Average Error: 43.7 → 0.8
Time: 29.8s
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 r50664 = x;
        double r50665 = exp(r50664);
        double r50666 = -r50664;
        double r50667 = exp(r50666);
        double r50668 = r50665 + r50667;
        double r50669 = 2.0;
        double r50670 = r50668 / r50669;
        double r50671 = y;
        double r50672 = cos(r50671);
        double r50673 = r50670 * r50672;
        double r50674 = r50665 - r50667;
        double r50675 = r50674 / r50669;
        double r50676 = sin(r50671);
        double r50677 = r50675 * r50676;
        double r50678 = /* ERROR: no complex support in C */;
        double r50679 = /* ERROR: no complex support in C */;
        return r50679;
}
↓
double f(double x, double y) {
        double r50680 = 0.3333333333333333;
        double r50681 = x;
        double r50682 = 3.0;
        double r50683 = pow(r50681, r50682);
        double r50684 = r50680 * r50683;
        double r50685 = 0.016666666666666666;
        double r50686 = 5.0;
        double r50687 = pow(r50681, r50686);
        double r50688 = r50685 * r50687;
        double r50689 = 2.0;
        double r50690 = r50689 * r50681;
        double r50691 = r50688 + r50690;
        double r50692 = r50684 + r50691;
        double r50693 = 2.0;
        double r50694 = r50692 / r50693;
        double r50695 = y;
        double r50696 = sin(r50695);
        double r50697 = r50694 * r50696;
        return r50697;
}

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. 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 2019294 
(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)))))