Average Error: 43.6 → 0.8
Time: 38.7s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\frac{1}{60} \cdot {x}^{5} + \left(\left(x + x\right) + \left(\left(x \cdot x\right) \cdot \frac{1}{3}\right) \cdot x\right)}{2} \cdot \sin y i\right))\]
\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{\frac{1}{60} \cdot {x}^{5} + \left(\left(x + x\right) + \left(\left(x \cdot x\right) \cdot \frac{1}{3}\right) \cdot x\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r993662 = x;
        double r993663 = exp(r993662);
        double r993664 = -r993662;
        double r993665 = exp(r993664);
        double r993666 = r993663 + r993665;
        double r993667 = 2.0;
        double r993668 = r993666 / r993667;
        double r993669 = y;
        double r993670 = cos(r993669);
        double r993671 = r993668 * r993670;
        double r993672 = r993663 - r993665;
        double r993673 = r993672 / r993667;
        double r993674 = sin(r993669);
        double r993675 = r993673 * r993674;
        double r993676 = /* ERROR: no complex support in C */;
        double r993677 = /* ERROR: no complex support in C */;
        return r993677;
}
↓
double f(double x, double y) {
        double r993678 = x;
        double r993679 = exp(r993678);
        double r993680 = -r993678;
        double r993681 = exp(r993680);
        double r993682 = r993679 + r993681;
        double r993683 = 2.0;
        double r993684 = r993682 / r993683;
        double r993685 = y;
        double r993686 = cos(r993685);
        double r993687 = r993684 * r993686;
        double r993688 = 0.016666666666666666;
        double r993689 = 5.0;
        double r993690 = pow(r993678, r993689);
        double r993691 = r993688 * r993690;
        double r993692 = r993678 + r993678;
        double r993693 = r993678 * r993678;
        double r993694 = 0.3333333333333333;
        double r993695 = r993693 * r993694;
        double r993696 = r993695 * r993678;
        double r993697 = r993692 + r993696;
        double r993698 = r993691 + r993697;
        double r993699 = r993698 / r993683;
        double r993700 = sin(r993685);
        double r993701 = r993699 * r993700;
        double r993702 = /* ERROR: no complex support in C */;
        double r993703 = /* ERROR: no complex support in C */;
        return r993703;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.6

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

  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

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2019153 
(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)))))