Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, B

Average Error: 0.4 → 0.4

Time: 16.4s

Precision: 64

Math

\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]

↓

\[\mathsf{fma}\left(3, y, \left(\frac{\frac{0.3333333333333333148296162562473909929395}{\sqrt{x}}}{\sqrt{x}} - 3\right) + \left(\left(-3\right) + 3\right)\right) \cdot \sqrt{x}\]

C

double f(double x, double y) {
        double r348354 = 3.0;
        double r348355 = x;
        double r348356 = sqrt(r348355);
        double r348357 = r348354 * r348356;
        double r348358 = y;
        double r348359 = 1.0;
        double r348360 = 9.0;
        double r348361 = r348355 * r348360;
        double r348362 = r348359 / r348361;
        double r348363 = r348358 + r348362;
        double r348364 = r348363 - r348359;
        double r348365 = r348357 * r348364;
        return r348365;
}

↓

double f(double x, double y) {
        double r348366 = 3.0;
        double r348367 = y;
        double r348368 = 0.3333333333333333;
        double r348369 = x;
        double r348370 = sqrt(r348369);
        double r348371 = r348368 / r348370;
        double r348372 = r348371 / r348370;
        double r348373 = r348372 - r348366;
        double r348374 = -r348366;
        double r348375 = r348374 + r348366;
        double r348376 = r348373 + r348375;
        double r348377 = fma(r348366, r348367, r348376);
        double r348378 = r348377 * r348370;
        return r348378;
}

Error

Bits error versus x

Bits error versus y

Target

Original	0.4
Target	0.4
Herbie	0.4

\[3 \cdot \left(y \cdot \sqrt{x} + \left(\frac{1}{x \cdot 9} - 1\right) \cdot \sqrt{x}\right)\]

Derivation

1. Initial program 0.4

   \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]
2. Using strategy rm

3. Applied associate-*l* 0.4

   \[\leadsto \color{blue}{3 \cdot \left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)}\]
4. Using strategy rm

5. Applied pow1 0.4

   \[\leadsto 3 \cdot \left(\sqrt{x} \cdot \color{blue}{{\left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)}^{1}}\right)\]

6. Applied pow1 0.4

   \[\leadsto 3 \cdot \left(\color{blue}{{\left(\sqrt{x}\right)}^{1}} \cdot {\left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)}^{1}\right)\]

7. Applied pow-prod-down 0.4

   \[\leadsto 3 \cdot \color{blue}{{\left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)}^{1}}\]

8. Applied pow1 0.4

   \[\leadsto \color{blue}{{3}^{1}} \cdot {\left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)}^{1}\]

9. Applied pow-prod-down 0.4

   \[\leadsto \color{blue}{{\left(3 \cdot \left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)\right)}^{1}}\]

10. Simplified 0.4

    \[\leadsto {\color{blue}{\left(\left(3 \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right) \cdot \sqrt{x}\right)}}^{1}\]

11. Taylor expanded around 0 0.4

    \[\leadsto {\left(\color{blue}{\left(\left(3 \cdot y + 0.3333333333333333148296162562473909929395 \cdot \frac{1}{x}\right) - 3\right)} \cdot \sqrt{x}\right)}^{1}\]

12. Simplified 0.4

    \[\leadsto {\left(\color{blue}{\mathsf{fma}\left(3, y, \frac{0.3333333333333333148296162562473909929395}{x} - 3\right)} \cdot \sqrt{x}\right)}^{1}\]
13. Using strategy rm

14. Applied add-cube-cbrt 0.4

    \[\leadsto {\left(\mathsf{fma}\left(3, y, \frac{0.3333333333333333148296162562473909929395}{x} - \color{blue}{\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \sqrt[3]{3}}\right) \cdot \sqrt{x}\right)}^{1}\]

15. Applied add-sqr-sqrt 0.4

    \[\leadsto {\left(\mathsf{fma}\left(3, y, \frac{0.3333333333333333148296162562473909929395}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}} - \left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \sqrt[3]{3}\right) \cdot \sqrt{x}\right)}^{1}\]

16. Applied *-un-lft-identity 0.4

    \[\leadsto {\left(\mathsf{fma}\left(3, y, \frac{\color{blue}{1 \cdot 0.3333333333333333148296162562473909929395}}{\sqrt{x} \cdot \sqrt{x}} - \left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \sqrt[3]{3}\right) \cdot \sqrt{x}\right)}^{1}\]

17. Applied times-frac 0.4

    \[\leadsto {\left(\mathsf{fma}\left(3, y, \color{blue}{\frac{1}{\sqrt{x}} \cdot \frac{0.3333333333333333148296162562473909929395}{\sqrt{x}}} - \left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \sqrt[3]{3}\right) \cdot \sqrt{x}\right)}^{1}\]

18. Applied prod-diff 0.4

    \[\leadsto {\left(\mathsf{fma}\left(3, y, \color{blue}{\mathsf{fma}\left(\frac{1}{\sqrt{x}}, \frac{0.3333333333333333148296162562473909929395}{\sqrt{x}}, -\sqrt[3]{3} \cdot \left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{3}, \sqrt[3]{3} \cdot \sqrt[3]{3}, \sqrt[3]{3} \cdot \left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right)\right)}\right) \cdot \sqrt{x}\right)}^{1}\]

19. Simplified 0.4

    \[\leadsto {\left(\mathsf{fma}\left(3, y, \color{blue}{\left(\frac{\frac{0.3333333333333333148296162562473909929395}{\sqrt{x}}}{\sqrt{x}} - 3\right)} + \mathsf{fma}\left(-\sqrt[3]{3}, \sqrt[3]{3} \cdot \sqrt[3]{3}, \sqrt[3]{3} \cdot \left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right)\right)\right) \cdot \sqrt{x}\right)}^{1}\]

20. Simplified 0.4

    \[\leadsto {\left(\mathsf{fma}\left(3, y, \left(\frac{\frac{0.3333333333333333148296162562473909929395}{\sqrt{x}}}{\sqrt{x}} - 3\right) + \color{blue}{\left(\left(-3\right) + 3\right)}\right) \cdot \sqrt{x}\right)}^{1}\]

21. Final simplification 0.4

    \[\leadsto \mathsf{fma}\left(3, y, \left(\frac{\frac{0.3333333333333333148296162562473909929395}{\sqrt{x}}}{\sqrt{x}} - 3\right) + \left(\left(-3\right) + 3\right)\right) \cdot \sqrt{x}\]

Reproduce

herbie shell --seed 2019212 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (* 3 (+ (* y (sqrt x)) (* (- (/ 1 (* x 9)) 1) (sqrt x))))

  (* (* 3 (sqrt x)) (- (+ y (/ 1 (* x 9))) 1)))