Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D

Average Error: 0.2 → 0.3

Time: 13.7s

Precision: 64

Math

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

↓

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

C

double f(double x, double y) {
        double r250976 = 1.0;
        double r250977 = x;
        double r250978 = 9.0;
        double r250979 = r250977 * r250978;
        double r250980 = r250976 / r250979;
        double r250981 = r250976 - r250980;
        double r250982 = y;
        double r250983 = 3.0;
        double r250984 = sqrt(r250977);
        double r250985 = r250983 * r250984;
        double r250986 = r250982 / r250985;
        double r250987 = r250981 - r250986;
        return r250987;
}

↓

double f(double x, double y) {
        double r250988 = 1.0;
        double r250989 = x;
        double r250990 = r250988 / r250989;
        double r250991 = 9.0;
        double r250992 = r250990 / r250991;
        double r250993 = r250988 - r250992;
        double r250994 = 1.0;
        double r250995 = 3.0;
        double r250996 = cbrt(r250995);
        double r250997 = r250996 * r250996;
        double r250998 = r250994 / r250997;
        double r250999 = cbrt(r250996);
        double r251000 = r250994 / r250999;
        double r251001 = r251000 / r250999;
        double r251002 = y;
        double r251003 = r251002 / r250999;
        double r251004 = sqrt(r250989);
        double r251005 = r251003 / r251004;
        double r251006 = r251001 * r251005;
        double r251007 = r250998 * r251006;
        double r251008 = r250993 - r251007;
        return r251008;
}

Error

Bits error versus x

Bits error versus y

Target

Original	0.2
Target	0.2
Herbie	0.3

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

Derivation

1. Initial program 0.2

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

3. Applied associate-/r* 0.2

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

5. Applied associate-/r* 0.2

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

7. Applied *-un-lft-identity 0.2

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

8. Applied sqrt-prod 0.2

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

9. Applied add-cube-cbrt 0.2

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

10. Applied *-un-lft-identity 0.2

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

11. Applied times-frac 0.3

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

12. Applied times-frac 0.3

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

13. Simplified 0.3

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

15. Applied *-un-lft-identity 0.3

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

16. Applied sqrt-prod 0.3

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

17. Applied add-cube-cbrt 0.3

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

18. Applied *-un-lft-identity 0.3

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

19. Applied times-frac 0.3

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

20. Applied times-frac 0.3

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

21. Simplified 0.3

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

22. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019326 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D"
  :precision binary64

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

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