Average Error: 43.8 → 0.9
Time: 14.3s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}{2} \cdot \sin y i\right))\]
\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \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 i\right))
double f(double x, double y) {
        double r47489 = x;
        double r47490 = exp(r47489);
        double r47491 = -r47489;
        double r47492 = exp(r47491);
        double r47493 = r47490 + r47492;
        double r47494 = 2.0;
        double r47495 = r47493 / r47494;
        double r47496 = y;
        double r47497 = cos(r47496);
        double r47498 = r47495 * r47497;
        double r47499 = r47490 - r47492;
        double r47500 = r47499 / r47494;
        double r47501 = sin(r47496);
        double r47502 = r47500 * r47501;
        double r47503 = /* ERROR: no complex support in C */;
        double r47504 = /* ERROR: no complex support in C */;
        return r47504;
}
↓
double f(double x, double y) {
        double r47505 = x;
        double r47506 = exp(r47505);
        double r47507 = -r47505;
        double r47508 = exp(r47507);
        double r47509 = r47506 + r47508;
        double r47510 = 2.0;
        double r47511 = r47509 / r47510;
        double r47512 = y;
        double r47513 = cos(r47512);
        double r47514 = r47511 * r47513;
        double r47515 = 0.3333333333333333;
        double r47516 = 3.0;
        double r47517 = pow(r47505, r47516);
        double r47518 = 0.016666666666666666;
        double r47519 = 5.0;
        double r47520 = pow(r47505, r47519);
        double r47521 = 2.0;
        double r47522 = r47521 * r47505;
        double r47523 = fma(r47518, r47520, r47522);
        double r47524 = fma(r47515, r47517, r47523);
        double r47525 = r47524 / r47510;
        double r47526 = sin(r47512);
        double r47527 = r47525 * r47526;
        double r47528 = /* ERROR: no complex support in C */;
        double r47529 = /* ERROR: no complex support in C */;
        return r47529;
}

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. Taylor expanded around 0 0.9

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

  3. Simplified0.9

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

  4. Final simplification0.9

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

Reproduce

herbie shell --seed 2019353 +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)))))