Average Error: 43.3 → 0.7
Time: 33.0s
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 r53722 = x;
        double r53723 = exp(r53722);
        double r53724 = -r53722;
        double r53725 = exp(r53724);
        double r53726 = r53723 + r53725;
        double r53727 = 2.0;
        double r53728 = r53726 / r53727;
        double r53729 = y;
        double r53730 = cos(r53729);
        double r53731 = r53728 * r53730;
        double r53732 = r53723 - r53725;
        double r53733 = r53732 / r53727;
        double r53734 = sin(r53729);
        double r53735 = r53733 * r53734;
        double r53736 = /* ERROR: no complex support in C */;
        double r53737 = /* ERROR: no complex support in C */;
        return r53737;
}
↓
double f(double x, double y) {
        double r53738 = 0.3333333333333333;
        double r53739 = x;
        double r53740 = 3.0;
        double r53741 = pow(r53739, r53740);
        double r53742 = r53738 * r53741;
        double r53743 = 0.016666666666666666;
        double r53744 = 5.0;
        double r53745 = pow(r53739, r53744);
        double r53746 = r53743 * r53745;
        double r53747 = 2.0;
        double r53748 = r53747 * r53739;
        double r53749 = r53746 + r53748;
        double r53750 = r53742 + r53749;
        double r53751 = 2.0;
        double r53752 = r53750 / r53751;
        double r53753 = y;
        double r53754 = sin(r53753);
        double r53755 = r53752 * r53754;
        return r53755;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.3

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

  2. Simplified43.3

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

  3. Taylor expanded around 0 0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019326 
(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)))))