Average Error: 0.1 → 0.1
Time: 4.3s
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 r96372 = 1.0;
        double r96373 = x;
        double r96374 = r96372 - r96373;
        double r96375 = y;
        double r96376 = sqrt(r96373);
        double r96377 = r96375 * r96376;
        double r96378 = r96374 + r96377;
        return r96378;
}
↓
double f(double x, double y) {
        double r96379 = x;
        double r96380 = sqrt(r96379);
        double r96381 = y;
        double r96382 = 1.0;
        double r96383 = r96382 - r96379;
        double r96384 = fma(r96380, r96381, r96383);
        return r96384;
}

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 2020046 +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))))