Average Error: 0.0 → 0.0
Time: 23.8s
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 r1314829 = x;
        double r1314830 = exp(r1314829);
        double r1314831 = -r1314829;
        double r1314832 = exp(r1314831);
        double r1314833 = r1314830 + r1314832;
        double r1314834 = 2.0;
        double r1314835 = r1314833 / r1314834;
        double r1314836 = y;
        double r1314837 = cos(r1314836);
        double r1314838 = r1314835 * r1314837;
        double r1314839 = r1314830 - r1314832;
        double r1314840 = r1314839 / r1314834;
        double r1314841 = sin(r1314836);
        double r1314842 = r1314840 * r1314841;
        double r1314843 = /* ERROR: no complex support in C */;
        double r1314844 = /* ERROR: no complex support in C */;
        return r1314844;
}
↓
double f(double x, double y) {
        double r1314845 = x;
        double r1314846 = exp(r1314845);
        double r1314847 = -r1314845;
        double r1314848 = exp(r1314847);
        double r1314849 = r1314846 + r1314848;
        double r1314850 = 2.0;
        double r1314851 = r1314849 / r1314850;
        double r1314852 = y;
        double r1314853 = cos(r1314852);
        double r1314854 = r1314851 * r1314853;
        double r1314855 = r1314846 - r1314848;
        double r1314856 = r1314855 / r1314850;
        double r1314857 = sin(r1314852);
        double r1314858 = r1314856 * r1314857;
        double r1314859 = /* ERROR: no complex support in C */;
        double r1314860 = /* ERROR: no complex support in C */;
        return r1314860;
}

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 2019138 
(FPCore (x y)
  :name "Euler formula real part (p55)"
  (re (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y)))))