Average Error: 0.1 → 0.1
Time: 16.0s
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 r85606 = 1.0;
        double r85607 = x;
        double r85608 = r85606 - r85607;
        double r85609 = y;
        double r85610 = sqrt(r85607);
        double r85611 = r85609 * r85610;
        double r85612 = r85608 + r85611;
        return r85612;
}
↓
double f(double x, double y) {
        double r85613 = y;
        double r85614 = x;
        double r85615 = sqrt(r85614);
        double r85616 = 1.0;
        double r85617 = r85616 - r85614;
        double r85618 = fma(r85613, r85615, r85617);
        return r85618;
}

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 2019303 +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))))