Average Error: 0.0 → 0.0
Time: 20.7s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{1}{2} \cdot \mathsf{fma}\left(\cos y, e^{x}, \frac{\cos y}{e^{x}}\right)\]
\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\frac{1}{2} \cdot \mathsf{fma}\left(\cos y, e^{x}, \frac{\cos y}{e^{x}}\right)
double f(double x, double y) {
        double r857455 = x;
        double r857456 = exp(r857455);
        double r857457 = -r857455;
        double r857458 = exp(r857457);
        double r857459 = r857456 + r857458;
        double r857460 = 2.0;
        double r857461 = r857459 / r857460;
        double r857462 = y;
        double r857463 = cos(r857462);
        double r857464 = r857461 * r857463;
        double r857465 = r857456 - r857458;
        double r857466 = r857465 / r857460;
        double r857467 = sin(r857462);
        double r857468 = r857466 * r857467;
        double r857469 = /* ERROR: no complex support in C */;
        double r857470 = /* ERROR: no complex support in C */;
        return r857470;
}
↓
double f(double x, double y) {
        double r857471 = 0.5;
        double r857472 = y;
        double r857473 = cos(r857472);
        double r857474 = x;
        double r857475 = exp(r857474);
        double r857476 = r857473 / r857475;
        double r857477 = fma(r857473, r857475, r857476);
        double r857478 = r857471 * r857477;
        return r857478;
}

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

    \[\leadsto \color{blue}{\frac{1}{2} \cdot \mathsf{fma}\left(\cos y, e^{x}, \frac{\cos y}{e^{x}}\right)}\]

  3. Final simplification0.0

    \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left(\cos y, e^{x}, \frac{\cos y}{e^{x}}\right)\]

Reproduce

herbie shell --seed 2019144 +o rules:numerics
(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)))))