Average Error: 0.0 → 0.0
Time: 8.6s
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 r491471 = x;
        double r491472 = exp(r491471);
        double r491473 = -r491471;
        double r491474 = exp(r491473);
        double r491475 = r491472 + r491474;
        double r491476 = 2.0;
        double r491477 = r491475 / r491476;
        double r491478 = y;
        double r491479 = cos(r491478);
        double r491480 = r491477 * r491479;
        double r491481 = r491472 - r491474;
        double r491482 = r491481 / r491476;
        double r491483 = sin(r491478);
        double r491484 = r491482 * r491483;
        double r491485 = /* ERROR: no complex support in C */;
        double r491486 = /* ERROR: no complex support in C */;
        return r491486;
}
↓
double f(double x, double y) {
        double r491487 = x;
        double r491488 = exp(r491487);
        double r491489 = y;
        double r491490 = cos(r491489);
        double r491491 = r491488 * r491490;
        double r491492 = r491490 / r491488;
        double r491493 = r491491 + r491492;
        double r491494 = 2.0;
        double r491495 = r491493 / r491494;
        return r491495;
}

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 2019124 
(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)))))