Average Error: 0.0 → 0.0
Time: 3.1s
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 r37636 = x;
        double r37637 = y;
        double r37638 = 1.0;
        double r37639 = r37637 - r37638;
        double r37640 = r37636 * r37639;
        double r37641 = 0.5;
        double r37642 = r37637 * r37641;
        double r37643 = r37640 - r37642;
        double r37644 = 0.918938533204673;
        double r37645 = r37643 + r37644;
        return r37645;
}
↓
double f(double x, double y) {
        double r37646 = y;
        double r37647 = 1.0;
        double r37648 = r37646 - r37647;
        double r37649 = x;
        double r37650 = 0.5;
        double r37651 = -r37646;
        double r37652 = 0.918938533204673;
        double r37653 = fma(r37650, r37651, r37652);
        double r37654 = fma(r37648, r37649, r37653);
        return r37654;
}

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 2019326 +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))