Average Error: 0.1 → 0.1
Time: 19.0s
Precision: 64
\[\left(1 - x\right) + y \cdot \sqrt{x}\]
↓
\[\mathsf{fma}\left(y, \sqrt{x}, 1\right) - x\]
\left(1 - x\right) + y \cdot \sqrt{x}
↓
\mathsf{fma}\left(y, \sqrt{x}, 1\right) - x
double f(double x, double y) {
        double r89653 = 1.0;
        double r89654 = x;
        double r89655 = r89653 - r89654;
        double r89656 = y;
        double r89657 = sqrt(r89654);
        double r89658 = r89656 * r89657;
        double r89659 = r89655 + r89658;
        return r89659;
}
↓
double f(double x, double y) {
        double r89660 = y;
        double r89661 = x;
        double r89662 = sqrt(r89661);
        double r89663 = 1.0;
        double r89664 = fma(r89660, r89662, r89663);
        double r89665 = r89664 - r89661;
        return r89665;
}

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

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E"
  (+ (- 1.0 x) (* y (sqrt x))))