Diagrams.Segment:$catParam from diagrams-lib-1.3.0.3, A

Average Error: 10.1 → 0.3

Time: 1.6s

Precision: 64

Math

\[\left(\left(x \cdot 3\right) \cdot x\right) \cdot y\]

↓

\[\left(3 \cdot \left(x \cdot y\right)\right) \cdot x\]

C

double code(double x, double y) {
	return (((x * 3.0) * x) * y);
}

↓

double code(double x, double y) {
	return ((3.0 * (x * y)) * x);
}

Error

Bits error versus x

Bits error versus y

Target

Original	10.1
Target	0.2
Herbie	0.3

\[\left(x \cdot 3\right) \cdot \left(x \cdot y\right)\]

Derivation

1. Initial program 10.1

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

3. Applied associate-*l* 0.2

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

5. Applied associate-*l* 0.3

   \[\leadsto \color{blue}{x \cdot \left(3 \cdot \left(x \cdot y\right)\right)}\]
6. Using strategy rm

7. Applied add-sqr-sqrt 0.6

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

8. Applied associate-*l* 0.5

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

9. Taylor expanded around 0 10.4

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

10. Simplified 0.3

    \[\leadsto \color{blue}{\left(3 \cdot \left(x \cdot y\right)\right) \cdot x}\]

11. Final simplification 0.3

    \[\leadsto \left(3 \cdot \left(x \cdot y\right)\right) \cdot x\]

Reproduce

herbie shell --seed 2020075 
(FPCore (x y)
  :name "Diagrams.Segment:$catParam from diagrams-lib-1.3.0.3, A"
  :precision binary64

  :herbie-target
  (* (* x 3) (* x y))

  (* (* (* x 3) x) y))