Numeric.Integration.TanhSinh:everywhere from integration-0.2.1

Average Error: 5.2 → 0.1

Time: 3.5s

Precision: 64

Math

\[x \cdot \left(1 + y \cdot y\right)\]

↓

\[\mathsf{hypot}\left(\sqrt{1}, y\right) \cdot \left(x \cdot \mathsf{hypot}\left(\sqrt{1}, y\right)\right)\]

C

double code(double x, double y) {
	return (x * (1.0 + (y * y)));
}

↓

double code(double x, double y) {
	return (hypot(sqrt(1.0), y) * (x * hypot(sqrt(1.0), y)));
}

Error

Bits error versus x

Bits error versus y

Target

Original	5.2
Target	0.1
Herbie	0.1

\[x + \left(x \cdot y\right) \cdot y\]

Derivation

1. Initial program 5.2

   \[x \cdot \left(1 + y \cdot y\right)\]
2. Using strategy rm

3. Applied add-sqr-sqrt 5.3

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

4. Applied associate-*r* 5.3

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

6. Applied add-sqr-sqrt 5.3

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

7. Applied sqrt-prod 5.4

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

8. Applied associate-*r* 5.3

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

9. Simplified 5.3

   \[\leadsto \color{blue}{\left(\left(x \cdot \sqrt{\sqrt{1 + y \cdot y}}\right) \cdot \mathsf{hypot}\left(\sqrt{1}, y\right)\right)} \cdot \sqrt{\sqrt{1 + y \cdot y}}\]
10. Using strategy rm

11. Applied pow1 5.3

    \[\leadsto \left(\left(x \cdot \sqrt{\sqrt{1 + y \cdot y}}\right) \cdot \mathsf{hypot}\left(\sqrt{1}, y\right)\right) \cdot \color{blue}{{\left(\sqrt{\sqrt{1 + y \cdot y}}\right)}^{1}}\]

12. Applied pow1 5.3

    \[\leadsto \left(\left(x \cdot \sqrt{\sqrt{1 + y \cdot y}}\right) \cdot \color{blue}{{\left(\mathsf{hypot}\left(\sqrt{1}, y\right)\right)}^{1}}\right) \cdot {\left(\sqrt{\sqrt{1 + y \cdot y}}\right)}^{1}\]

13. Applied pow1 5.3

    \[\leadsto \left(\left(x \cdot \color{blue}{{\left(\sqrt{\sqrt{1 + y \cdot y}}\right)}^{1}}\right) \cdot {\left(\mathsf{hypot}\left(\sqrt{1}, y\right)\right)}^{1}\right) \cdot {\left(\sqrt{\sqrt{1 + y \cdot y}}\right)}^{1}\]

14. Applied pow1 5.3

    \[\leadsto \left(\left(\color{blue}{{x}^{1}} \cdot {\left(\sqrt{\sqrt{1 + y \cdot y}}\right)}^{1}\right) \cdot {\left(\mathsf{hypot}\left(\sqrt{1}, y\right)\right)}^{1}\right) \cdot {\left(\sqrt{\sqrt{1 + y \cdot y}}\right)}^{1}\]

15. Applied pow-prod-down 5.3

    \[\leadsto \left(\color{blue}{{\left(x \cdot \sqrt{\sqrt{1 + y \cdot y}}\right)}^{1}} \cdot {\left(\mathsf{hypot}\left(\sqrt{1}, y\right)\right)}^{1}\right) \cdot {\left(\sqrt{\sqrt{1 + y \cdot y}}\right)}^{1}\]

16. Applied pow-prod-down 5.3

    \[\leadsto \color{blue}{{\left(\left(x \cdot \sqrt{\sqrt{1 + y \cdot y}}\right) \cdot \mathsf{hypot}\left(\sqrt{1}, y\right)\right)}^{1}} \cdot {\left(\sqrt{\sqrt{1 + y \cdot y}}\right)}^{1}\]

17. Applied pow-prod-down 5.3

    \[\leadsto \color{blue}{{\left(\left(\left(x \cdot \sqrt{\sqrt{1 + y \cdot y}}\right) \cdot \mathsf{hypot}\left(\sqrt{1}, y\right)\right) \cdot \sqrt{\sqrt{1 + y \cdot y}}\right)}^{1}}\]

18. Simplified 0.1

    \[\leadsto {\color{blue}{\left(\mathsf{hypot}\left(\sqrt{1}, y\right) \cdot \left(x \cdot \mathsf{hypot}\left(\sqrt{1}, y\right)\right)\right)}}^{1}\]

19. Final simplification 0.1

    \[\leadsto \mathsf{hypot}\left(\sqrt{1}, y\right) \cdot \left(x \cdot \mathsf{hypot}\left(\sqrt{1}, y\right)\right)\]

Reproduce

herbie shell --seed 2020049 +o rules:numerics
(FPCore (x y)
  :name "Numeric.Integration.TanhSinh:everywhere from integration-0.2.1"
  :precision binary64

  :herbie-target
  (+ x (* (* x y) y))

  (* x (+ 1 (* y y))))