Average Error: 43.8 → 0.7
Time: 10.5s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}{2} \cdot \sin y\]
\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 r59379 = x;
        double r59380 = exp(r59379);
        double r59381 = -r59379;
        double r59382 = exp(r59381);
        double r59383 = r59380 + r59382;
        double r59384 = 2.0;
        double r59385 = r59383 / r59384;
        double r59386 = y;
        double r59387 = cos(r59386);
        double r59388 = r59385 * r59387;
        double r59389 = r59380 - r59382;
        double r59390 = r59389 / r59384;
        double r59391 = sin(r59386);
        double r59392 = r59390 * r59391;
        double r59393 = /* ERROR: no complex support in C */;
        double r59394 = /* ERROR: no complex support in C */;
        return r59394;
}
↓
double f(double x, double y) {
        double r59395 = 0.3333333333333333;
        double r59396 = x;
        double r59397 = 3.0;
        double r59398 = pow(r59396, r59397);
        double r59399 = 0.016666666666666666;
        double r59400 = 5.0;
        double r59401 = pow(r59396, r59400);
        double r59402 = 2.0;
        double r59403 = r59402 * r59396;
        double r59404 = fma(r59399, r59401, r59403);
        double r59405 = fma(r59395, r59398, r59404);
        double r59406 = 2.0;
        double r59407 = r59405 / r59406;
        double r59408 = y;
        double r59409 = sin(r59408);
        double r59410 = r59407 * r59409;
        return r59410;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.8

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

  2. Simplified43.8

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

  3. Taylor expanded around 0 0.7

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

  4. Simplified0.7

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

  5. Final simplification0.7

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

Reproduce

herbie shell --seed 2020018 +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)))))