Average Error: 0.0 → 0.0
Time: 9.6s
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 r504555 = x;
        double r504556 = exp(r504555);
        double r504557 = -r504555;
        double r504558 = exp(r504557);
        double r504559 = r504556 + r504558;
        double r504560 = 2.0;
        double r504561 = r504559 / r504560;
        double r504562 = y;
        double r504563 = cos(r504562);
        double r504564 = r504561 * r504563;
        double r504565 = r504556 - r504558;
        double r504566 = r504565 / r504560;
        double r504567 = sin(r504562);
        double r504568 = r504566 * r504567;
        double r504569 = /* ERROR: no complex support in C */;
        double r504570 = /* ERROR: no complex support in C */;
        return r504570;
}
↓
double f(double x, double y) {
        double r504571 = x;
        double r504572 = exp(r504571);
        double r504573 = y;
        double r504574 = cos(r504573);
        double r504575 = r504572 * r504574;
        double r504576 = r504574 / r504572;
        double r504577 = r504575 + r504576;
        double r504578 = 2.0;
        double r504579 = r504577 / r504578;
        return r504579;
}

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 2019156 
(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)))))