Diagrams.Tangent:$catParam from diagrams-lib-1.3.0.3, E

Average Error: 0.2 → 0.3

Time: 2.8s

Precision: 64

Math

\[\left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x\]

↓

\[\left(3 \cdot x\right) \cdot \left(2 - x \cdot 3\right)\]

C

double code(double x) {
	return ((double) (((double) (3.0 * ((double) (2.0 - ((double) (x * 3.0)))))) * x));
}

↓

double code(double x) {
	return ((double) (((double) (3.0 * x)) * ((double) (2.0 - ((double) (x * 3.0))))));
}

Error

Bits error versus x

Target

Original	0.2
Target	0.2
Herbie	0.3

\[6 \cdot x - 9 \cdot \left(x \cdot x\right)\]

Derivation

1. Initial program 0.2

   \[\left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x\]
2. Using strategy rm

3. Applied associate-*l* 0.2

   \[\leadsto \color{blue}{3 \cdot \left(\left(2 - x \cdot 3\right) \cdot x\right)}\]
4. Using strategy rm

5. Applied add-cube-cbrt 0.2

   \[\leadsto \color{blue}{\left(\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \sqrt[3]{3}\right)} \cdot \left(\left(2 - x \cdot 3\right) \cdot x\right)\]

6. Applied associate-*l* 0.5

   \[\leadsto \color{blue}{\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \left(\sqrt[3]{3} \cdot \left(\left(2 - x \cdot 3\right) \cdot x\right)\right)}\]
7. Using strategy rm

8. Applied pow1 0.5

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

9. Applied pow1 0.5

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

10. Applied pow-prod-down 0.5

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

11. Applied pow1 0.5

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

12. Applied pow-prod-down 0.5

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

13. Applied pow1 0.5

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

14. Applied pow1 0.5

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

15. Applied pow-prod-down 0.5

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

16. Applied pow-prod-down 0.5

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

17. Simplified 0.3

    \[\leadsto {\color{blue}{\left(\left(3 \cdot x\right) \cdot \left(2 - x \cdot 3\right)\right)}}^{1}\]

18. Final simplification 0.3

    \[\leadsto \left(3 \cdot x\right) \cdot \left(2 - x \cdot 3\right)\]

Reproduce

herbie shell --seed 2020121 +o rules:numerics
(FPCore (x)
  :name "Diagrams.Tangent:$catParam from diagrams-lib-1.3.0.3, E"
  :precision binary64

  :herbie-target
  (- (* 6 x) (* 9 (* x x)))

  (* (* 3 (- 2 (* x 3))) x))