Average Error: 0.1 → 0.1
Time: 4.0s
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 r81335 = 1.0;
        double r81336 = x;
        double r81337 = r81335 - r81336;
        double r81338 = y;
        double r81339 = sqrt(r81336);
        double r81340 = r81338 * r81339;
        double r81341 = r81337 + r81340;
        return r81341;
}
↓
double f(double x, double y) {
        double r81342 = x;
        double r81343 = sqrt(r81342);
        double r81344 = y;
        double r81345 = 1.0;
        double r81346 = r81345 - r81342;
        double r81347 = fma(r81343, r81344, r81346);
        return r81347;
}

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 2019347 +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))))