Average Error: 43.5 → 0.9
Time: 32.7s
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 r53361 = x;
        double r53362 = exp(r53361);
        double r53363 = -r53361;
        double r53364 = exp(r53363);
        double r53365 = r53362 + r53364;
        double r53366 = 2.0;
        double r53367 = r53365 / r53366;
        double r53368 = y;
        double r53369 = cos(r53368);
        double r53370 = r53367 * r53369;
        double r53371 = r53362 - r53364;
        double r53372 = r53371 / r53366;
        double r53373 = sin(r53368);
        double r53374 = r53372 * r53373;
        double r53375 = /* ERROR: no complex support in C */;
        double r53376 = /* ERROR: no complex support in C */;
        return r53376;
}
↓
double f(double x, double y) {
        double r53377 = y;
        double r53378 = sin(r53377);
        double r53379 = 2.0;
        double r53380 = r53378 / r53379;
        double r53381 = 2.0;
        double r53382 = x;
        double r53383 = 0.3333333333333333;
        double r53384 = 3.0;
        double r53385 = pow(r53382, r53384);
        double r53386 = 5.0;
        double r53387 = pow(r53382, r53386);
        double r53388 = 0.016666666666666666;
        double r53389 = r53387 * r53388;
        double r53390 = fma(r53383, r53385, r53389);
        double r53391 = fma(r53381, r53382, r53390);
        double r53392 = r53380 * r53391;
        return r53392;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.5

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

  2. Simplified43.5

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

  3. Taylor expanded around 0 0.9

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

  4. Simplified0.9

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

  5. Final simplification0.9

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

Reproduce

herbie shell --seed 2019174 +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)))))