Average Error: 43.3 → 0.7
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 r5900684 = x;
        double r5900685 = exp(r5900684);
        double r5900686 = -r5900684;
        double r5900687 = exp(r5900686);
        double r5900688 = r5900685 + r5900687;
        double r5900689 = 2.0;
        double r5900690 = r5900688 / r5900689;
        double r5900691 = y;
        double r5900692 = cos(r5900691);
        double r5900693 = r5900690 * r5900692;
        double r5900694 = r5900685 - r5900687;
        double r5900695 = r5900694 / r5900689;
        double r5900696 = sin(r5900691);
        double r5900697 = r5900695 * r5900696;
        double r5900698 = /* ERROR: no complex support in C */;
        double r5900699 = /* ERROR: no complex support in C */;
        return r5900699;
}
↓
double f(double x, double y) {
        double r5900700 = x;
        double r5900701 = exp(r5900700);
        double r5900702 = -r5900700;
        double r5900703 = exp(r5900702);
        double r5900704 = r5900701 + r5900703;
        double r5900705 = 2.0;
        double r5900706 = r5900704 / r5900705;
        double r5900707 = y;
        double r5900708 = cos(r5900707);
        double r5900709 = r5900706 * r5900708;
        double r5900710 = 0.016666666666666666;
        double r5900711 = 5.0;
        double r5900712 = pow(r5900700, r5900711);
        double r5900713 = 0.3333333333333333;
        double r5900714 = r5900700 * r5900700;
        double r5900715 = fma(r5900713, r5900714, r5900705);
        double r5900716 = r5900715 * r5900700;
        double r5900717 = fma(r5900710, r5900712, r5900716);
        double r5900718 = r5900717 / r5900705;
        double r5900719 = sin(r5900707);
        double r5900720 = r5900718 * r5900719;
        double r5900721 = /* ERROR: no complex support in C */;
        double r5900722 = /* ERROR: no complex support in C */;
        return r5900722;
}

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

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019128 +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)))))