Average Error: 0.1 → 0.1
Time: 4.3s
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 r78679 = 1.0;
        double r78680 = x;
        double r78681 = r78679 - r78680;
        double r78682 = y;
        double r78683 = sqrt(r78680);
        double r78684 = r78682 * r78683;
        double r78685 = r78681 + r78684;
        return r78685;
}
↓
double f(double x, double y) {
        double r78686 = x;
        double r78687 = sqrt(r78686);
        double r78688 = y;
        double r78689 = 1.0;
        double r78690 = r78689 - r78686;
        double r78691 = fma(r78687, r78688, r78690);
        return r78691;
}

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 2020024 +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))))