Average Error: 0.0 → 0.0
Time: 20.5s
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 r697023 = x;
        double r697024 = exp(r697023);
        double r697025 = -r697023;
        double r697026 = exp(r697025);
        double r697027 = r697024 + r697026;
        double r697028 = 2.0;
        double r697029 = r697027 / r697028;
        double r697030 = y;
        double r697031 = cos(r697030);
        double r697032 = r697029 * r697031;
        double r697033 = r697024 - r697026;
        double r697034 = r697033 / r697028;
        double r697035 = sin(r697030);
        double r697036 = r697034 * r697035;
        double r697037 = /* ERROR: no complex support in C */;
        double r697038 = /* ERROR: no complex support in C */;
        return r697038;
}
↓
double f(double x, double y) {
        double r697039 = y;
        double r697040 = cos(r697039);
        double r697041 = x;
        double r697042 = exp(r697041);
        double r697043 = r697040 / r697042;
        double r697044 = r697042 * r697040;
        double r697045 = r697043 + r697044;
        double r697046 = 0.5;
        double r697047 = r697045 * r697046;
        return r697047;
}

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 2019149 
(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)))))