Average Error: 0.0 → 0.0
Time: 15.7s
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 r1776971 = x;
        double r1776972 = exp(r1776971);
        double r1776973 = -r1776971;
        double r1776974 = exp(r1776973);
        double r1776975 = r1776972 + r1776974;
        double r1776976 = 2.0;
        double r1776977 = r1776975 / r1776976;
        double r1776978 = y;
        double r1776979 = cos(r1776978);
        double r1776980 = r1776977 * r1776979;
        double r1776981 = r1776972 - r1776974;
        double r1776982 = r1776981 / r1776976;
        double r1776983 = sin(r1776978);
        double r1776984 = r1776982 * r1776983;
        double r1776985 = /* ERROR: no complex support in C */;
        double r1776986 = /* ERROR: no complex support in C */;
        return r1776986;
}
↓
double f(double x, double y) {
        double r1776987 = x;
        double r1776988 = exp(r1776987);
        double r1776989 = -r1776987;
        double r1776990 = exp(r1776989);
        double r1776991 = r1776988 + r1776990;
        double r1776992 = 2.0;
        double r1776993 = r1776991 / r1776992;
        double r1776994 = y;
        double r1776995 = cos(r1776994);
        double r1776996 = r1776993 * r1776995;
        double r1776997 = r1776988 - r1776990;
        double r1776998 = r1776997 / r1776992;
        double r1776999 = sin(r1776994);
        double r1777000 = r1776998 * r1776999;
        double r1777001 = /* ERROR: no complex support in C */;
        double r1777002 = /* ERROR: no complex support in C */;
        return r1777002;
}

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)))))