FullSimplify

Simplify expr harder than Simplify, trying a wider set of transformations including special-function identities. Provided by @enumeratio/analytic.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageFullSimplify✓↗

FullSimplify(expr)compute-engine's own simplify(), plus a few extra targeted passes.

Typed signatures

Type	Package	Applies
(expression) -> expression	analytic	always

Domain: Transformations

Details

• Starts from compute-engine's native simplify(), which already folds the Pythagorean identity and a double-angle product on its own.
• Adds exactly four passes: the Gamma functional equation Γ(a)/Γ(b) for a concrete integer a−b; the hyperbolic Pythagorean identity cosh²(u) − sinh²(u) = 1; denesting Sqrt(a + 2·Sqrt(b)) for concrete rational a, b with a rational discriminant; and ExpToTrig everywhere in the tree (which is how a (e^x − e^(−x))/2 shows up as Sinh(x), with no separate 'looks like a hyperbolic definition' rule needed).
• NOT a general simplifier: no broader special-function identity table beyond the one Gamma shift above, no general nested-radical denesting past the one quadratic form, and no trigonometric identities beyond what simplify() already has. An expression needing more than this list is left exactly as simplify() leaves it.

Examples

Basic #

The Pythagorean identity

live ↗

In$\mathrm{FullSimplify}(\sin(x)^2+\cos(x)^2)$Out$1$

Recognizes the definition of Sinh

live ↗

In$\mathrm{FullSimplify}(\frac{1}{2}(e^{x}-\exp(-x)))$Out$\sinh(x)$

Scope #

Folds the double-angle product

live ↗

In$\mathrm{FullSimplify}(2\sin(x)\cos(x))$Out$\sin(2x)$

The functional equation of Gamma

live ↗

In$\mathrm{FullSimplify}(\frac{Γ(x+1)}{Γ(x)})$Out$x$

Denests $\sqrt{5 + 2\sqrt6} = \sqrt2 + \sqrt3$

live ↗

In$\mathrm{FullSimplify}((\sqrt{2}+\sqrt{3})-\sqrt{5+2\sqrt{6}})$Out$0$

The hyperbolic Pythagorean identity

live ↗

In$\mathrm{FullSimplify}(\cosh(x)^2-\sinh(x)^2)$Out$1$

A difference of squares multiplies out to $x^{20}-1$

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In$\mathrm{FullSimplify}((x^{10}-1)(x^{10}+1))$Out$x^{20}-1$

A Machin-type identity collapses to exactly $\pi$

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In$\mathrm{FullSimplify}(16\arctan(\frac{1}{5})-4\arctan(\frac{1}{239}))$Out$π$

A trigonometric polynomial is only partly rewritten

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In$\mathrm{FullSimplify}(4\sin(x)\cos(x)+4\sin(x)^2\cos(x)^2+1)$Out$2\sin(2x)+4\sin(x)^2\cos(x)^2+1$

live ↗

In$\mathrm{FullSimplify}((1-i)\exp(i x)+(1+i)\exp(-i x))$Out$2\sin(x)+2\cos(x)$

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In$\mathrm{Assuming}(\mathrm{Re}(x)\gt0, \mathrm{FullSimplify}(\exp(\mathrm{GammaLn}(x))))$Out$Γ(x)$

ExpToTrig is kept only where it shortens the expression: $e^x$ is no simpler as $\cosh x+\sinh x$

live ↗

In$\mathrm{FullSimplify}(e^{x})$Out$e^{x}$

live ↗

In$\mathrm{Assuming}(\Re(x)\gt0, \mathrm{FullSimplify}(\exp(\mathrm{GammaLn}(x))))$Out$Γ(x)$

About

Examples	13
Checked against	Wolfram Language 13 agree
Crosswalk	Wolfram Language

See also: FunctionExpand, ExpToTrig