Average Error: 0.0 → 0.0
Time: 13.7s
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 r41383 = x;
        double r41384 = y;
        double r41385 = 1.0;
        double r41386 = r41384 - r41385;
        double r41387 = r41383 * r41386;
        double r41388 = 0.5;
        double r41389 = r41384 * r41388;
        double r41390 = r41387 - r41389;
        double r41391 = 0.918938533204673;
        double r41392 = r41390 + r41391;
        return r41392;
}
↓
double f(double x, double y) {
        double r41393 = y;
        double r41394 = 1.0;
        double r41395 = r41393 - r41394;
        double r41396 = x;
        double r41397 = 0.5;
        double r41398 = -r41393;
        double r41399 = 0.918938533204673;
        double r41400 = fma(r41397, r41398, r41399);
        double r41401 = fma(r41395, r41396, r41400);
        return r41401;
}

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 2019212 +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))