Average Error: 43.3 → 0.7
Time: 33.8s
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 r47471 = x;
        double r47472 = exp(r47471);
        double r47473 = -r47471;
        double r47474 = exp(r47473);
        double r47475 = r47472 + r47474;
        double r47476 = 2.0;
        double r47477 = r47475 / r47476;
        double r47478 = y;
        double r47479 = cos(r47478);
        double r47480 = r47477 * r47479;
        double r47481 = r47472 - r47474;
        double r47482 = r47481 / r47476;
        double r47483 = sin(r47478);
        double r47484 = r47482 * r47483;
        double r47485 = /* ERROR: no complex support in C */;
        double r47486 = /* ERROR: no complex support in C */;
        return r47486;
}
↓
double f(double x, double y) {
        double r47487 = 0.3333333333333333;
        double r47488 = x;
        double r47489 = 3.0;
        double r47490 = pow(r47488, r47489);
        double r47491 = 0.016666666666666666;
        double r47492 = 5.0;
        double r47493 = pow(r47488, r47492);
        double r47494 = 2.0;
        double r47495 = r47494 * r47488;
        double r47496 = fma(r47491, r47493, r47495);
        double r47497 = fma(r47487, r47490, r47496);
        double r47498 = 2.0;
        double r47499 = r47497 / r47498;
        double r47500 = y;
        double r47501 = sin(r47500);
        double r47502 = r47499 * r47501;
        return r47502;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.3

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

  2. Simplified43.3

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

  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 2019326 +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)))))