Average Error: 0.0 → 0.0
Time: 14.9s
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 r1917056 = x;
        double r1917057 = exp(r1917056);
        double r1917058 = -r1917056;
        double r1917059 = exp(r1917058);
        double r1917060 = r1917057 + r1917059;
        double r1917061 = 2.0;
        double r1917062 = r1917060 / r1917061;
        double r1917063 = y;
        double r1917064 = cos(r1917063);
        double r1917065 = r1917062 * r1917064;
        double r1917066 = r1917057 - r1917059;
        double r1917067 = r1917066 / r1917061;
        double r1917068 = sin(r1917063);
        double r1917069 = r1917067 * r1917068;
        double r1917070 = /* ERROR: no complex support in C */;
        double r1917071 = /* ERROR: no complex support in C */;
        return r1917071;
}
↓
double f(double x, double y) {
        double r1917072 = x;
        double r1917073 = exp(r1917072);
        double r1917074 = y;
        double r1917075 = cos(r1917074);
        double r1917076 = r1917073 * r1917075;
        double r1917077 = r1917075 / r1917073;
        double r1917078 = r1917076 + r1917077;
        double r1917079 = 2.0;
        double r1917080 = r1917078 / r1917079;
        return r1917080;
}

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 2019172 
(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)))))