Average Error: 0.0 → 0.0
Time: 960.0ms
Precision: 64
\[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673003\]
↓
\[\mathsf{fma}\left(x, y - 1, 0.918938533204673003\right) - y \cdot 0.5\]
\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673003
↓
\mathsf{fma}\left(x, y - 1, 0.918938533204673003\right) - y \cdot 0.5
double f(double x, double y) {
        double r58680 = x;
        double r58681 = y;
        double r58682 = 1.0;
        double r58683 = r58681 - r58682;
        double r58684 = r58680 * r58683;
        double r58685 = 0.5;
        double r58686 = r58681 * r58685;
        double r58687 = r58684 - r58686;
        double r58688 = 0.918938533204673;
        double r58689 = r58687 + r58688;
        return r58689;
}
↓
double f(double x, double y) {
        double r58690 = x;
        double r58691 = y;
        double r58692 = 1.0;
        double r58693 = r58691 - r58692;
        double r58694 = 0.918938533204673;
        double r58695 = fma(r58690, r58693, r58694);
        double r58696 = 0.5;
        double r58697 = r58691 * r58696;
        double r58698 = r58695 - r58697;
        return r58698;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 0.0

    \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673003\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, 0.918938533204673003\right) - \mathsf{fma}\left(x, 1, y \cdot 0.5\right)}\]
  3. Using strategy rm

  4. Applied fma-udef0.0

    \[\leadsto \mathsf{fma}\left(y, x, 0.918938533204673003\right) - \color{blue}{\left(x \cdot 1 + y \cdot 0.5\right)}\]

  5. Applied associate--r+0.0

    \[\leadsto \color{blue}{\left(\mathsf{fma}\left(y, x, 0.918938533204673003\right) - x \cdot 1\right) - y \cdot 0.5}\]

  6. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, y - 1, 0.918938533204673003\right)} - y \cdot 0.5\]

  7. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(x, y - 1, 0.918938533204673003\right) - y \cdot 0.5\]

Reproduce

herbie shell --seed 2020021 +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))