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

Average Error: 0.2 → 0.2

Time: 16.3s

Precision: 64

Math

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

↓

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

C

double f(double x, double y) {
        double r413553 = 1.0;
        double r413554 = x;
        double r413555 = 9.0;
        double r413556 = r413554 * r413555;
        double r413557 = r413553 / r413556;
        double r413558 = r413553 - r413557;
        double r413559 = y;
        double r413560 = 3.0;
        double r413561 = sqrt(r413554);
        double r413562 = r413560 * r413561;
        double r413563 = r413559 / r413562;
        double r413564 = r413558 - r413563;
        return r413564;
}

↓

double f(double x, double y) {
        double r413565 = 1.0;
        double r413566 = x;
        double r413567 = 9.0;
        double r413568 = r413566 * r413567;
        double r413569 = r413565 / r413568;
        double r413570 = r413565 - r413569;
        double r413571 = 1.0;
        double r413572 = 3.0;
        double r413573 = cbrt(r413572);
        double r413574 = cbrt(r413573);
        double r413575 = r413574 * r413574;
        double r413576 = r413573 * r413573;
        double r413577 = r413575 * r413576;
        double r413578 = r413571 / r413577;
        double r413579 = sqrt(r413566);
        double r413580 = y;
        double r413581 = r413580 / r413574;
        double r413582 = r413579 / r413581;
        double r413583 = r413578 / r413582;
        double r413584 = r413570 - r413583;
        return r413584;
}

Error

Bits error versus x

Bits error versus y

Target

Original	0.2
Target	0.2
Herbie	0.2

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

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 add-cube-cbrt 0.2

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

6. Applied *-un-lft-identity 0.2

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

7. Applied times-frac 0.3

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

8. Applied associate-/l* 0.3

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

10. Applied add-cube-cbrt 0.3

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

11. Applied *-un-lft-identity 0.3

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

12. Applied times-frac 0.3

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

13. Applied *-un-lft-identity 0.3

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

14. Applied sqrt-prod 0.3

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

15. Applied times-frac 0.3

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

16. Applied associate-/r* 0.2

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

17. Simplified 0.2

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

18. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019356 
(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)))))