Average Error: 0.1 → 0.1
Time: 4.9s
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 r90416 = 1.0;
        double r90417 = x;
        double r90418 = r90416 - r90417;
        double r90419 = y;
        double r90420 = sqrt(r90417);
        double r90421 = r90419 * r90420;
        double r90422 = r90418 + r90421;
        return r90422;
}
↓
double f(double x, double y) {
        double r90423 = x;
        double r90424 = sqrt(r90423);
        double r90425 = y;
        double r90426 = 1.0;
        double r90427 = r90426 - r90423;
        double r90428 = fma(r90424, r90425, r90427);
        return r90428;
}

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 2020035 +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))))