Average Error: 0.0 → 0.0
Time: 28.8s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\left(\frac{\cos y}{e^{x}} + e^{x} \cdot \cos y\right) \cdot \frac{1}{2}\]
\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\left(\frac{\cos y}{e^{x}} + e^{x} \cdot \cos y\right) \cdot \frac{1}{2}
double f(double x, double y) {
        double r944736 = x;
        double r944737 = exp(r944736);
        double r944738 = -r944736;
        double r944739 = exp(r944738);
        double r944740 = r944737 + r944739;
        double r944741 = 2.0;
        double r944742 = r944740 / r944741;
        double r944743 = y;
        double r944744 = cos(r944743);
        double r944745 = r944742 * r944744;
        double r944746 = r944737 - r944739;
        double r944747 = r944746 / r944741;
        double r944748 = sin(r944743);
        double r944749 = r944747 * r944748;
        double r944750 = /* ERROR: no complex support in C */;
        double r944751 = /* ERROR: no complex support in C */;
        return r944751;
}
↓
double f(double x, double y) {
        double r944752 = y;
        double r944753 = cos(r944752);
        double r944754 = x;
        double r944755 = exp(r944754);
        double r944756 = r944753 / r944755;
        double r944757 = r944755 * r944753;
        double r944758 = r944756 + r944757;
        double r944759 = 0.5;
        double r944760 = r944758 * r944759;
        return r944760;
}

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

    \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\frac{\cos y}{e^{x}} + e^{x} \cdot \cos y\right)}\]

  3. Final simplification0.0

    \[\leadsto \left(\frac{\cos y}{e^{x}} + e^{x} \cdot \cos y\right) \cdot \frac{1}{2}\]

Reproduce

herbie shell --seed 2019144 
(FPCore (x y)
  :name "Euler formula real part (p55)"
  (re (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y)))))