Average Error: 0.1 → 0.1
Time: 4.6s
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 r89302 = 1.0;
        double r89303 = x;
        double r89304 = r89302 - r89303;
        double r89305 = y;
        double r89306 = sqrt(r89303);
        double r89307 = r89305 * r89306;
        double r89308 = r89304 + r89307;
        return r89308;
}
↓
double f(double x, double y) {
        double r89309 = x;
        double r89310 = sqrt(r89309);
        double r89311 = y;
        double r89312 = 1.0;
        double r89313 = r89312 - r89309;
        double r89314 = fma(r89310, r89311, r89313);
        return r89314;
}

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 2019353 +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))))