Average Error: 43.4 → 0.8
Time: 30.0s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}{2} \cdot \sin y\]
\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\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
double f(double x, double y) {
        double r66549 = x;
        double r66550 = exp(r66549);
        double r66551 = -r66549;
        double r66552 = exp(r66551);
        double r66553 = r66550 + r66552;
        double r66554 = 2.0;
        double r66555 = r66553 / r66554;
        double r66556 = y;
        double r66557 = cos(r66556);
        double r66558 = r66555 * r66557;
        double r66559 = r66550 - r66552;
        double r66560 = r66559 / r66554;
        double r66561 = sin(r66556);
        double r66562 = r66560 * r66561;
        double r66563 = /* ERROR: no complex support in C */;
        double r66564 = /* ERROR: no complex support in C */;
        return r66564;
}
↓
double f(double x, double y) {
        double r66565 = 0.3333333333333333;
        double r66566 = x;
        double r66567 = 3.0;
        double r66568 = pow(r66566, r66567);
        double r66569 = 0.016666666666666666;
        double r66570 = 5.0;
        double r66571 = pow(r66566, r66570);
        double r66572 = 2.0;
        double r66573 = r66572 * r66566;
        double r66574 = fma(r66569, r66571, r66573);
        double r66575 = fma(r66565, r66568, r66574);
        double r66576 = 2.0;
        double r66577 = r66575 / r66576;
        double r66578 = y;
        double r66579 = sin(r66578);
        double r66580 = r66577 * r66579;
        return r66580;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.4

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

  2. Simplified43.4

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

  3. Taylor expanded around 0 0.8

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

  4. Simplified0.8

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

  5. Final simplification0.8

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

Reproduce

herbie shell --seed 2019198 +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)))))