Average Error: 43.9 → 0.8
Time: 34.7s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\mathsf{fma}\left(\frac{1}{60}, {x}^{5}, x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{3} + 2\right)\right)}{2} \cdot \sin y i\right))\]
\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}{60}, {x}^{5}, x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{3} + 2\right)\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r1279521 = x;
        double r1279522 = exp(r1279521);
        double r1279523 = -r1279521;
        double r1279524 = exp(r1279523);
        double r1279525 = r1279522 + r1279524;
        double r1279526 = 2.0;
        double r1279527 = r1279525 / r1279526;
        double r1279528 = y;
        double r1279529 = cos(r1279528);
        double r1279530 = r1279527 * r1279529;
        double r1279531 = r1279522 - r1279524;
        double r1279532 = r1279531 / r1279526;
        double r1279533 = sin(r1279528);
        double r1279534 = r1279532 * r1279533;
        double r1279535 = /* ERROR: no complex support in C */;
        double r1279536 = /* ERROR: no complex support in C */;
        return r1279536;
}
↓
double f(double x, double y) {
        double r1279537 = x;
        double r1279538 = exp(r1279537);
        double r1279539 = -r1279537;
        double r1279540 = exp(r1279539);
        double r1279541 = r1279538 + r1279540;
        double r1279542 = 2.0;
        double r1279543 = r1279541 / r1279542;
        double r1279544 = y;
        double r1279545 = cos(r1279544);
        double r1279546 = r1279543 * r1279545;
        double r1279547 = 0.016666666666666666;
        double r1279548 = 5.0;
        double r1279549 = pow(r1279537, r1279548);
        double r1279550 = r1279537 * r1279537;
        double r1279551 = 0.3333333333333333;
        double r1279552 = r1279550 * r1279551;
        double r1279553 = r1279552 + r1279542;
        double r1279554 = r1279537 * r1279553;
        double r1279555 = fma(r1279547, r1279549, r1279554);
        double r1279556 = r1279555 / r1279542;
        double r1279557 = sin(r1279544);
        double r1279558 = r1279556 * r1279557;
        double r1279559 = /* ERROR: no complex support in C */;
        double r1279560 = /* ERROR: no complex support in C */;
        return r1279560;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.9

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

  2. Taylor expanded around 0 0.8

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

  3. Simplified0.8

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(FPCore (x y)
  :name "Euler formula imaginary part (p55)"
  (im (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y)))))