Average Error: 43.9 → 0.8
Time: 40.3s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\mathsf{fma}\left(x, 2, \mathsf{fma}\left({x}^{5}, \frac{1}{60}, \frac{1}{3} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right)}{2} \cdot \sin y i\right))\]
\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(x, 2, \mathsf{fma}\left({x}^{5}, \frac{1}{60}, \frac{1}{3} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r1008407 = x;
        double r1008408 = exp(r1008407);
        double r1008409 = -r1008407;
        double r1008410 = exp(r1008409);
        double r1008411 = r1008408 + r1008410;
        double r1008412 = 2.0;
        double r1008413 = r1008411 / r1008412;
        double r1008414 = y;
        double r1008415 = cos(r1008414);
        double r1008416 = r1008413 * r1008415;
        double r1008417 = r1008408 - r1008410;
        double r1008418 = r1008417 / r1008412;
        double r1008419 = sin(r1008414);
        double r1008420 = r1008418 * r1008419;
        double r1008421 = /* ERROR: no complex support in C */;
        double r1008422 = /* ERROR: no complex support in C */;
        return r1008422;
}
↓
double f(double x, double y) {
        double r1008423 = x;
        double r1008424 = exp(r1008423);
        double r1008425 = -r1008423;
        double r1008426 = exp(r1008425);
        double r1008427 = r1008424 + r1008426;
        double r1008428 = 2.0;
        double r1008429 = r1008427 / r1008428;
        double r1008430 = y;
        double r1008431 = cos(r1008430);
        double r1008432 = r1008429 * r1008431;
        double r1008433 = 5.0;
        double r1008434 = pow(r1008423, r1008433);
        double r1008435 = 0.016666666666666666;
        double r1008436 = 0.3333333333333333;
        double r1008437 = r1008423 * r1008423;
        double r1008438 = r1008437 * r1008423;
        double r1008439 = r1008436 * r1008438;
        double r1008440 = fma(r1008434, r1008435, r1008439);
        double r1008441 = fma(r1008423, r1008428, r1008440);
        double r1008442 = r1008441 / r1008428;
        double r1008443 = sin(r1008430);
        double r1008444 = r1008442 * r1008443;
        double r1008445 = /* ERROR: no complex support in C */;
        double r1008446 = /* ERROR: no complex support in C */;
        return r1008446;
}

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

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

  4. Final simplification0.8

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

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)))))