Average Error: 0.1 → 0.1
Time: 12.2s
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 r105910 = 1.0;
        double r105911 = x;
        double r105912 = r105910 - r105911;
        double r105913 = y;
        double r105914 = sqrt(r105911);
        double r105915 = r105913 * r105914;
        double r105916 = r105912 + r105915;
        return r105916;
}
↓
double f(double x, double y) {
        double r105917 = y;
        double r105918 = x;
        double r105919 = sqrt(r105918);
        double r105920 = 1.0;
        double r105921 = r105920 - r105918;
        double r105922 = fma(r105917, r105919, r105921);
        return r105922;
}

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))))