Average Error: 0.0 → 0.0
Time: 23.1s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{e^{x} + e^{-x}}{2} \cdot \cos y\]
\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\frac{e^{x} + e^{-x}}{2} \cdot \cos y
double f(double x, double y) {
        double r44833 = x;
        double r44834 = exp(r44833);
        double r44835 = -r44833;
        double r44836 = exp(r44835);
        double r44837 = r44834 + r44836;
        double r44838 = 2.0;
        double r44839 = r44837 / r44838;
        double r44840 = y;
        double r44841 = cos(r44840);
        double r44842 = r44839 * r44841;
        double r44843 = r44834 - r44836;
        double r44844 = r44843 / r44838;
        double r44845 = sin(r44840);
        double r44846 = r44844 * r44845;
        double r44847 = /* ERROR: no complex support in C */;
        double r44848 = /* ERROR: no complex support in C */;
        return r44848;
}
↓
double f(double x, double y) {
        double r44849 = x;
        double r44850 = exp(r44849);
        double r44851 = -r44849;
        double r44852 = exp(r44851);
        double r44853 = r44850 + r44852;
        double r44854 = 2.0;
        double r44855 = r44853 / r44854;
        double r44856 = y;
        double r44857 = cos(r44856);
        double r44858 = r44855 * r44857;
        return r44858;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019323 +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)))))