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 r94498 = 1.0;
        double r94499 = x;
        double r94500 = r94498 - r94499;
        double r94501 = y;
        double r94502 = sqrt(r94499);
        double r94503 = r94501 * r94502;
        double r94504 = r94500 + r94503;
        return r94504;
}
↓
double f(double x, double y) {
        double r94505 = x;
        double r94506 = sqrt(r94505);
        double r94507 = y;
        double r94508 = 1.0;
        double r94509 = r94508 - r94505;
        double r94510 = fma(r94506, r94507, r94509);
        return r94510;
}

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 2020018 +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))))