Average Error: 0.1 → 0.1
Time: 4.2s
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 r89325 = 1.0;
        double r89326 = x;
        double r89327 = r89325 - r89326;
        double r89328 = y;
        double r89329 = sqrt(r89326);
        double r89330 = r89328 * r89329;
        double r89331 = r89327 + r89330;
        return r89331;
}
↓
double f(double x, double y) {
        double r89332 = x;
        double r89333 = sqrt(r89332);
        double r89334 = y;
        double r89335 = 1.0;
        double r89336 = r89335 - r89332;
        double r89337 = fma(r89333, r89334, r89336);
        return r89337;
}

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 2020064 +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))))