WOLFRAM

Wolfram Language & System Documentation Center
Wolfram Language Home Page »

FullSimplify

Updatedshow changeshide changes

✖
FullSimplify[expr]

tries a wide range of transformations on expr involving elementary and special functions and returns the simplest form it finds.

✖
FullSimplify[expr,assum]

does simplification using assumptions.

Details and Options

  • FullSimplify will always yield at least as simple a form as Simplify, but may take substantially longer.
  • FullSimplify uses RootReduce on expressions that involve Root objects.
  • FullSimplify does transformations on most kinds of special functions.
  • With assumptions of the form ForAll[vars,axioms], FullSimplify can simplify expressions and equations involving symbolic functions. »
  • You can specify default assumptions for FullSimplify using Assuming.
  • The following options can be given:
  • Assumptions $Assumptionsdefault assumptions to append to assum
    ComplexityFunction Automatichow to assess the complexity of each form generated
    ExcludedForms {}patterns specifying forms of subexpression that should not be touched
    TimeConstraint Infinityfor how many seconds to try doing any particular transformation
    TransformationFunctions Automaticfunctions to try in transforming the expression
    Trig Truewhether to do trigonometric as well as algebraic transformations
  • Assumptions can consist of equations, inequalities, domain specifications such as x∈Integers, and logical combinations of these.
  • With the setting TimeConstraint->{tloc,ttot}, at most tloc seconds are spent for any particular transformation, and at most ttot seconds are spent for all transformations before the best result is returned.
  • FullSimplify can be used with symbolic array expressions.

Examples

open allclose all

Basic Examples  (3)Summary of the most common use cases

Simplify an expression involving special functions:

In[1]:=1
✖
https://wolfram.com/xid/0i1l727rm-dey5e2
Out[1]=1

Simplify using assumptions:

In[1]:=1
✖
https://wolfram.com/xid/0i1l727rm-cv2mds
Out[1]=1

Prove a simple theorem from the assumption of associativity:

In[1]:=1
✖
https://wolfram.com/xid/0i1l727rm-d2ugjm
Out[1]=1

Scope  (9)Survey of the scope of standard use cases

Simplify polynomials:

In[1]:=1
✖
https://wolfram.com/xid/0i1l727rm-gn15it
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0i1l727rm-eoa13z
Out[2]=2

Simplify a hyperbolic expression to an exponential form:

In[1]:=1
✖
https://wolfram.com/xid/0i1l727rm-jau50d
Out[1]=1

Simplify an exponential expression to a trigonometric form:

In[1]:=1
✖
https://wolfram.com/xid/0i1l727rm-n4ltu
Out[1]=1

Simplify an algebraic number:

In[1]:=1
✖
https://wolfram.com/xid/0i1l727rm-wkkc7
Out[1]=1

Simplify transcendental numbers:

In[1]:=1
✖
https://wolfram.com/xid/0i1l727rm-g49yew
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0i1l727rm-org
Out[2]=2

Simplify expressions involving special functions:

In[1]:=1
✖
https://wolfram.com/xid/0i1l727rm-e6p52g
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0i1l727rm-cd3b29
Out[2]=2

Simplify expressions using assumptions:

In[1]:=1
✖
https://wolfram.com/xid/0i1l727rm-caiq55
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0i1l727rm-04wzf
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0i1l727rm-mye1i
Out[3]=3

Prove theorems based on axiom systems:

In[1]:=1
✖
https://wolfram.com/xid/0i1l727rm-n34
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0i1l727rm-vei
Out[2]=2

Any expression can be used as a variable:

In[3]:=3
✖
https://wolfram.com/xid/0i1l727rm-kjg
Out[3]=3

Variables not quantified in the axioms are treated as constants:

In[4]:=4
✖
https://wolfram.com/xid/0i1l727rm-q6h
Out[4]=4

Prove existence of right inverses assuming left identity and left inverses exist:

In[5]:=5
✖
https://wolfram.com/xid/0i1l727rm-gi5y5c
Out[5]=5

Simplify symbolic arrays expressions:

In[1]:=1
✖
https://wolfram.com/xid/0i1l727rm-m1ej8a
Out[1]=1

Options  (6)Common values & functionality for each option

Assumptions  (1)

Assumptions can be given both as an argument and as an option value:

In[1]:=1
✖
https://wolfram.com/xid/0i1l727rm-d7eor0
Out[1]=1

The default value of the Assumptions option is $Assumptions:

In[2]:=2
✖
https://wolfram.com/xid/0i1l727rm-f8iglz
Out[2]=2

When assumptions are given as an argument, $Assumptions is used as well:

In[3]:=3
✖
https://wolfram.com/xid/0i1l727rm-bgnpfp
Out[3]=3

Specifying assumptions as an option value prevents FullSimplify from using $Assumptions:

In[4]:=4
✖
https://wolfram.com/xid/0i1l727rm-bwql49
Out[4]=4

ComplexityFunction  (1)

By default, this expression is not simplified:

In[1]:=1
✖
https://wolfram.com/xid/0i1l727rm-g7ett
Out[1]=1

This complexity function makes ChebyshevT more expensive than other functions:

In[2]:=2
✖
https://wolfram.com/xid/0i1l727rm-bc51nt
In[3]:=3
✖
https://wolfram.com/xid/0i1l727rm-bh97x6
Out[3]=3

ExcludedForms  (1)

This gives a result in terms of Arg[x]:

In[1]:=1
✖
https://wolfram.com/xid/0i1l727rm-cz9t78
Out[1]=1

This specifies that Log[x] should not be transformed:

In[2]:=2
✖
https://wolfram.com/xid/0i1l727rm-o5gux6
Out[2]=2

TimeConstraint  (1)

This takes a long time due to expansion of trigonometric functions:

In[1]:=1
✖
https://wolfram.com/xid/0i1l727rm-2h197j
Out[1]=1

The most time‐consuming transformation is not the one that does the simplification:

In[2]:=2
✖
https://wolfram.com/xid/0i1l727rm-6jp5vq
Out[3]=3

With transformations restricted to 100 ms, the simplification does not happen:

In[4]:=4
✖
https://wolfram.com/xid/0i1l727rm-7au2ut
Out[5]=5

TransformationFunctions  (1)

By default, FullSimplify does not use Reduce:

In[1]:=1
✖
https://wolfram.com/xid/0i1l727rm-j0qym7
Out[1]=1

This makes FullSimplify use Reduce with respect to x over the real domain:

In[2]:=2
✖
https://wolfram.com/xid/0i1l727rm-bfueet
In[3]:=3
✖
https://wolfram.com/xid/0i1l727rm-g80zix
Out[3]=3

Trig  (1)

By default, FullSimplify uses trigonometric identities:

In[1]:=1
✖
https://wolfram.com/xid/0i1l727rm-gghype
Out[1]=1

With Trig->False, FullSimplify does not use trigonometric identities:

In[2]:=2
✖
https://wolfram.com/xid/0i1l727rm-esozdc
Out[2]=2

Applications  (6)Sample problems that can be solved with this function

Prove that a solution satisfies its equations:

In[1]:=1
✖
https://wolfram.com/xid/0i1l727rm-fp0ux8
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0i1l727rm-cp122e
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0i1l727rm-b3bll2
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0i1l727rm-fhq33y
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0i1l727rm-g8841z
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/0i1l727rm-lphq0e
Out[6]=6

Simplify expressions involving Mod:

In[1]:=1
✖
https://wolfram.com/xid/0i1l727rm-pr5
Out[1]=1

Prove that an operation g with associativity, left neutral element, and left inverse defines a group:

In[1]:=1
✖
https://wolfram.com/xid/0i1l727rm-jqe
Out[1]=1

Prove commutativity from Wolfram's minimal axiom for Boolean algebra:

In[1]:=1
✖
https://wolfram.com/xid/0i1l727rm-oua
Out[1]=1

Prove that a fixed-point combinator exists:

In[1]:=1
✖
https://wolfram.com/xid/0i1l727rm-qix
Out[1]=1

Prove a theorem about meet (⋁) and join (⋀):

In[1]:=1
✖
https://wolfram.com/xid/0i1l727rm-cta
Out[1]=1

Properties & Relations  (7)Properties of the function, and connections to other functions

The output is generically equivalent to the input:

In[1]:=1
✖
https://wolfram.com/xid/0i1l727rm-j9dxpy
In[2]:=2
✖
https://wolfram.com/xid/0i1l727rm-pn8gm
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0i1l727rm-cwgk4
Out[3]=3

FullSimplify uses a wider range of transformations than Simplify:

In[1]:=1
✖
https://wolfram.com/xid/0i1l727rm-sso
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0i1l727rm-tco
Out[2]=2

FullSimplify uses several expansion transformations, including Expand:

In[1]:=1
✖
https://wolfram.com/xid/0i1l727rm-ipw9wt
Out[1]=1

TrigExpand:

In[2]:=2
✖
https://wolfram.com/xid/0i1l727rm-lams5c
Out[2]=2

PiecewiseExpand:

In[3]:=3
✖
https://wolfram.com/xid/0i1l727rm-isg7ez
Out[3]=3

FunctionExpand:

In[4]:=4
✖
https://wolfram.com/xid/0i1l727rm-jirvp
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0i1l727rm-btimu3
Out[5]=5

LogicalExpand:

In[6]:=6
✖
https://wolfram.com/xid/0i1l727rm-zw0xn
Out[6]=6

PowerExpand makes special assumptions on input and is not used by FullSimplify:

In[1]:=1
✖
https://wolfram.com/xid/0i1l727rm-7tfrd
Out[1]=1

ComplexExpand assumes variables to be real and is also not used by FullSimplify:

In[2]:=2
✖
https://wolfram.com/xid/0i1l727rm-pucfc
Out[2]=2

FullSimplify uses several factoring transformations, including Factor:

In[1]:=1
✖
https://wolfram.com/xid/0i1l727rm-igj44u
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0i1l727rm-vb1k
Out[2]=2

FactorSquareFree:

In[3]:=3
✖
https://wolfram.com/xid/0i1l727rm-5kyud
Out[3]=3

TrigFactor:

In[4]:=4
✖
https://wolfram.com/xid/0i1l727rm-d2kyp
Out[4]=4

For algebraic numbers, RootReduce and ToRadicals are used:

In[1]:=1
✖
https://wolfram.com/xid/0i1l727rm-gi9hib
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0i1l727rm-jaud7k
Out[2]=2

For rational functions, Together and Apart are used:

In[1]:=1
✖
https://wolfram.com/xid/0i1l727rm-mjhtr
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0i1l727rm-gcqwo3
Out[2]=2

Possible Issues  (3)Common pitfalls and unexpected behavior

Some of the transformations used by FullSimplify are only generically correct:

In[1]:=1
✖
https://wolfram.com/xid/0i1l727rm-djctxy
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0i1l727rm-beqnz5
Out[2]=2

Results of simplification of singular expressions are uncertain:

In[1]:=1
✖
https://wolfram.com/xid/0i1l727rm-fo37ua
Out[1]=1

This result is caused by automatic evaluation:

In[2]:=2
✖
https://wolfram.com/xid/0i1l727rm-ff19v1
Out[2]=2

Results of simplification may depend on the names of symbols:

In[1]:=1
✖
https://wolfram.com/xid/0i1l727rm-hvqr6s
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0i1l727rm-mxr4vn
Out[2]=2

Neat Examples  (1)Surprising or curious use cases

FullSimplify knows about Fermat's last theorem:

In[1]:=1
✖
https://wolfram.com/xid/0i1l727rm-q9ghq
Out[1]=1
Wolfram Research (1996), FullSimplify, Wolfram Language function, https://reference.wolfram.com/language/ref/FullSimplify.html (updated 2025).
✖
Wolfram Research (1996), FullSimplify, Wolfram Language function, https://reference.wolfram.com/language/ref/FullSimplify.html (updated 2025).

Text

Wolfram Research (1996), FullSimplify, Wolfram Language function, https://reference.wolfram.com/language/ref/FullSimplify.html (updated 2025).

✖
Wolfram Research (1996), FullSimplify, Wolfram Language function, https://reference.wolfram.com/language/ref/FullSimplify.html (updated 2025).

CMS

Wolfram Language. 1996. "FullSimplify." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2025. https://reference.wolfram.com/language/ref/FullSimplify.html.

✖
Wolfram Language. 1996. "FullSimplify." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2025. https://reference.wolfram.com/language/ref/FullSimplify.html.

APA

Wolfram Language. (1996). FullSimplify. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FullSimplify.html

✖
Wolfram Language. (1996). FullSimplify. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FullSimplify.html

BibTeX

@misc{reference.wolfram_2025_fullsimplify, author="Wolfram Research", title="{FullSimplify}", year="2025", howpublished="\url{https://reference.wolfram.com/language/ref/FullSimplify.html}", note=[Accessed: 16-April-2025 ]}

✖
@misc{reference.wolfram_2025_fullsimplify, author="Wolfram Research", title="{FullSimplify}", year="2025", howpublished="\url{https://reference.wolfram.com/language/ref/FullSimplify.html}", note=[Accessed: 16-April-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_fullsimplify, organization={Wolfram Research}, title={FullSimplify}, year={2025}, url={https://reference.wolfram.com/language/ref/FullSimplify.html}, note=[Accessed: 16-April-2025 ]}

✖
@online{reference.wolfram_2025_fullsimplify, organization={Wolfram Research}, title={FullSimplify}, year={2025}, url={https://reference.wolfram.com/language/ref/FullSimplify.html}, note=[Accessed: 16-April-2025 ]}