Average Error: 43.7 → 0.8
Time: 20.4s
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 r42328 = x;
        double r42329 = exp(r42328);
        double r42330 = -r42328;
        double r42331 = exp(r42330);
        double r42332 = r42329 + r42331;
        double r42333 = 2.0;
        double r42334 = r42332 / r42333;
        double r42335 = y;
        double r42336 = cos(r42335);
        double r42337 = r42334 * r42336;
        double r42338 = r42329 - r42331;
        double r42339 = r42338 / r42333;
        double r42340 = sin(r42335);
        double r42341 = r42339 * r42340;
        double r42342 = /* ERROR: no complex support in C */;
        double r42343 = /* ERROR: no complex support in C */;
        return r42343;
}
↓
double f(double x, double y) {
        double r42344 = 0.3333333333333333;
        double r42345 = x;
        double r42346 = 3.0;
        double r42347 = pow(r42345, r42346);
        double r42348 = 0.016666666666666666;
        double r42349 = 5.0;
        double r42350 = pow(r42345, r42349);
        double r42351 = 2.0;
        double r42352 = r42351 * r42345;
        double r42353 = fma(r42348, r42350, r42352);
        double r42354 = fma(r42344, r42347, r42353);
        double r42355 = 2.0;
        double r42356 = r42354 / r42355;
        double r42357 = y;
        double r42358 = sin(r42357);
        double r42359 = r42356 * r42358;
        return r42359;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.7

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

  2. Simplified43.7

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

  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 2019351 +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)))))