\h 2SSKJr SSKJr SSKJrJrJrJrJ r J r J r J r SSK Jr SSKJrJrJrJrJrJrJr SSKJrJr SSKJr SSKJr SS KJrJ r J!r! SS K"J#r# SS K$J%r%J&r&J'r'J(r(J)r)J*r*J+r+J,r,J-r- SS K$J.r. SS K/J0r0 SSK1J2r2 SSK3J4r4J5r5J6r6J7r7 SSK8J9r9 SSK:J;r; SSKJ?r? SSK@JArA SSKBJCrC SSKDJErE SSKFJGrG /SSS4SjrH\2\04rISrJS0SjrKSrLSS.S jrMS!rNS"qOS#rPS$rQS%rRS&rSS'rTS(rUS0S)jrV\S0S*j5rWSS+.S,jrXS-rYS.rZS1S/jr[g")2) defaultdict)reduce)sympifyBasicSExpr factor_termsMulAdd bottom_up)cacheit) count_ops_mexpand FunctionClassexpand expand_mul _coeff_isneg Derivative)IInteger)igcd)_nodes)DummysymbolsWild) SYMPY_INTS) sincosexpcoshtanhsinhtancotcoth)atan2)HyperbolicFunction)TrigonometricFunction)Polyfactorcancelparallel_poly_from_expr)ZZ)PolificationFailed)groebner)cse)identity)greedy)iterable)debugFgrlexc^^^SmSmUUU4Sjn[S5mUR[RT5nT[R4/n[ U5R 5upx[ Xx/5uupn [SU R5 U"U RU5upn[SU 5 [SUS[U55 [S U S[U55 U(dU$[XU[S 9n[S [U5S[U55 S S KJn U (GaU R "[#U5R%U R56(Ga['X~U -S9R("U6n/nUR+5GHunn[#[ UU/5SR5nSnU(aSnU Hn['U5nUR-UR5(aM0UR "[#UR5R/U56(aMhSnUR1UR35R5 M U(aMUVs/sH nUU;dM UPM nnUR4Vs/sHBnUR "UR%UR56(dM2UR75PMD nnUR9[;[=U U5VVs/sH unnUU-PM snn6U"UU- UUUU[US9RU5-5 GM [?U6$U"U[U5X=U-U[US9RU5$![a Us$f=fs snfs snfs snnf)a Simplify trigonometric expressions using a groebner basis algorithm. Explanation =========== This routine takes a fraction involving trigonometric or hyperbolic expressions, and tries to simplify it. The primary metric is the total degree. Some attempts are made to choose the simplest possible expression of the minimal degree, but this is non-rigorous, and also very slow (see the ``quick=True`` option). If ``polynomial`` is set to True, instead of simplifying numerator and denominator together, this function just brings numerator and denominator into a canonical form. This is much faster, but has potentially worse results. However, if the input is a polynomial, then the result is guaranteed to be an equivalent polynomial of minimal degree. The most important option is hints. Its entries can be any of the following: - a natural number - a function - an iterable of the form (func, var1, var2, ...) - anything else, interpreted as a generator A number is used to indicate that the search space should be increased. A function is used to indicate that said function is likely to occur in a simplified expression. An iterable is used indicate that func(var1 + var2 + ...) is likely to occur in a simplified . An additional generator also indicates that it is likely to occur. (See examples below). This routine carries out various computationally intensive algorithms. The option ``quick=True`` can be used to suppress one particularly slow step (at the expense of potentially more complicated results, but never at the expense of increased total degree). Examples ======== >>> from sympy.abc import x, y >>> from sympy import sin, tan, cos, sinh, cosh, tanh >>> from sympy.simplify.trigsimp import trigsimp_groebner Suppose you want to simplify ``sin(x)*cos(x)``. Naively, nothing happens: >>> ex = sin(x)*cos(x) >>> trigsimp_groebner(ex) sin(x)*cos(x) This is because ``trigsimp_groebner`` only looks for a simplification involving just ``sin(x)`` and ``cos(x)``. You can tell it to also try ``2*x`` by passing ``hints=[2]``: >>> trigsimp_groebner(ex, hints=[2]) sin(2*x)/2 >>> trigsimp_groebner(sin(x)**2 - cos(x)**2, hints=[2]) -cos(2*x) Increasing the search space this way can quickly become expensive. A much faster way is to give a specific expression that is likely to occur: >>> trigsimp_groebner(ex, hints=[sin(2*x)]) sin(2*x)/2 Hyperbolic expressions are similarly supported: >>> trigsimp_groebner(sinh(2*x)/sinh(x)) 2*cosh(x) Note how no hints had to be passed, since the expression already involved ``2*x``. The tangent function is also supported. You can either pass ``tan`` in the hints, to indicate that tan should be tried whenever cosine or sine are, or you can pass a specific generator: >>> trigsimp_groebner(sin(x)/cos(x), hints=[tan]) tan(x) >>> trigsimp_groebner(sinh(x)/cosh(x), hints=[tanh(x)]) tanh(x) Finally, you can use the iterable form to suggest that angle sum formulae should be tried: >>> ex = (tan(x) + tan(y))/(1 - tan(x)*tan(y)) >>> trigsimp_groebner(ex, hints=[(tan, x, y)]) tan(x + y) c Sn///pCnUHn[U[[45(aUnM"[U[5(aUR U5 MJ[ U5(atUR USUSS45 UR [USSVs/sH oeS"U5PM snUS"[USS65/-5SR5 MUR U5 M XX44$s snf)z-Split hints into (n, funcs, iterables, gens).rN) isinstancerrrappendr3extendr,r gens)hintsnfuncs iterablesr<exs O/var/www/auris/envauris/lib/python3.13/site-packages/sympy/simplify/trigsimp.py parse_hints&trigsimp_groebner..parse_hintss !#R$A!j'233A}-- Q!  !A$!"/ 3&'e,eqT!We,!S!AB%[0A/BBDDEGGKtM A((-sC1c z/n[S5nUGHupE[[[[U5S-[U5S--S- /[[ [ [ U5S-[ U5S-- S- /4HupgpUS:XaXFU4;aUR"U 5 M'XH:Xa2UR"U"XP-5U"XP-5-U"XP-5- 5 M^XFU4;dMgU"XS-5RSS9RX05n UR"U"XP-5U - 5 M GM [[U55$)aF Build generators for our ideal. ``Terms`` is an iterable with elements of the form (fn, coeff), indicating that we have a generator fn(coeff*x). If any of the terms is trigonometric, sin(x) and cos(x) are guaranteed to appear in terms. Similarly for hyperbolic functions. For tan(n*x), sin(n*x) and cos(n*x) are guaranteed. yr8Ttrig) rrrr#r r"r!r:rsubslistset) rBtermsrrGfncoeffcstrelcns rC build_ideal&trigsimp_groebner..build_ideals  #JIB#sCFAIA $9A$=>4tAwzDGQJ'>'BC!E aA:"A,HHSMWHHQuwZ%' 2QuwZ?@q6\EG+++6;;AABHHR[2-.!ECF|c  >^^T""U5up#pE[SX4U45 [U5nURU5 [[U55n[[U55n[[U55n[[ [ [[[1nUVs/sH>nURU;dMURSR5UR4PM@ nnUVs/sHowRU;dMUPM n n/n 0n UH+uupmU RU /5RU T45 M- /nU R5GH^unnUVs/sHnUSPM nnUVs/sHnUSPM nn[!["U5n[%UU5VVs/sH unnUUU- 4PM nnn[UU-5m[ [[ /[[[/4HHunnn['U4SjUUU455(dM&TR)U5 TR)U5 MJ TH+mURU4Sj[+SUS-555 M- /nUHumnT[ :Xa.UR[U45 UR[ U45 T[[ 4;a![ T;aUR[ U45 T[:Xa.UR[U45 UR[U45 T[[4;dM[T;dMUR[U45 M URU5 U[-U6-nT "UU5nURU5 U RUVVs1sHunnU"UU-5iM snn5 GMa UHumnT[ :Xa!UR[U4[ U4/5 M1T[:Xa!UR[U4[U4/5 M\[/S[1U5-[2S9nT"[5U65R7SS 9R9[[%UU555nURT"[5U65U- 5 M T!U;a9URT!S -S-5 U R;T!5 U RT!5 XU 4$s snfs snfs snfs snfs snnfs snnf) z Analyse the generators ``gens``, using the hints ``hints``. The meaning of ``hints`` is described in the main docstring. Return a new list of generators, and also the ideal we should work with. z1n=%s funcs: %s iterables: %s extragens: %srr8c3,># UH oT;v M g7fN).0rBfss rC :trigsimp_groebner..analyse_gens..6s2 1Bw c3,># UH nTU4v M g7fr[r\)r]krOs rCr_r`:s>ob!Worazd:%iclsTrIrH)r4rLr;rMrrr#r"r r!funcargs as_coeff_mul setdefaultr:itemsrrzipanyaddranger rlenrr rrKremove)#r<r=r>r?r@ extragensallfuncsg trigtermsfreegensnewgenstrigdictrPvarreskeyvalrBfnsgcdrOvrNrQrRrSextrarrgdummysexprr^rVmyIrDs# ` @rC analyse_gens'trigsimp_groebner..analyse_genss.*5U);&) A+ -Dz ISZ Y( CIc4t4AE,A(*8affQi,,.7 , $>t!vvX'=At> ) LU"   R ( / / <!* (HC""%%A1Q4C%!$%A1Q4C%s#C03C > Wb!b!C%[ E>US[!B #sOdD$-?@1a2Aq 222FF1IFF1IA >eAq1uo>>EA9LL#q*LL#q*#s#r LL#q*:LL$+LL$+$%$"*LL$+ LL CI AAu%A JJqM NN7ABqsG7 8Y)^"HBSy  3+T{!;<t  4,t !=> #d)!3?3<(//T/:??SQUEV@WX 2c4j>D01" $; JJsAvz " OOC NN3 g%%W,?0&%>08s0S 1-S $S;SS*SSS# rz initial gens:zideal:z new gens:z -- lenz free gens:)orderr<domainzgroebner basis:r)ratsimpmodprime)r<r8TF)rr<quickr polynomial) rrKr ImaginaryUnitr+as_numer_denomr,r.r4r<ror/r-rLsympy.simplify.ratsimpr has_only_gensrM intersectionr)ejectrN issuperset differenceupdateexcludepolysas_exprr:r rkr ) rr=rrrrrKnumdenompnumpdenomoptidealrur<GrrymonomrPourgenschangedprBrealgensrsourGabrVrrDs @@@rCtrigsimp_groebnerrs f)(0d&L *C 99Q__c *D !// " #D,,.JC5slC /388$(59ET (E +tYD 2 ,)SY7  $r:A T!WiQ8 7F((#d)*@*@*MN3(]+1148IIKLE515%.A!DIIJGGAQA"--aff55??CK,B,B7,KLL"&qyy{'7'78 '$(84a1<4H8*+DAOOW%9%9!&&%AB AIIKDD JJsc(E.BC.BdaQT.BCD&uU{D,4E"2<>>Bd4jI J5(<Cy $q'd]z;;?4: F{  H9DCs0,M)& M;4M; 1N?N4N) M87M8c2^^SmUU4Sjm[UT5$)NcbURSURS:H$![a gf=f)NrF)rg IndexError)rBrGs rC check_args%_trigsimp_inverse..check_argss4 66!9q ) )  s ! ..c>[USS5nUb_[URSU"55(a<[U"5"S5[5(aURSRS$[U[5(aURup#[ U5(aT"[ U*U55*$[ U5(a#[ RT"[ X#*55- $T"X25(a[U[5(a$[U[5(aURS$[U[5(a8[U[5(a#[ RS- URS- $U$)Ninverserr8rH) getattrr9rgr(r&rrPirr)rvrsrGrBrfs rCr_trigsimp_inverse..fs B 4 ( MjQS9913q6#899771:??1% % b% 77DAA%A,''attaa o--!a%%*Q*<*<66!9$a%%*Q*<*<44!8affQi// rX)r )rrrs @@rC_trigsimp_inversers . R rXc x^^^ SSKJm [U5n[USS5nUbU"S0TD6$TR SS5nU(d7TR SS5 TR SS5 TR S S 5nOSnS m UU4S jS U U4SjU 4SjU4SjS.UnU"U5nU(a [ U5nU$)avReturns a reduced expression by using known trig identities. Parameters ========== inverse : bool, optional If ``inverse=True``, it will be assumed that a composition of inverse functions, such as sin and asin, can be cancelled in any order. For example, ``asin(sin(x))`` will yield ``x`` without checking whether x belongs to the set where this relation is true. The default is False. Default : True method : string, optional Specifies the method to use. Valid choices are: - ``'matching'``, default - ``'groebner'`` - ``'combined'`` - ``'fu'`` - ``'old'`` If ``'matching'``, simplify the expression recursively by targeting common patterns. If ``'groebner'``, apply an experimental groebner basis algorithm. In this case further options are forwarded to ``trigsimp_groebner``, please refer to its docstring. If ``'combined'``, it first runs the groebner basis algorithm with small default parameters, then runs the ``'matching'`` algorithm. If ``'fu'``, run the collection of trigonometric transformations described by Fu, et al. (see the :py:func:`~sympy.simplify.fu.fu` docstring). If ``'old'``, the original SymPy trig simplification function is run. opts : Optional keyword arguments passed to the method. See each method's function docstring for details. Examples ======== >>> from sympy import trigsimp, sin, cos, log >>> from sympy.abc import x >>> e = 2*sin(x)**2 + 2*cos(x)**2 >>> trigsimp(e) 2 Simplification occurs wherever trigonometric functions are located. >>> trigsimp(log(e)) log(2) Using ``method='groebner'`` (or ``method='combined'``) might lead to greater simplification. The old trigsimp routine can be accessed as with method ``method='old'``. >>> from sympy import coth, tanh >>> t = 3*tanh(x)**7 - 2/coth(x)**7 >>> trigsimp(t, method='old') == t True >>> trigsimp(t) tanh(x)**7 r)fu_eval_trigsimpNoldFdeep recursivemethodmatchingcj^^UU4SjmT"U5n[U[5(dU$[U40TD6$)Nc>UR(aU$URVs/sH nT"U5PM nnUR(dUR(aUVs/sHn[ U40TD6PM nnUR "U6$s snfs snfr[is_Atomrg is_Functionis_PowrrfrArBrgoptstraverses rCr0trigsimp..groebnersimp..traversemyy)*0AHQKD0}}>BCd)!4t4dC664= 1C BB )r9rr)exrnewrs ` @rC groebnersimptrigsimp..groebnersimps5 !rl#t$$J ---rXc>T"U40TD6$r[r\)rBrrs rCtrigsimp..,s ArXc[U5$r[)futrigrBs rCrr-svayrXc>T"U40TD6$r[r\)rBrrs rCrr.s|A66rXc2>[T"USS[/S95$NTrH)rr=)rr#)rBrs rCrr/svl1*.q#h'@ ArXc>[U40TD6$r[) trigsimp_old)rBrs rCrr1sa040rX)rrr/combinedrr\)sympy.simplify.furrrpopr) rrrrrr trigsimpfuncexpr_simplifiedrrs ` @@rCtrigsimprs~% 4=DT#3T:N!%%% ((5% C   d#(J/ .'(6A0 L#4(O+O< rXcn^SSKJnJn Sn[X5nU4Sjm[UT5nUR [ 5(aU"U5unmT"U"U55nUR [ 5(aU"U5nUR [5(aUR [5(aUnU$)z Simplifies exponential / trigonometric / hyperbolic functions. Examples ======== >>> from sympy import exptrigsimp, exp, cosh, sinh >>> from sympy.abc import z >>> exptrigsimp(exp(z) + exp(-z)) 2*cosh(z) >>> exptrigsimp(cosh(z) - sinh(z)) exp(-z) r) hyper_as_trigTR2icU/nUR"[6(a$URUR[55 URUR[ 55 [ US[06$)Nrz)has_trigsr:rewriterrminr)rAchoicess rCexp_trigexptrigsimp..exp_trigLsN# 55&> NN199S> *qyy~&G+++rXc>^ UR(dU$UR5up[U5S:aT "[U65[U6-$UR 5nUR 5n[ R4U 4Sjjm U[ RnUGHnUR(dM[UR5S:XdM2URSnT "URSU- 5upU (dMdX6n XF==U -ss'XY*U -S- :XaeU[ R==U-ss'SnUS:Xa!USU-[U S- 5-==U - ss'MUSU-[U S- 5-==U - ss'MUSU[ RU --- U *:Xa^USU[ RU --- US:XaXG*[U S- 5- ==U - ss'GMEXG*[U S- 5-==U - ss'GMdUSU[ RU ---==U - ss'XG==U - ss'GM [UVs/sH ofXF-PM sn6$s snf)Nr8cF>U[RLaU[R4$[U[5(d/UR (a+UR [R:Xa XR4$U[RLaT"U*[R*S9$g)N)sign)NN)rExp1Oner9rrbase)rrsignlogs rCr'exptrigsimp..f..signlogbsmqvv~QUU{"D#&&4;;499;NXX~%uAEE622!rXrHr)is_Mulargs_cncror as_powers_dictcopyrrris_Addrgr r"r!) rcommutative_partnoncommutative_partrvdnewdeercrQrrBmrrs @rCrexptrigsimp..fVsyyI02 - # $q (S*+,S2E-FF F!xxz uu "[AxxxCK1,FF1I!!&&)A+.F1 Aa<LB&LBqyQqSac]+q0+RT$qs)^,1,!d16619n,-!3Qaffai/0qyRQqS \*a/*RQqS \*a/*T!&&!)^+,1,GqLG58.AZ.//.s<I)rrrr rr'r(r)rrrrnewexprrArs @rC exptrigsimpr;s6,'G30h#G{{%&&W%1DG*{{())w- KKNN488A;; KrXT)firstc ^^UnU(GaSUR"[6(dU$[5R"UR"[6Vs/sHoDR PM sn6n[ U5S:aSSKJn U"U5nUR(aU"USS9=(d Un[U[5(aESnUR5H,nUn [U5nSTS'[U40TD6n X:XaU n X -nM. UnO_UR(aNUHFn UR!U 5upU (dMSTS'U [U 40TD6-nUR(aMF O UnTR#SS5nTR#S S5nTR#S S 5nS mS UU4SjU4SjS.UnU(aP[%U5unnU"USU5n['U5H$nUR)USUS5nU"UU5nM& UnOU"X5nTR+SS5(a[-U5nUU:wa [/SU5 U$s snf)a Reduces expression by using known trig identities. Notes ===== deep: - Apply trigsimp inside all objects with arguments recursive: - Use common subexpression elimination (cse()) and apply trigsimp recursively (this is quite expensive if the expression is large) method: - Determine the method to use. Valid choices are 'matching' (default), 'groebner', 'combined', 'fu' and 'futrig'. If 'matching', simplify the expression recursively by pattern matching. If 'groebner', apply an experimental groebner basis algorithm. In this case further options are forwarded to ``trigsimp_groebner``, please refer to its docstring. If 'combined', first run the groebner basis algorithm with small default parameters, then run the 'matching' algorithm. 'fu' runs the collection of trigonometric transformations described by Fu, et al. (see the `fu` docstring) while `futrig` runs a subset of Fu-transforms that mimic the behavior of `trigsimp`. compare: - show input and output from `trigsimp` and `futrig` when different, but returns the `trigsimp` value. Examples ======== >>> from sympy import trigsimp, sin, cos, log, cot >>> from sympy.abc import x >>> e = 2*sin(x)**2 + 2*cos(x)**2 >>> trigsimp(e, old=True) 2 >>> trigsimp(log(e), old=True) log(2*sin(x)**2 + 2*cos(x)**2) >>> trigsimp(log(e), deep=True, old=True) log(2) Using `method="groebner"` (or `"combined"`) can sometimes lead to a lot more simplification: >>> e = (-sin(x) + 1)/cos(x) + cos(x)/(-sin(x) + 1) >>> trigsimp(e, old=True) (1 - sin(x))/cos(x) + cos(x)/(1 - sin(x)) >>> trigsimp(e, method="groebner", old=True) 2/cos(x) >>> trigsimp(1/cot(x)**2, compare=True, old=True) futrig: tan(x)**2 cot(x)**(-2) r8r) separatevarsT)dictFrrrrrcJ^^UU4SjmU(aT"U5n[U40TD6$)Nc>UR(aU$URVs/sH nT"U5PM nnUR(dUR(aUVs/sHn[ U40TD6PM nnUR "U6$s snfs snfr[rrs rCr4trigsimp_old..groebnersimp..traverserr)r)rrrrs `@rCr"trigsimp_old..groebnersimps& ! 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MM!NsIcURUR:H=(ai UR[5=(a UR[5=(d1 UR[5=(a UR[5$)zHelper to tell whether ``a`` and ``b`` have the same sorts of symbols in them -- no need to test hyperbolic patterns against expressions that have no hyperbolics in them.)rfrr(r')rrs rC_dotrigr"s^ 66QVV  A #$E/D)E @  !?aee,>&?ArXNcH[S[S9upn[SSS9nU[U5U--[U5U-- U[ U5U--[U5[U54U[ U5U--[U5U--U[U5U--[U5[U54U[ U5U--[U5U--U[U5U--[U5[U54U[ U5U--[U5U-- U[U5U-- [U5[U54U[ U5U--[U5U-- U[U5U-- [U5[U54U[ U5U--[ U5U--U[U5[U54U[U5S-U--[U5S- U--U[U5S-*U--[U5S-[U5S- 4U[U5S-U--[U5S- U--U[U5S-*U--[U5S-[U5S- 4U[ U5U--[U5U-- U[U5U--[R[R4U[U5U--[U5U--U[ U5U--[R[R4U[U5U--[ U5U--U[U5U--[R[R4U[U5U--[ U5U-- U[U5U-- [R[R4U[U5U--[U5U-- U[ U5U-- [R[R4U[U5U--[U5U--U[R[R4U[U5[U5--S[U5[U5--- [X-5U-[R[R44nU[U5-[U5-U[U5-[U5--U-[X-5U-U-4U[U5-[U5-U[U5-[U5-- U-[X-5U-U-4U[U5-[U5-U[U5-[U5-- U-[X- 5U-U-4U[U5-[U5-U[U5-[U5--U-[X- 5U-U-4U[ U5-[U5-U[ U5-[U5--U-[ X-5U-U-4U[U5-[U5-U[ U5-[ U5--U-[X-5U-U-44nU[U5S--X[U5S--- 4U[ U5S--US[U5- S--U- 4U[ U5S--US[U5- S--U- 4U[X-5-U[U5[U5-[U5[U5---4U[X-5-U[U5[U5-[U5[U5-- -4U[ X-5-U[ U5[ U5-S[ U5[ U5-- - -4U[ U5S--U[U5S--U- 4U[U5S--XS[U5- S--- 4U[U5S--XS[ U5- S---4U[ X-5-U[ U5[U5-[ U5[U5---4U[X-5-U[U5[U5-[ U5[ U5---4U[X-5-U[U5[U5-S[U5[U5--- -44 nX[U5S--- U-U[U5S--U-[4XS[U5- S--- U-U*[ U5S--U-[4XS[U5- S--- U-U*[ U5S--U-[4X[U5S--- U-U*[ U5S--U-[4XS[U5- S--- U-U[U5S--U-[4XS[ U5- S---U-U[U5S--U-[ 4X-X-[U5S--- U-X-[U5S--U-[4X-X-S[U5- S--- U-U*U-[ U5S--U-[4X-X-S[U5- S--- U-U*U-[ U5S--U-[4X-X-[U5S--- U-U*U-[ U5S--U-[4X-X-S[U5- S--- U-X-[U5S--U-[4X-X-S[ U5- S---U-X-[U5S--U-[ 44 nXX#XEXg4q [$)Nza b crdr F) commutativer8rH) rrrrr#r$r"r r!rrr%_trigpat)rrrQr matchers_division matchers_addmatchers_identity artifactss rC _trigpatsr&,s6 g4(GA! Se$A 3q619SVQY #a&!) SVSV< 3q619SVQY #a&!) SVSV< 3q619SVQY #a&!) SVSV< 3q619SVQY #a&!) SVSV< 3q619SVQY #a&!) SVSV< 3q619SVQY 3q63q62 CFQJ? CFQJ? * A zAo s1vz3q6A: 7 CFQJ? CFQJ? * A zAo s1vz3q6A: 7 47A:d1gqj !DGQJ,quu= 47A:d1gqj !DGQJ,quu= 47A:d1gqj !DGQJ,quu= 47A:d1gqj !DGQJ,quu= 47A:d1gqj !DGQJ,quu= 47A:d1gqj !QUUAEE2 DGd1g  DGDGO 3 4 KM155!%% )'0 3q6#a&1SV8CF? *Q .AE 1 q0@A 3q6#a&1SV8CF? *Q .AE 1 q0@A 3q6#a&1SV8CF? *Q .AE 1 q0@A 3q6#a&1SV8CF? *Q .AE 1 q0@A 4747 QtAwYtAw. . 2DKMA4EF 4747 QtAwYtAw. . 2DKMA4EF L 3q619aCFAI+o& 3q619a3q6A o)* 3q619a3q6A o)* 3qu:q#a&Q-#a&Q-789 3qu:q#a&Q-#a&Q-789 3qu:q3q6CF?QQA->?@A 47A:qa!|a'( 47A:qaQi!^++, 47A:qaQi!^++, 4;4747?T!WT!W_<=> 4;4747?T!WT!W_<=> 4;DGd1g-DGDGO0CDEF( s1vqy[1 aA kAos3 #a&1}_ q 1"SVQY,"2C8 #a&1}_ q 1"SVQY,"2C8 tAwz\ A r$q'1*}q0$7 $q' A~   !1T!WaZ qsAc!fHq= 1 $qbd3q619nq&8#> qs47A:~  !A2a4Q ?Q#6= qsAd1gI>! !A %qs47A:~'94@ qsAd1gI>! !A %qs47A:~'94@!I&a-&H OrXc[[5n[[5n/nURHn U R(dU RX4;aU R 5upU R (dU R(aVU RU:XaXjRS==U - ss'MU RU:XaXzRS==U - ss'MURU 5 M [U5[U5-n Sn U (aqU R5nURU5nURU5nX"U5:Xa#URU"U5U"U5-5 Sn OXU'UX~'U (aMqU (dU$U(a5UR5upURU"U5U -5 U(aM5U(a5UR5upURU"U5U -5 U(aM5[U6$)zHelper for _match_div_rewrite. Replace f(b_)**c_*g(b_)**(rexp(c_)) with h(b)**rexph(c) if f(b_) and g(b_) are both positive or if c_ is an integer. rFT) rintrgrrf as_base_exp is_positive is_integerr:rMrpopitemr )rrrsrexphrexphfargsgargsrgrBrrAcommonhitrzfeges rC_replace_mul_fpowxgpowr6s  E  E D YY 88qvv!'==?DA}} 66Q;&&)$)$VVq[&&)$)$ AZ#e* $F C jjl YYs^ YYs^ b> KK#b ) *C#JEJ &   AcFAI %  AcFAI % :rXcU$r[r\rs rCrrsrXcU*$r[r\rs rCrrs1"rXc"[R$r[)rrrs rCrrsrXc$US:Xa&[U[[[[[ 5nU$US:Xa&[U[[[ [[ 5nU$US:Xa&[U[ [[ [[ 5nU$US:Xa&[U[[[[[5nU$US:Xa&[U[ [[[[5nU$US:Xa&[U[ [[ [[ 5nU$US:Xa&[U[[[[[ 5nU$US:Xa&[U[[[ [[ 5nU$US :Xa&[U[[[ [[ 5nU$US :Xa&[U[[[[[5nU$US :Xa&[U[[[[[5nU$US :Xa&[U[[[ [[ 5nU$g )zhelper for __trigsimprr8rH N) r6rr_midnr#_idnr$_oner"r r!r%)ris rC_match_div_rewriterHsAv%dC 3L KI a%dC #tF KC a%dC #t@ K= a%dC 3: K7 a%dC 34 K1 a%dC $. K) a%dD$ 4& K# a%dD$ $ K b%dD$ $ K b%dD$ 4  K b%dD$ 4  K b%dD$ $ KrXcLUR"[6(a [X5$U$r[)rr __trigsimp)rrs rCrrs  xx$%% KrXc j ^^^SSKJn [c [5 [ummp4pVpxUR(GayUR (dKUR 5up[[R"U 5U5[R"U 5-nGO[U5GH un upp[X 5(dM[X 5nUb UU:waUn OM6URU 5mT(dMPTRUS5(dMiTUR(dHUR!T5nUR"(dMUR!T5nUR"(dM[%UU4SjTTR'[([*555(aMU R!T5n O UR,(Ga/nUR.HnUR (d?UR 5up[R"U 5n [R"U 5nO[0R2n [UU5nUH-un nURU 5mTcMUR!T5n O UR5UU -5 M UUR.:wa [7U6n[9U[;U5[<S9nUR,(aUHun n[X 5(dMU"U5nUR?[*5(dM<URU 5mTbCTT;a=TT;a7[%UUU4SjTUR'[([*555(aMUR!T5n O UGHun nn[X 5(dM[ASU/S9nU R!TU5n UR!TU5nURU 5nSnU(dMeUU:wdMmUnUUS:Xd&UU*UUR.;dUUUU-S:XaMUU;aUUUU-UU-S:XaMUR!U5nURU 5nURCU[0RD5 U(dGM UU:waMGM OnUR(d)URF(dU(aEUR.(a4URH"UR.Vs/sHn[UU5PM sn6nUR>"[J6(d[LeUR'[N5nURQ[NUS 9nUU:Xa[Le[SU5nUU:wa[9UU5/[<S9nUR'[N5U- (dUnU$s snf![La U$f=f) zrecursive helper for trigsimpr)TR10iNc3L># UHoRSTT:Hv M g7frNrg)r]rrrys rCr___trigsimp.. s+H9G166!9A.9Gs!$)rzc3V># UHoRSTTTT4;v M g7frNrO)r]rrrrys rCr_rP0s7IH@G!q c!fc!f%55@Gs&)r)r)r)*rrLr!r&ris_commutativerrr _from_args enumeraterrHmatchrr+rKr*rlrr(r'rrgrrr:r rrrrrriZerorrfr TypeErrorrrr*)rrrLrQr r"r#r$r%comncrGpatternsimpok1ok2rokrgtermrra_trrrrArfnewrrys ` @@rCrJrJs( #+!Aq!! {{{""mmoGCS^^C0$7r8JJD09:K0L,,G3t--,T5&$& jj)33771a==q6,, XXc]!~~$ XXc]!~~$HA 13E9GHHH 99S>D;1M> {{{IID&&--/^^B'~~c*UUT4(D#4jj)?!;;s+D $5 KKR  499 :DtVD\y9D ;;#/t--T{88.//**W-C{18SSIH?B1v||13E@GIHFHFH!!;;s+D$0$$- GVR4))sRD)Cll1c*G[[C(F 7#AC!t S6Q;31Q499,#1 0B6afQqTkAaD0A5{{1~JJw' Q'!t $-.  t yytyyAy!9Q-yAB xx O JJsOll3Tl* !8Oc{ 3;sF3K(i8C #"D K%B   K s&VBV$$ V21V2)hyperc SSKJn [U5n[U[5(dU$UR (dU$Un[ U[5nU(a:UR[5(a U"U5upU"[ U[55nX:waEUR(a4UR SR(a[UR56nU$)aReturn simplified ``e`` using Fu-like transformations. This is not the "Fu" algorithm. This is called by default from ``trigsimp``. By default, hyperbolics subexpressions will be simplified, but this can be disabled by setting ``hyper=False``. Examples ======== >>> from sympy import trigsimp, tan, sinh, tanh >>> from sympy.simplify.trigsimp import futrig >>> from sympy.abc import x >>> trigsimp(1/tan(x)**2) tan(x)**(-2) >>> futrig(sinh(x)/tanh(x)) cosh(x) r)r)rrrr9rrgr _futrigrr'r is_Rationalr as_coeff_Mul)rArbkwargsrrrs rCrrgs(0 A a   66 C!WA )**Q i7# $xAHH!6!6 ! " HrXc!N^^^^^^^^^^SSKJnJmJnJmJnJmJnJnJ mJ mJ mJ nJ mJnJnJn Jn JmJm UR)[*5(dU$UR,(aUR/[*5upOSn U4Sjn Sm[0UUTU4SjT[0U4Sj/TU4SjU TUXTU4S jU [0U4S j/UU[0U/[0UU4S j/UU4S jUU4S j/UU4SjUU4Sj/U[0T/[0UU4Sj/UTT[0UU4Sj/4/n [3XS9"U5nU bX-nU$)zHelper for futrig.r)TR1TR2TR3rTR10LrLTR8TR6TR15TR16TR111TR5TRmorrieTR11_TR11TR14TR22TR12Nc>T"U5UR5[U5[UR5UR4$r[)rrrorgr)rBrms rCr_futrig..s*adAKKM6!9c!&&k188LrXc,UR[5$r[)rr(rs rCrr{saee12rXc&>[[UT5$r[_eapplyr*rBtrigss rCrr{'&!U+rXc&>[[UT5$r[rrrs rCrr{WXq%8rXc >[SUT5$)Nc4[UR55$r[)r*normal)rGs rCr+_futrig....sF188:$6rX)rrs rCrr{s'65ArXc&>[[UT5$r[r~rs rCrr{rrXc&>[[UT5$r[rrs rCrr{rrXc >T"T"U55$r[r\)rBrjrs rCrr{s T#a&\rXc2>[[T"U5T5$r[rr)rBrsrs rCrr{sgj#a&%8rXc2>[[T"U5T5$r[r)rBrprs rCrr{sgDGU,rXc2>[[T"U5T5$r[r)rBrors rCrr{swz3q659rXc2>[[T"U5T5$r[r)rBrqrs rCrr{swDGU,rXc2>[[T"U5T5$r[r)rBrxrs rCrr{sW Q(rXc2>[[T"U5T5$r[)rr )rBryrs rCrr{sW $q'5*rX) objective)rrirjrkrrlrmrLrnrorprqrrrsrtrurvrwrxryrr(rrr1r2)rArirkrlrLrnrrrtrurvrwrPLopstreermryrprqrjrxrrsrors @@@@@@@@@@rCrdrds3 55& ' 'xx##$9:q LD 2E    + 89 A    S+ 89  3 )* 8 , - : , -  4 ( ) S$ * +C# $ DJ t$Q'A  I HrXc[U[5(a[UR5$[U[5(dg[ SUR 55$)z@_eapply helper to tell whether ``e`` and all its args are Exprs.Fc38# UHn[U5v M g7fr[)_is_Expr)r]rGs rCr__is_Expr..s+Fqx{{Fs)r9rrrrallrg)rAs rCrrsE!Z   a   +AFF+ ++rXc [U[5(dU$[U5(dUR(dU"U5$UR"URVs/sH nUb U"U5(a [ X5OUPM" sn6$s snf)z`Apply ``func`` to ``e`` if all args are Exprs else only apply it to those args that *are* Exprs.)r9rrrgrfr)rfrAcondeis rCrrsy a  {{!&&Aw 66&&B#ld2hhR? s'B)Fr[)\ collectionsr functoolsr sympy.corerrrrr r r r sympy.core.cacher sympy.core.functionrrrrrrrsympy.core.numbersrrsympy.core.intfuncrsympy.core.sortingrsympy.core.symbolrrrsympy.external.gmpyrsympy.functionsrrrr r!r"r#r$r%r&%sympy.functions.elementary.hyperbolicr'(sympy.functions.elementary.trigonometricr( sympy.polysr)r*r+r,sympy.polys.domainsr-sympy.polys.polyerrorsr.sympy.polys.polytoolsr/sympy.simplify.cse_mainr0sympy.strategies.corer1sympy.strategies.treer2sympy.utilities.iterablesr3sympy.utilities.miscr4rrrrrrrr!r&r6rErDrFrHrrJrrdrrr\rXrCrs#---$GGG)#%22*KKK!DJEE"5*'*(.&"$E!&KF\ !3 4DiX[~!%EPA Qh)X)X ~ ~D( V< ~, rX