Average Error: 0.0 → 0.0
Time: 19.3s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\left(\frac{\cos y}{e^{x}} + e^{x} \cdot \cos y\right) \cdot \frac{1}{2}\]
\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 r834859 = x;
        double r834860 = exp(r834859);
        double r834861 = -r834859;
        double r834862 = exp(r834861);
        double r834863 = r834860 + r834862;
        double r834864 = 2.0;
        double r834865 = r834863 / r834864;
        double r834866 = y;
        double r834867 = cos(r834866);
        double r834868 = r834865 * r834867;
        double r834869 = r834860 - r834862;
        double r834870 = r834869 / r834864;
        double r834871 = sin(r834866);
        double r834872 = r834870 * r834871;
        double r834873 = /* ERROR: no complex support in C */;
        double r834874 = /* ERROR: no complex support in C */;
        return r834874;
}
↓
double f(double x, double y) {
        double r834875 = y;
        double r834876 = cos(r834875);
        double r834877 = x;
        double r834878 = exp(r834877);
        double r834879 = r834876 / r834878;
        double r834880 = r834878 * r834876;
        double r834881 = r834879 + r834880;
        double r834882 = 0.5;
        double r834883 = r834881 * r834882;
        return r834883;
}

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 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)))))