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

Average Error: 0.1 → 0.1

Time: 12.4s

Precision: 64

Math

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

↓

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

C

double f(double x, double y) {
        double r1748729 = 1.0;
        double r1748730 = x;
        double r1748731 = r1748729 - r1748730;
        double r1748732 = y;
        double r1748733 = sqrt(r1748730);
        double r1748734 = r1748732 * r1748733;
        double r1748735 = r1748731 + r1748734;
        return r1748735;
}

↓

double f(double x, double y) {
        double r1748736 = y;
        double r1748737 = x;
        double r1748738 = sqrt(r1748737);
        double r1748739 = 1.0;
        double r1748740 = r1748739 - r1748737;
        double r1748741 = fma(r1748736, r1748738, r1748740);
        return r1748741;
}

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

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

3. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019156 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E"
  (+ (- 1.0 x) (* y (sqrt x))))