Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E

Average Error: 0.1 → 0.1

Time: 17.9s

Precision: 64

Math

\[\left(1.0 - x\right) + y \cdot \sqrt{x}\]

↓

\[\mathsf{fma}\left(y, \sqrt{x}, 1.0\right) - x\]

C

double f(double x, double y) {
        double r4617910 = 1.0;
        double r4617911 = x;
        double r4617912 = r4617910 - r4617911;
        double r4617913 = y;
        double r4617914 = sqrt(r4617911);
        double r4617915 = r4617913 * r4617914;
        double r4617916 = r4617912 + r4617915;
        return r4617916;
}

↓

double f(double x, double y) {
        double r4617917 = y;
        double r4617918 = x;
        double r4617919 = sqrt(r4617918);
        double r4617920 = 1.0;
        double r4617921 = fma(r4617917, r4617919, r4617920);
        double r4617922 = r4617921 - r4617918;
        return r4617922;
}

Error

Bits error versus x

Bits error versus y

Derivation

1. Initial program 0.1

   \[\left(1.0 - x\right) + y \cdot \sqrt{x}\]

2. Simplified 0.1

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, \sqrt{x}, 1.0\right) - x}\]

3. Final simplification 0.1

   \[\leadsto \mathsf{fma}\left(y, \sqrt{x}, 1.0\right) - x\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E"
  (+ (- 1.0 x) (* y (sqrt x))))