Average Error: 43.6 → 0.8
Time: 1.7m
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\mathsf{fma}\left(\frac{1}{60}, \left({x}^{5}\right), \left(\mathsf{fma}\left(\frac{1}{3}, \left(x \cdot x\right), 2\right) \cdot x\right)\right)}{2} \cdot \sin y i\right))\]
\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(\frac{1}{60}, \left({x}^{5}\right), \left(\mathsf{fma}\left(\frac{1}{3}, \left(x \cdot x\right), 2\right) \cdot x\right)\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r6737751 = x;
        double r6737752 = exp(r6737751);
        double r6737753 = -r6737751;
        double r6737754 = exp(r6737753);
        double r6737755 = r6737752 + r6737754;
        double r6737756 = 2.0;
        double r6737757 = r6737755 / r6737756;
        double r6737758 = y;
        double r6737759 = cos(r6737758);
        double r6737760 = r6737757 * r6737759;
        double r6737761 = r6737752 - r6737754;
        double r6737762 = r6737761 / r6737756;
        double r6737763 = sin(r6737758);
        double r6737764 = r6737762 * r6737763;
        double r6737765 = /* ERROR: no complex support in C */;
        double r6737766 = /* ERROR: no complex support in C */;
        return r6737766;
}
↓
double f(double x, double y) {
        double r6737767 = x;
        double r6737768 = exp(r6737767);
        double r6737769 = -r6737767;
        double r6737770 = exp(r6737769);
        double r6737771 = r6737768 + r6737770;
        double r6737772 = 2.0;
        double r6737773 = r6737771 / r6737772;
        double r6737774 = y;
        double r6737775 = cos(r6737774);
        double r6737776 = r6737773 * r6737775;
        double r6737777 = 0.016666666666666666;
        double r6737778 = 5.0;
        double r6737779 = pow(r6737767, r6737778);
        double r6737780 = 0.3333333333333333;
        double r6737781 = r6737767 * r6737767;
        double r6737782 = fma(r6737780, r6737781, r6737772);
        double r6737783 = r6737782 * r6737767;
        double r6737784 = fma(r6737777, r6737779, r6737783);
        double r6737785 = r6737784 / r6737772;
        double r6737786 = sin(r6737774);
        double r6737787 = r6737785 * r6737786;
        double r6737788 = /* ERROR: no complex support in C */;
        double r6737789 = /* ERROR: no complex support in C */;
        return r6737789;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.6

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

  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

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2019124 +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)))))