Average Error: 0.0 → 0.0
Time: 13.8s
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 r1375048 = x;
        double r1375049 = exp(r1375048);
        double r1375050 = -r1375048;
        double r1375051 = exp(r1375050);
        double r1375052 = r1375049 + r1375051;
        double r1375053 = 2.0;
        double r1375054 = r1375052 / r1375053;
        double r1375055 = y;
        double r1375056 = cos(r1375055);
        double r1375057 = r1375054 * r1375056;
        double r1375058 = r1375049 - r1375051;
        double r1375059 = r1375058 / r1375053;
        double r1375060 = sin(r1375055);
        double r1375061 = r1375059 * r1375060;
        double r1375062 = /* ERROR: no complex support in C */;
        double r1375063 = /* ERROR: no complex support in C */;
        return r1375063;
}
↓
double f(double x, double y) {
        double r1375064 = y;
        double r1375065 = cos(r1375064);
        double r1375066 = x;
        double r1375067 = exp(r1375066);
        double r1375068 = r1375065 / r1375067;
        double r1375069 = r1375067 * r1375065;
        double r1375070 = r1375068 + r1375069;
        double r1375071 = 0.5;
        double r1375072 = r1375070 * r1375071;
        return r1375072;
}

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

    \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\frac{\cos y}{e^{x}} + \cos y \cdot e^{x}\right)}\]

  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 2019168 
(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)))))