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

Average Error: 0.1 → 0.1

Time: 19.5s

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 r4537504 = 1.0;
        double r4537505 = x;
        double r4537506 = r4537504 - r4537505;
        double r4537507 = y;
        double r4537508 = sqrt(r4537505);
        double r4537509 = r4537507 * r4537508;
        double r4537510 = r4537506 + r4537509;
        return r4537510;
}

↓

double f(double x, double y) {
        double r4537511 = y;
        double r4537512 = x;
        double r4537513 = sqrt(r4537512);
        double r4537514 = 1.0;
        double r4537515 = fma(r4537511, r4537513, r4537514);
        double r4537516 = r4537515 - r4537512;
        return r4537516;
}

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))))