Average Error: 0.0 → 0.0
Time: 12.3s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\cos y \cdot \frac{e^{x} + e^{-x}}{2}\]
\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\cos y \cdot \frac{e^{x} + e^{-x}}{2}
double f(double x, double y) {
        double r23032 = x;
        double r23033 = exp(r23032);
        double r23034 = -r23032;
        double r23035 = exp(r23034);
        double r23036 = r23033 + r23035;
        double r23037 = 2.0;
        double r23038 = r23036 / r23037;
        double r23039 = y;
        double r23040 = cos(r23039);
        double r23041 = r23038 * r23040;
        double r23042 = r23033 - r23035;
        double r23043 = r23042 / r23037;
        double r23044 = sin(r23039);
        double r23045 = r23043 * r23044;
        double r23046 = /* ERROR: no complex support in C */;
        double r23047 = /* ERROR: no complex support in C */;
        return r23047;
}
↓
double f(double x, double y) {
        double r23048 = y;
        double r23049 = cos(r23048);
        double r23050 = x;
        double r23051 = exp(r23050);
        double r23052 = -r23050;
        double r23053 = exp(r23052);
        double r23054 = r23051 + r23053;
        double r23055 = 2.0;
        double r23056 = r23054 / r23055;
        double r23057 = r23049 * r23056;
        return r23057;
}

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

    \[\leadsto \color{blue}{\cos y \cdot \frac{e^{x} + e^{-x}}{2}}\]

  3. Final simplification0.0

    \[\leadsto \cos y \cdot \frac{e^{x} + e^{-x}}{2}\]

Reproduce

herbie shell --seed 2020042 +o rules:numerics
(FPCore (x y)
  :name "Euler formula real part (p55)"
  :precision binary64
  (re (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y)))))