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

Average Error: 0.2 → 0.3

Time: 13.2s

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 r367918 = 1.0;
        double r367919 = x;
        double r367920 = 9.0;
        double r367921 = r367919 * r367920;
        double r367922 = r367918 / r367921;
        double r367923 = r367918 - r367922;
        double r367924 = y;
        double r367925 = 3.0;
        double r367926 = sqrt(r367919);
        double r367927 = r367925 * r367926;
        double r367928 = r367924 / r367927;
        double r367929 = r367923 - r367928;
        return r367929;
}

↓

double f(double x, double y) {
        double r367930 = 1.0;
        double r367931 = x;
        double r367932 = r367930 / r367931;
        double r367933 = 9.0;
        double r367934 = r367932 / r367933;
        double r367935 = r367930 - r367934;
        double r367936 = 1.0;
        double r367937 = 3.0;
        double r367938 = cbrt(r367937);
        double r367939 = r367938 * r367938;
        double r367940 = r367936 / r367939;
        double r367941 = cbrt(r367938);
        double r367942 = r367936 / r367941;
        double r367943 = r367942 / r367941;
        double r367944 = y;
        double r367945 = r367944 / r367941;
        double r367946 = sqrt(r367931);
        double r367947 = r367945 / r367946;
        double r367948 = r367943 * r367947;
        double r367949 = r367940 * r367948;
        double r367950 = r367935 - r367949;
        return r367950;
}

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)))))