Average Error: 0.1 → 0.1
Time: 11.0s
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 r106977 = 1.0;
        double r106978 = x;
        double r106979 = r106977 - r106978;
        double r106980 = y;
        double r106981 = sqrt(r106978);
        double r106982 = r106980 * r106981;
        double r106983 = r106979 + r106982;
        return r106983;
}
↓
double f(double x, double y) {
        double r106984 = y;
        double r106985 = x;
        double r106986 = sqrt(r106985);
        double r106987 = 1.0;
        double r106988 = r106987 - r106985;
        double r106989 = fma(r106984, r106986, r106988);
        return r106989;
}

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 2020042 +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))))