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

Average Error: 0.4 → 0.4

Time: 3.2s

Precision: binary64

Math

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

↓

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

C

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

↓

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

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. Simplified 0.4

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

4. Applied associate-/r* 0.4

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

6. Applied add-sqr-sqrt 0.4

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

7. Applied add-sqr-sqrt 0.4

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

8. Applied *-un-lft-identity 0.4

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

9. Applied times-frac 0.5

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

10. Applied times-frac 0.5

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

11. Simplified 0.4

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

12. Simplified 0.5

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

14. Applied pow1 0.5

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

15. Applied pow1 0.5

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

16. Applied pow-prod-down 0.5

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

17. Applied pow1 0.5

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

18. Applied pow-prod-down 0.5

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

19. Simplified 0.4

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

20. Final simplification 0.4

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

Reproduce

herbie shell --seed 2020196 
(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)))