Average Error: 0.0 → 0.0
Time: 23.7s
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 r456850 = x;
        double r456851 = exp(r456850);
        double r456852 = -r456850;
        double r456853 = exp(r456852);
        double r456854 = r456851 + r456853;
        double r456855 = 2.0;
        double r456856 = r456854 / r456855;
        double r456857 = y;
        double r456858 = cos(r456857);
        double r456859 = r456856 * r456858;
        double r456860 = r456851 - r456853;
        double r456861 = r456860 / r456855;
        double r456862 = sin(r456857);
        double r456863 = r456861 * r456862;
        double r456864 = /* ERROR: no complex support in C */;
        double r456865 = /* ERROR: no complex support in C */;
        return r456865;
}
↓
double f(double x, double y) {
        double r456866 = y;
        double r456867 = cos(r456866);
        double r456868 = x;
        double r456869 = exp(r456868);
        double r456870 = r456867 / r456869;
        double r456871 = r456869 * r456867;
        double r456872 = r456870 + r456871;
        double r456873 = 0.5;
        double r456874 = r456872 * r456873;
        return r456874;
}

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 2019146 
(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)))))