Average Error: 43.3 → 0.7
Time: 39.5s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\mathsf{fma}\left(2, x, \left(\mathsf{fma}\left(\frac{1}{60}, \left({x}^{5}\right), \left(\left(\left(x \cdot x\right) \cdot \frac{1}{3}\right) \cdot x\right)\right)\right)\right)}{2} \cdot \sin y i\right))\]
\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(2, x, \left(\mathsf{fma}\left(\frac{1}{60}, \left({x}^{5}\right), \left(\left(\left(x \cdot x\right) \cdot \frac{1}{3}\right) \cdot x\right)\right)\right)\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r1723802 = x;
        double r1723803 = exp(r1723802);
        double r1723804 = -r1723802;
        double r1723805 = exp(r1723804);
        double r1723806 = r1723803 + r1723805;
        double r1723807 = 2.0;
        double r1723808 = r1723806 / r1723807;
        double r1723809 = y;
        double r1723810 = cos(r1723809);
        double r1723811 = r1723808 * r1723810;
        double r1723812 = r1723803 - r1723805;
        double r1723813 = r1723812 / r1723807;
        double r1723814 = sin(r1723809);
        double r1723815 = r1723813 * r1723814;
        double r1723816 = /* ERROR: no complex support in C */;
        double r1723817 = /* ERROR: no complex support in C */;
        return r1723817;
}
↓
double f(double x, double y) {
        double r1723818 = x;
        double r1723819 = exp(r1723818);
        double r1723820 = -r1723818;
        double r1723821 = exp(r1723820);
        double r1723822 = r1723819 + r1723821;
        double r1723823 = 2.0;
        double r1723824 = r1723822 / r1723823;
        double r1723825 = y;
        double r1723826 = cos(r1723825);
        double r1723827 = r1723824 * r1723826;
        double r1723828 = 0.016666666666666666;
        double r1723829 = 5.0;
        double r1723830 = pow(r1723818, r1723829);
        double r1723831 = r1723818 * r1723818;
        double r1723832 = 0.3333333333333333;
        double r1723833 = r1723831 * r1723832;
        double r1723834 = r1723833 * r1723818;
        double r1723835 = fma(r1723828, r1723830, r1723834);
        double r1723836 = fma(r1723823, r1723818, r1723835);
        double r1723837 = r1723836 / r1723823;
        double r1723838 = sin(r1723825);
        double r1723839 = r1723837 * r1723838;
        double r1723840 = /* ERROR: no complex support in C */;
        double r1723841 = /* ERROR: no complex support in C */;
        return r1723841;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.3

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

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019132 +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)))))