Average Error: 0.0 → 0.0
Time: 3.6s
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 r31595 = x;
        double r31596 = y;
        double r31597 = 1.0;
        double r31598 = r31596 - r31597;
        double r31599 = r31595 * r31598;
        double r31600 = 0.5;
        double r31601 = r31596 * r31600;
        double r31602 = r31599 - r31601;
        double r31603 = 0.918938533204673;
        double r31604 = r31602 + r31603;
        return r31604;
}
↓
double f(double x, double y) {
        double r31605 = y;
        double r31606 = 1.0;
        double r31607 = r31605 - r31606;
        double r31608 = x;
        double r31609 = 0.5;
        double r31610 = -r31605;
        double r31611 = 0.918938533204673;
        double r31612 = fma(r31609, r31610, r31611);
        double r31613 = fma(r31607, r31608, r31612);
        return r31613;
}

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 2019208 +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))