Average Error: 0.1 → 0.1
Time: 17.2s
Precision: 64
\[\left(1 - x\right) + y \cdot \sqrt{x}\]
↓
\[\mathsf{fma}\left(y, \sqrt{x}, 1 - x\right)\]
\left(1 - x\right) + y \cdot \sqrt{x}
↓
\mathsf{fma}\left(y, \sqrt{x}, 1 - x\right)
double f(double x, double y) {
        double r136432 = 1.0;
        double r136433 = x;
        double r136434 = r136432 - r136433;
        double r136435 = y;
        double r136436 = sqrt(r136433);
        double r136437 = r136435 * r136436;
        double r136438 = r136434 + r136437;
        return r136438;
}
↓
double f(double x, double y) {
        double r136439 = y;
        double r136440 = x;
        double r136441 = sqrt(r136440);
        double r136442 = 1.0;
        double r136443 = r136442 - r136440;
        double r136444 = fma(r136439, r136441, r136443);
        return r136444;
}

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

    \[\leadsto \color{blue}{\mathsf{fma}\left(y, \sqrt{x}, 1 - x\right)}\]

  3. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(y, \sqrt{x}, 1 - x\right)\]

Reproduce

herbie shell --seed 2019325 +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))))