Average Error: 0.0 → 0.0
Time: 6.5s
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 r39160 = x;
        double r39161 = y;
        double r39162 = 1.0;
        double r39163 = r39161 - r39162;
        double r39164 = r39160 * r39163;
        double r39165 = 0.5;
        double r39166 = r39161 * r39165;
        double r39167 = r39164 - r39166;
        double r39168 = 0.918938533204673;
        double r39169 = r39167 + r39168;
        return r39169;
}
↓
double f(double x, double y) {
        double r39170 = y;
        double r39171 = 1.0;
        double r39172 = r39170 - r39171;
        double r39173 = x;
        double r39174 = 0.5;
        double r39175 = -r39170;
        double r39176 = 0.918938533204673;
        double r39177 = fma(r39174, r39175, r39176);
        double r39178 = fma(r39172, r39173, r39177);
        return r39178;
}

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 2019323 +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))