-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();
+static ex cos_eval(const ex & x)
+{
+ // 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();