Average Error: 43.0 → 0.8
Time: 28.5s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{\sin y}{2} \cdot \mathsf{fma}\left(2, x, \mathsf{fma}\left(\frac{1}{3}, {x}^{3}, {x}^{5} \cdot \frac{1}{60}\right)\right)\]
\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\frac{\sin y}{2} \cdot \mathsf{fma}\left(2, x, \mathsf{fma}\left(\frac{1}{3}, {x}^{3}, {x}^{5} \cdot \frac{1}{60}\right)\right)
double f(double x, double y) {
        double r52500 = x;
        double r52501 = exp(r52500);
        double r52502 = -r52500;
        double r52503 = exp(r52502);
        double r52504 = r52501 + r52503;
        double r52505 = 2.0;
        double r52506 = r52504 / r52505;
        double r52507 = y;
        double r52508 = cos(r52507);
        double r52509 = r52506 * r52508;
        double r52510 = r52501 - r52503;
        double r52511 = r52510 / r52505;
        double r52512 = sin(r52507);
        double r52513 = r52511 * r52512;
        double r52514 = /* ERROR: no complex support in C */;
        double r52515 = /* ERROR: no complex support in C */;
        return r52515;
}
↓
double f(double x, double y) {
        double r52516 = y;
        double r52517 = sin(r52516);
        double r52518 = 2.0;
        double r52519 = r52517 / r52518;
        double r52520 = 2.0;
        double r52521 = x;
        double r52522 = 0.3333333333333333;
        double r52523 = 3.0;
        double r52524 = pow(r52521, r52523);
        double r52525 = 5.0;
        double r52526 = pow(r52521, r52525);
        double r52527 = 0.016666666666666666;
        double r52528 = r52526 * r52527;
        double r52529 = fma(r52522, r52524, r52528);
        double r52530 = fma(r52520, r52521, r52529);
        double r52531 = r52519 * r52530;
        return r52531;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.0

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

  2. Simplified43.0

    \[\leadsto \color{blue}{\left(e^{x} - e^{-x}\right) \cdot \frac{\sin y}{2}}\]

  3. Taylor expanded around 0 0.8

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

  4. Simplified0.8

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

  5. Final simplification0.8

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

Reproduce

herbie shell --seed 2019195 +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)))))