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

Average Error: 0.4 → 0.4

Time: 11.6min

Precision: binary64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

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

C

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

↓

double code(double x, double y) {
	return ((double) (((double) (3.0 * ((double) (((double) (y + (1.0 / ((double) (x * 9.0))))) - 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 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 pow1 0.6

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

7. Applied pow1 0.6

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

8. Applied pow1 0.6

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

9. Applied pow-prod-down 0.6

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

10. Applied pow1 0.6

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

11. Applied pow1 0.6

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

12. Applied pow-prod-down 0.6

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

13. Applied pow-prod-down 0.6

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

14. Applied pow-prod-down 0.6

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

15. Simplified 0.4

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

16. Final simplification 0.4

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

Reproduce

herbie shell --seed 2020181 
(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)))