Average Error: 0.0 → 0.0
Time: 1.1s
Precision: 64
\[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673003\]
↓
\[\left(\mathsf{fma}\left(x, y, -\mathsf{fma}\left(1, x, 0.5 \cdot y\right)\right) + \mathsf{fma}\left(-0.5, y, 0.5 \cdot y\right)\right) + 0.918938533204673003\]
\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673003
↓
\left(\mathsf{fma}\left(x, y, -\mathsf{fma}\left(1, x, 0.5 \cdot y\right)\right) + \mathsf{fma}\left(-0.5, y, 0.5 \cdot y\right)\right) + 0.918938533204673003
double f(double x, double y) {
        double r54524 = x;
        double r54525 = y;
        double r54526 = 1.0;
        double r54527 = r54525 - r54526;
        double r54528 = r54524 * r54527;
        double r54529 = 0.5;
        double r54530 = r54525 * r54529;
        double r54531 = r54528 - r54530;
        double r54532 = 0.918938533204673;
        double r54533 = r54531 + r54532;
        return r54533;
}
↓
double f(double x, double y) {
        double r54534 = x;
        double r54535 = y;
        double r54536 = 1.0;
        double r54537 = 0.5;
        double r54538 = r54537 * r54535;
        double r54539 = fma(r54536, r54534, r54538);
        double r54540 = -r54539;
        double r54541 = fma(r54534, r54535, r54540);
        double r54542 = -r54537;
        double r54543 = fma(r54542, r54535, r54538);
        double r54544 = r54541 + r54543;
        double r54545 = 0.918938533204673;
        double r54546 = r54544 + r54545;
        return r54546;
}

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. Using strategy rm

  3. Applied prod-diff0.0

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2020057 +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))