}
// log(exp(t)) -> t (for real-valued t):
if (is_ex_the_function(x, exp)) {
- ex t=x.op(0);
+ ex t = x.op(0);
if (t.info(info_flags::real))
return t;
}
GINAC_ASSERT(diff_param==0);
// d/dx log(x) -> 1/x
- return power(x, -1);
+ return power(x, _ex_1());
}
REGISTER_FUNCTION(log, log_eval, log_evalf, log_diff, NULL);
static ex sin_eval(ex const & x)
{
- // sin(n*Pi/6) -> { 0 | +/-1/2 | +/-sqrt(3)/2 | +/-1 }
- ex SixExOverPi = _ex6()*x/Pi;
- if (SixExOverPi.info(info_flags::integer)) {
- numeric z = smod(ex_to_numeric(SixExOverPi),_num12());
- if (z.is_equal(_num_5())) // sin(7*Pi/6) -> -1/2
- return _ex_1_2();
- if (z.is_equal(_num_4())) // sin(8*Pi/6) -> -sqrt(3)/2
- return _ex_1_2()*power(_ex3(),_ex1_2());
- if (z.is_equal(_num_3())) // sin(9*Pi/6) -> -1
- return _ex_1();
- if (z.is_equal(_num_2())) // sin(10*Pi/6) -> -sqrt(3)/2
- return _ex_1_2()*power(_ex3(),_ex1_2());
- if (z.is_equal(_num_1())) // sin(11*Pi/6) -> -1/2
- return _ex_1_2();
- if (z.is_equal(_num0())) // sin(0) -> 0
- return _ex0();
- if (z.is_equal(_num1())) // sin(1*Pi/6) -> 1/2
- return _ex1_2();
- if (z.is_equal(_num2())) // sin(2*Pi/6) -> sqrt(3)/2
- return _ex1_2()*power(_ex3(),_ex1_2());
- if (z.is_equal(_num3())) // sin(3*Pi/6) -> 1
- return _ex1();
- if (z.is_equal(_num4())) // sin(4*Pi/6) -> sqrt(3)/2
- return _ex1_2()*power(_ex3(),_ex1_2());
- if (z.is_equal(_num5())) // sin(5*Pi/6) -> 1/2
- return _ex1_2();
- if (z.is_equal(_num6())) // sin(6*Pi/6) -> 0
+ // sin(n/d*Pi) -> { all known non-nested radicals }
+ ex SixtyExOverPi = _ex60()*x/Pi;
+ ex sign = _ex1();
+ if (SixtyExOverPi.info(info_flags::integer)) {
+ numeric z = mod(ex_to_numeric(SixtyExOverPi),_num120());
+ if (z>=_num60()) {
+ // wrap to interval [0, Pi)
+ z -= _num60();
+ sign = _ex_1();
+ }
+ if (z>_num30()) {
+ // wrap to interval [0, Pi/2)
+ z = _num60()-z;
+ }
+ if (z.is_equal(_num0())) // sin(0) -> 0
return _ex0();
+ if (z.is_equal(_num5())) // sin(Pi/12) -> sqrt(6)/4*(1-sqrt(3)/3)
+ return sign*_ex1_4()*power(_ex6(),_ex1_2())*(_ex1()+_ex_1_3()*power(_ex3(),_ex1_2()));
+ if (z.is_equal(_num6())) // sin(Pi/10) -> sqrt(5)/4-1/4
+ return sign*(_ex1_4()*power(_ex5(),_ex1_2())+_ex_1_4());
+ if (z.is_equal(_num10())) // sin(Pi/6) -> 1/2
+ return sign*_ex1_2();
+ if (z.is_equal(_num15())) // sin(Pi/4) -> sqrt(2)/2
+ return sign*_ex1_2()*power(_ex2(),_ex1_2());
+ if (z.is_equal(_num18())) // sin(3/10*Pi) -> sqrt(5)/4+1/4
+ return sign*(_ex1_4()*power(_ex5(),_ex1_2())+_ex1_4());
+ if (z.is_equal(_num20())) // sin(Pi/3) -> sqrt(3)/2
+ return sign*_ex1_2()*power(_ex3(),_ex1_2());
+ if (z.is_equal(_num25())) // sin(5/12*Pi) -> sqrt(6)/4*(1+sqrt(3)/3)
+ return sign*_ex1_4()*power(_ex6(),_ex1_2())*(_ex1()+_ex1_3()*power(_ex3(),_ex1_2()));
+ if (z.is_equal(_num30())) // sin(Pi/2) -> 1
+ return sign*_ex1();
}
if (is_ex_exactly_of_type(x, function)) {
- ex t=x.op(0);
+ ex t = x.op(0);
// sin(asin(x)) -> x
if (is_ex_the_function(x, asin))
return t;
static ex cos_eval(ex const & x)
{
- // cos(n*Pi/6) -> { 0 | +/-1/2 | +/-sqrt(3)/2 | +/-1 }
- ex SixExOverPi = _ex6()*x/Pi;
- if (SixExOverPi.info(info_flags::integer)) {
- numeric z = smod(ex_to_numeric(SixExOverPi),_num12());
- if (z.is_equal(_num_5())) // cos(7*Pi/6) -> -sqrt(3)/2
- return _ex_1_2()*power(_ex3(),_ex1_2());
- if (z.is_equal(_num_4())) // cos(8*Pi/6) -> -1/2
- return _ex_1_2();
- if (z.is_equal(_num_3())) // cos(9*Pi/6) -> 0
- return _ex0();
- if (z.is_equal(_num_2())) // cos(10*Pi/6) -> 1/2
- return _ex1_2();
- if (z.is_equal(_num_1())) // cos(11*Pi/6) -> sqrt(3)/2
- return _ex1_2()*power(_ex3(),_ex1_2());
- if (z.is_equal(_num0())) // cos(0) -> 1
- return _ex1();
- if (z.is_equal(_num1())) // cos(1*Pi/6) -> sqrt(3)/2
- return _ex1_2()*power(_ex3(),_ex1_2());
- if (z.is_equal(_num2())) // cos(2*Pi/6) -> 1/2
- return _ex1_2();
- if (z.is_equal(_num3())) // cos(3*Pi/6) -> 0
- return _ex0();
- if (z.is_equal(_num4())) // cos(4*Pi/6) -> -1/2
- return _ex_1_2();
- if (z.is_equal(_num5())) // cos(5*Pi/6) -> -sqrt(3)/2
- return _ex_1_2()*power(_ex3(),_ex1_2());
- if (z.is_equal(_num6())) // cos(6*Pi/6) -> -1
- return _ex_1();
+ // cos(n/d*Pi) -> { all known non-nested radicals }
+ ex SixtyExOverPi = _ex60()*x/Pi;
+ ex sign = _ex1();
+ if (SixtyExOverPi.info(info_flags::integer)) {
+ numeric z = mod(ex_to_numeric(SixtyExOverPi),_num120());
+ if (z>=_num60()) {
+ // wrap to interval [0, Pi)
+ z = _num120()-z;
+ }
+ if (z>=_num30()) {
+ // wrap to interval [0, Pi/2)
+ z = _num60()-z;
+ sign = _ex_1();
+ }
+ if (z.is_equal(_num0())) // cos(0) -> 1
+ return sign*_ex1();
+ if (z.is_equal(_num5())) // cos(Pi/12) -> sqrt(6)/4*(1+sqrt(3)/3)
+ return sign*_ex1_4()*power(_ex6(),_ex1_2())*(_ex1()+_ex1_3()*power(_ex3(),_ex1_2()));
+ if (z.is_equal(_num10())) // cos(Pi/6) -> sqrt(3)/2
+ return sign*_ex1_2()*power(_ex3(),_ex1_2());
+ if (z.is_equal(_num12())) // cos(Pi/5) -> sqrt(5)/4+1/4
+ return sign*(_ex1_4()*power(_ex5(),_ex1_2())+_ex1_4());
+ if (z.is_equal(_num15())) // cos(Pi/4) -> sqrt(2)/2
+ return sign*_ex1_2()*power(_ex2(),_ex1_2());
+ if (z.is_equal(_num20())) // cos(Pi/3) -> 1/2
+ return sign*_ex1_2();
+ if (z.is_equal(_num24())) // cos(2/5*Pi) -> sqrt(5)/4-1/4x
+ return sign*(_ex1_4()*power(_ex5(),_ex1_2())+_ex_1_4());
+ if (z.is_equal(_num25())) // cos(5/12*Pi) -> sqrt(6)/4*(1-sqrt(3)/3)
+ return sign*_ex1_4()*power(_ex6(),_ex1_2())*(_ex1()+_ex_1_3()*power(_ex3(),_ex1_2()));
+ if (z.is_equal(_num30())) // cos(Pi/2) -> 0
+ return sign*_ex0();
}
if (is_ex_exactly_of_type(x, function)) {
- ex t=x.op(0);
+ ex t = x.op(0);
// cos(acos(x)) -> x
if (is_ex_the_function(x, acos))
return t;
static ex tan_eval(ex const & x)
{
- // tan(n*Pi/6) -> { 0 | +/-sqrt(3) | +/-sqrt(3)/2 }
- ex SixExOverPi = _ex6()*x/Pi;
- if (SixExOverPi.info(info_flags::integer)) {
- numeric z = smod(ex_to_numeric(SixExOverPi),_num6());
- if (z.is_equal(_num_2())) // tan(4*Pi/6) -> -sqrt(3)
- return _ex_1()*power(_ex3(),_ex1_2());
- if (z.is_equal(_num_1())) // tan(5*Pi/6) -> -sqrt(3)/3
- return _ex_1_3()*power(_ex3(),_ex1_2());
- if (z.is_equal(_num0())) // tan(0) -> 0
+ // tan(n/d*Pi) -> { all known non-nested radicals }
+ ex SixtyExOverPi = _ex60()*x/Pi;
+ ex sign = _ex1();
+ if (SixtyExOverPi.info(info_flags::integer)) {
+ numeric z = mod(ex_to_numeric(SixtyExOverPi),_num60());
+ if (z>=_num60()) {
+ // wrap to interval [0, Pi)
+ z -= _num60();
+ }
+ if (z>=_num30()) {
+ // wrap to interval [0, Pi/2)
+ z = _num60()-z;
+ sign = _ex_1();
+ }
+ if (z.is_equal(_num0())) // tan(0) -> 0
return _ex0();
- if (z.is_equal(_num1())) // tan(1*Pi/6) -> sqrt(3)/3
- return _ex1_3()*power(_ex3(),_ex1_2());
- if (z.is_equal(_num2())) // tan(2*Pi/6) -> sqrt(3)
- return power(_ex3(),_ex1_2());
- if (z.is_equal(_num3())) // tan(3*Pi/6) -> infinity
+ if (z.is_equal(_num5())) // tan(Pi/12) -> 2-sqrt(3)
+ return sign*(_ex2()-power(_ex3(),_ex1_2()));
+ if (z.is_equal(_num10())) // tan(Pi/6) -> sqrt(3)/3
+ return sign*_ex1_3()*power(_ex3(),_ex1_2());
+ if (z.is_equal(_num15())) // tan(Pi/4) -> 1
+ return sign*_ex1();
+ if (z.is_equal(_num20())) // tan(Pi/3) -> sqrt(3)
+ return sign*power(_ex3(),_ex1_2());
+ if (z.is_equal(_num25())) // tan(5/12*Pi) -> 2+sqrt(3)
+ return sign*(power(_ex3(),_ex1_2())+_ex2());
+ if (z.is_equal(_num30())) // tan(Pi/2) -> infinity
throw (std::domain_error("tan_eval(): infinity"));
}
-
+
if (is_ex_exactly_of_type(x, function)) {
- ex t=x.op(0);
+ ex t = x.op(0);
// tan(atan(x)) -> x
if (is_ex_the_function(x, atan))
return t;
GINAC_ASSERT(diff_param==0);
// d/dx tan(x) -> 1+tan(x)^2;
- return (1+power(tan(x),_ex2()));
+ return (_ex1()+power(tan(x),_ex2()));
}
static ex tan_series(ex const & x, symbol const & s, ex const & point, int order)
return _ex0();
// acos(1/2) -> Pi/3
if (x.is_equal(_ex1_2()))
- return numeric(1,3)*Pi;
+ return _ex1_3()*Pi;
// acos(0) -> Pi/2
if (x.is_zero())
- return numeric(1,2)*Pi;
+ return _ex1_2()*Pi;
// acos(-1/2) -> 2/3*Pi
if (x.is_equal(_ex_1_2()))
return numeric(2,3)*Pi;
{
GINAC_ASSERT(diff_param==0);
- return power(1+x*x, -1);
+ // d/dx atan(x) -> 1/(1+x^2)
+ return power(_ex1()+power(x,_ex2()), _ex_1());
}
REGISTER_FUNCTION(atan, atan_eval, atan_evalf, atan_diff, NULL);
if (diff_param==0) {
// d/dy atan(y,x)
- return x*pow(pow(x,2)+pow(y,2),-1);
+ return x*power(power(x,_ex2())+power(y,_ex2()),_ex_1());
}
// d/dx atan(y,x)
- return -y*pow(pow(x,2)+pow(y,2),-1);
+ return -y*power(power(x,_ex2())+power(y,_ex2()),_ex_1());
}
REGISTER_FUNCTION(atan2, atan2_eval, atan2_evalf, atan2_diff, NULL);
return I*sin(x/I);
if (is_ex_exactly_of_type(x, function)) {
- ex t=x.op(0);
+ ex t = x.op(0);
// sinh(asinh(x)) -> x
if (is_ex_the_function(x, asinh))
return t;
return cos(x/I);
if (is_ex_exactly_of_type(x, function)) {
- ex t=x.op(0);
+ ex t = x.op(0);
// cosh(acosh(x)) -> x
if (is_ex_the_function(x, acosh))
return t;
return I*tan(x/I);
if (is_ex_exactly_of_type(x, function)) {
- ex t=x.op(0);
+ ex t = x.op(0);
// tanh(atanh(x)) -> x
if (is_ex_the_function(x, atanh))
return t;
GINAC_ASSERT(diff_param==0);
// d/dx asinh(x) -> 1/sqrt(1+x^2)
- return power(1+power(x,_ex2()),_ex_1_2());
+ return power(_ex1()+power(x,_ex2()),_ex_1_2());
}
REGISTER_FUNCTION(asinh, asinh_eval, asinh_evalf, asinh_diff, NULL);
git://www.ginac.de / ginac.git / blobdiff