Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, B

Average Error: 0.4 → 0.4

Time: 4.9s

Precision: 64

Math

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

↓

\[\left(\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot {\left({\left(\sqrt[3]{3}\right)}^{2} \cdot x\right)}^{\frac{1}{2}}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]

C

double f(double x, double y) {
        double r375981 = 3.0;
        double r375982 = x;
        double r375983 = sqrt(r375982);
        double r375984 = r375981 * r375983;
        double r375985 = y;
        double r375986 = 1.0;
        double r375987 = 9.0;
        double r375988 = r375982 * r375987;
        double r375989 = r375986 / r375988;
        double r375990 = r375985 + r375989;
        double r375991 = r375990 - r375986;
        double r375992 = r375984 * r375991;
        return r375992;
}

↓

double f(double x, double y) {
        double r375993 = 3.0;
        double r375994 = cbrt(r375993);
        double r375995 = r375994 * r375994;
        double r375996 = 2.0;
        double r375997 = pow(r375994, r375996);
        double r375998 = x;
        double r375999 = r375997 * r375998;
        double r376000 = 0.5;
        double r376001 = pow(r375999, r376000);
        double r376002 = r375995 * r376001;
        double r376003 = y;
        double r376004 = 1.0;
        double r376005 = 9.0;
        double r376006 = r375998 * r376005;
        double r376007 = r376004 / r376006;
        double r376008 = r376003 + r376007;
        double r376009 = r376008 - r376004;
        double r376010 = r376002 * r376009;
        return r376010;
}

Error

Bits error versus x

Bits error versus y

Target

Original	0.4
Target	0.4
Herbie	0.4

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

Derivation

1. Initial program 0.4

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

3. Applied add-cube-cbrt 0.4

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

4. Applied associate-*l* 0.6

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

6. Applied add-sqr-sqrt 0.6

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

7. Applied associate-*l* 0.5

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

9. Applied pow1 0.5

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

10. Applied sqrt-pow1 0.5

    \[\leadsto \left(\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \left(\sqrt{\sqrt[3]{3}} \cdot \left(\sqrt{\sqrt[3]{3}} \cdot \color{blue}{{x}^{\left(\frac{1}{2}\right)}}\right)\right)\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]

11. Applied pow1 0.5

    \[\leadsto \left(\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \left(\sqrt{\sqrt[3]{3}} \cdot \left(\sqrt{\color{blue}{{\left(\sqrt[3]{3}\right)}^{1}}} \cdot {x}^{\left(\frac{1}{2}\right)}\right)\right)\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]

12. Applied sqrt-pow1 0.5

    \[\leadsto \left(\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \left(\sqrt{\sqrt[3]{3}} \cdot \left(\color{blue}{{\left(\sqrt[3]{3}\right)}^{\left(\frac{1}{2}\right)}} \cdot {x}^{\left(\frac{1}{2}\right)}\right)\right)\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]

13. Applied pow-prod-down 0.5

    \[\leadsto \left(\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \left(\sqrt{\sqrt[3]{3}} \cdot \color{blue}{{\left(\sqrt[3]{3} \cdot x\right)}^{\left(\frac{1}{2}\right)}}\right)\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]

14. Applied pow1 0.5

    \[\leadsto \left(\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \left(\sqrt{\color{blue}{{\left(\sqrt[3]{3}\right)}^{1}}} \cdot {\left(\sqrt[3]{3} \cdot x\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]

15. Applied sqrt-pow1 0.5

    \[\leadsto \left(\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \left(\color{blue}{{\left(\sqrt[3]{3}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\sqrt[3]{3} \cdot x\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]

16. Applied pow-prod-down 0.6

    \[\leadsto \left(\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \color{blue}{{\left(\sqrt[3]{3} \cdot \left(\sqrt[3]{3} \cdot x\right)\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]

17. Simplified 0.4

    \[\leadsto \left(\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot {\color{blue}{\left({\left(\sqrt[3]{3}\right)}^{2} \cdot x\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]

18. Final simplification 0.4

    \[\leadsto \left(\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot {\left({\left(\sqrt[3]{3}\right)}^{2} \cdot x\right)}^{\frac{1}{2}}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]

Reproduce

herbie shell --seed 2019353 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

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

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