Average Error: 0.0 → 0.0
Time: 9.9s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r27478 = x;
        double r27479 = exp(r27478);
        double r27480 = -r27478;
        double r27481 = exp(r27480);
        double r27482 = r27479 + r27481;
        double r27483 = 2.0;
        double r27484 = r27482 / r27483;
        double r27485 = y;
        double r27486 = cos(r27485);
        double r27487 = r27484 * r27486;
        double r27488 = r27479 - r27481;
        double r27489 = r27488 / r27483;
        double r27490 = sin(r27485);
        double r27491 = r27489 * r27490;
        double r27492 = /* ERROR: no complex support in C */;
        double r27493 = /* ERROR: no complex support in C */;
        return r27493;
}
↓
double f(double x, double y) {
        double r27494 = x;
        double r27495 = exp(r27494);
        double r27496 = -r27494;
        double r27497 = exp(r27496);
        double r27498 = r27495 + r27497;
        double r27499 = 2.0;
        double r27500 = r27498 / r27499;
        double r27501 = y;
        double r27502 = cos(r27501);
        double r27503 = r27500 * r27502;
        double r27504 = r27495 - r27497;
        double r27505 = r27504 / r27499;
        double r27506 = sin(r27501);
        double r27507 = r27505 * r27506;
        double r27508 = /* ERROR: no complex support in C */;
        double r27509 = /* ERROR: no complex support in C */;
        return r27509;
}

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. Final simplification0.0

    \[\leadsto \Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]

Reproduce

herbie shell --seed 2019351 +o rules:numerics
(FPCore (x y)
  :name "Euler formula real part (p55)"
  :precision binary64
  (re (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y)))))