Average Error: 0.0 → 0.0
Time: 12.3s
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 r593860 = x;
        double r593861 = exp(r593860);
        double r593862 = -r593860;
        double r593863 = exp(r593862);
        double r593864 = r593861 + r593863;
        double r593865 = 2.0;
        double r593866 = r593864 / r593865;
        double r593867 = y;
        double r593868 = cos(r593867);
        double r593869 = r593866 * r593868;
        double r593870 = r593861 - r593863;
        double r593871 = r593870 / r593865;
        double r593872 = sin(r593867);
        double r593873 = r593871 * r593872;
        double r593874 = /* ERROR: no complex support in C */;
        double r593875 = /* ERROR: no complex support in C */;
        return r593875;
}
↓
double f(double x, double y) {
        double r593876 = x;
        double r593877 = exp(r593876);
        double r593878 = y;
        double r593879 = cos(r593878);
        double r593880 = r593877 * r593879;
        double r593881 = r593879 / r593877;
        double r593882 = r593880 + r593881;
        double r593883 = 2.0;
        double r593884 = r593882 / r593883;
        return r593884;
}

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 2019152 
(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)))))