The simplify(expr, constant) calling sequence is used to simplify expressions that are of type,constant and typically contain radicals or mathematical functions having exact numeric arguments.

The approach used consists of applying a set of simplification procedures (SP) in two steps:

1) The constant objects, in the form of sums, products, or function calls, are simplified using SP, starting with the inner most one and composing simplifications at each step. The routines can handle with reasonable efficiency products of fractional powers, exponentials, and various functions calls together with their multiplicative inverses.

2) When the resulting expression still contains radicals, the simplification procedure applied to each object in the previous step is applied once more, this time to the whole expression.

The simplification procedures SP used in the itemization above consist of analyzing the Real and Imaginary parts of the expression in order to conclude about possible cancellations or simpler representations. In addition, if the expression or subexpression can be evaluated numerically fast, using hardware floats, this information is used as a starting point to determine an exact rational value for the object, if that value exists.

The simplify/constant routines do take assumptions into account up to the point Re, Im, and abs take them.

The simplification of constant subexpressions entering input for simplify is performed automatically. However, to invoke this set of simplification routines, call simplify with the extra argument constant.

$\mathrm{e1}≔1+2\cos \left(\frac{2}{7}\mathrm{\pi }\right)-2\cos \left(\frac{3}{7}\mathrm{\pi }\right)-2\cos \left(\frac{1}{7}\mathrm{\pi }\right)$

$\mathrm{e1}≔1+2\cos \left(\frac{2\mathrm{\pi }}{7}\right)-2\cos \left(\frac{3\mathrm{\pi }}{7}\right)-2\cos \left(\frac{\mathrm{\pi }}{7}\right)$

$\mathrm{simplify}\left(\mathrm{e1}\,\mathrm{constant}\right)$

$0$

$\mathrm{abs}\left(\mathrm{e1}\right)$

$0$

$\mathrm{e2}≔{\left(-1\right)}^{\frac{6}{7}}-{\left(-1\right)}^{\frac{1}{7}}+{\left(-1\right)}^{\frac{4}{7}}+{\left(-1\right)}^{\frac{2}{7}}-{\left(-1\right)}^{\frac{3}{7}}+1-{\left(-1\right)}^{\frac{5}{7}}$

$\mathrm{e2}≔{\left(\mathrm{-1}\right)}^{\raisebox{1ex}{$6$}\!\left/ \!\raisebox{-1ex}{$7$}\right.}-{\left(\mathrm{-1}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$7$}\right.}+{\left(\mathrm{-1}\right)}^{\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$7$}\right.}+{\left(\mathrm{-1}\right)}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$7$}\right.}-{\left(\mathrm{-1}\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$7$}\right.}+1-{\left(\mathrm{-1}\right)}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$7$}\right.}$

$\mathrm{evalc}\left(\mathrm{e2}\right)$

$1+2\cos \left(\frac{2\mathrm{\pi }}{7}\right)-2\cos \left(\frac{3\mathrm{\pi }}{7}\right)-2\cos \left(\frac{\mathrm{\pi }}{7}\right)$

$\mathrm{simplify}\left(\mathrm{e2}\,\mathrm{constant}\right)$

$0$

$\mathrm{e3}≔\mathrm{e2}-\frac{17}{53}$

$\mathrm{e3}≔{\left(\mathrm{-1}\right)}^{\raisebox{1ex}{$6$}\!\left/ \!\raisebox{-1ex}{$7$}\right.}-{\left(\mathrm{-1}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$7$}\right.}+{\left(\mathrm{-1}\right)}^{\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$7$}\right.}+{\left(\mathrm{-1}\right)}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$7$}\right.}-{\left(\mathrm{-1}\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$7$}\right.}+\frac{36}{53}-{\left(\mathrm{-1}\right)}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$7$}\right.}$

$\mathrm{simplify}\left(\mathrm{e3}\,\mathrm{constant}\right)$

$-\frac{17}{53}$

$\mathrm{e4}≔\frac{\frac{1}{64}\mathrm{\Psi }\left(1\,\frac{1}{8}\right)\mathrm{\pi }\csc \left(\frac{1}{8}\mathrm{\pi }\right)}{\mathrm{\Gamma }\left(\frac{7}{8}\right)}+\frac{\frac{1}{64}{\mathrm{\Psi }\left(\frac{1}{8}\right)}^2\mathrm{\pi }\csc \left(\frac{1}{8}\mathrm{\pi }\right)}{\mathrm{\Gamma }\left(\frac{7}{8}\right)}-\frac{\frac{1}{64}\mathrm{\pi }\mathrm{\Psi }\left(1\,\frac{1}{8}\right)}{\sin \left(\frac{1}{8}\mathrm{\pi }\right)\mathrm{\Gamma }\left(\frac{7}{8}\right)}-\frac{\frac{1}{64}\mathrm{\pi }{\mathrm{\Psi }\left(\frac{1}{8}\right)}^2}{\sin \left(\frac{1}{8}\mathrm{\pi }\right)\mathrm{\Gamma }\left(\frac{7}{8}\right)}+1$

$\mathrm{e4}≔\frac{\mathrm{\Psi }\left(1\,\frac{1}{8}\right)\mathrm{\pi }\csc \left(\frac{\mathrm{\pi }}{8}\right)}{64\mathrm{\Gamma }\left(\frac{7}{8}\right)}+\frac{{\mathrm{\Psi }\left(\frac{1}{8}\right)}^2\mathrm{\pi }\csc \left(\frac{\mathrm{\pi }}{8}\right)}{64\mathrm{\Gamma }\left(\frac{7}{8}\right)}-\frac{\mathrm{\pi }\mathrm{\Psi }\left(1\,\frac{1}{8}\right)}{64\sin \left(\frac{\mathrm{\pi }}{8}\right)\mathrm{\Gamma }\left(\frac{7}{8}\right)}-\frac{\mathrm{\pi }{\mathrm{\Psi }\left(\frac{1}{8}\right)}^2}{64\sin \left(\frac{\mathrm{\pi }}{8}\right)\mathrm{\Gamma }\left(\frac{7}{8}\right)}+1$

$\mathrm{simplify}\left(\mathrm{e4}\,\mathrm{constant}\right)$

$1$

$\mathrm{e5}≔-\frac{1200}{7}{\cos \left(\frac{3}{7}\mathrm{\pi }\right)}^2\cos \left(\frac{1}{7}\mathrm{\pi }\right)+\frac{128}{7}{\cos \left(\frac{3}{7}\mathrm{\pi }\right)}^6-\frac{4}{7}\cos \left(\frac{3}{7}\mathrm{\pi }\right)\left(120\cos \left(\frac{3}{7}\mathrm{\pi }\right)\cos \left(\frac{1}{7}\mathrm{\pi }\right)-192{\cos \left(\frac{3}{7}\mathrm{\pi }\right)}^5\right)-\frac{1200}{7}{\cos \left(\frac{2}{7}\mathrm{\pi }\right)}^2\cos \left(\frac{3}{7}\mathrm{\pi }\right)+\frac{128}{7}{\cos \left(\frac{2}{7}\mathrm{\pi }\right)}^6+\frac{4}{7}\cos \left(\frac{2}{7}\mathrm{\pi }\right)\left(-120\cos \left(\frac{3}{7}\mathrm{\pi }\right)\cos \left(\frac{2}{7}\mathrm{\pi }\right)+192{\cos \left(\frac{2}{7}\mathrm{\pi }\right)}^5\right)+\frac{1200}{7}\cos \left(\frac{2}{7}\mathrm{\pi }\right){\cos \left(\frac{1}{7}\mathrm{\pi }\right)}^2+\frac{128}{7}{\cos \left(\frac{1}{7}\mathrm{\pi }\right)}^6-\frac{4}{7}\cos \left(\frac{1}{7}\mathrm{\pi }\right)\left(-120\cos \left(\frac{1}{7}\mathrm{\pi }\right)\cos \left(\frac{2}{7}\mathrm{\pi }\right)-192{\cos \left(\frac{1}{7}\mathrm{\pi }\right)}^5\right)$

$\mathrm{e5}≔-\frac{1200{\cos \left(\frac{3\mathrm{\pi }}{7}\right)}^2\cos \left(\frac{\mathrm{\pi }}{7}\right)}{7}+\frac{128{\cos \left(\frac{3\mathrm{\pi }}{7}\right)}^6}{7}-\frac{4\cos \left(\frac{3\mathrm{\pi }}{7}\right)\left(120\cos \left(\frac{3\mathrm{\pi }}{7}\right)\cos \left(\frac{\mathrm{\pi }}{7}\right)-192{\cos \left(\frac{3\mathrm{\pi }}{7}\right)}^5\right)}{7}-\frac{1200{\cos \left(\frac{2\mathrm{\pi }}{7}\right)}^2\cos \left(\frac{3\mathrm{\pi }}{7}\right)}{7}+\frac{128{\cos \left(\frac{2\mathrm{\pi }}{7}\right)}^6}{7}+\frac{4\cos \left(\frac{2\mathrm{\pi }}{7}\right)\left(-120\cos \left(\frac{3\mathrm{\pi }}{7}\right)\cos \left(\frac{2\mathrm{\pi }}{7}\right)+192{\cos \left(\frac{2\mathrm{\pi }}{7}\right)}^5\right)}{7}+\frac{1200\cos \left(\frac{2\mathrm{\pi }}{7}\right){\cos \left(\frac{\mathrm{\pi }}{7}\right)}^2}{7}+\frac{128{\cos \left(\frac{\mathrm{\pi }}{7}\right)}^6}{7}-\frac{4\cos \left(\frac{\mathrm{\pi }}{7}\right)\left(-120\cos \left(\frac{\mathrm{\pi }}{7}\right)\cos \left(\frac{2\mathrm{\pi }}{7}\right)-192{\cos \left(\frac{\mathrm{\pi }}{7}\right)}^5\right)}{7}$

$\mathrm{simplify}\left(\mathrm{e5}\,\mathrm{constant}\right)$

$166$

$\mathrm{e6}≔-\frac{1}{2}\mathrm{I}\exp \left(\mathrm{I}\right)\mathrm{Ei}\left(1\,\mathrm{I}\right)-\frac{1}{2}\mathrm{Ei}\left(1\,\mathrm{I}\right)\sin \left(1\right)+\frac{1}{2}\mathrm{I}\mathrm{Ei}\left(1\,\mathrm{I}\right)\cos \left(1\right)-\frac{1}{2}\mathrm{Ei}\left(1\,-\mathrm{I}\right)\sin \left(1\right)-\frac{1}{2}\mathrm{I}\mathrm{Ei}\left(1\,-\mathrm{I}\right)\cos \left(1\right)+\frac{1}{2}\mathrm{I}\exp \left(-\mathrm{I}\right)\mathrm{Ei}\left(1\,-\mathrm{I}\right)$

$\mathrm{e6}≔-\frac{\mathrm{I}{\ⅇ}^{\mathrm{I}}{\mathrm{Ei}}_1\left(\mathrm{I}\right)}{2}-\frac{{\mathrm{Ei}}_1\left(\mathrm{I}\right)\sin \left(1\right)}{2}+\frac{\mathrm{I}{\mathrm{Ei}}_1\left(\mathrm{I}\right)\cos \left(1\right)}{2}-\frac{{\mathrm{Ei}}_1\left(\mathrm{-I}\right)\sin \left(1\right)}{2}-\frac{\mathrm{I}{\mathrm{Ei}}_1\left(\mathrm{-I}\right)\cos \left(1\right)}{2}+\frac{\mathrm{I}{\ⅇ}^{\mathrm{-I}}{\mathrm{Ei}}_1\left(\mathrm{-I}\right)}{2}$

$\mathrm{\Re }\left(\mathrm{e6}\right)$

$-\frac{\mathrm{\Re }\left({\mathrm{Ei}}_1\left(\mathrm{I}\right)\sin \left(1\right)+{\mathrm{Ei}}_1\left(\mathrm{-I}\right)\sin \left(1\right)\right)}{2}+\frac{\mathrm{\Im }\left({\ⅇ}^{\mathrm{I}}{\mathrm{Ei}}_1\left(\mathrm{I}\right)-{\mathrm{Ei}}_1\left(\mathrm{I}\right)\cos \left(1\right)+{\mathrm{Ei}}_1\left(\mathrm{-I}\right)\cos \left(1\right)-{\ⅇ}^{\mathrm{-I}}{\mathrm{Ei}}_1\left(\mathrm{-I}\right)\right)}{2}$

$\mathrm{evalhf}\left(\mathrm{e6}\right)$

Error, unsupported type `indexed` in evalhf

$\mathrm{simplify}\left(\mathrm{e6}\,\mathrm{constant}\right)$

$0$

$\mathrm{e7}≔\frac{\frac{1}{2}{2}^{\frac{1}{2}}\left(-4{\left(1+2^{\frac{1}{2}}\right)}^{\frac{1}{2}}+2{\left(22^{\frac{1}{2}}-2\right)}^{\frac{1}{2}}\mathrm{\pi }+{\left(22^{\frac{1}{2}}-2\right)}^{\frac{1}{2}}\mathrm{\pi }{2}^{\frac{1}{2}}+2{\mathrm{\pi }}^{\frac{1}{2}}{\left(22^{\frac{1}{2}}-2\right)}^{\frac{1}{2}}+{\mathrm{\pi }}^{\frac{1}{2}}{2}^{\frac{1}{2}}{\left(22^{\frac{1}{2}}-2\right)}^{\frac{1}{2}}+2x{\left(22^{\frac{1}{2}}-2\right)}^{\frac{1}{2}}+x{2}^{\frac{1}{2}}{\left(22^{\frac{1}{2}}-2\right)}^{\frac{1}{2}}\right)}{{\left(22^{\frac{1}{2}}-2\right)}^{\frac{1}{2}}\left(2^{\frac{1}{2}}-1\right){\left(1+2^{\frac{1}{2}}\right)}^2}$

$\mathrm{e7}≔\frac{\sqrt{2}\left(-4\sqrt{1+\sqrt{2}}+2\sqrt{2\sqrt{2}-2}\mathrm{\pi }+\sqrt{2\sqrt{2}-2}\mathrm{\pi }\sqrt{2}+2\sqrt{\mathrm{\pi }}\sqrt{2\sqrt{2}-2}+\sqrt{\mathrm{\pi }}\sqrt{2}\sqrt{2\sqrt{2}-2}+2x\sqrt{2\sqrt{2}-2}+x\sqrt{2}\sqrt{2\sqrt{2}-2}\right)}{2\sqrt{2\sqrt{2}-2}\left(\sqrt{2}-1\right){\left(1+\sqrt{2}\right)}^2}$

$\mathrm{type}\left(\mathrm{e7}\,\mathrm{constant}\right)$

$\mathrm{false}$

$\mathrm{simplify}\left(\mathrm{e7}\right)$

$-2+\mathrm{\pi }+\sqrt{\mathrm{\pi }}+x$