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

Average Error: 0.3 → 0.2

Time: 5.3s

Precision: binary64

Cost: 448

Math

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

↓

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

FPCore

(FPCore (x y) :precision binary64 (* (* (* x 3.0) y) y))

↓

(FPCore (x y) :precision binary64 (* y (* x (* y 3.0))))

C

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

↓

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

Error

Bits error versus x

Bits error versus y

Target

Original	0.3
Target	0.2
Herbie	0.2

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

Alternatives

Alternative 1
Error	1.2
Cost	66176

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

Alternative 2
Error	1.2
Cost	46848

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

Alternative 3
Error	1.5
Cost	39936

\[\sqrt[3]{y \cdot \left(x \cdot \left(3 \cdot y\right)\right)} \cdot \left(\sqrt[3]{\sqrt[3]{y \cdot \left(x \cdot \left(3 \cdot y\right)\right)}} \cdot {\left(\sqrt[3]{\sqrt[3]{y \cdot \left(x \cdot \left(3 \cdot y\right)\right)}}\right)}^{5}\right)\]

Alternative 4
Error	22.5
Cost	39936

\[\sqrt[3]{y \cdot \left(x \cdot \left(3 \cdot y\right)\right)} \cdot \left(\sqrt{\sqrt[3]{y \cdot \left(x \cdot \left(3 \cdot y\right)\right)}} \cdot {\left(\sqrt{\sqrt[3]{y \cdot \left(x \cdot \left(3 \cdot y\right)\right)}}\right)}^{3}\right)\]

Alternative 5
Error	1.0
Cost	33472

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

Alternative 6
Error	0.9
Cost	33472

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

Alternative 7
Error	0.9
Cost	27072

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

Alternative 8
Error	0.9
Cost	20672

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

Alternative 9
Error	11.2
Cost	20672

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

Alternative 10
Error	0.9
Cost	20416

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

Alternative 11
Error	0.9
Cost	20224

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

Alternative 12
Error	0.9
Cost	19904

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

Alternative 13
Error	22.2
Cost	13760

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

Alternative 14
Error	27.1
Cost	13760

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

Alternative 15
Error	27.7
Cost	13632

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

Alternative 16
Error	32.2
Cost	13376

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

Alternative 17
Error	18.6
Cost	13312

\[y \cdot \sqrt[3]{{\left(\left(x \cdot 3\right) \cdot y\right)}^{3}}\]

Alternative 18
Error	29.3
Cost	13312

\[\sqrt[3]{{\left(\left(x \cdot 3\right) \cdot \left(y \cdot y\right)\right)}^{3}}\]

Alternative 19
Error	10.7
Cost	448

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

Alternative 20
Error	10.7
Cost	448

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

Alternative 21
Error	0.3
Cost	448

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

Alternative 22
Error	0.2
Cost	448

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

Alternative 23
Error	61.8
Cost	64

\[1\]

Alternative 24
Error	41.7
Cost	64

\[0\]

Alternative 25
Error	61.8
Cost	64

\[-1\]

Derivation

1. Initial program 0.3

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

3. Applied associate-*l*_binary64_19326 0.2

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

4. Simplified 0.2

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

5. Final simplification 0.2

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

Reproduce

herbie shell --seed 2021042 
(FPCore (x y)
  :name "Diagrams.Segment:$catParam from diagrams-lib-1.3.0.3, B"
  :precision binary64

  :herbie-target
  (* (* x (* 3.0 y)) y)

  (* (* (* x 3.0) y) y))