Average Error: 0.1 → 0.1
Time: 17.4s
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 r80683 = 1.0;
        double r80684 = x;
        double r80685 = r80683 - r80684;
        double r80686 = y;
        double r80687 = sqrt(r80684);
        double r80688 = r80686 * r80687;
        double r80689 = r80685 + r80688;
        return r80689;
}
↓
double f(double x, double y) {
        double r80690 = y;
        double r80691 = x;
        double r80692 = sqrt(r80691);
        double r80693 = 1.0;
        double r80694 = r80693 - r80691;
        double r80695 = fma(r80690, r80692, r80694);
        return r80695;
}

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 2019326 +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))))