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

Average Error: 0.4 → 0.4

Time: 5.7s

Precision: 64

Math

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

↓

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

C

double code(double x, double y) {
	return ((3.0 * sqrt(x)) * ((y + (1.0 / (x * 9.0))) - 1.0));
}

↓

double code(double x, double y) {
	return ((3.0 * ((((((pow(cbrt(1.0), 2.0) * cbrt(1.0)) / sqrt(x)) / 9.0) / sqrt(x)) + y) - 1.0)) * sqrt(x));
}

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 associate-*l* 0.4

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

5. Applied associate-/r* 0.4

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

7. Applied *-un-lft-identity 0.4

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

8. Applied add-sqr-sqrt 0.4

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

9. Applied add-cube-cbrt 0.4

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

10. Applied times-frac 0.5

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

11. Applied times-frac 0.4

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

12. Simplified 0.4

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

14. Applied pow1 0.4

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

15. Applied pow1 0.4

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

16. Applied pow-prod-down 0.4

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

17. Applied pow1 0.4

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

18. Applied pow-prod-down 0.4

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

19. Simplified 0.4

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

20. Final simplification 0.4

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

Reproduce

herbie shell --seed 2020057 +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)))