Average Error: 43.0 → 0.8
Time: 33.2s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \frac{1}{60} \cdot {x}^{5}\right)\right)}{2} \cdot \sin y i\right))\]
\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(2, x, \mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \frac{1}{60} \cdot {x}^{5}\right)\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r66528 = x;
        double r66529 = exp(r66528);
        double r66530 = -r66528;
        double r66531 = exp(r66530);
        double r66532 = r66529 + r66531;
        double r66533 = 2.0;
        double r66534 = r66532 / r66533;
        double r66535 = y;
        double r66536 = cos(r66535);
        double r66537 = r66534 * r66536;
        double r66538 = r66529 - r66531;
        double r66539 = r66538 / r66533;
        double r66540 = sin(r66535);
        double r66541 = r66539 * r66540;
        double r66542 = /* ERROR: no complex support in C */;
        double r66543 = /* ERROR: no complex support in C */;
        return r66543;
}
↓
double f(double x, double y) {
        double r66544 = x;
        double r66545 = exp(r66544);
        double r66546 = -r66544;
        double r66547 = exp(r66546);
        double r66548 = r66545 + r66547;
        double r66549 = 2.0;
        double r66550 = r66548 / r66549;
        double r66551 = y;
        double r66552 = cos(r66551);
        double r66553 = r66550 * r66552;
        double r66554 = 2.0;
        double r66555 = 0.3333333333333333;
        double r66556 = 3.0;
        double r66557 = pow(r66544, r66556);
        double r66558 = 0.016666666666666666;
        double r66559 = 5.0;
        double r66560 = pow(r66544, r66559);
        double r66561 = r66558 * r66560;
        double r66562 = fma(r66555, r66557, r66561);
        double r66563 = fma(r66554, r66544, r66562);
        double r66564 = r66563 / r66549;
        double r66565 = sin(r66551);
        double r66566 = r66564 * r66565;
        double r66567 = /* ERROR: no complex support in C */;
        double r66568 = /* ERROR: no complex support in C */;
        return r66568;
}

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. 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. Simplified0.8

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

  4. Final simplification0.8

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

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)))))