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

Average Error: 0.4 → 0.4

Time: 3.4s

Precision: binary64

Math

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

↓

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

C

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

↓

double code(double x, double y) {
	return ((double) (((double) (3.0 * ((double) (((double) (((double) (((double) (1.0 / ((double) sqrt(x)))) * ((double) (((double) (1.0 / 9.0)) / ((double) sqrt(x)))))) + y)) - 1.0)))) * ((double) 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 *-un-lft-identity 0.4

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

6. Applied times-frac 0.4

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

8. Applied pow1 0.4

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

9. Applied pow1 0.4

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

10. Applied pow-prod-down 0.4

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

11. Applied pow1 0.4

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

12. Applied pow-prod-down 0.4

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

13. Simplified 0.4

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

15. Applied add-sqr-sqrt 0.4

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

16. Applied *-un-lft-identity 0.4

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

17. Applied *-un-lft-identity 0.4

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

18. Applied times-frac 0.4

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

19. Applied times-frac 0.4

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

20. Simplified 0.4

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

21. Final simplification 0.4

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

Reproduce

herbie shell --seed 2020169 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (* 3.0 (+ (* y (sqrt x)) (* (- (/ 1.0 (* x 9.0)) 1.0) (sqrt x))))

  (* (* 3.0 (sqrt x)) (- (+ y (/ 1.0 (* x 9.0))) 1.0)))