Average Error: 0.0 → 0.0
Time: 5.3s
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 r33627 = x;
        double r33628 = y;
        double r33629 = 1.0;
        double r33630 = r33628 - r33629;
        double r33631 = r33627 * r33630;
        double r33632 = 0.5;
        double r33633 = r33628 * r33632;
        double r33634 = r33631 - r33633;
        double r33635 = 0.918938533204673;
        double r33636 = r33634 + r33635;
        return r33636;
}
↓
double f(double x, double y) {
        double r33637 = y;
        double r33638 = 1.0;
        double r33639 = r33637 - r33638;
        double r33640 = x;
        double r33641 = 0.5;
        double r33642 = -r33637;
        double r33643 = 0.918938533204673;
        double r33644 = fma(r33641, r33642, r33643);
        double r33645 = fma(r33639, r33640, r33644);
        return r33645;
}

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 2019325 +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)) 0.918938533204673))