Average Error: 0.0 → 0.0
Time: 8.4s
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 r431450 = x;
        double r431451 = exp(r431450);
        double r431452 = -r431450;
        double r431453 = exp(r431452);
        double r431454 = r431451 + r431453;
        double r431455 = 2.0;
        double r431456 = r431454 / r431455;
        double r431457 = y;
        double r431458 = cos(r431457);
        double r431459 = r431456 * r431458;
        double r431460 = r431451 - r431453;
        double r431461 = r431460 / r431455;
        double r431462 = sin(r431457);
        double r431463 = r431461 * r431462;
        double r431464 = /* ERROR: no complex support in C */;
        double r431465 = /* ERROR: no complex support in C */;
        return r431465;
}
↓
double f(double x, double y) {
        double r431466 = x;
        double r431467 = exp(r431466);
        double r431468 = y;
        double r431469 = cos(r431468);
        double r431470 = r431467 * r431469;
        double r431471 = r431469 / r431467;
        double r431472 = r431470 + r431471;
        double r431473 = 2.0;
        double r431474 = r431472 / r431473;
        return r431474;
}

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 2019121 
(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)))))