Average Error: 0.1 → 0.1
Time: 4.0s
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 r71783 = 1.0;
        double r71784 = x;
        double r71785 = r71783 - r71784;
        double r71786 = y;
        double r71787 = sqrt(r71784);
        double r71788 = r71786 * r71787;
        double r71789 = r71785 + r71788;
        return r71789;
}
↓
double f(double x, double y) {
        double r71790 = x;
        double r71791 = sqrt(r71790);
        double r71792 = y;
        double r71793 = 1.0;
        double r71794 = r71793 - r71790;
        double r71795 = fma(r71791, r71792, r71794);
        return r71795;
}

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 2019352 +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))))