Average Error: 0.0 → 0.0
Time: 10.2s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r27845 = x;
        double r27846 = exp(r27845);
        double r27847 = -r27845;
        double r27848 = exp(r27847);
        double r27849 = r27846 + r27848;
        double r27850 = 2.0;
        double r27851 = r27849 / r27850;
        double r27852 = y;
        double r27853 = cos(r27852);
        double r27854 = r27851 * r27853;
        double r27855 = r27846 - r27848;
        double r27856 = r27855 / r27850;
        double r27857 = sin(r27852);
        double r27858 = r27856 * r27857;
        double r27859 = /* ERROR: no complex support in C */;
        double r27860 = /* ERROR: no complex support in C */;
        return r27860;
}
↓
double f(double x, double y) {
        double r27861 = x;
        double r27862 = exp(r27861);
        double r27863 = -r27861;
        double r27864 = exp(r27863);
        double r27865 = r27862 + r27864;
        double r27866 = 2.0;
        double r27867 = r27865 / r27866;
        double r27868 = y;
        double r27869 = cos(r27868);
        double r27870 = r27867 * r27869;
        double r27871 = r27862 - r27864;
        double r27872 = r27871 / r27866;
        double r27873 = sin(r27868);
        double r27874 = r27872 * r27873;
        double r27875 = /* ERROR: no complex support in C */;
        double r27876 = /* ERROR: no complex support in C */;
        return r27876;
}

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. Final simplification0.0

    \[\leadsto \Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]

Reproduce

herbie shell --seed 2019351 
(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)))))