Average Error: 0.0 → 0.0
Time: 3.9s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{e^{x} + e^{-x}}{2} \cdot \cos y\]
\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\frac{e^{x} + e^{-x}}{2} \cdot \cos y
double f(double x, double y) {
        double r31317 = x;
        double r31318 = exp(r31317);
        double r31319 = -r31317;
        double r31320 = exp(r31319);
        double r31321 = r31318 + r31320;
        double r31322 = 2.0;
        double r31323 = r31321 / r31322;
        double r31324 = y;
        double r31325 = cos(r31324);
        double r31326 = r31323 * r31325;
        double r31327 = r31318 - r31320;
        double r31328 = r31327 / r31322;
        double r31329 = sin(r31324);
        double r31330 = r31328 * r31329;
        double r31331 = /* ERROR: no complex support in C */;
        double r31332 = /* ERROR: no complex support in C */;
        return r31332;
}
↓
double f(double x, double y) {
        double r31333 = x;
        double r31334 = exp(r31333);
        double r31335 = -r31333;
        double r31336 = exp(r31335);
        double r31337 = r31334 + r31336;
        double r31338 = 2.0;
        double r31339 = r31337 / r31338;
        double r31340 = y;
        double r31341 = cos(r31340);
        double r31342 = r31339 * r31341;
        return r31342;
}

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. Simplified0.0

    \[\leadsto \color{blue}{\frac{e^{x} + e^{-x}}{2} \cdot \cos y}\]

  3. Final simplification0.0

    \[\leadsto \frac{e^{x} + e^{-x}}{2} \cdot \cos y\]

Reproduce

herbie shell --seed 2020049 
(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)))))