Average Error: 0.0 → 0.0
Time: 4.8s
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 r57236 = x;
        double r57237 = exp(r57236);
        double r57238 = -r57236;
        double r57239 = exp(r57238);
        double r57240 = r57237 + r57239;
        double r57241 = 2.0;
        double r57242 = r57240 / r57241;
        double r57243 = y;
        double r57244 = cos(r57243);
        double r57245 = r57242 * r57244;
        double r57246 = r57237 - r57239;
        double r57247 = r57246 / r57241;
        double r57248 = sin(r57243);
        double r57249 = r57247 * r57248;
        double r57250 = /* ERROR: no complex support in C */;
        double r57251 = /* ERROR: no complex support in C */;
        return r57251;
}
↓
double f(double x, double y) {
        double r57252 = x;
        double r57253 = exp(r57252);
        double r57254 = -r57252;
        double r57255 = exp(r57254);
        double r57256 = r57253 + r57255;
        double r57257 = 2.0;
        double r57258 = r57256 / r57257;
        double r57259 = y;
        double r57260 = cos(r57259);
        double r57261 = r57258 * r57260;
        double r57262 = r57253 - r57255;
        double r57263 = r57262 / r57257;
        double r57264 = sin(r57259);
        double r57265 = r57263 * r57264;
        double r57266 = /* ERROR: no complex support in C */;
        double r57267 = /* ERROR: no complex support in C */;
        return r57267;
}

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 2020065 
(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)))))