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FindFormula
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FindFormula

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FindFormula[data]

finds a pure function that approximates data.

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FindFormula[data,x]

finds a symbolic function of the variable x that approximates data.

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FindFormula[data,x,n]

finds up to n functions that approximate data.

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FindFormula[data,x,n,prop]

returns up to n best functions associated with property prop.

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FindFormula[data,x,n,{prop1,prop2,…}]

returns up to n best functions associated with properties prop1, prop2, etc.

Details and Options

Examples

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Basic Examples  (2)Summary of the most common use cases

Make a table of values of the function x Sin[x]:

In[1]:=1
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https://wolfram.com/xid/0bdo9jv7e-qkdyju
Out[1]=1

FindFormula finds a formula that generates the data:

In[2]:=2
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https://wolfram.com/xid/0bdo9jv7e-g5ltz0
Out[2]=2

Plot the exponents of known Mersenne primes:

In[1]:=1
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https://wolfram.com/xid/0bdo9jv7e-o7olbk
Out[1]=1

Find the best simple function describing the data:

In[2]:=2
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https://wolfram.com/xid/0bdo9jv7e-tik57w
Out[2]=2

Visualize the fitted functions with the data:

In[3]:=3
✖
https://wolfram.com/xid/0bdo9jv7e-qbnvqy
Out[3]=3

Scope  (3)Survey of the scope of standard use cases

Generate data with normally distributed noise:

In[1]:=1
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https://wolfram.com/xid/0bdo9jv7e-e6veqe

Visualize the data:

In[2]:=2
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https://wolfram.com/xid/0bdo9jv7e-7qdhll
Out[2]=2

Find the first 5 best functions that approximate data:

In[3]:=3
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https://wolfram.com/xid/0bdo9jv7e-d69ty3
Out[3]=3

Visualize the fitted functions with the data:

In[4]:=4
✖
https://wolfram.com/xid/0bdo9jv7e-5nqq6f
Out[4]=4

Generate data with normally distributed noise:

In[1]:=1
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https://wolfram.com/xid/0bdo9jv7e-bcuepe

Visualize the data:

In[2]:=2
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https://wolfram.com/xid/0bdo9jv7e-8e5gp0
Out[2]=2

Visualize the dataset for the first 5 functions that approximate data:

In[3]:=3
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https://wolfram.com/xid/0bdo9jv7e-ltmh0x
Out[3]=3

Generate data with normally distributed noise:

In[1]:=1
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https://wolfram.com/xid/0bdo9jv7e-7pruse

Visualize the data:

In[2]:=2
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https://wolfram.com/xid/0bdo9jv7e-cow9oj
Out[2]=2

Look at the first 300 fits and plot their score as functions of the errors and complexity for different settings of SpecificityGoal:

In[3]:=3
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https://wolfram.com/xid/0bdo9jv7e-h6f505
In[4]:=4
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https://wolfram.com/xid/0bdo9jv7e-sfm1w0
Out[4]=4
In[5]:=5
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https://wolfram.com/xid/0bdo9jv7e-poy8j6
In[6]:=6
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https://wolfram.com/xid/0bdo9jv7e-48flom
Out[6]=6

Visualize the first fitted function with the data:

In[7]:=7
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https://wolfram.com/xid/0bdo9jv7e-disuev
Out[7]=7

Options  (4)Common values & functionality for each option

PerformanceGoal  (1)

Generate data with normally distributed noise:

In[1]:=1
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https://wolfram.com/xid/0bdo9jv7e-g90tbp

Visualize the data:

In[2]:=2
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https://wolfram.com/xid/0bdo9jv7e-wgjnz2
Out[2]=2

Find the best function that approximates data with its internal score:

In[4]:=4
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https://wolfram.com/xid/0bdo9jv7e-4tb92p
Out[4]=4

Find the best function that approximates data using PerformanceGoal with its internal score:

In[5]:=5
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https://wolfram.com/xid/0bdo9jv7e-o2sur8
Out[5]=5

Visualize the fitted functions with the data:

In[6]:=6
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https://wolfram.com/xid/0bdo9jv7e-s21izq
Out[6]=6

RandomSeeding  (1)

Generate data with normally distributed noise:

In[1]:=1
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https://wolfram.com/xid/0bdo9jv7e-zx1wwl

Compare different evaluations of FindFormula and notice how they differ:

In[2]:=2
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https://wolfram.com/xid/0bdo9jv7e-4boq0y
Out[2]=2

Use the option RandomSeeding to avoid having different results:

In[3]:=3
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https://wolfram.com/xid/0bdo9jv7e-p7gw51
Out[3]=3

SpecificityGoal  (1)

Generate data with normally distributed noise:

In[1]:=1
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https://wolfram.com/xid/0bdo9jv7e-lc5hp9

Visualize the data:

In[2]:=2
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https://wolfram.com/xid/0bdo9jv7e-ie4ell
Out[2]=2

Find the best functions that approximate data with their errors using different values of SpecificityGoal:

In[3]:=3
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https://wolfram.com/xid/0bdo9jv7e-5kfduk
Out[3]=3

Visualize the fitted functions with the data:

In[4]:=4
✖
https://wolfram.com/xid/0bdo9jv7e-293j9k
Out[4]=4

TargetFunctions  (1)

Generate data with normally distributed noise:

In[1]:=1
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https://wolfram.com/xid/0bdo9jv7e-shbafz

Visualize the data:

In[2]:=2
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https://wolfram.com/xid/0bdo9jv7e-ygeyk8
Out[2]=2

Find the best function that approximates data:

In[3]:=3
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https://wolfram.com/xid/0bdo9jv7e-c0k0xy
Out[3]=3

Find the best function that approximates data using TargetFunctions:

In[4]:=4
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https://wolfram.com/xid/0bdo9jv7e-bzdvzc
Out[4]=4

Visualize the fitted functions with the data:

In[5]:=5
✖
https://wolfram.com/xid/0bdo9jv7e-yaweeh
Out[5]=5

Applications  (3)Sample problems that can be solved with this function

Population Growth  (1)

Population growth in Poland:

In[1]:=1
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https://wolfram.com/xid/0bdo9jv7e-0q4edf
Out[1]=1

Find the best function that describes data:

In[2]:=2
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https://wolfram.com/xid/0bdo9jv7e-5uw4g3
Out[2]=2

Visualize the fitted function with the data:

In[3]:=3
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https://wolfram.com/xid/0bdo9jv7e-psohi7
Out[3]=3

Find a fit for the first 100 prime numbers:

In[4]:=4
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https://wolfram.com/xid/0bdo9jv7e-pqz6pn
In[5]:=5
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https://wolfram.com/xid/0bdo9jv7e-riq4p1
Out[5]=5

Compare the fit with the data and with the next 200 primes:

In[6]:=6
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https://wolfram.com/xid/0bdo9jv7e-4fduhq
In[7]:=7
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https://wolfram.com/xid/0bdo9jv7e-yc7wzh
Out[7]=7

Differential Equation  (1)

Find a fit for the numerical solution of a differential equation:

In[1]:=1
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https://wolfram.com/xid/0bdo9jv7e-rtv2bs
Out[4]=4
In[5]:=5
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https://wolfram.com/xid/0bdo9jv7e-66uuo5
Out[5]=5

Compare the fit with the data:

In[6]:=6
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https://wolfram.com/xid/0bdo9jv7e-8n9gxm
Out[6]=6

Orbital Mechanics  (1)

Plot the orbital periods of planets vs. their semimajor axes:

In[1]:=1
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https://wolfram.com/xid/0bdo9jv7e-5v2ul5
In[2]:=2
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https://wolfram.com/xid/0bdo9jv7e-worbyj
Out[2]=2

Find the best simple function describing the orbital radius in terms of the orbital period:

In[3]:=3
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https://wolfram.com/xid/0bdo9jv7e-t11533
Out[3]=3

Find the constant of proportionality:

In[4]:=4
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https://wolfram.com/xid/0bdo9jv7e-w70f76
Out[4]=4

Compare with the exact formula given by Kepler's third law:

In[5]:=5
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https://wolfram.com/xid/0bdo9jv7e-lc2jhq
Out[5]=5

The exact constant of proportionality has value:

In[6]:=6
✖
https://wolfram.com/xid/0bdo9jv7e-0s9gnb
Out[6]=6

Compare with the different values from the orbital data directly:

In[8]:=8
✖
https://wolfram.com/xid/0bdo9jv7e-2b8zaq
Out[8]=8
Wolfram Research (2015), FindFormula, Wolfram Language function, https://reference.wolfram.com/language/ref/FindFormula.html (updated 2017).
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Wolfram Research (2015), FindFormula, Wolfram Language function, https://reference.wolfram.com/language/ref/FindFormula.html (updated 2017).

Text

Wolfram Research (2015), FindFormula, Wolfram Language function, https://reference.wolfram.com/language/ref/FindFormula.html (updated 2017).

✖
Wolfram Research (2015), FindFormula, Wolfram Language function, https://reference.wolfram.com/language/ref/FindFormula.html (updated 2017).

CMS

Wolfram Language. 2015. "FindFormula." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2017. https://reference.wolfram.com/language/ref/FindFormula.html.

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Wolfram Language. 2015. "FindFormula." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2017. https://reference.wolfram.com/language/ref/FindFormula.html.

APA

Wolfram Language. (2015). FindFormula. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FindFormula.html

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Wolfram Language. (2015). FindFormula. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FindFormula.html

BibTeX

@misc{reference.wolfram_2025_findformula, author="Wolfram Research", title="{FindFormula}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/FindFormula.html}", note=[Accessed: 24-May-2025 ]}

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@misc{reference.wolfram_2025_findformula, author="Wolfram Research", title="{FindFormula}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/FindFormula.html}", note=[Accessed: 24-May-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_findformula, organization={Wolfram Research}, title={FindFormula}, year={2017}, url={https://reference.wolfram.com/language/ref/FindFormula.html}, note=[Accessed: 24-May-2025 ]}

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@online{reference.wolfram_2025_findformula, organization={Wolfram Research}, title={FindFormula}, year={2017}, url={https://reference.wolfram.com/language/ref/FindFormula.html}, note=[Accessed: 24-May-2025 ]}