Average Error: 43.2 → 0.8
Time: 50.0s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}{2} \cdot \sin y i\right))\]
\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}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r51709 = x;
        double r51710 = exp(r51709);
        double r51711 = -r51709;
        double r51712 = exp(r51711);
        double r51713 = r51710 + r51712;
        double r51714 = 2.0;
        double r51715 = r51713 / r51714;
        double r51716 = y;
        double r51717 = cos(r51716);
        double r51718 = r51715 * r51717;
        double r51719 = r51710 - r51712;
        double r51720 = r51719 / r51714;
        double r51721 = sin(r51716);
        double r51722 = r51720 * r51721;
        double r51723 = /* ERROR: no complex support in C */;
        double r51724 = /* ERROR: no complex support in C */;
        return r51724;
}
↓
double f(double x, double y) {
        double r51725 = x;
        double r51726 = exp(r51725);
        double r51727 = -r51725;
        double r51728 = exp(r51727);
        double r51729 = r51726 + r51728;
        double r51730 = 2.0;
        double r51731 = r51729 / r51730;
        double r51732 = y;
        double r51733 = cos(r51732);
        double r51734 = r51731 * r51733;
        double r51735 = 0.3333333333333333;
        double r51736 = 3.0;
        double r51737 = pow(r51725, r51736);
        double r51738 = 0.016666666666666666;
        double r51739 = 5.0;
        double r51740 = pow(r51725, r51739);
        double r51741 = 2.0;
        double r51742 = r51741 * r51725;
        double r51743 = fma(r51738, r51740, r51742);
        double r51744 = fma(r51735, r51737, r51743);
        double r51745 = r51744 / r51730;
        double r51746 = sin(r51732);
        double r51747 = r51745 * r51746;
        double r51748 = /* ERROR: no complex support in C */;
        double r51749 = /* ERROR: no complex support in C */;
        return r51749;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.2

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

  2. Taylor expanded around 0 0.8

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

  3. Simplified0.8

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (x y)
  :name "Euler formula imaginary part (p55)"
  :precision binary64
  (im (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y)))))