Average Error: 0.0 → 0.0
Time: 23.9s
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 r42925 = x;
        double r42926 = exp(r42925);
        double r42927 = -r42925;
        double r42928 = exp(r42927);
        double r42929 = r42926 + r42928;
        double r42930 = 2.0;
        double r42931 = r42929 / r42930;
        double r42932 = y;
        double r42933 = cos(r42932);
        double r42934 = r42931 * r42933;
        double r42935 = r42926 - r42928;
        double r42936 = r42935 / r42930;
        double r42937 = sin(r42932);
        double r42938 = r42936 * r42937;
        double r42939 = /* ERROR: no complex support in C */;
        double r42940 = /* ERROR: no complex support in C */;
        return r42940;
}
↓
double f(double x, double y) {
        double r42941 = x;
        double r42942 = exp(r42941);
        double r42943 = -r42941;
        double r42944 = exp(r42943);
        double r42945 = r42942 + r42944;
        double r42946 = 2.0;
        double r42947 = r42945 / r42946;
        double r42948 = y;
        double r42949 = cos(r42948);
        double r42950 = r42947 * r42949;
        double r42951 = r42942 - r42944;
        double r42952 = r42951 / r42946;
        double r42953 = sin(r42948);
        double r42954 = r42952 * r42953;
        double r42955 = /* ERROR: no complex support in C */;
        double r42956 = /* ERROR: no complex support in C */;
        return r42956;
}

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 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)))))