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

Average Error: 0.1 → 0.1

Time: 20.1s

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 r4843783 = 1.0;
        double r4843784 = x;
        double r4843785 = r4843783 - r4843784;
        double r4843786 = y;
        double r4843787 = sqrt(r4843784);
        double r4843788 = r4843786 * r4843787;
        double r4843789 = r4843785 + r4843788;
        return r4843789;
}

↓

double f(double x, double y) {
        double r4843790 = y;
        double r4843791 = x;
        double r4843792 = sqrt(r4843791);
        double r4843793 = 1.0;
        double r4843794 = fma(r4843790, r4843792, r4843793);
        double r4843795 = r4843794 - r4843791;
        return r4843795;
}

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 2019164 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E"
  (+ (- 1.0 x) (* y (sqrt x))))