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

Average Error: 0.2 → 0.1

Time: 1.2s

Precision: 64

Math

\[\frac{x \cdot x - 3}{6}\]

↓

\[x \cdot \frac{x}{6} - \frac{3}{6}\]

C

double f(double x) {
        double r52637 = x;
        double r52638 = r52637 * r52637;
        double r52639 = 3.0;
        double r52640 = r52638 - r52639;
        double r52641 = 6.0;
        double r52642 = r52640 / r52641;
        return r52642;
}

↓

double f(double x) {
        double r52643 = x;
        double r52644 = 6.0;
        double r52645 = r52643 / r52644;
        double r52646 = r52643 * r52645;
        double r52647 = 3.0;
        double r52648 = r52647 / r52644;
        double r52649 = r52646 - r52648;
        return r52649;
}

Error

Bits error versus x

Derivation

1. Initial program 0.2

   \[\frac{x \cdot x - 3}{6}\]
2. Using strategy rm

3. Applied div-sub 0.2

   \[\leadsto \color{blue}{\frac{x \cdot x}{6} - \frac{3}{6}}\]
4. Using strategy rm

5. Applied associate-/l* 0.1

   \[\leadsto \color{blue}{\frac{x}{\frac{6}{x}}} - \frac{3}{6}\]
6. Using strategy rm

7. Applied add-sqr-sqrt 32.8

   \[\leadsto \frac{x}{\frac{6}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}} - \frac{3}{6}\]

8. Applied *-un-lft-identity 32.8

   \[\leadsto \frac{x}{\frac{\color{blue}{1 \cdot 6}}{\sqrt{x} \cdot \sqrt{x}}} - \frac{3}{6}\]

9. Applied times-frac 32.8

   \[\leadsto \frac{x}{\color{blue}{\frac{1}{\sqrt{x}} \cdot \frac{6}{\sqrt{x}}}} - \frac{3}{6}\]

10. Applied add-sqr-sqrt 32.8

    \[\leadsto \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\frac{1}{\sqrt{x}} \cdot \frac{6}{\sqrt{x}}} - \frac{3}{6}\]

11. Applied times-frac 32.8

    \[\leadsto \color{blue}{\frac{\sqrt{x}}{\frac{1}{\sqrt{x}}} \cdot \frac{\sqrt{x}}{\frac{6}{\sqrt{x}}}} - \frac{3}{6}\]

12. Simplified 32.8

    \[\leadsto \color{blue}{x} \cdot \frac{\sqrt{x}}{\frac{6}{\sqrt{x}}} - \frac{3}{6}\]

13. Simplified 0.1

    \[\leadsto x \cdot \color{blue}{\frac{x}{6}} - \frac{3}{6}\]

14. Final simplification 0.1

    \[\leadsto x \cdot \frac{x}{6} - \frac{3}{6}\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, H"
  :precision binary64
  (/ (- (* x x) 3) 6))