--- title: "Simplify" language: "en" type: "Symbol" summary: "Simplify[expr] performs a sequence of algebraic and other transformations on expr and returns the simplest form it finds. Simplify[expr, assum] does simplification using assumptions." keywords: - algebraic simplification - artificial intelligence - assertions - assumptions - cleaning up expressions - combinatorial explosion - compress expression - concise form - constraints - efficiency - expressions - heuristics - number theory - predicates - rearrange expression - roots - shortest form - simplest forms - simplification - smallest form - speed - square roots - squash expression - transformations - trigonometric simplification - variables - combine - simplify canonical_url: "https://reference.wolfram.com/language/ref/Simplify.html" source: "Wolfram Language Documentation" related_guides: - title: "Formula Manipulation" link: "https://reference.wolfram.com/language/guide/FormulaManipulation.en.md" - title: "Manipulating Equations" link: "https://reference.wolfram.com/language/guide/ManipulatingEquations.en.md" - title: "Algebraic Transformations" link: "https://reference.wolfram.com/language/guide/AlgebraicTransformations.en.md" - title: "Assumptions and Domains" link: "https://reference.wolfram.com/language/guide/AssumptionsAndDomains.en.md" - title: "Finite Mathematics" link: "https://reference.wolfram.com/language/guide/FiniteMathematics.en.md" related_functions: - title: "FullSimplify" link: "https://reference.wolfram.com/language/ref/FullSimplify.en.md" - title: "Refine" link: "https://reference.wolfram.com/language/ref/Refine.en.md" - title: "Factor" link: "https://reference.wolfram.com/language/ref/Factor.en.md" - title: "Expand" link: "https://reference.wolfram.com/language/ref/Expand.en.md" - title: "TrigExpand" link: "https://reference.wolfram.com/language/ref/TrigExpand.en.md" - title: "PowerExpand" link: "https://reference.wolfram.com/language/ref/PowerExpand.en.md" - title: "ComplexExpand" link: "https://reference.wolfram.com/language/ref/ComplexExpand.en.md" - title: "PiecewiseExpand" link: "https://reference.wolfram.com/language/ref/PiecewiseExpand.en.md" - title: "Element" link: "https://reference.wolfram.com/language/ref/Element.en.md" - title: "FunctionExpand" link: "https://reference.wolfram.com/language/ref/FunctionExpand.en.md" - title: "ArraySimplify" link: "https://reference.wolfram.com/language/ref/ArraySimplify.en.md" - title: "ArrayExpand" link: "https://reference.wolfram.com/language/ref/ArrayExpand.en.md" - title: "Reduce" link: "https://reference.wolfram.com/language/ref/Reduce.en.md" - title: "Assuming" link: "https://reference.wolfram.com/language/ref/Assuming.en.md" - title: "TrigReduce" link: "https://reference.wolfram.com/language/ref/TrigReduce.en.md" - title: "TrigFactor" link: "https://reference.wolfram.com/language/ref/TrigFactor.en.md" related_tutorials: - title: "Simplifying Algebraic Expressions" link: "https://reference.wolfram.com/language/tutorial/AlgebraicCalculations.en.md#16626" - title: "Simplifying with Assumptions" link: "https://reference.wolfram.com/language/tutorial/AlgebraicCalculations.en.md#12157" - title: "Simplification" link: "https://reference.wolfram.com/language/tutorial/AlgebraicManipulation.en.md#1259" - title: "Putting Expressions into Different Forms" link: "https://reference.wolfram.com/language/tutorial/AlgebraicCalculations.en.md#21991" - title: "Using Assumptions" link: "https://reference.wolfram.com/language/tutorial/AlgebraicManipulation.en.md#31833" - title: "Implementation notes: Algebra and Calculus" link: "https://reference.wolfram.com/language/tutorial/SomeNotesOnInternalImplementation.en.md#20945" --- # Simplify Simplify[expr] performs a sequence of algebraic and other transformations on expr and returns the simplest form it finds. Simplify[expr, assum] does simplification using assumptions. ## Details and Options * ``Simplify`` tries expanding, factoring, and doing many other transformations on expressions, keeping track of the simplest form obtained. * ``Simplify`` can be used on equations, inequalities, and domain specifications. * Quantities that appear algebraically in inequalities are always assumed to be real. * ``FullSimplify`` does more extensive simplification than ``Simplify``. * You can specify default assumptions for ``Simplify`` using ``Assuming``. * The following options can be given: | | | | | :----------------------- | :------------ | :---------------------------------------------------------------- | | Assumptions | \$Assumptions | default assumptions to append to assum | | ComplexityFunction | Automatic | how to assess the complexity of each form generated | | TimeConstraint | 300 | how many seconds to try doing any particular transformation | | TransformationFunctions | Automatic | functions to try in transforming the expression | | Trig | True | whether 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. * ``Simplify`` can be used with symbolic array expressions. --- ## Examples (27) ### Basic Examples (3) ```wl In[1]:= D[Integrate[1 / (x ^ 3 + 1), x], x] Out[1]= (1/3 (1 + x)) - (-1 + 2 x/6 (1 - x + x^2)) + (2/3 (1 + (1/3) (-1 + 2 x)^2)) In[2]:= Simplify[%] Out[2]= (1/1 + x^3) ``` --- ```wl In[1]:= Simplify[Sin[x] ^ 2 + Cos[x] ^ 2] Out[1]= 1 ``` --- ``Simplify`` can get further if assumptions are made about ``x`` : ```wl In[1]:= Simplify[Sqrt[x ^ 2]] Out[1]= Sqrt[x^2] In[2]:= Simplify[Sqrt[x ^ 2], x > 0] Out[2]= x In[3]:= Simplify[Sqrt[x ^ 2], Element[x, Reals]] Out[3]= Abs[x] ``` ### Scope (5) Simplify a polynomial: ```wl In[1]:= Simplify[(x - 1)(x + 1)(x ^ 2 + 1) + 1] Out[1]= x^4 ``` Simplify a rational expression: ```wl In[2]:= Simplify[3 / (x + 3) + x / (x + 3)] Out[2]= 1 ``` Simplify a trigonometric expression: ```wl In[3]:= Simplify[2Tan[x] / (1 + Tan[x] ^ 2)] Out[3]= Sin[2 x] ``` Simplify an exponential expression: ```wl In[4]:= Simplify[(E ^ x - E ^ -x) / Sinh[x]] Out[4]= 2 ``` --- Simplify an equation: ```wl In[1]:= Simplify[2x - 4y + 6z - 10 == -8] Out[1]= x + 3 z == 1 + 2 y ``` --- Simplify expressions using assumptions: ```wl In[1]:= Simplify[1 / Sqrt[x] - Sqrt[1 / x], x > 0] Out[1]= 0 In[2]:= Simplify[Log[x ^ p], x > 0 && Element[p, Reals]] Out[2]= p Log[x] In[3]:= Simplify[Sin[n Pi], Element[n, Integers]] Out[3]= 0 ``` --- Use assumptions to prove inequalities: ```wl In[1]:= Simplify[x ^ 2 > 3, x > 2] Out[1]= True In[2]:= Simplify[Abs[x] < 2, x ^ 2 + y ^ 2 < 4] Out[2]= True ``` --- Simplify symbolic array expressions: ```wl In[1]:= Simplify[a.b + 2a.c] Out[1]= a.(b + 2 c) In[2]:= Simplify[m.Inverse[m], Element[m, Matrices[{n, n}]]] Out[2]= SymbolicIdentityArray[{n}] In[3]:= a = MatrixSymbol["a", {n, n}]; Simplify[Tr[Transpose[KroneckerProduct[Transpose[a], a]]]] Out[3]= Tr[MatrixSymbol["a", {n, n}]]^2 In[4]:= Simplify[D[(VectorSymbol["v", n] + x).(VectorSymbol["v", n] + x), x], Element[x, Complexes]] Out[4]= 2 Total[x + VectorSymbol["v", n]] ``` ### Options (10) #### Assumptions (3) Assumptions can be given both as an argument and as an option value: ```wl In[1]:= Simplify[Cos[k Pi] ^ m, Element[k, Integers], Assumptions -> Mod[m, 2] == 0] Out[1]= 1 ``` --- The default value of the ``Assumptions`` option is ``\$Assumptions`` : ```wl In[1]:= Assuming[x > 0, Simplify[Sqrt[x ^ 2]]] Out[1]= x In[2]:= Simplify[Sqrt[x ^ 2]] Out[2]= Sqrt[x^2] ``` --- When assumptions are given as an argument, ``\$Assumptions`` is used as well: ```wl In[1]:= Assuming[x > 0, Simplify[Sqrt[x ^ 2 y ^ 2], y < 0]] Out[1]= -x y ``` Specifying assumptions as an option value prevents ``Simplify`` from using ``\$Assumptions`` : ```wl In[2]:= Assuming[x > 0, Simplify[Sqrt[x ^ 2 y ^ 2], Assumptions -> y < 0]] Out[2]= -Sqrt[x^2] y ``` #### ComplexityFunction (2) The default ``ComplexityFunction`` counts the subexpressions and digits of integers: ```wl In[1]:= Simplify[100 Log[2]] Out[1]= 100 Log[2] ``` ``LeafCount`` counts only the number of subexpressions: ```wl In[2]:= Simplify[100 Log[2], ComplexityFunction -> LeafCount] Out[2]= Log[1267650600228229401496703205376] ``` --- With the default ``ComplexityFunction``, ``Abs[x]`` is simpler than the ``FullForm`` of ``-x`` : ```wl In[1]:= Simplify[Abs[x], x < 0] Out[1]= Abs[x] In[2]:= Map[FullForm, {Abs[x], -x}] Out[2]= {Abs[x], Times[-1, x]} ``` This complexity function makes ``Abs`` more expensive than ``Times`` : ```wl In[3]:= f[e_] := 100Count[e, _Abs, {0, Infinity}] + LeafCount[e] In[4]:= Simplify[Abs[x], x < 0, ComplexityFunction -> f] Out[4]= -x ``` #### ExcludedForms (1) This gives no simplification: ```wl In[1]:= Simplify[(x - 1)(x + 1)(x ^ 2 + 1)(x - 2) ^ 10] Out[1]= (-2 + x)^10 (-1 + x) (1 + x) (1 + x^2) ``` Excluding transformations of ``(x - 2) ^ 10`` allows ``Simplify`` to expand the remaining terms: ```wl In[2]:= Simplify[(x - 1)(x + 1)(x ^ 2 + 1)(x - 2) ^ 10, ExcludedForms -> {(_Plus) ^ _}] Out[2]= (-2 + x)^10 (-1 + x^4) ``` #### TimeConstraint (2) This takes a long time, due to trigonometric expansion, but does not yield a simplification: ```wl In[1]:= Simplify[2Sin[10x + 11y + 12z]Cos[10x + 10y + 10t]]//Timing Out[1]= {32.218, 2 Cos[10 (t + x + y)] Sin[10 x + 11 y + 12 z]} ``` Use ``TimeConstraint`` to limit the time spent on any single transformation: ```wl In[2]:= Simplify[2Sin[10x + 11y + 12z]Cos[10x + 10y + 10t], TimeConstraint -> 1]//Timing ``` Simplify::time: Time spent on a transformation exceeded 1 seconds, and the transformation was aborted. Increasing the value of TimeConstraint option may improve the result of simplification. Simplify::time: Time spent on a transformation exceeded 1 seconds, and the transformation was aborted. Increasing the value of TimeConstraint option may improve the result of simplification. ```wl Out[2]= {2.156, 2 Cos[10 (t + x + y)] Sin[10 x + 11 y + 12 z]} ``` --- A similar example, where the transformation yields a simplification: ```wl In[1]:= Simplify[2Sin[10x + 11y + 12z]Cos[10x + 10y + 10z]]//Timing Out[1]= {18.188, Sin[y + 2 z] + Sin[20 x + 21 y + 22 z]} ``` In this case, setting ``TimeConstraint`` prevents some simplification: ```wl In[2]:= Simplify[2Sin[10x + 11y + 12z]Cos[10x + 10y + 10z], TimeConstraint -> 1]//Timing ``` Simplify::time: Time spent on a transformation exceeded 1 seconds, and the transformation was aborted. Increasing the value of TimeConstraint option may improve the result of simplification. Simplify::time: Time spent on a transformation exceeded 1 seconds, and the transformation was aborted. Increasing the value of TimeConstraint option may improve the result of simplification. ```wl Out[2]= {2.766, 2 Cos[10 (x + y + z)] Sin[10 x + 11 y + 12 z]} ``` #### TransformationFunctions (1) Here ``Simplify`` uses ``t`` as the only transformation: ```wl In[1]:= t[e_] := PolynomialReduce[e, {s ^ 2 + c ^ 2 - 1}, {c, s}][[2]] In[2]:= Simplify[(c ^ 3 - s ^ 3) ^ 2 - (s ^ 3 + c ^ 3) ^ 2, TransformationFunctions -> {t}] Out[2]= -4 c s^3 + 4 c s^5 ``` Here ``Simplify`` uses both ``t`` and all built-in transformations: ```wl In[3]:= Simplify[(c ^ 3 - s ^ 3) ^ 2 - (s ^ 3 + c ^ 3) ^ 2, TransformationFunctions -> {Automatic, t}] Out[3]= 4 c s^3 (-1 + s^2) ``` #### Trig (1) By default, ``Simplify`` uses trigonometric identities: ```wl In[1]:= Simplify[4Sin[x] ^ 2Cos[x] ^ 2 + 4Sin[x]Cos[x] + 1] Out[1]= (Cos[x] + Sin[x])^4 ``` With ``Trig -> False``, ``Simplify`` does not use trigonometric identities: ```wl In[2]:= Simplify[4Sin[x] ^ 2Cos[x] ^ 2 + 4Sin[x]Cos[x] + 1, Trig -> False] Out[2]= (1 + 2 Cos[x] Sin[x])^2 ``` ### Applications (4) Prove that a solution satisfies its equations: ```wl In[1]:= Solve[x ^ 2 + x + 1 == 0, x] Out[1]= {{x -> -(-1)^1 / 3}, {x -> (-1)^2 / 3}} In[2]:= x ^ 2 + x + 1 == 0 /. %//Simplify Out[2]= {True, True} In[3]:= DSolve[y''[x] + y[x] == Exp[x], y, x] Out[3]= {{y -> Function[{x}, C[1] Cos[x] + C[2] Sin[x] + (1/2) E^x (Cos[x]^2 + Sin[x]^2)]}} In[4]:= y''[x] + y[x] == Exp[x] /. First[%]//Simplify Out[4]= True In[5]:= RSolve[y[k + 2] + y[k] == 2 ^ k, y, k] Out[5]= {{y -> Function[{k}, (2^k/5) + I^k C[1] + (-I)^k C[2]]}} In[6]:= y[k + 2] + y[k] == 2 ^ k /. First[%]//Simplify Out[6]= True ``` --- Show that the arithmetic mean is larger than the geometric one: ```wl In[1]:= Simplify[(x + y) / 2 >= Sqrt[x y], x >= 0 && y >= 0] Out[1]= True ``` --- This applies Fermat's little theorem: ```wl In[1]:= Simplify[Mod[a ^ p, p], a∈Integers && p∈Primes] Out[1]= Mod[a, p] ``` --- Prove commutativity from Wolfram's minimal axiom for Boolean algebra: ```wl In[1]:= Simplify[f[a, b] == f[b, a], ForAll[{p, q, r}, f[f[f[p, q], r], f[p, f[f[p, r], p]]] == r]] Out[1]= True ``` ### Properties & Relations (3) Use ``Assuming`` to propagate assumptions: ```wl In[1]:= Assuming[x > 0, Simplify[Sqrt[x ^ 2]]] Out[1]= x ``` --- Use ``FullSimplify`` to simplify expressions involving special functions: ```wl In[1]:= Simplify[Gamma[x] Gamma[1 - x]] Out[1]= Gamma[1 - x] Gamma[x] In[2]:= FullSimplify[%] Out[2]= π Csc[π x] ``` --- ``ArraySimplify`` performs only array transformations: ```wl In[1]:= v = VectorSymbol["v", n / 2 + m / 2]; In[2]:= ArraySimplify[v - Transpose[v]] Out[2]= SymbolicZerosArray[{(m/2) + (n/2)}] ``` ``Simplify`` performs other transformations as well: ```wl In[3]:= Simplify[v - Transpose[v]] Out[3]= SymbolicZerosArray[{(m + n/2)}] ``` ### Possible Issues (2) The Wolfram Language evaluates zero times a symbolic expression to zero: ```wl In[1]:= 0 symbolic Out[1]= 0 ``` This happens even if the symbolic expression is always infinite: ```wl In[2]:= hiddenzero = x ^ 2 + 2x + 1 - (x + 1) ^ 2; In[3]:= 0 / hiddenzero Out[3]= 0 ``` Because of this, results of simplification of expressions with singularities are uncertain: ```wl In[4]:= anotherhiddenzero = Sin[x] ^ 2 + Cos[x] ^ 2 - 1; In[5]:= betterhiddenzero = Gamma[x + 1] - x Gamma[x]; In[6]:= Map[Simplify, {hiddenzero / anotherhiddenzero, anotherhiddenzero / hiddenzero, hiddenzero / betterhiddenzero, betterhiddenzero / hiddenzero}] ``` Simplify::infd: Expression (-1+Cos[x]^2+Sin[x]^2)/(1+2 x+x^2-(1+x)^2) simplified to Indeterminate. Simplify::infd: Expression (-x Gamma[x]+Gamma[1+x])/(1+2 x+x^2-(1+x)^2) simplified to ComplexInfinity. ```wl Out[6]= {0, Indeterminate, 0, ComplexInfinity} ``` In this case, ``FullSimplify`` recognizes the zero: ```wl In[7]:= FullSimplify[betterhiddenzero] Out[7]= 0 ``` --- Results of simplification may depend on the names of symbols: ```wl In[1]:= Simplify[(1 - a ^ 2) / b ^ 2, a ^ 2 + b ^ 2 == 1] Out[1]= 1 In[2]:= Simplify[(1 - c ^ 2) / b ^ 2, c ^ 2 + b ^ 2 == 1] Out[2]= (1 - c^2/b^2) ``` ## See Also * [`FullSimplify`](https://reference.wolfram.com/language/ref/FullSimplify.en.md) * [`Refine`](https://reference.wolfram.com/language/ref/Refine.en.md) * [`Factor`](https://reference.wolfram.com/language/ref/Factor.en.md) * [`Expand`](https://reference.wolfram.com/language/ref/Expand.en.md) * [`TrigExpand`](https://reference.wolfram.com/language/ref/TrigExpand.en.md) * [`PowerExpand`](https://reference.wolfram.com/language/ref/PowerExpand.en.md) * [`ComplexExpand`](https://reference.wolfram.com/language/ref/ComplexExpand.en.md) * [`PiecewiseExpand`](https://reference.wolfram.com/language/ref/PiecewiseExpand.en.md) * [`Element`](https://reference.wolfram.com/language/ref/Element.en.md) * [`FunctionExpand`](https://reference.wolfram.com/language/ref/FunctionExpand.en.md) * [`ArraySimplify`](https://reference.wolfram.com/language/ref/ArraySimplify.en.md) * [`ArrayExpand`](https://reference.wolfram.com/language/ref/ArrayExpand.en.md) * [`Reduce`](https://reference.wolfram.com/language/ref/Reduce.en.md) * [`Assuming`](https://reference.wolfram.com/language/ref/Assuming.en.md) * [`TrigReduce`](https://reference.wolfram.com/language/ref/TrigReduce.en.md) * [`TrigFactor`](https://reference.wolfram.com/language/ref/TrigFactor.en.md) ## Tech Notes * [Simplifying Algebraic Expressions](https://reference.wolfram.com/language/tutorial/AlgebraicCalculations.en.md#16626) * [Simplifying with Assumptions](https://reference.wolfram.com/language/tutorial/AlgebraicCalculations.en.md#12157) * [Simplification](https://reference.wolfram.com/language/tutorial/AlgebraicManipulation.en.md#1259) * [Putting Expressions into Different Forms](https://reference.wolfram.com/language/tutorial/AlgebraicCalculations.en.md#21991) * [Using Assumptions](https://reference.wolfram.com/language/tutorial/AlgebraicManipulation.en.md#31833) * [Implementation notes: Algebra and Calculus](https://reference.wolfram.com/language/tutorial/SomeNotesOnInternalImplementation.en.md#20945) ## Related Guides * [Formula Manipulation](https://reference.wolfram.com/language/guide/FormulaManipulation.en.md) * [Manipulating Equations](https://reference.wolfram.com/language/guide/ManipulatingEquations.en.md) * [Algebraic Transformations](https://reference.wolfram.com/language/guide/AlgebraicTransformations.en.md) * [Assumptions and Domains](https://reference.wolfram.com/language/guide/AssumptionsAndDomains.en.md) * [Finite Mathematics](https://reference.wolfram.com/language/guide/FiniteMathematics.en.md) ## Related Links * [NKS\|Online](http://www.wolframscience.com/nks/search/?q=Simplify) * [A New Kind of Science](http://www.wolframscience.com/nks/) ## History * Introduced in 1988 (1.0) \| Updated in 1996 (3.0) ▪ 1999 (4.0) ▪ 2000 (4.1) ▪ 2002 (4.2) ▪ 2003 (5.0) ▪ [2014 (10.0)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn100.en.md) ▪ [2025 (14.2)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn142.en.md)