Average Error: 0.1 → 0.1
Time: 4.2s
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 r60451 = 1.0;
        double r60452 = x;
        double r60453 = r60451 - r60452;
        double r60454 = y;
        double r60455 = sqrt(r60452);
        double r60456 = r60454 * r60455;
        double r60457 = r60453 + r60456;
        return r60457;
}
↓
double f(double x, double y) {
        double r60458 = x;
        double r60459 = sqrt(r60458);
        double r60460 = y;
        double r60461 = 1.0;
        double r60462 = r60461 - r60458;
        double r60463 = fma(r60459, r60460, r60462);
        return r60463;
}

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 2019362 +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))))