Average Error: 0.0 → 0.0
Time: 15.9s
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 r1776974 = x;
        double r1776975 = exp(r1776974);
        double r1776976 = -r1776974;
        double r1776977 = exp(r1776976);
        double r1776978 = r1776975 + r1776977;
        double r1776979 = 2.0;
        double r1776980 = r1776978 / r1776979;
        double r1776981 = y;
        double r1776982 = cos(r1776981);
        double r1776983 = r1776980 * r1776982;
        double r1776984 = r1776975 - r1776977;
        double r1776985 = r1776984 / r1776979;
        double r1776986 = sin(r1776981);
        double r1776987 = r1776985 * r1776986;
        double r1776988 = /* ERROR: no complex support in C */;
        double r1776989 = /* ERROR: no complex support in C */;
        return r1776989;
}
↓
double f(double x, double y) {
        double r1776990 = x;
        double r1776991 = exp(r1776990);
        double r1776992 = -r1776990;
        double r1776993 = exp(r1776992);
        double r1776994 = r1776991 + r1776993;
        double r1776995 = 2.0;
        double r1776996 = r1776994 / r1776995;
        double r1776997 = y;
        double r1776998 = cos(r1776997);
        double r1776999 = r1776996 * r1776998;
        double r1777000 = r1776991 - r1776993;
        double r1777001 = r1777000 / r1776995;
        double r1777002 = sin(r1776997);
        double r1777003 = r1777001 * r1777002;
        double r1777004 = /* ERROR: no complex support in C */;
        double r1777005 = /* ERROR: no complex support in C */;
        return r1777005;
}

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 
(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)))))