Average Error: 0.1 → 0.1
Time: 3.8s
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 r95731 = 1.0;
        double r95732 = x;
        double r95733 = r95731 - r95732;
        double r95734 = y;
        double r95735 = sqrt(r95732);
        double r95736 = r95734 * r95735;
        double r95737 = r95733 + r95736;
        return r95737;
}
↓
double f(double x, double y) {
        double r95738 = x;
        double r95739 = sqrt(r95738);
        double r95740 = y;
        double r95741 = 1.0;
        double r95742 = r95741 - r95738;
        double r95743 = fma(r95739, r95740, r95742);
        return r95743;
}

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 2020100 +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))))