Average Error: 43.5 → 0.9
Time: 37.9s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\mathsf{fma}\left({x}^{5}, \frac{1}{60}, x \cdot \left(2 + \frac{1}{3} \cdot \left(x \cdot x\right)\right)\right)}{2} \cdot \sin y i\right))\]
\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({x}^{5}, \frac{1}{60}, x \cdot \left(2 + \frac{1}{3} \cdot \left(x \cdot x\right)\right)\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r2081712 = x;
        double r2081713 = exp(r2081712);
        double r2081714 = -r2081712;
        double r2081715 = exp(r2081714);
        double r2081716 = r2081713 + r2081715;
        double r2081717 = 2.0;
        double r2081718 = r2081716 / r2081717;
        double r2081719 = y;
        double r2081720 = cos(r2081719);
        double r2081721 = r2081718 * r2081720;
        double r2081722 = r2081713 - r2081715;
        double r2081723 = r2081722 / r2081717;
        double r2081724 = sin(r2081719);
        double r2081725 = r2081723 * r2081724;
        double r2081726 = /* ERROR: no complex support in C */;
        double r2081727 = /* ERROR: no complex support in C */;
        return r2081727;
}
↓
double f(double x, double y) {
        double r2081728 = x;
        double r2081729 = exp(r2081728);
        double r2081730 = -r2081728;
        double r2081731 = exp(r2081730);
        double r2081732 = r2081729 + r2081731;
        double r2081733 = 2.0;
        double r2081734 = r2081732 / r2081733;
        double r2081735 = y;
        double r2081736 = cos(r2081735);
        double r2081737 = r2081734 * r2081736;
        double r2081738 = 5.0;
        double r2081739 = pow(r2081728, r2081738);
        double r2081740 = 0.016666666666666666;
        double r2081741 = 2.0;
        double r2081742 = 0.3333333333333333;
        double r2081743 = r2081728 * r2081728;
        double r2081744 = r2081742 * r2081743;
        double r2081745 = r2081741 + r2081744;
        double r2081746 = r2081728 * r2081745;
        double r2081747 = fma(r2081739, r2081740, r2081746);
        double r2081748 = r2081747 / r2081733;
        double r2081749 = sin(r2081735);
        double r2081750 = r2081748 * r2081749;
        double r2081751 = /* ERROR: no complex support in C */;
        double r2081752 = /* ERROR: no complex support in C */;
        return r2081752;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.5

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

  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. Simplified0.9

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

  4. Final simplification0.9

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

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(FPCore (x y)
  :name "Euler formula imaginary part (p55)"
  (im (complex (* (/ (+ (exp x) (exp (- x))) 2.0) (cos y)) (* (/ (- (exp x) (exp (- x))) 2.0) (sin y)))))