Average Error: 43.8 → 0.8
Time: 31.0s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2} \cdot \sin y\]
\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2} \cdot \sin y
double f(double x, double y) {
        double r53590 = x;
        double r53591 = exp(r53590);
        double r53592 = -r53590;
        double r53593 = exp(r53592);
        double r53594 = r53591 + r53593;
        double r53595 = 2.0;
        double r53596 = r53594 / r53595;
        double r53597 = y;
        double r53598 = cos(r53597);
        double r53599 = r53596 * r53598;
        double r53600 = r53591 - r53593;
        double r53601 = r53600 / r53595;
        double r53602 = sin(r53597);
        double r53603 = r53601 * r53602;
        double r53604 = /* ERROR: no complex support in C */;
        double r53605 = /* ERROR: no complex support in C */;
        return r53605;
}
↓
double f(double x, double y) {
        double r53606 = 0.3333333333333333;
        double r53607 = x;
        double r53608 = 3.0;
        double r53609 = pow(r53607, r53608);
        double r53610 = r53606 * r53609;
        double r53611 = 0.016666666666666666;
        double r53612 = 5.0;
        double r53613 = pow(r53607, r53612);
        double r53614 = r53611 * r53613;
        double r53615 = 2.0;
        double r53616 = r53615 * r53607;
        double r53617 = r53614 + r53616;
        double r53618 = r53610 + r53617;
        double r53619 = 2.0;
        double r53620 = r53618 / r53619;
        double r53621 = y;
        double r53622 = sin(r53621);
        double r53623 = r53620 * r53622;
        return r53623;
}

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. Simplified43.8

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

  3. Taylor expanded around 0 0.8

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

  4. Final simplification0.8

    \[\leadsto \frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2} \cdot \sin y\]

Reproduce

herbie shell --seed 2019347 
(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)))))