Average Error: 43.6 → 0.7
Time: 12.8s
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 r68682 = x;
        double r68683 = exp(r68682);
        double r68684 = -r68682;
        double r68685 = exp(r68684);
        double r68686 = r68683 + r68685;
        double r68687 = 2.0;
        double r68688 = r68686 / r68687;
        double r68689 = y;
        double r68690 = cos(r68689);
        double r68691 = r68688 * r68690;
        double r68692 = r68683 - r68685;
        double r68693 = r68692 / r68687;
        double r68694 = sin(r68689);
        double r68695 = r68693 * r68694;
        double r68696 = /* ERROR: no complex support in C */;
        double r68697 = /* ERROR: no complex support in C */;
        return r68697;
}
↓
double f(double x, double y) {
        double r68698 = x;
        double r68699 = exp(r68698);
        double r68700 = -r68698;
        double r68701 = exp(r68700);
        double r68702 = r68699 + r68701;
        double r68703 = 2.0;
        double r68704 = r68702 / r68703;
        double r68705 = y;
        double r68706 = cos(r68705);
        double r68707 = r68704 * r68706;
        double r68708 = 0.3333333333333333;
        double r68709 = 3.0;
        double r68710 = pow(r68698, r68709);
        double r68711 = 0.016666666666666666;
        double r68712 = 5.0;
        double r68713 = pow(r68698, r68712);
        double r68714 = 2.0;
        double r68715 = r68714 * r68698;
        double r68716 = fma(r68711, r68713, r68715);
        double r68717 = fma(r68708, r68710, r68716);
        double r68718 = r68717 / r68703;
        double r68719 = sin(r68705);
        double r68720 = r68718 * r68719;
        double r68721 = /* ERROR: no complex support in C */;
        double r68722 = /* ERROR: no complex support in C */;
        return r68722;
}

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 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019346 +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)))))