Average Error: 0.1 → 0.1

Time: 5.5s

Precision: binary64

Cost: 704

\[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right)\]

↓

\[3 - \left(x \cdot \left(x \cdot -9\right) + x \cdot 12\right)\]

3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right)

↓

3 - \left(x \cdot \left(x \cdot -9\right) + x \cdot 12\right)

(FPCore (x) :precision binary64 (* 3.0 (+ (- (* (* x 3.0) x) (* x 4.0)) 1.0)))

↓

(FPCore (x) :precision binary64 (- 3.0 (+ (* x (* x -9.0)) (* x 12.0))))

double code(double x) {
	return 3.0 * ((((x * 3.0) * x) - (x * 4.0)) + 1.0);
}

↓

double code(double x) {
	return 3.0 - ((x * (x * -9.0)) + (x * 12.0));
}

Error

The X axis uses an exponential scale — Bits error versus `x`

Try it out

In
Out

Enter valid numbers for all inputs

Target

Original	0.1
Target	0.1
Herbie	0.1

\[3 + \left(\left(9 \cdot x\right) \cdot x - 12 \cdot x\right)\]

Alternatives

Alternative 1
Error	0.5
Cost	21312

\[\left(3 \cdot \left(\sqrt[3]{1 + x \cdot \left(3 \cdot x + -4\right)} \cdot \sqrt[3]{1 + x \cdot \left(3 \cdot x + -4\right)}\right)\right) \cdot \sqrt[3]{1 + \left(x \cdot \left(3 \cdot x\right) - x \cdot 4\right)}\]

Alternative 2
Error	0.4
Cost	20800

\[3 - \sqrt[3]{x \cdot \left(x \cdot -9 + 12\right)} \cdot \left(\sqrt[3]{x \cdot \left(x \cdot -9 + 12\right)} \cdot \sqrt[3]{x \cdot \left(x \cdot -9 + 12\right)}\right)\]

Alternative 3
Error	0.5
Cost	20544

\[3 - \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \frac{144 - x \cdot \left(x \cdot 81\right)}{x \cdot 9 + 12}\right)\]

Alternative 4
Error	0.5
Cost	20544

\[3 - \sqrt[3]{x \cdot -9 + 12} \cdot \left(x \cdot \left(\sqrt[3]{x \cdot -9 + 12} \cdot \sqrt[3]{x \cdot -9 + 12}\right)\right)\]

Alternative 5
Error	0.2
Cost	20160

\[\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \left(\left(1 + x \cdot \left(3 \cdot x + -4\right)\right) \cdot \sqrt[3]{3}\right)\]

Alternative 6
Error	0.4
Cost	20032

\[3 - \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \left(x \cdot -9 + 12\right)\right)\]

Alternative 7
Error	0.9
Cost	13632

\[\sqrt{3} \cdot \left(\left(1 + x \cdot \left(3 \cdot x + -4\right)\right) \cdot \sqrt{3}\right)\]

Alternative 8
Error	7.1
Cost	8576

\[3 - \left(x \cdot \frac{144 - x \cdot \left(x \cdot 81\right)}{1728 - {\left(x \cdot -9\right)}^{3}}\right) \cdot \left(144 + \left(\left(x \cdot 9\right) \cdot \left(x \cdot 9\right) + \left(x \cdot -9\right) \cdot 12\right)\right)\]

Alternative 9
Error	0.2
Cost	1088

\[3 - x \cdot \frac{144 - x \cdot \left(x \cdot 81\right)}{x \cdot 9 + 12}\]

Alternative 10
Error	7.3
Cost	1088

\[3 - \frac{x \cdot \left(144 - x \cdot \left(x \cdot 81\right)\right)}{x \cdot 9 + 12}\]

Alternative 11
Error	42.8
Cost	960

\[3 - x \cdot \frac{144 - x \cdot \left(x \cdot 81\right)}{x \cdot 9}\]

Alternative 12
Error	0.1
Cost	832

\[3 \cdot \left(1 + \left(x \cdot \left(3 \cdot x\right) - x \cdot 4\right)\right)\]

Alternative 13
Error	0.1
Cost	704

\[3 + 3 \cdot \left(x \cdot \left(3 \cdot x + -4\right)\right)\]

Alternative 14
Error	2.1
Cost	576

\[3 \cdot \left(x \cdot \left(3 \cdot x\right) + 1\right)\]

Alternative 15
Error	0.1
Cost	576

\[3 + x \cdot \left(x \cdot 9 + -12\right)\]

Alternative 16
Error	41.2
Cost	448

\[x \cdot \left(x \cdot 9 + -12\right)\]

Alternative 17
Error	2.1
Cost	448

\[3 - x \cdot \left(x \cdot -9\right)\]

Alternative 18
Error	41.5
Cost	320

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

Alternative 19
Error	21.4
Cost	320

\[3 - x \cdot 12\]

Alternative 20
Error	41.5
Cost	320

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

Alternative 21
Error	21.6
Cost	64

\[3\]

Alternative 22
Error	55.3
Cost	64

\[1\]

Alternative 23
Error	62.2
Cost	64

\[0\]

Alternative 24
Error	63.1
Cost	64

\[-1\]

Error

Derivation

Initial program 0.1
\[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right)\]
Simplified0.1
\[\leadsto \color{blue}{3 - x \cdot \left(12 + x \cdot -9\right)}\]
Using strategy rm
Applied distribute-rgt-in_binary64_150320.1
\[\leadsto 3 - \color{blue}{\left(12 \cdot x + \left(x \cdot -9\right) \cdot x\right)}\]
Simplified0.1
\[\leadsto 3 - \left(\color{blue}{x \cdot 12} + \left(x \cdot -9\right) \cdot x\right)\]
Simplified0.1
\[\leadsto 3 - \left(x \cdot 12 + \color{blue}{x \cdot \left(x \cdot -9\right)}\right)\]
Simplified0.1
\[\leadsto \color{blue}{3 - \left(x \cdot \left(x \cdot -9\right) + x \cdot 12\right)}\]
Final simplification0.1
\[\leadsto 3 - \left(x \cdot \left(x \cdot -9\right) + x \cdot 12\right)\]

Reproduce

herbie shell --seed 2021022 
(FPCore (x)
  :name "Diagrams.Tangent:$catParam from diagrams-lib-1.3.0.3, D"
  :precision binary64

  :herbie-target
  (+ 3.0 (- (* (* 9.0 x) x) (* 12.0 x)))

  (* 3.0 (+ (- (* (* x 3.0) x) (* x 4.0)) 1.0)))