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

Average Error: 0.4 → 0.4

Time: 4.6s

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 r427698 = 3.0;
        double r427699 = x;
        double r427700 = sqrt(r427699);
        double r427701 = r427698 * r427700;
        double r427702 = y;
        double r427703 = 1.0;
        double r427704 = 9.0;
        double r427705 = r427699 * r427704;
        double r427706 = r427703 / r427705;
        double r427707 = r427702 + r427706;
        double r427708 = r427707 - r427703;
        double r427709 = r427701 * r427708;
        return r427709;
}

↓

double f(double x, double y) {
        double r427710 = 3.0;
        double r427711 = cbrt(r427710);
        double r427712 = r427711 * r427711;
        double r427713 = 2.0;
        double r427714 = pow(r427711, r427713);
        double r427715 = x;
        double r427716 = r427714 * r427715;
        double r427717 = 0.5;
        double r427718 = pow(r427716, r427717);
        double r427719 = r427712 * r427718;
        double r427720 = y;
        double r427721 = 1.0;
        double r427722 = 9.0;
        double r427723 = r427715 * r427722;
        double r427724 = r427721 / r427723;
        double r427725 = r427720 + r427724;
        double r427726 = r427725 - r427721;
        double r427727 = r427719 * r427726;
        return r427727;
}

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 
(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)))