Average Error: 43.7 → 0.8
Time: 13.1s
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 r45523 = x;
        double r45524 = exp(r45523);
        double r45525 = -r45523;
        double r45526 = exp(r45525);
        double r45527 = r45524 + r45526;
        double r45528 = 2.0;
        double r45529 = r45527 / r45528;
        double r45530 = y;
        double r45531 = cos(r45530);
        double r45532 = r45529 * r45531;
        double r45533 = r45524 - r45526;
        double r45534 = r45533 / r45528;
        double r45535 = sin(r45530);
        double r45536 = r45534 * r45535;
        double r45537 = /* ERROR: no complex support in C */;
        double r45538 = /* ERROR: no complex support in C */;
        return r45538;
}
↓
double f(double x, double y) {
        double r45539 = x;
        double r45540 = exp(r45539);
        double r45541 = -r45539;
        double r45542 = exp(r45541);
        double r45543 = r45540 + r45542;
        double r45544 = 2.0;
        double r45545 = r45543 / r45544;
        double r45546 = y;
        double r45547 = cos(r45546);
        double r45548 = r45545 * r45547;
        double r45549 = 0.3333333333333333;
        double r45550 = 3.0;
        double r45551 = pow(r45539, r45550);
        double r45552 = 0.016666666666666666;
        double r45553 = 5.0;
        double r45554 = pow(r45539, r45553);
        double r45555 = 2.0;
        double r45556 = r45555 * r45539;
        double r45557 = fma(r45552, r45554, r45556);
        double r45558 = fma(r45549, r45551, r45557);
        double r45559 = r45558 / r45544;
        double r45560 = sin(r45546);
        double r45561 = r45559 * r45560;
        double r45562 = /* ERROR: no complex support in C */;
        double r45563 = /* ERROR: no complex support in C */;
        return r45563;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.7

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

  2. Taylor expanded around 0 0.8

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

  3. Simplified0.8

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2020021 +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)))))