Average Error: 0.0 → 0.0
Time: 16.4s
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 r338406 = x;
        double r338407 = exp(r338406);
        double r338408 = -r338406;
        double r338409 = exp(r338408);
        double r338410 = r338407 + r338409;
        double r338411 = 2.0;
        double r338412 = r338410 / r338411;
        double r338413 = y;
        double r338414 = cos(r338413);
        double r338415 = r338412 * r338414;
        double r338416 = r338407 - r338409;
        double r338417 = r338416 / r338411;
        double r338418 = sin(r338413);
        double r338419 = r338417 * r338418;
        double r338420 = /* ERROR: no complex support in C */;
        double r338421 = /* ERROR: no complex support in C */;
        return r338421;
}
↓
double f(double x, double y) {
        double r338422 = 0.5;
        double r338423 = y;
        double r338424 = cos(r338423);
        double r338425 = x;
        double r338426 = exp(r338425);
        double r338427 = r338424 / r338426;
        double r338428 = fma(r338424, r338426, r338427);
        double r338429 = r338422 * r338428;
        return r338429;
}

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 2019146 +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)))))