Average Error: 43.1 → 0.8
Time: 37.5s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{1}{3}, x \cdot x, 2\right), \frac{1}{60} \cdot {x}^{5}\right)}{2} \cdot \sin y i\right))\]
\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{\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{1}{3}, x \cdot x, 2\right), \frac{1}{60} \cdot {x}^{5}\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r2093823 = x;
        double r2093824 = exp(r2093823);
        double r2093825 = -r2093823;
        double r2093826 = exp(r2093825);
        double r2093827 = r2093824 + r2093826;
        double r2093828 = 2.0;
        double r2093829 = r2093827 / r2093828;
        double r2093830 = y;
        double r2093831 = cos(r2093830);
        double r2093832 = r2093829 * r2093831;
        double r2093833 = r2093824 - r2093826;
        double r2093834 = r2093833 / r2093828;
        double r2093835 = sin(r2093830);
        double r2093836 = r2093834 * r2093835;
        double r2093837 = /* ERROR: no complex support in C */;
        double r2093838 = /* ERROR: no complex support in C */;
        return r2093838;
}
↓
double f(double x, double y) {
        double r2093839 = x;
        double r2093840 = exp(r2093839);
        double r2093841 = -r2093839;
        double r2093842 = exp(r2093841);
        double r2093843 = r2093840 + r2093842;
        double r2093844 = 2.0;
        double r2093845 = r2093843 / r2093844;
        double r2093846 = y;
        double r2093847 = cos(r2093846);
        double r2093848 = r2093845 * r2093847;
        double r2093849 = 0.3333333333333333;
        double r2093850 = r2093839 * r2093839;
        double r2093851 = fma(r2093849, r2093850, r2093844);
        double r2093852 = 0.016666666666666666;
        double r2093853 = 5.0;
        double r2093854 = pow(r2093839, r2093853);
        double r2093855 = r2093852 * r2093854;
        double r2093856 = fma(r2093839, r2093851, r2093855);
        double r2093857 = r2093856 / r2093844;
        double r2093858 = sin(r2093846);
        double r2093859 = r2093857 * r2093858;
        double r2093860 = /* ERROR: no complex support in C */;
        double r2093861 = /* ERROR: no complex support in C */;
        return r2093861;
}

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 0.8

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

  3. Simplified0.8

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2019164 +o rules:numerics
(FPCore (x y)
  :name "Euler formula imaginary part (p55)"
  (im (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y)))))