Average Error: 0.0 → 0.0
Time: 4.9s
Precision: 64
\[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.9189385332046730026078762421093415468931\]
↓
\[\mathsf{fma}\left(y, x, -\mathsf{fma}\left(1, x, 0.5 \cdot y\right)\right) + 0.9189385332046730026078762421093415468931\]
\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.9189385332046730026078762421093415468931
↓
\mathsf{fma}\left(y, x, -\mathsf{fma}\left(1, x, 0.5 \cdot y\right)\right) + 0.9189385332046730026078762421093415468931
double f(double x, double y) {
        double r47118 = x;
        double r47119 = y;
        double r47120 = 1.0;
        double r47121 = r47119 - r47120;
        double r47122 = r47118 * r47121;
        double r47123 = 0.5;
        double r47124 = r47119 * r47123;
        double r47125 = r47122 - r47124;
        double r47126 = 0.918938533204673;
        double r47127 = r47125 + r47126;
        return r47127;
}
↓
double f(double x, double y) {
        double r47128 = y;
        double r47129 = x;
        double r47130 = 1.0;
        double r47131 = 0.5;
        double r47132 = r47131 * r47128;
        double r47133 = fma(r47130, r47129, r47132);
        double r47134 = -r47133;
        double r47135 = fma(r47128, r47129, r47134);
        double r47136 = 0.918938533204673;
        double r47137 = r47135 + r47136;
        return r47137;
}

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. Taylor expanded around 0 0.0

    \[\leadsto \color{blue}{\left(x \cdot y - \left(1 \cdot x + 0.5 \cdot y\right)\right)} + 0.9189385332046730026078762421093415468931\]

  3. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, -\mathsf{fma}\left(1, x, y \cdot 0.5\right)\right)} + 0.9189385332046730026078762421093415468931\]

  4. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(y, x, -\mathsf{fma}\left(1, x, 0.5 \cdot y\right)\right) + 0.9189385332046730026078762421093415468931\]

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, A"
  (+ (- (* x (- y 1.0)) (* y 0.5)) 0.918938533204673))