Average Error: 0.1 → 0.1
Time: 4.5s
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 r74543 = 1.0;
        double r74544 = x;
        double r74545 = r74543 - r74544;
        double r74546 = y;
        double r74547 = sqrt(r74544);
        double r74548 = r74546 * r74547;
        double r74549 = r74545 + r74548;
        return r74549;
}
↓
double f(double x, double y) {
        double r74550 = x;
        double r74551 = sqrt(r74550);
        double r74552 = y;
        double r74553 = 1.0;
        double r74554 = r74553 - r74550;
        double r74555 = fma(r74551, r74552, r74554);
        return r74555;
}

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 2020001 +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))))