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 r126428 = 1.0;
        double r126429 = x;
        double r126430 = r126428 - r126429;
        double r126431 = y;
        double r126432 = sqrt(r126429);
        double r126433 = r126431 * r126432;
        double r126434 = r126430 + r126433;
        return r126434;
}
↓
double f(double x, double y) {
        double r126435 = x;
        double r126436 = sqrt(r126435);
        double r126437 = y;
        double r126438 = 1.0;
        double r126439 = r126438 - r126435;
        double r126440 = fma(r126436, r126437, r126439);
        return r126440;
}

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 2020062 +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))))