Average Error: 0.0 → 0.0
Time: 5.8s
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 r45291 = x;
        double r45292 = y;
        double r45293 = 1.0;
        double r45294 = r45292 - r45293;
        double r45295 = r45291 * r45294;
        double r45296 = 0.5;
        double r45297 = r45292 * r45296;
        double r45298 = r45295 - r45297;
        double r45299 = 0.918938533204673;
        double r45300 = r45298 + r45299;
        return r45300;
}
↓
double f(double x, double y) {
        double r45301 = y;
        double r45302 = 1.0;
        double r45303 = r45301 - r45302;
        double r45304 = x;
        double r45305 = 0.5;
        double r45306 = -r45301;
        double r45307 = 0.918938533204673;
        double r45308 = fma(r45305, r45306, r45307);
        double r45309 = fma(r45303, r45304, r45308);
        return r45309;
}

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 2019209 +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))