Average Error: 0.0 → 0.0
Time: 4.2s
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 r12688 = x;
        double r12689 = y;
        double r12690 = 1.0;
        double r12691 = r12689 - r12690;
        double r12692 = r12688 * r12691;
        double r12693 = 0.5;
        double r12694 = r12689 * r12693;
        double r12695 = r12692 - r12694;
        double r12696 = 0.918938533204673;
        double r12697 = r12695 + r12696;
        return r12697;
}
↓
double f(double x, double y) {
        double r12698 = y;
        double r12699 = 1.0;
        double r12700 = r12698 - r12699;
        double r12701 = x;
        double r12702 = 0.5;
        double r12703 = -r12698;
        double r12704 = 0.918938533204673;
        double r12705 = fma(r12702, r12703, r12704);
        double r12706 = fma(r12700, r12701, r12705);
        return r12706;
}

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 2019315 +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))