Average Error: 0.0 → 0.0
Time: 17.9s
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 r1352796 = x;
        double r1352797 = exp(r1352796);
        double r1352798 = -r1352796;
        double r1352799 = exp(r1352798);
        double r1352800 = r1352797 + r1352799;
        double r1352801 = 2.0;
        double r1352802 = r1352800 / r1352801;
        double r1352803 = y;
        double r1352804 = cos(r1352803);
        double r1352805 = r1352802 * r1352804;
        double r1352806 = r1352797 - r1352799;
        double r1352807 = r1352806 / r1352801;
        double r1352808 = sin(r1352803);
        double r1352809 = r1352807 * r1352808;
        double r1352810 = /* ERROR: no complex support in C */;
        double r1352811 = /* ERROR: no complex support in C */;
        return r1352811;
}
↓
double f(double x, double y) {
        double r1352812 = x;
        double r1352813 = exp(r1352812);
        double r1352814 = -r1352812;
        double r1352815 = exp(r1352814);
        double r1352816 = r1352813 + r1352815;
        double r1352817 = 2.0;
        double r1352818 = r1352816 / r1352817;
        double r1352819 = y;
        double r1352820 = cos(r1352819);
        double r1352821 = r1352818 * r1352820;
        double r1352822 = r1352813 - r1352815;
        double r1352823 = r1352822 / r1352817;
        double r1352824 = sin(r1352819);
        double r1352825 = r1352823 * r1352824;
        double r1352826 = /* ERROR: no complex support in C */;
        double r1352827 = /* ERROR: no complex support in C */;
        return r1352827;
}

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 2019163 +o rules:numerics
(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)))))