Average Error: 0.1 → 0.1
Time: 3.5s
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 r101847 = 1.0;
        double r101848 = x;
        double r101849 = r101847 - r101848;
        double r101850 = y;
        double r101851 = sqrt(r101848);
        double r101852 = r101850 * r101851;
        double r101853 = r101849 + r101852;
        return r101853;
}
↓
double f(double x, double y) {
        double r101854 = x;
        double r101855 = sqrt(r101854);
        double r101856 = y;
        double r101857 = 1.0;
        double r101858 = r101857 - r101854;
        double r101859 = fma(r101855, r101856, r101858);
        return r101859;
}

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 2020021 +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))))