Average Error: 43.1 → 0.9
Time: 13.1s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2} \cdot \sin y i\right))\]
\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r49844 = x;
        double r49845 = exp(r49844);
        double r49846 = -r49844;
        double r49847 = exp(r49846);
        double r49848 = r49845 + r49847;
        double r49849 = 2.0;
        double r49850 = r49848 / r49849;
        double r49851 = y;
        double r49852 = cos(r49851);
        double r49853 = r49850 * r49852;
        double r49854 = r49845 - r49847;
        double r49855 = r49854 / r49849;
        double r49856 = sin(r49851);
        double r49857 = r49855 * r49856;
        double r49858 = /* ERROR: no complex support in C */;
        double r49859 = /* ERROR: no complex support in C */;
        return r49859;
}
↓
double f(double x, double y) {
        double r49860 = x;
        double r49861 = exp(r49860);
        double r49862 = -r49860;
        double r49863 = exp(r49862);
        double r49864 = r49861 + r49863;
        double r49865 = 2.0;
        double r49866 = r49864 / r49865;
        double r49867 = y;
        double r49868 = cos(r49867);
        double r49869 = r49866 * r49868;
        double r49870 = 0.3333333333333333;
        double r49871 = 3.0;
        double r49872 = pow(r49860, r49871);
        double r49873 = r49870 * r49872;
        double r49874 = 0.016666666666666666;
        double r49875 = 5.0;
        double r49876 = pow(r49860, r49875);
        double r49877 = r49874 * r49876;
        double r49878 = 2.0;
        double r49879 = r49878 * r49860;
        double r49880 = r49877 + r49879;
        double r49881 = r49873 + r49880;
        double r49882 = r49881 / r49865;
        double r49883 = sin(r49867);
        double r49884 = r49882 * r49883;
        double r49885 = /* ERROR: no complex support in C */;
        double r49886 = /* ERROR: no complex support in C */;
        return r49886;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.1

    \[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]

  2. Taylor expanded around 0 0.9

    \[\leadsto \Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\color{blue}{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}}{2} \cdot \sin y i\right))\]

  3. Final simplification0.9

    \[\leadsto \Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2} \cdot \sin y i\right))\]

Reproduce

herbie shell --seed 2020039 
(FPCore (x y)
  :name "Euler formula imaginary part (p55)"
  :precision binary64
  (im (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y)))))