Average Error: 0.0 → 0.0
Time: 24.8s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{\frac{\cos y}{e^{x}} + e^{x} \cdot \cos y}{2}\]
\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\frac{\frac{\cos y}{e^{x}} + e^{x} \cdot \cos y}{2}
double f(double x, double y) {
        double r7283705 = x;
        double r7283706 = exp(r7283705);
        double r7283707 = -r7283705;
        double r7283708 = exp(r7283707);
        double r7283709 = r7283706 + r7283708;
        double r7283710 = 2.0;
        double r7283711 = r7283709 / r7283710;
        double r7283712 = y;
        double r7283713 = cos(r7283712);
        double r7283714 = r7283711 * r7283713;
        double r7283715 = r7283706 - r7283708;
        double r7283716 = r7283715 / r7283710;
        double r7283717 = sin(r7283712);
        double r7283718 = r7283716 * r7283717;
        double r7283719 = /* ERROR: no complex support in C */;
        double r7283720 = /* ERROR: no complex support in C */;
        return r7283720;
}
↓
double f(double x, double y) {
        double r7283721 = y;
        double r7283722 = cos(r7283721);
        double r7283723 = x;
        double r7283724 = exp(r7283723);
        double r7283725 = r7283722 / r7283724;
        double r7283726 = r7283724 * r7283722;
        double r7283727 = r7283725 + r7283726;
        double r7283728 = 2.0;
        double r7283729 = r7283727 / r7283728;
        return r7283729;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 0.0

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

  2. Simplified0.0

    \[\leadsto \color{blue}{\frac{\frac{\cos y}{e^{x}} + e^{x} \cdot \cos y}{2}}\]

  3. Final simplification0.0

    \[\leadsto \frac{\frac{\cos y}{e^{x}} + e^{x} \cdot \cos y}{2}\]

Reproduce

herbie shell --seed 2019173 
(FPCore (x y)
  :name "Euler formula real part (p55)"
  (re (complex (* (/ (+ (exp x) (exp (- x))) 2.0) (cos y)) (* (/ (- (exp x) (exp (- x))) 2.0) (sin y)))))