Average Error: 0.1 → 0.1
Time: 4.7s
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 r69607 = 1.0;
        double r69608 = x;
        double r69609 = r69607 - r69608;
        double r69610 = y;
        double r69611 = sqrt(r69608);
        double r69612 = r69610 * r69611;
        double r69613 = r69609 + r69612;
        return r69613;
}
↓
double f(double x, double y) {
        double r69614 = x;
        double r69615 = sqrt(r69614);
        double r69616 = y;
        double r69617 = 1.0;
        double r69618 = r69617 - r69614;
        double r69619 = fma(r69615, r69616, r69618);
        return r69619;
}

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 2020056 +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))))