Average Error: 0.0 → 0.0
Time: 13.1s
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 r853478 = x;
        double r853479 = exp(r853478);
        double r853480 = -r853478;
        double r853481 = exp(r853480);
        double r853482 = r853479 + r853481;
        double r853483 = 2.0;
        double r853484 = r853482 / r853483;
        double r853485 = y;
        double r853486 = cos(r853485);
        double r853487 = r853484 * r853486;
        double r853488 = r853479 - r853481;
        double r853489 = r853488 / r853483;
        double r853490 = sin(r853485);
        double r853491 = r853489 * r853490;
        double r853492 = /* ERROR: no complex support in C */;
        double r853493 = /* ERROR: no complex support in C */;
        return r853493;
}
↓
double f(double x, double y) {
        double r853494 = x;
        double r853495 = exp(r853494);
        double r853496 = y;
        double r853497 = cos(r853496);
        double r853498 = r853495 * r853497;
        double r853499 = r853497 / r853495;
        double r853500 = r853498 + r853499;
        double r853501 = 2.0;
        double r853502 = r853500 / r853501;
        return r853502;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019171 
(FPCore (x y)
  :name "Euler formula real part (p55)"
  (re (complex (* (/ (+ (exp x) (exp (- x))) 2.0) (cos y)) (* (/ (- (exp x) (exp (- x))) 2.0) (sin y)))))