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

Average Error: 0.1 → 0.1

Time: 22.1s

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 r3786820 = 1.0;
        double r3786821 = x;
        double r3786822 = r3786820 - r3786821;
        double r3786823 = y;
        double r3786824 = sqrt(r3786821);
        double r3786825 = r3786823 * r3786824;
        double r3786826 = r3786822 + r3786825;
        return r3786826;
}

↓

double f(double x, double y) {
        double r3786827 = y;
        double r3786828 = x;
        double r3786829 = sqrt(r3786828);
        double r3786830 = 1.0;
        double r3786831 = fma(r3786827, r3786829, r3786830);
        double r3786832 = r3786831 - r3786828;
        return r3786832;
}

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 2019158 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E"
  (+ (- 1.0 x) (* y (sqrt x))))