Average Error: 0.1 → 0.1
Time: 3.7s
Precision: 64
\[\left(1 - x\right) + y \cdot \sqrt{x}\]
↓
\[1 + \mathsf{fma}\left(\sqrt{x}, y, -x\right)\]
\left(1 - x\right) + y \cdot \sqrt{x}
↓
1 + \mathsf{fma}\left(\sqrt{x}, y, -x\right)
double f(double x, double y) {
        double r107364 = 1.0;
        double r107365 = x;
        double r107366 = r107364 - r107365;
        double r107367 = y;
        double r107368 = sqrt(r107365);
        double r107369 = r107367 * r107368;
        double r107370 = r107366 + r107369;
        return r107370;
}
↓
double f(double x, double y) {
        double r107371 = 1.0;
        double r107372 = x;
        double r107373 = sqrt(r107372);
        double r107374 = y;
        double r107375 = -r107372;
        double r107376 = fma(r107373, r107374, r107375);
        double r107377 = r107371 + r107376;
        return r107377;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 0.1

    \[\left(1 - x\right) + y \cdot \sqrt{x}\]
  2. Using strategy rm

  3. Applied sub-neg0.1

    \[\leadsto \color{blue}{\left(1 + \left(-x\right)\right)} + y \cdot \sqrt{x}\]

  4. Applied associate-+l+0.1

    \[\leadsto \color{blue}{1 + \left(\left(-x\right) + y \cdot \sqrt{x}\right)}\]

  5. Simplified0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2019354 +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))))