Average Error: 0.1 → 0.1
Time: 15.7s
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 r64339 = 1.0;
        double r64340 = x;
        double r64341 = r64339 - r64340;
        double r64342 = y;
        double r64343 = sqrt(r64340);
        double r64344 = r64342 * r64343;
        double r64345 = r64341 + r64344;
        return r64345;
}
↓
double f(double x, double y) {
        double r64346 = y;
        double r64347 = x;
        double r64348 = sqrt(r64347);
        double r64349 = 1.0;
        double r64350 = r64349 - r64347;
        double r64351 = fma(r64346, r64348, r64350);
        return r64351;
}

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 2019304 +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))))