Average Error: 0.0 → 0.0
Time: 975.0ms
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 r56021 = x;
        double r56022 = y;
        double r56023 = 1.0;
        double r56024 = r56022 - r56023;
        double r56025 = r56021 * r56024;
        double r56026 = 0.5;
        double r56027 = r56022 * r56026;
        double r56028 = r56025 - r56027;
        double r56029 = 0.918938533204673;
        double r56030 = r56028 + r56029;
        return r56030;
}
↓
double f(double x, double y) {
        double r56031 = x;
        double r56032 = y;
        double r56033 = 1.0;
        double r56034 = r56032 - r56033;
        double r56035 = 0.918938533204673;
        double r56036 = fma(r56031, r56034, r56035);
        double r56037 = 0.5;
        double r56038 = r56032 * r56037;
        double r56039 = r56036 - r56038;
        return r56039;
}

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 2020047 +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))