Average Error: 0.0 → 0.0
Time: 36.5s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{e^{x} + e^{-x}}{2} \cdot \cos y\]
\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 r40845 = x;
        double r40846 = exp(r40845);
        double r40847 = -r40845;
        double r40848 = exp(r40847);
        double r40849 = r40846 + r40848;
        double r40850 = 2.0;
        double r40851 = r40849 / r40850;
        double r40852 = y;
        double r40853 = cos(r40852);
        double r40854 = r40851 * r40853;
        double r40855 = r40846 - r40848;
        double r40856 = r40855 / r40850;
        double r40857 = sin(r40852);
        double r40858 = r40856 * r40857;
        double r40859 = /* ERROR: no complex support in C */;
        double r40860 = /* ERROR: no complex support in C */;
        return r40860;
}
↓
double f(double x, double y) {
        double r40861 = x;
        double r40862 = exp(r40861);
        double r40863 = -r40861;
        double r40864 = exp(r40863);
        double r40865 = r40862 + r40864;
        double r40866 = 2.0;
        double r40867 = r40865 / r40866;
        double r40868 = y;
        double r40869 = cos(r40868);
        double r40870 = r40867 * r40869;
        return r40870;
}

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 2019291 
(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)))))