(14)-30 299 y Fp(truncate\(x\))450
2540 408 y Fr(Among)30 b(the)g(in)m(tegers)i(b)s(et)m(w)m(een)f(0)g(and)e
2541 Fp(x)h Fr(\(inclusiv)m(e\))i(the)f(one)g(nearest)f(to)i
2542 Fp(x)p Fr(.)-30 566 y Fp(round\(x\))96 b Fr(The)32 b(in)m(teger)i
2543 (nearest)g(to)f Fp(x)p Fr(.)48 b(If)32 b Fp(x)g Fr(is)h(exactly)i
2544 (halfw)m(a)m(y)e(b)s(et)m(w)m(een)h(t)m(w)m(o)g(in)m(tegers,)h(c)m(ho)s
2545 (ose)e(the)g(ev)m(en)450 676 y(one.)-30 833 y(These)d(functions)g(ha)m
2546 (v)m(e)i(di\013eren)m(t)e(adv)-5 b(an)m(tages:)-30 967
2547 y Fp(floor)44 b Fr(and)i Fp(ceiling)e Fr(are)i(translation)h(in)m(v)-5
2548 b(arian)m(t:)73 b Fp(floor\(x+n\))27 b(=)j(floor\(x\))e(+)i(n)46
2549 b Fr(and)f Fp(ceiling\(x+n\))27 b(=)-30 1076 y(ceiling\(x\))h(+)i(n)g
2550 Fr(for)g(ev)m(ery)h Fp(x)f Fr(and)g(ev)m(ery)h(in)m(teger)h
2551 Fp(n)p Fr(.)-30 1210 y(On)51 b(the)h(other)h(hand,)j
2552 Fp(truncate)50 b Fr(and)h Fp(round)g Fr(are)h(symmetric:)83
2553 b Fp(truncate\(-x\))27 b(=)j(-truncate\(x\))49 b Fr(and)-30
2554 1319 y Fp(round\(-x\))28 b(=)i(-round\(x\))p Fr(,)44
2555 b(and)f(furthermore)e Fp(round)g Fr(is)j(un)m(biased:)65
2556 b(on)44 b(the)f(\\a)m(v)m(erage",)50 b(it)44 b(rounds)d(do)m(wn)-30
2557 1429 y(exactly)32 b(as)f(often)g(as)f(it)h(rounds)e(up.)-30
2558 1563 y(The)h(functions)g(are)h(related)g(lik)m(e)h(this:)150
2559 1696 y Fp(ceiling\(m/n\))27 b(=)j(floor\(\(m+n-1\)/n\))c(=)k
2560 (floor\(\(m-1\)/n\)+1)f Fr(for)j(rational)j(n)m(um)m(b)s(ers)30
2561 b Fp(m/n)i Fr(\()p Fp(m)p Fr(,)i Fp(n)f Fr(in)m(te-)150
2562 1806 y(gers,)e Fp(n>)p Fr(0\),)g(and)150 1939 y Fp(truncate\(x\))c(=)j
2563 (sign\(x\))f(*)h(floor\(abs\(x\)\))-30 2097 y Fr(Eac)m(h)f(of)f(the)g
2564 (classes)h Fp(cl_R)p Fr(,)e Fp(cl_RA)p Fr(,)h Fp(cl_F)p
2565 Fr(,)f Fp(cl_SF)p Fr(,)h Fp(cl_FF)p Fr(,)f Fp(cl_DF)p
2566 Fr(,)g Fp(cl_LF)g Fr(de\014nes)g(the)h(follo)m(wing)h(op)s(erations:)
2567 -30 2254 y Fp(cl_I)g(floor1)g(\(const)g Fl(t)m(yp)s(e)5
2568 b Fp(&)30 b(x\))450 2364 y Fr(Returns)f Fp(floor\(x\))p
2569 Fr(.)-30 2521 y Fp(cl_I)g(ceiling1)f(\(const)h Fl(t)m(yp)s(e)5
2570 b Fp(&)30 b(x\))450 2631 y Fr(Returns)f Fp(ceiling\(x\))p
2571 Fr(.)-30 2789 y Fp(cl_I)g(truncate1)f(\(const)h Fl(t)m(yp)s(e)5
2572 b Fp(&)30 b(x\))450 2898 y Fr(Returns)f Fp(truncate\(x\))p
2573 Fr(.)-30 3056 y Fp(cl_I)g(round1)g(\(const)g Fl(t)m(yp)s(e)5
2574 b Fp(&)30 b(x\))450 3165 y Fr(Returns)f Fp(round\(x\))p
2575 Fr(.)-30 3323 y(Eac)m(h)43 b(of)g(the)f(classes)i Fp(cl_R)p
2576 Fr(,)h Fp(cl_RA)p Fr(,)f Fp(cl_I)p Fr(,)h Fp(cl_F)p Fr(,)f
2577 Fp(cl_SF)p Fr(,)g Fp(cl_FF)p Fr(,)h Fp(cl_DF)p Fr(,)f
2578 Fp(cl_LF)d Fr(de\014nes)h(the)g(follo)m(wing)-30 3432
2579 y(op)s(erations:)-30 3590 y Fp(cl_I)29 b(floor1)g(\(const)g
2580 Fl(t)m(yp)s(e)5 b Fp(&)30 b(x,)g(const)f Fl(t)m(yp)s(e)5
2581 b Fp(&)30 b(y\))450 3700 y Fr(Returns)f Fp(floor\(x/y\))p
2582 Fr(.)-30 3857 y Fp(cl_I)g(ceiling1)f(\(const)h Fl(t)m(yp)s(e)5
2583 b Fp(&)30 b(x,)g(const)f Fl(t)m(yp)s(e)5 b Fp(&)30 b(y\))450
2584 3967 y Fr(Returns)f Fp(ceiling\(x/y\))p Fr(.)-30 4124
2585 y Fp(cl_I)g(truncate1)f(\(const)h Fl(t)m(yp)s(e)5 b Fp(&)30
2586 b(x,)g(const)f Fl(t)m(yp)s(e)5 b Fp(&)30 b(y\))450 4234
2587 y Fr(Returns)f Fp(truncate\(x/y\))p Fr(.)-30 4391 y Fp(cl_I)g(round1)g
2588 (\(const)g Fl(t)m(yp)s(e)5 b Fp(&)30 b(x,)g(const)f Fl(t)m(yp)s(e)5
2589 b Fp(&)30 b(y\))450 4501 y Fr(Returns)f Fp(round\(x/y\))p
2590 Fr(.)-30 4658 y(These)41 b(functions)g(are)h(called)h(`)p
2591 Fp(floor1)p Fr(',)48 b(.)23 b(.)f(.)52 b(here)42 b(instead)f(of)h(`)p
2592 Fp(floor)p Fr(',)49 b(.)22 b(.)h(.)11 b(,)44 b(b)s(ecause)e(on)f(some)g
2593 (systems,)-30 4768 y(system)30 b(dep)s(enden)m(t)f(include)h(\014les)g
2594 (de\014ne)g(`)p Fp(floor)p Fr(')f(and)h(`)p Fp(ceiling)p
2595 Fr(')f(as)i(macros.)-30 4902 y(In)h(man)m(y)g(cases,)j(one)e(needs)g(b)
2596 s(oth)f(the)h(quotien)m(t)i(and)d(the)h(remainder)e(of)j(a)f(division.)
2597 49 b(It)33 b(is)g(more)f(e\016cien)m(t)i(to)-30 5011
2598 y(compute)h(b)s(oth)h(at)h(the)f(same)g(time)f(than)h(to)h(p)s(erform)d
2599 (t)m(w)m(o)j(divisions,)h(one)e(for)g(quotien)m(t)i(and)d(the)h(next)h
2600 (one)-30 5121 y(for)28 b(the)g(remainder.)38 b(The)27
2601 b(follo)m(wing)j(functions)d(therefore)i(return)d(a)j(structure)e(con)m
2602 (taining)i(b)s(oth)f(the)g(quotien)m(t)-30 5230 y(and)g(the)g
2603 (remainder.)38 b(The)28 b(su\016x)f(`)p Fp(2)p Fr(')h(indicates)h(the)g
2604 (n)m(um)m(b)s(er)c(of)k(\\return)e(v)-5 b(alues".)41
2605 b(The)27 b(remainder)g(is)h(de\014ned)-30 5340 y(as)j(follo)m(ws:)p
2608 15 16 bop -30 -116 a Fr(Chapter)30 b(4:)41 b(F)-8 b(unctions)31
2609 b(on)f(n)m(um)m(b)s(ers)2523 b(15)45 299 y Fq(\017)60
2610 b Fr(for)30 b(the)h(computation)f(of)h Fp(quotient)d(=)i(floor\(x\))p
2611 Fr(,)e Fp(remainder)g(=)i(x)g(-)g(quotient)p Fr(,)45
2612 441 y Fq(\017)60 b Fr(for)30 b(the)h(computation)f(of)h
2613 Fp(quotient)d(=)i(floor\(x,y\))p Fr(,)e Fp(remainder)f(=)j(x)g(-)g
2614 (quotient*y)p Fr(,)-30 624 y(and)g(similarly)g(for)g(the)g(other)h
2615 (three)g(op)s(erations.)-30 774 y(Eac)m(h)e(of)f(the)g(classes)h
2616 Fp(cl_R)p Fr(,)e Fp(cl_RA)p Fr(,)h Fp(cl_F)p Fr(,)f Fp(cl_SF)p
2617 Fr(,)h Fp(cl_FF)p Fr(,)f Fp(cl_DF)p Fr(,)g Fp(cl_LF)g
2618 Fr(de\014nes)g(the)h(follo)m(wing)h(op)s(erations:)-30
2619 957 y Fp(struct)g Fl(t)m(yp)s(e)5 b Fp(_div_t)29 b({)h(cl_I)f
2620 (quotient;)f Fl(t)m(yp)s(e)35 b Fp(remainder;)28 b(};)-30
2621 1066 y Fl(t)m(yp)s(e)5 b Fp(_div_t)29 b(floor2)g(\(const)f
2622 Fl(t)m(yp)s(e)5 b Fp(&)31 b(x\))-30 1176 y Fl(t)m(yp)s(e)5
2623 b Fp(_div_t)29 b(ceiling2)f(\(const)h Fl(t)m(yp)s(e)5
2624 b Fp(&)30 b(x\))-30 1286 y Fl(t)m(yp)s(e)5 b Fp(_div_t)29
2625 b(truncate2)f(\(const)h Fl(t)m(yp)s(e)5 b Fp(&)30 b(x\))-30
2626 1395 y Fl(t)m(yp)s(e)5 b Fp(_div_t)29 b(round2)g(\(const)f
2627 Fl(t)m(yp)s(e)5 b Fp(&)31 b(x\))-30 1545 y Fr(Eac)m(h)43
2628 b(of)g(the)f(classes)i Fp(cl_R)p Fr(,)h Fp(cl_RA)p Fr(,)f
2629 Fp(cl_I)p Fr(,)h Fp(cl_F)p Fr(,)f Fp(cl_SF)p Fr(,)g Fp(cl_FF)p
2630 Fr(,)h Fp(cl_DF)p Fr(,)f Fp(cl_LF)d Fr(de\014nes)h(the)g(follo)m(wing)
2631 -30 1655 y(op)s(erations:)-30 1838 y Fp(struct)29 b Fl(t)m(yp)s(e)5
2632 b Fp(_div_t)29 b({)h(cl_I)f(quotient;)f Fl(t)m(yp)s(e)35
2633 b Fp(remainder;)28 b(};)-30 1947 y Fl(t)m(yp)s(e)5 b
2634 Fp(_div_t)29 b(floor2)g(\(const)f Fl(t)m(yp)s(e)5 b Fp(&)31
2635 b(x,)e(const)g Fl(t)m(yp)s(e)5 b Fp(&)31 b(y\))-30 2057
2636 y Fl(t)m(yp)s(e)5 b Fp(_div_t)29 b(ceiling2)f(\(const)h
2637 Fl(t)m(yp)s(e)5 b Fp(&)30 b(x,)g(const)f Fl(t)m(yp)s(e)5
2638 b Fp(&)30 b(y\))-30 2166 y Fl(t)m(yp)s(e)5 b Fp(_div_t)29
2639 b(truncate2)f(\(const)h Fl(t)m(yp)s(e)5 b Fp(&)30 b(x,)g(const)f
2640 Fl(t)m(yp)s(e)5 b Fp(&)30 b(y\))-30 2276 y Fl(t)m(yp)s(e)5
2641 b Fp(_div_t)29 b(round2)g(\(const)f Fl(t)m(yp)s(e)5 b
2642 Fp(&)31 b(x,)e(const)g Fl(t)m(yp)s(e)5 b Fp(&)31 b(y\))-30
2643 2426 y Fr(Sometimes,)d(one)g(w)m(an)m(ts)h(the)f(quotien)m(t)h(as)g(a)f
2644 (\015oating-p)s(oin)m(t)i(n)m(um)m(b)s(er)c(\(of)i(the)h(same)e(format)
2645 h(as)g(the)g(argumen)m(t,)-30 2536 y(if)i(the)h(argumen)m(t)f(is)g(a)h
2646 (\015oat\))g(instead)g(of)f(as)h(an)f(in)m(teger.)42
2647 b(The)30 b(pre\014x)g(`)p Fp(f)p Fr(')g(indicates)h(this.)-30
2648 2686 y(Eac)m(h)g(of)g(the)f(classes)i Fp(cl_F)p Fr(,)d
2649 Fp(cl_SF)p Fr(,)g Fp(cl_FF)p Fr(,)g Fp(cl_DF)p Fr(,)h
2650 Fp(cl_LF)e Fr(de\014nes)i(the)h(follo)m(wing)g(op)s(erations:)-30
2651 2868 y Fl(t)m(yp)s(e)36 b Fp(ffloor)28 b(\(const)h Fl(t)m(yp)s(e)5
2652 b Fp(&)30 b(x\))-30 2978 y Fl(t)m(yp)s(e)36 b Fp(fceiling)28
2653 b(\(const)g Fl(t)m(yp)s(e)5 b Fp(&)31 b(x\))-30 3088
2654 y Fl(t)m(yp)s(e)36 b Fp(ftruncate)27 b(\(const)i Fl(t)m(yp)s(e)5
2655 b Fp(&)31 b(x\))-30 3197 y Fl(t)m(yp)s(e)36 b Fp(fround)28
2656 b(\(const)h Fl(t)m(yp)s(e)5 b Fp(&)30 b(x\))-30 3347
2657 y Fr(and)g(similarly)g(for)g(class)h Fp(cl_R)p Fr(,)e(but)h(with)g
2658 (return)f(t)m(yp)s(e)i Fp(cl_F)p Fr(.)-30 3497 y(The)f(class)h
2659 Fp(cl_R)e Fr(de\014nes)h(the)g(follo)m(wing)i(op)s(erations:)-30
2660 3680 y Fp(cl_F)d(ffloor)g(\(const)g Fl(t)m(yp)s(e)5 b
2661 Fp(&)30 b(x,)g(const)f Fl(t)m(yp)s(e)5 b Fp(&)30 b(y\))-30
2662 3790 y(cl_F)f(fceiling)f(\(const)h Fl(t)m(yp)s(e)5 b
2663 Fp(&)30 b(x,)g(const)f Fl(t)m(yp)s(e)5 b Fp(&)30 b(y\))-30
2664 3899 y(cl_F)f(ftruncate)f(\(const)h Fl(t)m(yp)s(e)5 b
2665 Fp(&)30 b(x,)g(const)f Fl(t)m(yp)s(e)5 b Fp(&)30 b(y\))-30
2666 4009 y(cl_F)f(fround)g(\(const)g Fl(t)m(yp)s(e)5 b Fp(&)30
2667 b(x,)g(const)f Fl(t)m(yp)s(e)5 b Fp(&)30 b(y\))-30 4159
2668 y Fr(These)g(functions)f(also)i(exist)g(in)e(v)m(ersions)h(whic)m(h)g
2669 (return)f(b)s(oth)g(the)h(quotien)m(t)h(and)e(the)i(remainder.)39
2670 b(The)29 b(su\016x)-30 4269 y(`)p Fp(2)p Fr(')i(indicates)g(this.)-30
2671 4419 y(Eac)m(h)g(of)g(the)f(classes)i Fp(cl_F)p Fr(,)d
2672 Fp(cl_SF)p Fr(,)g Fp(cl_FF)p Fr(,)g Fp(cl_DF)p Fr(,)h
2673 Fp(cl_LF)e Fr(de\014nes)i(the)h(follo)m(wing)g(op)s(erations:)-30
2674 4601 y Fp(struct)e Fl(t)m(yp)s(e)5 b Fp(_fdiv_t)28 b({)i
2675 Fl(t)m(yp)s(e)36 b Fp(quotient;)28 b Fl(t)m(yp)s(e)36
2676 b Fp(remainder;)27 b(};)-30 4711 y Fl(t)m(yp)s(e)5 b
2677 Fp(_fdiv_t)29 b(ffloor2)f(\(const)h Fl(t)m(yp)s(e)5 b
2678 Fp(&)30 b(x\))-30 4821 y Fl(t)m(yp)s(e)5 b Fp(_fdiv_t)29
2679 b(fceiling2)f(\(const)g Fl(t)m(yp)s(e)5 b Fp(&)31 b(x\))-30
2680 4930 y Fl(t)m(yp)s(e)5 b Fp(_fdiv_t)29 b(ftruncate2)e(\(const)i
2681 Fl(t)m(yp)s(e)5 b Fp(&)30 b(x\))-30 5040 y Fl(t)m(yp)s(e)5
2682 b Fp(_fdiv_t)29 b(fround2)f(\(const)h Fl(t)m(yp)s(e)5
2683 b Fp(&)30 b(x\))-30 5190 y Fr(and)g(similarly)g(for)g(class)h
2684 Fp(cl_R)p Fr(,)e(but)h(with)g(quotien)m(t)i(t)m(yp)s(e)e
2685 Fp(cl_F)p Fr(.)-30 5340 y(The)g(class)h Fp(cl_R)e Fr(de\014nes)h(the)g
2686 (follo)m(wing)i(op)s(erations:)p eop
2688 16 17 bop -30 -116 a Fr(Chapter)30 b(4:)41 b(F)-8 b(unctions)31
2689 b(on)f(n)m(um)m(b)s(ers)2523 b(16)-30 299 y Fp(struct)29
2690 b Fl(t)m(yp)s(e)5 b Fp(_fdiv_t)28 b({)i(cl_F)g(quotient;)d(cl_R)j
2691 (remainder;)d(};)-30 408 y Fl(t)m(yp)s(e)5 b Fp(_fdiv_t)29
2692 b(ffloor2)f(\(const)h Fl(t)m(yp)s(e)5 b Fp(&)30 b(x,)g(const)f
2693 Fl(t)m(yp)s(e)5 b Fp(&)30 b(y\))-30 518 y Fl(t)m(yp)s(e)5
2694 b Fp(_fdiv_t)29 b(fceiling2)f(\(const)g Fl(t)m(yp)s(e)5
2695 b Fp(&)31 b(x,)e(const)g Fl(t)m(yp)s(e)5 b Fp(&)31 b(y\))-30
2696 628 y Fl(t)m(yp)s(e)5 b Fp(_fdiv_t)29 b(ftruncate2)e(\(const)i
2697 Fl(t)m(yp)s(e)5 b Fp(&)30 b(x,)g(const)f Fl(t)m(yp)s(e)5
2698 b Fp(&)30 b(y\))-30 737 y Fl(t)m(yp)s(e)5 b Fp(_fdiv_t)29
2699 b(fround2)f(\(const)h Fl(t)m(yp)s(e)5 b Fp(&)30 b(x,)g(const)f
2700 Fl(t)m(yp)s(e)5 b Fp(&)30 b(y\))-30 868 y Fr(Other)g(applications)h
2701 (need)f(only)g(the)h(remainder)e(of)h(a)h(division.)40
2702 b(The)30 b(remainder)f(of)h(`)p Fp(floor)p Fr(')f(and)h(`)p
2703 Fp(ffloor)p Fr(')f(is)-30 978 y(called)k(`)p Fp(mod)p
2704 Fr(')e(\(abbreviation)h(of)g(\\mo)s(dulo"\).)43 b(The)31
2705 b(remainder)f(`)p Fp(truncate)p Fr(')g(and)h(`)p Fp(ftruncate)p
2706 Fr(')e(is)j(called)g(`)p Fp(rem)p Fr(')-30 1087 y(\(abbreviation)f(of)g
2707 (\\remainder"\).)45 1218 y Fq(\017)60 b Fp(mod\(x,y\))28
2708 b(=)i(floor2\(x,y\).remainder)25 b(=)30 b(x)g(-)g(floor\(x/y\)*y)45
2709 1349 y Fq(\017)60 b Fp(rem\(x,y\))28 b(=)i(truncate2\(x,y\).remainder)
2710 24 b(=)30 b(x)g(-)g(truncate\(x/y\)*y)-30 1501 y Fr(If)j
2711 Fp(x)f Fr(and)h Fp(y)f Fr(are)i(b)s(oth)e Fp(>)p Fr(=)g(0,)j
2712 Fp(mod\(x,y\))28 b(=)i(rem\(x,y\))e(>=)i(0)p Fr(.)48
2713 b(In)32 b(general,)j Fp(mod\(x,y\))c Fr(has)i(the)g(sign)g(of)g
2714 Fp(y)g Fr(or)g(is)-30 1611 y(zero,)e(and)f Fp(rem\(x,y\))e
2715 Fr(has)i(the)h(sign)f(of)h Fp(x)f Fr(or)g(is)h(zero.)-30
2716 1742 y(The)f(classes)h Fp(cl_R)p Fr(,)f Fp(cl_I)f Fr(de\014ne)h(the)g
2717 (follo)m(wing)i(op)s(erations:)-30 1894 y Fl(t)m(yp)s(e)k
2718 Fp(mod)29 b(\(const)g Fl(t)m(yp)s(e)5 b Fp(&)30 b(x,)g(const)f
2719 Fl(t)m(yp)s(e)5 b Fp(&)30 b(y\))-30 2004 y Fl(t)m(yp)s(e)36
2720 b Fp(rem)29 b(\(const)g Fl(t)m(yp)s(e)5 b Fp(&)30 b(x,)g(const)f
2721 Fl(t)m(yp)s(e)5 b Fp(&)30 b(y\))-30 2229 y Fs(4.7)68
2722 b(Ro)t(ots)-30 2418 y Fr(Eac)m(h)31 b(of)g(the)f(classes)i
2723 Fp(cl_R)p Fr(,)d Fp(cl_F)p Fr(,)g Fp(cl_SF)p Fr(,)h Fp(cl_FF)p
2724 Fr(,)f Fp(cl_DF)p Fr(,)g Fp(cl_LF)g Fr(de\014nes)g(the)i(follo)m(wing)h
2725 (op)s(eration:)-30 2570 y Fl(t)m(yp)s(e)k Fp(sqrt)29
2726 b(\(const)g Fl(t)m(yp)s(e)5 b Fp(&)30 b(x\))450 2680
2727 y(x)e Fr(m)m(ust)g(b)s(e)g Fp(>)p Fr(=)h(0.)40 b(This)28
2728 b(function)h(returns)e(the)i(square)g(ro)s(ot)g(of)g
2729 Fp(x)p Fr(,)g(normalized)f(to)i(b)s(e)e Fp(>)p Fr(=)g(0.)41
2730 b(If)28 b Fp(x)450 2789 y Fr(is)g(the)g(square)f(of)h(a)g(rational)h(n)
2731 m(um)m(b)s(er,)d Fp(sqrt\(x\))g Fr(will)i(b)s(e)f(a)h(rational)h(n)m
2732 (um)m(b)s(er,)d(else)j(it)f(will)g(return)450 2899 y(a)j(\015oating-p)s
2733 (oin)m(t)g(appro)m(ximation.)-30 3051 y(The)f(classes)h
2734 Fp(cl_RA)p Fr(,)e Fp(cl_I)h Fr(de\014ne)f(the)i(follo)m(wing)h(op)s
2735 (eration:)-30 3203 y Fp(cl_boolean)c(sqrtp)h(\(const)f
2736 Fl(t)m(yp)s(e)5 b Fp(&)31 b(x,)e Fl(t)m(yp)s(e)5 b Fp(*)31
2737 b(root\))450 3313 y Fr(This)e(tests)i(whether)e Fp(x)g
2738 Fr(is)h(a)h(p)s(erfect)e(square.)41 b(If)29 b(so,)i(it)f(returns)f
2739 (true)g(and)h(the)g(exact)h(square)f(ro)s(ot)450 3423
2740 y(in)g Fp(*root)p Fr(,)f(else)i(it)g(returns)f(false.)-30
2741 3575 y(F)-8 b(urthermore,)30 b(for)g(in)m(tegers,)i(similarly:)-30
2742 3727 y Fp(cl_boolean)c(isqrt)h(\(const)f Fl(t)m(yp)s(e)5
2743 b Fp(&)31 b(x,)e Fl(t)m(yp)s(e)5 b Fp(*)31 b(root\))450
2744 3837 y(x)j Fr(should)g(b)s(e)f Fp(>)p Fr(=)h(0.)53 b(This)34
2745 b(function)g(sets)h Fp(*root)e Fr(to)i Fp(floor\(sqrt\(x\)\))c
2746 Fr(and)j(returns)f(the)i(same)450 3946 y(v)-5 b(alue)31
2747 b(as)f Fp(sqrtp)p Fr(:)40 b(the)30 b(b)s(o)s(olean)h(v)-5
2748 b(alue)31 b Fp(\(expt\(*root,2\))26 b(==)k(x\))p Fr(.)-30
2749 4098 y(F)-8 b(or)31 b Fp(n)p Fr(th)f(ro)s(ots,)h(the)f(classes)i
2750 Fp(cl_RA)p Fr(,)d Fp(cl_I)g Fr(de\014ne)h(the)g(follo)m(wing)i(op)s
2751 (eration:)-30 4251 y Fp(cl_boolean)c(rootp)h(\(const)f
2752 Fl(t)m(yp)s(e)5 b Fp(&)31 b(x,)e(const)g(cl_I&)g(n,)h
2753 Fl(t)m(yp)s(e)5 b Fp(*)30 b(root\))450 4360 y(x)38 b
2754 Fr(m)m(ust)g(b)s(e)f Fp(>)p Fr(=)h(0.)66 b Fp(n)38 b
2755 Fr(m)m(ust)f(b)s(e)h Fp(>)g Fr(0.)66 b(This)37 b(tests)j(whether)e
2756 Fp(x)g Fr(is)g(an)h Fp(n)p Fr(th)f(p)s(o)m(w)m(er)g(of)h(a)g(rational)
2757 450 4470 y(n)m(um)m(b)s(er.)f(If)30 b(so,)h(it)g(returns)e(true)h(and)g
2758 (the)h(exact)h(ro)s(ot)e(in)g Fp(*root)p Fr(,)g(else)h(it)g(returns)e
2759 (false.)-30 4622 y(The)h(only)g(square)g(ro)s(ot)h(function)f(whic)m(h)
2760 g(accepts)i(negativ)m(e)h(n)m(um)m(b)s(ers)28 b(is)i(the)h(one)f(for)g
2761 (class)h Fp(cl_N)p Fr(:)-30 4774 y Fp(cl_N)e(sqrt)g(\(const)g(cl_N&)g
2762 (z\))450 4884 y Fr(Returns)44 b(the)h(square)f(ro)s(ot)h(of)g
2763 Fp(z)p Fr(,)j(as)d(de\014ned)f(b)m(y)g(the)h(form)m(ula)f
2764 Fp(sqrt\(z\))28 b(=)i(exp\(log\(z\)/2\))p Fr(.)450 4994
2765 y(Con)m(v)m(ersion)j(to)g(a)f(\015oating-p)s(oin)m(t)i(t)m(yp)s(e)e(or)
2766 h(to)g(a)f(complex)g(n)m(um)m(b)s(er)e(are)j(done)f(if)g(necessary)-8
2767 b(.)47 b(The)450 5103 y(range)25 b(of)f(the)h(result)f(is)g(the)h(righ)
2768 m(t)g(half)f(plane)g Fp(realpart\(sqrt\(z\)\))i(>=)k(0)24
2769 b Fr(including)f(the)i(p)s(ositiv)m(e)450 5213 y(imaginary)k(axis)h
2770 (and)f(0,)h(but)f(excluding)g(the)h(negativ)m(e)i(imaginary)d(axis.)40
2771 b(The)29 b(result)h(is)f(an)g(exact)450 5322 y(n)m(um)m(b)s(er)f(only)j
2772 (if)f Fp(z)g Fr(is)g(an)h(exact)g(n)m(um)m(b)s(er.)p
2775 17 18 bop -30 -116 a Fr(Chapter)30 b(4:)41 b(F)-8 b(unctions)31
2776 b(on)f(n)m(um)m(b)s(ers)2523 b(17)-30 299 y Fs(4.8)68
2777 b(T)-11 b(ranscenden)l(tal)45 b(functions)-30 495 y Fr(The)30
2778 b(transcenden)m(tal)i(functions)f(return)f(an)g(exact)j(result)e(if)f
2779 (the)i(argumen)m(t)e(is)h(exact)h(and)f(the)g(result)f(is)h(exact)-30
2780 605 y(as)j(w)m(ell.)54 b(Otherwise)34 b(they)g(m)m(ust)f(return)g
2781 (inexact)j(n)m(um)m(b)s(ers)c(ev)m(en)j(if)f(the)g(argumen)m(t)g(is)g
2782 (exact.)54 b(F)-8 b(or)35 b(example,)-30 714 y Fp(cos\(0\))29
2783 b(=)h(1)g Fr(returns)f(the)h(rational)i(n)m(um)m(b)s(er)c
2784 Fp(1)p Fr(.)-30 946 y Fn(4.8.1)63 b(Exp)s(onen)m(tial)40
2785 b(and)h(logarithmic)g(functions)-30 1169 y Fp(cl_R)29
2786 b(exp)h(\(const)e(cl_R&)h(x\))-30 1278 y(cl_N)g(exp)h(\(const)e(cl_N&)h
2787 (x\))450 1388 y Fr(Returns)35 b(the)i(exp)s(onen)m(tial)g(function)f
2788 (of)g Fp(x)p Fr(.)58 b(This)36 b(is)g Fp(e^x)f Fr(where)h
2789 Fp(e)g Fr(is)g(the)h(base)f(of)g(the)h(natural)450 1498
2790 y(logarithms.)k(The)29 b(range)i(of)g(the)f(result)h(is)f(the)h(en)m
2791 (tire)g(complex)f(plane)g(excluding)h(0.)-30 1661 y Fp(cl_R)e(ln)h
2792 (\(const)f(cl_R&)g(x\))450 1770 y(x)h Fr(m)m(ust)f(b)s(e)h
2793 Fp(>)g Fr(0.)41 b(Returns)30 b(the)g(\(natural\))h(logarithm)g(of)f(x.)
2794 -30 1933 y Fp(cl_N)f(log)h(\(const)e(cl_N&)h(x\))450
2795 2043 y Fr(Returns)g(the)g(\(natural\))h(logarithm)g(of)f(x.)41
2796 b(If)29 b Fp(x)g Fr(is)g(real)h(and)f(p)s(ositiv)m(e,)i(this)e(is)h
2797 Fp(ln\(x\))p Fr(.)39 b(In)29 b(general,)450 2153 y Fp(log\(x\))g(=)h
2798 (log\(abs\(x\)\))d(+)j(i*phase\(x\))p Fr(.)35 b(The)21
2799 b(range)i(of)f(the)g(result)f(is)h(the)g(strip)g(in)f(the)h(complex)450
2800 2262 y(plane)30 b Fp(-pi)g(<)g(imagpart\(log\(x\)\))c(<=)k(pi)p
2801 Fr(.)-30 2425 y Fp(cl_R)f(phase)g(\(const)g(cl_N&)g(x\))450
2802 2535 y Fr(Returns)41 b(the)i(angle)h(part)e(of)g Fp(x)g
2803 Fr(in)h(its)f(p)s(olar)g(represen)m(tation)i(as)f(a)f(complex)g(n)m(um)
2804 m(b)s(er.)75 b(That)450 2644 y(is,)41 b Fp(phase\(x\))29
2805 b(=)h(atan\(realpart\(x\),imagpa)o(rt\()o(x\)\))p Fr(.)60
2806 b(This)38 b(is)h(also)h(the)f(imaginary)g(part)g(of)450
2807 2754 y Fp(log\(x\))p Fr(.)52 b(The)35 b(range)g(of)g(the)g(result)g(is)
2808 g(the)g(in)m(terv)-5 b(al)36 b Fp(-pi)29 b(<)h(phase\(x\))e(<=)i(pi)p
2809 Fr(.)54 b(The)34 b(result)h(will)450 2864 y(b)s(e)30
2810 b(an)g(exact)i(n)m(um)m(b)s(er)c(only)i(if)h Fp(zerop\(x\))d
2811 Fr(or)i(if)h Fp(x)f Fr(is)g(real)h(and)f(p)s(ositiv)m(e.)-30
2812 3027 y Fp(cl_R)f(log)h(\(const)e(cl_R&)h(a,)h(const)f(cl_R&)g(b\))450
2813 3136 y(a)35 b Fr(and)g Fp(b)g Fr(m)m(ust)f(b)s(e)h Fp(>)g
2814 Fr(0.)56 b(Returns)35 b(the)g(logarithm)h(of)f Fp(a)g
2815 Fr(with)h(resp)s(ect)f(to)h(base)g Fp(b)p Fr(.)55 b Fp(log\(a,b\))28
2816 b(=)450 3246 y(ln\(a\)/ln\(b\))p Fr(.)38 b(The)30 b(result)g(can)g(b)s
2817 (e)g(exact)i(only)f(if)f Fp(a)g(=)g(1)g Fr(or)g(if)h
2818 Fp(a)f Fr(and)g Fp(b)g Fr(are)g(b)s(oth)g(rational.)-30
2819 3409 y Fp(cl_N)f(log)h(\(const)e(cl_N&)h(a,)h(const)f(cl_N&)g(b\))450
2820 3518 y Fr(Returns)g(the)i(logarithm)f(of)h Fp(a)f Fr(with)g(resp)s(ect)
2821 g(to)i(base)e Fp(b)p Fr(.)40 b Fp(log\(a,b\))29 b(=)h
2822 (log\(a\)/log\(b\))p Fr(.)-30 3682 y Fp(cl_N)f(expt)g(\(const)g(cl_N&)g
2823 (x,)h(const)f(cl_N&)g(y\))450 3791 y Fr(Exp)s(onen)m(tiation:)42
2824 b(Returns)29 b Fp(x^y)g(=)h(exp\(y*log\(x\)\))p Fr(.)-30
2825 3956 y(The)g(constan)m(t)i(e)e(=)g(exp\(1\))i(=)e(2.71828)6
2826 b(.)24 b(.)f(.)41 b(is)31 b(returned)e(b)m(y)h(the)h(follo)m(wing)h
2827 (functions:)-30 4121 y Fp(cl_F)d(exp1)g(\(cl_float_format_t)d(f\))450
2828 4231 y Fr(Returns)j(e)i(as)g(a)g(\015oat)g(of)f(format)g
2829 Fp(f)p Fr(.)-30 4394 y Fp(cl_F)f(exp1)g(\(const)g(cl_F&)g(y\))450
2830 4503 y Fr(Returns)g(e)i(in)f(the)h(\015oat)g(format)f(of)g
2831 Fp(y)p Fr(.)-30 4666 y Fp(cl_F)f(exp1)g(\(void\))450
2832 4776 y Fr(Returns)g(e)i(as)g(a)g(\015oat)g(of)f(format)g
2833 Fp(default_float_format)p Fr(.)-30 5007 y Fn(4.8.2)63
2834 b(T)-10 b(rigonometric)41 b(functions)-30 5230 y Fp(cl_R)29
2835 b(sin)h(\(const)e(cl_R&)h(x\))450 5340 y Fr(Returns)g
2836 Fp(sin\(x\))p Fr(.)39 b(The)30 b(range)h(of)g(the)f(result)g(is)h(the)f
2837 (in)m(terv)-5 b(al)32 b Fp(-1)e(<=)f(sin\(x\))g(<=)h(1)p
2840 18 19 bop -30 -116 a Fr(Chapter)30 b(4:)41 b(F)-8 b(unctions)31
2841 b(on)f(n)m(um)m(b)s(ers)2523 b(18)-30 299 y Fp(cl_N)29
2842 b(sin)h(\(const)e(cl_N&)h(z\))450 408 y Fr(Returns)g
2843 Fp(sin\(z\))p Fr(.)39 b(The)30 b(range)h(of)g(the)f(result)g(is)h(the)f
2844 (en)m(tire)i(complex)e(plane.)-30 562 y Fp(cl_R)f(cos)h(\(const)e
2845 (cl_R&)h(x\))450 671 y Fr(Returns)g Fp(cos\(x\))p Fr(.)39
2846 b(The)30 b(range)h(of)g(the)f(result)g(is)h(the)f(in)m(terv)-5
2847 b(al)32 b Fp(-1)e(<=)f(cos\(x\))g(<=)h(1)p Fr(.)-30 825
2848 y Fp(cl_N)f(cos)h(\(const)e(cl_N&)h(x\))450 934 y Fr(Returns)g
2849 Fp(cos\(z\))p Fr(.)39 b(The)30 b(range)h(of)g(the)f(result)g(is)h(the)f
2850 (en)m(tire)i(complex)e(plane.)-30 1088 y Fp(struct)f(cos_sin_t)f({)i
2851 (cl_R)f(cos;)g(cl_R)g(sin;)g(};)-30 1198 y(cos_sin_t)f(cos_sin)g
2852 (\(const)h(cl_R&)g(x\))450 1307 y Fr(Returns)20 b(b)s(oth)h
2853 Fp(sin\(x\))f Fr(and)h Fp(cos\(x\))p Fr(.)36 b(This)20
2854 b(is)i(more)e(e\016cien)m(t)j(than)e(computing)g(them)f(separately)-8
2855 b(.)450 1417 y(The)30 b(relation)h Fp(cos^2)e(+)h(sin^2)f(=)h(1)g
2856 Fr(will)h(hold)f(only)h(appro)m(ximately)-8 b(.)-30 1570
2857 y Fp(cl_R)29 b(tan)h(\(const)e(cl_R&)h(x\))-30 1680 y(cl_N)g(tan)h
2858 (\(const)e(cl_N&)h(x\))450 1789 y Fr(Returns)g Fp(tan\(x\))g(=)h
2859 (sin\(x\)/cos\(x\))p Fr(.)-30 1943 y Fp(cl_N)f(cis)h(\(const)e(cl_R&)h
2860 (x\))-30 2052 y(cl_N)g(cis)h(\(const)e(cl_N&)h(x\))450
2861 2162 y Fr(Returns)35 b Fp(exp\(i*x\))p Fr(.)54 b(The)35
2862 b(name)f(`)p Fp(cis)p Fr(')h(means)g(\\cos)i Fp(+)e Fr(i)h(sin",)h(b)s
2863 (ecause)e Fp(e^\(i*x\))29 b(=)h(cos\(x\))e(+)450 2271
2864 y(i*sin\(x\))p Fr(.)-30 2425 y Fp(cl_N)h(asin)g(\(const)g(cl_N&)g(z\))
2865 450 2534 y Fr(Returns)j Fp(arcsin\(z\))p Fr(.)44 b(This)32
2866 b(is)h(de\014ned)e(as)i Fp(arcsin\(z\))27 b(=)j
2867 (log\(iz+sqrt\(1-z^2\)\)/i)d Fr(and)32 b(sat-)450 2644
2868 y(is\014es)d Fp(arcsin\(-z\))e(=)j(-arcsin\(z\))p Fr(.)37
2869 b(The)29 b(range)g(of)g(the)g(result)f(is)h(the)g(strip)f(in)g(the)h
2870 (complex)g(do-)450 2754 y(main)34 b Fp(-pi/2)29 b(<=)h
2871 (realpart\(arcsin\(z\)\))25 b(<=)30 b(pi/2)p Fr(,)35
2872 b(excluding)g(the)h(n)m(um)m(b)s(ers)c(with)j Fp(realpart)450
2873 2863 y(=)30 b(-pi/2)f Fr(and)h Fp(imagpart)e(<)i(0)g
2874 Fr(and)g(the)g(n)m(um)m(b)s(ers)e(with)i Fp(realpart)e(=)i(pi/2)g
2875 Fr(and)f Fp(imagpart)f(>)j(0)p Fr(.)-30 3017 y Fp(cl_N)e(acos)g
2876 (\(const)g(cl_N&)g(z\))450 3126 y Fr(Returns)101 b Fp(arccos\(z\))p
2877 Fr(.)252 b(This)101 b(is)h(de\014ned)e(as)i Fp(arccos\(z\))28
2878 b(=)i(pi/2)f(-)h(arcsin\(z\))e(=)450 3236 y(log\(z+i*sqrt\(1-z^2\)\)/i)
2879 53 b Fr(and)58 b(satis\014es)h Fp(arccos\(-z\))28 b(=)i(pi)g(-)g
2880 (arccos\(z\))p Fr(.)123 b(The)59 b(range)450 3345 y(of)48
2881 b(the)g(result)g(is)g(the)h(strip)e(in)h(the)g(complex)g(domain)f
2882 Fp(0)30 b(<=)f(realpart\(arcsin\(z\)\))d(<=)j(pi)p Fr(,)450
2883 3455 y(excluding)46 b(the)g(n)m(um)m(b)s(ers)d(with)i
2884 Fp(realpart)28 b(=)i(0)46 b Fr(and)f Fp(imagpart)28 b(<)i(0)45
2885 b Fr(and)g(the)h(n)m(um)m(b)s(ers)d(with)450 3565 y Fp(realpart)28
2886 b(=)i(pi)g Fr(and)g Fp(imagpart)e(>)i(0)p Fr(.)-30 3718
2887 y Fp(cl_R)f(atan)g(\(const)g(cl_R&)g(x,)h(const)f(cl_R&)g(y\))450
2888 3828 y Fr(Returns)41 b(the)i(angle)g(of)g(the)f(p)s(olar)g(represen)m
2889 (tation)h(of)g(the)f(complex)g(n)m(um)m(b)s(er)e Fp(x+iy)p
2890 Fr(.)75 b(This)42 b(is)450 3937 y Fp(atan\(y/x\))31 b
2891 Fr(if)i Fp(x>0)p Fr(.)49 b(The)33 b(range)h(of)f(the)h(result)f(is)h
2892 (the)g(in)m(terv)-5 b(al)34 b Fp(-pi)c(<)g(atan\(x,y\))d(<=)j(pi)p
2893 Fr(.)49 b(The)450 4047 y(result)23 b(will)g(b)s(e)f(an)g(exact)i(n)m
2894 (um)m(b)s(er)d(only)i(if)f Fp(x)30 b(>)g(0)23 b Fr(and)f
2895 Fp(y)g Fr(is)h(the)g(exact)h Fp(0)p Fr(.)38 b(W)-10 b(ARNING:)23
2896 b(In)f(Common)450 4156 y(Lisp,)30 b(this)g(function)g(is)h(called)g(as)
2897 g Fp(\(atan)e(y)h(x\))p Fr(,)g(with)g(rev)m(ersed)h(order)e(of)i
2898 (argumen)m(ts.)-30 4310 y Fp(cl_R)e(atan)g(\(const)g(cl_R&)g(x\))450
2899 4419 y Fr(Returns)40 b Fp(arctan\(x\))p Fr(.)68 b(This)40
2900 b(is)h(the)g(same)f(as)g Fp(atan\(1,x\))p Fr(.)69 b(The)40
2901 b(range)h(of)g(the)g(result)f(is)h(the)450 4529 y(in)m(terv)-5
2902 b(al)38 b Fp(-pi/2)29 b(<)h(atan\(x\))e(<)i(pi/2)p Fr(.)59
2903 b(The)36 b(result)g(will)h(b)s(e)f(an)h(exact)h(n)m(um)m(b)s(er)c(only)
2904 j(if)g Fp(x)f Fr(is)h(the)450 4639 y(exact)32 b Fp(0)p
2905 Fr(.)-30 4792 y Fp(cl_N)d(atan)g(\(const)g(cl_N&)g(z\))450
2906 4902 y Fr(Returns)51 b Fp(arctan\(z\))p Fr(.)102 b(This)52
2907 b(is)f(de\014ned)g(as)h Fp(arctan\(z\))28 b(=)i
2908 (\(log\(1+iz\)-log\(1-iz\)\))25 b(/)30 b(2i)450 5011
2909 y Fr(and)44 b(satis\014es)h Fp(arctan\(-z\))28 b(=)i(-arctan\(z\))p
2910 Fr(.)80 b(The)44 b(range)h(of)g(the)g(result)f(is)h(the)f(strip)g(in)h
2911 (the)450 5121 y(complex)28 b(domain)g Fp(-pi/2)h(<=)g
2912 (realpart\(arctan\(z\)\))d(<=)j(pi/2)p Fr(,)f(excluding)h(the)g(n)m(um)
2913 m(b)s(ers)d(with)450 5230 y Fp(realpart)i(=)i(-pi/2)49
2914 b Fr(and)i Fp(imagpart)28 b(>=)i(0)50 b Fr(and)g(the)h(n)m(um)m(b)s
2915 (ers)d(with)j Fp(realpart)28 b(=)i(pi/2)50 b Fr(and)450
2916 5340 y Fp(imagpart)28 b(<=)i(0)p Fr(.)p eop
2918 19 20 bop -30 -116 a Fr(Chapter)30 b(4:)41 b(F)-8 b(unctions)31
2919 b(on)f(n)m(um)m(b)s(ers)2523 b(19)-30 299 y(Arc)m(himedes')30
2920 b(constan)m(t)i(pi)e(=)g(3.14)6 b(.)23 b(.)f(.)42 b(is)30
2921 b(returned)g(b)m(y)g(the)g(follo)m(wing)i(functions:)-30
2922 469 y Fp(cl_F)d(pi)h(\(cl_float_format_t)25 b(f\))450
2923 578 y Fr(Returns)k(pi)i(as)f(a)h(\015oat)g(of)f(format)g
2924 Fp(f)p Fr(.)-30 745 y Fp(cl_F)f(pi)h(\(const)f(cl_F&)g(y\))450
2925 855 y Fr(Returns)g(pi)i(in)f(the)g(\015oat)h(format)f(of)g
2926 Fp(y)p Fr(.)-30 1021 y Fp(cl_F)f(pi)h(\(void\))450 1131
2927 y Fr(Returns)f(pi)i(as)f(a)h(\015oat)g(of)f(format)g
2928 Fp(default_float_format)p Fr(.)-30 1369 y Fn(4.8.3)63
2929 b(Hyp)s(erb)s(olic)42 b(functions)-30 1597 y Fp(cl_R)29
2930 b(sinh)g(\(const)g(cl_R&)g(x\))450 1707 y Fr(Returns)g
2931 Fp(sinh\(x\))p Fr(.)-30 1873 y Fp(cl_N)g(sinh)g(\(const)g(cl_N&)g(z\))
2932 450 1983 y Fr(Returns)g Fp(sinh\(z\))p Fr(.)39 b(The)30
2933 b(range)h(of)f(the)h(result)f(is)h(the)f(en)m(tire)h(complex)g(plane.)
2934 -30 2149 y Fp(cl_R)e(cosh)g(\(const)g(cl_R&)g(x\))450
2935 2259 y Fr(Returns)g Fp(cosh\(x\))p Fr(.)39 b(The)30 b(range)h(of)f(the)
2936 h(result)f(is)h(the)f(in)m(terv)-5 b(al)32 b Fp(cosh\(x\))c(>=)i(1)p
2937 Fr(.)-30 2425 y Fp(cl_N)f(cosh)g(\(const)g(cl_N&)g(z\))450
2938 2535 y Fr(Returns)g Fp(cosh\(z\))p Fr(.)39 b(The)30 b(range)h(of)f(the)
2939 h(result)f(is)h(the)f(en)m(tire)h(complex)g(plane.)-30
2940 2701 y Fp(struct)e(cosh_sinh_t)e({)j(cl_R)f(cosh;)g(cl_R)g(sinh;)g(};)
2941 -30 2811 y(cosh_sinh_t)e(cosh_sinh)h(\(const)h(cl_R&)g(x\))450
2942 2921 y Fr(Returns)i(b)s(oth)g Fp(sinh\(x\))f Fr(and)h
2943 Fp(cosh\(x\))p Fr(.)43 b(This)31 b(is)h(more)f(e\016cien)m(t)i(than)f
2944 (computing)f(them)f(sepa-)450 3030 y(rately)-8 b(.)42
2945 b(The)30 b(relation)i Fp(cosh^2)c(-)i(sinh^2)f(=)h(1)g
2946 Fr(will)h(hold)f(only)g(appro)m(ximately)-8 b(.)-30 3197
2947 y Fp(cl_R)29 b(tanh)g(\(const)g(cl_R&)g(x\))-30 3306
2948 y(cl_N)g(tanh)g(\(const)g(cl_N&)g(x\))450 3416 y Fr(Returns)g
2949 Fp(tanh\(x\))g(=)h(sinh\(x\)/cosh\(x\))p Fr(.)-30 3582
2950 y Fp(cl_N)f(asinh)g(\(const)g(cl_N&)g(z\))450 3692 y
2951 Fr(Returns)h Fp(arsinh\(z\))p Fr(.)40 b(This)31 b(is)g(de\014ned)f(as)h
2952 Fp(arsinh\(z\))d(=)i(log\(z+sqrt\(1+z^2\)\))c Fr(and)k(satis\014es)450
2953 3802 y Fp(arsinh\(-z\))e(=)i(-arsinh\(z\))p Fr(.)36 b(The)26
2954 b(range)h(of)g(the)g(result)f(is)h(the)g(strip)f(in)g(the)h(complex)f
2955 (domain)f Fp(-)450 3911 y(pi/2)k(<=)h(imagpart\(arsinh\(z\)\))25
2956 b(<=)30 b(pi/2)p Fr(,)22 b(excluding)g(the)g(n)m(um)m(b)s(ers)d(with)i
2957 Fp(imagpart)28 b(=)i(-pi/2)450 4021 y Fr(and)g Fp(realpart)e(>)i(0)g
2958 Fr(and)g(the)g(n)m(um)m(b)s(ers)e(with)i Fp(imagpart)e(=)i(pi/2)g
2959 Fr(and)f Fp(realpart)g(<)h(0)p Fr(.)-30 4187 y Fp(cl_N)f(acosh)g
2960 (\(const)g(cl_N&)g(z\))450 4297 y Fr(Returns)35 b Fp(arcosh\(z\))p
2961 Fr(.)53 b(This)35 b(is)h(de\014ned)e(as)i Fp(arcosh\(z\))28
2962 b(=)i(2*log\(sqrt\(\(z+1\)/2\)+sqr)o(t\(\(z)o(-)450 4406
2963 y(1\)/2\)\))p Fr(.)114 b(The)55 b(range)h(of)g(the)g(result)f(is)g(the)
2964 h(half-strip)f(in)h(the)f(complex)g(domain)g Fp(-pi)29
2965 b(<)450 4516 y(imagpart\(arcosh\(z\)\))c(<=)30 b(pi,)f
2966 (realpart\(arcosh\(z\)\))c(>=)30 b(0)p Fr(,)99 b(excluding)85
2967 b(the)g(n)m(um)m(b)s(ers)450 4626 y(with)30 b Fp(realpart)e(=)i(0)g
2968 Fr(and)g Fp(-pi)f(<)i(imagpart)d(<)i(0)p Fr(.)-30 4792
2969 y Fp(cl_N)f(atanh)g(\(const)g(cl_N&)g(z\))450 4902 y
2970 Fr(Returns)24 b Fp(artanh\(z\))p Fr(.)37 b(This)24 b(is)h(de\014ned)f
2971 (as)h Fp(artanh\(z\))j(=)i(\(log\(1+z\)-log\(1-z\)\))25
2972 b(/)30 b(2)25 b Fr(and)f(sat-)450 5011 y(is\014es)29
2973 b Fp(artanh\(-z\))e(=)j(-artanh\(z\))p Fr(.)37 b(The)29
2974 b(range)g(of)g(the)g(result)f(is)h(the)g(strip)f(in)g(the)h(complex)g
2975 (do-)450 5121 y(main)34 b Fp(-pi/2)29 b(<=)h(imagpart\(artanh\(z\)\))25
2976 b(<=)30 b(pi/2)p Fr(,)35 b(excluding)g(the)h(n)m(um)m(b)s(ers)c(with)j
2977 Fp(imagpart)450 5230 y(=)30 b(-pi/2)j Fr(and)h Fp(realpart)28
2978 b(<=)i(0)k Fr(and)g(the)h(n)m(um)m(b)s(ers)d(with)i Fp(imagpart)28
2979 b(=)i(pi/2)k Fr(and)f Fp(realpart)c(>=)450 5340 y(0)p
2982 20 21 bop -30 -116 a Fr(Chapter)30 b(4:)41 b(F)-8 b(unctions)31
2983 b(on)f(n)m(um)m(b)s(ers)2523 b(20)-30 299 y Fn(4.8.4)63
2984 b(Euler)40 b(gamma)-30 498 y Fr(Euler's)30 b(constan)m(t)i(C)e(=)g
2985 (0.577)6 b(.)24 b(.)e(.)41 b(is)31 b(returned)e(b)m(y)h(the)h(follo)m
2986 (wing)h(functions:)-30 666 y Fp(cl_F)d(eulerconst)f
2987 (\(cl_float_format_t)d(f\))450 776 y Fr(Returns)k(Euler's)i(constan)m
2988 (t)g(as)g(a)g(\015oat)g(of)f(format)g Fp(f)p Fr(.)-30
2989 942 y Fp(cl_F)f(eulerconst)f(\(const)g(cl_F&)h(y\))450
2990 1051 y Fr(Returns)g(Euler's)i(constan)m(t)g(in)f(the)h(\015oat)g
2991 (format)f(of)g Fp(y)p Fr(.)-30 1217 y Fp(cl_F)f(eulerconst)f(\(void\))
2992 450 1326 y Fr(Returns)h(Euler's)i(constan)m(t)g(as)g(a)g(\015oat)g(of)f
2993 (format)g Fp(default_float_format)p Fr(.)-30 1495 y(Catalan's)i
2994 (constan)m(t)f(G)g(=)f(0.915)6 b(.)24 b(.)e(.)42 b(is)30
2995 b(returned)f(b)m(y)h(the)h(follo)m(wing)h(functions:)-30
2996 1664 y Fp(cl_F)d(catalanconst)e(\(cl_float_format_t)f(f\))450
2997 1773 y Fr(Returns)j(Catalan's)j(constan)m(t)g(as)e(a)h(\015oat)g(of)g
2998 (format)e Fp(f)p Fr(.)-30 1939 y Fp(cl_F)g(catalanconst)e(\(const)i
2999 (cl_F&)g(y\))450 2048 y Fr(Returns)g(Catalan's)j(constan)m(t)g(in)e
3000 (the)g(\015oat)h(format)f(of)h Fp(y)p Fr(.)-30 2214 y
3001 Fp(cl_F)e(catalanconst)e(\(void\))450 2324 y Fr(Returns)i(Catalan's)j
3002 (constan)m(t)g(as)e(a)h(\015oat)g(of)g(format)e Fp
3003 (default_float_format)p Fr(.)-30 2560 y Fn(4.8.5)63 b(Riemann)39
3004 b(zeta)-30 2759 y Fr(Riemann's)29 b(zeta)j(function)e(at)h(an)g(in)m
3005 (tegral)h(p)s(oin)m(t)e Fp(s>1)f Fr(is)i(returned)e(b)m(y)h(the)h
3006 (follo)m(wing)h(functions:)-30 2927 y Fp(cl_F)d(zeta)g(\(int)h(s,)f
3007 (cl_float_format_t)d(f\))450 3037 y Fr(Returns)j(Riemann's)h(zeta)i
3008 (function)e(at)h Fp(s)f Fr(as)g(a)h(\015oat)g(of)g(format)e
3009 Fp(f)p Fr(.)-30 3203 y Fp(cl_F)g(zeta)g(\(int)h(s,)f(const)g(cl_F&)g
3010 (y\))450 3312 y Fr(Returns)g(Riemann's)h(zeta)i(function)e(at)h
3011 Fp(s)f Fr(in)g(the)g(\015oat)h(format)f(of)h Fp(y)p Fr(.)-30
3012 3478 y Fp(cl_F)e(zeta)g(\(int)h(s\))450 3587 y Fr(Returns)f(Riemann's)h
3013 (zeta)i(function)e(at)h Fp(s)f Fr(as)g(a)h(\015oat)g(of)g(format)e
3014 Fp(default_float_format)p Fr(.)-30 3860 y Fs(4.9)68 b(F)-11
3015 b(unctions)44 b(on)h(in)l(tegers)-30 4155 y Fn(4.9.1)63
3016 b(Logical)40 b(functions)-30 4354 y Fr(In)m(tegers,)31
3017 b(when)d(view)m(ed)j(as)f(in)f(t)m(w)m(o's)i(complemen)m(t)e(notation,)
3018 i(can)f(b)s(e)f(though)m(t)h(as)g(in\014nite)f(bit)h(strings)f(where)
3019 -30 4463 y(the)i(bits')f(v)-5 b(alues)31 b(ev)m(en)m(tually)h(are)f
3020 (constan)m(t.)42 b(F)-8 b(or)31 b(example,)401 4598 y
3021 Fp(17)47 b(=)h(......00010001)401 4702 y(-6)f(=)h(......11111010)-30
3022 4843 y Fr(The)33 b(logical)j(op)s(erations)e(view)g(in)m(tegers)h(as)f
3023 (suc)m(h)f(bit)h(strings)f(and)h(op)s(erate)g(on)f(eac)m(h)i(of)f(the)g
3024 (bit)g(p)s(ositions)f(in)-30 4952 y(parallel.)-30 5121
3025 y Fp(cl_I)c(lognot)g(\(const)g(cl_I&)f(x\))-30 5230 y(cl_I)h(operator)f
3026 (~)i(\(const)f(cl_I&)g(x\))450 5340 y Fr(Logical)j(not,)f(lik)m(e)h
3027 Fp(~x)e Fr(in)g(C.)g(This)g(is)g(the)h(same)e(as)i Fp(-1-x)p
3030 21 22 bop -30 -116 a Fr(Chapter)30 b(4:)41 b(F)-8 b(unctions)31
3031 b(on)f(n)m(um)m(b)s(ers)2523 b(21)-30 299 y Fp(cl_I)29
3032 b(logand)g(\(const)g(cl_I&)f(x,)i(const)f(cl_I&)g(y\))-30
3033 408 y(cl_I)g(operator)f(&)i(\(const)f(cl_I&)g(x,)h(const)f(cl_I&)g(y\))
3034 450 518 y Fr(Logical)j(and,)e(lik)m(e)i Fp(x)e(&)g(y)g
3035 Fr(in)g(C.)-30 673 y Fp(cl_I)f(logior)g(\(const)g(cl_I&)f(x,)i(const)f
3036 (cl_I&)g(y\))-30 783 y(cl_I)g(operator)f(|)i(\(const)f(cl_I&)g(x,)h
3037 (const)f(cl_I&)g(y\))450 892 y Fr(Logical)j(\(inclusiv)m(e\))g(or,)f
3038 (lik)m(e)h Fp(x)e(|)g(y)g Fr(in)g(C.)-30 1047 y Fp(cl_I)f(logxor)g
3039 (\(const)g(cl_I&)f(x,)i(const)f(cl_I&)g(y\))-30 1157
3040 y(cl_I)g(operator)f(^)i(\(const)f(cl_I&)g(x,)h(const)f(cl_I&)g(y\))450
3041 1267 y Fr(Exclusiv)m(e)i(or,)g(lik)m(e)h Fp(x)e(^)g(y)g
3042 Fr(in)g(C.)-30 1422 y Fp(cl_I)f(logeqv)g(\(const)g(cl_I&)f(x,)i(const)f
3043 (cl_I&)g(y\))450 1531 y Fr(Bit)m(wise)j(equiv)-5 b(alence,)32
3044 b(lik)m(e)g Fp(~\(x)d(^)h(y\))g Fr(in)g(C.)-30 1686 y
3045 Fp(cl_I)f(lognand)g(\(const)f(cl_I&)h(x,)h(const)f(cl_I&)g(y\))450
3046 1796 y Fr(Bit)m(wise)j(not)f(and,)f(lik)m(e)h Fp(~\(x)f(&)g(y\))g
3047 Fr(in)g(C.)-30 1951 y Fp(cl_I)f(lognor)g(\(const)g(cl_I&)f(x,)i(const)f
3048 (cl_I&)g(y\))450 2060 y Fr(Bit)m(wise)j(not)f(or,)f(lik)m(e)i
3049 Fp(~\(x)d(|)h(y\))g Fr(in)g(C.)-30 2216 y Fp(cl_I)f(logandc1)f(\(const)
3050 h(cl_I&)g(x,)h(const)f(cl_I&)g(y\))450 2325 y Fr(Logical)j(and,)e
3051 (complemen)m(ting)g(the)h(\014rst)e(argumen)m(t,)h(lik)m(e)i
3052 Fp(~x)e(&)g(y)g Fr(in)g(C.)-30 2480 y Fp(cl_I)f(logandc2)f(\(const)h
3053 (cl_I&)g(x,)h(const)f(cl_I&)g(y\))450 2590 y Fr(Logical)j(and,)e
3054 (complemen)m(ting)g(the)h(second)f(argumen)m(t,)g(lik)m(e)i
3055 Fp(x)e(&)g(~y)g Fr(in)g(C.)-30 2745 y Fp(cl_I)f(logorc1)g(\(const)f
3056 (cl_I&)h(x,)h(const)f(cl_I&)g(y\))450 2854 y Fr(Logical)j(or,)f
3057 (complemen)m(ting)f(the)g(\014rst)g(argumen)m(t,)g(lik)m(e)i
3058 Fp(~x)d(|)i(y)f Fr(in)g(C.)-30 3009 y Fp(cl_I)f(logorc2)g(\(const)f
3059 (cl_I&)h(x,)h(const)f(cl_I&)g(y\))450 3119 y Fr(Logical)j(or,)f
3060 (complemen)m(ting)f(the)g(second)h(argumen)m(t,)f(lik)m(e)i
3061 Fp(x)e(|)g(~y)f Fr(in)h(C.)-30 3274 y(These)g(op)s(erations)h(are)g
3062 (all)g(a)m(v)-5 b(ailable)33 b(though)d(the)g(function)-30
3063 3429 y Fp(cl_I)f(boole)g(\(cl_boole)f(op,)h(const)g(cl_I&)g(x,)h(const)
3064 f(cl_I&)g(y\))-30 3561 y Fr(where)35 b Fp(op)f Fr(m)m(ust)g(ha)m(v)m(e)
3065 j(one)e(of)g(the)h(16)g(v)-5 b(alues)35 b(\(eac)m(h)i(one)e(stands)g
3066 (for)g(a)g(function)g(whic)m(h)g(com)m(bines)g(t)m(w)m(o)h(bits)-30
3067 3671 y(in)m(to)31 b(one)g(bit\):)42 b Fp(boole_clr)p
3068 Fr(,)28 b Fp(boole_set)p Fr(,)g Fp(boole_1)p Fr(,)h Fp(boole_2)p
3069 Fr(,)f Fp(boole_c1)p Fr(,)h Fp(boole_c2)p Fr(,)f Fp(boole_and)p
3070 Fr(,)h Fp(boole_)-30 3781 y(ior)p Fr(,)45 b Fp(boole_xor)p
3071 Fr(,)f Fp(boole_eqv)p Fr(,)g Fp(boole_nand)p Fr(,)f Fp(boole_nor)p
3072 Fr(,)h Fp(boole_andc1)p Fr(,)f Fp(boole_andc2)p Fr(,)g
3073 Fp(boole_orc1)p Fr(,)-30 3890 y Fp(boole_orc2)p Fr(.)-30
3074 4023 y(Other)30 b(functions)g(that)h(view)f(in)m(tegers)i(as)f(bit)f
3075 (strings:)-30 4178 y Fp(cl_boolean)e(logtest)g(\(const)h(cl_I&)g(x,)g
3076 (const)g(cl_I&)g(y\))450 4287 y Fr(Returns)g(true)i(if)f(some)g(bit)g
3077 (is)h(set)f(in)h(b)s(oth)e Fp(x)h Fr(and)g Fp(y)p Fr(,)g(i.e.)42
3078 b(if)30 b Fp(logand\(x,y\))d(!=)j(0)p Fr(.)-30 4442 y
3079 Fp(cl_boolean)e(logbitp)g(\(const)h(cl_I&)g(n,)g(const)g(cl_I&)g(x\))
3080 450 4552 y Fr(Returns)g(true)i(if)f(the)h Fp(n)p Fr(th)e(bit)i(\(from)e
3081 (the)i(righ)m(t\))g(of)f Fp(x)h Fr(is)f(set.)41 b(Bit)32
3082 b(0)e(is)h(the)f(least)i(signi\014can)m(t)f(bit.)-30
3083 4707 y Fp(uintL)e(logcount)f(\(const)h(cl_I&)g(x\))450
3084 4816 y Fr(Returns)g(the)h(n)m(um)m(b)s(er)d(of)j(one)h(bits)e(in)h
3085 Fp(x)p Fr(,)g(if)f Fp(x)h(>)p Fr(=)f(0,)i(or)e(the)h(n)m(um)m(b)s(er)e
3086 (of)i(zero)h(bits)e(in)h Fp(x)p Fr(,)g(if)f Fp(x)h(<)f
3087 Fr(0.)-30 4972 y(The)h(follo)m(wing)i(functions)e(op)s(erate)h(on)f(in)
3088 m(terv)-5 b(als)31 b(of)g(bits)f(in)g(in)m(tegers.)42
3089 b(The)30 b(t)m(yp)s(e)210 5098 y Fp(struct)46 b(cl_byte)g({)h(uintL)g
3090 (size;)f(uintL)g(position;)g(};)-30 5230 y Fr(represen)m(ts)41
3091 b(the)h(bit)f(in)m(terv)-5 b(al)43 b(con)m(taining)g(the)e(bits)g
3092 Fp(position)6 b Fr(.)20 b(.)i(.)12 b Fp(position+size-1)37
3093 b Fr(of)k(an)g(in)m(teger.)75 b(The)-30 5340 y(constructor)31
3094 b Fp(cl_byte\(size,position\))24 b Fr(constructs)31 b(a)f
3095 Fp(cl_byte)p Fr(.)p eop
3097 22 23 bop -30 -116 a Fr(Chapter)30 b(4:)41 b(F)-8 b(unctions)31
3098 b(on)f(n)m(um)m(b)s(ers)2523 b(22)-30 299 y Fp(cl_I)29
3099 b(ldb)h(\(const)e(cl_I&)h(n,)h(const)f(cl_byte&)f(b\))450
3100 408 y Fr(extracts)k(the)e(bits)g(of)h Fp(n)f Fr(describ)s(ed)f(b)m(y)h
3101 (the)h(bit)f(in)m(terv)-5 b(al)32 b Fp(b)e Fr(and)g(returns)f(them)g
3102 (as)i(a)f(nonnegativ)m(e)450 518 y(in)m(teger)i(with)e
3103 Fp(b.size)e Fr(bits.)-30 670 y Fp(cl_boolean)g(ldb_test)g(\(const)g
3104 (cl_I&)h(n,)h(const)f(cl_byte&)f(b\))450 780 y Fr(Returns)h(true)i(if)f
3105 (some)g(bit)g(describ)s(ed)f(b)m(y)i(the)f(bit)h(in)m(terv)-5
3106 b(al)31 b Fp(b)f Fr(is)h(set)g(in)f Fp(n)p Fr(.)-30 932
3107 y Fp(cl_I)f(dpb)h(\(const)e(cl_I&)h(newbyte,)f(const)h(cl_I&)g(n,)h
3108 (const)f(cl_byte&)f(b\))450 1041 y Fr(Returns)j Fp(n)p
3109 Fr(,)h(with)f(the)h(bits)f(describ)s(ed)g(b)m(y)g(the)h(bit)g(in)m
3110 (terv)-5 b(al)32 b Fp(b)g Fr(replaced)g(b)m(y)f Fp(newbyte)p
3111 Fr(.)43 b(Only)31 b(the)450 1151 y(lo)m(w)m(est)h Fp(b.size)d
3112 Fr(bits)h(of)h Fp(newbyte)d Fr(are)j(relev)-5 b(an)m(t.)-30
3113 1303 y(The)33 b(functions)f Fp(ldb)h Fr(and)f Fp(dpb)g
3114 Fr(implicitly)i(shift.)49 b(The)32 b(follo)m(wing)j(functions)e(are)g
3115 (their)g(coun)m(terparts)h(without)-30 1412 y(shifting:)-30
3116 1564 y Fp(cl_I)29 b(mask_field)f(\(const)g(cl_I&)h(n,)h(const)f
3117 (cl_byte&)f(b\))450 1674 y Fr(returns)35 b(an)g(in)m(teger)j(with)d
3118 (the)h(bits)g(describ)s(ed)f(b)m(y)h(the)g(bit)g(in)m(terv)-5
3119 b(al)37 b Fp(b)e Fr(copied)i(from)d(the)i(corre-)450
3120 1784 y(sp)s(onding)29 b(bits)h(in)g Fp(n)p Fr(,)g(the)h(other)g(bits)f
3121 (zero.)-30 1936 y Fp(cl_I)f(deposit_field)e(\(const)i(cl_I&)g(newbyte,)
3122 f(const)h(cl_I&)g(n,)g(const)g(cl_byte&)f(b\))450 2045
3123 y Fr(returns)e(an)h(in)m(teger)h(where)f(the)g(bits)g(describ)s(ed)f(b)
3124 m(y)h(the)h(bit)f(in)m(terv)-5 b(al)28 b Fp(b)f Fr(come)g(from)e
3125 Fp(newbyte)h Fr(and)450 2155 y(the)31 b(other)f(bits)g(come)h(from)e
3126 Fp(n)p Fr(.)-30 2307 y(The)h(follo)m(wing)i(relations)f(hold:)150
3127 2438 y Fp(ldb)e(\(n,)h(b\))g(=)g(mask_field\(n,)d(b\))i(>>)h
3128 (b.position)p Fr(,)150 2568 y Fp(dpb)f(\(newbyte,)f(n,)i(b\))g(=)g
3129 (deposit_field)d(\(newbyte)h(<<)h(b.position,)f(n,)i(b\))p
3130 Fr(,)150 2699 y Fp(deposit_field\(newbyte,n,)o(b\))24
3131 b(=)30 b(n)g(^)g(mask_field\(n,b\))c(^)31 b(mask_field\(new_byte,b\))p
3132 Fr(.)-30 2851 y(The)41 b(follo)m(wing)h(op)s(erations)g(on)f(in)m
3133 (tegers)i(as)e(bit)h(strings)f(are)g(e\016cien)m(t)i(shortcuts)e(for)g
3134 (common)f(arithmetic)-30 2961 y(op)s(erations:)-30 3113
3135 y Fp(cl_boolean)28 b(oddp)h(\(const)f(cl_I&)h(x\))450
3136 3222 y Fr(Returns)g(true)i(if)f(the)h(least)g(signi\014can)m(t)g(bit)g
3137 (of)f Fp(x)g Fr(is)h(1.)41 b(Equiv)-5 b(alen)m(t)31 b(to)g
3138 Fp(mod\(x,2\))d(!=)i(0)p Fr(.)-30 3374 y Fp(cl_boolean)e(evenp)h
3139 (\(const)f(cl_I&)h(x\))450 3484 y Fr(Returns)g(true)i(if)f(the)h(least)
3140 g(signi\014can)m(t)g(bit)g(of)f Fp(x)g Fr(is)h(0.)41
3141 b(Equiv)-5 b(alen)m(t)31 b(to)g Fp(mod\(x,2\))d(==)i(0)p
3142 Fr(.)-30 3636 y Fp(cl_I)f(operator)f(<<)i(\(const)f(cl_I&)g(x,)g(const)
3143 g(cl_I&)g(n\))450 3746 y Fr(Shifts)h Fp(x)g Fr(b)m(y)g
3144 Fp(n)g Fr(bits)g(to)h(the)g(left.)41 b Fp(n)30 b Fr(should)g(b)s(e)f
3145 Fp(>)p Fr(=0.)41 b(Equiv)-5 b(alen)m(t)31 b(to)g Fp(x)f(*)g
3146 (expt\(2,n\))p Fr(.)-30 3898 y Fp(cl_I)f(operator)f(>>)i(\(const)f
3147 (cl_I&)g(x,)g(const)g(cl_I&)g(n\))450 4007 y Fr(Shifts)e
3148 Fp(x)g Fr(b)m(y)h Fp(n)f Fr(bits)h(to)g(the)g(righ)m(t.)41
3149 b Fp(n)27 b Fr(should)g(b)s(e)g Fp(>)p Fr(=0.)40 b(Bits)28
3150 b(shifted)g(out)g(to)g(the)g(righ)m(t)g(are)g(thro)m(wn)450
3151 4117 y(a)m(w)m(a)m(y)-8 b(.)43 b(Equiv)-5 b(alen)m(t)31
3152 b(to)g Fp(floor\(x)e(/)h(expt\(2,n\)\))p Fr(.)-30 4269
3153 y Fp(cl_I)f(ash)h(\(const)e(cl_I&)h(x,)h(const)f(cl_I&)g(y\))450
3154 4378 y Fr(Shifts)37 b Fp(x)g Fr(b)m(y)h Fp(y)f Fr(bits)h(to)g(the)g
3155 (left)h(\(if)f Fp(y>)p Fr(=0\))f(or)h(b)m(y)g Fp(-y)f
3156 Fr(bits)g(to)i(the)f(righ)m(t)g(\(if)g Fp(y<)p Fr(=0\).)63
3157 b(In)37 b(other)450 4488 y(w)m(ords,)30 b(this)g(returns)g
3158 Fp(floor\(x)e(*)i(expt\(2,y\)\))p Fr(.)-30 4640 y Fp(uintL)f
3159 (integer_length)d(\(const)j(cl_I&)g(x\))450 4750 y Fr(Returns)44
3160 b(the)i(n)m(um)m(b)s(er)d(of)j(bits)f(\(excluding)h(the)g(sign)f(bit\))
3161 h(needed)f(to)h(represen)m(t)f Fp(x)g Fr(in)g(t)m(w)m(o's)450
3162 4859 y(complemen)m(t)31 b(notation.)45 b(This)30 b(is)i(the)g(smallest)
3163 f(n)g Fp(>)p Fr(=)g(0)h(suc)m(h)f(that)h(-2)p Fp(^)p
3164 Fr(n)f Fp(<)p Fr(=)g(x)g Fp(<)g Fr(2)p Fp(^)p Fr(n.)44
3165 b(If)31 b(x)g Fp(>)g Fr(0,)450 4969 y(this)f(is)h(the)f(unique)g(n)g
3166 Fp(>)g Fr(0)g(suc)m(h)g(that)h(2)p Fp(^)p Fr(\(n-1\))h
3167 Fp(<)p Fr(=)e(x)g Fp(<)g Fr(2)p Fp(^)p Fr(n.)-30 5121
3168 y Fp(uintL)f(ord2)g(\(const)g(cl_I&)g(x\))450 5230 y(x)e
3169 Fr(m)m(ust)f(b)s(e)h(non-zero.)40 b(This)27 b(function)g(returns)f(the)
3170 i(n)m(um)m(b)s(er)d(of)j(0)g(bits)f(at)h(the)g(righ)m(t)g(of)f
3171 Fp(x)g Fr(in)g(t)m(w)m(o's)450 5340 y(complemen)m(t)j(notation.)42
3172 b(This)29 b(is)i(the)f(largest)i(n)e Fp(>)p Fr(=)f(0)i(suc)m(h)f(that)h
3173 (2)p Fp(^)p Fr(n)f(divides)g Fp(x)p Fr(.)p eop
3175 23 24 bop -30 -116 a Fr(Chapter)30 b(4:)41 b(F)-8 b(unctions)31
3176 b(on)f(n)m(um)m(b)s(ers)2523 b(23)-30 299 y Fp(uintL)29
3177 b(power2p)f(\(const)h(cl_I&)g(x\))450 408 y(x)f Fr(m)m(ust)e(b)s(e)i
3178 Fp(>)f Fr(0.)41 b(This)27 b(function)g(c)m(hec)m(ks)j(whether)d
3179 Fp(x)h Fr(is)g(a)g(p)s(o)m(w)m(er)g(of)g(2.)40 b(If)28
3180 b Fp(x)f Fr(=)h(2)p Fp(^)p Fr(\(n-1\),)i(it)e(returns)450
3181 518 y(n.)40 b(Else)31 b(it)g(returns)e(0.)41 b(\(See)31
3182 b(also)g(the)g(function)f Fp(logp)p Fr(.\))-30 736 y
3183 Fn(4.9.2)63 b(Num)m(b)s(er)40 b(theoretic)h(functions)-30
3184 949 y Fp(uint32)29 b(gcd)g(\(uint32)g(a,)g(uint32)g(b\))-30
3185 1058 y(cl_I)g(gcd)h(\(const)e(cl_I&)h(a,)h(const)f(cl_I&)g(b\))450
3186 1168 y Fr(This)h(function)g(returns)f(the)h(greatest)j(common)28
3187 b(divisor)i(of)h Fp(a)f Fr(and)g Fp(b)p Fr(,)g(normalized)g(to)h(b)s(e)
3188 f Fp(>)p Fr(=)f(0.)-30 1323 y Fp(cl_I)g(xgcd)g(\(const)g(cl_I&)g(a,)h
3189 (const)f(cl_I&)g(b,)g(cl_I*)g(u,)h(cl_I*)f(v\))450 1433
3190 y Fr(This)34 b(function)h(\(\\extended)h(gcd"\))g(returns)e(the)h
3191 (greatest)i(common)d(divisor)h Fp(g)f Fr(of)i Fp(a)e
3192 Fr(and)h Fp(b)g Fr(and)450 1542 y(at)30 b(the)g(same)f(time)g(the)h
3193 (represen)m(tation)g(of)g Fp(g)f Fr(as)h(an)g(in)m(tegral)h(linear)f
3194 (com)m(bination)g(of)f Fp(a)g Fr(and)g Fp(b)p Fr(:)40
3195 b Fp(u)450 1652 y Fr(and)32 b Fp(v)g Fr(with)g Fp(u*a+v*b)d(=)h(g)p
3196 Fr(,)j Fp(g)f(>)p Fr(=)g(0.)48 b Fp(u)32 b Fr(and)g Fp(v)g
3197 Fr(will)h(b)s(e)f(normalized)g(to)h(b)s(e)f(of)h(smallest)f(p)s
3198 (ossible)450 1761 y(absolute)d(v)-5 b(alue,)29 b(in)e(the)h(follo)m
3199 (wing)h(sense:)40 b(If)27 b Fp(a)h Fr(and)f Fp(b)g Fr(are)i(non-zero,)g
3200 (and)e Fp(abs\(a\))i(!=)g(abs\(b\))p Fr(,)e Fp(u)450
3201 1871 y Fr(and)j Fp(v)g Fr(will)h(satisfy)f(the)h(inequalities)h
3202 Fp(abs\(u\))c(<=)i(abs\(b\)/\(2*g\))p Fr(,)d Fp(abs\(v\))i(<=)h
3203 (abs\(a\)/\(2*g\))p Fr(.)-30 2026 y Fp(cl_I)f(lcm)h(\(const)e(cl_I&)h
3204 (a,)h(const)f(cl_I&)g(b\))450 2136 y Fr(This)h(function)g(returns)f
3205 (the)h(least)i(common)d(m)m(ultiple)h(of)g Fp(a)g Fr(and)g
3206 Fp(b)p Fr(,)g(normalized)g(to)h(b)s(e)f Fp(>)p Fr(=)g(0.)-30
3207 2291 y Fp(cl_boolean)e(logp)h(\(const)f(cl_I&)h(a,)h(const)f(cl_I&)g
3208 (b,)h(cl_RA*)e(l\))-30 2400 y(cl_boolean)g(logp)h(\(const)f(cl_RA&)h
3209 (a,)h(const)f(cl_RA&)f(b,)i(cl_RA*)f(l\))450 2510 y(a)38
3210 b Fr(m)m(ust)g(b)s(e)f Fp(>)i Fr(0.)65 b Fp(b)38 b Fr(m)m(ust)f(b)s(e)h
3211 Fp(>)p Fr(0)h(and)f(!=)g(1.)66 b(If)38 b(log\(a,b\))i(is)f(rational)g
3212 (n)m(um)m(b)s(er,)g(this)f(function)450 2619 y(returns)29
3213 b(true)h(and)g(sets)h(*l)g(=)f(log\(a,b\),)j(else)e(it)g(returns)e
3214 (false.)-30 2837 y Fn(4.9.3)63 b(Com)m(binatorial)40
3215 b(functions)-30 3050 y Fp(cl_I)29 b(factorial)f(\(uintL)h(n\))450
3216 3160 y(n)h Fr(m)m(ust)f(b)s(e)h(a)h(small)f(in)m(teger)h
3217 Fp(>)p Fr(=)f(0.)41 b(This)30 b(function)g(returns)f(the)i(factorial)h
3218 Fp(n)p Fr(!)40 b(=)30 b Fp(1*2*...)n(*n)p Fr(.)-30 3315
3219 y Fp(cl_I)f(doublefactorial)d(\(uintL)j(n\))450 3425
3220 y(n)48 b Fr(m)m(ust)e(b)s(e)i(a)g(small)f(in)m(teger)j
3221 Fp(>)p Fr(=)d(0.)94 b(This)47 b(function)h(returns)e(the)i
3222 (doublefactorial)i Fp(n)p Fr(!!)93 b(=)450 3534 y Fp(1*3*...)n(*n)30
3223 b Fr(or)g Fp(n)p Fr(!!)41 b(=)30 b Fp(2*4*...)n(*n)p
3224 Fr(,)g(resp)s(ectiv)m(ely)-8 b(.)-30 3689 y Fp(cl_I)29
3225 b(binomial)f(\(uintL)h(n,)h(uintL)f(k\))450 3799 y(n)40
3226 b Fr(and)g Fp(k)g Fr(m)m(ust)g(b)s(e)f(small)h(in)m(tegers)i
3227 Fp(>)p Fr(=)e(0.)71 b(This)40 b(function)g(returns)g(the)g(binomial)g
3228 (co)s(e\016cien)m(t)450 3839 y Fk(\000)488 3868 y Fj(n)490
3229 3940 y(k)529 3839 y Fk(\001)592 3908 y Fr(=)798 3873
3230 y Fj(n)p Fi(!)p 698 3888 261 4 v 698 3940 a Fj(n)p Fi(!\()p
3231 Fj(n)p Fh(\000)p Fj(k)q Fi(\)!)999 3908 y Fr(for)30 b(0)h
3232 Fp(<)p Fr(=)e(k)i Fp(<)p Fr(=)e(n,)h(0)h(else.)-30 4159
3233 y Fs(4.10)68 b(F)-11 b(unctions)44 b(on)h(\015oating-p)t(oin)l(t)h(n)l
3234 (um)l(b)t(ers)-30 4350 y Fr(Recall)31 b(that)f(a)f(\015oating-p)s(oin)m
3235 (t)i(n)m(um)m(b)s(er)c(consists)j(of)f(a)h(sign)f Fp(s)p
3236 Fr(,)h(an)f(exp)s(onen)m(t)g Fp(e)g Fr(and)g(a)g(man)m(tissa)h
3237 Fp(m)p Fr(.)40 b(The)28 b(v)-5 b(alue)-30 4459 y(of)31
3238 b(the)f(n)m(um)m(b)s(er)e(is)j Fp(\(-1\)^s)d(*)i(2^e)g(*)g(m)p
3239 Fr(.)-30 4592 y(Eac)m(h)h(of)g(the)f(classes)i Fp(cl_F)p
3240 Fr(,)d Fp(cl_SF)p Fr(,)g Fp(cl_FF)p Fr(,)g Fp(cl_DF)p
3241 Fr(,)h Fp(cl_LF)e Fr(de\014nes)i(the)h(follo)m(wing)g(op)s(erations.)
3242 -30 4747 y Fl(t)m(yp)s(e)36 b Fp(scale_float)27 b(\(const)i
3243 Fl(t)m(yp)s(e)5 b Fp(&)30 b(x,)g(sintL)f(delta\))-30
3244 4856 y Fl(t)m(yp)s(e)36 b Fp(scale_float)27 b(\(const)i
3245 Fl(t)m(yp)s(e)5 b Fp(&)30 b(x,)g(const)f(cl_I&)g(delta\))450
3246 4966 y Fr(Returns)40 b Fp(x*2^delta)p Fr(.)70 b(This)40
3247 b(is)h(more)f(e\016cien)m(t)j(than)d(an)h(explicit)i(m)m(ultiplication)
3248 e(b)s(ecause)g(it)450 5075 y(copies)31 b Fp(x)f Fr(and)g(mo)s(di\014es)
3249 f(the)h(exp)s(onen)m(t.)-30 5230 y(The)36 b(follo)m(wing)j(functions)d
3250 (pro)m(vide)h(an)g(abstract)h(in)m(terface)g(to)g(the)f(underlying)f
3251 (represen)m(tation)i(of)f(\015oating-)-30 5340 y(p)s(oin)m(t)30
3252 b(n)m(um)m(b)s(ers.)p eop
3254 24 25 bop -30 -116 a Fr(Chapter)30 b(4:)41 b(F)-8 b(unctions)31
3255 b(on)f(n)m(um)m(b)s(ers)2523 b(24)-30 299 y Fp(sintL)29
3256 b(float_exponent)d(\(const)j Fl(t)m(yp)s(e)5 b Fp(&)30
3257 b(x\))450 408 y Fr(Returns)j(the)g(exp)s(onen)m(t)h Fp(e)f
3258 Fr(of)h Fp(x)p Fr(.)50 b(F)-8 b(or)34 b Fp(x)c(=)g(0.0)p
3259 Fr(,)k(this)f(is)h(0.)50 b(F)-8 b(or)35 b Fp(x)e Fr(non-zero,)i(this)f
3260 (is)f(the)h(unique)450 518 y(in)m(teger)e(with)e Fp(2^\(e-1\))e(<=)i
3261 (abs\(x\))f(<)h(2^e)p Fr(.)-30 701 y Fp(sintL)f(float_radix)e(\(const)i
3262 Fl(t)m(yp)s(e)5 b Fp(&)30 b(x\))450 811 y Fr(Returns)f(the)i(base)g(of)
3263 f(the)h(\015oating-p)s(oin)m(t)g(represen)m(tation.)42
3264 b(This)30 b(is)g(alw)m(a)m(ys)i Fp(2)p Fr(.)-30 994 y
3265 Fl(t)m(yp)s(e)k Fp(float_sign)27 b(\(const)i Fl(t)m(yp)s(e)5
3266 b Fp(&)30 b(x\))450 1103 y Fr(Returns)f(the)i(sign)f
3267 Fp(s)g Fr(of)h Fp(x)f Fr(as)h(a)f(\015oat.)42 b(The)30
3268 b(v)-5 b(alue)31 b(is)f(1)h(for)f Fp(x)g(>)p Fr(=)g(0,)h(-1)g(for)f
3269 Fp(x)g(<)g Fr(0.)-30 1287 y Fp(uintL)f(float_digits)e(\(const)i
3270 Fl(t)m(yp)s(e)5 b Fp(&)30 b(x\))450 1396 y Fr(Returns)f(the)h(n)m(um)m
3271 (b)s(er)e(of)i(man)m(tissa)g(bits)g(in)g(the)g(\015oating-p)s(oin)m(t)h
3272 (represen)m(tation)g(of)f Fp(x)p Fr(,)g(including)450
3273 1506 y(the)h(hidden)e(bit.)40 b(The)30 b(v)-5 b(alue)31
3274 b(only)g(dep)s(ends)d(on)i(the)h(t)m(yp)s(e)f(of)h Fp(x)p
3275 Fr(,)f(not)h(on)f(its)h(v)-5 b(alue.)-30 1689 y Fp(uintL)29
3276 b(float_precision)d(\(const)j Fl(t)m(yp)s(e)5 b Fp(&)30
3277 b(x\))450 1798 y Fr(Returns)c(the)h(n)m(um)m(b)s(er)d(of)j
3278 (signi\014can)m(t)h(man)m(tissa)f(bits)g(in)f(the)h(\015oating-p)s(oin)
3279 m(t)h(represen)m(tation)g(of)f Fp(x)p Fr(.)450 1908 y(Since)j
3280 (denormalized)e(n)m(um)m(b)s(ers)g(are)h(not)h(supp)s(orted,)e(this)i
3281 (is)f(the)h(same)f(as)h Fp(float_digits\(x\))25 b Fr(if)450
3282 2018 y Fp(x)30 b Fr(is)g(non-zero,)i(and)d(0)i(if)f Fp(x)h
3283 Fr(=)f(0.)-30 2213 y(The)f(complete)g(in)m(ternal)h(represen)m(tation)h
3284 (of)e(a)g(\015oat)h(is)g(enco)s(ded)e(in)h(the)h(t)m(yp)s(e)f
3285 Fp(decoded_float)d Fr(\(or)j Fp(decoded_)-30 2322 y(sfloat)p
3286 Fr(,)g Fp(decoded_ffloat)p Fr(,)e Fp(decoded_dfloat)p
3287 Fr(,)f Fp(decoded_lfloat)p Fr(,)h(resp)s(ectiv)m(ely\),)32
3288 b(de\014ned)e(b)m(y)210 2475 y Fp(struct)46 b(decoded_)p
3289 Fl(t)m(yp)s(e)5 b Fp(float)45 b({)592 2578 y Fl(t)m(yp)s(e)53
3290 b Fp(mantissa;)45 b(cl_I)i(exponent;)e Fl(t)m(yp)s(e)53
3291 b Fp(sign;)210 2682 y(};)-30 2840 y Fr(and)30 b(returned)f(b)m(y)h(the)
3292 h(function)-30 3035 y Fp(decoded_)p Fl(t)m(yp)s(e)5 b
3293 Fp(float)27 b(decode_float)g(\(const)i Fl(t)m(yp)s(e)5
3294 b Fp(&)30 b(x\))450 3145 y Fr(F)-8 b(or)26 b Fp(x)e Fr(non-zero,)j
3295 (this)d(returns)g Fp(\(-1\)^s)p Fr(,)h Fp(e)p Fr(,)g
3296 Fp(m)g Fr(with)f Fp(x)30 b(=)g(\(-1\)^s)f(*)h(2^e)g(*)g(m)24
3297 b Fr(and)g Fp(0.5)30 b(<=)g(m)g(<)g(1.0)p Fr(.)450 3255
3298 y(F)-8 b(or)35 b Fp(x)f Fr(=)h(0,)h(it)f(returns)e Fp(\(-1\)^s)p
3299 Fr(=1,)h Fp(e)p Fr(=0,)i Fp(m)p Fr(=0.)53 b Fp(e)34 b
3300 Fr(is)g(the)h(same)f(as)h(returned)e(b)m(y)i(the)f(function)450
3301 3364 y Fp(float_exponent)p Fr(.)-30 3559 y(A)c(complete)h(deco)s(ding)g
3302 (in)f(terms)f(of)i(in)m(tegers)g(is)g(pro)m(vided)f(as)g(t)m(yp)s(e)210
3303 3712 y Fp(struct)46 b(cl_idecoded_float)d({)592 3815
3304 y(cl_I)j(mantissa;)g(cl_I)g(exponent;)g(cl_I)g(sign;)210
3305 3919 y(};)-30 4077 y Fr(b)m(y)30 b(the)h(follo)m(wing)h(function:)-30
3306 4272 y Fp(cl_idecoded_float)26 b(integer_decode_float)e(\(const)29
3307 b Fl(t)m(yp)s(e)5 b Fp(&)31 b(x\))450 4382 y Fr(F)-8
3308 b(or)38 b Fp(x)e Fr(non-zero,)k(this)d(returns)f Fp(\(-1\)^s)p
3309 Fr(,)h Fp(e)p Fr(,)h Fp(m)f Fr(with)g Fp(x)30 b(=)g(\(-1\)^s)e(*)i(2^e)
3310 g(*)g(m)37 b Fr(and)f Fp(m)h Fr(an)g(in)m(teger)450 4491
3311 y(with)e Fp(float_digits\(x\))30 b Fr(bits.)54 b(F)-8
3312 b(or)36 b Fp(x)e Fr(=)h(0,)h(it)g(returns)d Fp(\(-1\)^s)p
3313 Fr(=1,)i Fp(e)p Fr(=0,)h Fp(m)p Fr(=0.)54 b(W)-10 b(ARNING:)450
3314 4601 y(The)27 b(exp)s(onen)m(t)h Fp(e)f Fr(is)h(not)f(the)h(same)f(as)h
3315 (the)g(one)g(returned)e(b)m(y)h(the)h(functions)f Fp(decode_float)e
3316 Fr(and)450 4711 y Fp(float_exponent)p Fr(.)-30 4906 y(Some)k(other)i
3317 (function,)f(implemen)m(ted)f(only)i(for)f(class)h Fp(cl_F)p
3318 Fr(:)-30 5101 y Fp(cl_F)e(float_sign)f(\(const)g(cl_F&)h(x,)h(const)f
3319 (cl_F&)g(y\))450 5210 y Fr(This)d(returns)g(a)i(\015oating)g(p)s(oin)m
3320 (t)f(n)m(um)m(b)s(er)e(whose)h(precision)i(and)e(absolute)i(v)-5
3321 b(alue)28 b(is)f(that)h(of)f Fp(y)g Fr(and)450 5320 y(whose)j(sign)h
3322 (is)f(that)h(of)f Fp(x)p Fr(.)41 b(If)30 b Fp(x)g Fr(is)g(zero,)i(it)f
3323 (is)f(treated)h(as)g(p)s(ositiv)m(e.)42 b(Same)29 b(for)h
3326 25 26 bop -30 -116 a Fr(Chapter)30 b(4:)41 b(F)-8 b(unctions)31
3327 b(on)f(n)m(um)m(b)s(ers)2523 b(25)-30 299 y Fs(4.11)68
3328 b(Con)l(v)l(ersion)46 b(functions)-30 612 y Fn(4.11.1)63
3329 b(Con)m(v)m(ersion)41 b(to)g(\015oating-p)s(oin)m(t)h(n)m(um)m(b)s(ers)
3330 -30 821 y Fr(The)30 b(t)m(yp)s(e)g Fp(cl_float_format_t)c
3331 Fr(describ)s(es)k(a)g(\015oating-p)s(oin)m(t)i(format.)-30
3332 1003 y Fp(cl_float_format_t)26 b(cl_float_format)g(\(uintL)j(n\))450
3333 1113 y Fr(Returns)45 b(the)g(smallest)h(\015oat)g(format)f(whic)m(h)h
3334 (guaran)m(tees)g(at)h(least)g Fp(n)e Fr(decimal)g(digits)h(in)g(the)450
3335 1222 y(man)m(tissa)30 b(\(after)i(the)e(decimal)g(p)s(oin)m(t\).)-30
3336 1397 y Fp(cl_float_format_t)c(cl_float_format)g(\(const)j(cl_F&)f(x\))
3337 450 1507 y Fr(Returns)h(the)i(\015oating)g(p)s(oin)m(t)g(format)e(of)i
3338 Fp(x)p Fr(.)-30 1682 y Fp(cl_float_format_t)26 b(default_float_format)
3339 450 1792 y Fr(Global)d(v)-5 b(ariable:)37 b(the)23 b(default)f(\015oat)
3340 g(format)g(used)f(when)g(con)m(v)m(erting)j(rational)f(n)m(um)m(b)s
3341 (ers)c(to)k(\015oats.)-30 1974 y(T)-8 b(o)30 b(con)m(v)m(ert)h(a)f
3342 (real)g(n)m(um)m(b)s(er)d(to)j(a)g(\015oat,)g(eac)m(h)h(of)e(the)h(t)m
3343 (yp)s(es)f Fp(cl_R)p Fr(,)g Fp(cl_F)p Fr(,)f Fp(cl_I)p
3344 Fr(,)h Fp(cl_RA)p Fr(,)g Fp(int)p Fr(,)f Fp(unsigned)h(int)p
3345 Fr(,)-30 2084 y Fp(float)p Fr(,)g Fp(double)g Fr(de\014nes)g(the)i
3346 (follo)m(wing)h(op)s(erations:)-30 2267 y Fp(cl_F)d(cl_float)f(\(const)
3347 h Fl(t)m(yp)s(e)5 b Fp(&x,)30 b(cl_float_format_t)25
3348 b(f\))450 2376 y Fr(Returns)k Fp(x)h Fr(as)h(a)g(\015oat)g(of)f(format)
3349 g Fp(f)p Fr(.)-30 2551 y Fp(cl_F)f(cl_float)f(\(const)h
3350 Fl(t)m(yp)s(e)5 b Fp(&x,)30 b(const)f(cl_F&)g(y\))450
3351 2661 y Fr(Returns)g Fp(x)h Fr(in)h(the)f(\015oat)h(format)f(of)g
3352 Fp(y)p Fr(.)-30 2836 y Fp(cl_F)f(cl_float)f(\(const)h
3353 Fl(t)m(yp)s(e)5 b Fp(&x\))450 2945 y Fr(Returns)24 b
3354 Fp(x)g Fr(as)h(a)g(\015oat)h(of)f(format)f Fp(default_float_format)19
3355 b Fr(if)24 b(it)i(is)e(an)h(exact)h(n)m(um)m(b)s(er,)e(or)h
3356 Fp(x)f Fr(itself)450 3055 y(if)30 b(it)h(is)g(already)g(a)f(\015oat.)
3357 -30 3238 y(Of)g(course,)h(con)m(v)m(erting)h(a)f(n)m(um)m(b)s(er)d(to)j
3358 (a)f(\015oat)i(can)e(lose)h(precision.)-30 3388 y(Ev)m(ery)g
3359 (\015oating-p)s(oin)m(t)g(format)f(has)g(some)g(c)m(haracteristic)j(n)m
3360 (um)m(b)s(ers:)-30 3571 y Fp(cl_F)c(most_positive_float)c
3361 (\(cl_float_format_t)h(f\))450 3680 y Fr(Returns)j(the)i(largest)h
3362 (\(most)e(p)s(ositiv)m(e\))i(\015oating)f(p)s(oin)m(t)f(n)m(um)m(b)s
3363 (er)e(in)i(\015oat)h(format)f Fp(f)p Fr(.)-30 3855 y
3364 Fp(cl_F)f(most_negative_float)c(\(cl_float_format_t)h(f\))450
3365 3965 y Fr(Returns)j(the)i(smallest)g(\(most)f(negativ)m(e\))j
3366 (\015oating)e(p)s(oin)m(t)f(n)m(um)m(b)s(er)e(in)i(\015oat)h(format)f
3367 Fp(f)p Fr(.)-30 4140 y Fp(cl_F)f(least_positive_float)c
3368 (\(cl_float_format_t)g(f\))450 4249 y Fr(Returns)g(the)h(least)h(p)s
3369 (ositiv)m(e)g(\015oating)g(p)s(oin)m(t)e(n)m(um)m(b)s(er)f(\(i.e.)40
3370 b Fp(>)26 b Fr(0)g(but)f(closest)i(to)g(0\))g(in)e(\015oat)i(format)450
3371 4359 y Fp(f)p Fr(.)-30 4534 y Fp(cl_F)i(least_negative_float)c
3372 (\(cl_float_format_t)g(f\))450 4644 y Fr(Returns)e(the)i(least)h
3373 (negativ)m(e)g(\015oating)g(p)s(oin)m(t)e(n)m(um)m(b)s(er)e(\(i.e.)40
3374 b Fp(<)24 b Fr(0)h(but)e(closest)j(to)f(0\))g(in)g(\015oat)g(format)450
3375 4753 y Fp(f)p Fr(.)-30 4928 y Fp(cl_F)k(float_epsilon)e
3376 (\(cl_float_format_t)e(f\))450 5038 y Fr(Returns)k(the)i(smallest)g
3377 (\015oating)g(p)s(oin)m(t)f(n)m(um)m(b)s(er)e(e)j Fp(>)f
3378 Fr(0)h(suc)m(h)f(that)h Fp(1+e)e(!=)h(1)p Fr(.)-30 5213
3379 y Fp(cl_F)f(float_negative_epsilon)24 b(\(cl_float_format_t)i(f\))450
3380 5322 y Fr(Returns)j(the)i(smallest)g(\015oating)g(p)s(oin)m(t)f(n)m(um)
3381 m(b)s(er)e(e)j Fp(>)f Fr(0)h(suc)m(h)f(that)h Fp(1-e)e(!=)h(1)p
3384 26 27 bop -30 -116 a Fr(Chapter)30 b(4:)41 b(F)-8 b(unctions)31
3385 b(on)f(n)m(um)m(b)s(ers)2523 b(26)-30 299 y Fn(4.11.2)63
3386 b(Con)m(v)m(ersion)41 b(to)g(rational)g(n)m(um)m(b)s(ers)-30
3387 488 y Fr(Eac)m(h)31 b(of)g(the)f(classes)i Fp(cl_R)p
3388 Fr(,)d Fp(cl_RA)p Fr(,)g Fp(cl_F)g Fr(de\014nes)h(the)g(follo)m(wing)i
3389 (op)s(eration:)-30 639 y Fp(cl_RA)d(rational)f(\(const)h
3390 Fl(t)m(yp)s(e)5 b Fp(&)30 b(x\))450 749 y Fr(Returns)j(the)h(v)-5
3391 b(alue)35 b(of)f Fp(x)f Fr(as)h(an)g(exact)i(n)m(um)m(b)s(er.)49
3392 b(If)33 b Fp(x)h Fr(is)g(already)g(an)g(exact)h(n)m(um)m(b)s(er,)e
3393 (this)h(is)g Fp(x)p Fr(.)450 858 y(If)g Fp(x)g Fr(is)g(a)g
3394 (\015oating-p)s(oin)m(t)i(n)m(um)m(b)s(er,)d(the)h(v)-5
3395 b(alue)35 b(is)f(a)h(rational)g(n)m(um)m(b)s(er)d(whose)i(denominator)f
3396 (is)i(a)450 968 y(p)s(o)m(w)m(er)30 b(of)h(2.)-30 1119
3397 y(In)f(order)g(to)h(con)m(v)m(ert)h(bac)m(k,)f(sa)m(y)-8
3398 b(,)32 b Fp(\(cl_F\)\(cl_R\)"1/3")25 b Fr(to)31 b Fp(1/3)p
3399 Fr(,)f(there)h(is)f(the)h(function)-30 1271 y Fp(cl_RA)e(rationalize)e
3400 (\(const)i(cl_R&)g(x\))450 1380 y Fr(If)36 b Fp(x)f Fr(is)h(a)h
3401 (\015oating-p)s(oin)m(t)g(n)m(um)m(b)s(er,)f(it)g(actually)i(represen)m
3402 (ts)e(an)g(in)m(terv)-5 b(al)37 b(of)f(real)h(n)m(um)m(b)s(ers,)f(and)
3403 450 1490 y(this)d(function)g(returns)e(the)i(rational)i(n)m(um)m(b)s
3404 (er)30 b(with)j(smallest)g(denominator)f(\(and)h(smallest)g(n)m(u-)450
3405 1600 y(merator,)e(in)f(magnitude\))g(whic)m(h)g(lies)i(in)e(this)h(in)m
3406 (terv)-5 b(al.)42 b(If)30 b Fp(x)h Fr(is)f(already)i(an)e(exact)i(n)m
3407 (um)m(b)s(er,)d(this)450 1709 y(function)h(returns)f
3408 Fp(x)p Fr(.)-30 1861 y(If)h Fp(x)g Fr(is)g(an)m(y)h(\015oat,)g(one)g
3409 (has)150 1991 y Fp(cl_float\(rational\(x\),x\))24 b(=)30
3410 b(x)150 2122 y(cl_float\(rationalize\(x\),)o(x\))24 b(=)30
3411 b(x)-30 2367 y Fs(4.12)68 b(Random)46 b(n)l(um)l(b)t(er)e(generators)
3412 -30 2556 y Fr(A)j(random)e(generator)j(is)f(a)h(mac)m(hine)e(whic)m(h)h
3413 (pro)s(duces)e(\(pseudo-\)random)h(n)m(um)m(b)s(ers.)88
3414 b(The)46 b(include)h(\014le)-30 2666 y Fp(<cln/random.h>)40
3415 b Fr(de\014nes)j(a)h(class)h Fp(random_state)c Fr(whic)m(h)i(con)m
3416 (tains)j(the)e(state)h(of)f(a)g(random)f(generator.)-30
3417 2775 y(If)29 b(y)m(ou)g(mak)m(e)g(a)g(cop)m(y)h(of)f(the)h(random)d(n)m
3418 (um)m(b)s(er)g(generator,)j(the)g(original)g(one)f(and)f(the)i(cop)m(y)
3419 f(will)h(pro)s(duce)e(the)-30 2885 y(same)i(sequence)h(of)f(random)f(n)
3420 m(um)m(b)s(ers.)-30 3015 y(The)j(follo)m(wing)i(functions)d(return)h
3421 (\(pseudo-\)random)e(n)m(um)m(b)s(ers)g(in)i(di\013eren)m(t)h(formats.)
3422 45 b(Calling)33 b(one)g(of)f(these)-30 3125 y(mo)s(di\014es)d(the)h
3423 (state)i(of)e(the)h(random)e(n)m(um)m(b)s(er)f(generator)j(in)f(a)h
3424 (complicated)g(but)f(deterministic)g(w)m(a)m(y)-8 b(.)-30
3425 3255 y(The)30 b(global)i(v)-5 b(ariable)210 3380 y Fp(random_state)44
3426 b(default_random_state)-30 3511 y Fr(con)m(tains)29 b(a)f(default)f
3427 (random)f(n)m(um)m(b)s(er)f(generator.)41 b(It)28 b(is)f(used)g(when)g
3428 (the)g(functions)h(b)s(elo)m(w)f(are)h(called)h(without)-30
3429 3620 y Fp(random_state)e Fr(argumen)m(t.)-30 3772 y Fp(uint32)i
3430 (random32)f(\(random_state&)e(randomstate\))-30 3881
3431 y(uint32)j(random32)f(\(\))450 3991 y Fr(Returns)h(a)i(random)e
3432 (unsigned)g(32-bit)j(n)m(um)m(b)s(er.)38 b(All)31 b(bits)f(are)h
3433 (equally)g(random.)-30 4143 y Fp(cl_I)e(random_I)f(\(random_state&)f
3434 (randomstate,)g(const)i(cl_I&)g(n\))-30 4252 y(cl_I)g(random_I)f
3435 (\(const)h(cl_I&)g(n\))450 4362 y(n)i Fr(m)m(ust)e(b)s(e)h(an)h(in)m
3436 (teger)h Fp(>)f Fr(0.)42 b(This)31 b(function)f(returns)g(a)h(random)e
3437 (in)m(teger)j Fp(x)f Fr(in)f(the)h(range)g Fp(0)g(<=)e(x)450
3438 4471 y(<)h(n)p Fr(.)-30 4623 y Fp(cl_F)f(random_F)f(\(random_state&)f
3439 (randomstate,)g(const)i(cl_F&)g(n\))-30 4732 y(cl_F)g(random_F)f
3440 (\(const)h(cl_F&)g(n\))450 4842 y(n)d Fr(m)m(ust)f(b)s(e)h(a)h(\015oat)
3441 g Fp(>)f Fr(0.)40 b(This)26 b(function)g(returns)f(a)i(random)d
3442 (\015oating-p)s(oin)m(t)k(n)m(um)m(b)s(er)c(of)j(the)g(same)450
3443 4952 y(format)j(as)g Fp(n)g Fr(in)g(the)h(range)g Fp(0)f(<=)f(x)i(<)f
3444 (n)p Fr(.)-30 5103 y Fp(cl_R)f(random_R)f(\(random_state&)f
3445 (randomstate,)g(const)i(cl_R&)g(n\))-30 5213 y(cl_R)g(random_R)f
3446 (\(const)h(cl_R&)g(n\))450 5322 y Fr(Beha)m(v)m(es)j(lik)m(e)g
3447 Fp(random_I)c Fr(if)i Fp(n)g Fr(is)h(an)f(in)m(teger)i(and)e(lik)m(e)h
3448 Fp(random_F)d Fr(if)j Fp(n)f Fr(is)g(a)h(\015oat.)p eop
3450 27 28 bop -30 -116 a Fr(Chapter)30 b(4:)41 b(F)-8 b(unctions)31
3451 b(on)f(n)m(um)m(b)s(ers)2523 b(27)-30 299 y Fs(4.13)68
3452 b(Obfuscating)46 b(op)t(erators)-30 491 y Fr(The)36 b(mo)s(difying)g
3453 (C/C)p Fp(++)g Fr(op)s(erators)h Fp(+=)p Fr(,)h Fp(-=)p
3454 Fr(,)g Fp(*=)p Fr(,)g Fp(/=)p Fr(,)g Fp(&=)p Fr(,)g Fp(|=)p
3455 Fr(,)g Fp(^=)p Fr(,)h Fp(<<=)p Fr(,)e Fp(>>=)f Fr(are)i(not)f(a)m(v)-5
3456 b(ailable)39 b(b)m(y)e(default)-30 601 y(b)s(ecause)45
3457 b(their)f(use)g(tends)h(to)g(mak)m(e)g(programs)e(unreadable.)83
3458 b(It)44 b(is)h(trivial)h(to)f(get)h(a)m(w)m(a)m(y)g(without)f(them.)-30
3459 711 y(Ho)m(w)m(ev)m(er,)33 b(if)d(y)m(ou)h(feel)g(that)g(y)m(ou)g
3460 (absolutely)g(need)f(these)h(op)s(erators)f(to)h(get)h(happ)m(y)-8
3461 b(,)30 b(then)g(add)210 839 y Fp(#define)46 b(WANT_OBFUSCATING_OPERATO)
3462 o(RS)-30 974 y Fr(to)37 b(the)g(b)s(eginning)f(of)h(y)m(our)g(source)g
3463 (\014les,)h(b)s(efore)f(the)g(inclusion)f(of)h(an)m(y)g(CLN)f(include)h
3464 (\014les.)60 b(This)36 b(\015ag)h(will)-30 1083 y(enable)31
3465 b(the)f(follo)m(wing)i(op)s(erators:)-30 1218 y(F)-8
3466 b(or)31 b(the)g(classes)g Fp(cl_N)p Fr(,)f Fp(cl_R)p
3467 Fr(,)f Fp(cl_RA)p Fr(,)g Fp(cl_F)p Fr(,)h Fp(cl_SF)p
3468 Fr(,)f Fp(cl_FF)p Fr(,)g Fp(cl_DF)p Fr(,)g Fp(cl_LF)p
3469 Fr(:)-30 1377 y Fl(t)m(yp)s(e)5 b Fp(&)30 b(operator)e(+=)i(\()p
3470 Fl(t)m(yp)s(e)5 b Fp(&,)30 b(const)f Fl(t)m(yp)s(e)5
3471 b Fp(&\))-30 1487 y Fl(t)m(yp)s(e)g Fp(&)30 b(operator)e(-=)i(\()p
3472 Fl(t)m(yp)s(e)5 b Fp(&,)30 b(const)f Fl(t)m(yp)s(e)5
3473 b Fp(&\))-30 1597 y Fl(t)m(yp)s(e)g Fp(&)30 b(operator)e(*=)i(\()p
3474 Fl(t)m(yp)s(e)5 b Fp(&,)30 b(const)f Fl(t)m(yp)s(e)5
3475 b Fp(&\))-30 1706 y Fl(t)m(yp)s(e)g Fp(&)30 b(operator)e(/=)i(\()p
3476 Fl(t)m(yp)s(e)5 b Fp(&,)30 b(const)f Fl(t)m(yp)s(e)5
3477 b Fp(&\))-30 1841 y Fr(F)-8 b(or)31 b(the)g(class)g Fp(cl_I)p
3478 Fr(:)-30 2000 y Fl(t)m(yp)s(e)5 b Fp(&)30 b(operator)e(+=)i(\()p
3479 Fl(t)m(yp)s(e)5 b Fp(&,)30 b(const)f Fl(t)m(yp)s(e)5
3480 b Fp(&\))-30 2110 y Fl(t)m(yp)s(e)g Fp(&)30 b(operator)e(-=)i(\()p
3481 Fl(t)m(yp)s(e)5 b Fp(&,)30 b(const)f Fl(t)m(yp)s(e)5
3482 b Fp(&\))-30 2219 y Fl(t)m(yp)s(e)g Fp(&)30 b(operator)e(*=)i(\()p
3483 Fl(t)m(yp)s(e)5 b Fp(&,)30 b(const)f Fl(t)m(yp)s(e)5
3484 b Fp(&\))-30 2329 y Fl(t)m(yp)s(e)g Fp(&)30 b(operator)e(&=)i(\()p
3485 Fl(t)m(yp)s(e)5 b Fp(&,)30 b(const)f Fl(t)m(yp)s(e)5
3486 b Fp(&\))-30 2438 y Fl(t)m(yp)s(e)g Fp(&)30 b(operator)e(|=)i(\()p
3487 Fl(t)m(yp)s(e)5 b Fp(&,)30 b(const)f Fl(t)m(yp)s(e)5
3488 b Fp(&\))-30 2548 y Fl(t)m(yp)s(e)g Fp(&)30 b(operator)e(^=)i(\()p
3489 Fl(t)m(yp)s(e)5 b Fp(&,)30 b(const)f Fl(t)m(yp)s(e)5
3490 b Fp(&\))-30 2658 y Fl(t)m(yp)s(e)g Fp(&)30 b(operator)e(<<=)i(\()p
3491 Fl(t)m(yp)s(e)5 b Fp(&,)30 b(const)f Fl(t)m(yp)s(e)5
3492 b Fp(&\))-30 2767 y Fl(t)m(yp)s(e)g Fp(&)30 b(operator)e(>>=)i(\()p
3493 Fl(t)m(yp)s(e)5 b Fp(&,)30 b(const)f Fl(t)m(yp)s(e)5
3494 b Fp(&\))-30 2902 y Fr(F)-8 b(or)31 b(the)g(classes)g
3495 Fp(cl_N)p Fr(,)f Fp(cl_R)p Fr(,)f Fp(cl_RA)p Fr(,)g Fp(cl_I)p
3496 Fr(,)h Fp(cl_F)p Fr(,)f Fp(cl_SF)p Fr(,)g Fp(cl_FF)p
3497 Fr(,)h Fp(cl_DF)p Fr(,)f Fp(cl_LF)p Fr(:)-30 3061 y Fl(t)m(yp)s(e)5
3498 b Fp(&)30 b(operator)e(++)i(\()p Fl(t)m(yp)s(e)5 b Fp(&)30
3499 b(x\))450 3171 y Fr(The)g(pre\014x)f(op)s(erator)i Fp(++x)p
3500 Fr(.)-30 3330 y Fp(void)e(operator)f(++)i(\()p Fl(t)m(yp)s(e)5
3501 b Fp(&)30 b(x,)g(int\))450 3440 y Fr(The)g(p)s(ost\014x)f(op)s(erator)i
3502 Fp(x++)p Fr(.)-30 3599 y Fl(t)m(yp)s(e)5 b Fp(&)30 b(operator)e(--)i
3503 (\()p Fl(t)m(yp)s(e)5 b Fp(&)30 b(x\))450 3709 y Fr(The)g(pre\014x)f
3504 (op)s(erator)i Fp(--x)p Fr(.)-30 3868 y Fp(void)e(operator)f(--)i(\()p
3505 Fl(t)m(yp)s(e)5 b Fp(&)30 b(x,)g(int\))450 3978 y Fr(The)g(p)s
3506 (ost\014x)f(op)s(erator)i Fp(x--)p Fr(.)-30 4137 y(Note)37
3507 b(that)g(b)m(y)f(using)g(these)h(obfuscating)f(op)s(erators,)i(y)m(ou)f
3508 (w)m(ouldn't)f(gain)h(e\016ciency:)53 b(In)35 b(CLN)h(`)p
3509 Fp(x)30 b(+=)g(y;)p Fr(')36 b(is)-30 4247 y(exactly)c(the)f(same)e(as)i
3510 (`)p Fp(x)f(=)g(x+y;)p Fr(',)g(not)h(more)e(e\016cien)m(t.)p
3513 28 29 bop -30 -116 a Fr(Chapter)30 b(5:)41 b(Input/Output)2844
3514 b(28)-30 299 y Fo(5)80 b(Input/Output)-30 758 y Fs(5.1)68
3515 b(In)l(ternal)46 b(and)f(prin)l(ted)g(represen)l(tation)-30
3516 973 y Fr(All)31 b(computations)f(deal)h(with)f(the)h(in)m(ternal)g
3517 (represen)m(tations)g(of)g(the)f(n)m(um)m(b)s(ers.)-30
3518 1130 y(Ev)m(ery)36 b(n)m(um)m(b)s(er)d(has)j(an)f(external)i(represen)m
3519 (tation)g(as)f(a)g(sequence)g(of)g(ASCI)s(I)e(c)m(haracters.)58
3520 b(Sev)m(eral)37 b(external)-30 1239 y(represen)m(tations)31
3521 b(ma)m(y)f(denote)h(the)g(same)f(n)m(um)m(b)s(er,)e(for)i(example,)g
3522 Fp(")p Fr(20.0)p Fp(")i Fr(and)d Fp(")p Fr(20.000)p Fp(")p
3523 Fr(.)-30 1397 y(Con)m(v)m(erting)34 b(an)f(in)m(ternal)h(to)g(an)e
3524 (external)i(represen)m(tation)h(is)e(called)h(\\prin)m(ting",)h(con)m
3525 (v)m(erting)f(an)f(external)h(to)-30 1506 y(an)24 b(in)m(ternal)h
3526 (represen)m(tation)g(is)f(called)h(\\reading".)40 b(In)24
3527 b(CLN,)f(it)i(is)f(alw)m(a)m(ys)i(true)e(that)h(con)m(v)m(ersion)g(of)f
3528 (an)g(in)m(ternal)-30 1616 y(to)j(an)e(external)i(represen)m(tation)g
3529 (and)e(then)h(bac)m(k)g(to)h(an)e(in)m(ternal)i(represen)m(tation)g
3530 (will)f(yield)g(the)g(same)f(in)m(ternal)-30 1725 y(represen)m(tation.)
3531 42 b(Sym)m(b)s(olically:)f Fp(read\(print\(x\)\))26 b(==)k(x)p
3532 Fr(.)40 b(This)30 b(is)g(called)i(\\prin)m(t-read)e(consistency".)-30
3533 1882 y(Di\013eren)m(t)i(t)m(yp)s(es)e(of)h(n)m(um)m(b)s(ers)d(ha)m(v)m
3534 (e)j(di\013eren)m(t)g(external)g(represen)m(tations)h(\(case)f(is)g
3535 (insigni\014can)m(t\):)-30 2076 y(In)m(tegers)167 b(External)35
3536 b(represen)m(tation:)50 b Fl(sign)p Fp({)p Fl(digit)r
3537 Fp(}+)p Fr(.)j(The)34 b(reader)g(also)h(accepts)h(the)f(Common)d(Lisp)h
3538 (syn-)450 2185 y(taxes)43 b Fl(sign)p Fp({)p Fl(digit)r
3539 Fp(}+.)g Fr(with)f(a)h(trailing)g(dot)g(for)f(decimal)g(in)m(tegers)i
3540 (and)e(the)h Fp(#)p Fl(n)p Fp(R)p Fr(,)h Fp(#b)p Fr(,)h
3541 Fp(#o)p Fr(,)h Fp(#x)450 2295 y Fr(pre\014xes.)-30 2477
3542 y(Rational)32 b(n)m(um)m(b)s(ers)450 2586 y(External)26
3543 b(represen)m(tation:)39 b Fl(sign)p Fp({)p Fl(digit)r
3544 Fp(}+/{)p Fl(digit)r Fp(}+)p Fr(.)g(The)25 b Fp(#)p Fl(n)p
3545 Fp(R)p Fr(,)h Fp(#b)p Fr(,)g Fp(#o)p Fr(,)g Fp(#x)f Fr(pre\014xes)g
3546 (are)h(allo)m(w)m(ed)450 2696 y(here)k(as)h(w)m(ell.)-30
3547 2878 y(Floating-p)s(oin)m(t)i(n)m(um)m(b)s(ers)450 2988
3548 y(External)58 b(represen)m(tation:)96 b Fl(sign)p Fp({)p
3549 Fl(digit)r Fp(})p Fr(*)p Fl(exp)s(onen)m(t)60 b Fr(or)d
3550 Fl(sign)p Fp({)p Fl(digit)r Fp(})p Fr(*)p Fp(.{)p Fl(digit)r
3551 Fp(})p Fr(*)p Fl(exp)s(onen)m(t)k Fr(or)450 3097 y Fl(sign)p
3552 Fp({)p Fl(digit)r Fp(})p Fr(*)p Fp(.{)p Fl(digit)r Fp(}+)p
3553 Fr(.)45 b(A)31 b(precision)h(sp)s(eci\014er)f(of)h(the)f(form)p
3554 2693 3097 28 4 v 63 w Fl(prec)37 b Fr(ma)m(y)31 b(b)s(e)g(app)s(ended.)
3555 42 b(There)450 3207 y(m)m(ust)g(b)s(e)h(at)h(least)h(one)e(digit)h(in)f
3556 (the)h(non-exp)s(onen)m(t)f(part.)80 b(The)42 b(exp)s(onen)m(t)i(has)f
3557 (the)g(syn)m(tax)450 3316 y Fl(expmark)m(er)36 b(expsign)30
3558 b Fp({)p Fl(digit)r Fp(}+)p Fr(.)41 b(The)30 b(exp)s(onen)m(t)g(mark)m
3559 (er)g(is)630 3462 y(`)p Fp(s)p Fr(')h(for)f(short-\015oats,)630
3560 3608 y(`)p Fp(f)p Fr(')h(for)f(single-\015oats,)630 3754
3561 y(`)p Fp(d)p Fr(')h(for)f(double-\015oats,)630 3900 y(`)p
3562 Fp(L)p Fr(')h(for)f(long-\015oats,)450 4082 y(or)23 b(`)p
3563 Fp(e)p Fr(',)h(whic)m(h)f(denotes)g(a)g(default)f(\015oat)i(format.)37
3564 b(The)22 b(precision)h(sp)s(ecifying)g(su\016x)e(has)i(the)f(syn)m(tax)
3565 p 450 4191 V 477 4191 a Fl(prec)48 b Fr(where)41 b Fl(prec)48
3566 b Fr(denotes)42 b(the)g(n)m(um)m(b)s(er)e(of)i(v)-5 b(alid)43
3567 b(man)m(tissa)f(digits)g(\(in)h(decimal,)i(excluding)450
3568 4301 y(leading)31 b(zero)s(es\),)h(cf.)41 b(also)31 b(function)f(`)p
3569 Fp(cl_float_format)p Fr('.)-30 4483 y(Complex)f(n)m(um)m(b)s(ers)450
3570 4592 y(External)i(represen)m(tation:)630 4738 y(In)36
3571 b(algebraic)j(notation:)54 b Fl(realpart)r Fp(+)p Fl(imagpart)r
3572 Fp(i)p Fr(.)60 b(Of)37 b(course,)h(if)f Fl(imagpart)i
3573 Fr(is)e(negativ)m(e,)k(its)630 4848 y(prin)m(ted)28 b(represen)m
3574 (tation)i(b)s(egins)e(with)g(a)h(`)p Fp(-)p Fr(',)g(and)f(the)g(`)p
3575 Fp(+)p Fr(')h(b)s(et)m(w)m(een)g Fl(realpart)i Fr(and)d
3576 Fl(imagpart)630 4957 y Fr(ma)m(y)41 b(b)s(e)h(omitted.)75
3577 b(Note)43 b(that)g(this)e(notation)j(cannot)e(b)s(e)f(used)g(when)g
3578 (the)h Fl(imagpart)i Fr(is)630 5067 y(rational)c(and)e(the)i(rational)g
3579 (n)m(um)m(b)s(er's)c(base)j(is)g Fp(>)p Fr(18,)j(b)s(ecause)d(the)g(`)p
3580 Fp(i)p Fr(')g(is)g(then)g(read)f(as)i(a)630 5177 y(digit.)630
3581 5322 y(In)30 b(Common)e(Lisp)h(notation:)42 b Fp(#C\()p
3582 Fl(realpart)32 b(imagpart)r Fp(\))p Fr(.)p eop
3584 29 30 bop -30 -116 a Fr(Chapter)30 b(5:)41 b(Input/Output)2844
3585 b(29)-30 299 y Fs(5.2)68 b(Input)44 b(functions)-30 486
3586 y Fr(Including)25 b Fp(<cln/io.h>)d Fr(de\014nes)j(a)h(t)m(yp)s(e)f
3587 Fp(cl_istream)p Fr(,)f(whic)m(h)i(is)f(the)h(t)m(yp)s(e)f(of)h(the)g
3588 (\014rst)e(argumen)m(t)h(to)i(all)f(input)-30 595 y(functions.)40
3589 b Fp(cl_istream)28 b Fr(is)i(the)h(same)f(as)g Fp(std::istream&)p
3590 Fr(.)-30 724 y(The)g(v)-5 b(ariable)150 852 y Fp(cl_istream)28
3591 b(stdin)-30 1000 y Fr(con)m(tains)k(the)e(standard)g(input)f(stream.)
3592 -30 1129 y(These)h(are)h(the)f(simple)g(input)f(functions:)-30
3593 1276 y Fp(int)g(freadchar)f(\(cl_istream)g(stream\))450
3594 1386 y Fr(Reads)37 b(a)f(c)m(haracter)j(from)c Fp(stream)p
3595 Fr(.)57 b(Returns)35 b Fp(cl_EOF)g Fr(\(not)i(a)g(`)p
3596 Fp(char)p Fr('!\))59 b(if)36 b(the)h(end)f(of)h(stream)450
3597 1495 y(w)m(as)31 b(encoun)m(tered)g(or)f(an)g(error)g(o)s(ccurred.)-30
3598 1643 y Fp(int)f(funreadchar)f(\(cl_istream)f(stream,)h(int)i(c\))450
3599 1753 y Fr(Puts)41 b(bac)m(k)h Fp(c)f Fr(on)m(to)i Fp(stream)p
3600 Fr(.)72 b Fp(c)41 b Fr(m)m(ust)f(b)s(e)h(the)h(result)f(of)g(the)h
3601 (last)g Fp(freadchar)d Fr(op)s(eration)j(on)450 1862
3602 y Fp(stream)p Fr(.)-30 2010 y(Eac)m(h)53 b(of)f(the)g(classes)i
3603 Fp(cl_N)p Fr(,)i Fp(cl_R)p Fr(,)h Fp(cl_RA)p Fr(,)f Fp(cl_I)p
3604 Fr(,)h Fp(cl_F)p Fr(,)f Fp(cl_SF)p Fr(,)h Fp(cl_FF)p
3605 Fr(,)f Fp(cl_DF)p Fr(,)h Fp(cl_LF)50 b Fr(de\014nes,)57
3606 b(in)-30 2119 y Fp(<cln/)p Fl(t)m(yp)s(e)5 b Fp(_io.h>)p
3607 Fr(,)28 b(the)i(follo)m(wing)i(input)e(function:)-30
3608 2267 y Fp(cl_istream)e(operator>>)f(\(cl_istream)g(stream,)i
3609 Fl(t)m(yp)s(e)5 b Fp(&)30 b(result\))450 2377 y Fr(Reads)g(a)h(n)m(um)m
3610 (b)s(er)d(from)h Fp(stream)g Fr(and)h(stores)g(it)h(in)f(the)h
3611 Fp(result)p Fr(.)-30 2524 y(The)f(most)g(\015exible)g(input)g
3612 (functions,)g(de\014ned)f(in)h Fp(<cln/)p Fl(t)m(yp)s(e)5
3613 b Fp(_io.h>)p Fr(,)28 b(are)j(the)f(follo)m(wing:)-30
3614 2672 y Fp(cl_N)f(read_complex)e(\(cl_istream)h(stream,)g(const)h
3615 (cl_read_flags&)d(flags\))-30 2781 y(cl_R)j(read_real)f(\(cl_istream)f
3616 (stream,)i(const)g(cl_read_flags&)d(flags\))-30 2891
3617 y(cl_F)j(read_float)f(\(cl_istream)f(stream,)h(const)h(cl_read_flags&)e
3618 (flags\))-30 3001 y(cl_RA)i(read_rational)e(\(cl_istream)g(stream,)h
3619 (const)h(cl_read_flags&)e(flags\))-30 3110 y(cl_I)i(read_integer)e
3620 (\(cl_istream)h(stream,)g(const)h(cl_read_flags&)d(flags\))450
3621 3220 y Fr(Reads)i(a)g(n)m(um)m(b)s(er)e(from)g Fp(stream)p
3622 Fr(.)38 b(The)27 b Fp(flags)g Fr(are)h(parameters)f(whic)m(h)h
3623 (a\013ect)h(the)f(input)f(syn)m(tax.)450 3329 y(Whitespace)32
3624 b(b)s(efore)e(the)g(n)m(um)m(b)s(er)e(is)j(silen)m(tly)h(skipp)s(ed.)
3625 -30 3477 y Fp(cl_N)d(read_complex)e(\(const)i(cl_read_flags&)d(flags,)j
3626 (const)g(char)g(*)h(string,)f(const)g(char)g(*)-30 3587
3627 y(string_limit,)e(const)i(char)g(*)h(*)g(end_of_parse\))-30
3628 3696 y(cl_R)f(read_real)f(\(const)h(cl_read_flags&)d(flags,)j(const)g
3629 (char)g(*)h(string,)e(const)h(char)h(*)-30 3806 y(string_limit,)d
3630 (const)i(char)g(*)h(*)g(end_of_parse\))-30 3915 y(cl_F)f(read_float)f
3631 (\(const)g(cl_read_flags&)f(flags,)i(const)g(char)g(*)h(string,)e
3632 (const)h(char)g(*)-30 4025 y(string_limit,)e(const)i(char)g(*)h(*)g
3633 (end_of_parse\))-30 4134 y(cl_RA)f(read_rational)e(\(const)h
3634 (cl_read_flags&)f(flags,)h(const)h(char)h(*)g(string,)e(const)h(char)g
3635 (*)-30 4244 y(string_limit,)e(const)i(char)g(*)h(*)g(end_of_parse\))-30
3636 4354 y(cl_I)f(read_integer)e(\(const)i(cl_read_flags&)d(flags,)j(const)
3637 g(char)g(*)h(string,)f(const)g(char)g(*)-30 4463 y(string_limit,)e
3638 (const)i(char)g(*)h(*)g(end_of_parse\))450 4573 y Fr(Reads)k(a)h(n)m
3639 (um)m(b)s(er)d(from)g(a)j(string)f(in)g(memory)-8 b(.)50
3640 b(The)34 b Fp(flags)e Fr(are)j(parameters)e(whic)m(h)h(a\013ect)i(the)
3641 450 4682 y(input)31 b(syn)m(tax.)46 b(The)31 b(string)h(starts)g(at)h
3642 Fp(string)d Fr(and)i(ends)f(at)h Fp(string_limit)d Fr(\(exclusiv)m(e)34
3643 b(limit\).)450 4792 y Fp(string_limit)29 b Fr(ma)m(y)i(also)i(b)s(e)f
3644 Fp(NULL)p Fr(,)f(denoting)i(the)f(en)m(tire)h(string,)g(i.e.)47
3645 b(equiv)-5 b(alen)m(t)33 b(to)g Fp(string_)450 4902 y(limit)c(=)h
3646 (string)f(+)h(strlen\(string\))p Fr(.)75 b(If)42 b Fp(end_of_parse)e
3647 Fr(is)j Fp(NULL)p Fr(,)j(the)d(string)g(in)g(memory)450
3648 5011 y(m)m(ust)30 b(con)m(tain)j(exactly)g(one)f(n)m(um)m(b)s(er)d(and)
3649 i(nothing)g(more,)g(else)i(a)f(fatal)g(error)f(will)h(b)s(e)f
3650 (signalled.)450 5121 y(If)37 b Fp(end_of_parse)d Fr(is)k(not)g
3651 Fp(NULL)p Fr(,)g Fp(*end_of_parse)c Fr(will)k(b)s(e)f(assigned)h(a)g(p)
3652 s(oin)m(ter)g(past)f(the)h(last)450 5230 y(parsed)32
3653 b(c)m(haracter)h(\(i.e.)48 b Fp(string_limit)29 b Fr(if)j(nothing)g
3654 (came)g(after)h(the)g(n)m(um)m(b)s(er\).)44 b(Whitespace)34
3655 b(is)450 5340 y(not)d(allo)m(w)m(ed.)p eop
3657 30 31 bop -30 -116 a Fr(Chapter)30 b(5:)41 b(Input/Output)2844
3658 b(30)-30 299 y(The)30 b(structure)g Fp(cl_read_flags)d
3659 Fr(con)m(tains)k(the)g(follo)m(wing)g(\014elds:)-30 461
3660 y Fp(cl_read_syntax_t)26 b(syntax)450 570 y Fr(The)i(p)s(ossible)f
3661 (results)h(of)g(the)h(read)f(op)s(eration.)40 b(P)m(ossible)29
3662 b(v)-5 b(alues)28 b(are)h Fp(syntax_number)p Fr(,)c Fp(syntax_)450
3663 680 y(real)p Fr(,)49 b Fp(syntax_rational)p Fr(,)e Fp(syntax_integer)p
3664 Fr(,)g Fp(syntax_float)p Fr(,)g Fp(syntax_sfloat)p Fr(,)g
3665 Fp(syntax_)450 789 y(ffloat)p Fr(,)29 b Fp(syntax_dfloat)p
3666 Fr(,)e Fp(syntax_lfloat)p Fr(.)-30 950 y Fp(cl_read_lsyntax_t)f
3667 (lsyntax)450 1060 y Fr(Sp)s(eci\014es)31 b(the)g(language-dep)s(enden)m
3668 (t)h(syn)m(tax)g(v)-5 b(arian)m(t)32 b(for)f(the)h(read)f(op)s
3669 (eration.)43 b(P)m(ossible)33 b(v)-5 b(alues)450 1169
3670 y(are)450 1330 y Fp(lsyntax_standard)930 1440 y Fr(accept)32
3671 b(standard)d(algebraic)j(notation)g(only)-8 b(,)31 b(no)g(complex)f(n)m
3672 (um)m(b)s(ers,)450 1601 y Fp(lsyntax_algebraic)930 1710
3673 y Fr(accept)i(the)e(algebraic)i(notation)g Fl(x)6 b Fp(+)p
3674 Fl(y)i Fp(i)30 b Fr(for)g(complex)g(n)m(um)m(b)s(ers,)450
3675 1871 y Fp(lsyntax_commonlisp)930 1981 y Fr(accept)49
3676 b(the)g Fp(#b)p Fr(,)j Fp(#o)p Fr(,)g Fp(#x)47 b Fr(syn)m(taxes)i(for)f
3677 (binary)-8 b(,)52 b(o)s(ctal,)i(hexadecimal)48 b(n)m(um)m(b)s(ers,)930
3678 2090 y Fp(#)p Fl(base)5 b Fp(R)25 b Fr(for)f(rational)i(n)m(um)m(b)s
3679 (ers)d(in)h(a)i(giv)m(en)g(base,)g Fp(#c\()p Fl(realpart)32
3680 b(imagpart)r Fp(\))24 b Fr(for)h(com-)930 2200 y(plex)30
3681 b(n)m(um)m(b)s(ers,)450 2361 y Fp(lsyntax_all)930 2471
3682 y Fr(accept)i(all)f(of)g(these)f(extensions.)-30 2631
3683 y Fp(unsigned)e(int)i(rational_base)450 2741 y Fr(The)g(base)g(in)g
3684 (whic)m(h)h(rational)g(n)m(um)m(b)s(ers)d(are)j(read.)-30
3685 2902 y Fp(cl_float_format_t)26 b(float_flags.default_flo)o(at_f)o(orm)o
3686 (at)450 3012 y Fr(The)k(\015oat)h(format)f(used)f(when)h(reading)g
3687 (\015oats)h(with)f(exp)s(onen)m(t)g(mark)m(er)g(`)p Fp(e)p
3688 Fr('.)-30 3172 y Fp(cl_float_format_t)c(float_flags.default_lfl)o(oat_)
3689 o(for)o(mat)450 3282 y Fr(The)k(\015oat)h(format)f(used)f(when)h
3690 (reading)g(\015oats)h(with)f(exp)s(onen)m(t)g(mark)m(er)g(`)p
3691 Fp(l)p Fr('.)-30 3443 y Fp(cl_boolean)e(float_flags.mantissa_de)o(pen)o
3692 (dent)o(_flo)o(at_)o(form)o(at)450 3553 y Fr(When)h(this)g(\015ag)h(is)
3693 f(true,)g(\015oats)h(sp)s(eci\014ed)e(with)h(more)g(digits)g(than)g
3694 (corresp)s(onding)g(to)g(the)h(exp)s(o-)450 3662 y(nen)m(t)g(mark)m(er)
3695 e(they)i(con)m(tain,)h(but)e(without)p 2018 3662 28 4
3696 v 62 w Fl(nnn)e Fr(su\016x,)i(will)h(get)h(a)e(precision)h(corresp)s
3697 (onding)e(to)450 3772 y(their)i(n)m(um)m(b)s(er)e(of)j(signi\014can)m
3698 (t)g(digits.)-30 4033 y Fs(5.3)68 b(Output)45 b(functions)-30
3699 4227 y Fr(Including)40 b Fp(<cln/io.h>)f Fr(de\014nes)h(a)i(t)m(yp)s(e)
3700 f Fp(cl_ostream)p Fr(,)g(whic)m(h)g(is)h(the)f(t)m(yp)s(e)g(of)h(the)f
3701 (\014rst)f(argumen)m(t)h(to)h(all)-30 4337 y(output)30
3702 b(functions.)40 b Fp(cl_ostream)28 b Fr(is)i(the)h(same)f(as)g
3703 Fp(std::ostream&)p Fr(.)-30 4473 y(The)g(v)-5 b(ariable)150
3704 4609 y Fp(cl_ostream)28 b(stdout)-30 4770 y Fr(con)m(tains)k(the)e
3705 (standard)g(output)g(stream.)-30 4906 y(The)g(v)-5 b(ariable)150
3706 5042 y Fp(cl_ostream)28 b(stderr)-30 5204 y Fr(con)m(tains)k(the)e
3707 (standard)g(error)g(output)g(stream.)-30 5340 y(These)g(are)h(the)f
3708 (simple)g(output)g(functions:)p eop
3710 31 32 bop -30 -116 a Fr(Chapter)30 b(5:)41 b(Input/Output)2844
3711 b(31)-30 299 y Fp(void)29 b(fprintchar)f(\(cl_ostream)f(stream,)h(char)
3712 i(c\))450 408 y Fr(Prin)m(ts)g(the)h(c)m(haracter)h Fp(x)e
3713 Fr(literally)i(on)e(the)h Fp(stream)p Fr(.)-30 574 y
3714 Fp(void)e(fprint)g(\(cl_ostream)e(stream,)i(const)f(char)i(*)g
3715 (string\))450 683 y Fr(Prin)m(ts)g(the)h Fp(string)d
3716 Fr(literally)33 b(on)d(the)g Fp(stream)p Fr(.)-30 848
3717 y Fp(void)f(fprintdecimal)e(\(cl_ostream)g(stream,)i(int)g(x\))-30
3718 958 y(void)g(fprintdecimal)e(\(cl_ostream)g(stream,)i(const)g(cl_I&)g
3719 (x\))450 1068 y Fr(Prin)m(ts)h(the)h(in)m(teger)h Fp(x)e
3720 Fr(in)g(decimal)g(on)g(the)h Fp(stream)p Fr(.)-30 1233
3721 y Fp(void)e(fprintbinary)e(\(cl_ostream)h(stream,)g(const)h(cl_I&)g
3722 (x\))450 1342 y Fr(Prin)m(ts)h(the)h(in)m(teger)h Fp(x)e
3723 Fr(in)g(binary)f(\(base)i(2,)g(without)g(pre\014x\))e(on)h(the)h
3724 Fp(stream)p Fr(.)-30 1508 y Fp(void)e(fprintoctal)f(\(cl_ostream)f
3725 (stream,)h(const)h(cl_I&)g(x\))450 1617 y Fr(Prin)m(ts)h(the)h(in)m
3726 (teger)h Fp(x)e Fr(in)g(o)s(ctal)h(\(base)g(8,)g(without)g(pre\014x\))e
3727 (on)i(the)f Fp(stream)p Fr(.)-30 1782 y Fp(void)f(fprinthexadecimal)d
3728 (\(cl_ostream)h(stream,)i(const)f(cl_I&)h(x\))450 1892
3729 y Fr(Prin)m(ts)h(the)h(in)m(teger)h Fp(x)e Fr(in)g(hexadecimal)g
3730 (\(base)h(16,)h(without)e(pre\014x\))g(on)g(the)h Fp(stream)p
3731 Fr(.)-30 2060 y(Eac)m(h)53 b(of)f(the)g(classes)i Fp(cl_N)p
3732 Fr(,)i Fp(cl_R)p Fr(,)h Fp(cl_RA)p Fr(,)f Fp(cl_I)p Fr(,)h
3733 Fp(cl_F)p Fr(,)f Fp(cl_SF)p Fr(,)h Fp(cl_FF)p Fr(,)f
3734 Fp(cl_DF)p Fr(,)h Fp(cl_LF)50 b Fr(de\014nes,)57 b(in)-30
3735 2170 y Fp(<cln/)p Fl(t)m(yp)s(e)5 b Fp(_io.h>)p Fr(,)28
3736 b(the)i(follo)m(wing)i(output)e(functions:)-30 2338 y
3737 Fp(void)f(fprint)g(\(cl_ostream)e(stream,)i(const)f Fl(t)m(yp)s(e)5
3738 b Fp(&)31 b(x\))-30 2447 y(cl_ostream)d(operator<<)f(\(cl_ostream)g
3739 (stream,)i(const)g Fl(t)m(yp)s(e)5 b Fp(&)30 b(x\))450
3740 2557 y Fr(Prin)m(ts)42 b(the)g(n)m(um)m(b)s(er)e Fp(x)i
3741 Fr(on)g(the)g Fp(stream)p Fr(.)75 b(The)41 b(output)h(ma)m(y)g(dep)s
3742 (end)e(on)i(the)h(global)g(prin)m(ter)450 2667 y(settings)36
3743 b(in)g(the)f(v)-5 b(ariable)37 b Fp(default_print_flags)p
3744 Fr(.)51 b(The)35 b Fp(ostream)e Fr(\015ags)j(and)f(settings)h
3745 (\(\015ags,)450 2776 y(width)30 b(and)f(lo)s(cale\))k(are)e(ignored.)
3746 -30 2944 y(The)f(most)g(\015exible)g(output)g(function,)g(de\014ned)g
3747 (in)g Fp(<cln/)p Fl(t)m(yp)s(e)5 b Fp(_io.h>)p Fr(,)28
3748 b(are)i(the)h(follo)m(wing:)210 3079 y Fp(void)47 b(print_complex)92
3749 b(\(cl_ostream)44 b(stream,)i(const)g(cl_print_flags&)e(flags,)1212
3750 3183 y(const)j(cl_N&)f(z\);)210 3286 y(void)h(print_real)236
3751 b(\(cl_ostream)44 b(stream,)i(const)g(cl_print_flags&)e(flags,)1212
3752 3390 y(const)j(cl_R&)f(z\);)210 3494 y(void)h(print_float)188
3753 b(\(cl_ostream)44 b(stream,)i(const)g(cl_print_flags&)e(flags,)1212
3754 3598 y(const)j(cl_F&)f(z\);)210 3702 y(void)h(print_rational)d
3755 (\(cl_ostream)g(stream,)i(const)g(cl_print_flags&)e(flags,)1212
3756 3805 y(const)j(cl_RA&)f(z\);)210 3909 y(void)h(print_integer)92
3757 b(\(cl_ostream)44 b(stream,)i(const)g(cl_print_flags&)e(flags,)1212
3758 4013 y(const)j(cl_I&)f(z\);)-30 4153 y Fr(Prin)m(ts)30
3759 b(the)h(n)m(um)m(b)s(er)d Fp(x)i Fr(on)g(the)h Fp(stream)p
3760 Fr(.)39 b(The)30 b Fp(flags)f Fr(are)h(parameters)g(whic)m(h)g
3761 (a\013ect)i(the)f(output.)-30 4293 y(The)f(structure)g(t)m(yp)s(e)g
3762 Fp(cl_print_flags)d Fr(con)m(tains)k(the)g(follo)m(wing)h(\014elds:)-30
3763 4462 y Fp(unsigned)c(int)i(rational_base)450 4571 y Fr(The)g(base)g(in)
3764 g(whic)m(h)h(rational)g(n)m(um)m(b)s(ers)d(are)j(prin)m(ted.)40
3765 b(Default)32 b(is)e Fp(10)p Fr(.)-30 4736 y Fp(cl_boolean)e
3766 (rational_readably)450 4846 y Fr(If)34 b(this)h(\015ag)h(is)f(true,)h
3767 (rational)g(n)m(um)m(b)s(ers)c(are)k(prin)m(ted)e(with)h(radix)f(sp)s
3768 (eci\014ers)h(in)f(Common)f(Lisp)450 4956 y(syn)m(tax)e(\()p
3769 Fp(#)p Fl(n)p Fp(R)f Fr(or)g Fp(#b)g Fr(or)g Fp(#o)g
3770 Fr(or)g Fp(#x)g Fr(pre\014xes,)g(trailing)h(dot\).)41
3771 b(Default)32 b(is)e(false.)-30 5121 y Fp(cl_boolean)e(float_readably)
3772 450 5230 y Fr(If)34 b(this)g(\015ag)h(is)f(true,)i(t)m(yp)s(e)e(sp)s
3773 (eci\014c)g(exp)s(onen)m(t)h(mark)m(ers)e(ha)m(v)m(e)j(precedence)f(o)m
3774 (v)m(er)g('E'.)g(Default)h(is)450 5340 y(false.)p eop
3776 32 33 bop -30 -116 a Fr(Chapter)30 b(5:)41 b(Input/Output)2844
3777 b(32)-30 299 y Fp(cl_float_format_t)26 b(default_float_format)450
3778 408 y Fr(Floating)40 b(p)s(oin)m(t)d(n)m(um)m(b)s(ers)e(of)j(this)f
3779 (format)g(will)h(b)s(e)f(prin)m(ted)g(using)g(the)h('E')g(exp)s(onen)m
3780 (t)f(mark)m(er.)450 518 y(Default)31 b(is)g Fp(cl_float_format_ffloat)p
3781 Fr(.)-30 677 y Fp(cl_boolean)d(complex_readably)450 787
3782 y Fr(If)41 b(this)g(\015ag)g(is)h(true,)i(complex)c(n)m(um)m(b)s(ers)f
3783 (will)j(b)s(e)e(prin)m(ted)h(using)g(the)g(Common)e(Lisp)h(syn)m(tax)
3784 450 897 y Fp(#C\()p Fl(realpart)32 b(imagpart)r Fp(\))p
3785 Fr(.)40 b(Default)32 b(is)e(false.)-30 1056 y Fp(cl_string)e
3786 (univpoly_varname)450 1166 y Fr(Univ)-5 b(ariate)37 b(p)s(olynomials)d
3787 (with)i(no)f(explicit)i(indeterminate)e(name)f(will)i(b)s(e)f(prin)m
3788 (ted)g(using)g(this)450 1275 y(v)-5 b(ariable)31 b(name.)40
3789 b(Default)31 b(is)g Fp("x")p Fr(.)-30 1435 y(The)26 b(global)i(v)-5
3790 b(ariable)27 b Fp(default_print_flags)21 b Fr(con)m(tains)28
3791 b(the)f(default)f(v)-5 b(alues,)28 b(used)e(b)m(y)g(the)h(function)f
3792 Fp(fprint)p Fr(.)p eop
3794 33 34 bop -30 -116 a Fr(Chapter)30 b(6:)41 b(Rings)3180
3795 b(33)-30 299 y Fo(6)80 b(Rings)-30 533 y Fr(CLN)30 b(has)g(a)h(class)g
3796 (of)g(abstract)g(rings.)1403 662 y Fp(Ring)1308 765 y(cl_ring)1212
3797 869 y(<cln/ring.h>)-30 1004 y Fr(Rings)f(can)h(b)s(e)f(compared)f(for)h
3798 (equalit)m(y:)-30 1163 y Fp(bool)f(operator==)f(\(const)g(cl_ring&,)g
3799 (const)h(cl_ring&\))-30 1273 y(bool)g(operator!=)f(\(const)g(cl_ring&,)
3800 g(const)h(cl_ring&\))450 1382 y Fr(These)h(compare)g(t)m(w)m(o)i(rings)
3801 e(for)g(equalit)m(y)-8 b(.)-30 1542 y(Giv)m(en)31 b(a)g(ring)f
3802 Fp(R)p Fr(,)g(the)h(follo)m(wing)h(mem)m(b)s(ers)27 b(can)k(b)s(e)f
3803 (used.)-30 1701 y Fp(void)f(R->fprint)f(\(cl_ostream)f(stream,)i(const)
3804 g(cl_ring_element&)d(x\))-30 1811 y(cl_boolean)i(R->equal)g(\(const)g
3805 (cl_ring_element&)e(x,)k(const)f(cl_ring_element&)d(y\))-30
3806 1920 y(cl_ring_element)g(R->zero)j(\(\))-30 2030 y(cl_boolean)f
3807 (R->zerop)g(\(const)g(cl_ring_element&)e(x\))-30 2139
3808 y(cl_ring_element)g(R->plus)j(\(const)f(cl_ring_element&)e(x,)k(const)f
3809 (cl_ring_element&)d(y\))-30 2249 y(cl_ring_element)g(R->minus)i
3810 (\(const)h(cl_ring_element&)d(x,)k(const)f(cl_ring_element&)d(y\))-30
3811 2359 y(cl_ring_element)g(R->uminus)i(\(const)h(cl_ring_element&)d(x\))
3812 -30 2468 y(cl_ring_element)g(R->one)j(\(\))-30 2578 y(cl_ring_element)d
3813 (R->canonhom)h(\(const)i(cl_I&)g(x\))-30 2687 y(cl_ring_element)d
3814 (R->mul)j(\(const)g(cl_ring_element&)c(x,)30 b(const)f
3815 (cl_ring_element&)d(y\))-30 2797 y(cl_ring_element)g(R->square)i
3816 (\(const)h(cl_ring_element&)d(x\))-30 2907 y(cl_ring_element)g
3817 (R->expt_pos)h(\(const)i(cl_ring_element&)d(x,)k(const)f(cl_I&)g(y\))
3818 -30 3041 y Fr(The)h(follo)m(wing)i(rings)e(are)g(built-in.)-30
3819 3200 y Fp(cl_null_ring)d(cl_0_ring)450 3310 y Fr(The)j(n)m(ull)g(ring,)
3820 h(con)m(taining)g(only)g(zero.)-30 3469 y Fp(cl_complex_ring)26
3821 b(cl_C_ring)450 3579 y Fr(The)k(ring)g(of)h(complex)f(n)m(um)m(b)s
3822 (ers.)38 b(This)29 b(corresp)s(onds)g(to)j(the)e(t)m(yp)s(e)h
3823 Fp(cl_N)p Fr(.)-30 3738 y Fp(cl_real_ring)c(cl_R_ring)450
3824 3848 y Fr(The)j(ring)g(of)h(real)g(n)m(um)m(b)s(ers.)38
3825 b(This)29 b(corresp)s(onds)g(to)i(the)g(t)m(yp)s(e)g
3826 Fp(cl_R)p Fr(.)-30 4007 y Fp(cl_rational_ring)26 b(cl_RA_ring)450
3827 4117 y Fr(The)k(ring)g(of)h(rational)g(n)m(um)m(b)s(ers.)38
3828 b(This)30 b(corresp)s(onds)f(to)i(the)g(t)m(yp)s(e)f
3829 Fp(cl_RA)p Fr(.)-30 4276 y Fp(cl_integer_ring)c(cl_I_ring)450
3830 4386 y Fr(The)k(ring)g(of)h(in)m(tegers.)41 b(This)30
3831 b(corresp)s(onds)f(to)i(the)g(t)m(yp)s(e)f Fp(cl_I)p
3832 Fr(.)-30 4545 y(T)m(yp)s(e)g(tests)h(can)g(b)s(e)e(p)s(erformed)f(for)i
3833 (an)m(y)h(of)g Fp(cl_C_ring)p Fr(,)d Fp(cl_R_ring)p Fr(,)g
3834 Fp(cl_RA_ring)p Fr(,)g Fp(cl_I_ring)p Fr(:)-30 4705 y
3835 Fp(cl_boolean)g(instanceof)f(\(const)i(cl_number&)e(x,)j(const)f
3836 (cl_number_ring&)d(R\))450 4814 y Fr(T)-8 b(ests)31 b(whether)f(the)g
3837 (giv)m(en)i(n)m(um)m(b)s(er)c(is)i(an)g(elemen)m(t)h(of)g(the)f(n)m(um)
3838 m(b)s(er)e(ring)i(R.)p eop
3840 34 35 bop -30 -116 a Fr(Chapter)30 b(7:)41 b(Mo)s(dular)30
3841 b(in)m(tegers)2730 b(34)-30 299 y Fo(7)80 b(Mo)t(dular)55
3842 b(in)l(tegers)-30 657 y Fs(7.1)68 b(Mo)t(dular)45 b(in)l(teger)h(rings)
3843 -30 850 y Fr(CLN)29 b(implemen)m(ts)f(mo)s(dular)f(in)m(tegers,)k(i.e.)
3844 41 b(in)m(tegers)31 b(mo)s(dulo)c(a)j(\014xed)e(in)m(teger)j(N.)f(The)e
3845 (mo)s(dulus)f(is)i(explicitly)-30 960 y(part)37 b(of)h(ev)m(ery)g(mo)s
3846 (dular)e(in)m(teger.)63 b(CLN)37 b(do)s(esn't)g(allo)m(w)i(y)m(ou)f(to)
3847 g(\(acciden)m(tally\))j(mix)36 b(elemen)m(ts)i(of)f(di\013eren)m(t)-30
3848 1069 y(mo)s(dular)25 b(rings,)j(e.g.)41 b Fp(\(3)30 b(mod)f(4\))h(+)g
3849 (\(2)g(mod)f(5\))e Fr(will)h(result)f(in)g(a)h(run)m(time)e(error.)40
3850 b(\(Ideally)28 b(one)g(w)m(ould)f(imagine)-30 1179 y(a)k(generic)h
3851 (data)f(t)m(yp)s(e)g Fp(cl_MI\(N\))p Fr(,)e(but)h(C)p
3852 Fp(++)g Fr(do)s(esn't)h(ha)m(v)m(e)h(generic)g(t)m(yp)s(es.)42
3853 b(So)30 b(one)h(has)g(to)h(liv)m(e)g(with)e(run)m(time)-30
3854 1288 y(c)m(hec)m(ks.\))-30 1423 y(The)g(class)h(of)g(mo)s(dular)d(in)m
3855 (teger)k(rings)e(is)1403 1552 y Fp(Ring)1308 1656 y(cl_ring)1212
3856 1760 y(<cln/ring.h>)1451 1863 y(|)1451 1967 y(|)1021
3857 2071 y(Modular)46 b(integer)g(ring)1165 2175 y(cl_modint_ring)1069
3858 2279 y(<cln/modinteger.h>)-30 2413 y Fr(and)30 b(the)g(class)h(of)g
3859 (all)g(mo)s(dular)e(in)m(tegers)i(\(elemen)m(ts)g(of)g(mo)s(dular)d(in)
3860 m(teger)k(rings\))e(is)1165 2542 y Fp(Modular)45 b(integer)1403
3861 2646 y(cl_MI)1117 2750 y(<cln/modinteger.h>)-30 2885
3862 y Fr(Mo)s(dular)30 b(in)m(teger)i(rings)e(are)g(constructed)h(using)f
3863 (the)h(function)-30 3044 y Fp(cl_modint_ring)26 b(find_modint_ring)g
3864 (\(const)j(cl_I&)g(N\))450 3154 y Fr(This)f(function)g(returns)f(the)h
3865 (mo)s(dular)e(ring)j(`)p Fp(Z/NZ)p Fr('.)39 b(It)28 b(tak)m(es)i(care)f
3866 (of)g(\014nding)e(out)h(ab)s(out)g(sp)s(ecial)450 3264
3867 y(cases)e(of)e Fp(N)p Fr(,)i(lik)m(e)g(p)s(o)m(w)m(ers)f(of)g(t)m(w)m
3868 (o)h(and)e(o)s(dd)f(n)m(um)m(b)s(ers)g(for)h(whic)m(h)h(Mon)m(tgomery)g
3869 (m)m(ultiplication)h(will)450 3373 y(b)s(e)g(a)h(win,)g(and)f
3870 (precomputes)f(an)m(y)i(necessary)g(auxiliary)h(data)f(for)f(computing)
3871 g(mo)s(dulo)f Fp(N)p Fr(.)39 b(There)450 3483 y(is)27
3872 b(a)h(cac)m(he)g(table)g(of)g(rings,)f(indexed)g(b)m(y)g
3873 Fp(N)g Fr(\(or,)h(more)e(precisely)-8 b(,)29 b(b)m(y)f
3874 Fp(abs\(N\))p Fr(\).)38 b(This)26 b(ensures)g(that)450
3875 3592 y(the)31 b(precomputation)e(costs)j(are)e(reduced)g(to)h(a)g
3876 (minim)m(um.)-30 3752 y(Mo)s(dular)f(in)m(teger)i(rings)e(can)g(b)s(e)g
3877 (compared)f(for)i(equalit)m(y:)-30 3912 y Fp(bool)e(operator==)f
3878 (\(const)g(cl_modint_ring&,)e(const)j(cl_modint_ring&\))-30
3879 4022 y(bool)g(operator!=)f(\(const)g(cl_modint_ring&,)e(const)j
3880 (cl_modint_ring&\))450 4131 y Fr(These)42 b(compare)g(t)m(w)m(o)i(mo)s
3881 (dular)c(in)m(teger)k(rings)e(for)g(equalit)m(y)-8 b(.)78
3882 b(Tw)m(o)43 b(di\013eren)m(t)g(calls)g(to)g Fp(find_)450
3883 4241 y(modint_ring)c Fr(with)i(the)h(same)f(argumen)m(t)g(necessarily)h
3884 (return)f(the)h(same)f(ring)g(b)s(ecause)h(it)g(is)450
3885 4350 y(memoized)29 b(in)h(the)h(cac)m(he)h(table.)-30
3886 4608 y Fs(7.2)68 b(F)-11 b(unctions)44 b(on)h(mo)t(dular)g(in)l(tegers)
3887 -30 4801 y Fr(Giv)m(en)31 b(a)g(mo)s(dular)d(in)m(teger)k(ring)e
3888 Fp(R)p Fr(,)h(the)f(follo)m(wing)i(mem)m(b)s(ers)c(can)i(b)s(e)g(used.)
3889 -30 4961 y Fp(cl_I)f(R->modulus)450 5071 y Fr(This)h(is)g(the)h(ring's)
3890 f(mo)s(dulus,)e(normalized)i(to)h(b)s(e)e(nonnegativ)m(e:)43
3891 b Fp(abs\(N\))p Fr(.)-30 5230 y Fp(cl_MI)29 b(R->zero\(\))450
3892 5340 y Fr(This)h(returns)f Fp(0)h(mod)f(N)p Fr(.)p eop
3894 35 36 bop -30 -116 a Fr(Chapter)30 b(7:)41 b(Mo)s(dular)30
3895 b(in)m(tegers)2730 b(35)-30 299 y Fp(cl_MI)29 b(R->one\(\))450
3896 408 y Fr(This)h(returns)f Fp(1)h(mod)f(N)p Fr(.)-30 576
3897 y Fp(cl_MI)g(R->canonhom)e(\(const)i(cl_I&)g(x\))450
3898 686 y Fr(This)h(returns)f Fp(x)h(mod)f(N)p Fr(.)-30 853
3899 y Fp(cl_I)g(R->retract)f(\(const)g(cl_MI&)h(x\))450 963
3900 y Fr(This)c(is)h(a)h(partial)f(in)m(v)m(erse)h(function)f(to)h
3901 Fp(R->canonhom)p Fr(.)36 b(It)26 b(returns)f(the)h(standard)f(represen)
3902 m(tativ)m(e)450 1072 y(\()p Fp(>=0)p Fr(,)30 b Fp(<N)p
3903 Fr(\))g(of)h Fp(x)p Fr(.)-30 1240 y Fp(cl_MI)e
3904 (R->random\(random_state&)24 b(randomstate\))-30 1349
3905 y(cl_MI)29 b(R->random\(\))450 1459 y Fr(This)h(returns)f(a)h(random)f
3906 (in)m(teger)j(mo)s(dulo)d Fp(N)p Fr(.)-30 1630 y(The)h(follo)m(wing)i
3907 (op)s(erations)e(are)h(de\014ned)e(on)i(mo)s(dular)d(in)m(tegers.)-30
3908 1802 y Fp(cl_modint_ring)e(x.ring)j(\(\))450 1912 y Fr(Returns)g(the)i
3909 (ring)f(to)h(whic)m(h)f(the)h(mo)s(dular)d(in)m(teger)k
3910 Fp(x)e Fr(b)s(elongs.)-30 2079 y Fp(cl_MI)f(operator+)f(\(const)g
3911 (cl_MI&,)h(const)g(cl_MI&\))450 2189 y Fr(Returns)36
3912 b(the)h(sum)d(of)j(t)m(w)m(o)h(mo)s(dular)d(in)m(tegers.)60
3913 b(One)37 b(of)f(the)h(argumen)m(ts)f(ma)m(y)g(also)i(b)s(e)e(a)h(plain)
3914 450 2298 y(in)m(teger.)-30 2466 y Fp(cl_MI)29 b(operator-)f(\(const)g
3915 (cl_MI&,)h(const)g(cl_MI&\))450 2575 y Fr(Returns)37
3916 b(the)h(di\013erence)g(of)g(t)m(w)m(o)h(mo)s(dular)d(in)m(tegers.)65
3917 b(One)37 b(of)h(the)g(argumen)m(ts)f(ma)m(y)h(also)g(b)s(e)g(a)450
3918 2685 y(plain)30 b(in)m(teger.)-30 2852 y Fp(cl_MI)f(operator-)f
3919 (\(const)g(cl_MI&\))450 2962 y Fr(Returns)h(the)i(negativ)m(e)i(of)d(a)
3920 h(mo)s(dular)d(in)m(teger.)-30 3130 y Fp(cl_MI)h(operator*)f(\(const)g
3921 (cl_MI&,)h(const)g(cl_MI&\))450 3239 y Fr(Returns)e(the)g(pro)s(duct)f
3922 (of)i(t)m(w)m(o)h(mo)s(dular)c(in)m(tegers.)41 b(One)27
3923 b(of)h(the)f(argumen)m(ts)g(ma)m(y)g(also)h(b)s(e)f(a)h(plain)450
3924 3349 y(in)m(teger.)-30 3516 y Fp(cl_MI)h(square)g(\(const)f(cl_MI&\))
3925 450 3626 y Fr(Returns)h(the)i(square)f(of)h(a)f(mo)s(dular)f(in)m
3926 (teger.)-30 3793 y Fp(cl_MI)g(recip)g(\(const)g(cl_MI&)f(x\))450
3927 3903 y Fr(Returns)g(the)i(recipro)s(cal)g Fp(x^-1)e Fr(of)i(a)f(mo)s
3928 (dular)f(in)m(teger)i Fp(x)p Fr(.)40 b Fp(x)29 b Fr(m)m(ust)g(b)s(e)f
3929 (coprime)h(to)h(the)f(mo)s(dulus,)450 4012 y(otherwise)i(an)f(error)g
3930 (message)h(is)f(issued.)-30 4180 y Fp(cl_MI)f(div)g(\(const)g(cl_MI&)g
3931 (x,)h(const)e(cl_MI&)h(y\))450 4290 y Fr(Returns)36 b(the)g(quotien)m
3932 (t)i Fp(x*y^-1)d Fr(of)h(t)m(w)m(o)i(mo)s(dular)c(in)m(tegers)k
3933 Fp(x)p Fr(,)g Fp(y)p Fr(.)59 b Fp(y)36 b Fr(m)m(ust)f(b)s(e)h(coprime)g
3934 (to)h(the)450 4399 y(mo)s(dulus,)28 b(otherwise)j(an)f(error)g(message)
3935 g(is)h(issued.)-30 4567 y Fp(cl_MI)e(expt_pos)f(\(const)h(cl_MI&)f(x,)i
3936 (const)f(cl_I&)g(y\))450 4676 y(y)h Fr(m)m(ust)f(b)s(e)h
3937 Fp(>)g Fr(0.)41 b(Returns)30 b Fp(x^y)p Fr(.)-30 4844
3938 y Fp(cl_MI)f(expt)g(\(const)g(cl_MI&)f(x,)i(const)f(cl_I&)g(y\))450
3939 4953 y Fr(Returns)h Fp(x^y)p Fr(.)42 b(If)30 b Fp(y)h
3940 Fr(is)g(negativ)m(e,)i Fp(x)e Fr(m)m(ust)f(b)s(e)g(coprime)g(to)i(the)f
3941 (mo)s(dulus,)e(else)i(an)g(error)g(message)450 5063 y(is)f(issued.)-30
3942 5230 y Fp(cl_MI)f(operator<<)f(\(const)g(cl_MI&)h(x,)h(const)f(cl_I&)f
3943 (y\))450 5340 y Fr(Returns)h Fp(x*2^y)p Fr(.)p eop
3945 36 37 bop -30 -116 a Fr(Chapter)30 b(7:)41 b(Mo)s(dular)30
3946 b(in)m(tegers)2730 b(36)-30 299 y Fp(cl_MI)29 b(operator>>)f(\(const)g
3947 (cl_MI&)h(x,)h(const)f(cl_I&)f(y\))450 408 y Fr(Returns)k
3948 Fp(x*2^-y)p Fr(.)45 b(When)33 b Fp(y)f Fr(is)g(p)s(ositiv)m(e,)j(the)e
3949 (mo)s(dulus)c(m)m(ust)j(b)s(e)g(o)s(dd,)g(or)h(an)f(error)g(message)h
3950 (is)450 518 y(issued.)-30 677 y Fp(bool)c(operator==)f(\(const)g
3951 (cl_MI&,)h(const)g(cl_MI&\))-30 787 y(bool)g(operator!=)f(\(const)g
3952 (cl_MI&,)h(const)g(cl_MI&\))450 897 y Fr(Compares)21
3953 b(t)m(w)m(o)j(mo)s(dular)c(in)m(tegers,)25 b(b)s(elonging)e(to)g(the)f
3954 (same)g(mo)s(dular)e(in)m(teger)k(ring,)g(for)e(equalit)m(y)-8
3955 b(.)-30 1056 y Fp(cl_boolean)28 b(zerop)h(\(const)f(cl_MI&)h(x\))450
3956 1166 y Fr(Returns)g(true)i(if)f Fp(x)g Fr(is)g Fp(0)g(mod)g(N)p
3957 Fr(.)-30 1325 y(The)g(follo)m(wing)i(output)e(functions)g(are)g
3958 (de\014ned)g(\(see)h(also)g(the)g(c)m(hapter)g(on)f(input/output\).)-30
3959 1484 y Fp(void)f(fprint)g(\(cl_ostream)e(stream,)i(const)f(cl_MI&)h
3960 (x\))-30 1594 y(cl_ostream)f(operator<<)f(\(cl_ostream)g(stream,)i
3961 (const)g(cl_MI&)f(x\))450 1704 y Fr(Prin)m(ts)40 b(the)h(mo)s(dular)d
3962 (in)m(teger)k Fp(x)d Fr(on)i(the)f Fp(stream)p Fr(.)69
3963 b(The)40 b(output)g(ma)m(y)f(dep)s(end)g(on)h(the)h(global)450
3964 1813 y(prin)m(ter)30 b(settings)h(in)f(the)h(v)-5 b(ariable)31
3965 b Fp(default_print_flags)p Fr(.)p eop
3967 37 38 bop -30 -116 a Fr(Chapter)30 b(8:)41 b(Sym)m(b)s(olic)29
3968 b(data)i(t)m(yp)s(es)2596 b(37)-30 299 y Fo(8)80 b(Sym)l(b)t(olic)54
3969 b(data)g(t)l(yp)t(es)-30 513 y Fr(CLN)30 b(implemen)m(ts)f(t)m(w)m(o)j
3970 (sym)m(b)s(olic)d(\(non-n)m(umeric\))h(data)h(t)m(yp)s(es:)41
3971 b(strings)30 b(and)g(sym)m(b)s(ols.)-30 758 y Fs(8.1)68
3972 b(Strings)-30 946 y Fr(The)30 b(class)1260 1071 y Fp(String)1212
3973 1175 y(cl_string)1117 1278 y(<cln/string.h>)-30 1409
3974 y Fr(implemen)m(ts)f(imm)m(utable)g(strings.)-30 1539
3975 y(Strings)h(are)h(constructed)f(through)g(the)h(follo)m(wing)g
3976 (constructors:)-30 1691 y Fp(cl_string)d(\(const)h(char)g(*)h(s\))450
3977 1800 y Fr(Returns)f(an)i(imm)m(utable)e(cop)m(y)i(of)f(the)h
3978 (\(zero-terminated\))h(C)e(string)g Fp(s)p Fr(.)-30 1951
3979 y Fp(cl_string)e(\(const)h(char)g(*)h(ptr,)f(unsigned)f(long)h(len\))
3980 450 2061 y Fr(Returns)36 b(an)h(imm)m(utable)f(cop)m(y)i(of)f(the)g
3981 Fp(len)f Fr(c)m(haracters)j(at)e Fp(ptr[0])p Fr(,)43
3982 b(.)22 b(.)h(.)11 b(,)39 b Fp(ptr[len-1])p Fr(.)58 b(NUL)450
3983 2171 y(c)m(haracters)32 b(are)f(allo)m(w)m(ed.)-30 2322
3984 y(The)f(follo)m(wing)i(functions)e(are)g(a)m(v)-5 b(ailable)33
3985 b(on)d(strings:)-30 2473 y Fp(operator)e(=)450 2583 y
3986 Fr(Assignmen)m(t)i(from)f Fp(cl_string)f Fr(and)h Fp(const)g(char)h(*)p
3987 Fr(.)-30 2734 y Fp(s.length\(\))-30 2844 y(strlen\(s\))450
3988 2953 y Fr(Returns)f(the)i(length)g(of)f(the)h(string)f
3989 Fp(s)p Fr(.)-30 3104 y Fp(s[i])288 b Fr(Returns)28 b(the)i
3990 Fp(i)p Fr(th)f(c)m(haracter)i(of)e(the)g(string)h Fp(s)p
3991 Fr(.)40 b Fp(i)29 b Fr(m)m(ust)f(b)s(e)h(in)g(the)g(range)h
3992 Fp(0)g(<=)f(i)h(<)g(s.length\(\))p Fr(.)-30 3256 y Fp(bool)f(equal)g
3993 (\(const)g(cl_string&)e(s1,)j(const)f(cl_string&)e(s2\))450
3994 3365 y Fr(Compares)38 b(t)m(w)m(o)i(strings)f(for)f(equalit)m(y)-8
3995 b(.)69 b(One)38 b(of)h(the)g(argumen)m(ts)g(ma)m(y)f(also)i(b)s(e)e(a)i
3996 (plain)e Fp(const)450 3475 y(char)29 b(*)p Fr(.)-30 3720
3997 y Fs(8.2)68 b(Sym)l(b)t(ols)-30 3908 y Fr(Sym)m(b)s(ols)31
3998 b(are)i(uniqui\014ed)d(strings:)45 b(all)34 b(sym)m(b)s(ols)d(with)h
3999 (the)g(same)g(name)g(are)h(shared.)46 b(This)32 b(means)f(that)i(com-)
4000 -30 4018 y(parison)38 b(of)g(t)m(w)m(o)i(sym)m(b)s(ols)d(is)h(fast)h
4001 (\(e\013ectiv)m(ely)j(just)37 b(a)i(p)s(oin)m(ter)f(comparison\),)i
4002 (whereas)f(comparison)e(of)i(t)m(w)m(o)-30 4128 y(strings)32
4003 b(m)m(ust)f(in)g(the)i(w)m(orst)f(case)h(w)m(alk)g(b)s(oth)e(strings)h
4004 (un)m(til)g(their)g(end.)45 b(Sym)m(b)s(ols)30 b(are)j(used,)f(for)f
4005 (example,)i(as)-30 4237 y(tags)e(for)g(prop)s(erties,)f(as)g(names)f
4006 (of)i(v)-5 b(ariables)31 b(in)f(p)s(olynomial)g(rings,)g(etc.)-30
4007 4368 y(Sym)m(b)s(ols)f(are)h(constructed)h(through)f(the)g(follo)m
4008 (wing)i(constructor:)-30 4519 y Fp(cl_symbol)c(\(const)h(cl_string&)e
4009 (s\))450 4629 y Fr(Lo)s(oks)j(up)g(or)g(creates)i(a)f(new)e(sym)m(b)s
4010 (ol)h(with)g(a)g(giv)m(en)i(name.)-30 4780 y(The)e(follo)m(wing)i(op)s
4011 (erations)e(are)h(a)m(v)-5 b(ailable)33 b(on)d(sym)m(b)s(ols:)-30
4012 4931 y Fp(cl_string)e(\(const)h(cl_symbol&)e(sym\))450
4013 5041 y Fr(Con)m(v)m(ersion)k(to)g Fp(cl_string)p Fr(:)38
4014 b(Returns)30 b(the)g(string)h(whic)m(h)f(names)f(the)i(sym)m(b)s(ol)e
4015 Fp(sym)p Fr(.)-30 5192 y Fp(bool)g(equal)g(\(const)g(cl_symbol&)e
4016 (sym1,)i(const)g(cl_symbol&)f(sym2\))450 5301 y Fr(Compares)h(t)m(w)m
4017 (o)j(sym)m(b)s(ols)d(for)h(equalit)m(y)-8 b(.)43 b(This)29
4018 b(is)i(v)m(ery)f(fast.)p eop
4020 38 39 bop -30 -116 a Fr(Chapter)30 b(9:)41 b(Univ)-5
4021 b(ariate)32 b(p)s(olynomials)2483 b(38)-30 299 y Fo(9)80
4022 b(Univ)-9 b(ariate)56 b(p)t(olynomials)-30 652 y Fs(9.1)68
4023 b(Univ)-7 b(ariate)47 b(p)t(olynomial)f(rings)-30 844
4024 y Fr(CLN)33 b(implemen)m(ts)f(univ)-5 b(ariate)35 b(p)s(olynomials)e
4025 (\(p)s(olynomials)g(in)g(one)h(v)-5 b(ariable\))35 b(o)m(v)m(er)g(an)f
4026 (arbitrary)f(ring.)50 b(The)-30 953 y(indeterminate)34
4027 b(v)-5 b(ariable)36 b(ma)m(y)e(b)s(e)h(either)g(unnamed)d(\(and)j(will)
4028 g(b)s(e)f(prin)m(ted)h(according)g(to)h Fp(default_print_)-30
4029 1063 y(flags.univpoly_varname)p Fr(,)h(whic)m(h)j(defaults)g(to)h(`)p
4030 Fp(x)p Fr('\))g(or)f(carry)h(a)f(giv)m(en)h(name.)70
4031 b(The)40 b(base)g(ring)g(and)g(the)-30 1172 y(indeterminate)31
4032 b(are)h(explicitly)g(part)f(of)h(ev)m(ery)g(p)s(olynomial.)42
4033 b(CLN)31 b(do)s(esn't)g(allo)m(w)i(y)m(ou)e(to)h(\(acciden)m(tally\))j
4034 (mix)-30 1282 y(elemen)m(ts)23 b(of)g(di\013eren)m(t)h(p)s(olynomial)e
4035 (rings,)i(e.g.)40 b Fp(\(a^2+1\))28 b(*)i(\(b^3-1\))21
4036 b Fr(will)i(result)g(in)g(a)g(run)m(time)f(error.)38
4037 b(\(Ideally)-30 1391 y(this)30 b(should)g(return)f(a)i(m)m(ultiv)-5
4038 b(ariate)31 b(p)s(olynomial,)f(but)g(they)g(are)h(not)g(y)m(et)g
4039 (implemen)m(ted)e(in)h(CLN.\))-30 1525 y(The)g(classes)h(of)g(univ)-5
4040 b(ariate)31 b(p)s(olynomial)f(rings)f(are)1499 1653 y
4041 Fp(Ring)1403 1757 y(cl_ring)1308 1861 y(<cln/ring.h>)1546
4042 1965 y(|)1546 2068 y(|)1021 2172 y(Univariate)45 b(polynomial)g(ring)
4043 1260 2276 y(cl_univpoly_ring)1260 2380 y(<cln/univpoly.h>)1546
4044 2484 y(|)735 2587 y(+----------------+------)o(----)o(----)o(---)o(--+)
4045 735 2691 y(|)763 b(|)907 b(|)258 2795 y(Complex)46 b(polynomial)f(ring)
4046 189 b(|)i(Modular)46 b(integer)g(polynomial)f(ring)258
4047 2899 y(cl_univpoly_complex_rin)o(g)137 b(|)382 b
4048 (cl_univpoly_modint_ring)258 3002 y(<cln/univpoly_complex.h)o(>)137
4049 b(|)382 b(<cln/univpoly_modint.h>)1546 3106 y(|)735 3210
4050 y(+----------------+)735 3314 y(|)763 b(|)353 3418 y(Real)47
4051 b(polynomial)e(ring)237 b(|)353 3521 y(cl_univpoly_real_ring)185
4052 b(|)353 3625 y(<cln/univpoly_real.h>)g(|)1546 3729 y(|)735
4053 3833 y(+----------------+)735 3936 y(|)763 b(|)258 4040
4054 y(Rational)45 b(polynomial)g(ring)142 b(|)258 4144 y
4055 (cl_univpoly_rational_ri)o(ng)89 b(|)258 4248 y
4056 (<cln/univpoly_rational.)o(h>)g(|)1546 4352 y(|)735 4455
4057 y(+----------------+)735 4559 y(|)258 4663 y(Integer)46
4058 b(polynomial)f(ring)258 4767 y(cl_univpoly_integer_rin)o(g)258
4059 4870 y(<cln/univpoly_integer.h)o(>)-30 5004 y Fr(and)30
4060 b(the)g(corresp)s(onding)g(classes)h(of)g(univ)-5 b(ariate)31
4061 b(p)s(olynomials)e(are)1117 5132 y Fp(Univariate)45 b(polynomial)1451
4062 5236 y(cl_UP)1260 5340 y(<cln/univpoly.h>)p eop
4064 39 40 bop -30 -116 a Fr(Chapter)30 b(9:)41 b(Univ)-5
4065 b(ariate)32 b(p)s(olynomials)2483 b(39)1546 299 y Fp(|)735
4066 403 y(+----------------+------)o(----)o(----)o(---)o(--+)735
4067 506 y(|)763 b(|)907 b(|)353 610 y(Complex)46 b(polynomial)331
4068 b(|)286 b(Modular)46 b(integer)g(polynomial)592 714 y(cl_UP_N)618
4069 b(|)764 b(cl_UP_MI)258 818 y(<cln/univpoly_complex.h)o(>)137
4070 b(|)382 b(<cln/univpoly_modint.h>)1546 922 y(|)735 1025
4071 y(+----------------+)735 1129 y(|)763 b(|)449 1233 y(Real)46
4072 b(polynomial)379 b(|)592 1337 y(cl_UP_R)618 b(|)305 1440
4073 y(<cln/univpoly_real.h>)233 b(|)1546 1544 y(|)735 1648
4074 y(+----------------+)735 1752 y(|)763 b(|)353 1856 y(Rational)46
4075 b(polynomial)283 b(|)592 1959 y(cl_UP_RA)570 b(|)258
4076 2063 y(<cln/univpoly_rational.)o(h>)89 b(|)1546 2167
4077 y(|)735 2271 y(+----------------+)735 2374 y(|)353 2478
4078 y(Integer)46 b(polynomial)592 2582 y(cl_UP_I)258 2686
4079 y(<cln/univpoly_integer.h)o(>)-30 3545 y Fr(Univ)-5 b(ariate)32
4080 b(p)s(olynomial)d(rings)h(are)h(constructed)g(using)f(the)g(functions)
4081 -30 4792 y Fp(cl_univpoly_ring)c(find_univpoly_ring)f(\(const)k
4082 (cl_ring&)f(R\))-30 4902 y(cl_univpoly_ring)e(find_univpoly_ring)f
4083 (\(const)k(cl_ring&)f(R,)i(const)f(cl_symbol&)e(varname\))450
4084 5011 y Fr(This)40 b(function)h(returns)f(the)h(p)s(olynomial)g(ring)g
4085 (`)p Fp(R[X])p Fr(',)i(unnamed)c(or)i(named.)71 b Fp(R)41
4086 b Fr(ma)m(y)g(b)s(e)f(an)450 5121 y(arbitrary)f(ring.)68
4087 b(This)39 b(function)g(tak)m(es)i(care)f(of)g(\014nding)e(out)h(ab)s
4088 (out)h(sp)s(ecial)g(cases)g(of)g Fp(R)p Fr(,)h(suc)m(h)450
4089 5230 y(as)33 b(the)g(rings)g(of)g(complex)f(n)m(um)m(b)s(ers,)g(real)h
4090 (n)m(um)m(b)s(ers,)f(rational)i(n)m(um)m(b)s(ers,)d(in)m(tegers,)k(or)e
4091 (mo)s(dular)450 5340 y(in)m(teger)i(rings.)48 b(There)33
4092 b(is)g(a)h(cac)m(he)g(table)g(of)g(rings,)f(indexed)g(b)m(y)g
4093 Fp(R)g Fr(and)f Fp(varname)p Fr(.)48 b(This)32 b(ensures)p
4096 40 41 bop -30 -116 a Fr(Chapter)30 b(9:)41 b(Univ)-5
4097 b(ariate)32 b(p)s(olynomials)2483 b(40)450 299 y(that)28
4098 b(t)m(w)m(o)i(calls)f(of)e(this)h(function)g(with)f(the)h(same)f
4099 (argumen)m(ts)h(will)g(return)e(the)i(same)g(p)s(olynomial)450
4100 408 y(ring.)-30 547 y Fp(cl_univpoly_complex_ring)c(find_univpoly_ring)
4101 h(\(const)k(cl_complex_ring&)d(R\))-30 657 y(cl_univpoly_complex_ring)e
4102 (find_univpoly_ring)h(\(const)k(cl_complex_ring&)d(R,)j(const)-30
4103 766 y(cl_symbol&)f(varname\))-30 876 y(cl_univpoly_real_ring)d
4104 (find_univpoly_ring)g(\(const)k(cl_real_ring&)d(R\))-30
4105 986 y(cl_univpoly_real_ring)f(find_univpoly_ring)g(\(const)k
4106 (cl_real_ring&)d(R,)k(const)f(cl_symbol&)-30 1095 y(varname\))-30
4107 1205 y(cl_univpoly_rational_rin)o(g)24 b(find_univpoly_ring)i(\(const)j
4108 (cl_rational_ring&)c(R\))-30 1314 y(cl_univpoly_rational_rin)o(g)f
4109 (find_univpoly_ring)i(\(const)j(cl_rational_ring&)c(R,)30
4110 b(const)-30 1424 y(cl_symbol&)e(varname\))-30 1533 y
4111 (cl_univpoly_integer_ring)c(find_univpoly_ring)h(\(const)k
4112 (cl_integer_ring&)d(R\))-30 1643 y(cl_univpoly_integer_ring)e
4113 (find_univpoly_ring)h(\(const)k(cl_integer_ring&)d(R,)j(const)-30
4114 1753 y(cl_symbol&)f(varname\))-30 1862 y(cl_univpoly_modint_ring)c
4115 (find_univpoly_ring)h(\(const)k(cl_modint_ring&)d(R\))-30
4116 1972 y(cl_univpoly_modint_ring)e(find_univpoly_ring)h(\(const)k
4117 (cl_modint_ring&)d(R,)k(const)f(cl_symbol&)-30 2081 y(varname\))450
4118 2191 y Fr(These)24 b(functions)g(are)h(equiv)-5 b(alen)m(t)26
4119 b(to)g(the)e(general)i Fp(find_univpoly_ring)p Fr(,)21
4120 b(only)k(the)f(return)g(t)m(yp)s(e)450 2301 y(is)30 b(more)g(sp)s
4121 (eci\014c,)g(according)i(to)f(the)f(base)h(ring's)f(t)m(yp)s(e.)-30
4122 2579 y Fs(9.2)68 b(F)-11 b(unctions)44 b(on)h(univ)-7
4123 b(ariate)46 b(p)t(olynomials)-30 2780 y Fr(Giv)m(en)31
4124 b(a)g(univ)-5 b(ariate)31 b(p)s(olynomial)f(ring)g Fp(R)p
4125 Fr(,)g(the)h(follo)m(wing)h(mem)m(b)s(ers)27 b(can)k(b)s(e)f(used.)-30
4126 2952 y Fp(cl_ring)e(R->basering\(\))450 3061 y Fr(This)i(returns)f(the)
4127 h(base)h(ring,)f(as)h(passed)f(to)h(`)p Fp(find_univpoly_ring)p
4128 Fr('.)-30 3229 y Fp(cl_UP)e(R->zero\(\))450 3339 y Fr(This)h(returns)f
4129 Fp(0)h(in)g(R)p Fr(,)g(a)h(p)s(olynomial)e(of)i(degree)g(-1.)-30
4130 3506 y Fp(cl_UP)e(R->one\(\))450 3616 y Fr(This)h(returns)f
4131 Fp(1)h(in)g(R)p Fr(,)g(a)h(p)s(olynomial)e(of)i(degree)g
4132 Fp(<)p Fr(=)f(0.)-30 3784 y Fp(cl_UP)f(R->canonhom)e(\(const)i(cl_I&)g
4133 (x\))450 3893 y Fr(This)h(returns)f Fp(x)h(in)g(R)p Fr(,)g(a)h(p)s
4134 (olynomial)e(of)i(degree)g Fp(<)p Fr(=)f(0.)-30 4061
4135 y Fp(cl_UP)f(R->monomial)e(\(const)i(cl_ring_element&)d(x,)k(uintL)f
4136 (e\))450 4171 y Fr(This)h(returns)f(a)h(sparse)g(p)s(olynomial:)41
4137 b Fp(x)30 b(*)g(X^e)p Fr(,)f(where)h Fp(X)g Fr(is)h(the)f
4138 (indeterminate.)-30 4339 y Fp(cl_UP)f(R->create)f(\(sintL)g(degree\))
4139 450 4448 y Fr(Creates)d(a)f(new)f(p)s(olynomial)g(with)h(a)g(giv)m(en)h
4140 (degree.)39 b(The)24 b(zero)g(p)s(olynomial)g(has)f(degree)i
4141 Fp(-1)p Fr(.)38 b(After)450 4558 y(creating)26 b(the)f(p)s(olynomial,)g
4142 (y)m(ou)g(should)e(put)h(in)g(the)h(co)s(e\016cien)m(ts,)j(using)c(the)
4143 h Fp(set_coeff)d Fr(mem)m(b)s(er)450 4667 y(function,)30
4144 b(and)g(then)g(call)i(the)e Fp(finalize)f Fr(mem)m(b)s(er)e(function.)
4145 -30 4839 y(The)j(follo)m(wing)i(are)e(the)h(only)g(destructiv)m(e)g(op)
4146 s(erations)g(on)f(univ)-5 b(ariate)31 b(p)s(olynomials.)-30
4147 5011 y Fp(void)e(set_coeff)f(\(cl_UP&)g(x,)i(uintL)f(index,)g(const)g
4148 (cl_ring_element&)d(y\))450 5121 y Fr(This)32 b(c)m(hanges)i(the)f(co)s
4149 (e\016cien)m(t)i(of)e Fp(X^index)e Fr(in)i Fp(x)g Fr(to)g(b)s(e)g
4150 Fp(y)p Fr(.)48 b(After)33 b(c)m(hanging)h(a)f(p)s(olynomial)g(and)450
4151 5230 y(b)s(efore)40 b(applying)h(an)m(y)g Fp(")p Fr(normal)p
4152 Fp(")e Fr(op)s(eration)i(on)g(it,)j(y)m(ou)d(should)f(call)i(its)f
4153 Fp(finalize)e Fr(mem)m(b)s(er)450 5340 y(function.)p
4156 41 42 bop -30 -116 a Fr(Chapter)30 b(9:)41 b(Univ)-5
4157 b(ariate)32 b(p)s(olynomials)2483 b(41)-30 299 y Fp(void)29
4158 b(finalize)f(\(cl_UP&)h(x\))450 408 y Fr(This)j(function)h(marks)f(the)
4159 h(endp)s(oin)m(t)f(of)i(destructiv)m(e)g(mo)s(di\014cations)e(of)h(a)h
4160 (p)s(olynomial.)48 b(It)33 b(nor-)450 518 y(malizes)c(the)f(in)m
4161 (ternal)h(represen)m(tation)h(so)e(that)h(subsequen)m(t)f(computations)
4162 g(ha)m(v)m(e)i(less)e(o)m(v)m(erhead.)450 628 y(Doing)f(normal)e
4163 (computations)g(on)h(unnormalized)f(p)s(olynomials)g(ma)m(y)g(pro)s
4164 (duce)g(wrong)g(results)h(or)450 737 y(crash)k(the)h(program.)-30
4165 887 y(The)f(follo)m(wing)i(op)s(erations)e(are)h(de\014ned)e(on)i(univ)
4166 -5 b(ariate)31 b(p)s(olynomials.)-30 1036 y Fp(cl_univpoly_ring)26
4167 b(x.ring)j(\(\))450 1146 y Fr(Returns)g(the)i(ring)f(to)h(whic)m(h)f
4168 (the)h(univ)-5 b(ariate)31 b(p)s(olynomial)f Fp(x)g Fr(b)s(elongs.)-30
4169 1295 y Fp(cl_UP)f(operator+)f(\(const)g(cl_UP&,)h(const)g(cl_UP&\))450
4170 1405 y Fr(Returns)g(the)i(sum)d(of)j(t)m(w)m(o)h(univ)-5
4171 b(ariate)31 b(p)s(olynomials.)-30 1554 y Fp(cl_UP)e(operator-)f
4172 (\(const)g(cl_UP&,)h(const)g(cl_UP&\))450 1664 y Fr(Returns)g(the)i
4173 (di\013erence)g(of)f(t)m(w)m(o)i(univ)-5 b(ariate)31
4174 b(p)s(olynomials.)-30 1813 y Fp(cl_UP)e(operator-)f(\(const)g(cl_UP&\))
4175 450 1923 y Fr(Returns)h(the)i(negativ)m(e)i(of)d(a)h(univ)-5
4176 b(ariate)31 b(p)s(olynomial.)-30 2072 y Fp(cl_UP)e(operator*)f(\(const)
4177 g(cl_UP&,)h(const)g(cl_UP&\))450 2182 y Fr(Returns)i(the)i(pro)s(duct)d
4178 (of)j(t)m(w)m(o)g(univ)-5 b(ariate)33 b(p)s(olynomials.)45
4179 b(One)32 b(of)g(the)g(argumen)m(ts)g(ma)m(y)f(also)j(b)s(e)450
4180 2291 y(a)d(plain)f(in)m(teger)i(or)e(an)g(elemen)m(t)h(of)g(the)f(base)
4181 h(ring.)-30 2441 y Fp(cl_UP)e(square)g(\(const)f(cl_UP&\))450
4182 2550 y Fr(Returns)h(the)i(square)f(of)h(a)f(univ)-5 b(ariate)31
4183 b(p)s(olynomial.)-30 2700 y Fp(cl_UP)e(expt_pos)f(\(const)h(cl_UP&)f
4184 (x,)i(const)f(cl_I&)g(y\))450 2809 y(y)h Fr(m)m(ust)f(b)s(e)h
4185 Fp(>)g Fr(0.)41 b(Returns)30 b Fp(x^y)p Fr(.)-30 2959
4186 y Fp(bool)f(operator==)f(\(const)g(cl_UP&,)h(const)g(cl_UP&\))-30
4187 3068 y(bool)g(operator!=)f(\(const)g(cl_UP&,)h(const)g(cl_UP&\))450
4188 3178 y Fr(Compares)23 b(t)m(w)m(o)i(univ)-5 b(ariate)25
4189 b(p)s(olynomials,)g(b)s(elonging)g(to)g(the)f(same)g(univ)-5
4190 b(ariate)24 b(p)s(olynomial)g(ring,)450 3288 y(for)30
4191 b(equalit)m(y)-8 b(.)-30 3437 y Fp(cl_boolean)28 b(zerop)h(\(const)f
4192 (cl_UP&)h(x\))450 3547 y Fr(Returns)g(true)i(if)f Fp(x)g
4193 Fr(is)g Fp(0)g(in)g(R)p Fr(.)-30 3696 y Fp(sintL)f(degree)g(\(const)f
4194 (cl_UP&)h(x\))450 3806 y Fr(Returns)g(the)i(degree)g(of)g(the)f(p)s
4195 (olynomial.)40 b(The)30 b(zero)h(p)s(olynomial)f(has)g(degree)h
4196 Fp(-1)p Fr(.)-30 3955 y Fp(cl_ring_element)26 b(coeff)j(\(const)g
4197 (cl_UP&)f(x,)i(uintL)f(index\))450 4065 y Fr(Returns)g(the)i(co)s
4198 (e\016cien)m(t)h(of)f Fp(X^index)d Fr(in)i(the)h(p)s(olynomial)f
4199 Fp(x)p Fr(.)-30 4214 y Fp(cl_ring_element)c(x)k(\(const)f
4200 (cl_ring_element&)d(y\))450 4324 y Fr(Ev)-5 b(aluation:)42
4201 b(If)29 b Fp(x)h Fr(is)h(a)f(p)s(olynomial)g(and)f Fp(y)h
4202 Fr(b)s(elongs)g(to)h(the)g(base)f(ring,)g(then)g(`)p
4203 Fp(x\(y\))p Fr(')g(returns)f(the)450 4433 y(v)-5 b(alue)31
4204 b(of)f(the)h(substitution)f(of)h Fp(y)f Fr(in)m(to)h
4205 Fp(x)p Fr(.)-30 4583 y Fp(cl_UP)e(deriv)g(\(const)g(cl_UP&)f(x\))450
4206 4692 y Fr(Returns)h(the)i(deriv)-5 b(ativ)m(e)32 b(of)e(the)h(p)s
4207 (olynomial)f Fp(x)g Fr(with)g(resp)s(ect)g(to)h(the)g(indeterminate)f
4208 Fp(X)p Fr(.)-30 4842 y(The)g(follo)m(wing)i(output)e(functions)g(are)g
4209 (de\014ned)g(\(see)h(also)g(the)g(c)m(hapter)g(on)f(input/output\).)-30
4210 4991 y Fp(void)f(fprint)g(\(cl_ostream)e(stream,)i(const)f(cl_UP&)h
4211 (x\))-30 5101 y(cl_ostream)f(operator<<)f(\(cl_ostream)g(stream,)i
4212 (const)g(cl_UP&)f(x\))450 5210 y Fr(Prin)m(ts)d(the)f(univ)-5
4213 b(ariate)26 b(p)s(olynomial)e Fp(x)g Fr(on)h(the)f Fp(stream)p
4214 Fr(.)38 b(The)24 b(output)g(ma)m(y)g(dep)s(end)f(on)i(the)g(global)450
4215 5320 y(prin)m(ter)30 b(settings)h(in)f(the)h(v)-5 b(ariable)31
4216 b Fp(default_print_flags)p Fr(.)p eop
4218 42 43 bop -30 -116 a Fr(Chapter)30 b(9:)41 b(Univ)-5
4219 b(ariate)32 b(p)s(olynomials)2483 b(42)-30 299 y Fs(9.3)68
4220 b(Sp)t(ecial)45 b(p)t(olynomials)-30 491 y Fr(The)30
4221 b(follo)m(wing)i(functions)e(return)f(sp)s(ecial)i(p)s(olynomials.)-30
4222 651 y Fp(cl_UP_I)d(tschebychev)g(\(sintL)g(n\))450 760
4223 y Fr(Returns)h(the)i(n-th)f(Cheb)m(yshev)g(p)s(olynomial)f(\(n)h
4224 Fp(>)p Fr(=)g(0\).)-30 920 y Fp(cl_UP_I)e(hermite)h(\(sintL)f(n\))450
4225 1029 y Fr(Returns)h(the)i(n-th)f(Hermite)g(p)s(olynomial)g(\(n)g
4226 Fp(>)p Fr(=)g(0\).)-30 1189 y Fp(cl_UP_RA)e(legendre)g(\(sintL)h(n\))
4227 450 1298 y Fr(Returns)g(the)i(n-th)f(Legendre)g(p)s(olynomial)g(\(n)g
4228 Fp(>)p Fr(=)g(0\).)-30 1458 y Fp(cl_UP_I)e(laguerre)h(\(sintL)f(n\))450
4229 1567 y Fr(Returns)h(the)i(n-th)f(Laguerre)h(p)s(olynomial)e(\(n)h
4230 Fp(>)p Fr(=)g(0\).)-30 1727 y(Information)e(ho)m(w)h(to)h(deriv)m(e)g
4231 (the)g(di\013eren)m(tial)g(equation)h(satis\014ed)e(b)m(y)g(eac)m(h)h
4232 (of)g(these)g(p)s(olynomials)e(from)g(their)-30 1836
4233 y(de\014nition)i(can)h(b)s(e)e(found)g(in)i(the)f Fp(doc/polynomial/)c
4234 Fr(directory)-8 b(.)p eop
4236 43 44 bop -30 -116 a Fr(Chapter)30 b(10:)41 b(In)m(ternals)3010
4237 b(43)-30 299 y Fo(10)80 b(In)l(ternals)-30 651 y Fs(10.1)68
4238 b(Wh)l(y)45 b(C)p Fg(++)f Fs(?)-30 843 y Fr(Using)31
4239 b(C)p Fp(++)e Fr(as)i(an)f(implemen)m(tation)g(language)i(pro)m(vides)
4240 45 977 y Fq(\017)60 b Fr(E\016ciency:)41 b(It)31 b(compiles)f(to)h(mac)
4241 m(hine)f(co)s(de.)45 1111 y Fq(\017)60 b Fr(P)m(ortabilit)m(y:)49
4242 b(It)33 b(runs)f(on)h(all)h(platforms)f(supp)s(orting)e(a)j(C)p
4243 Fp(++)e Fr(compiler.)50 b(Because)34 b(of)g(the)f(a)m(v)-5
4244 b(ailabilit)m(y)37 b(of)150 1220 y(GNU)j(C)p Fp(++)p
4245 Fr(,)h(this)f(includes)f(all)h(curren)m(tly)g(used)e(32-bit)j(and)e
4246 (64-bit)i(platforms,)g(indep)s(enden)m(tly)d(of)i(the)150
4247 1330 y(qualit)m(y)32 b(of)e(the)h(v)m(endor's)f(C)p Fp(++)f
4248 Fr(compiler.)45 1464 y Fq(\017)60 b Fr(T)m(yp)s(e)30
4249 b(safet)m(y:)43 b(The)30 b(C)p Fp(++)g Fr(compilers)g(kno)m(ws)g(ab)s
4250 (out)h(the)f(n)m(um)m(b)s(er)f(t)m(yp)s(es)h(and)h(complains)f(if,)h
4251 (for)f(example,)150 1573 y(y)m(ou)44 b(try)f(to)i(assign)f(a)g(\015oat)
4252 g(to)g(an)g(in)m(teger)h(v)-5 b(ariable.)81 b(Ho)m(w)m(ev)m(er,)50
4253 b(a)44 b(dra)m(wbac)m(k)g(is)f(that)h(C)p Fp(++)f Fr(do)s(esn't)150
4254 1683 y(kno)m(w)33 b(ab)s(out)h(generic)g(t)m(yp)s(es,)g(hence)g(a)g
4255 (restriction)g(lik)m(e)h(that)f Fp(operator+)28 b(\(const)g(cl_MI&,)h
4256 (const)g(cl_)150 1792 y(MI&\))j Fr(requires)h(that)g(b)s(oth)g(argumen)
4257 m(ts)f(b)s(elong)h(to)h(the)g(same)e(mo)s(dular)f(ring)i(cannot)h(b)s
4258 (e)f(expressed)f(as)i(a)150 1902 y(compile-time)c(information.)45
4259 2036 y Fq(\017)60 b Fr(Algebraic)37 b(syn)m(tax:)52 b(The)35
4260 b(elemen)m(tary)h(op)s(erations)g Fp(+)p Fr(,)g Fp(-)p
4261 Fr(,)h Fp(*)p Fr(,)g Fp(=)p Fr(,)g Fp(==)p Fr(,)f(...)57
4262 b(can)36 b(b)s(e)f(used)f(in)i(in\014x)e(notation,)150
4263 2145 y(whic)m(h)c(is)h(more)e(con)m(v)m(enien)m(t)k(than)d(Lisp)f
4264 (notation)j(`)p Fp(\(+)e(x)g(y\))p Fr(')g(or)g(C)g(notation)i(`)p
4265 Fp(add\(x,y,&z\))p Fr('.)-30 2303 y(With)24 b(these)f(language)h
4266 (features,)i(there)d(is)g(no)g(need)g(for)f(t)m(w)m(o)j(separate)f
4267 (languages,)i(one)d(for)g(the)g(implemen)m(tation)-30
4268 2413 y(of)44 b(the)g(library)f(and)g(one)h(in)g(whic)m(h)f(the)h
4269 (library's)g(users)f(can)h(program.)79 b(This)43 b(means)g(that)h(a)g
4270 (protot)m(yp)s(e)-30 2523 y(implemen)m(tation)c(of)g(an)g(algorithm)g
4271 (can)h(b)s(e)f(in)m(tegrated)h(in)m(to)h(the)e(library)g(immediately)f
4272 (after)i(it)g(has)f(b)s(een)-30 2632 y(tested)g(and)e(debugged.)65
4273 b(No)39 b(need)g(to)g(rewrite)g(it)h(in)e(a)h(lo)m(w-lev)m(el)j
4274 (language)e(after)f(ha)m(ving)g(protot)m(yp)s(ed)g(in)g(a)-30
4275 2742 y(high-lev)m(el)32 b(language.)-30 2997 y Fs(10.2)68
4276 b(Memory)45 b(e\016ciency)-30 3189 y Fr(In)30 b(order)g(to)h(sa)m(v)m
4277 (e)h(memory)c(allo)s(cations,)33 b(CLN)d(implemen)m(ts:)45
4278 3323 y Fq(\017)60 b Fr(Ob)5 b(ject)31 b(sharing:)40 b(An)30
4279 b(op)s(eration)h(lik)m(e)g Fp(x+0)f Fr(returns)f Fp(x)h
4280 Fr(without)g(cop)m(ying)i(it.)45 3457 y Fq(\017)60 b
4281 Fr(Garbage)29 b(collection:)42 b(A)28 b(reference)h(coun)m(ting)g(mec)m
4282 (hanism)d(mak)m(es)i(sure)f(that)i(an)m(y)f(n)m(um)m(b)s(er)d(ob)5
4283 b(ject's)29 b(stor-)150 3566 y(age)j(is)e(freed)g(immediately)g(when)f
4284 (the)i(last)g(reference)g(to)g(the)f(ob)5 b(ject)32 b(is)e(gone.)45
4285 3700 y Fq(\017)60 b Fr(Small)35 b(in)m(tegers)h(are)g(represen)m(ted)f
4286 (as)h(immediate)e(v)-5 b(alues)36 b(instead)g(of)f(p)s(oin)m(ters)h(to)
4287 g(heap)f(allo)s(cated)i(stor-)150 3810 y(age.)51 b(This)33
4288 b(means)f(that)i(in)m(tegers)g Fp(>)d(-2^29)p Fr(,)h
4289 Fp(<)f(2^29)h Fr(don't)h(consume)f(heap)h(memory)-8 b(,)33
4290 b(unless)g(they)g(w)m(ere)150 3919 y(explicitly)f(allo)s(cated)g(on)f
4291 (the)f(heap.)-30 4174 y Fs(10.3)68 b(Sp)t(eed)45 b(e\016ciency)-30
4292 4366 y Fr(Sp)s(eed)29 b(e\016ciency)j(is)e(obtained)h(b)m(y)f(the)h
4293 (com)m(bination)f(of)h(the)f(follo)m(wing)i(tric)m(ks)f(and)f
4294 (algorithms:)45 4500 y Fq(\017)60 b Fr(Small)43 b(in)m(tegers,)49
4295 b(b)s(eing)43 b(represen)m(ted)h(as)g(immediate)f(v)-5
4296 b(alues,)48 b(don't)43 b(require)h(memory)e(access,)48
4297 b(just)c(a)150 4610 y(couple)31 b(of)f(instructions)h(for)f(eac)m(h)h
4298 (elemen)m(tary)g(op)s(eration.)45 4744 y Fq(\017)60 b
4299 Fr(The)31 b(k)m(ernel)h(of)f(CLN)g(has)g(b)s(een)g(written)g(in)g
4300 (assem)m(bly)g(language)i(for)e(some)g(CPUs)f(\()p Fp(i386)p
4301 Fr(,)h Fp(m68k)p Fr(,)g Fp(sparc)p Fr(,)150 4853 y Fp(mips)p
4302 Fr(,)e Fp(arm)p Fr(\).)45 4987 y Fq(\017)60 b Fr(On)38
4303 b(all)h(CPUs,)h(CLN)e(ma)m(y)g(b)s(e)g(con\014gured)g(to)h(use)f(the)g
4304 (sup)s(ere\016cien)m(t)h(lo)m(w-lev)m(el)i(routines)d(from)f(GNU)150
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4306 b Fr(F)-8 b(or)32 b(large)g(n)m(um)m(b)s(ers,)e(CLN)g(uses,)i(instead)f
4307 (of)h(the)f(standard)f Fp(O\(N^2\))g Fr(algorithm,)i(the)f(Karatsuba)g
4308 (m)m(ulti-)150 5340 y(plication,)h(whic)m(h)e(is)h(an)f
4309 Ff(O)s Fr(\()p Ff(N)1217 5307 y Fi(1)p Fj(:)p Fi(6)1307
4310 5340 y Fr(\))h(algorithm.)p eop
4312 44 45 bop -30 -116 a Fr(Chapter)30 b(10:)41 b(In)m(ternals)3010
4313 b(44)45 299 y Fq(\017)60 b Fr(F)-8 b(or)50 b(v)m(ery)g(large)g(n)m(um)m
4314 (b)s(ers)d(\(more)i(than)g(12000)j(decimal)d(digits\),)55
4315 b(CLN)49 b(uses)g(Sc)m(h\177)-45 b(onhage-Strassen)150
4316 408 y(m)m(ultiplication,)32 b(whic)m(h)e(is)g(an)g(asymptotically)i
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4318 b Fr(These)32 b(fast)g(m)m(ultiplication)i(algorithms)e(also)h(giv)m(e)
4319 g(impro)m(v)m(emen)m(ts)f(in)g(the)g(sp)s(eed)f(of)i(division)f(and)f
4320 (radix)150 653 y(con)m(v)m(ersion.)-30 910 y Fs(10.4)68
4321 b(Garbage)46 b(collection)-30 1103 y Fr(All)33 b(the)g(n)m(um)m(b)s(er)
4322 d(classes)j(are)g(reference)g(coun)m(t)g(classes:)46
4323 b(They)31 b(only)i(con)m(tain)h(a)f(p)s(oin)m(ter)f(to)h(an)f(ob)5
4324 b(ject)34 b(in)e(the)-30 1212 y(heap.)40 b(Up)s(on)27
4325 b(construction,)j(assignmen)m(t)e(and)g(destruction)h(of)f(n)m(um)m(b)s
4326 (er)e(ob)5 b(jects,)30 b(only)f(the)f(ob)5 b(jects')29
4327 b(reference)-30 1322 y(coun)m(t)i(are)g(manipulated.)-30
4328 1456 y(Memory)k(o)s(ccupied)h(b)m(y)f(n)m(um)m(b)s(er)f(ob)5
4329 b(jects)36 b(are)h(automatically)g(reclaimed)f(as)g(so)s(on)f(as)h
4330 (their)g(reference)g(coun)m(t)-30 1566 y(drops)29 b(to)i(zero.)-30
4331 1700 y(F)-8 b(or)34 b(n)m(um)m(b)s(er)e(rings,)i(another)g(strategy)h
4332 (is)f(implemen)m(ted:)46 b(There)33 b(is)h(a)g(cac)m(he)h(of,)g(for)f
4333 (example,)g(the)g(mo)s(dular)-30 1810 y(in)m(teger)29
4334 b(rings.)39 b(A)27 b(mo)s(dular)f(in)m(teger)j(ring)e(is)g(destro)m(y)m
4335 (ed)h(only)g(if)f(its)h(reference)g(coun)m(t)g(dropp)s(ed)d(to)j(zero)g
4336 (and)f(the)-30 1919 y(cac)m(he)36 b(is)f(ab)s(out)f(to)h(b)s(e)f
4337 (resized.)54 b(The)34 b(e\013ect)i(of)e(this)h(strategy)h(is)e(that)h
4338 (recen)m(tly)h(used)e(rings)g(remain)g(cac)m(hed,)-30
4339 2029 y(whereas)c(undue)f(memory)f(consumption)h(through)h(cac)m(hed)h
4340 (rings)f(is)h(a)m(v)m(oided.)p eop
4342 45 46 bop -30 -116 a Fr(Chapter)30 b(11:)41 b(Using)31
4343 b(the)f(library)2681 b(45)-30 299 y Fo(11)80 b(Using)54
4344 b(the)f(library)-30 542 y Fr(F)-8 b(or)47 b(the)g(follo)m(wing)g
4345 (discussion,)j(w)m(e)d(will)g(assume)e(that)h(y)m(ou)h(ha)m(v)m(e)h
4346 (installed)f(the)f(CLN)g(source)g(in)g Fp($CLN_)-30 652
4347 y(DIR)41 b Fr(and)g(built)h(it)g(in)g Fp($CLN_TARGETDIR)p
4348 Fr(.)71 b(F)-8 b(or)43 b(example,)h(for)e(me)f(it's)h
4349 Fp(CLN_DIR="$HOME/cln")37 b Fr(and)k Fp(CLN_)-30 762
4350 y(TARGETDIR="$HOME/cln/lin)o(uxel)o(f")p Fr(.)53 b(Y)-8
4351 b(ou)38 b(migh)m(t)e(de\014ne)g(these)h(as)g(en)m(vironmen)m(t)f(v)-5
4352 b(ariables,)40 b(or)c(directly)-30 871 y(substitute)30
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4354 b(Compiler)46 b(options)-30 1332 y Fr(Un)m(til)31 b(y)m(ou)g(ha)m(v)m
4355 (e)h(installed)f(CLN)f(in)g(a)h(public)e(place,)j(the)e(follo)m(wing)i
4356 (options)f(are)g(needed:)-30 1470 y(When)f(y)m(ou)h(compile)f(CLN)g
4357 (application)i(co)s(de,)f(add)e(the)i(\015ags)353 1601
4358 y Fp(-I$CLN_DIR/include)43 b(-I$CLN_TARGETDIR/include)-30
4359 1739 y Fr(to)29 b(the)g(C)p Fp(++)e Fr(compiler's)h(command)f(line)h
4360 (\()p Fp(make)g Fr(v)-5 b(ariable)29 b(CFLA)m(GS)g(or)f(CXXFLA)m(GS\).)
4361 i(When)e(y)m(ou)h(link)f(CLN)-30 1849 y(application)k(co)s(de)e(to)h
4362 (form)e(an)i(executable,)h(add)e(the)g(\015ags)353 1980
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4364 Fp(++)e Fr(compiler's)h(command)e(line)j(\()p Fp(make)f
4365 Fr(v)-5 b(ariable)31 b(LIBS\).)-30 2256 y(If)62 b(y)m(ou)h(did)f(a)h
4366 Fp(make)29 b(install)p Fr(,)69 b(the)62 b(include)h(\014les)f(are)h
4367 (installed)g(in)g(a)f(public)g(directory)h(\(normally)-30
4368 2365 y Fp(/usr/local/include)p Fr(\),)45 b(hence)g(y)m(ou)h(don't)g
4369 (need)f(sp)s(ecial)h(\015ags)g(for)f(compiling.)85 b(The)45
4370 b(library)g(has)h(b)s(een)-30 2475 y(installed)i(to)g(a)f(public)f
4371 (directory)i(as)f(w)m(ell)h(\(normally)f Fp(/usr/local/lib)p
4372 Fr(\),)h(hence)f(when)f(linking)h(a)g(CLN)-30 2584 y(application)32
4373 b(it)f(is)f(su\016cien)m(t)h(to)g(giv)m(e)h(the)e(\015ag)h
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4378 3155 y(p)s(oten)m(tial)f(name)e(clashes)h(with)f(other)h(C)p
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4380 (application,)j(y)m(ou)d(will)f(ha)m(v)m(e)i(to)-30 3264
4381 y(man)m(ually)30 b(p)s(ort)g(it)h(to)g(the)f(new)g(sc)m(heme.)41
4382 b(The)29 b(follo)m(wing)j(principles)e(will)h(help)f(during)f(the)i
4383 (transition:)45 3402 y Fq(\017)60 b Fr(All)35 b(headers)f(are)h(no)m(w)
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4385 b(of)h(including)f Fp(cl_)p Fl(something)8 b Fp(.h)p
4386 Fr(,)33 b(include)150 3512 y Fp(cln/)p Fl(something)8
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4391 (um)m(b)s(er)d(t)m(yp)s(es,)j(\(cl)p 1404 3757 28 4 v
4392 34 w(N,)f(cl)p 1625 3757 V 34 w(I,)g(cl)p 1811 3757 V
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4396 950 3867 V 33 w(UP)p 1113 3867 V 33 w Fl(t)m(yp)s(e)5
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4398 (not)h(b)s(e)f(mnemonic)e(enough)150 3977 y(once)d(the)g(namespace)f
4399 Fp(cln)f Fr(is)i(imp)s(orted.)39 b(Ev)m(en)30 b(in)g(a)h(namespace)f(w)
4400 m(e)h(fa)m(v)m(or)g Fp(cl_N)f Fr(o)m(v)m(er)h Fp(N)p
4401 Fr(.\))45 4113 y Fq(\017)60 b Fr(All)30 b(public)e Fe(functions)37
4402 b Fr(that)29 b(had)g(b)m(y)g(a)g Fp(cl_)f Fr(in)h(their)g(name)f(still)
4403 i(carry)e(that)i Fp(cl_)e Fr(if)h(it)h(is)f(in)m(trinsic)g(part)g(of)
4404 150 4222 y(a)i(t)m(yp)s(ename)e(\(as)i(in)f Fp(cl_I_to_int)e(\(\))p
4405 Fr(\).)-30 4386 y(When)d(dev)m(eloping)h(other)f(libraries,)h(please)g
4406 (k)m(eep)f(in)g(mind)e(not)i(to)g(imp)s(ort)f(the)h(namespace)f
4407 Fp(cln)g Fr(in)h(one)g(of)g(y)m(our)-30 4496 y(public)e(header)g
4408 (\014les)h(b)m(y)g(sa)m(ying)g Fp(using)29 b(namespace)f(cln;)p
4409 Fr(.)37 b(This)23 b(w)m(ould)h(propagate)g(to)h(other)f(applications)h
4410 (and)-30 4605 y(can)31 b(cause)g(name)e(clashes)i(there.)-30
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4419 408 y Fr(The)30 b(class)h(cl)p 920 408 28 4 v 33 w(n)m(um)m(b)s(er.)-30
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4424 (ers.)-30 1121 y Fp(<cln/float.h>)450 1230 y Fr(F)-8
4425 b(unctions)31 b(for)f(class)h(cl)p 1288 1230 V 34 w(F,)f(the)h
4426 (\015oats.)-30 1395 y Fp(<cln/sfloat.h>)450 1504 y Fr(F)-8
4427 b(unctions)31 b(for)f(class)h(cl)p 1288 1504 V 34 w(SF,)f(the)h
4428 (short-\015oats.)-30 1669 y Fp(<cln/ffloat.h>)450 1778
4429 y Fr(F)-8 b(unctions)31 b(for)f(class)h(cl)p 1288 1778
4430 V 34 w(FF,)g(the)f(single-\015oats.)-30 1943 y Fp(<cln/dfloat.h>)450
4431 2052 y Fr(F)-8 b(unctions)31 b(for)f(class)h(cl)p 1288
4432 2052 V 34 w(DF,)g(the)f(double-\015oats.)-30 2217 y Fp(<cln/lfloat.h>)
4433 450 2326 y Fr(F)-8 b(unctions)31 b(for)f(class)h(cl)p
4434 1288 2326 V 34 w(LF,)f(the)h(long-\015oats.)-30 2491
4435 y Fp(<cln/rational.h>)450 2600 y Fr(F)-8 b(unctions)31
4436 b(for)f(class)h(cl)p 1288 2600 V 34 w(RA,)f(the)h(rational)g(n)m(um)m
4437 (b)s(ers.)-30 2765 y Fp(<cln/integer.h>)450 2874 y Fr(F)-8
4438 b(unctions)31 b(for)f(class)h(cl)p 1288 2874 V 34 w(I,)f(the)h(in)m
4439 (tegers.)-30 3039 y Fp(<cln/io.h>)450 3148 y Fr(Input/Output.)-30
4440 3313 y Fp(<cln/complex_io.h>)450 3422 y Fr(Input/Output)d(for)j(class)g
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4442 3587 y Fp(<cln/real_io.h>)450 3696 y Fr(Input/Output)e(for)j(class)g
4443 (cl)p 1461 3696 V 33 w(R,)g(the)f(real)h(n)m(um)m(b)s(ers.)-30
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4446 y Fp(<cln/sfloat_io.h>)450 4244 y Fr(Input/Output)d(for)j(class)g(cl)p
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4461 28 4 v 33 w(I,)f(the)h(in)m(tegers.)-30 573 y Fp(<cln/input.h>)450
4462 682 y Fr(Flags)h(for)e(customizing)g(input)g(op)s(erations.)-30
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4467 y Fp(<cln/condition.h>)450 1778 y Fr(Conditions/exceptions.)-30
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4469 y Fp(<cln/symbol.h>)450 2326 y Fr(Sym)m(b)s(ols.)-30
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4474 f(n)m(um)m(b)s(ers.)-30 3587 y Fp(<cln/real_ring.h>)450
4475 3696 y Fr(The)g(ring)g(of)h(real)g(n)m(um)m(b)s(ers.)-30
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4478 4244 y Fr(The)f(ring)g(of)h(in)m(tegers.)-30 4408 y Fp
4479 (<cln/numtheory.h>)450 4518 y Fr(Num)m(b)s(er)d(threory)i(functions.)
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4481 (tegers.)-30 4956 y Fp(<cln/V.h>)450 5066 y Fr(V)-8 b(ectors.)-30
4482 5230 y Fp(<cln/GV.h>)450 5340 y Fr(General)31 b(v)m(ectors.)p
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4487 408 y Fr(General)31 b(v)m(ectors)h(o)m(v)m(er)g(cl)p
4488 1363 408 28 4 v 33 w(n)m(um)m(b)s(er.)-30 573 y Fp(<cln/GV_complex.h>)
4489 450 682 y Fr(General)f(v)m(ectors)h(o)m(v)m(er)g(cl)p
4490 1363 682 V 33 w(N.)-30 847 y Fp(<cln/GV_real.h>)450 956
4491 y Fr(General)f(v)m(ectors)h(o)m(v)m(er)g(cl)p 1363 956
4492 V 33 w(R.)-30 1121 y Fp(<cln/GV_rational.h>)450 1230
4493 y Fr(General)f(v)m(ectors)h(o)m(v)m(er)g(cl)p 1363 1230
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4495 y Fr(General)f(v)m(ectors)h(o)m(v)m(er)g(cl)p 1363 1504
4496 V 33 w(I.)-30 1669 y Fp(<cln/GV_modinteger.h>)450 1778
4497 y Fr(General)f(v)m(ectors)h(of)f(mo)s(dular)d(in)m(tegers.)-30
4498 1943 y Fp(<cln/SV.h>)450 2052 y Fr(Simple)h(v)m(ectors.)-30
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4500 (er)g(cl)p 1322 2326 V 33 w(n)m(um)m(b)s(er.)-30 2491
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4502 g(cl)p 1322 2600 V 33 w(N.)-30 2765 y Fp(<cln/SV_real.h>)450
4503 2874 y Fr(Simple)d(v)m(ectors)j(o)m(v)m(er)g(cl)p 1322
4504 2874 V 33 w(R.)-30 3039 y Fp(<cln/SV_rational.h>)450
4505 3148 y Fr(Simple)d(v)m(ectors)j(o)m(v)m(er)g(cl)p 1322
4506 3148 V 33 w(RA.)-30 3313 y Fp(<cln/SV_integer.h>)450
4507 3422 y Fr(Simple)d(v)m(ectors)j(o)m(v)m(er)g(cl)p 1322
4508 3422 V 33 w(I.)-30 3587 y Fp(<cln/SV_ringelt.h>)450 3696
4509 y Fr(Simple)d(v)m(ectors)j(of)e(general)i(ring)e(elemen)m(ts.)-30
4510 3861 y Fp(<cln/univpoly.h>)450 3970 y Fr(Univ)-5 b(ariate)32
4511 b(p)s(olynomials.)-30 4134 y Fp(<cln/univpoly_integer.h>)450
4512 4244 y Fr(Univ)-5 b(ariate)32 b(p)s(olynomials)d(o)m(v)m(er)j(the)f(in)
4513 m(tegers.)-30 4408 y Fp(<cln/univpoly_rational.h)o(>)450
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4516 4792 y Fr(Univ)-5 b(ariate)32 b(p)s(olynomials)d(o)m(v)m(er)j(the)f
4517 (real)g(n)m(um)m(b)s(ers.)-30 4956 y Fp(<cln/univpoly_complex.h>)450
4518 5066 y Fr(Univ)-5 b(ariate)32 b(p)s(olynomials)d(o)m(v)m(er)j(the)f
4519 (complex)f(n)m(um)m(b)s(ers.)-30 5230 y Fp(<cln/univpoly_modint.h>)450
4520 5340 y Fr(Univ)-5 b(ariate)32 b(p)s(olynomials)d(o)m(v)m(er)j(mo)s
4521 (dular)c(in)m(teger)k(rings.)p eop
4523 49 50 bop -30 -116 a Fr(Chapter)30 b(11:)41 b(Using)31
4524 b(the)f(library)2681 b(49)-30 299 y Fp(<cln/timing.h>)450
4525 408 y Fr(Timing)29 b(facilities.)-30 576 y Fp(<cln/cln.h>)450
4526 686 y Fr(Includes)h(all)h(of)f(the)h(ab)s(o)m(v)m(e.)-30
4527 965 y Fs(11.4)68 b(An)44 b(Example)-30 1166 y Fr(A)30
4528 b(function)h(whic)m(h)f(computes)f(the)i(n)m(th)f(Fib)s(onacci)h(n)m
4529 (um)m(b)s(er)d(can)j(b)s(e)f(written)g(as)h(follo)m(ws.)210
4530 1303 y Fp(#include)46 b(<cln/integer.h>)210 1407 y(#include)g
4531 (<cln/real.h>)210 1511 y(using)g(namespace)g(cln;)210
4532 1718 y(//)h(Returns)f(F_n,)h(computed)e(as)i(the)g(nearest)f(integer)g
4533 (to)210 1822 y(//)h(\(\(1+sqrt\(5\)\)/2\)^n/sqrt\(5\))o(.)42
4534 b(Assume)k(n>=0.)210 1926 y(const)g(cl_I)h(fibonacci)e(\(int)i(n\))210
4535 2030 y({)592 2133 y(//)g(Need)g(a)g(precision)e(of)j
4536 (\(\(1+sqrt\(5\)\)/2\)^-n.)592 2237 y(cl_float_format_t)43
4537 b(prec)k(=)g(cl_float_format\(\(int\)\(0.)o(2089)o(8764)o(1*n)o(+5\)\))
4538 o(;)592 2341 y(cl_R)f(sqrt5)h(=)g(sqrt\(cl_float\(5,prec\)\);)592
4539 2445 y(cl_R)f(phi)h(=)h(\(1+sqrt5\)/2;)592 2548 y(return)e(round1\()g
4540 (expt\(phi,n\)/sqrt5)d(\);)210 2652 y(})-30 2795 y Fr(Let's)31
4541 b(explain)g(what)f(is)g(going)i(on)e(in)g(detail.)-30
4542 2938 y(The)k(include)h(\014le)f Fp(<cln/integer.h>)d
4543 Fr(is)j(necessary)h(b)s(ecause)g(the)g(t)m(yp)s(e)g Fp(cl_I)e
4544 Fr(is)i(used)f(in)g(the)h(function,)h(and)-30 3048 y(the)31
4545 b(include)g(\014le)g Fp(<cln/real.h>)c Fr(is)k(needed)g(for)f(the)i(t)m
4546 (yp)s(e)f Fp(cl_R)e Fr(and)i(the)g(\015oating)h(p)s(oin)m(t)f(n)m(um)m
4547 (b)s(er)d(functions.)-30 3157 y(The)37 b(order)g(of)h(the)f(include)h
4548 (\014les)f(do)s(es)g(not)h(matter.)62 b(In)37 b(order)g(not)h(to)g
4549 (write)g(out)g Fp(cln::)p Fl(fo)s(o)i Fr(w)m(e)e(can)g(safely)-30
4550 3267 y(imp)s(ort)29 b(the)h(whole)h(namespace)f Fp(cln)p
4551 Fr(.)-30 3410 y(Then)23 b(comes)i(the)f(function)h(declaration.)40
4552 b(The)24 b(argumen)m(t)g(is)g(an)h Fp(int)p Fr(,)g(the)f(result)h(an)f
4553 (in)m(teger.)40 b(The)24 b(return)g(t)m(yp)s(e)-30 3520
4554 y(is)33 b(de\014ned)e(as)i(`)p Fp(const)c(cl_I)p Fr(',)j(not)h(simply)e
4555 (`)p Fp(cl_I)p Fr(',)i(b)s(ecause)f(that)h(allo)m(ws)h(the)f(compiler)f
4556 (to)h(detect)h(t)m(yp)s(os)f(lik)m(e)-30 3629 y(`)p Fp(fibonacci\(n\))
4557 27 b(=)j(100)p Fr('.)40 b(It)29 b(w)m(ould)f(b)s(e)h(p)s(ossible)f(to)i
4558 (declare)f(the)h(return)d(t)m(yp)s(e)i(as)g Fp(const)g(cl_R)f
4559 Fr(\(real)i(n)m(um)m(b)s(er\))-30 3739 y(or)36 b(ev)m(en)g
4560 Fp(const)29 b(cl_N)35 b Fr(\(complex)h(n)m(um)m(b)s(er\).)55
4561 b(W)-8 b(e)37 b(use)f(the)g(most)f(sp)s(ecialized)i(p)s(ossible)e
4562 (return)g(t)m(yp)s(e)h(b)s(ecause)-30 3848 y(functions)26
4563 b(whic)m(h)g(call)h(`)p Fp(fibonacci)p Fr(')d(will)j(b)s(e)e(able)i(to)
4564 g(pro\014t)f(from)e(the)j(compiler's)f(t)m(yp)s(e)g(analysis:)39
4565 b(Adding)26 b(t)m(w)m(o)-30 3958 y(in)m(tegers)g(is)e(sligh)m(tly)i
4566 (more)e(e\016cien)m(t)i(than)e(adding)g(the)h(same)f(ob)5
4567 b(jects)25 b(declared)g(as)g(complex)f(n)m(um)m(b)s(ers,)g(b)s(ecause)
4568 -30 4068 y(it)34 b(needs)f(less)h(t)m(yp)s(e)f(dispatc)m(h.)51
4569 b(Also,)35 b(when)d(linking)i(to)g(CLN)f(as)h(a)g(non-shared)e(library)
4570 -8 b(,)35 b(this)e(minimizes)f(the)-30 4177 y(size)f(of)g(the)f
4571 (resulting)h(executable)h(program.)-30 4320 y(The)f(result)g(will)h(b)s
4572 (e)f(computed)f(as)i(expt\(phi,n\)/sqrt\(5\),)h(rounded)d(to)j(the)e
4573 (nearest)h(in)m(teger.)46 b(In)31 b(order)g(to)h(get)-30
4574 4430 y(a)c(correct)h(result,)g(the)f(absolute)h(error)e(should)g(b)s(e)
4575 g(less)i(than)e(1/2,)j(i.e.)41 b(the)28 b(relativ)m(e)i(error)e(should)
4576 f(b)s(e)g(less)h(than)-30 4539 y(sqrt\(5\)/\(2*expt\(phi,n\)\).)65
4577 b(T)-8 b(o)38 b(this)g(end,)h(the)e(\014rst)g(line)h(computes)f(a)h
4578 (\015oating)g(p)s(oin)m(t)g(precision)g(for)f(sqrt\(5\))-30
4579 4649 y(and)30 b(phi.)-30 4792 y(Then)41 b(sqrt\(5\))h(is)g(computed)e
4580 (b)m(y)i(\014rst)e(con)m(v)m(erting)k(the)e(in)m(teger)h(5)f(to)g(a)g
4581 (\015oating)g(p)s(oin)m(t)g(n)m(um)m(b)s(er)d(and)i(than)-30
4582 4902 y(taking)i(the)f(square)g(ro)s(ot.)76 b(The)42 b(con)m(v)m(erse,)k
4583 (\014rst)c(taking)h(the)f(square)g(ro)s(ot)g(of)g(5,)k(and)c(then)f
4584 (con)m(v)m(erting)j(to)-30 5011 y(the)c(desired)f(precision,)j(w)m
4585 (ould)e(not)g(w)m(ork)f(in)h(CLN:)f(The)g(square)h(ro)s(ot)g(w)m(ould)f
4586 (b)s(e)g(computed)f(to)j(a)f(default)-30 5121 y(precision)g(\(normally)
4587 f(single-\015oat)j(precision\),)h(and)c(the)h(follo)m(wing)i(con)m(v)m
4588 (ersion)f(could)f(not)g(help)f(ab)s(out)h(the)-30 5230
4589 y(lac)m(king)h(accuracy)-8 b(.)68 b(This)39 b(is)g(b)s(ecause)g(CLN)g
4590 (is)g(not)g(a)g(sym)m(b)s(olic)g(computer)f(algebra)i(system)e(and)h
4591 (do)s(es)g(not)-30 5340 y(represen)m(t)30 b(sqrt\(5\))i(in)e(a)h(non-n)
4592 m(umeric)d(w)m(a)m(y)-8 b(.)p eop
4594 50 51 bop -30 -116 a Fr(Chapter)30 b(11:)41 b(Using)31
4595 b(the)f(library)2681 b(50)-30 299 y(The)29 b(t)m(yp)s(e)i
4596 Fp(cl_R)d Fr(for)i(sqrt5)g(and,)g(in)g(the)g(follo)m(wing)h(line,)g
4597 (phi)e(is)h(the)h(only)f(p)s(ossible)f(c)m(hoice.)43
4598 b(Y)-8 b(ou)30 b(cannot)h(write)-30 408 y Fp(cl_F)26
4599 b Fr(b)s(ecause)h(the)g(C)p Fp(++)g Fr(compiler)f(can)i(only)f(infer)g
4600 (that)g Fp(cl_float\(5,prec\))c Fr(is)k(a)h(real)f(n)m(um)m(b)s(er.)38
4601 b(Y)-8 b(ou)27 b(cannot)-30 518 y(write)k Fp(cl_N)e Fr(b)s(ecause)h(a)h
4602 (`)p Fp(round1)p Fr(')e(do)s(es)h(not)h(exist)g(for)f(general)h
4603 (complex)f(n)m(um)m(b)s(ers.)-30 651 y(When)21 b(the)g(function)g
4604 (returns,)h(all)g(the)f(lo)s(cal)h(v)-5 b(ariables)22
4605 b(in)f(the)g(function)g(are)g(automatically)i(reclaimed)e(\(garbage)-30
4606 761 y(collected\).)43 b(Only)30 b(the)h(result)f(surviv)m(es)g(and)g
4607 (gets)h(passed)f(to)h(the)g(caller.)-30 894 y(The)d(\014le)h
4608 Fp(fibonacci.cc)c Fr(in)j(the)h(sub)s(directory)e Fp(examples)f
4609 Fr(con)m(tains)k(this)f(implemen)m(tation)f(together)i(with)e(an)-30
4610 1003 y(ev)m(en)j(faster)g(algorithm.)-30 1257 y Fs(11.5)68
4611 b(Debugging)46 b(supp)t(ort)-30 1448 y Fr(When)30 b(debugging)g(a)h
4612 (CLN)f(application)i(with)e(GNU)h Fp(gdb)p Fr(,)f(t)m(w)m(o)h
4613 (facilities)i(are)e(a)m(v)-5 b(ailable)33 b(from)28 b(the)j(library:)45
4614 1581 y Fq(\017)60 b Fr(The)36 b(library)g(do)s(es)g(t)m(yp)s(e)h(c)m
4615 (hec)m(ks,)j(range)d(c)m(hec)m(ks,)i(consistency)f(c)m(hec)m(ks)g(at)f
4616 (man)m(y)f(places.)60 b(When)36 b(one)h(of)150 1691 y(these)44
4617 b(fails,)j(the)c(function)g Fp(cl_abort\(\))e Fr(is)i(called.)81
4618 b(Its)43 b(default)g(implemen)m(tation)g(is)g(to)h(p)s(erform)d(an)150
4619 1800 y Fp(exit\(1\))p Fr(,)32 b(so)i(y)m(ou)g(w)m(on't)g(ha)m(v)m(e)h
4620 (a)f(core)g(dump.)47 b(But)34 b(for)f(debugging,)h(it)g(is)g(b)s(est)f
4621 (to)h(set)g(a)g(breakp)s(oin)m(t)f(at)150 1910 y(this)d(function:)390
4622 2037 y Fp(\(gdb\))46 b(break)h(cl_abort)150 2170 y Fr(When)30
4623 b(this)g(breakp)s(oin)m(t)h(is)f(hit,)h(lo)s(ok)g(at)g(the)f(stac)m
4624 (k's)i(bac)m(ktrace:)390 2298 y Fp(\(gdb\))46 b(where)45
4625 2431 y Fq(\017)60 b Fr(The)32 b(debugger's)g(normal)g
4626 Fp(print)f Fr(command)f(do)s(esn't)j(kno)m(w)f(ab)s(out)h(CLN's)f(t)m
4627 (yp)s(es)g(and)g(therefore)h(prin)m(ts)150 2540 y(mostly)j(useless)g
4628 (hexadecimal)h(addresses.)58 b(CLN)35 b(o\013ers)i(a)g(function)f
4629 Fp(cl_print)p Fr(,)f(callable)k(from)34 b(the)j(de-)150
4630 2650 y(bugger,)29 b(for)f(prin)m(ting)g(n)m(um)m(b)s(er)d(ob)5
4631 b(jects.)41 b(In)28 b(order)f(to)i(get)g(this)f(function,)h(y)m(ou)f
4632 (ha)m(v)m(e)i(to)f(de\014ne)e(the)h(macro)150 2760 y(`)p
4633 Fp(CL_DEBUG)p Fr(')21 b(and)h(then)g(include)g(all)i(the)f(header)f
4634 (\014les)h(for)f(whic)m(h)g(y)m(ou)h(w)m(an)m(t)h Fp(cl_print)c
4635 Fr(debugging)i(supp)s(ort.)150 2869 y(F)-8 b(or)31 b(example:)390
4636 2996 y Fp(#define)46 b(CL_DEBUG)390 3100 y(#include)g(<cln/string.h>)
4637 150 3233 y Fr(No)m(w,)30 b(if)e(y)m(ou)g(ha)m(v)m(e)i(in)d(y)m(our)h
4638 (program)f(a)i(v)-5 b(ariable)29 b Fp(cl_string)f(s)p
4639 Fr(,)g(and)g(insp)s(ect)g(it)g(under)f Fp(gdb)p Fr(,)h(the)g(output)150
4640 3343 y(ma)m(y)i(lo)s(ok)h(lik)m(e)h(this:)390 3470 y
4641 Fp(\(gdb\))46 b(print)h(s)390 3574 y($7)g(=)h({<cl_gcpointer>)43
4642 b(=)48 b({)f(=)h({pointer)d(=)j(0x8055b60,)d(heappointer)f(=)k
4643 (0x8055b60,)485 3678 y(word)f(=)h(134568800}},)c(})390
4644 3782 y(\(gdb\))i(call)h(cl_print\(s\))390 3885 y(\(cl_string\))e("")390
4645 3989 y($8)i(=)h(134568800)150 4122 y Fr(Note)38 b(that)f(the)g(output)f
4646 (of)h Fp(cl_print)d Fr(go)s(es)j(to)g(the)g(program's)f(error)g
4647 (output,)i(not)e(to)i(gdb's)e(standard)150 4232 y(output.)150
4648 4365 y(Note,)43 b(ho)m(w)m(ev)m(er,)f(that)d(the)g(ab)s(o)m(v)m(e)i
4649 (facilit)m(y)f(do)s(es)f(not)g(w)m(ork)g(with)f(all)i(CLN)f(t)m(yp)s
4650 (es,)i(only)e(with)f(n)m(um)m(b)s(er)150 4475 y(ob)5
4651 b(jects)27 b(and)e(similar.)38 b(Therefore)26 b(CLN)f(o\013ers)i(a)f
4652 (mem)m(b)s(er)d(function)j Fp(debug_print\(\))c Fr(on)j(all)i(CLN)f(t)m
4653 (yp)s(es.)150 4584 y(The)h(same)g(macro)g(`)p Fp(CL_DEBUG)p
4654 Fr(')f(is)h(needed)h(for)f(this)g(mem)m(b)s(er)e(function)i(to)i(b)s(e)
4655 e(implemen)m(ted.)38 b(Under)27 b Fp(gdb)p Fr(,)150 4694
4656 y(y)m(ou)k(call)g(it)g(lik)m(e)h(this:)390 4821 y Fp(\(gdb\))46
4657 b(print)h(s)390 4925 y($7)g(=)h({<cl_gcpointer>)43 b(=)48
4658 b({)f(=)h({pointer)d(=)j(0x8055b60,)d(heappointer)f(=)k(0x8055b60,)485
4659 5029 y(word)f(=)h(134568800}},)c(})390 5132 y(\(gdb\))i(call)h
4660 (s.debug_print\(\))390 5236 y(\(cl_string\))e("")390
4661 5340 y(\(gdb\))h(define)g(cprint)p eop
4663 51 52 bop -30 -116 a Fr(Chapter)30 b(11:)41 b(Using)31
4664 b(the)f(library)2681 b(51)390 299 y Fp(>call)46 b
4665 (\($1\).debug_print\(\))390 403 y(>end)390 506 y(\(gdb\))g(cprint)g(s)
4666 390 610 y(\(cl_string\))f("")150 745 y Fr(Unfortunately)-8
4667 b(,)31 b(this)g(feature)f(do)s(es)g(not)h(seem)f(to)h(w)m(ork)f(under)f
4668 (all)i(circumstances.)p eop
4670 52 53 bop -30 -116 a Fr(Chapter)30 b(12:)41 b(Customizing)2863
4671 b(52)-30 299 y Fo(12)80 b(Customizing)-30 656 y Fs(12.1)68
4672 b(Error)46 b(handling)-30 848 y Fr(When)30 b(a)h(fatal)g(error)f(o)s
4673 (ccurs,)g(an)g(error)g(message)g(is)g(output)g(to)h(the)f(standard)g
4674 (error)g(output)f(stream,)h(and)g(the)-30 958 y(function)c
4675 Fp(cl_abort)e Fr(is)i(called.)41 b(The)26 b(default)g(v)m(ersion)h(of)g
4676 (this)f(function)g(\(pro)m(vided)g(in)g(the)h(library\))f(terminates)
4677 -30 1068 y(the)i(application.)41 b(T)-8 b(o)29 b(catc)m(h)h(suc)m(h)d
4678 (a)i(fatal)g(error,)f(y)m(ou)h(need)e(to)i(de\014ne)f(the)g(function)f
4679 Fp(cl_abort)f Fr(y)m(ourself,)j(with)-30 1177 y(the)i(protot)m(yp)s(e)
4680 210 1306 y Fp(#include)46 b(<cln/abort.h>)210 1410 y(void)h(cl_abort)e
4681 (\(void\);)-30 1544 y Fr(This)30 b(function)g(m)m(ust)f(not)i(return)e
4682 (con)m(trol)j(to)f(its)f(caller.)-30 1802 y Fs(12.2)68
4683 b(Floating-p)t(oin)l(t)47 b(under\015o)l(w)-30 1994 y
4684 Fr(Floating)32 b(p)s(oin)m(t)e(under\015o)m(w)e(denotes)j(the)f
4685 (situation)h(when)e(a)h(\015oating-p)s(oin)m(t)h(n)m(um)m(b)s(er)d(is)i
4686 (to)h(b)s(e)e(created)i(whic)m(h)-30 2104 y(is)k(so)g(close)g(to)h
4687 Fp(0)e Fr(that)h(its)g(exp)s(onen)m(t)g(is)g(to)s(o)g(lo)m(w)g(to)h(b)s
4688 (e)e(represen)m(ted)g(in)m(ternally)-8 b(.)55 b(By)35
4689 b(default,)h(this)f(causes)g(a)-30 2213 y(fatal)d(error.)40
4690 b(If)30 b(y)m(ou)h(set)f(the)h(global)h(v)-5 b(ariable)210
4691 2342 y Fp(cl_boolean)45 b(cl_inhibit_floating_poin)o(t_un)o(derf)o(low)
4692 -30 2477 y Fr(to)40 b Fp(cl_true)p Fr(,)f(the)g(error)g(will)g(b)s(e)f
4693 (inhibited,)j(and)d(a)h(\015oating-p)s(oin)m(t)i(zero)e(will)h(b)s(e)e
4694 (generated)i(instead.)66 b(The)-30 2586 y(default)31
4695 b(v)-5 b(alue)30 b(of)h Fp(cl_inhibit_floating_poin)o(t_un)o(der)o
4696 (flow)24 b Fr(is)30 b Fp(cl_false)p Fr(.)-30 2844 y Fs(12.3)68
4697 b(Customizing)46 b(I/O)-30 3036 y Fr(The)31 b(output)g(of)g(the)g
4698 (function)g Fp(fprint)f Fr(ma)m(y)h(b)s(e)f(customized)h(b)m(y)g(c)m
4699 (hanging)i(the)e(v)-5 b(alue)32 b(of)f(the)h(global)g(v)-5
4700 b(ariable)-30 3146 y Fp(default_print_flags)p Fr(.)-30
4701 3403 y Fs(12.4)68 b(Customizing)46 b(the)f(memory)h(allo)t(cator)-30
4702 3596 y Fr(Ev)m(ery)28 b(memory)e(allo)s(cation)k(of)e(CLN)g(is)g(done)f
4703 (through)g(the)h(function)g(p)s(oin)m(ter)g Fp(malloc_hook)p
4704 Fr(.)37 b(F)-8 b(reeing)29 b(of)f(this)-30 3705 y(memory)33
4705 b(is)j(done)f(through)g(the)g(function)g(p)s(oin)m(ter)h
4706 Fp(free_hook)p Fr(.)53 b(The)35 b(default)g(v)m(ersions)h(of)f(these)h
4707 (functions,)-30 3815 y(pro)m(vided)27 b(in)h(the)f(library)-8
4708 b(,)29 b(call)g Fp(malloc)d Fr(and)h Fp(free)f Fr(and)h(c)m(hec)m(k)i
4709 (the)f Fp(malloc)e Fr(result)h(against)i Fp(NULL)p Fr(.)39
4710 b(If)27 b(y)m(ou)h(w)m(an)m(t)-30 3924 y(to)i(pro)m(vide)f(another)h
4711 (memory)d(allo)s(cator,)32 b(y)m(ou)e(need)f(to)h(de\014ne)e(the)i(v)-5
4712 b(ariables)30 b Fp(malloc_hook)c Fr(and)i Fp(free_hook)-30
4713 4034 y Fr(y)m(ourself,)j(lik)m(e)h(this:)210 4163 y Fp(#include)46
4714 b(<cln/malloc.h>)210 4267 y(namespace)f(cln)i({)592 4370
4715 y(void*)f(\(*malloc_hook\))e(\(size_t)i(size\))g(=)i(...)o(;)592
4716 4474 y(void)e(\(*free_hook\))f(\(void*)h(ptr\))285 b(=)48
4717 b(...)o(;)210 4578 y(})-30 4712 y Fr(The)30 b Fp(cl_malloc_hook)c
4718 Fr(function)k(m)m(ust)g(not)g(return)g(a)g Fp(NULL)g
4719 Fr(p)s(oin)m(ter.)-30 4847 y(It)e(is)f(not)h(p)s(ossible)f(to)h(c)m
4720 (hange)h(the)e(memory)f(allo)s(cator)j(at)f(run)m(time,)f(b)s(ecause)h
4721 (it)g(is)f(already)h(called)h(at)f(program)-30 4956 y(startup)i(b)m(y)g
4722 (the)h(constructors)f(of)h(some)f(global)h(v)-5 b(ariables.)p
4725 53 54 bop -30 -116 a Fr(Index)3646 b(53)-30 299 y Fo(Index)-30
4726 614 y Fs(A)-30 737 y Fd(abs)26 b(\(\))12 b Fc(.)h(.)f(.)h(.)f(.)g(.)h
4727 (.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
4728 g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
4729 (.)g(.)38 b Fb(12)-30 829 y(abstract)26 b(class)g Fc(.)12
4730 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
4731 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
4732 g(.)50 b Fb(6)-30 921 y Fd(acos)27 b(\(\))10 b Fc(.)j(.)f(.)g(.)h(.)f
4733 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)
4734 h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
4735 (.)h(.)36 b Fb(18)-30 1013 y Fd(acosh)27 b(\(\))9 b Fc(.)j(.)h(.)f(.)g
4736 (.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
4737 g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
4738 (.)f(.)35 b Fb(19)-30 1105 y(adv)n(o)r(cacy)24 b Fc(.)12
4739 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
4740 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)
4741 g(.)h(.)f(.)g(.)50 b Fb(43)-30 1197 y(Arc)n(himedes')25
4742 b(constan)n(t)17 b Fc(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
4743 f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)43
4744 b Fb(19)-30 1289 y Fd(As\(\)\(\))6 b Fc(.)14 b(.)e(.)g(.)h(.)f(.)g(.)h
4745 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
4746 h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
4747 (.)g(.)32 b Fb(9)-30 1381 y Fd(ash)26 b(\(\))12 b Fc(.)h(.)f(.)h(.)f(.)
4748 g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
4749 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
4750 h(.)f(.)g(.)38 b Fb(22)-30 1473 y Fd(asin)8 b Fc(.)13
4751 b(.)g(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
4752 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
4753 g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)34 b Fb(18)-30 1565
4754 y Fd(asin)27 b(\(\))10 b Fc(.)j(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
4755 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
4756 h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)36
4757 b Fb(18)-30 1657 y Fd(asinh)27 b(\(\))9 b Fc(.)j(.)h(.)f(.)g(.)h(.)f(.)
4758 g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
4759 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)35
4760 b Fb(19)-30 1749 y Fd(atan)8 b Fc(.)13 b(.)g(.)f(.)g(.)g(.)h(.)f(.)g(.)
4761 h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
4762 (.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
4763 f(.)34 b Fb(18)-30 1841 y Fd(atan)27 b(\(\))10 b Fc(.)j(.)f(.)g(.)h(.)f
4764 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)
4765 h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
4766 (.)h(.)36 b Fb(18)-30 1933 y Fd(atanh)27 b(\(\))9 b Fc(.)j(.)h(.)f(.)g
4767 (.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
4768 g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
4769 (.)f(.)35 b Fb(19)-30 2180 y Fs(B)-30 2303 y Fd(basering)27
4770 b(\(\))d Fc(.)12 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
4771 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
4772 f(.)g(.)h(.)f(.)g(.)49 b Fb(40)-30 2395 y Fd(binomial)27
4773 b(\(\))d Fc(.)12 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
4774 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
4775 f(.)g(.)h(.)f(.)g(.)49 b Fb(23)-30 2487 y Fd(boole)27
4776 b(\(\))9 b Fc(.)j(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
4777 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
4778 g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)35 b Fb(21)-30 2579
4779 y Fd(boole_1)23 b Fc(.)13 b(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
4780 (.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
4781 f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)48 b Fb(21)-30
4782 2671 y Fd(boole_2)23 b Fc(.)13 b(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
4783 f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
4784 (.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)48
4785 b Fb(21)-30 2763 y Fd(boole_and)21 b Fc(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g
4786 (.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
4787 f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)45
4788 b Fb(21)-30 2855 y Fd(boole_andc1)16 b Fc(.)f(.)d(.)g(.)h(.)f(.)g(.)h
4789 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
4790 h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)42 b
4791 Fb(21)-30 2947 y Fd(boole_andc2)16 b Fc(.)f(.)d(.)g(.)h(.)f(.)g(.)h(.)f
4792 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
4793 f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)42 b Fb(21)-30
4794 3039 y Fd(boole_c1)22 b Fc(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
4795 (.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
4796 g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)46 b
4797 Fb(21)-30 3131 y Fd(boole_c2)22 b Fc(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g(.)h
4798 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
4799 h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)46
4800 b Fb(21)-30 3222 y Fd(boole_clr)21 b Fc(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g
4801 (.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
4802 f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)45
4803 b Fb(21)-30 3314 y Fd(boole_eqv)21 b Fc(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g
4804 (.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
4805 f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)45
4806 b Fb(21)-30 3406 y Fd(boole_nand)17 b Fc(.)e(.)d(.)h(.)f(.)g(.)h(.)f(.)
4807 g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
4808 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)43
4809 b Fb(21)-30 3498 y Fd(boole_nor)21 b Fc(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g
4810 (.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
4811 f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)45
4812 b Fb(21)-30 3590 y Fd(boole_orc1)17 b Fc(.)e(.)d(.)h(.)f(.)g(.)h(.)f(.)
4813 g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
4814 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)43
4815 b Fb(21)-30 3682 y Fd(boole_orc2)17 b Fc(.)e(.)d(.)h(.)f(.)g(.)h(.)f(.)
4816 g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
4817 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)43
4818 b Fb(21)-30 3774 y Fd(boole_set)21 b Fc(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g
4819 (.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
4820 f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)45
4821 b Fb(21)-30 3866 y Fd(boole_xor)21 b Fc(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g
4822 (.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
4823 f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)45
4824 b Fb(21)-30 4114 y Fs(C)-30 4236 y Fd(canonhom)27 b(\(\))8
4825 b Fc(.)13 b(.)g(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
4826 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)34
4827 b Fb(33,)26 b(35,)h(40)-30 4328 y(Catalan's)h(constan)n(t)10
4828 b Fc(.)i(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
4829 f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)36
4830 b Fb(20)-30 4420 y Fd(catalanconst)28 b(\(\))17 b Fc(.)c(.)f(.)h(.)f(.)
4831 g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
4832 (.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)43 b Fb(20)-30
4833 4512 y Fd(ceiling1)27 b(\(\))d Fc(.)12 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
4834 h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
4835 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)49 b Fb(14)-30
4836 4604 y Fd(ceiling2)27 b(\(\))d Fc(.)12 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
4837 h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
4838 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)49 b Fb(15)-30
4839 4696 y(Cheb)n(yshev)25 b(p)r(olynomial)16 b Fc(.)c(.)h(.)f(.)g(.)h(.)f
4840 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)
4841 h(.)f(.)g(.)h(.)42 b Fb(42)-30 4788 y Fd(cis)26 b(\(\))12
4842 b Fc(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)
4843 f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
4844 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)38 b Fb(18)-30 4880
4845 y Fd(cl_abort)27 b(\(\))d Fc(.)12 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
4846 (.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
4847 h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)49 b Fb(52)-30 4972
4848 y Fd(cl_byte)23 b Fc(.)13 b(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
4849 (.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
4850 f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)48 b Fb(21)-30
4851 5064 y Fd(CL_DEBUG)22 b Fc(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
4852 (.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
4853 g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)46 b
4854 Fb(50)-30 5156 y Fd(cl_DF)8 b Fc(.)13 b(.)f(.)h(.)f(.)g(.)h(.)f(.)g(.)h
4855 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)
4856 g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
4857 (.)33 b Fb(7)-30 5248 y Fd(cl_DF_fdiv_t)15 b Fc(.)g(.)d(.)g(.)h(.)f(.)g
4858 (.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
4859 f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)40 b
4860 Fb(15)-30 5340 y Fd(cl_F)22 b Fc(.)12 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
4861 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)
4862 g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)47
4863 b Fb(6,)26 b(7)2032 614 y Fd(cl_F_fdiv_t)16 b Fc(.)f(.)d(.)h(.)f(.)g(.)
4864 h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f
4865 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)42
4866 b Fb(15)2032 707 y Fd(cl_FF)8 b Fc(.)13 b(.)g(.)f(.)g(.)h(.)f(.)g(.)h
4867 (.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
4868 g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h
4869 (.)f(.)34 b Fb(7)2032 799 y Fd(cl_FF_fdiv_t)15 b Fc(.)g(.)d(.)g(.)h(.)f
4870 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
4871 f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)40
4872 b Fb(15)2032 891 y Fd(cl_float)28 b(\(\))23 b Fc(.)12
4873 b(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
4874 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
4875 h(.)48 b Fb(25)2032 984 y Fd(cl_float_format)29 b(\(\))13
4876 b Fc(.)g(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)
4877 h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)38
4878 b Fb(25)2032 1076 y Fd(cl_float_format_t)7 b Fc(.)17
4879 b(.)12 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
4880 g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)33
4881 b Fb(25)2032 1168 y Fd(cl_I_to_int)28 b(\(\))20 b Fc(.)13
4882 b(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
4883 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)46
4884 b Fb(9)2032 1261 y Fd(cl_I_to_long)29 b(\(\))18 b Fc(.)12
4885 b(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
4886 (.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)44
4887 b Fb(9)2032 1353 y Fd(cl_I_to_uint)29 b(\(\))18 b Fc(.)12
4888 b(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
4889 (.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)44
4890 b Fb(9)2032 1445 y Fd(cl_I_to_ulong)29 b(\(\))17 b Fc(.)12
4891 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
4892 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)43
4893 b Fb(9)2032 1538 y Fd(cl_idecoded_float)7 b Fc(.)17 b(.)12
4894 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g
4895 (.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)33 b Fb(24)2032
4896 1630 y Fd(cl_LF)8 b Fc(.)13 b(.)g(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g
4897 (.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
4898 g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)34
4899 b Fb(7)2032 1722 y Fd(cl_LF_fdiv_t)15 b Fc(.)g(.)d(.)g(.)h(.)f(.)g(.)h
4900 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
4901 h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)40 b Fb(15)2032
4902 1815 y Fd(cl_modint_ring)12 b Fc(.)j(.)e(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
4903 g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
4904 (.)f(.)g(.)h(.)f(.)g(.)h(.)37 b Fb(34)2032 1907 y Fd(cl_N)9
4905 b Fc(.)k(.)g(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
4906 h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f
4907 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)35 b Fb(6)2032
4908 1999 y Fd(cl_number)22 b Fc(.)13 b(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g
4909 (.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
4910 g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)46 b
4911 Fb(6)2032 2092 y Fd(cl_R)9 b Fc(.)k(.)g(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
4912 (.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
4913 g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
4914 (.)35 b Fb(6)2032 2184 y Fd(cl_R_fdiv_t)16 b Fc(.)f(.)d(.)h(.)f(.)g(.)h
4915 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)
4916 g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)42
4917 b Fb(15)2032 2276 y Fd(cl_RA)8 b Fc(.)13 b(.)g(.)f(.)g(.)h(.)f(.)g(.)h
4918 (.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
4919 g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h
4920 (.)f(.)34 b Fb(6)2032 2369 y Fd(cl_SF)8 b Fc(.)13 b(.)g(.)f(.)g(.)h(.)f
4921 (.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
4922 h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
4923 (.)g(.)h(.)f(.)34 b Fb(7)2032 2461 y Fd(cl_SF_fdiv_t)15
4924 b Fc(.)g(.)d(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
4925 f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g
4926 (.)h(.)40 b Fb(15)2032 2553 y Fd(cl_string)21 b Fc(.)12
4927 b(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
4928 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)
4929 h(.)f(.)g(.)45 b Fb(37)2032 2646 y Fd(cl_symbol)21 b
4930 Fc(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
4931 (.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)
4932 f(.)g(.)h(.)f(.)g(.)45 b Fb(37)2032 2738 y Fd(coeff)27
4933 b(\(\))9 b Fc(.)k(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
4934 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
4935 h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)35 b Fb(41)2032 2830
4936 y Fd(compare)27 b(\(\))6 b Fc(.)13 b(.)g(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
4937 g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
4938 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)32 b Fb(13)2032
4939 2923 y(comparison)24 b Fc(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
4940 g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
4941 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)49 b Fb(13)2032
4942 3015 y(compatibilit)n(y)10 b Fc(.)i(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
4943 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
4944 f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)36 b Fb(45)2032 3107
4945 y(compiler)26 b(options)9 b Fc(.)j(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
4946 (.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)
4947 f(.)g(.)h(.)f(.)g(.)h(.)34 b Fb(45)2032 3200 y Fd(complex)27
4948 b(\(\))6 b Fc(.)13 b(.)g(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
4949 g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
4950 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)32 b Fb(12)2032 3292 y(complex)25
4951 b(n)n(um)n(b)r(er)20 b Fc(.)12 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
4952 g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h
4953 (.)f(.)g(.)h(.)47 b Fb(6,)27 b(8)2032 3384 y Fd(conjugate)h(\(\))22
4954 b Fc(.)12 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
4955 (.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
4956 h(.)f(.)47 b Fb(12)2032 3477 y(con)n(v)n(ersion)15 b
4957 Fc(.)e(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
4958 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
4959 h(.)f(.)41 b Fb(8,)26 b(25)2032 3569 y Fd(cos)h(\(\))12
4960 b Fc(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
4961 g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h
4962 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)38 b Fb(18)2032 3661
4963 y Fd(cos_sin)27 b(\(\))6 b Fc(.)13 b(.)g(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
4964 g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
4965 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)32 b Fb(18)2032
4966 3754 y Fd(cos_sin_t)21 b Fc(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
4967 (.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
4968 g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)45 b Fb(18)2032
4969 3846 y Fd(cosh)27 b(\(\))10 b Fc(.)j(.)f(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
4970 f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
4971 (.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)36
4972 b Fb(19)2032 3938 y Fd(cosh_sinh)28 b(\(\))22 b Fc(.)12
4973 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g
4974 (.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
4975 47 b Fb(19)2032 4031 y Fd(cosh_sinh_t)16 b Fc(.)f(.)d(.)h(.)f(.)g(.)h
4976 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)
4977 g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)42
4978 b Fb(19)2032 4123 y Fd(create)27 b(\(\))8 b Fc(.)13 b(.)f(.)g(.)g(.)h
4979 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
4980 h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)33
4981 b Fb(40)2032 4215 y(customizing)15 b Fc(.)e(.)f(.)g(.)h(.)f(.)g(.)h(.)f
4982 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)
4983 h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)41 b
4984 Fb(52)2032 4478 y Fs(D)2032 4601 y Fd(debug_print)28
4985 b(\(\))18 b Fc(.)c(.)e(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
4986 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
4987 h(.)f(.)44 b Fb(50)2032 4694 y(debugging)7 b Fc(.)12
4988 b(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
4989 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
4990 h(.)f(.)g(.)h(.)32 b Fb(50)2032 4786 y Fd(decode_float)d(\(\))17
4991 b Fc(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
4992 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)43
4993 b Fb(24)2032 4878 y Fd(decoded_dfloat)12 b Fc(.)j(.)e(.)f(.)g(.)h(.)f
4994 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)
4995 h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)37 b Fb(24)2032
4996 4971 y Fd(decoded_ffloat)12 b Fc(.)j(.)e(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
4997 g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
4998 (.)f(.)g(.)h(.)f(.)g(.)h(.)37 b Fb(24)2032 5063 y Fd(decoded_float)13
4999 b Fc(.)j(.)c(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
5000 h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
5001 (.)39 b Fb(24)2032 5155 y Fd(decoded_lfloat)12 b Fc(.)j(.)e(.)f(.)g(.)h
5002 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)
5003 g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)37 b Fb(24)2032
5004 5248 y Fd(decoded_sfloat)12 b Fc(.)j(.)e(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
5005 g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
5006 (.)f(.)g(.)h(.)f(.)g(.)h(.)37 b Fb(24)2032 5340 y Fd
5007 (default_float_format)26 b Fc(.)12 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
5008 f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)47
5011 54 55 bop -30 -116 a Fr(Index)3646 b(54)-30 299 y Fd
5012 (default_print_flags)27 b Fc(.)12 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
5013 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
5014 48 b Fb(52)-30 391 y Fd(default_random_state)26 b Fc(.)12
5015 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g
5016 (.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)47 b Fb(26)-30 483
5017 y Fd(degree)27 b(\(\))8 b Fc(.)k(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
5018 g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
5019 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)33 b Fb(41)-30
5020 575 y Fd(denominator)28 b(\(\))18 b Fc(.)13 b(.)g(.)f(.)g(.)h(.)f(.)g
5021 (.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
5022 f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)44 b Fb(12)-30 667
5023 y Fd(deposit_field)29 b(\(\))16 b Fc(.)c(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)
5024 g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
5025 (.)g(.)h(.)f(.)g(.)h(.)41 b Fb(22)-30 759 y Fd(deriv)27
5026 b(\(\))9 b Fc(.)j(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
5027 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
5028 g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)35 b Fb(41)-30 851
5029 y Fd(div)26 b(\(\))12 b Fc(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
5030 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
5031 f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)38
5032 b Fb(35)-30 943 y Fd(double_approx)29 b(\(\))17 b Fc(.)12
5033 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
5034 (.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)42
5035 b Fb(9)-30 1035 y Fd(doublefactorial)29 b(\(\))13 b Fc(.)f(.)h(.)f(.)g
5036 (.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
5037 g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)39 b Fb(23)-30 1127 y
5038 Fd(dpb)26 b(\(\))12 b Fc(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
5039 g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
5040 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)38
5041 b Fb(22)-30 1389 y Fs(E)-30 1512 y Fd(equal)27 b(\(\))20
5042 b Fc(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f
5043 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
5044 f(.)g(.)h(.)45 b Fb(33,)27 b(37)-30 1604 y Fd(equal_hashcode)i(\(\))14
5045 b Fc(.)f(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)
5046 h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)40
5047 b Fb(13)-30 1696 y(Euler's)27 b(constan)n(t)15 b Fc(.)d(.)h(.)f(.)g(.)h
5048 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
5049 h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)41 b Fb(20)-30
5050 1788 y Fd(eulerconst)28 b(\(\))20 b Fc(.)13 b(.)f(.)g(.)g(.)h(.)f(.)g
5051 (.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
5052 g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)46 b Fb(20)-30
5053 1880 y Fd(evenp)27 b(\(\))9 b Fc(.)j(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
5054 g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
5055 (.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)35
5056 b Fb(22)-30 1972 y(exact)26 b(n)n(um)n(b)r(er)20 b Fc(.)13
5057 b(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f
5058 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
5059 f(.)48 b Fb(6)-30 2064 y Fd(exp)26 b(\(\))12 b Fc(.)h(.)f(.)h(.)f(.)g
5060 (.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
5061 f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
5062 (.)f(.)g(.)38 b Fb(17)-30 2156 y Fd(exp1)27 b(\(\))10
5063 b Fc(.)j(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
5064 f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
5065 (.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)36 b Fb(17)-30 2248 y
5066 Fd(expt)27 b(\(\))14 b Fc(.)e(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
5067 (.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
5068 g(.)g(.)h(.)f(.)g(.)h(.)40 b Fb(11,)26 b(17,)h(35)-30
5069 2340 y Fd(expt_pos)g(\(\))20 b Fc(.)12 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
5070 h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
5071 (.)45 b Fb(11,)27 b(33,)f(35,)h(41)-30 2432 y Fd(exquo)g(\(\))9
5072 b Fc(.)j(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
5073 h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f
5074 (.)g(.)h(.)f(.)g(.)h(.)f(.)35 b Fb(11)-30 2694 y Fs(F)-30
5075 2817 y Fd(factorial)28 b(\(\))21 b Fc(.)13 b(.)f(.)g(.)h(.)f(.)g(.)h(.)
5076 f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g
5077 (.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)47 b Fb(23)-30
5078 2909 y Fd(fceiling)27 b(\(\))d Fc(.)12 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
5079 h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
5080 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)49 b Fb(15)-30
5081 3001 y Fd(fceiling2)28 b(\(\))21 b Fc(.)13 b(.)f(.)g(.)h(.)f(.)g(.)h(.)
5082 f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g
5083 (.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)47 b Fb(15)-30
5084 3093 y Fd(ffloor)27 b(\(\))8 b Fc(.)k(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
5085 (.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
5086 h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)33
5087 b Fb(15)-30 3185 y Fd(ffloor2)27 b(\(\))6 b Fc(.)13 b(.)f(.)h(.)f(.)g
5088 (.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)
5089 f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)32
5090 b Fb(15)-30 3277 y(Fib)r(onacci)27 b(n)n(um)n(b)r(er)18
5091 b Fc(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
5092 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)46
5093 b Fb(49)-30 3369 y Fd(finalize)27 b(\(\))d Fc(.)12 b(.)g(.)h(.)f(.)g(.)
5094 h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
5095 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)49
5096 b Fb(41)-30 3461 y Fd(find_modint_ring)29 b(\(\))11 b
5097 Fc(.)i(.)g(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
5098 (.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)37 b Fb(34)-30
5099 3553 y Fd(find_univpoly_ring)30 b(\(\))8 b Fc(.)13 b(.)f(.)g(.)h(.)f(.)
5100 g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h
5101 (.)f(.)g(.)h(.)34 b Fb(40)-30 3645 y Fd(float_approx)28
5102 b(\(\))18 b Fc(.)13 b(.)f(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
5103 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)
5104 h(.)f(.)g(.)44 b Fb(9)-30 3737 y Fd(float_digits)28 b(\(\))17
5105 b Fc(.)c(.)f(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
5106 g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)43
5107 b Fb(24)-30 3829 y Fd(float_epsilon)29 b(\(\))16 b Fc(.)c(.)g(.)g(.)h
5108 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
5109 h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)41 b Fb(25)-30
5110 3921 y Fd(float_exponent)29 b(\(\))14 b Fc(.)f(.)f(.)g(.)h(.)f(.)g(.)h
5111 (.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
5112 g(.)h(.)f(.)g(.)h(.)f(.)40 b Fb(24)-30 4013 y Fd
5113 (float_negative_epsilon)31 b(\(\))21 b Fc(.)12 b(.)g(.)h(.)f(.)g(.)h(.)
5114 f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)47
5115 b Fb(25)-30 4105 y Fd(float_precision)29 b(\(\))13 b
5116 Fc(.)f(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
5117 (.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)39
5118 b Fb(24)-30 4197 y Fd(float_radix)28 b(\(\))18 b Fc(.)13
5119 b(.)g(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h
5120 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)44
5121 b Fb(24)-30 4289 y Fd(float_sign)28 b(\(\))20 b Fc(.)13
5122 b(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
5123 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)46
5124 b Fb(24)-30 4381 y(\015oating-p)r(oin)n(t)26 b(n)n(um)n(b)r(er)12
5125 b Fc(.)e(.)j(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
5126 g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)38
5127 b Fb(7)-30 4473 y Fd(floor1)27 b(\(\))8 b Fc(.)k(.)h(.)f(.)g(.)h(.)f(.)
5128 g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
5129 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)33
5130 b Fb(14)-30 4565 y Fd(floor2)27 b(\(\))8 b Fc(.)k(.)h(.)f(.)g(.)h(.)f
5131 (.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
5132 h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)33
5133 b Fb(15)-30 4657 y Fd(fprint)27 b(\(\))11 b Fc(.)i(.)f(.)g(.)h(.)f(.)g
5134 (.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)
5135 f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)37 b Fb(33,)26 b(36,)h(41)-30
5136 4749 y Fd(free_hook)h(\(\))21 b Fc(.)13 b(.)f(.)g(.)h(.)f(.)g(.)h(.)f
5137 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)
5138 h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)47 b Fb(52)-30
5139 4841 y Fd(fround)27 b(\(\))8 b Fc(.)k(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
5140 (.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
5141 h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)33
5142 b Fb(15)-30 4933 y Fd(fround2)27 b(\(\))6 b Fc(.)13 b(.)f(.)h(.)f(.)g
5143 (.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)
5144 f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)32
5145 b Fb(15)-30 5024 y Fd(ftruncate)c(\(\))21 b Fc(.)13 b(.)f(.)g(.)h(.)f
5146 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
5147 f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)47
5148 b Fb(15)2032 299 y Fd(ftruncate2)28 b(\(\))20 b Fc(.)13
5149 b(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
5150 (.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)46
5151 b Fb(15)2032 545 y Fs(G)2032 667 y Fb(garbage)27 b(collection)14
5152 b Fc(.)h(.)d(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)
5153 h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)40 b
5154 Fb(43,)27 b(44)2032 759 y Fd(gcd)g(\(\))12 b Fc(.)g(.)h(.)f(.)g(.)g(.)h
5155 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
5156 h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
5157 (.)g(.)38 b Fb(23)2032 851 y(GMP)8 b Fc(.)13 b(.)g(.)f(.)g(.)h(.)f(.)g
5158 (.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
5159 f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
5160 (.)33 b Fb(1,)27 b(4)2032 1110 y Fs(H)2032 1232 y Fb(header)f(\014les)7
5161 b Fc(.)13 b(.)f(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
5162 (.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
5163 h(.)f(.)g(.)h(.)f(.)33 b Fb(45)2032 1324 y Fd(hermite)27
5164 b(\(\))6 b Fc(.)13 b(.)g(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
5165 g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
5166 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)32 b Fb(42)2032 1416 y(Hermite)25
5167 b(p)r(olynomial)7 b Fc(.)12 b(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g
5168 (.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
5169 g(.)h(.)32 b Fb(42)2032 1675 y Fs(I)2032 1797 y Fd(imagpart)c(\(\))23
5170 b Fc(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
5171 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
5172 f(.)g(.)h(.)48 b Fb(12)2032 1889 y(include)26 b(\014les)17
5173 b Fc(.)c(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)
5174 h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
5175 (.)h(.)f(.)g(.)43 b Fb(45)2032 1980 y(Input/Output)14
5176 b Fc(.)c(.)i(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
5177 g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h
5178 (.)f(.)40 b Fb(28)2032 2072 y(installation)9 b Fc(.)14
5179 b(.)e(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g
5180 (.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
5181 g(.)h(.)f(.)g(.)35 b Fb(4)2032 2164 y Fd(instanceof)28
5182 b(\(\))20 b Fc(.)13 b(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
5183 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)
5184 g(.)h(.)f(.)g(.)46 b Fb(33)2032 2256 y(in)n(teger)10
5185 b Fc(.)j(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
5186 h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
5187 (.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)36 b Fb(6)2032
5188 2347 y Fd(integer_decode_float)30 b(\(\))24 b Fc(.)13
5189 b(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
5190 (.)g(.)h(.)f(.)g(.)h(.)49 b Fb(24)2032 2439 y Fd(integer_length)29
5191 b(\(\))14 b Fc(.)f(.)f(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
5192 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)40
5193 b Fb(22)2032 2531 y Fd(isqrt)27 b(\(\))9 b Fc(.)k(.)f(.)g(.)h(.)f(.)g
5194 (.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
5195 g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)35
5196 b Fb(16)2032 2792 y Fs(L)2032 2914 y Fd(laguerre)28 b(\(\))23
5197 b Fc(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
5198 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
5199 f(.)g(.)h(.)48 b Fb(42)2032 3006 y(Laguerre)27 b(p)r(olynomial)13
5200 b Fc(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
5201 f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)38
5202 b Fb(42)2032 3098 y Fd(lcm)27 b(\(\))12 b Fc(.)g(.)h(.)f(.)g(.)g(.)h(.)
5203 f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
5204 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
5205 g(.)38 b Fb(23)2032 3190 y Fd(ldb)27 b(\(\))12 b Fc(.)g(.)h(.)f(.)g(.)g
5206 (.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
5207 g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
5208 (.)f(.)g(.)38 b Fb(22)2032 3281 y Fd(ldb_test)28 b(\(\))23
5209 b Fc(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
5210 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
5211 f(.)g(.)h(.)48 b Fb(22)2032 3373 y Fd(least_negative_float)30
5212 b(\(\))24 b Fc(.)13 b(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
5213 (.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)49 b Fb(25)2032
5214 3465 y Fd(least_positive_float)30 b(\(\))24 b Fc(.)13
5215 b(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
5216 (.)g(.)h(.)f(.)g(.)h(.)49 b Fb(25)2032 3557 y(Legende)26
5217 b(p)r(olynomial)e Fc(.)12 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g
5218 (.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
5219 49 b Fb(42)2032 3648 y Fd(legendre)28 b(\(\))23 b Fc(.)12
5220 b(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
5221 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
5222 h(.)48 b Fb(42)2032 3740 y Fd(length)27 b(\(\))8 b Fc(.)13
5223 b(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
5224 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)
5225 h(.)f(.)g(.)h(.)33 b Fb(37)2032 3832 y Fd(ln)26 b(\(\))13
5226 b Fc(.)g(.)g(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
5227 h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
5228 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)39 b Fb(17)2032
5229 3924 y Fd(log)27 b(\(\))12 b Fc(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f
5230 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
5231 f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)38
5232 b Fb(17)2032 4015 y Fd(logand)27 b(\(\))8 b Fc(.)13 b(.)f(.)g(.)g(.)h
5233 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
5234 h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)33
5235 b Fb(21)2032 4107 y Fd(logandc1)28 b(\(\))23 b Fc(.)12
5236 b(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
5237 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
5238 h(.)48 b Fb(21)2032 4199 y Fd(logandc2)28 b(\(\))23 b
5239 Fc(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
5240 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
5241 f(.)g(.)h(.)48 b Fb(21)2032 4291 y Fd(logbitp)27 b(\(\))6
5242 b Fc(.)13 b(.)g(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f
5243 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
5244 f(.)g(.)h(.)f(.)g(.)32 b Fb(21)2032 4382 y Fd(logcount)c(\(\))23
5245 b Fc(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
5246 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
5247 f(.)g(.)h(.)48 b Fb(21)2032 4474 y Fd(logeqv)27 b(\(\))8
5248 b Fc(.)13 b(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
5249 (.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
5250 g(.)g(.)h(.)f(.)g(.)h(.)33 b Fb(21)2032 4566 y Fd(logior)27
5251 b(\(\))8 b Fc(.)13 b(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
5252 h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
5253 (.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)33 b Fb(21)2032 4658
5254 y Fd(lognand)27 b(\(\))6 b Fc(.)13 b(.)g(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
5255 g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
5256 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)32 b Fb(21)2032
5257 4749 y Fd(lognor)27 b(\(\))8 b Fc(.)13 b(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)
5258 f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
5259 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)33
5260 b Fb(21)2032 4841 y Fd(lognot)27 b(\(\))8 b Fc(.)13 b(.)f(.)g(.)g(.)h
5261 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
5262 h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)33
5263 b Fb(20)2032 4933 y Fd(logorc1)27 b(\(\))6 b Fc(.)13
5264 b(.)g(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h
5265 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
5266 h(.)f(.)g(.)32 b Fb(21)2032 5024 y Fd(logorc2)27 b(\(\))6
5267 b Fc(.)13 b(.)g(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f
5268 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
5269 f(.)g(.)h(.)f(.)g(.)32 b Fb(21)p eop
5271 55 56 bop -30 -116 a Fr(Index)3646 b(55)-30 299 y Fd(logp)27
5272 b(\(\))10 b Fc(.)j(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
5273 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
5274 h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)36 b Fb(23)-30
5275 391 y Fd(logtest)27 b(\(\))6 b Fc(.)13 b(.)f(.)h(.)f(.)g(.)h(.)f(.)g(.)
5276 h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f
5277 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)32
5278 b Fb(21)-30 483 y Fd(logxor)27 b(\(\))8 b Fc(.)k(.)h(.)f(.)g(.)h(.)f(.)
5279 g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
5280 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)33
5281 b Fb(21)-30 747 y Fs(M)-30 870 y Fd(make)9 b Fc(.)k(.)g(.)f(.)g(.)h(.)f
5282 (.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
5283 h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f
5284 (.)g(.)h(.)f(.)g(.)35 b Fb(3)-30 962 y Fd(malloc_hook)28
5285 b(\(\))18 b Fc(.)13 b(.)g(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
5286 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
5287 f(.)g(.)h(.)44 b Fb(52)-30 1054 y Fd(mask_field)28 b(\(\))20
5288 b Fc(.)13 b(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
5289 (.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
5290 g(.)46 b Fb(22)-30 1147 y Fd(max)26 b(\(\))12 b Fc(.)h(.)f(.)h(.)f(.)g
5291 (.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
5292 f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
5293 (.)f(.)g(.)38 b Fb(13)-30 1239 y Fd(min)26 b(\(\))12
5294 b Fc(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)
5295 f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
5296 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)38 b Fb(13)-30 1331
5297 y Fd(minus)27 b(\(\))9 b Fc(.)j(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
5298 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
5299 h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)35 b
5300 Fb(33)-30 1423 y Fd(minus1)27 b(\(\))8 b Fc(.)k(.)h(.)f(.)g(.)h(.)f(.)g
5301 (.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
5302 f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)33
5303 b Fb(11)-30 1515 y Fd(minusp)27 b(\(\))8 b Fc(.)k(.)h(.)f(.)g(.)h(.)f
5304 (.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
5305 h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)33
5306 b Fb(13)-30 1607 y Fd(mod)26 b(\(\))12 b Fc(.)h(.)f(.)h(.)f(.)g(.)h(.)f
5307 (.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
5308 h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
5309 (.)38 b Fb(16)-30 1700 y(mo)r(difying)26 b(op)r(erators)c
5310 Fc(.)13 b(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g
5311 (.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)47
5312 b Fb(27)-30 1792 y(mo)r(dular)25 b(in)n(teger)20 b Fc(.)12
5313 b(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
5314 (.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)45
5315 b Fb(34)-30 1884 y Fd(modulus)23 b Fc(.)13 b(.)f(.)g(.)h(.)f(.)g(.)h(.)
5316 f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g
5317 (.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)48
5318 b Fb(34)-30 1976 y Fd(monomial)27 b(\(\))d Fc(.)12 b(.)g(.)h(.)f(.)g(.)
5319 h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
5320 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)49
5321 b Fb(40)-30 2068 y(Mon)n(tgomery)25 b(m)n(ultiplication)9
5322 b Fc(.)k(.)f(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
5323 f(.)g(.)h(.)f(.)g(.)h(.)f(.)35 b Fb(34)-30 2161 y Fd
5324 (most_negative_float)30 b(\(\))7 b Fc(.)13 b(.)f(.)g(.)g(.)h(.)f(.)g(.)
5325 h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
5326 (.)33 b Fb(25)-30 2253 y Fd(most_positive_float)d(\(\))7
5327 b Fc(.)13 b(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
5328 (.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)33 b Fb(25)-30
5329 2345 y Fd(mul)26 b(\(\))12 b Fc(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
5330 (.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
5331 h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)38
5332 b Fb(33)-30 2594 y Fs(N)-30 2717 y Fb(namespace)10 b
5333 Fc(.)i(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
5334 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)
5335 h(.)f(.)36 b Fb(2,)26 b(45)-30 2809 y Fd(numerator)i(\(\))21
5336 b Fc(.)13 b(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
5337 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
5338 g(.)h(.)47 b Fb(12)-30 3058 y Fs(O)-30 3181 y Fd(oddp)27
5339 b(\(\))10 b Fc(.)j(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
5340 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
5341 h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)36 b Fb(22)-30
5342 3273 y Fd(one)26 b(\(\))15 b Fc(.)e(.)g(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
5343 (.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
5344 f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)42 b Fb(33,)26 b(35,)h(40)-30
5345 3365 y Fd(operator)g(!=)f(\(\))c Fc(.)13 b(.)f(.)g(.)g(.)h(.)f(.)g(.)h
5346 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)47
5347 b Fb(13,)27 b(34,)f(36,)h(41)-30 3457 y Fd(operator)g(&)f(\(\))9
5348 b Fc(.)k(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
5349 h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
5350 (.)h(.)34 b Fb(21)-30 3550 y Fd(operator)27 b(&=)f(\(\))7
5351 b Fc(.)13 b(.)g(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
5352 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
5353 f(.)33 b Fb(27)-30 3642 y Fd(operator)27 b(\(\))f(\(\))7
5354 b Fc(.)13 b(.)g(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
5355 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
5356 f(.)33 b Fb(41)-30 3734 y Fd(operator)27 b(*)f(\(\))12
5357 b Fc(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)
5358 f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)38 b
5359 Fb(11,)26 b(35,)h(41)-30 3826 y Fd(operator)g(*=)f(\(\))7
5360 b Fc(.)13 b(.)g(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
5361 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
5362 f(.)33 b Fb(27)-30 3918 y Fd(operator)27 b(-)f(\(\))12
5363 b Fc(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)
5364 f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)38 b
5365 Fb(11,)26 b(35,)h(41)-30 4010 y Fd(operator)g(--)f(\(\))7
5366 b Fc(.)13 b(.)g(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
5367 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
5368 f(.)33 b Fb(27)-30 4103 y Fd(operator)27 b(-=)f(\(\))7
5369 b Fc(.)13 b(.)g(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
5370 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
5371 f(.)33 b Fb(27)-30 4195 y Fd(operator)27 b(/)f(\(\))9
5372 b Fc(.)k(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
5373 h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
5374 (.)h(.)34 b Fb(11)-30 4287 y Fd(operator)27 b(/=)f(\(\))7
5375 b Fc(.)13 b(.)g(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
5376 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
5377 f(.)33 b Fb(27)-30 4379 y Fd(operator)27 b(==)f(\(\))c
5378 Fc(.)13 b(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
5379 (.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)47 b Fb(13,)27 b(34,)f(36,)h(41)-30
5380 4471 y Fd(operator)g([])f(\(\))7 b Fc(.)13 b(.)g(.)f(.)g(.)h(.)f(.)g(.)
5381 h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
5382 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)33 b Fb(37)-30
5383 4564 y Fd(operator)27 b(|)f(\(\))9 b Fc(.)k(.)f(.)g(.)h(.)f(.)g(.)g(.)h
5384 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
5385 h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)34 b Fb(21)-30
5386 4656 y Fd(operator)27 b(|=)f(\(\))7 b Fc(.)13 b(.)g(.)f(.)g(.)h(.)f(.)g
5387 (.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
5388 f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)33 b Fb(27)-30
5389 4748 y Fd(operator)27 b(~)f(\(\))9 b Fc(.)k(.)f(.)g(.)h(.)f(.)g(.)g(.)h
5390 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
5391 h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)34 b Fb(20)-30
5392 4840 y Fd(operator)27 b(+)f(\(\))12 b Fc(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)
5393 g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
5394 (.)f(.)g(.)h(.)38 b Fb(11,)26 b(35,)h(41)-30 4932 y Fd(operator)g(+=)f
5395 (\(\))7 b Fc(.)13 b(.)g(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
5396 (.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
5397 g(.)h(.)f(.)33 b Fb(27)-30 5024 y Fd(operator)27 b(++)f(\(\))7
5398 b Fc(.)13 b(.)g(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
5399 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
5400 f(.)33 b Fb(27)2032 299 y Fd(operator)28 b(>)d(\(\))9
5401 b Fc(.)k(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
5402 f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g
5403 (.)h(.)34 b Fb(13)2032 391 y Fd(operator)28 b(>=)e(\(\))7
5404 b Fc(.)13 b(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g
5405 (.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
5406 g(.)33 b Fb(13)2032 483 y Fd(operator)28 b(>>)e(\(\))18
5407 b Fc(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
5408 (.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)43
5409 b Fb(22,)27 b(36)2032 576 y Fd(operator)h(>>=)e(\(\))e
5410 Fc(.)13 b(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
5411 (.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)50
5412 b Fb(27)2032 668 y Fd(operator)28 b(^)d(\(\))9 b Fc(.)k(.)f(.)g(.)h(.)f
5413 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
5414 f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)34
5415 b Fb(21)2032 760 y Fd(operator)28 b(^=)e(\(\))7 b Fc(.)13
5416 b(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
5417 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)33
5418 b Fb(27)2032 853 y Fd(operator)28 b(<)d(\(\))9 b Fc(.)k(.)f(.)g(.)h(.)f
5419 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
5420 f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)34
5421 b Fb(13)2032 945 y Fd(operator)28 b(<=)e(\(\))7 b Fc(.)13
5422 b(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
5423 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)33
5424 b Fb(13)2032 1037 y Fd(operator)28 b(<<)e(\(\))21 b Fc(.)13
5425 b(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
5426 (.)h(.)f(.)g(.)h(.)f(.)47 b Fb(22,)27 b(35,)f(36,)h(41)2032
5427 1129 y Fd(operator)h(<<=)e(\(\))e Fc(.)13 b(.)f(.)g(.)g(.)h(.)f(.)g(.)h
5428 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
5429 h(.)f(.)g(.)h(.)f(.)g(.)g(.)50 b Fb(27)2032 1222 y Fd(ord2)27
5430 b(\(\))10 b Fc(.)j(.)f(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
5431 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
5432 f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)36 b Fb(22)2032
5433 1471 y Fs(P)2032 1594 y Fd(phase)27 b(\(\))9 b Fc(.)k(.)f(.)g(.)h(.)f
5434 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
5435 f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
5436 (.)35 b Fb(17)2032 1686 y(pi)18 b Fc(.)13 b(.)f(.)g(.)h(.)f(.)g(.)h(.)f
5437 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
5438 h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
5439 (.)h(.)f(.)g(.)44 b Fb(19)2032 1779 y Fd(pi)26 b(\(\))13
5440 b Fc(.)g(.)g(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
5441 h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
5442 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)39 b Fb(19)2032
5443 1871 y Fd(plus)27 b(\(\))10 b Fc(.)j(.)f(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
5444 f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
5445 (.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)36
5446 b Fb(33)2032 1963 y Fd(plus1)27 b(\(\))9 b Fc(.)k(.)f(.)g(.)h(.)f(.)g
5447 (.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
5448 g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)35
5449 b Fb(11)2032 2055 y Fd(plusp)27 b(\(\))9 b Fc(.)k(.)f(.)g(.)h(.)f(.)g
5450 (.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
5451 g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)35
5452 b Fb(13)2032 2148 y(p)r(olynomial)10 b Fc(.)j(.)f(.)g(.)h(.)f(.)g(.)g
5453 (.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
5454 g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)36
5455 b Fb(38)2032 2240 y(p)r(ortabilit)n(y)17 b Fc(.)c(.)f(.)h(.)f(.)g(.)h
5456 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
5457 h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)43
5458 b Fb(43)2032 2332 y Fd(power2p)27 b(\(\))6 b Fc(.)13
5459 b(.)g(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h
5460 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
5461 h(.)f(.)g(.)32 b Fb(23)2032 2425 y(prin)n(ting)8 b Fc(.)k(.)g(.)h(.)f
5462 (.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
5463 h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f
5464 (.)g(.)h(.)33 b Fb(28)2032 2686 y Fs(R)2032 2810 y Fd(random)27
5465 b(\(\))8 b Fc(.)13 b(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
5466 h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
5467 (.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)33 b Fb(35)2032 2902
5468 y Fd(random_F)28 b(\(\))23 b Fc(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h
5469 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
5470 h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)48 b Fb(26)2032
5471 2994 y Fd(random_I)28 b(\(\))23 b Fc(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g(.)g
5472 (.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
5473 g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)48 b Fb(26)2032
5474 3087 y Fd(random_R)28 b(\(\))23 b Fc(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g(.)g
5475 (.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
5476 g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)48 b Fb(26)2032
5477 3179 y Fd(random_state)15 b Fc(.)g(.)d(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
5478 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
5479 h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)40 b Fb(26)2032 3271
5480 y Fd(random32)28 b(\(\))23 b Fc(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h
5481 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
5482 h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)48 b Fb(26)2032
5483 3364 y Fd(rational)28 b(\(\))23 b Fc(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g(.)g
5484 (.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
5485 g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)48 b Fb(26)2032
5486 3456 y(rational)27 b(n)n(um)n(b)r(er)17 b Fc(.)11 b(.)h(.)g(.)h(.)f(.)g
5487 (.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
5488 g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)44 b Fb(6)2032
5489 3548 y Fd(rationalize)28 b(\(\))18 b Fc(.)c(.)e(.)g(.)h(.)f(.)g(.)g(.)h
5490 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
5491 h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)44 b Fb(26)2032 3640
5492 y(reading)17 b Fc(.)c(.)g(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
5493 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
5494 f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)43 b Fb(28)2032
5495 3733 y(real)27 b(n)n(um)n(b)r(er)13 b Fc(.)d(.)i(.)h(.)f(.)g(.)h(.)f(.)
5496 g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
5497 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)39
5498 b Fb(6)2032 3825 y Fd(realpart)28 b(\(\))23 b Fc(.)12
5499 b(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
5500 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
5501 h(.)48 b Fb(12)2032 3917 y Fd(recip)27 b(\(\))20 b Fc(.)13
5502 b(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
5503 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)
5504 h(.)45 b Fb(11,)27 b(35)2032 4009 y(reference)g(coun)n(ting)17
5505 b Fc(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
5506 (.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)43
5507 b Fb(43)2032 4102 y Fd(rem)27 b(\(\))12 b Fc(.)g(.)h(.)f(.)g(.)g(.)h(.)
5508 f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
5509 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
5510 g(.)38 b Fb(16)2032 4194 y(represen)n(tation)12 b Fc(.)h(.)f(.)g(.)h(.)
5511 f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g
5512 (.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)37
5513 b Fb(28)2032 4286 y Fd(retract)27 b(\(\))6 b Fc(.)13
5514 b(.)g(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h
5515 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
5516 h(.)f(.)g(.)32 b Fb(35)2032 4379 y(Riemann's)25 b(zeta)11
5517 b Fc(.)i(.)g(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
5518 h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
5519 (.)37 b Fb(20)2032 4471 y(ring)21 b Fc(.)12 b(.)h(.)f(.)g(.)g(.)h(.)f
5520 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
5521 f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g
5522 (.)h(.)f(.)46 b Fb(34)2032 4563 y Fd(ring)27 b(\(\))21
5523 b Fc(.)13 b(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
5524 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
5525 g(.)h(.)f(.)g(.)47 b Fb(35,)27 b(41)2032 4655 y Fd(rootp)g(\(\))9
5526 b Fc(.)k(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
5527 f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
5528 (.)h(.)f(.)g(.)h(.)f(.)g(.)35 b Fb(16)2032 4748 y Fd(round1)27
5529 b(\(\))8 b Fc(.)13 b(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
5530 h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
5531 (.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)33 b Fb(14)2032 4840
5532 y Fd(round2)27 b(\(\))8 b Fc(.)13 b(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g
5533 (.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
5534 g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)33 b
5535 Fb(15)2032 4932 y(rounding)10 b Fc(.)i(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
5536 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)
5537 g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)35
5538 b Fb(13)2032 5024 y(rounding)26 b(error)7 b Fc(.)13 b(.)f(.)h(.)f(.)g
5539 (.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)
5540 f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)34
5543 56 57 bop -30 -116 a Fr(Index)3646 b(56)-30 299 y Fb(Rubik's)25
5544 b(cub)r(e)14 b Fc(.)e(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
5545 (.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)
5546 f(.)g(.)h(.)f(.)g(.)h(.)f(.)40 b Fb(8)-30 548 y Fs(S)-30
5547 672 y Fd(scale_float)28 b(\(\))18 b Fc(.)13 b(.)g(.)f(.)g(.)h(.)f(.)g
5548 (.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
5549 f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)44 b Fb(23)-30 764
5550 y(Sc)n(h\177)-38 b(onhage-Strassen)26 b(m)n(ultiplication)14
5551 b Fc(.)e(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)41
5552 b Fb(1,)26 b(44)-30 857 y Fd(sed)10 b Fc(.)j(.)g(.)f(.)g(.)h(.)f(.)g(.)
5553 h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
5554 (.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
5555 f(.)g(.)h(.)f(.)36 b Fb(3)-30 949 y Fd(set_coeff)28 b(\(\))21
5556 b Fc(.)13 b(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
5557 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
5558 g(.)h(.)47 b Fb(40)-30 1042 y Fd(signum)27 b(\(\))8 b
5559 Fc(.)k(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g
5560 (.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
5561 g(.)h(.)f(.)g(.)h(.)33 b Fb(12)-30 1134 y Fd(sin)26 b(\(\))12
5562 b Fc(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)
5563 f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
5564 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)38 b Fb(17)-30 1226
5565 y Fd(sinh)27 b(\(\))10 b Fc(.)j(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
5566 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
5567 h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)36
5568 b Fb(19)-30 1319 y Fd(sqrt)27 b(\(\))10 b Fc(.)j(.)f(.)g(.)h(.)f(.)g(.)
5569 h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f
5570 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
5571 36 b Fb(16)-30 1411 y Fd(sqrtp)27 b(\(\))9 b Fc(.)j(.)h(.)f(.)g(.)h(.)f
5572 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
5573 f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)35
5574 b Fb(16)-30 1503 y Fd(square)27 b(\(\))22 b Fc(.)12 b(.)h(.)f(.)g(.)h
5575 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
5576 h(.)f(.)g(.)h(.)f(.)g(.)48 b Fb(11,)27 b(33,)f(35,)h(41)-30
5577 1596 y(string)8 b Fc(.)13 b(.)g(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
5578 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
5579 h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)34
5580 b Fb(37)-30 1688 y Fd(strlen)27 b(\(\))8 b Fc(.)k(.)h(.)f(.)g(.)h(.)f
5581 (.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
5582 h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)33
5583 b Fb(37)-30 1781 y(sym)n(b)r(ol)22 b Fc(.)12 b(.)g(.)g(.)h(.)f(.)g(.)h
5584 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
5585 h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)48
5586 b Fb(37)-30 1873 y(sym)n(b)r(olic)25 b(t)n(yp)r(e)17
5587 b Fc(.)12 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h
5588 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
5589 h(.)f(.)43 b Fb(37)-30 2135 y Fs(T)-30 2259 y Fd(tan)26
5590 b(\(\))12 b Fc(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f
5591 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
5592 f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)38 b Fb(18)-30
5593 2351 y Fd(tanh)27 b(\(\))10 b Fc(.)j(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
5594 h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f
5595 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)36
5596 b Fb(19)-30 2444 y Fd(The\(\)\(\))24 b Fc(.)13 b(.)f(.)g(.)h(.)f(.)g(.)
5597 h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
5598 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
5599 48 b Fb(9)2032 299 y(transcenden)n(tal)26 b(functions)c
5600 Fc(.)12 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
5601 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)47 b Fb(17)2032
5602 390 y Fd(truncate1)28 b(\(\))22 b Fc(.)12 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g
5603 (.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
5604 f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)47 b Fb(14)2032
5605 482 y Fd(truncate2)28 b(\(\))22 b Fc(.)12 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g
5606 (.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
5607 f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)47 b Fb(15)2032
5608 573 y Fd(tschebychev)28 b(\(\))18 b Fc(.)c(.)e(.)g(.)h(.)f(.)g(.)g(.)h
5609 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
5610 h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)44 b Fb(42)2032 832 y
5611 Fs(U)2032 954 y Fd(uminus)27 b(\(\))8 b Fc(.)13 b(.)f(.)g(.)g(.)h(.)f
5612 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
5613 f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)33
5614 b Fb(33)2032 1045 y(under\015o)n(w)14 b Fc(.)e(.)g(.)h(.)f(.)g(.)h(.)f
5615 (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
5616 f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)40
5617 b Fb(52)2032 1136 y(univ)l(ariate)26 b(p)r(olynomial)11
5618 b Fc(.)i(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
5619 h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)37 b
5620 Fb(38)2032 1393 y Fs(W)2032 1515 y Fd(WANT_OBFUSCATING_OPERATORS)13
5621 b Fc(.)18 b(.)12 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
5622 (.)h(.)f(.)g(.)h(.)f(.)38 b Fb(27)2032 1759 y Fs(X)2032
5623 1880 y Fd(xgcd)27 b(\(\))10 b Fc(.)j(.)f(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
5624 f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
5625 (.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)36
5626 b Fb(23)2032 2139 y Fs(Z)2032 2261 y Fd(zero)27 b(\(\))14
5627 b Fc(.)e(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
5628 h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
5629 (.)40 b Fb(33,)26 b(34,)h(40)2032 2352 y Fd(zerop)g(\(\))d
5630 Fc(.)12 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
5631 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)49
5632 b Fb(13,)27 b(33,)f(36,)h(41)2032 2444 y Fd(zeta)g(\(\))10
5633 b Fc(.)j(.)f(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)
5634 f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
5635 (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)36 b Fb(20)p eop
5637 -1 58 bop 3905 -116 a Fr(i)-30 299 y Fo(T)-13 b(able)54
5638 b(of)g(Con)l(ten)l(ts)-30 641 y Fs(1)135 b(In)l(tro)t(duction)39
5639 b Fa(.)20 b(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f
5640 (.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)
5641 h(.)f(.)84 b Fs(1)-30 911 y(2)135 b(Installation)13 b
5642 Fa(.)22 b(.)d(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f
5643 (.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)
5644 f(.)g(.)h(.)f(.)58 b Fs(3)269 1048 y Fr(2.1)92 b(Prerequisites)24
5645 b Fe(.)15 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
5646 (.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
5647 h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
5648 (.)53 b Fr(3)568 1157 y(2.1.1)93 b(C)p Fp(++)29 b Fr(compiler)22
5649 b Fe(.)15 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
5650 (.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
5651 g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)52 b Fr(3)568 1267
5652 y(2.1.2)93 b(Mak)m(e)31 b(utilit)m(y)22 b Fe(.)17 b(.)e(.)g(.)g(.)g(.)g
5653 (.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)
5654 g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
5655 (.)g(.)g(.)52 b Fr(3)568 1377 y(2.1.3)93 b(Sed)29 b(utilit)m(y)16
5656 b Fe(.)g(.)f(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
5657 g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
5658 (.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)45 b Fr(3)269
5659 1486 y(2.2)92 b(Building)30 b(the)h(library)13 b Fe(.)h(.)h(.)g(.)g(.)g
5660 (.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
5661 g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
5662 (.)g(.)g(.)h(.)f(.)g(.)42 b Fr(3)568 1596 y(2.2.1)93
5663 b(Using)30 b(the)h(GNU)g(MP)f(Library)c Fe(.)15 b(.)g(.)g(.)h(.)f(.)g
5664 (.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
5665 g(.)g(.)g(.)g(.)g(.)57 b Fr(4)269 1705 y(2.3)92 b(Installing)31
5666 b(the)g(library)17 b Fe(.)d(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
5667 (.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
5668 g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)47
5669 b Fr(4)269 1815 y(2.4)92 b(Cleaning)31 b(up)17 b Fe(.)d(.)h(.)g(.)g(.)g
5670 (.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
5671 g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
5672 (.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)47
5673 b Fr(5)-30 2057 y Fs(3)135 b(Ordinary)45 b(n)l(um)l(b)t(er)f(t)l(yp)t
5674 (es)29 b Fa(.)20 b(.)g(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h
5675 (.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)74 b Fs(6)269
5676 2194 y Fr(3.1)92 b(Exact)31 b(n)m(um)m(b)s(ers)22 b Fe(.)13
5677 b(.)i(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
5678 (.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
5679 g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)52
5680 b Fr(6)269 2304 y(3.2)92 b(Floating-p)s(oin)m(t)33 b(n)m(um)m(b)s(ers)
5681 23 b Fe(.)15 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
5682 (.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)
5683 g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)55 b Fr(7)269 2413
5684 y(3.3)92 b(Complex)29 b(n)m(um)m(b)s(ers)c Fe(.)15 b(.)g(.)g(.)g(.)g(.)
5685 g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f
5686 (.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
5687 g(.)g(.)g(.)g(.)g(.)g(.)57 b Fr(8)269 2523 y(3.4)92 b(Con)m(v)m
5688 (ersions)20 b Fe(.)c(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
5689 g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
5690 (.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
5691 g(.)g(.)g(.)g(.)50 b Fr(8)-30 2765 y Fs(4)135 b(F)-11
5692 b(unctions)44 b(on)h(n)l(um)l(b)t(ers)37 b Fa(.)19 b(.)h(.)f(.)h(.)f(.)
5693 h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h
5694 (.)f(.)h(.)82 b Fs(10)269 2902 y Fr(4.1)92 b(Constructing)30
5695 b(n)m(um)m(b)s(ers)d Fe(.)15 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
5696 (.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
5697 g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)59
5698 b Fr(10)568 3012 y(4.1.1)93 b(Constructing)30 b(in)m(tegers)10
5699 b Fe(.)16 b(.)f(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
5700 (.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
5701 g(.)40 b Fr(10)568 3122 y(4.1.2)93 b(Constructing)30
5702 b(rational)h(n)m(um)m(b)s(ers)20 b Fe(.)13 b(.)i(.)g(.)g(.)g(.)g(.)g(.)
5703 g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
5704 (.)49 b Fr(10)568 3231 y(4.1.3)93 b(Constructing)30 b(\015oating-p)s
5705 (oin)m(t)h(n)m(um)m(b)s(ers)14 b Fe(.)f(.)i(.)g(.)g(.)g(.)g(.)g(.)g(.)g
5706 (.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)43 b Fr(10)568
5707 3341 y(4.1.4)93 b(Constructing)30 b(complex)g(n)m(um)m(b)s(ers)11
5708 b Fe(.)h(.)j(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)
5709 g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)40 b Fr(10)269 3450
5710 y(4.2)92 b(Elemen)m(tary)30 b(functions)25 b Fe(.)15
5711 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g
5712 (.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
5713 g(.)g(.)g(.)g(.)g(.)g(.)g(.)55 b Fr(11)269 3560 y(4.3)92
5714 b(Elemen)m(tary)30 b(rational)i(functions)c Fe(.)15 b(.)g(.)g(.)g(.)g
5715 (.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
5716 g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)57 b
5717 Fr(12)269 3670 y(4.4)92 b(Elemen)m(tary)30 b(complex)g(functions)19
5718 b Fe(.)c(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
5719 g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
5720 (.)49 b Fr(12)269 3779 y(4.5)92 b(Comparisons)21 b Fe(.)14
5721 b(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
5722 (.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
5723 g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)51
5724 b Fr(13)269 3889 y(4.6)92 b(Rounding)29 b(functions)18
5725 b Fe(.)d(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
5726 g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g
5727 (.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)47 b Fr(13)269
5728 3998 y(4.7)92 b(Ro)s(ots)11 b Fe(.)k(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)
5729 g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
5730 (.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)
5731 g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)40
5732 b Fr(16)269 4108 y(4.8)92 b(T)-8 b(ranscenden)m(tal)31
5733 b(functions)17 b Fe(.)e(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
5734 (.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)
5735 f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)46 b Fr(17)568 4218
5736 y(4.8.1)93 b(Exp)s(onen)m(tial)30 b(and)g(logarithmic)h(functions)17
5737 b Fe(.)e(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
5738 h(.)f(.)46 b Fr(17)568 4327 y(4.8.2)93 b(T)-8 b(rigonometric)30
5739 b(functions)c Fe(.)15 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
5740 (.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
5741 g(.)55 b Fr(17)568 4437 y(4.8.3)93 b(Hyp)s(erb)s(olic)29
5742 b(functions)23 b Fe(.)15 b(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g
5743 (.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
5744 g(.)g(.)g(.)g(.)g(.)53 b Fr(19)568 4546 y(4.8.4)93 b(Euler)29
5745 b(gamma)20 b Fe(.)14 b(.)i(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
5746 (.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
5747 g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)49 b Fr(20)568
5748 4656 y(4.8.5)93 b(Riemann)29 b(zeta)13 b Fe(.)j(.)f(.)g(.)g(.)g(.)g(.)g
5749 (.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)
5750 g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)42
5751 b Fr(20)269 4765 y(4.9)92 b(F)-8 b(unctions)31 b(on)f(in)m(tegers)c
5752 Fe(.)16 b(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
5753 (.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
5754 g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)54 b Fr(20)568 4875
5755 y(4.9.1)93 b(Logical)32 b(functions)13 b Fe(.)h(.)h(.)g(.)g(.)g(.)g(.)g
5756 (.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
5757 g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)42
5758 b Fr(20)568 4985 y(4.9.2)93 b(Num)m(b)s(er)28 b(theoretic)k(functions)
5759 20 b Fe(.)14 b(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g
5760 (.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)49
5761 b Fr(23)568 5094 y(4.9.3)93 b(Com)m(binatorial)30 b(functions)21
5762 b Fe(.)15 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
5763 (.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)50
5764 b Fr(23)269 5204 y(4.10)92 b(F)-8 b(unctions)31 b(on)g(\015oating-p)s
5765 (oin)m(t)g(n)m(um)m(b)s(ers)25 b Fe(.)15 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g
5766 (.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)
5767 g(.)g(.)56 b Fr(23)269 5313 y(4.11)92 b(Con)m(v)m(ersion)31
5768 b(functions)12 b Fe(.)j(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
5769 (.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)
5770 g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)41 b
5773 -2 59 bop 3879 -116 a Fr(ii)568 83 y(4.11.1)93 b(Con)m(v)m(ersion)31
5774 b(to)g(\015oating-p)s(oin)m(t)h(n)m(um)m(b)s(ers)17 b
5775 Fe(.)c(.)i(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
5776 (.)g(.)47 b Fr(25)568 193 y(4.11.2)93 b(Con)m(v)m(ersion)31
5777 b(to)g(rational)h(n)m(um)m(b)s(ers)21 b Fe(.)15 b(.)g(.)g(.)g(.)g(.)g
5778 (.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
5779 53 b Fr(26)269 302 y(4.12)92 b(Random)29 b(n)m(um)m(b)s(er)f
5780 (generators)e Fe(.)15 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
5781 (.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
5782 h(.)f(.)g(.)g(.)g(.)53 b Fr(26)269 412 y(4.13)92 b(Obfuscating)30
5783 b(op)s(erators)g Fe(.)15 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
5784 (.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
5785 g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)59 b Fr(27)-30
5786 652 y Fs(5)135 b(Input/Output)26 b Fa(.)19 b(.)h(.)f(.)g(.)h(.)f(.)h(.)
5787 f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)g
5788 (.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)71 b Fs(28)269 789
5789 y Fr(5.1)92 b(In)m(ternal)31 b(and)e(prin)m(ted)h(represen)m(tation)g
5790 Fe(.)15 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g
5791 (.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)58
5792 b Fr(28)269 898 y(5.2)92 b(Input)29 b(functions)14 b
5793 Fe(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
5794 (.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
5795 g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)44
5796 b Fr(29)269 1008 y(5.3)92 b(Output)29 b(functions)21
5797 b Fe(.)15 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
5798 (.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)
5799 g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)50
5800 b Fr(30)-30 1248 y Fs(6)135 b(Rings)22 b Fa(.)e(.)f(.)h(.)f(.)h(.)f(.)h
5801 (.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)
5802 f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)67
5803 b Fs(33)-30 1516 y(7)135 b(Mo)t(dular)44 b(in)l(tegers)27
5804 b Fa(.)22 b(.)d(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f
5805 (.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)72
5806 b Fs(34)269 1653 y Fr(7.1)92 b(Mo)s(dular)30 b(in)m(teger)i(rings)18
5807 b Fe(.)c(.)h(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
5808 g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
5809 (.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)47 b Fr(34)269 1762
5810 y(7.2)92 b(F)-8 b(unctions)31 b(on)f(mo)s(dular)e(in)m(tegers)16
5811 b Fe(.)h(.)e(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
5812 g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
5813 (.)46 b Fr(34)-30 2002 y Fs(8)135 b(Sym)l(b)t(olic)45
5814 b(data)g(t)l(yp)t(es)28 b Fa(.)21 b(.)e(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h
5815 (.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)
5816 73 b Fs(37)269 2139 y Fr(8.1)92 b(Strings)9 b Fe(.)14
5817 b(.)h(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
5818 (.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
5819 g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
5820 (.)g(.)g(.)g(.)38 b Fr(37)269 2249 y(8.2)92 b(Sym)m(b)s(ols)24
5821 b Fe(.)15 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
5822 (.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)
5823 g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
5824 (.)g(.)g(.)g(.)55 b Fr(37)-30 2489 y Fs(9)135 b(Univ)-7
5825 b(ariate)46 b(p)t(olynomials)40 b Fa(.)19 b(.)h(.)f(.)g(.)h(.)f(.)h(.)f
5826 (.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)83
5827 b Fs(38)269 2626 y Fr(9.1)92 b(Univ)-5 b(ariate)32 b(p)s(olynomial)d
5828 (rings)c Fe(.)15 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
5829 (.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
5830 g(.)g(.)g(.)g(.)g(.)54 b Fr(38)269 2736 y(9.2)92 b(F)-8
5831 b(unctions)31 b(on)f(univ)-5 b(ariate)31 b(p)s(olynomials)d
5832 Fe(.)15 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
5833 (.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)58
5834 b Fr(40)269 2845 y(9.3)92 b(Sp)s(ecial)31 b(p)s(olynomials)13
5835 b Fe(.)h(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)
5836 g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
5837 (.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)43 b Fr(42)-30
5838 3085 y Fs(10)135 b(In)l(ternals)20 b Fa(.)g(.)g(.)f(.)g(.)h(.)f(.)h(.)f
5839 (.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)
5840 f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)64 b
5841 Fs(43)269 3222 y Fr(10.1)92 b(Wh)m(y)31 b(C)p Fp(++)f
5842 Fr(?)d Fe(.)15 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
5843 g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
5844 (.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
5845 g(.)57 b Fr(43)269 3332 y(10.2)92 b(Memory)30 b(e\016ciency)25
5846 b Fe(.)15 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
5847 (.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
5848 g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)53 b Fr(43)269
5849 3442 y(10.3)92 b(Sp)s(eed)30 b(e\016ciency)e Fe(.)15
5850 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g
5851 (.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
5852 g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)57 b Fr(43)269
5853 3551 y(10.4)92 b(Garbage)32 b(collection)14 b Fe(.)k(.)d(.)g(.)g(.)g(.)
5854 g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g
5855 (.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
5856 g(.)g(.)g(.)g(.)44 b Fr(44)-30 3791 y Fs(11)135 b(Using)46
5857 b(the)f(library)25 b Fa(.)20 b(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h
5858 (.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)
5859 h(.)69 b Fs(45)269 3928 y Fr(11.1)92 b(Compiler)30 b(options)21
5860 b Fe(.)15 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
5861 (.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)
5862 g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)50 b
5863 Fr(45)269 4038 y(11.2)92 b(Compatibilit)m(y)31 b(to)g(old)g(CLN)f(v)m
5864 (ersions)17 b Fe(.)e(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
5865 g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)46
5866 b Fr(45)269 4147 y(11.3)92 b(Include)30 b(\014les)18
5867 b Fe(.)d(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)
5868 g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
5869 (.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)47
5870 b Fr(45)269 4257 y(11.4)92 b(An)30 b(Example)d Fe(.)15
5871 b(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
5872 (.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
5873 h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)57
5874 b Fr(49)269 4367 y(11.5)92 b(Debugging)32 b(supp)s(ort)23
5875 b Fe(.)15 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g
5876 (.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
5877 g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)54 b Fr(50)-30
5878 4607 y Fs(12)135 b(Customizing)39 b Fa(.)19 b(.)h(.)f(.)h(.)f(.)h(.)f
5879 (.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)
5880 h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)83 b Fs(52)269 4744
5881 y Fr(12.1)92 b(Error)30 b(handling)24 b Fe(.)15 b(.)g(.)g(.)g(.)g(.)h
5882 (.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
5883 g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f
5884 (.)g(.)g(.)g(.)g(.)g(.)g(.)54 b Fr(52)269 4853 y(12.2)92
5885 b(Floating-p)s(oin)m(t)33 b(under\015o)m(w)19 b Fe(.)14
5886 b(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g
5887 (.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
5888 g(.)g(.)g(.)49 b Fr(52)269 4963 y(12.3)92 b(Customizing)30
5889 b(I/O)25 b Fe(.)15 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
5890 g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g
5891 (.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)54
5892 b Fr(52)269 5072 y(12.4)92 b(Customizing)30 b(the)h(memory)d(allo)s
5893 (cator)16 b Fe(.)i(.)d(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
5894 (.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)46
5895 b Fr(52)-30 5313 y Fs(Index)30 b Fa(.)19 b(.)g(.)h(.)f(.)h(.)f(.)h(.)f
5896 (.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)
5897 h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f
5898 (.)h(.)74 b Fs(53)p eop
5901 userdict /end-hook known{end-hook}if
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