Average Error: 0.1 → 0.1
Time: 4.4s
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 r76303 = 1.0;
        double r76304 = x;
        double r76305 = r76303 - r76304;
        double r76306 = y;
        double r76307 = sqrt(r76304);
        double r76308 = r76306 * r76307;
        double r76309 = r76305 + r76308;
        return r76309;
}
↓
double f(double x, double y) {
        double r76310 = x;
        double r76311 = sqrt(r76310);
        double r76312 = y;
        double r76313 = 1.0;
        double r76314 = r76313 - r76310;
        double r76315 = fma(r76311, r76312, r76314);
        return r76315;
}

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 2020025 +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))))