Average Error: 0.0 → 0.0
Time: 5.1s
Precision: 64
\[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.9189385332046730026078762421093415468931\]
↓
\[\mathsf{fma}\left(y - 1, x, \mathsf{fma}\left(0.5, -y, 0.9189385332046730026078762421093415468931\right)\right)\]
\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.9189385332046730026078762421093415468931
↓
\mathsf{fma}\left(y - 1, x, \mathsf{fma}\left(0.5, -y, 0.9189385332046730026078762421093415468931\right)\right)
double f(double x, double y) {
        double r16561 = x;
        double r16562 = y;
        double r16563 = 1.0;
        double r16564 = r16562 - r16563;
        double r16565 = r16561 * r16564;
        double r16566 = 0.5;
        double r16567 = r16562 * r16566;
        double r16568 = r16565 - r16567;
        double r16569 = 0.918938533204673;
        double r16570 = r16568 + r16569;
        return r16570;
}
↓
double f(double x, double y) {
        double r16571 = y;
        double r16572 = 1.0;
        double r16573 = r16571 - r16572;
        double r16574 = x;
        double r16575 = 0.5;
        double r16576 = -r16571;
        double r16577 = 0.918938533204673;
        double r16578 = fma(r16575, r16576, r16577);
        double r16579 = fma(r16573, r16574, r16578);
        return r16579;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 0.0

    \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.9189385332046730026078762421093415468931\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(y - 1, x, \mathsf{fma}\left(0.5, -y, 0.9189385332046730026078762421093415468931\right)\right)}\]

  3. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(y - 1, x, \mathsf{fma}\left(0.5, -y, 0.9189385332046730026078762421093415468931\right)\right)\]

Reproduce

herbie shell --seed 2019306 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (+ (- (* x (- y 1)) (* y 0.5)) 0.918938533204673003))