Average Error: 0.0 → 0.0
Time: 11.3s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{1}{2} \cdot \mathsf{fma}\left(\left(\cos y\right), \left(e^{x}\right), \left(\frac{\cos y}{e^{x}}\right)\right)\]
\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\frac{1}{2} \cdot \mathsf{fma}\left(\left(\cos y\right), \left(e^{x}\right), \left(\frac{\cos y}{e^{x}}\right)\right)
double f(double x, double y) {
        double r899512 = x;
        double r899513 = exp(r899512);
        double r899514 = -r899512;
        double r899515 = exp(r899514);
        double r899516 = r899513 + r899515;
        double r899517 = 2.0;
        double r899518 = r899516 / r899517;
        double r899519 = y;
        double r899520 = cos(r899519);
        double r899521 = r899518 * r899520;
        double r899522 = r899513 - r899515;
        double r899523 = r899522 / r899517;
        double r899524 = sin(r899519);
        double r899525 = r899523 * r899524;
        double r899526 = /* ERROR: no complex support in C */;
        double r899527 = /* ERROR: no complex support in C */;
        return r899527;
}
↓
double f(double x, double y) {
        double r899528 = 0.5;
        double r899529 = y;
        double r899530 = cos(r899529);
        double r899531 = x;
        double r899532 = exp(r899531);
        double r899533 = r899530 / r899532;
        double r899534 = fma(r899530, r899532, r899533);
        double r899535 = r899528 * r899534;
        return r899535;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019130 +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)))))