Average Error: 0.0 → 0.0
Time: 20.9s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\left(\cos y \cdot \sqrt{\mathsf{fma}\left(\frac{1}{2}, \left(e^{x}\right), \left(\frac{\frac{1}{2}}{e^{x}}\right)\right)}\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{2}, \left(e^{x}\right), \left(\frac{\frac{1}{2}}{e^{x}}\right)\right)}\]
\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\left(\cos y \cdot \sqrt{\mathsf{fma}\left(\frac{1}{2}, \left(e^{x}\right), \left(\frac{\frac{1}{2}}{e^{x}}\right)\right)}\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{2}, \left(e^{x}\right), \left(\frac{\frac{1}{2}}{e^{x}}\right)\right)}
double f(double x, double y) {
        double r321517 = x;
        double r321518 = exp(r321517);
        double r321519 = -r321517;
        double r321520 = exp(r321519);
        double r321521 = r321518 + r321520;
        double r321522 = 2.0;
        double r321523 = r321521 / r321522;
        double r321524 = y;
        double r321525 = cos(r321524);
        double r321526 = r321523 * r321525;
        double r321527 = r321518 - r321520;
        double r321528 = r321527 / r321522;
        double r321529 = sin(r321524);
        double r321530 = r321528 * r321529;
        double r321531 = /* ERROR: no complex support in C */;
        double r321532 = /* ERROR: no complex support in C */;
        return r321532;
}
↓
double f(double x, double y) {
        double r321533 = y;
        double r321534 = cos(r321533);
        double r321535 = 0.5;
        double r321536 = x;
        double r321537 = exp(r321536);
        double r321538 = r321535 / r321537;
        double r321539 = fma(r321535, r321537, r321538);
        double r321540 = sqrt(r321539);
        double r321541 = r321534 * r321540;
        double r321542 = r321541 * r321540;
        return r321542;
}

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

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

  4. Applied add-sqr-sqrt0.0

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

  5. Applied associate-*l*0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019129 +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)))))