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

Average Error: 0.4 → 0.4

Time: 4.5s

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 r476469 = 3.0;
        double r476470 = x;
        double r476471 = sqrt(r476470);
        double r476472 = r476469 * r476471;
        double r476473 = y;
        double r476474 = 1.0;
        double r476475 = 9.0;
        double r476476 = r476470 * r476475;
        double r476477 = r476474 / r476476;
        double r476478 = r476473 + r476477;
        double r476479 = r476478 - r476474;
        double r476480 = r476472 * r476479;
        return r476480;
}

↓

double f(double x, double y) {
        double r476481 = 3.0;
        double r476482 = cbrt(r476481);
        double r476483 = r476482 * r476482;
        double r476484 = 2.0;
        double r476485 = pow(r476482, r476484);
        double r476486 = x;
        double r476487 = r476485 * r476486;
        double r476488 = 0.5;
        double r476489 = pow(r476487, r476488);
        double r476490 = r476483 * r476489;
        double r476491 = y;
        double r476492 = 1.0;
        double r476493 = 9.0;
        double r476494 = r476486 * r476493;
        double r476495 = r476492 / r476494;
        double r476496 = r476491 + r476495;
        double r476497 = r476496 - r476492;
        double r476498 = r476490 * r476497;
        return r476498;
}

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)))