Average Error: 0.0 → 0.0
Time: 11.9s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{e^{x} \cdot \cos y + \frac{\cos y}{e^{x}}}{2}\]
\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} \cdot \cos y + \frac{\cos y}{e^{x}}}{2}
double f(double x, double y) {
        double r208331 = x;
        double r208332 = exp(r208331);
        double r208333 = -r208331;
        double r208334 = exp(r208333);
        double r208335 = r208332 + r208334;
        double r208336 = 2.0;
        double r208337 = r208335 / r208336;
        double r208338 = y;
        double r208339 = cos(r208338);
        double r208340 = r208337 * r208339;
        double r208341 = r208332 - r208334;
        double r208342 = r208341 / r208336;
        double r208343 = sin(r208338);
        double r208344 = r208342 * r208343;
        double r208345 = /* ERROR: no complex support in C */;
        double r208346 = /* ERROR: no complex support in C */;
        return r208346;
}
↓
double f(double x, double y) {
        double r208347 = x;
        double r208348 = exp(r208347);
        double r208349 = y;
        double r208350 = cos(r208349);
        double r208351 = r208348 * r208350;
        double r208352 = r208350 / r208348;
        double r208353 = r208351 + r208352;
        double r208354 = 2.0;
        double r208355 = r208353 / r208354;
        return r208355;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019128 
(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)))))