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

Average Error: 0.2 → 0.2

Time: 17.2s

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 r435181 = 1.0;
        double r435182 = x;
        double r435183 = 9.0;
        double r435184 = r435182 * r435183;
        double r435185 = r435181 / r435184;
        double r435186 = r435181 - r435185;
        double r435187 = y;
        double r435188 = 3.0;
        double r435189 = sqrt(r435182);
        double r435190 = r435188 * r435189;
        double r435191 = r435187 / r435190;
        double r435192 = r435186 - r435191;
        return r435192;
}

↓

double f(double x, double y) {
        double r435193 = 1.0;
        double r435194 = x;
        double r435195 = 9.0;
        double r435196 = r435194 * r435195;
        double r435197 = r435193 / r435196;
        double r435198 = r435193 - r435197;
        double r435199 = 1.0;
        double r435200 = 3.0;
        double r435201 = cbrt(r435200);
        double r435202 = cbrt(r435201);
        double r435203 = r435202 * r435202;
        double r435204 = r435201 * r435201;
        double r435205 = r435203 * r435204;
        double r435206 = r435199 / r435205;
        double r435207 = sqrt(r435194);
        double r435208 = y;
        double r435209 = r435208 / r435202;
        double r435210 = r435207 / r435209;
        double r435211 = r435206 / r435210;
        double r435212 = r435198 - r435211;
        return r435212;
}

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 +o rules:numerics
(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)))))