X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Fnumeric.cpp;h=55d5d556526e3038f47684e53a665527791cb9e8;hp=3398f60db6aca79a82bd5b27244a03ebe8b70cf9;hb=aaaa74e1033fdd659b4420835b348c72f5fab536;hpb=e5362a33f72613b324b3714524a8c2e5f7b7f46f

diff --git a/ginac/numeric.cpp b/ginac/numeric.cpp
index 3398f60d..55d5d556 100644
--- a/ginac/numeric.cpp
+++ b/ginac/numeric.cpp
@@ -7,7 +7,7 @@
  *  of special functions or implement the interface to the bignum package. */
 
 /*
- *  GiNaC Copyright (C) 1999-2001 Johannes Gutenberg University Mainz, Germany
+ *  GiNaC Copyright (C) 1999-2002 Johannes Gutenberg University Mainz, Germany
  *
  *  This program is free software; you can redistribute it and/or modify
  *  it under the terms of the GNU General Public License as published by
@@ -29,19 +29,13 @@
 #include <vector>
 #include <stdexcept>
 #include <string>
-
-#if defined(HAVE_SSTREAM)
 #include <sstream>
-#elif defined(HAVE_STRSTREAM)
-#include <strstream>
-#else
-#error Need either sstream or strstream
-#endif
 
 #include "numeric.h"
 #include "ex.h"
+#include "print.h"
 #include "archive.h"
-#include "debugmsg.h"
+#include "tostring.h"
 #include "utils.h"
 
 // CLN should pollute the global namespace as little as possible.  Hence, we
@@ -63,106 +57,76 @@
 #include <cln/complex_ring.h>
 #include <cln/numtheory.h>
 
-#ifndef NO_NAMESPACE_GINAC
 namespace GiNaC {
-#endif  // ndef NO_NAMESPACE_GINAC
 
 GINAC_IMPLEMENT_REGISTERED_CLASS(numeric, basic)
 
 //////////
-// default constructor, destructor, copy constructor assignment
-// operator and helpers
+// default ctor, dtor, copy ctor, assignment operator and helpers
 //////////
 
-// public
-
 /** default ctor. Numerically it initializes to an integer zero. */
 numeric::numeric() : basic(TINFO_numeric)
 {
-	debugmsg("numeric default constructor", LOGLEVEL_CONSTRUCT);
 	value = cln::cl_I(0);
-	calchash();
-	setflag(status_flags::evaluated |
-	        status_flags::expanded |
-	        status_flags::hash_calculated);
+	setflag(status_flags::evaluated | status_flags::expanded);
 }
 
-// protected
-
-void numeric::copy(const numeric & other)
+void numeric::copy(const numeric &other)
 {
-	basic::copy(other);
+	inherited::copy(other);
 	value = other.value;
 }
 
-void numeric::destroy(bool call_parent)
-{
-	if (call_parent) basic::destroy(call_parent);
-}
+DEFAULT_DESTROY(numeric)
 
 //////////
-// other constructors
+// other ctors
 //////////
 
 // public
 
 numeric::numeric(int i) : basic(TINFO_numeric)
 {
-	debugmsg("numeric constructor from int",LOGLEVEL_CONSTRUCT);
 	// Not the whole int-range is available if we don't cast to long
 	// first.  This is due to the behaviour of the cl_I-ctor, which
-	// emphasizes efficiency.  However, if the integer is small enough, 
-	// i.e. satisfies cl_immediate_p(), we save space and dereferences by
-	// using an immediate type:
-	if (cln::cl_immediate_p(i))
+	// emphasizes efficiency.  However, if the integer is small enough
+	// we save space and dereferences by using an immediate type.
+	// (C.f. <cln/object.h>)
+	if (i < (1U<<cl_value_len-1))
 		value = cln::cl_I(i);
 	else
 		value = cln::cl_I((long) i);
-	calchash();
-	setflag(status_flags::evaluated |
-	        status_flags::expanded |
-	        status_flags::hash_calculated);
+	setflag(status_flags::evaluated | status_flags::expanded);
 }
 
 
 numeric::numeric(unsigned int i) : basic(TINFO_numeric)
 {
-	debugmsg("numeric constructor from uint",LOGLEVEL_CONSTRUCT);
 	// Not the whole uint-range is available if we don't cast to ulong
 	// first.  This is due to the behaviour of the cl_I-ctor, which
-	// emphasizes efficiency.  However, if the integer is small enough, 
-	// i.e. satisfies cl_immediate_p(), we save space and dereferences by
-	// using an immediate type:
-	if (cln::cl_immediate_p(i))
+	// emphasizes efficiency.  However, if the integer is small enough
+	// we save space and dereferences by using an immediate type.
+	// (C.f. <cln/object.h>)
+	if (i < (1U<<cl_value_len-1))
 		value = cln::cl_I(i);
 	else
 		value = cln::cl_I((unsigned long) i);
-	calchash();
-	setflag(status_flags::evaluated |
-	        status_flags::expanded |
-	        status_flags::hash_calculated);
+	setflag(status_flags::evaluated | status_flags::expanded);
 }
 
 
 numeric::numeric(long i) : basic(TINFO_numeric)
 {
-	debugmsg("numeric constructor from long",LOGLEVEL_CONSTRUCT);
 	value = cln::cl_I(i);
-	calchash();
-	setflag(status_flags::evaluated |
-	        status_flags::expanded |
-	        status_flags::hash_calculated);
+	setflag(status_flags::evaluated | status_flags::expanded);
 }
 
 
 numeric::numeric(unsigned long i) : basic(TINFO_numeric)
 {
-	debugmsg("numeric constructor from ulong",LOGLEVEL_CONSTRUCT);
 	value = cln::cl_I(i);
-	calchash();
-	setflag(status_flags::evaluated |
-	        status_flags::expanded |
-	        status_flags::hash_calculated);
+	setflag(status_flags::evaluated | status_flags::expanded);
 }
 
 /** Ctor for rational numerics a/b.
@@ -170,68 +134,68 @@ numeric::numeric(unsigned long i) : basic(TINFO_numeric)
  *  @exception overflow_error (division by zero) */
 numeric::numeric(long numer, long denom) : basic(TINFO_numeric)
 {
-	debugmsg("numeric constructor from long/long",LOGLEVEL_CONSTRUCT);
 	if (!denom)
 		throw std::overflow_error("division by zero");
 	value = cln::cl_I(numer) / cln::cl_I(denom);
-	calchash();
-	setflag(status_flags::evaluated |
-	        status_flags::expanded |
-	        status_flags::hash_calculated);
+	setflag(status_flags::evaluated | status_flags::expanded);
 }
 
 
 numeric::numeric(double d) : basic(TINFO_numeric)
 {
-	debugmsg("numeric constructor from double",LOGLEVEL_CONSTRUCT);
 	// We really want to explicitly use the type cl_LF instead of the
 	// more general cl_F, since that would give us a cl_DF only which
 	// will not be promoted to cl_LF if overflow occurs:
 	value = cln::cl_float(d, cln::default_float_format);
-	calchash();
-	setflag(status_flags::evaluated |
-	        status_flags::expanded |
-	        status_flags::hash_calculated);
+	setflag(status_flags::evaluated | status_flags::expanded);
 }
 
+
 /** ctor from C-style string.  It also accepts complex numbers in GiNaC
  *  notation like "2+5*I". */
 numeric::numeric(const char *s) : basic(TINFO_numeric)
 {
-	debugmsg("numeric constructor from string",LOGLEVEL_CONSTRUCT);
 	cln::cl_N ctorval = 0;
 	// parse complex numbers (functional but not completely safe, unfortunately
 	// std::string does not understand regexpese):
 	// ss should represent a simple sum like 2+5*I
-	std::string ss(s);
-	// make it safe by adding explicit sign
+	std::string ss = s;
+	std::string::size_type delim;
+
+	// make this implementation safe by adding explicit sign
 	if (ss.at(0) != '+' && ss.at(0) != '-' && ss.at(0) != '#')
 		ss = '+' + ss;
-	std::string::size_type delim;
+
+	// We use 'E' as exponent marker in the output, but some people insist on
+	// writing 'e' at input, so let's substitute them right at the beginning:
+	while ((delim = ss.find("e"))!=std::string::npos)
+		ss.replace(delim,1,"E");
+
+	// main parser loop:
 	do {
 		// chop ss into terms from left to right
 		std::string term;
 		bool imaginary = false;
 		delim = ss.find_first_of(std::string("+-"),1);
 		// Do we have an exponent marker like "31.415E-1"?  If so, hop on!
-		if ((delim != std::string::npos) && (ss.at(delim-1) == 'E'))
+		if (delim!=std::string::npos && ss.at(delim-1)=='E')
 			delim = ss.find_first_of(std::string("+-"),delim+1);
 		term = ss.substr(0,delim);
-		if (delim != std::string::npos)
+		if (delim!=std::string::npos)
 			ss = ss.substr(delim);
 		// is the term imaginary?
-		if (term.find("I") != std::string::npos) {
+		if (term.find("I")!=std::string::npos) {
 			// erase 'I':
-			term = term.replace(term.find("I"),1,"");
+			term.erase(term.find("I"),1);
 			// erase '*':
-			if (term.find("*") != std::string::npos)
-				term = term.replace(term.find("*"),1,"");
+			if (term.find("*")!=std::string::npos)
+				term.erase(term.find("*"),1);
 			// correct for trivial +/-I without explicit factor on I:
-			if (term.size() == 1)
-				term += "1";
+			if (term.size()==1)
+				term += '1';
 			imaginary = true;
 		}
-		if (term.find(".") != std::string::npos) {
+		if (term.find('.')!=std::string::npos || term.find('E')!=std::string::npos) {
 			// CLN's short type cl_SF is not very useful within the GiNaC
 			// framework where we are mainly interested in the arbitrary
 			// precision type cl_LF.  Hence we go straight to the construction
@@ -242,70 +206,50 @@ numeric::numeric(const char *s) : basic(TINFO_numeric)
 			// 31.4E-1   -->   31.4e-1_<Digits>
 			// and s on.
 			// No exponent marker?  Let's add a trivial one.
-			if (term.find("E") == std::string::npos)
+			if (term.find("E")==std::string::npos)
 				term += "E0";
 			// E to lower case
 			term = term.replace(term.find("E"),1,"e");
 			// append _<Digits> to term
-#if defined(HAVE_SSTREAM)
-			std::ostringstream buf;
-			buf << unsigned(Digits) << std::ends;
-			term += "_" + buf.str();
-#else
-			char buf[14];
-			std::ostrstream(buf,sizeof(buf)) << unsigned(Digits) << std::ends;
-			term += "_" + string(buf);
-#endif
+			term += "_" + ToString((unsigned)Digits);
 			// construct float using cln::cl_F(const char *) ctor.
 			if (imaginary)
 				ctorval = ctorval + cln::complex(cln::cl_I(0),cln::cl_F(term.c_str()));
 			else
 				ctorval = ctorval + cln::cl_F(term.c_str());
 		} else {
-			// not a floating point number...
+			// this is not a floating point number...
 			if (imaginary)
 				ctorval = ctorval + cln::complex(cln::cl_I(0),cln::cl_R(term.c_str()));
 			else
 				ctorval = ctorval + cln::cl_R(term.c_str());
 		}
-	} while(delim != std::string::npos);
+	} while (delim != std::string::npos);
 	value = ctorval;
-	calchash();
-	setflag(status_flags::evaluated |
-			status_flags::expanded |
-			status_flags::hash_calculated);
+	setflag(status_flags::evaluated | status_flags::expanded);
 }
 
+
 /** Ctor from CLN types.  This is for the initiated user or internal use
  *  only. */
-numeric::numeric(const cln::cl_N & z) : basic(TINFO_numeric)
+numeric::numeric(const cln::cl_N &z) : basic(TINFO_numeric)
 {
-	debugmsg("numeric constructor from cl_N", LOGLEVEL_CONSTRUCT);
 	value = z;
-	calchash();
-	setflag(status_flags::evaluated |
-	        status_flags::expanded |
-	        status_flags::hash_calculated);
+	setflag(status_flags::evaluated | status_flags::expanded);
 }
 
 //////////
 // archiving
 //////////
 
-/** Construct object from archive_node. */
 numeric::numeric(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
 {
-	debugmsg("numeric constructor from archive_node", LOGLEVEL_CONSTRUCT);
 	cln::cl_N ctorval = 0;
 
 	// Read number as string
 	std::string str;
 	if (n.find_string("number", str)) {
-#ifdef HAVE_SSTREAM
 		std::istringstream s(str);
-#else
-		std::istrstream s(str.c_str(), str.size() + 1);
-#endif
 		cln::cl_idecoded_float re, im;
 		char c;
 		s.get(c);
@@ -327,30 +271,15 @@ numeric::numeric(const archive_node &n, const lst &sym_lst) : inherited(n, sym_l
 		}
 	}
 	value = ctorval;
-	calchash();
-	setflag(status_flags::evaluated |
-	        status_flags::expanded |
-	        status_flags::hash_calculated);
-}
-
-/** Unarchive the object. */
-ex numeric::unarchive(const archive_node &n, const lst &sym_lst)
-{
-	return (new numeric(n, sym_lst))->setflag(status_flags::dynallocated);
+	setflag(status_flags::evaluated | status_flags::expanded);
 }
 
-/** Archive the object. */
 void numeric::archive(archive_node &n) const
 {
 	inherited::archive(n);
 
 	// Write number as string
-#ifdef HAVE_SSTREAM
 	std::ostringstream s;
-#else
-	char buf[1024];
-	std::ostrstream s(buf, 1024);
-#endif
 	if (this->is_crational())
 		s << cln::the<cln::cl_N>(value);
 	else {
@@ -368,28 +297,15 @@ void numeric::archive(archive_node &n) const
 			s << im.sign << " " << im.mantissa << " " << im.exponent;
 		}
 	}
-#ifdef HAVE_SSTREAM
 	n.add_string("number", s.str());
-#else
-	s << ends;
-	std::string str(buf);
-	n.add_string("number", str);
-#endif
 }
 
+DEFAULT_UNARCHIVE(numeric)
+
 //////////
-// functions overriding virtual functions from bases classes
+// functions overriding virtual functions from base classes
 //////////
 
-// public
-
-basic * numeric::duplicate() const
-{
-	debugmsg("numeric duplicate", LOGLEVEL_DUPLICATE);
-	return new numeric(*this);
-}
-
-
 /** Helper function to print a real number in a nicer way than is CLN's
  *  default.  Instead of printing 42.0L0 this just prints 42.0 to ostream os
  *  and instead of 3.99168L7 it prints 3.99168E7.  This is fine in GiNaC as
@@ -397,146 +313,140 @@ basic * numeric::duplicate() const
  *  want to visibly distinguish from cl_LF.
  *
  *  @see numeric::print() */
-static void print_real_number(std::ostream & os, const cln::cl_R & num)
+static void print_real_number(const print_context & c, const cln::cl_R &x)
 {
 	cln::cl_print_flags ourflags;
-	if (cln::instanceof(num, cln::cl_RA_ring)) {
-		// case 1: integer or rational, nothing special to do:
-		cln::print_real(os, ourflags, num);
+	if (cln::instanceof(x, cln::cl_RA_ring)) {
+		// case 1: integer or rational
+		if (cln::instanceof(x, cln::cl_I_ring) ||
+		    !is_a<print_latex>(c)) {
+			cln::print_real(c.s, ourflags, x);
+		} else {  // rational output in LaTeX context
+			c.s << "\\frac{";
+			cln::print_real(c.s, ourflags, cln::numerator(cln::the<cln::cl_RA>(x)));
+			c.s << "}{";
+			cln::print_real(c.s, ourflags, cln::denominator(cln::the<cln::cl_RA>(x)));
+			c.s << '}';
+		}
 	} else {
 		// case 2: float
 		// make CLN believe this number has default_float_format, so it prints
 		// 'E' as exponent marker instead of 'L':
-		ourflags.default_float_format = cln::float_format(cln::the<cln::cl_F>(num));
-		cln::print_real(os, ourflags, num);
+		ourflags.default_float_format = cln::float_format(cln::the<cln::cl_F>(x));
+		cln::print_real(c.s, ourflags, x);
 	}
-	return;
 }
 
 /** This method adds to the output so it blends more consistently together
  *  with the other routines and produces something compatible to ginsh input.
  *  
  *  @see print_real_number() */
-void numeric::print(std::ostream & os, unsigned upper_precedence) const
-{
-	debugmsg("numeric print", LOGLEVEL_PRINT);
-	cln::cl_R r = cln::realpart(cln::the<cln::cl_N>(value));
-	cln::cl_R i = cln::imagpart(cln::the<cln::cl_N>(value));
-	if (cln::zerop(i)) {
-		// case 1, real:  x  or  -x
-		if ((precedence<=upper_precedence) && (!this->is_nonneg_integer())) {
-			os << "(";
-			print_real_number(os, r);
-			os << ")";
+void numeric::print(const print_context & c, unsigned level) const
+{
+	if (is_a<print_tree>(c)) {
+
+		c.s << std::string(level, ' ') << cln::the<cln::cl_N>(value)
+		    << " (" << class_name() << ")"
+		    << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
+		    << std::endl;
+
+	} else if (is_a<print_csrc>(c)) {
+
+		std::ios::fmtflags oldflags = c.s.flags();
+		c.s.setf(std::ios::scientific);
+		if (this->is_rational() && !this->is_integer()) {
+			if (compare(_num0) > 0) {
+				c.s << "(";
+				if (is_a<print_csrc_cl_N>(c))
+					c.s << "cln::cl_F(\"" << numer().evalf() << "\")";
+				else
+					c.s << numer().to_double();
+			} else {
+				c.s << "-(";
+				if (is_a<print_csrc_cl_N>(c))
+					c.s << "cln::cl_F(\"" << -numer().evalf() << "\")";
+				else
+					c.s << -numer().to_double();
+			}
+			c.s << "/";
+			if (is_a<print_csrc_cl_N>(c))
+				c.s << "cln::cl_F(\"" << denom().evalf() << "\")";
+			else
+				c.s << denom().to_double();
+			c.s << ")";
 		} else {
-			print_real_number(os, r);
+			if (is_a<print_csrc_cl_N>(c))
+				c.s << "cln::cl_F(\"" << evalf() << "\")";
+			else
+				c.s << to_double();
 		}
+		c.s.flags(oldflags);
+
 	} else {
-		if (cln::zerop(r)) {
-			// case 2, imaginary:  y*I  or  -y*I
-			if ((precedence<=upper_precedence) && (i < 0)) {
-				if (i == -1) {
-					os << "(-I)";
-				} else {
-					os << "(";
-					print_real_number(os, i);
-					os << "*I)";
-				}
+		const std::string par_open  = is_a<print_latex>(c) ? "{(" : "(";
+		const std::string par_close = is_a<print_latex>(c) ? ")}" : ")";
+		const std::string imag_sym  = is_a<print_latex>(c) ? "i" : "I";
+		const std::string mul_sym   = is_a<print_latex>(c) ? " " : "*";
+		const cln::cl_R r = cln::realpart(cln::the<cln::cl_N>(value));
+		const cln::cl_R i = cln::imagpart(cln::the<cln::cl_N>(value));
+		if (is_a<print_python_repr>(c))
+			c.s << class_name() << "('";
+		if (cln::zerop(i)) {
+			// case 1, real:  x  or  -x
+			if ((precedence() <= level) && (!this->is_nonneg_integer())) {
+				c.s << par_open;
+				print_real_number(c, r);
+				c.s << par_close;
 			} else {
-				if (i == 1) {
-					os << "I";
-				} else {
-					if (i == -1) {
-						os << "-I";
-					} else {
-						print_real_number(os, i);
-						os << "*I";
-					}
-				}
+				print_real_number(c, r);
 			}
 		} else {
-			// case 3, complex:  x+y*I  or  x-y*I  or  -x+y*I  or  -x-y*I
-			if (precedence <= upper_precedence)
-				os << "(";
-			print_real_number(os, r);
-			if (i < 0) {
-				if (i == -1) {
-					os << "-I";
-				} else {
-					print_real_number(os, i);
-					os << "*I";
+			if (cln::zerop(r)) {
+				// case 2, imaginary:  y*I  or  -y*I
+				if (i==1)
+					c.s << imag_sym;
+				else {
+					if (precedence()<=level)
+						c.s << par_open;
+					if (i == -1)
+						c.s << "-" << imag_sym;
+					else {
+						print_real_number(c, i);
+						c.s << mul_sym+imag_sym;
+					}
+					if (precedence()<=level)
+						c.s << par_close;
 				}
 			} else {
-				if (i == 1) {
-					os << "+I";
+				// case 3, complex:  x+y*I  or  x-y*I  or  -x+y*I  or  -x-y*I
+				if (precedence() <= level)
+					c.s << par_open;
+				print_real_number(c, r);
+				if (i < 0) {
+					if (i == -1) {
+						c.s << "-"+imag_sym;
+					} else {
+						print_real_number(c, i);
+						c.s << mul_sym+imag_sym;
+					}
 				} else {
-					os << "+";
-					print_real_number(os, i);
-					os << "*I";
+					if (i == 1) {
+						c.s << "+"+imag_sym;
+					} else {
+						c.s << "+";
+						print_real_number(c, i);
+						c.s << mul_sym+imag_sym;
+					}
 				}
+				if (precedence() <= level)
+					c.s << par_close;
 			}
-			if (precedence <= upper_precedence)
-				os << ")";
-		}
-	}
-}
-
-
-void numeric::printraw(std::ostream & os) const
-{
-	// The method printraw doesn't do much, it simply uses CLN's operator<<()
-	// for output, which is ugly but reliable. e.g: 2+2i
-	debugmsg("numeric printraw", LOGLEVEL_PRINT);
-	os << "numeric(" << cln::the<cln::cl_N>(value) << ")";
-}
-
-
-void numeric::printtree(std::ostream & os, unsigned indent) const
-{
-	debugmsg("numeric printtree", LOGLEVEL_PRINT);
-	os << std::string(indent,' ') << cln::the<cln::cl_N>(value)
-	   << " (numeric): "
-	   << "hash=" << hashvalue
-	   << " (0x" << std::hex << hashvalue << std::dec << ")"
-	   << ", flags=" << flags << std::endl;
-}
-
-
-void numeric::printcsrc(std::ostream & os, unsigned type, unsigned upper_precedence) const
-{
-	debugmsg("numeric print csrc", LOGLEVEL_PRINT);
-	std::ios::fmtflags oldflags = os.flags();
-	os.setf(std::ios::scientific);
-	if (this->is_rational() && !this->is_integer()) {
-		if (compare(_num0()) > 0) {
-			os << "(";
-			if (type == csrc_types::ctype_cl_N)
-				os << "cln::cl_F(\"" << numer().evalf() << "\")";
-			else
-				os << numer().to_double();
-		} else {
-			os << "-(";
-			if (type == csrc_types::ctype_cl_N)
-				os << "cln::cl_F(\"" << -numer().evalf() << "\")";
-			else
-				os << -numer().to_double();
 		}
-		os << "/";
-		if (type == csrc_types::ctype_cl_N)
-			os << "cln::cl_F(\"" << denom().evalf() << "\")";
-		else
-			os << denom().to_double();
-		os << ")";
-	} else {
-		if (type == csrc_types::ctype_cl_N)
-			os << "cln::cl_F(\"" << evalf() << "\")";
-		else
-			os << to_double();
+		if (is_a<print_python_repr>(c))
+			c.s << "')";
 	}
-	os.flags(oldflags);
 }
 
-
 bool numeric::info(unsigned inf) const
 {
 	switch (inf) {
@@ -582,17 +492,32 @@ bool numeric::info(unsigned inf) const
 	return false;
 }
 
+int numeric::degree(const ex & s) const
+{
+	return 0;
+}
+
+int numeric::ldegree(const ex & s) const
+{
+	return 0;
+}
+
+ex numeric::coeff(const ex & s, int n) const
+{
+	return n==0 ? *this : _ex0;
+}
+
 /** Disassemble real part and imaginary part to scan for the occurrence of a
  *  single number.  Also handles the imaginary unit.  It ignores the sign on
  *  both this and the argument, which may lead to what might appear as funny
  *  results:  (2+I).has(-2) -> true.  But this is consistent, since we also
  *  would like to have (-2+I).has(2) -> true and we want to think about the
  *  sign as a multiplicative factor. */
-bool numeric::has(const ex & other) const
+bool numeric::has(const ex &other) const
 {
-	if (!is_exactly_of_type(*other.bp, numeric))
+	if (!is_ex_exactly_of_type(other, numeric))
 		return false;
-	const numeric & o = static_cast<numeric &>(const_cast<basic &>(*other.bp));
+	const numeric &o = ex_to<numeric>(other);
 	if (this->is_equal(o) || this->is_equal(-o))
 		return true;
 	if (o.imag().is_zero())  // e.g. scan for 3 in -3*I
@@ -629,35 +554,26 @@ ex numeric::evalf(int level) const
 {
 	// level can safely be discarded for numeric objects.
 	return numeric(cln::cl_float(1.0, cln::default_float_format) *
-				   (cln::the<cln::cl_N>(value)));
+	               (cln::the<cln::cl_N>(value)));
 }
 
 // protected
 
-/** Implementation of ex::diff() for a numeric. It always returns 0.
- *
- *  @see ex::diff */
-ex numeric::derivative(const symbol & s) const
-{
-	return _ex0();
-}
-
-
-int numeric::compare_same_type(const basic & other) const
+int numeric::compare_same_type(const basic &other) const
 {
-	GINAC_ASSERT(is_exactly_of_type(other, numeric));
-	const numeric & o = static_cast<numeric &>(const_cast<basic &>(other));
+	GINAC_ASSERT(is_exactly_a<numeric>(other));
+	const numeric &o = static_cast<const numeric &>(other);
 	
 	return this->compare(o);
 }
 
 
-bool numeric::is_equal_same_type(const basic & other) const
+bool numeric::is_equal_same_type(const basic &other) const
 {
-	GINAC_ASSERT(is_exactly_of_type(other,numeric));
-	const numeric *o = static_cast<const numeric *>(&other);
+	GINAC_ASSERT(is_exactly_a<numeric>(other));
+	const numeric &o = static_cast<const numeric &>(other);
 	
-	return this->is_equal(*o);
+	return this->is_equal(o);
 }
 
 
@@ -666,6 +582,7 @@ unsigned numeric::calchash(void) const
 	// Use CLN's hashcode.  Warning: It depends only on the number's value, not
 	// its type or precision (i.e. a true equivalence relation on numbers).  As
 	// a consequence, 3 and 3.0 share the same hashvalue.
+	setflag(status_flags::hash_calculated);
 	return (hashvalue = cln::equal_hashcode(cln::the<cln::cl_N>(value)) | 0x80000000U);
 }
 
@@ -683,14 +600,13 @@ unsigned numeric::calchash(void) const
 // public
 
 /** Numerical addition method.  Adds argument to *this and returns result as
- *  a new numeric object. */
-const numeric numeric::add(const numeric & other) const
+ *  a numeric object. */
+const numeric numeric::add(const numeric &other) const
 {
 	// Efficiency shortcut: trap the neutral element by pointer.
-	static const numeric * _num0p = &_num0();
-	if (this==_num0p)
+	if (this==_num0_p)
 		return other;
-	else if (&other==_num0p)
+	else if (&other==_num0_p)
 		return *this;
 	
 	return numeric(cln::the<cln::cl_N>(value)+cln::the<cln::cl_N>(other.value));
@@ -698,22 +614,21 @@ const numeric numeric::add(const numeric & other) const
 
 
 /** Numerical subtraction method.  Subtracts argument from *this and returns
- *  result as a new numeric object. */
-const numeric numeric::sub(const numeric & other) const
+ *  result as a numeric object. */
+const numeric numeric::sub(const numeric &other) const
 {
 	return numeric(cln::the<cln::cl_N>(value)-cln::the<cln::cl_N>(other.value));
 }
 
 
 /** Numerical multiplication method.  Multiplies *this and argument and returns
- *  result as a new numeric object. */
-const numeric numeric::mul(const numeric & other) const
+ *  result as a numeric object. */
+const numeric numeric::mul(const numeric &other) const
 {
 	// Efficiency shortcut: trap the neutral element by pointer.
-	static const numeric * _num1p = &_num1();
-	if (this==_num1p)
+	if (this==_num1_p)
 		return other;
-	else if (&other==_num1p)
+	else if (&other==_num1_p)
 		return *this;
 	
 	return numeric(cln::the<cln::cl_N>(value)*cln::the<cln::cl_N>(other.value));
@@ -721,10 +636,10 @@ const numeric numeric::mul(const numeric & other) const
 
 
 /** Numerical division method.  Divides *this by argument and returns result as
- *  a new numeric object.
+ *  a numeric object.
  *
  *  @exception overflow_error (division by zero) */
-const numeric numeric::div(const numeric & other) const
+const numeric numeric::div(const numeric &other) const
 {
 	if (cln::zerop(cln::the<cln::cl_N>(other.value)))
 		throw std::overflow_error("numeric::div(): division by zero");
@@ -732,11 +647,12 @@ const numeric numeric::div(const numeric & other) const
 }
 
 
-const numeric numeric::power(const numeric & other) const
+/** Numerical exponentiation.  Raises *this to the power given as argument and
+ *  returns result as a numeric object. */
+const numeric numeric::power(const numeric &other) const
 {
 	// Efficiency shortcut: trap the neutral exponent by pointer.
-	static const numeric * _num1p = &_num1();
-	if (&other==_num1p)
+	if (&other==_num1_p)
 		return *this;
 	
 	if (cln::zerop(cln::the<cln::cl_N>(value))) {
@@ -747,19 +663,18 @@ const numeric numeric::power(const numeric & other) const
 		else if (cln::minusp(cln::realpart(cln::the<cln::cl_N>(other.value))))
 			throw std::overflow_error("numeric::eval(): division by zero");
 		else
-			return _num0();
+			return _num0;
 	}
 	return numeric(cln::expt(cln::the<cln::cl_N>(value),cln::the<cln::cl_N>(other.value)));
 }
 
 
-const numeric & numeric::add_dyn(const numeric & other) const
+const numeric &numeric::add_dyn(const numeric &other) const
 {
 	// Efficiency shortcut: trap the neutral element by pointer.
-	static const numeric * _num0p = &_num0();
-	if (this==_num0p)
+	if (this==_num0_p)
 		return other;
-	else if (&other==_num0p)
+	else if (&other==_num0_p)
 		return *this;
 	
 	return static_cast<const numeric &>((new numeric(cln::the<cln::cl_N>(value)+cln::the<cln::cl_N>(other.value)))->
@@ -767,20 +682,19 @@ const numeric & numeric::add_dyn(const numeric & other) const
 }
 
 
-const numeric & numeric::sub_dyn(const numeric & other) const
+const numeric &numeric::sub_dyn(const numeric &other) const
 {
 	return static_cast<const numeric &>((new numeric(cln::the<cln::cl_N>(value)-cln::the<cln::cl_N>(other.value)))->
 										setflag(status_flags::dynallocated));
 }
 
 
-const numeric & numeric::mul_dyn(const numeric & other) const
+const numeric &numeric::mul_dyn(const numeric &other) const
 {
 	// Efficiency shortcut: trap the neutral element by pointer.
-	static const numeric * _num1p = &_num1();
-	if (this==_num1p)
+	if (this==_num1_p)
 		return other;
-	else if (&other==_num1p)
+	else if (&other==_num1_p)
 		return *this;
 	
 	return static_cast<const numeric &>((new numeric(cln::the<cln::cl_N>(value)*cln::the<cln::cl_N>(other.value)))->
@@ -788,7 +702,7 @@ const numeric & numeric::mul_dyn(const numeric & other) const
 }
 
 
-const numeric & numeric::div_dyn(const numeric & other) const
+const numeric &numeric::div_dyn(const numeric &other) const
 {
 	if (cln::zerop(cln::the<cln::cl_N>(other.value)))
 		throw std::overflow_error("division by zero");
@@ -797,11 +711,10 @@ const numeric & numeric::div_dyn(const numeric & other) const
 }
 
 
-const numeric & numeric::power_dyn(const numeric & other) const
+const numeric &numeric::power_dyn(const numeric &other) const
 {
 	// Efficiency shortcut: trap the neutral exponent by pointer.
-	static const numeric * _num1p=&_num1();
-	if (&other==_num1p)
+	if (&other==_num1_p)
 		return *this;
 	
 	if (cln::zerop(cln::the<cln::cl_N>(value))) {
@@ -812,44 +725,44 @@ const numeric & numeric::power_dyn(const numeric & other) const
 		else if (cln::minusp(cln::realpart(cln::the<cln::cl_N>(other.value))))
 			throw std::overflow_error("numeric::eval(): division by zero");
 		else
-			return _num0();
+			return _num0;
 	}
 	return static_cast<const numeric &>((new numeric(cln::expt(cln::the<cln::cl_N>(value),cln::the<cln::cl_N>(other.value))))->
 	                                     setflag(status_flags::dynallocated));
 }
 
 
-const numeric & numeric::operator=(int i)
+const numeric &numeric::operator=(int i)
 {
 	return operator=(numeric(i));
 }
 
 
-const numeric & numeric::operator=(unsigned int i)
+const numeric &numeric::operator=(unsigned int i)
 {
 	return operator=(numeric(i));
 }
 
 
-const numeric & numeric::operator=(long i)
+const numeric &numeric::operator=(long i)
 {
 	return operator=(numeric(i));
 }
 
 
-const numeric & numeric::operator=(unsigned long i)
+const numeric &numeric::operator=(unsigned long i)
 {
 	return operator=(numeric(i));
 }
 
 
-const numeric & numeric::operator=(double d)
+const numeric &numeric::operator=(double d)
 {
 	return operator=(numeric(d));
 }
 
 
-const numeric & numeric::operator=(const char * s)
+const numeric &numeric::operator=(const char * s)
 {
 	return operator=(numeric(s));
 }
@@ -868,7 +781,7 @@ const numeric numeric::inverse(void) const
  *  csgn(x)==0 for x==0, csgn(x)==1 for Re(x)>0 or Re(x)=0 and Im(x)>0,
  *  csgn(x)==-1 for Re(x)<0 or Re(x)=0 and Im(x)<0.
  *
- *  @see numeric::compare(const numeric & other) */
+ *  @see numeric::compare(const numeric &other) */
 int numeric::csgn(void) const
 {
 	if (cln::zerop(cln::the<cln::cl_N>(value)))
@@ -895,7 +808,7 @@ int numeric::csgn(void) const
  *
  *  @return csgn(*this-other)
  *  @see numeric::csgn(void) */
-int numeric::compare(const numeric & other) const
+int numeric::compare(const numeric &other) const
 {
 	// Comparing two real numbers?
 	if (cln::instanceof(value, cln::cl_R_ring) &&
@@ -913,7 +826,7 @@ int numeric::compare(const numeric & other) const
 }
 
 
-bool numeric::is_equal(const numeric & other) const
+bool numeric::is_equal(const numeric &other) const
 {
 	return cln::equal(cln::the<cln::cl_N>(value),cln::the<cln::cl_N>(other.value));
 }
@@ -1003,15 +916,15 @@ bool numeric::is_real(void) const
 }
 
 
-bool numeric::operator==(const numeric & other) const
+bool numeric::operator==(const numeric &other) const
 {
-	return equal(cln::the<cln::cl_N>(value), cln::the<cln::cl_N>(other.value));
+	return cln::equal(cln::the<cln::cl_N>(value), cln::the<cln::cl_N>(other.value));
 }
 
 
-bool numeric::operator!=(const numeric & other) const
+bool numeric::operator!=(const numeric &other) const
 {
-	return !equal(cln::the<cln::cl_N>(value), cln::the<cln::cl_N>(other.value));
+	return !cln::equal(cln::the<cln::cl_N>(value), cln::the<cln::cl_N>(other.value));
 }
 
 
@@ -1048,7 +961,7 @@ bool numeric::is_crational(void) const
 /** Numerical comparison: less.
  *
  *  @exception invalid_argument (complex inequality) */ 
-bool numeric::operator<(const numeric & other) const
+bool numeric::operator<(const numeric &other) const
 {
 	if (this->is_real() && other.is_real())
 		return (cln::the<cln::cl_R>(value) < cln::the<cln::cl_R>(other.value));
@@ -1059,7 +972,7 @@ bool numeric::operator<(const numeric & other) const
 /** Numerical comparison: less or equal.
  *
  *  @exception invalid_argument (complex inequality) */ 
-bool numeric::operator<=(const numeric & other) const
+bool numeric::operator<=(const numeric &other) const
 {
 	if (this->is_real() && other.is_real())
 		return (cln::the<cln::cl_R>(value) <= cln::the<cln::cl_R>(other.value));
@@ -1070,7 +983,7 @@ bool numeric::operator<=(const numeric & other) const
 /** Numerical comparison: greater.
  *
  *  @exception invalid_argument (complex inequality) */ 
-bool numeric::operator>(const numeric & other) const
+bool numeric::operator>(const numeric &other) const
 {
 	if (this->is_real() && other.is_real())
 		return (cln::the<cln::cl_R>(value) > cln::the<cln::cl_R>(other.value));
@@ -1081,7 +994,7 @@ bool numeric::operator>(const numeric & other) const
 /** Numerical comparison: greater or equal.
  *
  *  @exception invalid_argument (complex inequality) */  
-bool numeric::operator>=(const numeric & other) const
+bool numeric::operator>=(const numeric &other) const
 {
 	if (this->is_real() && other.is_real())
 		return (cln::the<cln::cl_R>(value) >= cln::the<cln::cl_R>(other.value));
@@ -1179,16 +1092,16 @@ const numeric numeric::numer(void) const
 const numeric numeric::denom(void) const
 {
 	if (this->is_integer())
-		return _num1();
+		return _num1;
 	
-	if (instanceof(value, cln::cl_RA_ring))
+	if (cln::instanceof(value, cln::cl_RA_ring))
 		return numeric(cln::denominator(cln::the<cln::cl_RA>(value)));
 	
 	if (!this->is_real()) {  // complex case, handle Q(i):
 		const cln::cl_RA r = cln::the<cln::cl_RA>(cln::realpart(cln::the<cln::cl_N>(value)));
 		const cln::cl_RA i = cln::the<cln::cl_RA>(cln::imagpart(cln::the<cln::cl_N>(value)));
 		if (cln::instanceof(r, cln::cl_I_ring) && cln::instanceof(i, cln::cl_I_ring))
-			return _num1();
+			return _num1;
 		if (cln::instanceof(r, cln::cl_I_ring) && cln::instanceof(i, cln::cl_RA_ring))
 			return numeric(cln::denominator(i));
 		if (cln::instanceof(r, cln::cl_RA_ring) && cln::instanceof(i, cln::cl_I_ring))
@@ -1197,7 +1110,7 @@ const numeric numeric::denom(void) const
 			return numeric(cln::lcm(cln::denominator(r), cln::denominator(i)));
 	}
 	// at least one float encountered
-	return _num1();
+	return _num1;
 }
 
 
@@ -1215,28 +1128,20 @@ int numeric::int_length(void) const
 		return 0;
 }
 
-
-//////////
-// static member variables
-//////////
-
-// protected
-
-unsigned numeric::precedence = 30;
-
 //////////
 // global constants
 //////////
 
 /** Imaginary unit.  This is not a constant but a numeric since we are
- *  natively handing complex numbers anyways. */
+ *  natively handing complex numbers anyways, so in each expression containing
+ *  an I it is automatically eval'ed away anyhow. */
 const numeric I = numeric(cln::complex(cln::cl_I(0),cln::cl_I(1)));
 
 
 /** Exponential function.
  *
  *  @return  arbitrary precision numerical exp(x). */
-const numeric exp(const numeric & x)
+const numeric exp(const numeric &x)
 {
 	return cln::exp(x.to_cl_N());
 }
@@ -1247,7 +1152,7 @@ const numeric exp(const numeric & x)
  *  @param z complex number
  *  @return  arbitrary precision numerical log(x).
  *  @exception pole_error("log(): logarithmic pole",0) */
-const numeric log(const numeric & z)
+const numeric log(const numeric &z)
 {
 	if (z.is_zero())
 		throw pole_error("log(): logarithmic pole",0);
@@ -1258,7 +1163,7 @@ const numeric log(const numeric & z)
 /** Numeric sine (trigonometric function).
  *
  *  @return  arbitrary precision numerical sin(x). */
-const numeric sin(const numeric & x)
+const numeric sin(const numeric &x)
 {
 	return cln::sin(x.to_cl_N());
 }
@@ -1267,7 +1172,7 @@ const numeric sin(const numeric & x)
 /** Numeric cosine (trigonometric function).
  *
  *  @return  arbitrary precision numerical cos(x). */
-const numeric cos(const numeric & x)
+const numeric cos(const numeric &x)
 {
 	return cln::cos(x.to_cl_N());
 }
@@ -1276,7 +1181,7 @@ const numeric cos(const numeric & x)
 /** Numeric tangent (trigonometric function).
  *
  *  @return  arbitrary precision numerical tan(x). */
-const numeric tan(const numeric & x)
+const numeric tan(const numeric &x)
 {
 	return cln::tan(x.to_cl_N());
 }
@@ -1285,7 +1190,7 @@ const numeric tan(const numeric & x)
 /** Numeric inverse sine (trigonometric function).
  *
  *  @return  arbitrary precision numerical asin(x). */
-const numeric asin(const numeric & x)
+const numeric asin(const numeric &x)
 {
 	return cln::asin(x.to_cl_N());
 }
@@ -1294,7 +1199,7 @@ const numeric asin(const numeric & x)
 /** Numeric inverse cosine (trigonometric function).
  *
  *  @return  arbitrary precision numerical acos(x). */
-const numeric acos(const numeric & x)
+const numeric acos(const numeric &x)
 {
 	return cln::acos(x.to_cl_N());
 }
@@ -1305,11 +1210,11 @@ const numeric acos(const numeric & x)
  *  @param z complex number
  *  @return atan(z)
  *  @exception pole_error("atan(): logarithmic pole",0) */
-const numeric atan(const numeric & x)
+const numeric atan(const numeric &x)
 {
 	if (!x.is_real() &&
 	    x.real().is_zero() &&
-	    abs(x.imag()).is_equal(_num1()))
+	    abs(x.imag()).is_equal(_num1))
 		throw pole_error("atan(): logarithmic pole",0);
 	return cln::atan(x.to_cl_N());
 }
@@ -1320,7 +1225,7 @@ const numeric atan(const numeric & x)
  *  @param x real number
  *  @param y real number
  *  @return atan(y/x) */
-const numeric atan(const numeric & y, const numeric & x)
+const numeric atan(const numeric &y, const numeric &x)
 {
 	if (x.is_real() && y.is_real())
 		return cln::atan(cln::the<cln::cl_R>(x.to_cl_N()),
@@ -1333,7 +1238,7 @@ const numeric atan(const numeric & y, const numeric & x)
 /** Numeric hyperbolic sine (trigonometric function).
  *
  *  @return  arbitrary precision numerical sinh(x). */
-const numeric sinh(const numeric & x)
+const numeric sinh(const numeric &x)
 {
 	return cln::sinh(x.to_cl_N());
 }
@@ -1342,7 +1247,7 @@ const numeric sinh(const numeric & x)
 /** Numeric hyperbolic cosine (trigonometric function).
  *
  *  @return  arbitrary precision numerical cosh(x). */
-const numeric cosh(const numeric & x)
+const numeric cosh(const numeric &x)
 {
 	return cln::cosh(x.to_cl_N());
 }
@@ -1351,7 +1256,7 @@ const numeric cosh(const numeric & x)
 /** Numeric hyperbolic tangent (trigonometric function).
  *
  *  @return  arbitrary precision numerical tanh(x). */
-const numeric tanh(const numeric & x)
+const numeric tanh(const numeric &x)
 {
 	return cln::tanh(x.to_cl_N());
 }
@@ -1360,7 +1265,7 @@ const numeric tanh(const numeric & x)
 /** Numeric inverse hyperbolic sine (trigonometric function).
  *
  *  @return  arbitrary precision numerical asinh(x). */
-const numeric asinh(const numeric & x)
+const numeric asinh(const numeric &x)
 {
 	return cln::asinh(x.to_cl_N());
 }
@@ -1369,7 +1274,7 @@ const numeric asinh(const numeric & x)
 /** Numeric inverse hyperbolic cosine (trigonometric function).
  *
  *  @return  arbitrary precision numerical acosh(x). */
-const numeric acosh(const numeric & x)
+const numeric acosh(const numeric &x)
 {
 	return cln::acosh(x.to_cl_N());
 }
@@ -1378,14 +1283,14 @@ const numeric acosh(const numeric & x)
 /** Numeric inverse hyperbolic tangent (trigonometric function).
  *
  *  @return  arbitrary precision numerical atanh(x). */
-const numeric atanh(const numeric & x)
+const numeric atanh(const numeric &x)
 {
 	return cln::atanh(x.to_cl_N());
 }
 
 
-/*static cln::cl_N Li2_series(const ::cl_N & x,
-                            const ::float_format_t & prec)
+/*static cln::cl_N Li2_series(const ::cl_N &x,
+                            const ::float_format_t &prec)
 {
 	// Note: argument must be in the unit circle
 	// This is very inefficient unless we have fast floating point Bernoulli
@@ -1412,8 +1317,8 @@ const numeric atanh(const numeric & x)
 
 /** Numeric evaluation of Dilogarithm within circle of convergence (unit
  *  circle) using a power series. */
-static cln::cl_N Li2_series(const cln::cl_N & x,
-                            const cln::float_format_t & prec)
+static cln::cl_N Li2_series(const cln::cl_N &x,
+                            const cln::float_format_t &prec)
 {
 	// Note: argument must be in the unit circle
 	cln::cl_N aug, acc;
@@ -1431,8 +1336,8 @@ static cln::cl_N Li2_series(const cln::cl_N & x,
 }
 
 /** Folds Li2's argument inside a small rectangle to enhance convergence. */
-static cln::cl_N Li2_projection(const cln::cl_N & x,
-                                const cln::float_format_t & prec)
+static cln::cl_N Li2_projection(const cln::cl_N &x,
+                                const cln::float_format_t &prec)
 {
 	const cln::cl_R re = cln::realpart(x);
 	const cln::cl_R im = cln::imagpart(x);
@@ -1457,10 +1362,10 @@ static cln::cl_N Li2_projection(const cln::cl_N & x,
  *  continuous with quadrant IV.
  *
  *  @return  arbitrary precision numerical Li2(x). */
-const numeric Li2(const numeric & x)
+const numeric Li2(const numeric &x)
 {
 	if (x.is_zero())
-		return _num0();
+		return _num0;
 	
 	// what is the desired float format?
 	// first guess: default format
@@ -1487,7 +1392,7 @@ const numeric Li2(const numeric & x)
 
 /** Numeric evaluation of Riemann's Zeta function.  Currently works only for
  *  integer arguments. */
-const numeric zeta(const numeric & x)
+const numeric zeta(const numeric &x)
 {
 	// A dirty hack to allow for things like zeta(3.0), since CLN currently
 	// only knows about integer arguments and zeta(3).evalf() automatically
@@ -1499,50 +1404,35 @@ const numeric zeta(const numeric & x)
 		if (cln::zerop(x.to_cl_N()-aux))
 			return cln::zeta(aux);
 	}
-	std::clog << "zeta(" << x
-			  << "): Does anybody know good way to calculate this numerically?"
-			  << std::endl;
-	return numeric(0);
+	throw dunno();
 }
 
 
 /** The Gamma function.
  *  This is only a stub! */
-const numeric lgamma(const numeric & x)
+const numeric lgamma(const numeric &x)
 {
-	std::clog << "lgamma(" << x
-	          << "): Does anybody know good way to calculate this numerically?"
-	          << std::endl;
-	return numeric(0);
+	throw dunno();
 }
-const numeric tgamma(const numeric & x)
+const numeric tgamma(const numeric &x)
 {
-	std::clog << "tgamma(" << x
-	          << "): Does anybody know good way to calculate this numerically?"
-	          << std::endl;
-	return numeric(0);
+	throw dunno();
 }
 
 
 /** The psi function (aka polygamma function).
  *  This is only a stub! */
-const numeric psi(const numeric & x)
+const numeric psi(const numeric &x)
 {
-	std::clog << "psi(" << x
-	          << "): Does anybody know good way to calculate this numerically?"
-	          << std::endl;
-	return numeric(0);
+	throw dunno();
 }
 
 
 /** The psi functions (aka polygamma functions).
  *  This is only a stub! */
-const numeric psi(const numeric & n, const numeric & x)
+const numeric psi(const numeric &n, const numeric &x)
 {
-	std::clog << "psi(" << n << "," << x
-	          << "): Does anybody know good way to calculate this numerically?"
-	          << std::endl;
-	return numeric(0);
+	throw dunno();
 }
 
 
@@ -1550,7 +1440,7 @@ const numeric psi(const numeric & n, const numeric & x)
  *
  *  @param n  integer argument >= 0
  *  @exception range_error (argument must be integer >= 0) */
-const numeric factorial(const numeric & n)
+const numeric factorial(const numeric &n)
 {
 	if (!n.is_nonneg_integer())
 		throw std::range_error("numeric::factorial(): argument must be integer >= 0");
@@ -1564,10 +1454,10 @@ const numeric factorial(const numeric & n)
  *  @param n  integer argument >= -1
  *  @return n!! == n * (n-2) * (n-4) * ... * ({1|2}) with 0!! == (-1)!! == 1
  *  @exception range_error (argument must be integer >= -1) */
-const numeric doublefactorial(const numeric & n)
+const numeric doublefactorial(const numeric &n)
 {
-	if (n == numeric(-1))
-		return _num1();
+	if (n.is_equal(_num_1))
+		return _num1;
 	
 	if (!n.is_nonneg_integer())
 		throw std::range_error("numeric::doublefactorial(): argument must be integer >= -1");
@@ -1580,16 +1470,16 @@ const numeric doublefactorial(const numeric & n)
  *  integer n and k and positive n this is the number of ways of choosing k
  *  objects from n distinct objects.  If n is negative, the formula
  *  binomial(n,k) == (-1)^k*binomial(k-n-1,k) is used to compute the result. */
-const numeric binomial(const numeric & n, const numeric & k)
+const numeric binomial(const numeric &n, const numeric &k)
 {
 	if (n.is_integer() && k.is_integer()) {
 		if (n.is_nonneg_integer()) {
-			if (k.compare(n)!=1 && k.compare(_num0())!=-1)
+			if (k.compare(n)!=1 && k.compare(_num0)!=-1)
 				return numeric(cln::binomial(n.to_int(),k.to_int()));
 			else
-				return _num0();
+				return _num0;
 		} else {
-			return _num_1().power(k)*binomial(k-n-_num1(),k);
+			return _num_1.power(k)*binomial(k-n-_num1,k);
 		}
 	}
 	
@@ -1603,11 +1493,11 @@ const numeric binomial(const numeric & n, const numeric & k)
  *
  *  @return the nth Bernoulli number (a rational number).
  *  @exception range_error (argument must be integer >= 0) */
-const numeric bernoulli(const numeric & nn)
+const numeric bernoulli(const numeric &nn)
 {
 	if (!nn.is_integer() || nn.is_negative())
 		throw std::range_error("numeric::bernoulli(): argument must be integer >= 0");
-	
+
 	// Method:
 	//
 	// The Bernoulli numbers are rational numbers that may be computed using
@@ -1631,45 +1521,61 @@ const numeric bernoulli(const numeric & nn)
 	// But if somebody works with the n'th Bernoulli number she is likely to
 	// also need all previous Bernoulli numbers. So we need a complete remember
 	// table and above divide and conquer algorithm is not suited to build one
-	// up.  The code below is adapted from Pari's function bernvec().
+	// up.  The formula below accomplishes this.  It is a modification of the
+	// defining formula above but the computation of the binomial coefficients
+	// is carried along in an inline fashion.  It also honors the fact that
+	// B_n is zero when n is odd and greater than 1.
 	// 
 	// (There is an interesting relation with the tangent polynomials described
-	// in `Concrete Mathematics', which leads to a program twice as fast as our
-	// implementation below, but it requires storing one such polynomial in
+	// in `Concrete Mathematics', which leads to a program a little faster as
+	// our implementation below, but it requires storing one such polynomial in
 	// addition to the remember table.  This doubles the memory footprint so
 	// we don't use it.)
-	
+
+	const unsigned n = nn.to_int();
+
 	// the special cases not covered by the algorithm below
-	if (nn.is_equal(_num1()))
-		return _num_1_2();
-	if (nn.is_odd())
-		return _num0();
-	
+	if (n & 1)
+		return (n==1) ? _num_1_2 : _num0;
+	if (!n)
+		 return _num1;
+
 	// store nonvanishing Bernoulli numbers here
 	static std::vector< cln::cl_RA > results;
-	static int highest_result = 0;
-	// algorithm not applicable to B(0), so just store it
-	if (results.size()==0)
-		results.push_back(cln::cl_RA(1));
-	
-	int n = nn.to_long();
-	for (int i=highest_result; i<n/2; ++i) {
-		cln::cl_RA B = 0;
-		long n = 8;
-		long m = 5;
-		long d1 = i;
-		long d2 = 2*i-1;
-		for (int j=i; j>0; --j) {
-			B = cln::cl_I(n*m) * (B+results[j]) / (d1*d2);
-			n += 4;
-			m += 2;
-			d1 -= 1;
-			d2 -= 2;
-		}
-		B = (1 - ((B+1)/(2*i+3))) / (cln::cl_I(1)<<(2*i+2));
-		results.push_back(B);
-		++highest_result;
+	static unsigned next_r = 0;
+
+	// algorithm not applicable to B(2), so just store it
+	if (!next_r) {
+		results.push_back(); // results[0] is not used
+		results.push_back(cln::recip(cln::cl_RA(6)));
+		next_r = 4;
+	}
+	if (n<next_r)
+		return results[n/2];
+
+	results.reserve(n/2 + 1);
+	for (unsigned p=next_r; p<=n;  p+=2) {
+		cln::cl_I  c = 1;  // seed for binonmial coefficients
+		cln::cl_RA b = cln::cl_RA(1-p)/2;
+		const unsigned p3 = p+3;
+		const unsigned pm = p-2;
+		unsigned i, k, p_2;
+		// test if intermediate unsigned int can be represented by immediate
+		// objects by CLN (i.e. < 2^29 for 32 Bit machines, see <cln/object.h>)
+		if (p < (1UL<<cl_value_len/2)) {
+			for (i=2, k=1, p_2=p/2; i<=pm; i+=2, ++k, --p_2) {
+				c = cln::exquo(c * ((p3-i) * p_2), (i-1)*k);
+				b = b + c*results[k];
+			}
+		} else {
+			for (i=2, k=1, p_2=p/2; i<=pm; i+=2, ++k, --p_2) {
+				c = cln::exquo((c * (p3-i)) * p_2, cln::cl_I(i-1)*k);
+				b = b + c*results[k];
+			}
+ 		}
+		results.push_back(-b/(p+1));
 	}
+	next_r = n+2;
 	return results[n/2];
 }
 
@@ -1680,7 +1586,7 @@ const numeric bernoulli(const numeric & nn)
  *  @param n an integer
  *  @return the nth Fibonacci number F(n) (an integer number)
  *  @exception range_error (argument must be an integer) */
-const numeric fibonacci(const numeric & n)
+const numeric fibonacci(const numeric &n)
 {
 	if (!n.is_integer())
 		throw std::range_error("numeric::fibonacci(): argument must be integer");
@@ -1701,7 +1607,7 @@ const numeric fibonacci(const numeric & n)
 	// hence
 	//      F(2n+2) = F(n+1)*(2*F(n) + F(n+1))
 	if (n.is_zero())
-		return _num0();
+		return _num0;
 	if (n.is_negative())
 		if (n.is_even())
 			return -fibonacci(-n);
@@ -1747,13 +1653,13 @@ const numeric abs(const numeric& x)
  *
  *  @return a mod b in the range [0,abs(b)-1] with sign of b if both are
  *  integer, 0 otherwise. */
-const numeric mod(const numeric & a, const numeric & b)
+const numeric mod(const numeric &a, const numeric &b)
 {
 	if (a.is_integer() && b.is_integer())
 		return cln::mod(cln::the<cln::cl_I>(a.to_cl_N()),
 		                cln::the<cln::cl_I>(b.to_cl_N()));
 	else
-		return _num0();
+		return _num0;
 }
 
 
@@ -1761,14 +1667,14 @@ const numeric mod(const numeric & a, const numeric & b)
  *  Equivalent to Maple's mods.
  *
  *  @return a mod b in the range [-iquo(abs(m)-1,2), iquo(abs(m),2)]. */
-const numeric smod(const numeric & a, const numeric & b)
+const numeric smod(const numeric &a, const numeric &b)
 {
 	if (a.is_integer() && b.is_integer()) {
 		const cln::cl_I b2 = cln::ceiling1(cln::the<cln::cl_I>(b.to_cl_N()) >> 1) - 1;
 		return cln::mod(cln::the<cln::cl_I>(a.to_cl_N()) + b2,
 		                cln::the<cln::cl_I>(b.to_cl_N())) - b2;
 	} else
-		return _num0();
+		return _num0;
 }
 
 
@@ -1778,13 +1684,13 @@ const numeric smod(const numeric & a, const numeric & b)
  *  sign of a or is zero.
  *
  *  @return remainder of a/b if both are integer, 0 otherwise. */
-const numeric irem(const numeric & a, const numeric & b)
+const numeric irem(const numeric &a, const numeric &b)
 {
 	if (a.is_integer() && b.is_integer())
 		return cln::rem(cln::the<cln::cl_I>(a.to_cl_N()),
 		                cln::the<cln::cl_I>(b.to_cl_N()));
 	else
-		return _num0();
+		return _num0;
 }
 
 
@@ -1795,7 +1701,7 @@ const numeric irem(const numeric & a, const numeric & b)
  *
  *  @return remainder of a/b and quotient stored in q if both are integer,
  *  0 otherwise. */
-const numeric irem(const numeric & a, const numeric & b, numeric & q)
+const numeric irem(const numeric &a, const numeric &b, numeric &q)
 {
 	if (a.is_integer() && b.is_integer()) {
 		const cln::cl_I_div_t rem_quo = cln::truncate2(cln::the<cln::cl_I>(a.to_cl_N()),
@@ -1803,8 +1709,8 @@ const numeric irem(const numeric & a, const numeric & b, numeric & q)
 		q = rem_quo.quotient;
 		return rem_quo.remainder;
 	} else {
-		q = _num0();
-		return _num0();
+		q = _num0;
+		return _num0;
 	}
 }
 
@@ -1813,13 +1719,13 @@ const numeric irem(const numeric & a, const numeric & b, numeric & q)
  *  Equivalent to Maple's iquo as far as sign conventions are concerned.
  *  
  *  @return truncated quotient of a/b if both are integer, 0 otherwise. */
-const numeric iquo(const numeric & a, const numeric & b)
+const numeric iquo(const numeric &a, const numeric &b)
 {
 	if (a.is_integer() && b.is_integer())
-		return truncate1(cln::the<cln::cl_I>(a.to_cl_N()),
-	                     cln::the<cln::cl_I>(b.to_cl_N()));
+		return cln::truncate1(cln::the<cln::cl_I>(a.to_cl_N()),
+	                          cln::the<cln::cl_I>(b.to_cl_N()));
 	else
-		return _num0();
+		return _num0;
 }
 
 
@@ -1829,7 +1735,7 @@ const numeric iquo(const numeric & a, const numeric & b)
  *
  *  @return truncated quotient of a/b and remainder stored in r if both are
  *  integer, 0 otherwise. */
-const numeric iquo(const numeric & a, const numeric & b, numeric & r)
+const numeric iquo(const numeric &a, const numeric &b, numeric &r)
 {
 	if (a.is_integer() && b.is_integer()) {
 		const cln::cl_I_div_t rem_quo = cln::truncate2(cln::the<cln::cl_I>(a.to_cl_N()),
@@ -1837,8 +1743,8 @@ const numeric iquo(const numeric & a, const numeric & b, numeric & r)
 		r = rem_quo.remainder;
 		return rem_quo.quotient;
 	} else {
-		r = _num0();
-		return _num0();
+		r = _num0;
+		return _num0;
 	}
 }
 
@@ -1847,13 +1753,13 @@ const numeric iquo(const numeric & a, const numeric & b, numeric & r)
  *   
  *  @return  The GCD of two numbers if both are integer, a numerical 1
  *  if they are not. */
-const numeric gcd(const numeric & a, const numeric & b)
+const numeric gcd(const numeric &a, const numeric &b)
 {
 	if (a.is_integer() && b.is_integer())
 		return cln::gcd(cln::the<cln::cl_I>(a.to_cl_N()),
 		                cln::the<cln::cl_I>(b.to_cl_N()));
 	else
-		return _num1();
+		return _num1;
 }
 
 
@@ -1861,7 +1767,7 @@ const numeric gcd(const numeric & a, const numeric & b)
  *   
  *  @return  The LCM of two numbers if both are integer, the product of those
  *  two numbers if they are not. */
-const numeric lcm(const numeric & a, const numeric & b)
+const numeric lcm(const numeric &a, const numeric &b)
 {
 	if (a.is_integer() && b.is_integer())
 		return cln::lcm(cln::the<cln::cl_I>(a.to_cl_N()),
@@ -1879,21 +1785,21 @@ const numeric lcm(const numeric & a, const numeric & b)
  *  @return square root of z. Branch cut along negative real axis, the negative
  *  real axis itself where imag(z)==0 and real(z)<0 belongs to the upper part
  *  where imag(z)>0. */
-const numeric sqrt(const numeric & z)
+const numeric sqrt(const numeric &z)
 {
 	return cln::sqrt(z.to_cl_N());
 }
 
 
 /** Integer numeric square root. */
-const numeric isqrt(const numeric & x)
+const numeric isqrt(const numeric &x)
 {
 	if (x.is_integer()) {
 		cln::cl_I root;
 		cln::isqrt(cln::the<cln::cl_I>(x.to_cl_N()), &root);
 		return root;
 	} else
-		return _num0();
+		return _num0;
 }
 
 
@@ -1918,13 +1824,15 @@ ex CatalanEvalf(void)
 }
 
 
+/** _numeric_digits default ctor, checking for singleton invariance. */
 _numeric_digits::_numeric_digits()
   : digits(17)
 {
 	// It initializes to 17 digits, because in CLN float_format(17) turns out
 	// to be 61 (<64) while float_format(18)=65.  The reason is we want to
 	// have a cl_LF instead of cl_SF, cl_FF or cl_DF.
-	assert(!too_late);
+	if (too_late)
+		throw(std::runtime_error("I told you not to do instantiate me!"));
 	too_late = true;
 	cln::default_float_format = cln::float_format(17);
 }
@@ -1948,14 +1856,13 @@ _numeric_digits::operator long()
 
 
 /** Append global Digits object to ostream. */
-void _numeric_digits::print(std::ostream & os) const
+void _numeric_digits::print(std::ostream &os) const
 {
-	debugmsg("_numeric_digits print", LOGLEVEL_PRINT);
 	os << digits;
 }
 
 
-std::ostream& operator<<(std::ostream& os, const _numeric_digits & e)
+std::ostream& operator<<(std::ostream &os, const _numeric_digits &e)
 {
 	e.print(os);
 	return os;
@@ -1974,6 +1881,4 @@ bool _numeric_digits::too_late = false;
  *  assignment from C++ unsigned ints and evaluated like any built-in type. */
 _numeric_digits Digits;
 
-#ifndef NO_NAMESPACE_GINAC
 } // namespace GiNaC
-#endif // ndef NO_NAMESPACE_GINAC