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

Average Error: 0.2 → 0.2

Time: 13.4s

Precision: 64

Math

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

↓

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

C

double f(double x, double y) {
        double r395240 = 1.0;
        double r395241 = x;
        double r395242 = 9.0;
        double r395243 = r395241 * r395242;
        double r395244 = r395240 / r395243;
        double r395245 = r395240 - r395244;
        double r395246 = y;
        double r395247 = 3.0;
        double r395248 = sqrt(r395241);
        double r395249 = r395247 * r395248;
        double r395250 = r395246 / r395249;
        double r395251 = r395245 - r395250;
        return r395251;
}

↓

double f(double x, double y) {
        double r395252 = 1.0;
        double r395253 = x;
        double r395254 = 9.0;
        double r395255 = r395253 * r395254;
        double r395256 = r395252 / r395255;
        double r395257 = r395252 - r395256;
        double r395258 = 1.0;
        double r395259 = 3.0;
        double r395260 = cbrt(r395259);
        double r395261 = r395260 * r395260;
        double r395262 = cbrt(r395261);
        double r395263 = r395262 * r395261;
        double r395264 = r395258 / r395263;
        double r395265 = sqrt(r395253);
        double r395266 = y;
        double r395267 = cbrt(r395260);
        double r395268 = r395266 / r395267;
        double r395269 = r395265 / r395268;
        double r395270 = r395264 / r395269;
        double r395271 = r395257 - r395270;
        return r395271;
}

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 0.2

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

8. Applied associate-/l* 0.2

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

10. Applied add-cube-cbrt 0.2

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

11. Applied cbrt-prod 0.2

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

12. Applied *-un-lft-identity 0.2

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

13. Applied times-frac 0.3

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

14. Applied *-un-lft-identity 0.3

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

15. Applied times-frac 0.3

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

16. Applied associate-/r* 0.2

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

17. Simplified 0.2

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

18. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019304 
(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)))))