Average Error: 43.4 → 0.8
Time: 10.3s
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 r48420 = x;
        double r48421 = exp(r48420);
        double r48422 = -r48420;
        double r48423 = exp(r48422);
        double r48424 = r48421 + r48423;
        double r48425 = 2.0;
        double r48426 = r48424 / r48425;
        double r48427 = y;
        double r48428 = cos(r48427);
        double r48429 = r48426 * r48428;
        double r48430 = r48421 - r48423;
        double r48431 = r48430 / r48425;
        double r48432 = sin(r48427);
        double r48433 = r48431 * r48432;
        double r48434 = /* ERROR: no complex support in C */;
        double r48435 = /* ERROR: no complex support in C */;
        return r48435;
}
↓
double f(double x, double y) {
        double r48436 = 0.3333333333333333;
        double r48437 = x;
        double r48438 = 3.0;
        double r48439 = pow(r48437, r48438);
        double r48440 = 0.016666666666666666;
        double r48441 = 5.0;
        double r48442 = pow(r48437, r48441);
        double r48443 = 2.0;
        double r48444 = r48443 * r48437;
        double r48445 = fma(r48440, r48442, r48444);
        double r48446 = fma(r48436, r48439, r48445);
        double r48447 = 2.0;
        double r48448 = r48446 / r48447;
        double r48449 = y;
        double r48450 = sin(r48449);
        double r48451 = r48448 * r48450;
        return r48451;
}

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 2020060 +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)))))