Average Error: 0.0 → 0.0
Time: 9.3s
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 r49206 = x;
        double r49207 = y;
        double r49208 = 1.0;
        double r49209 = r49207 - r49208;
        double r49210 = r49206 * r49209;
        double r49211 = 0.5;
        double r49212 = r49207 * r49211;
        double r49213 = r49210 - r49212;
        double r49214 = 0.918938533204673;
        double r49215 = r49213 + r49214;
        return r49215;
}
↓
double f(double x, double y) {
        double r49216 = y;
        double r49217 = 1.0;
        double r49218 = r49216 - r49217;
        double r49219 = x;
        double r49220 = 0.5;
        double r49221 = -r49216;
        double r49222 = 0.918938533204673;
        double r49223 = fma(r49220, r49221, r49222);
        double r49224 = fma(r49218, r49219, r49223);
        return r49224;
}

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 2019347 +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))