Average Error: 43.1 → 0.8
Time: 32.4s
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 r46705 = x;
        double r46706 = exp(r46705);
        double r46707 = -r46705;
        double r46708 = exp(r46707);
        double r46709 = r46706 + r46708;
        double r46710 = 2.0;
        double r46711 = r46709 / r46710;
        double r46712 = y;
        double r46713 = cos(r46712);
        double r46714 = r46711 * r46713;
        double r46715 = r46706 - r46708;
        double r46716 = r46715 / r46710;
        double r46717 = sin(r46712);
        double r46718 = r46716 * r46717;
        double r46719 = /* ERROR: no complex support in C */;
        double r46720 = /* ERROR: no complex support in C */;
        return r46720;
}
↓
double f(double x, double y) {
        double r46721 = 0.3333333333333333;
        double r46722 = x;
        double r46723 = 3.0;
        double r46724 = pow(r46722, r46723);
        double r46725 = r46721 * r46724;
        double r46726 = 0.016666666666666666;
        double r46727 = 5.0;
        double r46728 = pow(r46722, r46727);
        double r46729 = r46726 * r46728;
        double r46730 = 2.0;
        double r46731 = r46730 * r46722;
        double r46732 = r46729 + r46731;
        double r46733 = r46725 + r46732;
        double r46734 = 2.0;
        double r46735 = r46733 / r46734;
        double r46736 = y;
        double r46737 = sin(r46736);
        double r46738 = r46735 * r46737;
        return r46738;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.1

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

  2. Simplified43.1

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

  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 2019303 
(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)))))