Average Error: 0.1 → 0.1
Time: 4.5s
Precision: 64
\[\left(1 - x\right) + y \cdot \sqrt{x}\]
↓
\[\mathsf{fma}\left(\sqrt{x}, y, 1 - x\right)\]
\left(1 - x\right) + y \cdot \sqrt{x}
↓
\mathsf{fma}\left(\sqrt{x}, y, 1 - x\right)
double f(double x, double y) {
        double r97863 = 1.0;
        double r97864 = x;
        double r97865 = r97863 - r97864;
        double r97866 = y;
        double r97867 = sqrt(r97864);
        double r97868 = r97866 * r97867;
        double r97869 = r97865 + r97868;
        return r97869;
}
↓
double f(double x, double y) {
        double r97870 = x;
        double r97871 = sqrt(r97870);
        double r97872 = y;
        double r97873 = 1.0;
        double r97874 = r97873 - r97870;
        double r97875 = fma(r97871, r97872, r97874);
        return r97875;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 0.1

    \[\left(1 - x\right) + y \cdot \sqrt{x}\]

  2. Simplified0.1

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2020057 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E"
  :precision binary64
  (+ (- 1 x) (* y (sqrt x))))