Average Error: 0.0 → 0.0
Time: 9.9s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{e^{x} \cdot \cos y + \frac{\cos y}{e^{x}}}{2}\]
\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\frac{e^{x} \cdot \cos y + \frac{\cos y}{e^{x}}}{2}
double f(double x, double y) {
        double r932793 = x;
        double r932794 = exp(r932793);
        double r932795 = -r932793;
        double r932796 = exp(r932795);
        double r932797 = r932794 + r932796;
        double r932798 = 2.0;
        double r932799 = r932797 / r932798;
        double r932800 = y;
        double r932801 = cos(r932800);
        double r932802 = r932799 * r932801;
        double r932803 = r932794 - r932796;
        double r932804 = r932803 / r932798;
        double r932805 = sin(r932800);
        double r932806 = r932804 * r932805;
        double r932807 = /* ERROR: no complex support in C */;
        double r932808 = /* ERROR: no complex support in C */;
        return r932808;
}
↓
double f(double x, double y) {
        double r932809 = x;
        double r932810 = exp(r932809);
        double r932811 = y;
        double r932812 = cos(r932811);
        double r932813 = r932810 * r932812;
        double r932814 = r932812 / r932810;
        double r932815 = r932813 + r932814;
        double r932816 = 2.0;
        double r932817 = r932815 / r932816;
        return r932817;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019171 
(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)))))