Average Error: 0.1 → 0.1
Time: 13.0s
Precision: 64
\[\left(1 - x\right) + y \cdot \sqrt{x}\]
↓
\[\mathsf{fma}\left(y, \sqrt{x}, 1 - x\right)\]
\left(1 - x\right) + y \cdot \sqrt{x}
↓
\mathsf{fma}\left(y, \sqrt{x}, 1 - x\right)
double f(double x, double y) {
        double r63540 = 1.0;
        double r63541 = x;
        double r63542 = r63540 - r63541;
        double r63543 = y;
        double r63544 = sqrt(r63541);
        double r63545 = r63543 * r63544;
        double r63546 = r63542 + r63545;
        return r63546;
}
↓
double f(double x, double y) {
        double r63547 = y;
        double r63548 = x;
        double r63549 = sqrt(r63548);
        double r63550 = 1.0;
        double r63551 = r63550 - r63548;
        double r63552 = fma(r63547, r63549, r63551);
        return r63552;
}

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

    \[\leadsto \color{blue}{\mathsf{fma}\left(y, \sqrt{x}, 1 - x\right)}\]

  3. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(y, \sqrt{x}, 1 - x\right)\]

Reproduce

herbie shell --seed 2019208 +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))))