Average Error: 0.0 → 0.0
Time: 10.5s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
\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 r26779 = x;
        double r26780 = exp(r26779);
        double r26781 = -r26779;
        double r26782 = exp(r26781);
        double r26783 = r26780 + r26782;
        double r26784 = 2.0;
        double r26785 = r26783 / r26784;
        double r26786 = y;
        double r26787 = cos(r26786);
        double r26788 = r26785 * r26787;
        double r26789 = r26780 - r26782;
        double r26790 = r26789 / r26784;
        double r26791 = sin(r26786);
        double r26792 = r26790 * r26791;
        double r26793 = /* ERROR: no complex support in C */;
        double r26794 = /* ERROR: no complex support in C */;
        return r26794;
}
↓
double f(double x, double y) {
        double r26795 = x;
        double r26796 = exp(r26795);
        double r26797 = -r26795;
        double r26798 = exp(r26797);
        double r26799 = r26796 + r26798;
        double r26800 = 2.0;
        double r26801 = r26799 / r26800;
        double r26802 = y;
        double r26803 = cos(r26802);
        double r26804 = r26801 * r26803;
        double r26805 = r26796 - r26798;
        double r26806 = r26805 / r26800;
        double r26807 = sin(r26802);
        double r26808 = r26806 * r26807;
        double r26809 = /* ERROR: no complex support in C */;
        double r26810 = /* ERROR: no complex support in C */;
        return r26810;
}

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 2020042 
(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)))))