+ j 4^RIHt^RIHt^RIHtHtHtHtH t H t H t H t ^RI Ht^RIHtHtHtHtHtHtHt^RIHtHt^RIHt^RIHt^R IHtH t H!t!^R I"H#t#^R I$H%t%H&t&H't'H(t(H)t)H*t*H+t+H,t,H-t-^R I$H.t.^R I/H0t0^RI1H2t2^RI3H4t4H5t5H6t6H7t7^RI8H9t9^RI:H;t;^RIH?t?^RI@HAtA^RIBHCtC^RIDHEtE^RIFHGtG.RRR3RltH]2]03tIRtJR0RltKRtLRR/R ltMR!tNR"sOR#tPR$tQR%tRR&tSR'tTR(tUR0R)ltV]R0R*l4tWR+R/R,ltXR-tYR.tZR1R/lt[R"#)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)debugFgrlexchaaaRoRoVVV3Rlp\R4oVP\PS4pS\P3.p\ V4P 4wrx\ Wx.4wwrp \RT P4T!T PT4wrp\RT 4\RTR\T44\R T R\T44T'gT#\YT\R 7p\R \T4R\T44^R IHpT 'Ed3T P !\#T4P%T P4!'Ed\'Y~T ,R 7P(!T!p.pTP+4EFwpp\#\ TT.4^,P4pRpT'dRpT Fp\'T4pTP-TP4'dK1TP !\#TP4P/T4!'dKkRpTP1TP34P4K KTUu.uFpTT9gK TNK ppTP4Uu.uFDpTP !TP%TP4!'gK4TP74NKF ppTP9\;\=T T4UUu.uFwppTT,NK upp!T!TT, TTTT\TR7PT4,4EK \?T!#T!T\T4Y=T,T\TR7PT4# \dTu#i;iuupiuupiuuppi)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 6^p...rCpVEFp\V\\34'dTpK$\V\4'dVP V4KM\ V4'dVP V^,VR,34TP \VR,Uu.uFqe^,!V4NK upV^,!\VR,!4.,4^,P4KVP V4EK WW43#uupi)z-Split hints into (n, funcs, iterables, gens).:NN) isinstancerrrappendr3extendr,r gens)hintsnfuncs iterablesr<exs& X/home/luca/screen-checking/.venv/lib/python3.14/site-packages/sympy/simplify/trigsimp.py parse_hints&trigsimp_groebner..parse_hintss !#R$A!j'233A}-- Q!  !A$"/ 3&'e,eqT!We,!S!B%[0A/BBDDEGGKtM A((-s$Dc .p\R4pVEF[wrE\\\\V4^,\V4^,,^, .\\ \ \ V4^,\ V4^,, ^, .3FwrgrV^8XdWFV39dVP!V 4K)WH8XdGVP!V!WP,4V!WP,4,V!WP,4, 4KuWFV39gKV!WS,4PRR7PW04p VP!V!WP,4V , 4K EK^ \\V44#)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. yTtrig) 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  <aaS"!V4wr#rE\RW4V34\V4pVPV4\\V44p\\V44p\\V44p\\ \ \\\0pVUu.uFCpVPV9gKVP^,P4VP3NKE ppVUu.uFqwPV9gKVNK p p.p /p VF+wwroV PV .4PV S34K- .pV P4EFwppVUu.uF pV^,NK ppVUu.uF pV^,NK pp\!\"V4p\%VV4UUu.uFwppVVV, 3NK ppp\VV,4o\ \\ .\\\.3Fqwppp\&;QJd"V3RlVVV34F 'gK RM RM!V3RlVVV344'gKOSP)V4SP)V4Ks SF/oVPV3Rl\+^V^,444K1 .pVFwopS\ 8Xd/VP\V34VP\ V34S\\ 39d#\ S9dVP\ V34S\8Xd/VP\V34VP\V34S\\39gK\S9gKVP\V34K VPV4V\-V!,pS !VV4pVPV4T PVUUu0uFwppV!VV,4kK upp4EK VFwopS\ 8Xd"VP\V3\ V3.4K2S\8Xd"VP\V3\V3.4K^\/R\1V4,\2R7pS!\5V!4P7RR7P9\\%VV444pVPS!\5V!4V, 4K S!V9dBVPS!^,^,4V P;S!4V PS!4WV 3#uupiuupiuupiuupiuuppiuuppi) 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: %sc3,<"TF qS9xK R#5iN).0rBfss& rC :trigsimp_groebner..analyse_gens..6s2 1Bw TFc3,<"TF pSV3xK R#5irZr[)r\krNs& rCr^r_:s>ob!Wor`zd:%iclsrH)r4rKr;rLrrr#r"r r!funcargs as_coeff_mul setdefaultr:itemsrrzipanyaddranger rlenrr rrJremove)#r<r=r>r?r@ extragensallfuncsg trigtermsfreegensnewgenstrigdictrOvarreskeyvalrBfnsgcdrNvrMrPrQrRextrarrfdummysexprr]rUmyIrDs#&& ` @rC analyse_gens'trigsimp_groebner..analyse_genssJ*5U);&) A+ -Dz ISZ Y( CIc4t4AE,A(*8affQi,,.7 , $>t!vvX'=AAt> ) LU"   R ( / / <!* (HC""%%A1Q44C%!$%A1Q44C%s#C03C > Wb!b!C%[ E>US[!B #sOdD$-?@1a32Aq 23332Aq 222FF1IFF1IA >eAq1uo>>EA9LL#q*LL#q*#s#r LL#q*:LL$+LL$+$%$"*LL$+ LL CI AAu%A JJqM NN7ABqsG7 8Y)^"HBSy  3+T{!;<t  4,t !=> #d)!3?3<(//T/:??SQUEV@WX 2c4j>D01" $; JJsAvz " OOC NN3 g%%W,?0&%>08s0T011T0(T5?T5T:1T?$UU rz initial gens:zideal:z new gens:z -- lenz free gens:)orderr<domainzgroebner basis:)ratsimpmodprime)r<TF)rr<quickr polynomial) rrJr ImaginaryUnitr+as_numer_denomr,r.r4r<rnr/r-rKsympy.simplify.ratsimpr has_only_gensrL intersectionr)ejectrM issuperset differenceupdateexcludepolysas_exprr:r rjr ) rr=rrrrrJnumdenompnumpdenomoptidealrtr<GrrxmonomrOourgenschangedprBrealgensrrourGabrUrrDs &&&&& @@@rCtrigsimp_groebnerrs f)(0d&L *C 99Q__c *D !// " #D,,.JC5slC /388$(59ET (E +tYD 2 ,)SY7  $r:A T!WiQ8 7xF((#d)*@*@*MNN3(]+1148IIKLE515%.A!DIIJGGAQA"--aff55??CK,B,B7,KLL"&qyy{'7'78 $(84a1<4H8*+DAOOW%9%9!&&%AB AIIKDD JJsc(E.BC.BdaQTT.BCD&uU{D,4E"2<>>Bd4jI J5(<Cy $q'd]z;;?4: F{  H9DCs0,N6 N$N$1N)N)N. N! N!c2aaRoVV3Rlo\VS4#)cvVP^,VP^,8H# \dR#i;i)rF)rf IndexError)rBrGs&&rC check_args%_trigsimp_inverse..check_argss4 66!9q ) )  s &) 88cX<\VRR4pVen\VP^,V!44'dF\V!4!^4\4'd%VP^,P^,#\V\4'EdVPwr#\ V4'dS!\ V)V44)#\ V4'd(\ PS!\ W#)44, #S!W24'd\V\4'd*\V\4'dVP^,#\V\4'dF\V\4'd0\ P^, VP^,, #V#)inverseN) getattrr9rfr(r&rrPirr)rvrrrGrBrfs& rCr_trigsimp_inverse..fs B 4 ( MjQS9913q6#899771:??1% % b% 77DAA%A,''attaa o--!a%%*Q*<*<66!9$a%%*Q*<*<44!8affQi// rW)r )rrrs&@@rC_trigsimp_inversers . R rWc aaa ^RIHo\V4p\VRR4pVe V!R/SB#SP RR4pV'g8SP RR4SP RR4SP RR 4pMRpR o R VV3R lR R RV V3RlRV 3RlRV3Rl/V,pV!V4pV'd \ V4pV#)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 )fu_eval_trigsimpNoldFdeep recursivemethodmatchingclaaVV3RloS!V4p\V\4'gV#\V3/SB#)c <VP'dV#VPUu.uF pS!V4NK ppVP'gVP'dVUu.uFp\ V3/SBNK ppVP !V!#uupiuupirZis_Atomrf is_Functionis_PowrrerArBrfoptstraverses& rCr0trigsimp..groebnersimp..traversesyyy)*0AHQKD0}}}>BCd)!4t4dC664= 1C BB )r9rr)exrnewrs&l @rC groebnersimptrigsimp..groebnersimps5 !rl#t$$J ---rWrc<S!V3/SB#rZr[)rBrrs&rCtrigsimp..,s ArWc\V4#rZ)futrigrBs&rCrr-svayrWr/c<S!V3/SB#rZr[)rBrrs&rCrr.s|A66rWcombinedc8<\S!VR^\.R74#T)rr=)rr#)rBrs&rCrr/svl1*.q#h'@ ArWc<\V3/SB#rZ) trigsimp_old)rBrs&rCrr1sa040rWr[)sympy.simplify.furrrpopr) rrrrrr trigsimpfuncexpr_simplifiedrrs &&l @@rCtrigsimprs~% 4=DT#3T:N!%%% ((5% C   d#(J/ . &(6A 0 L#4(O+O< rWcva^RIHpHpRp\W4pV3Rlo\VS4pVP \ 4'dV!V4wpoS!V!V44pVP \ 4'd V!V4pVP \4'dVP \4'dTpV#)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) ) hyper_as_trigTR2icV.pVP!\'d%VPVP\44VPVP\ 44\ VR\/#ry)has_trigsr:rewriterrminr)rAchoicess& rCexp_trigexptrigsimp..exp_trigLsN# 55&> NN199S> *qyy~&G+++rWc4<a VP'gV#VP4wr\V4^8dS !\V!4\V!,#VP 4pVP 4p\ P3V 3Rllo V\ P,pVEFEpVP'gK\VP4^8XgK4VP^,pS !VP^,V, 4wrV 'gKsW6,p WF;;,V ,uu&WY)V ,^, 8XdV\ P;;,V,uu&^pV^8Xd7V^V,\V ^, 4,;;,V , uu&EKVRV,\V ^, 4,;;,V , uu&EK<V^V\ PV ,,, ,V )8XdV^V\ PV ,,, V^8Xd0WG)\V ^, 4, ;;,V , uu&EKWG)\V ^, 4,;;,V , uu&EKV^V\ PV ,,,;;,V , uu&WG;;,V , uu&EKH \VUu.uFqfWF,,NK up!#uupi)r8cX<V\PJdV\P3#\V\4'g2VP 'd-VP \P8XdWP3#V\PJdS!V)\P)R7#R#))sign)NN)rExp1Oner9rrbase)rrsignlogs&&rCr'exptrigsimp..f..signlogbsrqvv~QUU{"D#&&4;;;499;NXX~%uAEE622!!rW)is_Mulargs_cncrnr as_powers_dictcopyrrris_Addrfr r"r!) rcommutative_partnoncommutative_partrvdnewdeerbrPrrBmrrs & @rCrexptrigsimp..fVsyyyI02 - # $q (S*+,S2E-FF F!xxz uu "[AxxxCK1,FF1I!!&&)A+.F1 Aa<LB&LBqyQqSac]+q0+RT$qs)^,1,!d16619n,-!3Qaffai/0qyRQqS \*a/*RQqS \*a/*T!&&!)^+,1,GqLG58.AZZ.//.s9L)rrrr rr'r(r)rrrrnewexprrArs& @rC exptrigsimpr;s6,'G30h#G{{%&&W%1DG*{{())w- KKNN488A;; KrWfirstTc aaTpV'EdhVP!\'gV#\4P!VP!\Uu.uFqDP NK up!p\ V4^8d^RIHpV!V4pVP'dV!VRR7;'gTp\V\4'dK^pVP4F1pTp \V4pRSR&\V3/SBp W8XdT p W ,pK3 TpMfVP'dUVFLp VP!V 4wrV 'gK RSR&V \V 3/SB,pVP'dKLM TpSP#RR4pSP#RR4pSP#RR 4pR oR R R VV3R lRV3Rl/V,pV'd]\%V4wppV!V^,V4p\'V4F,pVP)V^,V^,4pV!VV4pK. TpMV!W4pSP+RR4'd\-V4pVV8wd \/RV4V#uupi)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) ) separatevarsT)dictFrrrrrcLaaVV3RloV'd S!V4p\V3/SB#)c <VP'dV#VPUu.uF pS!V4NK ppVP'gVP'dVUu.uFp\ V3/SBNK ppVP !V!#uupiuupirZrrs& rCr4trigsimp_old..groebnersimp..traverserr)r)rrrrs&&l@rCr"trigsimp_old..groebnersimps& ! 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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 $qbd3q619nq&8#> qs47A:~  !A2a4Q ?Q#6= qsAd1gI>! !A %qs47A:~'94@ qsAd1gI>! !A %qs47A:~'94@!I&a-&H OrWcV\\4p\\4p.pVPFp V P'gV PW39dV P 4wrV P 'gV P'dqV PV8Xd(WjP^,;;,V , uu&KV PV8Xd(WzP^,;;,V , uu&KVPV 4K \V4\V4,p Rp V 'dqV P4pVPV4pVPV4pW!V48Xd)VPV!V4V!V4,4Rp KnWV&VW~&KxV 'gV#V'd3VP4wrVPV!V4V ,4K:V'd3VP4wrVPV!V4V ,4K:\V!#)zHelper for _match_div_rewrite. Replace f(b_)**c_*g(b_)**(rexp(c_)) with h(b)**rexph(c) if f(b_) and g(b_) are both positive or if c_ is an integer. FT) rintrfrre as_base_exp is_positive is_integerr:rLrpopitemr )rrrrrexphrexphfargsgargsrfrBrrAcommonhitryfeges&&&&&& rC_replace_mul_fpowxgpowr6s  E  E D YY 888qvv!'==?DA}}} 66Q;&&)$)$VVq[&&)$)$ AZ#e* $F C jjl YYs^ YYs^ b> KK#b ) *C#JEJ   AcFAI  AcFAI :rWcV#rZr[rs&rCrrsrWcV)#rZr[rs&rCrrs1"rWc"\P#rZ)rrrs&rCrrsrWc>V^8Xd'\V\\\\\ 4pV#V^8Xd'\V\\\ \\ 4pV#V^8Xd'\V\ \\ \\ 4pV#V^8Xd'\V\\\\\4pV#V^8Xd'\V\ \\\\4pV#V^8Xd'\V\ \\ \\ 4pV#V^8Xd'\V\\\\\ 4pV#V^ 8Xd'\V\\\ \\ 4pV#V^ 8Xd'\V\\\ \\ 4pV#V^ 8Xd'\V\\\\\4pV#V^ 8Xd'\V\\\\\4pV#V^ 8Xd'\V\\\ \\ 4pV#R#)zhelper for __trigsimpN) r6rr_midnr#_idnr$_oner"r r!r%)ris&&rC_match_div_rewriter?sAv%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$ $ KrWcNVP!\'d \W4#V#rZ)rr __trigsimp)rrs&&rCrrs  xx$%% KrWc P aaa^RIHp\f \4\woor4rVrxVP'EdVP 'gPVP 4wr\\P!V 4V4\P!V 4,pEMu\V4EFewp wrr\W 4'gK\W 4pVeVV8wdTpEM2K:VPV 4oS'gKUSPV^4'gKoSV,P'gKVP!S4pVP"'gKVP!S4pVP"'gK\$;QJd?VV3RlSS,P'\(\*44F 'gK RM4 RM0!VV3RlSS,P'\(\*444'dEKUV P!S4pM VP,'EdO.pVP.FpVP 'g@VP 4wr\P!V 4p \P!V 4pM\0P2p \VV4pVF.wp pVPV 4oSfKVP!S4pM VP5VV ,4K VVP.8wd%\7V!p\9V\;V4\<R7pVP,'dVFwp p\W 4'gKV!V4pVP?\*4'gK>VPV 4oSeSS9dSS9d\$;QJd@VVV3RlSV,P'\(\*44F 'gK RM5 RM1!VVV3RlSV,P'\(\*444'dKVP!S4pM VEF5wp pp\W 4'gK\ARV.R 7pV P!SV4p VP!SV4pVPV 4pRpV'gKjVV8wgKsTpVV,^8Xg=VV,)VV,P.9gVV,VV,,^8XdKVV9d-VV,VV,,VV,,^8XdKVP!V4pVPV 4pVPCV\0PD4K MrVP'g-VPF'gV'dGVP.'d5VPH!VP.Uu.uFp\VV4NK up!pVP>!\K'g\LhVP'\N4pVPQ\NVR 7pVV8Xd\Lh\SV4pVV8wd\9V\SV4.\<R7pVP'\N4V, 'gTpV#uupi \LdT#i;i) zrecursive helper for trigsimp)TR10iNc3\<"TF!qP^,SS,8HxK# R#5irNrf)r\rrrxs& rCr^__trigsimp.. s+H9G166!9A.9Gs),TFrc3n<"TF*qP^,SS,SS,39xK, R#5irErF)r\rrrrxs& rCr^rG0s7IH@G!q c!fc!f%55@Gs25r)r)r)*rrCr!r&ris_commutativerrr _from_args enumeraterr?matchrr+rJr*rkrr(r'rrfrrr:r rrrrrrhZerorrer TypeErrorrrr*)rrrCrPr r"r#r$r%comncr>patternsimpok1ok2rokrftermrra_trrrrArfnewrrxs&& ` @@rCrArAs!( #+!Aq!! {{{"""mmoGCS^^C0$7r8JJD09:K0L,,G3t--,T5&$& jj)33771a==q6,,, XXc]!~~~$ XXc]!~~~$sHA 13E9GHsssHA 13E9GHHH 99S>D;1M> {{{IID&&&--/^^B'~~c*UUT4(D#4jj)?!;;s+D $5 KKR  499 :DtVD\y9D ;;;#/t--T{88.//**W-C{18SSSIH?B1v||13E@GIHSSSIH?B1v||13E@GIHFHFH!!;;s+D$0$$- GVR4))sRD)Cll1c*G[[C(F 7#AC!t S6Q;31Q499,#1 0B6afQqTkAaD0A5{{1~JJw' Q'+$-.  t yytyyAy!9Q-yAB xx O JJsOll3Tl* !8Oc{ 3;sF3K(i8C #""D K%B   K s Z%B'Z Z Z%$Z%hyperc ^RIHp\V4p\V\4'gV#VP 'gV#Tp\ V\4pV'd<VP\4'd!V!V4wrV!\ V\44pW8wdMVP'd;VP ^,P'd\VP4!pV#)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)rrrr9rrfr _futrigrr'r is_Rationalr as_coeff_Mul)rArYkwargsrrrs&$, rCrrgs(0 A a   666 C!WA )**Q i7# $xAHHH!6!6!6 ! " HrWc!baaaaaaaaaa^RIHpHoHpHoHpHoHpHpH oH oH oH pH oHpHpHp Hp HoHoVP)\*4'gV#VP,'dVP/\*4wrMRp V3Rlp Ro\0VVSV3RlS\0V3Rl.SV3RlV SVWSV3RlV \0V3R l.VV\0V.\0VV3R l.VV3R lVV3R l.VV3R lVV3Rl.V\0S.\0VV3Rl.VSS\0VV3Rl.3.p \3WR7!V4pV e W,pV#)zHelper for futrig.)TR1TR2TR3rTR10LrCTR8TR6TR15TR16TR111TR5TRmorrieTR11_TR11TR14TR22TR12Nc<S!V4VP4\V4\VP4VP3#rZ)rrrnrfr)rBrds&rCr_futrig..s*adAKKM6!9c!&&k188LrWc,VP\4#rZ)rr(rs&rCrrrsaee12rWc&<\\VS4#rZ_eapplyr*rBtrigss&rCrrr'&!U+rWc&<\\VS4#rZrvrrws&rCrrrWXq%8rWc <\RVS4#)c4\VP44#rZ)r*normal)r>s&rCr+_futrig....sF188:$6rW)rvrws&rCrrrs'65ArWc&<\\VS4#rZrurws&rCrrrryrWc&<\\VS4#rZr{rws&rCrrrr|rWc <S!S!V44#rZr[)rBrars&rCrrrs T#a&\rWc2<\\S!V4S4#rZrvr)rBrjrxs&rCrrrsgj#a&%8rWc2<\\S!V4S4#rZr)rBrgrxs&rCrrrsgDGU,rWc2<\\S!V4S4#rZr)rBrfrxs&rCrrrswz3q659rWc2<\\S!V4S4#rZr)rBrhrxs&rCrrrswDGU,rWc2<\\S!V4S4#rZr)rBrorxs&rCrrrsW Q(rWc2<\\S!V4S4#rZ)rvr )rBrprxs&rCrrrsW $q'5*rW) objective)rr`rarbrrcrdrCrerfrgrhrirjrkrlrmrnrorprr(rrr1r2)rAr`rbrcrCrerirkrlrmrnrOLopstreerdrprgrhrarorrjrfrxs& @@@@@@@@@@rCr[r[s5 55& ' 'xxx##$9:q LD 2E    + 89 A    S+ 89  3 )* 8 , - : , -  4 ( ) S$ * +C# $ DJ t$Q'A  I HrWc\V\4'd\VP4#\V\4'gR#\ ;QJd&RVP 4F 'dK R# R#!RVP 44#)z@_eapply helper to tell whether ``e`` and all its args are Exprs.Fc38"TFp\V4xK R#5irZ)_is_Expr)r\r>s& rCr^_is_Expr..s+Fqx{{FsT)r9rrrrallrf)rAs&rCrrsa!Z   a   3+AFF+33+3+3+AFF+ ++rWc  \V\4'gV#\V4'gVP'g V!V4#VP!VPUu.uF"pVeV!V4'd \ W4MTNK$ up!#uupi)z`Apply ``func`` to ``e`` if all args are Exprs else only apply it to those args that *are* Exprs.)r9rrrfrerv)rerAcondeis&&& rCrvrvs{ a  {{!&&&Aw 66&&B#ld2hhR? s B 5B )FrZ)\ 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$r%r&%sympy.functions.elementary.hyperbolicr'(sympy.functions.elementary.trigonometricr( sympy.polysr)r*r+r,sympy.polys.domainsr-sympy.polys.polyerrorsr.sympy.polys.polytoolsr/sympy.simplify.cse_mainr0sympy.strategies.corer1sympy.strategies.treer2sympy.utilities.iterablesr3sympy.utilities.miscr4rrrrrrrr!r&r6r<r;r=r?rrArr[rrvr[rWrCrs#---$GGG)#%22*KKK!DJEE"5*'*(.&"$E!&KF\ !3 4DiX[~EEPA Qh)X)X ~ ~D( t( V< ~, rW