Average Error: 0.0 → 0.0
Time: 17.4s
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 r713973 = x;
        double r713974 = exp(r713973);
        double r713975 = -r713973;
        double r713976 = exp(r713975);
        double r713977 = r713974 + r713976;
        double r713978 = 2.0;
        double r713979 = r713977 / r713978;
        double r713980 = y;
        double r713981 = cos(r713980);
        double r713982 = r713979 * r713981;
        double r713983 = r713974 - r713976;
        double r713984 = r713983 / r713978;
        double r713985 = sin(r713980);
        double r713986 = r713984 * r713985;
        double r713987 = /* ERROR: no complex support in C */;
        double r713988 = /* ERROR: no complex support in C */;
        return r713988;
}
↓
double f(double x, double y) {
        double r713989 = x;
        double r713990 = exp(r713989);
        double r713991 = y;
        double r713992 = cos(r713991);
        double r713993 = r713990 * r713992;
        double r713994 = r713992 / r713990;
        double r713995 = r713993 + r713994;
        double r713996 = 2.0;
        double r713997 = r713995 / r713996;
        return r713997;
}

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 2019153 
(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)))))