*h&SSKJr SSKJr SSKJr SSKJr SSKJ r J r J r SSK J r SSKJr SSKJrJr SS KJr SS KJr SS KJr SS KJr SS KJr SSKJr SSKJ r SSK!J"r"J#r#J$r$J%r%J&r&J'r'J(r( SSK)J*r*J+r+J,r,J-r-J.r.J/r/J0r0J1r1 SSK2J3r3 SSK4J5r5 SSK6J7r7 SSK8J9r9J:r: SSK;JSr?S@Sjr@SrASrBSrCSASjrDSASjrES rFSBS!jrGS"rHSBS#jrIS$rJSCS&jrKS'rLSBS(jrMS)rNS*rOS+rPSBS,jrQSAS-jrRSAS.jrSS/rTSAS0jrUS1rVS2rW\<(a<\X"\Y"\:\=\>\?\A\B\D\E\F\G\H\I\K\M\O\@\P\Q\R\S\N\T\U455ur=r>r?rArBrDrErFrGrHrIrKrMrOr@rPrQrRrSrNrTrU\D\=4\E\=4\9/rZ\K\D\=4\E\=4\=/4r[\P\G\=4\P\G\J\=4\9/r\\B\J4\9/r]\B\A\B\M\B\O\B\=4r^\B\A\I\B\A\K4\D\F\K\B4\\\Z\H\[\B\H\H\]4\9/r_S34S4jr`SDS5jraS6R5rc\d"\X"\e"\c\X"\Y"\f"5R\c55555rh\S75ri\S85rj\S95rkS@S:jrlS;rmS<rnS=roS>rpS?rqg%)E) defaultdict)Add)cacheit)Expr)Factors gcd_terms factor_terms) expand_mul)Mul)piI)Pow)S)ordered)Dummy)sympify bottom_up)binomial)coshsinhtanhcothsechcschHyperbolicFunction)cossintancotseccscsqrtTrigonometricFunction) perfect_power)factor)greedy)identitydebug) SYMPY_DEBUGcZUR5R5R5$)zwSimplification of rational polynomials, trying to simplify the expression, e.g. combine things like 3*x + 2*x, etc.... )normalr&expandrvs N/home/mande/repo/quber/.venv/lib/python3.13/site-packages/sympy/simplify/fu.pyTR0r1 s" 99;    & & ((cSn[X5$)zReplace sec, csc with 1/cos, 1/sin Examples ======== >>> from sympy.simplify.fu import TR1, sec, csc >>> from sympy.abc import x >>> TR1(2*csc(x) + sec(x)) 1/cos(x) + 2/sin(x) c[U[5(a+URSn[R[ U5- $[U[ 5(a+URSn[R[U5- $U$Nr) isinstancer!argsrOnerr"rr/as r0fTR1..f5s_ b#   A55Q<  C  A55Q<  r2rr/r;s r0TR1r>)s R r2cSn[X5$)aReplace tan and cot with sin/cos and cos/sin Examples ======== >>> from sympy.simplify.fu import TR2 >>> from sympy.abc import x >>> from sympy import tan, cot, sin, cos >>> TR2(tan(x)) sin(x)/cos(x) >>> TR2(cot(x)) cos(x)/sin(x) >>> TR2(tan(tan(x) - sin(x)/cos(x))) 0 c[U[5(a&URSn[U5[ U5- $[U[ 5(a&URSn[ U5[U5- $U$r5)r6rr7rrr r9s r0r;TR2..fSs_ b#   Aq6#a&= C  Aq6#a&=  r2rr=s r0TR2rBAs$ R r2c&^U4Sjn[X5$)ayConverts ratios involving sin and cos as follows:: sin(x)/cos(x) -> tan(x) sin(x)/(cos(x) + 1) -> tan(x/2) if half=True Examples ======== >>> from sympy.simplify.fu import TR2i >>> from sympy.abc import x, a >>> from sympy import sin, cos >>> TR2i(sin(x)/cos(x)) tan(x) Powers of the numerator and denominator are also recognized >>> TR2i(sin(x)**2/(cos(x) + 1)**2, half=True) tan(x/2)**2 The transformation does not take place unless assumptions allow (i.e. the base must be positive or the exponent must be an integer for both numerator and denominator) >>> TR2i(sin(x)**a/(cos(x) + 1)**a) sin(x)**a/(cos(x) + 1)**a c >^ UR(dU$UR5upUR(dUR(aU$U 4Sjm UR5n[ UR 55Vs/sH'nT "X1U5(aMX1R U54PM) nnU(dU$UR5n[ UR 55Vs/sH'nT "X2U5(aMX2R U54PM) nnU(dU$U U 4SjnU"X5 U"X%5 /nUGHFn[U[5(a[URSSS9nX;a@X(X:Xa6UR[URS5X-5 S=X'X('MuT (aQSU-n X;aEX)X:Xa9UR[URSS- 5X-5 S=X'X)'MMMM[U[5(ad[URSSS9nX;aEX(X:Xa8UR[URS5X*-5 S=X'X('GMBGMEGMHT (dGMRUR(dGMfURS[RLdGM[URS[5(dGM[URSRSSS9nX;dGMX(X:XdGMX(R (dUR"(dGMUR[URSS- 5X*-5 S=X'X('GMI U(a[%XqR'5V V s/sHupU (dMX-PM sn n -6[%UR'5V V s/sHupU (dMX-PM sn n 6- nU[%UV V s/sH upX-PM sn n 6[%UV V s/sH upX-PM sn n 6- -nU$s snfs snfs sn n fs sn n fs sn n fs sn n f)Nc>>UR=(d UR=(aw UR[[4;=(dW T=(aN UR =(a; [ UR5S:=(a [SUR55$)Nc3n# UH+n[S[R"U555v M- g7f)c3# UHAn[U[5=(d% UR=(a UR[Lv MC g7fN)r6ris_Powbase).0ais r0 8TR2i..f..ok...s7,*B#2s+Kryy/KRWW^K*sA A N)anyr make_args)rLr:s r0rN.TR2i..f..ok..s7=5;,--*,,,5;s35) is_integer is_positivefuncrris_Addlenr7rP)kehalfs r0okTR2i..f..oksx.?3*$>*=*=AFF q *==56VV==  @r2c>/nUHgnUR(dM[UR5S:dM1T(a [U5O [ U5nXC:wdMUUR X445 Mi U(al[ U5Hunup4X XBU'M [U6R5nUH1nXX#-nT"X65(aX`U'MUR X645 M3 AggN) rVrWr7r&r append enumerater as_powers_dict) dddonenewkrXknewivrZr[s r0 factorize"TR2i..f..factorizesD888AFF a(,6!9,q/Dy QI.  $-dOLAy"G%4Dz002AtwA!xx ! aV,  r2rFevaluater_rF)is_Mulas_numer_denomis_Atomrblistkeyspopr6rrr7r`rrVrr8rSrTr items)r/nrcrXndonerdritr:a1brYr[rZs @r0r;TR2i..f{styyI  " 99 I @    (,QVVXJ1baDk!UU1XJI   (,QVVXJ1baDk!UU1XJI & !! A!S!!q E26adadlHHS^QT12"&&AD14QBw15AD=#affQik"2QT!9:'++qu$1w As##q E26adadlHHS^adU23"&&AD14+6!(((qvvayAEE'9qvvay#..q q)E:6adadl HHS1-u45"&&AD14-0 qWWYqwwy6ytqAdady678B #//06Nqt6N1OO OB AK Kn=6/6Ns<=P:P:!P?9P? Q 0Q  Q 'Q Q Qr)r/rZr;s ` r0TR2irz_s8Sj R r2cZ^SSKJm U4SjnURSS5n[X5$)aInduced formula: example sin(-a) = -sin(a) Examples ======== >>> from sympy.simplify.fu import TR3 >>> from sympy.abc import x, y >>> from sympy import pi >>> from sympy import cos >>> TR3(cos(y - x*(y - x))) cos(x*(x - y) + y) >>> cos(pi/2 + x) -sin(x) >>> cos(30*pi/2 + x) -cos(x) rsignsimpc \>[U[5(dU$URT"URS55n[U[5(dU$URS[R S- - R [R S- URS- R s=LaSLawO U$[[[[[[[[[[[[0nU[U5"[R S- URS- 5nU$)NrrFT)r6r$rUr7rPirTrrrr r!r"type)r/fmapr}s r0r;TR3..fs"344I WWXbggaj) *"344I GGAJa  , ,a"''!*1D0Q0Q YUY Y c3S#sCc3ODd2hQ 34B r2c"[U[5$rI)r6r$xs r0TR3..s *Q 56r2c*URSS5$)Nc@UR=(a UR$rI) is_numberrmrts r0r'TR3....sakk.ahh.r2c4UR"UR6$rIrUr7rs r0rrsaffaffor2replacers r0rrs!)) . %'r2)sympy.simplify.simplifyr}rr)r/r;r}s @r0TR3rs2$1  6 ' (B R r2c*URSS5$)aTIdentify values of special angles. a= 0 pi/6 pi/4 pi/3 pi/2 ---------------------------------------------------- sin(a) 0 1/2 sqrt(2)/2 sqrt(3)/2 1 cos(a) 1 sqrt(3)/2 sqrt(2)/2 1/2 0 tan(a) 0 sqt(3)/3 1 sqrt(3) -- Examples ======== >>> from sympy import pi >>> from sympy import cos, sin, tan, cot >>> for s in (0, pi/6, pi/4, pi/3, pi/2): ... print('%s %s %s %s' % (cos(s), sin(s), tan(s), cot(s))) ... 1 0 0 zoo sqrt(3)/2 1/2 sqrt(3)/3 sqrt(3) sqrt(2)/2 sqrt(2)/2 1 1 1/2 sqrt(3)/2 sqrt(3) sqrt(3)/3 0 1 zoo 0 c[U[5=(a8 URS[- =nR=(a UR S;$)Nr)r_rFr)r6r$r7r is_Rationalq)rrs r0rTR4..sD q/ 0 Eq " _Q ) ) E./cc_.D Er2cURURSR"URSR65$r5rrs r0rr s+ FF166!9>>166!9>>2 3r2rr.s r0TR4rs 0 :: E 4  55r2c6^^^^^UUUUU4Sjn[X5$)aHelper for TR5 and TR6 to replace f**2 with h(g**2) Options ======= max : controls size of exponent that can appear on f e.g. if max=4 then f**4 will be changed to h(g**2)**2. pow : controls whether the exponent must be a perfect power of 2 e.g. if pow=True (and max >= 6) then f**6 will not be changed but f**8 will be changed to h(g**2)**4 >>> from sympy.simplify.fu import _TR56 as T >>> from sympy.abc import x >>> from sympy import sin, cos >>> h = lambda x: 1 - x >>> T(sin(x)**3, sin, cos, h, 4, False) (1 - cos(x)**2)*sin(x) >>> T(sin(x)**6, sin, cos, h, 6, False) (1 - cos(x)**2)**3 >>> T(sin(x)**6, sin, cos, h, 6, True) sin(x)**6 >>> T(sin(x)**8, sin, cos, h, 10, True) (1 - cos(x)**2)**4 c>UR(aURRT:XdU$URR(dU$URS:S:XaU$URT:S:XaU$URS:XaU$URS:Xa(T"T"URR S5S-5$URS-S:XaZURS-nT"URR S5T"T"URR S5S-5U--$URS:XaSnOZT(d&URS-(aU$URS-nO-[ UR5nU(dU$URS-nT"T"URR S5S-5U-$)NrTr_rFr)rJrKrUexpis_realr7r%)r/rYpr;ghmaxpows r0_f_TR56.._f>sg  bgglla/Ivv~~I FFQJ4 I FFSLT !I 66Q;I 66Q;Qrww||A'*+ +vvzQFFAIa)!Abggll1o,>,A*BA*EEE166A:IFFAI!"&&)IFFAIQrww||A'*+Q. .r2r)r/r;rrrrrs ````` r0_TR56r$s4!/!/F R r2c .[U[[SXS9$)aMReplacement of sin**2 with 1 - cos(x)**2. See _TR56 docstring for advanced use of ``max`` and ``pow``. Examples ======== >>> from sympy.simplify.fu import TR5 >>> from sympy.abc import x >>> from sympy import sin >>> TR5(sin(x)**2) 1 - cos(x)**2 >>> TR5(sin(x)**-2) # unchanged sin(x)**(-2) >>> TR5(sin(x)**4) (1 - cos(x)**2)**2 c SU- $r^rs r0rTR5..vQr2rr)rrrr/rrs r0TR5rd$ S#C AAr2c .[U[[SXS9$)aLReplacement of cos**2 with 1 - sin(x)**2. See _TR56 docstring for advanced use of ``max`` and ``pow``. Examples ======== >>> from sympy.simplify.fu import TR6 >>> from sympy.abc import x >>> from sympy import cos >>> TR6(cos(x)**2) 1 - sin(x)**2 >>> TR6(cos(x)**-2) #unchanged cos(x)**(-2) >>> TR6(cos(x)**4) (1 - sin(x)**2)**2 c SU- $r^rrs r0rTR6..rr2r)rrrrs r0TR6ryrr2cSn[X5$)zLowering the degree of cos(x)**2. Examples ======== >>> from sympy.simplify.fu import TR7 >>> from sympy.abc import x >>> from sympy import cos >>> TR7(cos(x)**2) cos(2*x)/2 + 1/2 >>> TR7(cos(x)**2 + 1) cos(2*x)/2 + 3/2 cUR(a.URR[:XaURS:XdU$S[SURR S-5-S- $)NrFr_r)rJrKrUrrr7r.s r0r;TR7..fsM bggllc1bffkIC"'',,q/)**A--r2rr=s r0TR7rs . R r2c&^U4Sjn[X5$)aAConverting products of ``cos`` and/or ``sin`` to a sum or difference of ``cos`` and or ``sin`` terms. Examples ======== >>> from sympy.simplify.fu import TR8 >>> from sympy import cos, sin >>> TR8(cos(2)*cos(3)) cos(5)/2 + cos(1)/2 >>> TR8(cos(2)*sin(3)) sin(5)/2 + sin(1)/2 >>> TR8(sin(2)*sin(3)) -cos(5)/2 + cos(1)/2 cZ >UR(dmUR(aZURR[[ 4;a6UR R(dURR(dU$T (aUR5Vs/sHn[U5PM snup#[USS9n[USS9nXB:wdXS:wa[XE- 5nUR(akURSR(aM[UR5S:Xa4URSR (a[#UR%56nU$[/[ /S/0n["R&"U5GH nUR[[ 4;a,U[)U5R+URS5 MJUR(aUR R,(aUR S:arURR[[ 4;aNU[)UR5R/URRS/UR -5 MUSR+U5 GM U[nU[ n U(aU (d [U5S:d[U 5S:dU$USn[1[U5[U 55n[3U5HPnU R55n UR55n UR+[ X-5[ X- 5-S- 5 MR [U5S:a^UR55n UR55n UR+[ X-5[ X- 5-S- 5 [U5S:aM^U(a(UR+[ UR5555 [U 5S:a_U R55n U R55n UR+[ X-5*[ X- 5-S- 5 [U 5S:aM_U (a(UR+[ U R5555 [[[#U655$s snf)NFfirstrrFr_)rmrJrKrUrrrrSrTrnr TR8rr7rrWrVr as_coeff_MulrQrr` is_Integerextendminrangerr) r/rgrtrcnewnnewdr7r:csrwa2rs r0r;TR8..fs: II II GGLLS#J & VV  "''"5"5I +-+<+<+>?+>aJqM+>?DAq&Dq&DyDIty)99!7!7BGG )bggaj.?.?boo/0BIRb$+r"Avv#s#T!W $$QVVAY/((quu//AEEAIFFKKC:-T!&&\"))166;;q>*:155*@AT !!!$# I Ia3q6A:Q!IDz AA qABB KKRWBG 4a7 8!fqjBB KKRWBG 4a7 8!fqj KKAEEG %!fqjBB KK#bg,RW5q8 9!fqj KKAEEG %:c4j)**Y@sR(rr/rr;s ` r0rrs"5+n R r2cSn[X5$)a'Sum of ``cos`` or ``sin`` terms as a product of ``cos`` or ``sin``. Examples ======== >>> from sympy.simplify.fu import TR9 >>> from sympy import cos, sin >>> TR9(cos(1) + cos(2)) 2*cos(1/2)*cos(3/2) >>> TR9(cos(1) + 2*sin(1) + 2*sin(2)) cos(1) + 4*sin(3/2)*cos(1/2) If no change is made by TR9, no re-arrangement of the expression will be made. For example, though factoring of common term is attempted, if the factored expression was not changed, the original expression will be returned: >>> TR9(cos(3) + cos(3)*cos(2)) cos(3) + cos(2)*cos(3) cR^UR(dU$SU4Sjjm[UT5$)Nc>UR(dU$[[UR55n[ U5S:waSn[ [ U55HRnX$nUcM [ US-[ U55H*nX&nUcM XW-nT"U5n X:wdMXU'SX&'Sn MP MT U(a:[ UV s/sH o(dM U PM sn 6nUR(aT"U5nU$[U6n U (dU$U uppnnU(acX:Xa+X-S-[UU-S- 5-[UU- S- 5-$U S:aUUnnSU -[UU-S- 5-[UU- S- 5-$X:Xa+X-S-[UU-S- 5-[UU- S- 5-$U S:aUUnnSU -[UU-S- 5-[UU- S- 5-$s sn f)NrFFr_Tr rVrprr7rWrr trig_splitrr)r/rr7hitrgrMjajwasnewrsplitgcdn1n2r:rxiscosdos r0rTR9..f..do s99 ()D4yA~s4y)ABz "1q5#d)4!W:$ g g:&)G&*DG"&C!5 *D7DbBrD78ByyV %E ', $CRAu86!8CQ N23Aqy>AA6aqA#vc1q5!)n,S!a%^;;86!8CQ N23Aqy>AA6aqAuS!a%^+CQ N::18s > G GT)rVprocess_common_addends)r/rs @r0r;TR9..fs$yyI< ;|&b"--r2rr=s r0TR9rs.B.H R r2c&^U4Sjn[X5$)awSeparate sums in ``cos`` and ``sin``. Examples ======== >>> from sympy.simplify.fu import TR10 >>> from sympy.abc import a, b, c >>> from sympy import cos, sin >>> TR10(cos(a + b)) -sin(a)*sin(b) + cos(a)*cos(b) >>> TR10(sin(a + b)) sin(a)*cos(b) + sin(b)*cos(a) >>> TR10(sin(a + b + c)) (-sin(a)*sin(b) + cos(a)*cos(b))*sin(c) + (sin(a)*cos(b) + sin(b)*cos(a))*cos(c) c^>UR[[4;aU$URnURSnUR(GabT(a[ [ UR55nO[ UR5nUR5n[R"U5nUR(aU[:Xa?[U5[[U5SS9-[U5[[U5SS9--$[U5[[U5SS9-[U5[[U5SS9-- $U[:Xa/[U5[U5-[U5[U5--$[U5[U5-[U5[U5-- $U$)NrFr) rUrrr7rVrprrrr _from_argsTR10)r/r;argr7r:rxrs r0r;TR10..fasG 773* $I GGggaj :::GCHH-.CHH~ At$Axx8q6$s1vU";;AtCF%8899q6$s1vU";;AtCF%88998q6#a&=3q6#a&=88q6#a&=3q6#a&=88 r2rrs ` r0rrOs$6 R r2cSn[X5$)a^Sum of products to function of sum. Examples ======== >>> from sympy.simplify.fu import TR10i >>> from sympy import cos, sin, sqrt >>> from sympy.abc import x >>> TR10i(cos(1)*cos(3) + sin(1)*sin(3)) cos(2) >>> TR10i(cos(1)*sin(3) + sin(1)*cos(3) + cos(3)) cos(3) + sin(4) >>> TR10i(sqrt(2)*cos(x)*x + sqrt(6)*sin(x)*x) 2*sqrt(2)*x*sin(x + pi/6) c ^ UR(dU$SU 4Sjjm [UT S5nUR(Ga[[5nURHnSnUR (awURHgnUR (dMUR[RLdM5URR(dMRXRU5 Sn O U(aMU[RRU5 M /nUHn[5U-[54HnXa;dM [![#X55H{nXUcM [![#X55HTnXUcM [%XUXU-5n T "U 5n X:wdM4URU 5 SXU'SXU' My M} M M U(aL[%UUR'5V V s/sH$n [%U V s/sH o(dM U PM sn 6PM& sn n -6nO T "U5nU$UR(aGMU$s sn fs sn n f)Nc>UR(dU$[[UR55n[ U5S:waSn[ [ U55HRnX$nUcM [ US-[ U55H*nX&nUcM XW-nT"U5n X:wdMXU'SX&'Sn MP MT U(a:[ UV s/sH o(dM U PM sn 6nUR(aT"U5nU$[USS06n U (dU$U uppnnU(a+X-n X:XaU [UU- 5-$U [UU-5-$X-n X:XaU [UU-5-$U [UU- 5-$s sn f)NrFFr_Ttwor)r/rr7rrgrMrrrrrrrrrr:rxsamers r0rTR10i..f..doso99 ()D4yA~s4y)ABz "1q5#d)4!W:$ g g:&)G&*DG"&C!5 *D7DbBrD78ByyV /$/E &+ #CRAtf8s1q5z>)3q1u:~%f8s1q5z>)3q1u:~%-8s > E) E)c>[[UR55$rI)tupler free_symbolsrs r0r"TR10i..f..seGANN$;.fsyyI6 &p$ <> iii%EWW88ff999166)9 " 2 2 2!I,,Q/"#C! % s!%%L''*D (1*ik2Az!&s58}!5A$x{2 (%*3ux=%9#(8A;#6$,&)%(1+ *C&D&(g#&:$(KK$426EHQK26EHQK$)&:"63 4"\\^#-+$'a(>a2a(>#?+#--/V QiiiP )?#-s9I  I I I I rr=s r0TR10irs$iV R r2Nc&^U4Sjn[X5$)aFunction of double angle to product. The ``base`` argument can be used to indicate what is the un-doubled argument, e.g. if 3*pi/7 is the base then cosine and sine functions with argument 6*pi/7 will be replaced. Examples ======== >>> from sympy.simplify.fu import TR11 >>> from sympy import cos, sin, pi >>> from sympy.abc import x >>> TR11(sin(2*x)) 2*sin(x)*cos(x) >>> TR11(cos(2*x)) -sin(x)**2 + cos(x)**2 >>> TR11(sin(4*x)) 4*(-sin(x)**2 + cos(x)**2)*sin(x)*cos(x) >>> TR11(sin(4*x/3)) 4*(-sin(x/3)**2 + cos(x/3)**2)*sin(x/3)*cos(x/3) If the arguments are simply integers, no change is made unless a base is provided: >>> TR11(cos(2)) cos(2) >>> TR11(cos(4), 2) -sin(2)**2 + cos(2)**2 There is a subtle issue here in that autosimplification will convert some higher angles to lower angles >>> cos(6*pi/7) + cos(3*pi/7) -cos(pi/7) + cos(3*pi/7) The 6*pi/7 angle is now pi/7 but can be targeted with TR11 by supplying the 3*pi/7 base: >>> TR11(_, 3*pi/7) -sin(3*pi/7)**2 + cos(3*pi/7)**2 + cos(3*pi/7) cL>UR[[4;aU$T(aURnU"TS-5n[RnUR (aUR 5up2UR[[4;aU$URSURS:Xa8[T5n[T5nU[LaUS-US-- U- $SU-U-U- $U$URSR(dURSR SS9upFURS-S:XapURS-U-UR- n[[U55n[[U55nUR[:Xa SU-U-nU$US-US-- nU$)NrFrT)rational) rUrrrr8rmrr7 is_NumberrrTR11) r/r;rvcorrmrrKs r0r;TR11..f)sk 773* $I A$q& ABxx(vvc3Z' wwqzQVVAY&II8qD1a4K++Q3q58OI%%771:**D*9DAssQw!|cc1fQhqsslSNSN77c>1QB A1B r2r)r/rKr;s ` r0rrsT!F R r2cSn[X5$)a Helper for TR11 to find half-arguments for sin in factors of num/den that appear in cos or sin factors in the den/num. Examples ======== >>> from sympy.simplify.fu import TR11, _TR11 >>> from sympy import cos, sin >>> from sympy.abc import x >>> TR11(sin(x/3)/(cos(x/6))) sin(x/3)/cos(x/6) >>> _TR11(sin(x/3)/(cos(x/6))) 2*sin(x/6) >>> TR11(sin(x/6)/(sin(x/3))) sin(x/6)/sin(x/3) >>> _TR11(sin(x/6)/(sin(x/3))) 1/(2*cos(x/6)) c[U[5(dU$Sn[XR55up#SnU"XU5nU"XU5nU$)NcD[[5n[R"U5HvnUR 5up4UR (dM(US:dM0UR [[4;dMLU[U5RURS5 Mx U$r5) rsetr rQ as_base_exprrUrrraddr7)flatr7firxrYs r0 sincos_args%_TR11..f..sincos_argshsss#DmmD)~~'<<.f..handle_matchts\! AvC=(Dc]*D"^%%d+&Ir2)r6rmaprn)r/r rrrs r0r;_TR11..fdsU"d##I !.?.?.AB  " 1 " 1 r2rr=s r0_TR11rOs*#J R r2c&^U4Sjn[X5$)zSeparate sums in ``tan``. Examples ======== >>> from sympy.abc import x, y >>> from sympy import tan >>> from sympy.simplify.fu import TR12 >>> TR12(tan(x + y)) (tan(x) + tan(y))/(-tan(x)*tan(y) + 1) c>UR[:XdU$URSnUR(aT(a[ [ UR55nO[ UR5nUR 5n[R"U5nUR(a[[U5SS9nO [U5n[U5U-S[U5U-- - $U$)NrFrr_) rUrr7rVrprrrrrTR12)r/rr7r:rxtbrs r0r;TR12..fsww#~Iggaj ::GCHH-.CHH~ At$Axx#a&.VFRK!c!fRi-0 0 r2rrs ` r0rrs& R r2cSn[X5$)aCombine tan arguments as (tan(y) + tan(x))/(tan(x)*tan(y) - 1) -> -tan(x + y). Examples ======== >>> from sympy.simplify.fu import TR12i >>> from sympy import tan >>> from sympy.abc import a, b, c >>> ta, tb, tc = [tan(i) for i in (a, b, c)] >>> TR12i((ta + tb)/(-ta*tb + 1)) tan(a + b) >>> TR12i((ta + tb)/(ta*tb - 1)) -tan(a + b) >>> TR12i((-ta - tb)/(ta*tb - 1)) tan(a + b) >>> eq = (ta + tb)/(-ta*tb + 1)**2*(-3*ta - 3*tc)/(2*(ta*tc - 1)) >>> TR12i(eq.expand()) -3*tan(a + b)*tan(a + c)/(2*(tan(a) + tan(b) - 1)) cN UR(d$UR(dUR(dU$UR5upUR(aUR(dU$0nSn[ [ R"U55n[U5GHupgU"U5nU(aKUup[U RV s/sHoRSPM sn 6n [RX<'XU'M`UR(aM[U5nUR(a/URUR5 [RXV'MMUR(dMURR(dUR R"(dGM U"UR 5nU(aTUup[U RV s/sHoRSPM sn 6n URX<'XR-XV'GMw[U5nUR(dGMURUR5 [RXV'GM U(dU$Sn[ [ R"[%U555n Sn[U 5GHupoU"U5nU(GdBU"U*5nU(a[R&X'GO0UR(aK[U5nUR(a-U RUR5 [RX'MUR(aURR(dUR R"(axU"UR 5nU(a[RX'Oa[U5nUR(a-U RUR5 [RX'GMUGMX[RX'Sn[UV s/sHoRSPM sn 6n X<nUR)[R5nUbU(aUX<'OUR+U 5 X==[-U 5*-ss'GM U(ap[ U 6[ U6- [ UR/5VVVs/sH8unn[URVs/sHn[-U5PM sn6S- U-PM: snnn6- nU$s sn fs sn fs sn fs snfs snnnf)Nc[U5nU(ajUup#nU[RLaQUR(a?[ UR 5S:Xa%[ SUR 55(aX#4$ggggg)NrFc3B# UHn[U[5v M g7frI)r6r)rLrs r0rN/TR12i..f..ok..sA&BJr3//&s) as_f_sign_1r NegativeOnermrWr7all)dirrr;rs r0r[TR12i..f..oksjBAa %!((s166{a7GA!&&AAA4KB8H(%r2rcUR(aW[UR5S:Xa=URup[U[5(a[U[5(aX4$ggggr )rVrWr7r6r)nir:rxs r0r[r$sRyyS\Q.wwa%%*Q*<*<4K+=%/yr2FTr_)rVrmrJrnr7rpr rQrarrr8r&rrrSrKrTr r!extract_additivelyrrrrs)r/rtrcdokr[d_argsrgr#rrrv_rn_argsrr&ednewedrYr:s r0r;TR12i..fs RYY"))I  "vvQVVI cmmA&'v&EA2AQVV4V&&)V45q yyBZ99MM"''* !FI 1 1RWW5H5H5HrwwKDA8AffQi89AVVCF !66 FIByyy bgg.$%EE 1'2I cmmLO45v&EA2AsG ! FIyy#BZ99"MM"''2()FI FF--1D1DrwwK()FI!'B!yy & bgg 6,-EE $ EE C+AffQi+,AB))!%%0E "CFGGAJ I#a& IK'N fc6l*3=@YY[1J=HTQ36 !8( &1A8(3)+,3-/021=H1J,KKB U59^,8(1Js*T T T T T4T T rr=s r0TR12ir/s*aF R r2cSn[X5$)zChange products of ``tan`` or ``cot``. Examples ======== >>> from sympy.simplify.fu import TR13 >>> from sympy import tan, cot >>> TR13(tan(3)*tan(2)) -tan(2)/tan(5) - tan(3)/tan(5) + 1 >>> TR13(cot(3)*cot(2)) cot(2)*cot(5) + 1 + cot(3)*cot(5) c UR(dU$[/[/S/0n[R"U5H]nUR [[4;a,U[ U5RURS5 MIUSRU5 M_ U[nU[n[U5S:a[U5S:aU$USn[U5S:avUR5nUR5nURS[U5[XV-5- [U5[XV-5- -- 5 [U5S:aMvU(a(UR[UR555 [U5S:avUR5nUR5nURS[U5[XV-5--[U5[XV-5--5 [U5S:aMvU(a(UR[UR555 [U6$NrrFr_) rmrr r rQrUrr`r7rWrr)r/r7r:rvrt1t2s r0r;TR13..f8syyIRb$+r"Avv#s#T!W $$QVVAY/T !!!$ # I I q6A:#a&1*IDz!fqjBB KKSWS\1CGCL4HHI J!fqj KKAEEG %!fqjBB KKCGCL003r73rw<3GG H!fqj KKAEEG %Dzr2rr=s r0TR13r6*s< R r2c,^SU4Sjjm[UT5$)a Returns cos(x)*cos(2*x)*...*cos(2**(k-1)*x) -> sin(2**k*x)/(2**k*sin(x)) Examples ======== >>> from sympy.simplify.fu import TRmorrie, TR8, TR3 >>> from sympy.abc import x >>> from sympy import Mul, cos, pi >>> TRmorrie(cos(x)*cos(2*x)) sin(4*x)/(4*sin(x)) >>> TRmorrie(7*Mul(*[cos(x) for x in range(10)])) 7*sin(12)*sin(16)*cos(5)*cos(7)*cos(9)/(64*sin(1)*sin(3)) Sometimes autosimplification will cause a power to be not recognized. e.g. in the following, cos(4*pi/7) automatically simplifies to -cos(3*pi/7) so only 2 of the 3 terms are recognized: >>> TRmorrie(cos(pi/7)*cos(2*pi/7)*cos(4*pi/7)) -sin(3*pi/7)*cos(3*pi/7)/(4*sin(pi/7)) A touch by TR8 resolves the expression to a Rational >>> TR8(_) -1/8 In this case, if eq is unsimplified, the answer is obtained directly: >>> eq = cos(pi/9)*cos(2*pi/9)*cos(3*pi/9)*cos(4*pi/9) >>> TRmorrie(eq) 1/16 But if angles are made canonical with TR3 then the answer is not simplified without further work: >>> TR3(eq) sin(pi/18)*cos(pi/9)*cos(2*pi/9)/2 >>> TRmorrie(_) sin(pi/18)*sin(4*pi/9)/(8*sin(pi/9)) >>> TR8(_) cos(7*pi/18)/(16*sin(pi/9)) >>> TR3(_) 1/16 The original expression would have resolve to 1/16 directly with TR8, however: >>> TR8(eq) 1/16 References ========== .. [1] https://en.wikipedia.org/wiki/Morrie%27s_law c @>UR(dU$U(a%UR5up#T"US5T"US5- $[[5n0n/nURHnUR 5upU R (aM[U[5(a8URSR5upXKRU 5 XU'MsURU5 M /n UGHjn XKnUR5 U(dM!Sn US=pX;aU S- n US-nX;aMU S:a[SU -U-U -5SU -- [X-5- nSn/n[U 5H@nUS-n[X-SS9nURU5 [UUU=(d UU5nMB [U 5HInUR5n[X-SS9nUU==U-ss'UU(aM8UR!U5 MK U RUU-5 O3[URS5U -5nURXU-5 U(aGMIGMm U (a6[#X-UV V s/sHoU Hn [X-SS9PM M sn n -6nU$s sn n f)Nrr_rFFrk)rmrnrrpr7rrr6rrr`sortrrrrrr r )r/rrtrcr7cossotherrrxrYrr:rrXcccinewargtakeccsrgkeyr;s r0r;TRmorrie..fsjyyI $$&DAQ71Q7? "4 A==?DA|| 1c 2 2q ..0r"Q QAA FFH!A$gFA!GBgq5 Ab^AqD0RT:FDC"1Xa!!$7 2"49d.?d3i@ & #1X WWY!!$7S T) #CyyHHRL & JJvt|,AEE!HQJALLG,5!> s{26&I26QQ1AC%(($&IIKB &Is/"J rrr=s @r0TRmorrierCYsv7r R r2c&^U4Sjn[X5$)aConvert factored powers of sin and cos identities into simpler expressions. Examples ======== >>> from sympy.simplify.fu import TR14 >>> from sympy.abc import x, y >>> from sympy import cos, sin >>> TR14((cos(x) - 1)*(cos(x) + 1)) -sin(x)**2 >>> TR14((sin(x) - 1)*(sin(x) + 1)) -cos(x)**2 >>> p1 = (cos(x) + 1)*(cos(x) - 1) >>> p2 = (cos(y) - 1)*2*(cos(y) + 1) >>> p3 = (3*(cos(y) - 1))*(3*(cos(y) + 1)) >>> TR14(p1*p2*p3*(x - 1)) -18*(x - 1)*sin(x)**2*sin(y)**4 c >UR(dU$T(aIUR5upU[RLa$[ USS9n[ USS9nX1:wdXB:waX4- nU$/n/nUR HnUR (aJUR5upU R(d$UR(dURU5 M[UnO[Rn [U5n U (aU SR[[4;a:U [RLaURU5 OURXy-5 MU upn URXRXX45 M [![#U55n[%U5n[!['S55=nun nppU(Ga#UR)S5nU(GaUSnUU R(Ga0UU R(GaUU UU :XGa UU UU :wGaUR)S5n[+UU UU 5nUU U:wa5UVs/sHnUUPM nnUU ==U-ss'UR-SU5 O=UU U:wa4UVs/sHnUUPM nnUU ==U-ss'UR-SU5 [/UU [5(a[nO[nURUU *UU -U"UU R S5S--U-5 GMjOUU UU :XaUU UU :XaUU UU :waxUR)S5nUU n[/UU [5(a[nO[nURUU *UU -U"UU R S5S--U-5 GMURUUUU -5 U(aGM#[%U5U:wa[1U6nU$s snfs snf)NFrr_rrrF)rmrnrr8TR14r7rJrrSrTr`r rUrrrrprrWrrrrinsertr6r )r/rtrcrrr;processr:rxrYrrr;sinotherrqrvABr?rgremrs r0r;TR14..fs}yyI $$&DA~AU+AU+9 B Axx}}  LLOEEAA! #s3:LLOLL&HA" NNA{{A"8 9#(ww'(U&*%(^3"1aB AAAJQ4>>>adnnntqt|R5AbE> ' AA#&qtQqT?D !tt|59&:TqtT&: #A$ 'q# 6!"159&:TqtT&: #A$ 'q# 6)!A$44$'$'!LL1Q4%!*Qqtyy|_a5G*G$)NO$qTQqT\tqt|R5AbE> ' AA#$Q4D)!A$44$'$'!LL1Q4%!*Qqtyy|_a5G*G$)NO$ LL1qt $Yg\ u: eB E';';s O1O6rrs ` r0rFrFs,^@ R r2c*^^UU4Sjn[X5$)zConvert sin(x)**-2 to 1 + cot(x)**2. See _TR56 docstring for advanced use of ``max`` and ``pow``. Examples ======== >>> from sympy.simplify.fu import TR15 >>> from sympy.abc import x >>> from sympy import sin >>> TR15(1 - 1/sin(x)**2) -cot(x)**2 c 4>[U[5(a[UR[5(dU$URnUS-S:Xa([ URUS--5UR- $SU- n[ U[[STTS9nX2:waUnU$)NrFr_c SU-$r^rrs r0r!TR15..f..b!a%r2r)r6rrKrrTR15rr r/rYiar:rrs r0r;TR15..fY2s## 277C(@(@I FF q5A:!a%()"''1 1 rT "c3Sc B 7B r2rr/rrr;s `` r0rTrTI  R r2c*^^UU4Sjn[X5$)zConvert cos(x)**-2 to 1 + tan(x)**2. See _TR56 docstring for advanced use of ``max`` and ``pow``. Examples ======== >>> from sympy.simplify.fu import TR16 >>> from sympy.abc import x >>> from sympy import cos >>> TR16(1 - 1/cos(x)**2) -tan(x)**2 c 4>[U[5(a[UR[5(dU$URnUS-S:Xa([ URUS--5UR- $SU- n[ U[[STTS9nX2:waUnU$)NrFr_c SU-$r^rrs r0r!TR16..f..rSr2r)r6rrKrrrTrrrUs r0r;TR16..fzrXr2rrYs `` r0TR16r`jrZr2cSn[X5$)a Convert f(x)**-i to g(x)**i where either ``i`` is an integer or the base is positive and f, g are: tan, cot; sin, csc; or cos, sec. Examples ======== >>> from sympy.simplify.fu import TR111 >>> from sympy.abc import x >>> from sympy import tan >>> TR111(1 - 1/tan(x)**2) 1 - cot(x)**2 c[U[5(aQURR(d8URR (aURR (dU$[UR[5(a0[URRS5UR*-$[UR[5(a0[URRS5UR*-$[UR[5(a0[URRS5UR*-$U$r5)r6rrKrTrrS is_negativerr r7rr"rr!r.s r0r;TR111..fs r3   WW BFF$5$5"&&:L:LI bggs # #rww||A'"&&0 0  % %rww||A'"&&0 0  % %rww||A'"&&0 0 r2rr=s r0TR111res  R r2c*^^UU4Sjn[X5$)a<Convert tan(x)**2 to sec(x)**2 - 1 and cot(x)**2 to csc(x)**2 - 1. See _TR56 docstring for advanced use of ``max`` and ``pow``. Examples ======== >>> from sympy.simplify.fu import TR22 >>> from sympy.abc import x >>> from sympy import tan, cot >>> TR22(1 + tan(x)**2) sec(x)**2 >>> TR22(1 + cot(x)**2) csc(x)**2 c >[U[5(a$URR[[ 4;dU$[ U[ [STTS9n[ U[[STTS9nU$)Nc US- $r^rrs r0r!TR22..f..1q5r2rc US- $r^rrs r0rrirjr2) r6rrKrUr rrr!r"rs r0r;TR22..fsU2s## c (BI 2sCcs C 2sCcs C r2rrYs `` r0TR22rms$ R r2cSn[X5$)aConvert sin(x)**n and cos(x)**n with positive n to sums. Examples ======== >>> from sympy.simplify.fu import TRpower >>> from sympy.abc import x >>> from sympy import cos, sin >>> TRpower(sin(x)**6) -15*cos(2*x)/32 + 3*cos(4*x)/16 - cos(6*x)/32 + 5/16 >>> TRpower(sin(x)**3*cos(2*x)**4) (3*sin(x)/4 - sin(3*x)/4)*(cos(4*x)/2 + cos(8*x)/8 + 3/8) References ========== .. [1] https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Power-reduction_formulae c [U[5(a%[UR[[45(dU$UR 5upUR SnUR(GagUR(GaUUR(af[U[5(aQSSU- -[[US-S- 5Vs/sH#n[X$5[ USU-- U-5-PM% sn6-nGOUR(a[U[5(aSSU- -[RUS- S- --[[US-S- 5Vs/sH7n[X$5[RU--[USU-- U-5-PM9 sn6-nGOUR(ab[U[5(aMSSU- -[[US- 5Vs/sH#n[X$5[ USU-- U-5-PM% sn6-nOUR(a[U[5(awSSU- -[RUS- --[[US- 5Vs/sH7n[X$5[RU--[ USU-- U-5-PM9 sn6-nUR(aUSU*-[X"S- 5-- nU$s snfs snfs snfs snfr2)r6rrKrrrr7rrTis_oddrrrrr!is_even)r/rxrtrrXs r0r;TRpower..fs2s## 277S#J(G(GI~~ FF1I <<Fa^MM1$>%%(!ac'1%5>6?O=Q8RRz!S111Xc"1Q3Z$)'%-QN3AaC{3C$C'$)**z!S111Xammac223?DQqSz9K?I!:B!MM1$:%%(!ac'1%5:6?I9K4LLyya1"ghqA#... $/=Q$)9Ks*K >K *K# *>K( rr=s r0TRpowerrss*, R r2c>[UR[55$)zReturn count of trigonometric functions in expression. Examples ======== >>> from sympy.simplify.fu import L >>> from sympy.abc import x >>> from sympy import cos, sin >>> L(cos(x)+sin(x)) 2 )rcountr$r.s r0Lrvs RXX+ , --r2c8[U5UR54$rI)rv count_opsrs r0rr*sadAKKM2r2c [[U5n[[U5nUn[U5n[ U[ 5(d1UR "URVs/sH n[XQS9PM sn6$[U5nUR[[5(aFU"U5nU"U5U"U5:aUnUR[[5(a [U5nUR[[5(a(U"U5n[![#U55n[%X@Xg/US9n[%['U5XS9$s snf)aAttempt to simplify expression by using transformation rules given in the algorithm by Fu et al. :func:`fu` will try to minimize the objective function ``measure``. By default this first minimizes the number of trig terms and then minimizes the number of total operations. Examples ======== >>> from sympy.simplify.fu import fu >>> from sympy import cos, sin, tan, pi, S, sqrt >>> from sympy.abc import x, y, a, b >>> fu(sin(50)**2 + cos(50)**2 + sin(pi/6)) 3/2 >>> fu(sqrt(6)*cos(x) + sqrt(2)*sin(x)) 2*sqrt(2)*sin(x + pi/3) CTR1 example >>> eq = sin(x)**4 - cos(y)**2 + sin(y)**2 + 2*cos(x)**2 >>> fu(eq) cos(x)**4 - 2*cos(y)**2 + 2 CTR2 example >>> fu(S.Half - cos(2*x)/2) sin(x)**2 CTR3 example >>> fu(sin(a)*(cos(b) - sin(b)) + cos(a)*(sin(b) + cos(b))) sqrt(2)*sin(a + b + pi/4) CTR4 example >>> fu(sqrt(3)*cos(x)/2 + sin(x)/2) sin(x + pi/3) Example 1 >>> fu(1-sin(2*x)**2/4-sin(y)**2-cos(x)**4) -cos(x)**2 + cos(y)**2 Example 2 >>> fu(cos(4*pi/9)) sin(pi/18) >>> fu(cos(pi/9)*cos(2*pi/9)*cos(3*pi/9)*cos(4*pi/9)) 1/16 Example 3 >>> fu(tan(7*pi/18)+tan(5*pi/18)-sqrt(3)*tan(5*pi/18)*tan(7*pi/18)) -sqrt(3) Objective function example >>> fu(sin(x)/cos(x)) # default objective function tan(x) >>> fu(sin(x)/cos(x), measure=lambda x: -x.count_ops()) # maximize op count sin(x)/cos(x) References ========== .. [1] https://www.sciencedirect.com/science/article/pii/S0895717706001609 )measure)rA)r'RL1RL2rr6rrUr7fur>hasrr rBrrrrCrrz)r/rzfRL1fRL2rr:rv1rv2s r0r}r}*sL #w D #w D C B b$  wwAAA/ABB RB vvc32h CL72; &B 66#s  RB vvc32h(3-  #3$' 2 tBx ))BsD>cr[[5nU(aVURHEnUR5upeUS:aU*nU*nXFU(aU"U5OS4R U5 MG OPU(a>URH-nU[ R U"U54R U5 M/ O [S5e/nSnUHan XIn U upk[U 5S:a1[U SS06n U"U 5n X:waU n SnUR Xl-5 MKUR XjS-5 Mc U(a[U6nU$)zApply ``do`` to addends of ``rv`` that (if ``key1=True``) share at least a common absolute value of their coefficient and the value of ``key2`` when applied to the argument. If ``key1`` is False ``key2`` must be supplied and will be the only key applied. rr_zmust have at least one keyFrlT) rrpr7rr`rr8 ValueErrorrWr)r/rkey2key1abscr:rr7rrXrhr*rYrs r0rrs( t D A>>#DA1uBB T!W!, - 4 4Q 7  A !%%a! " ) )! ,566 D C  G q6A:Q''AQ%Cx KK  KKA$  $Z Ir2z~ TR0 TR1 TR2 TR3 TR4 TR5 TR6 TR7 TR8 TR9 TR10 TR10i TR11 TR12 TR13 L TR2i TRmorrie TR12i TR14 TR15 TR16 TR111 TR22c[S5$r r#rr2r0_ROOT2r 7Nr2c[S5$)Nrrrr2r0rrrr2cS[S5- $)Nr_rrrr2r0rrs T!W9r2c h^X4Vs/sHn[U5PM snupURU5upEURU5R5nS=px[R UR ;a#UR[R 5nU*nO@[R UR ;a"UR[R 5nU*nXE4Vs/sHo3R5PM snupSn U "X5n U cgU upn U "X5n U cgU upnU (dU(dU (a"[U [5(a XUXU 4uppnnXpU(daU =(d U nU=(d Un[UUR5(dgXgUURSURS[U[54$U (dU(dU (aU(aU (aU(a[XR5[UUR5LagX4Vs1sHnURiM snm[U4SjUU455(dgXgXRSU RS[XR54$U (aU (d!U(aU(dU(a U cU bUcUcgU =(d U nU=(d UnURUR:wagU (d[Rn U(d[RnXLa(U[5-nXgUURS[ S- S4$X- [#5:Xa#USU--nXgUURS[ S - S4$X- [%5:Xa#USU --nXgUURS[ S - S4$gs snfs snfs snf) aReturn the gcd, s1, s2, a1, a2, bool where If two is False (default) then:: a + b = gcd*(s1*f(a1) + s2*f(a2)) where f = cos if bool else sin else: if bool, a + b was +/- cos(a1)*cos(a2) +/- sin(a1)*sin(a2) and equals n1*gcd*cos(a - b) if n1 == n2 else n1*gcd*cos(a + b) else a + b was +/- cos(a1)*sin(a2) +/- sin(a1)*cos(a2) and equals n1*gcd*sin(a + b) if n1 = n2 else n1*gcd*sin(b - a) Examples ======== >>> from sympy.simplify.fu import trig_split >>> from sympy.abc import x, y, z >>> from sympy import cos, sin, sqrt >>> trig_split(cos(x), cos(y)) (1, 1, 1, x, y, True) >>> trig_split(2*cos(x), -2*cos(y)) (2, 1, -1, x, y, True) >>> trig_split(cos(x)*sin(y), cos(y)*sin(y)) (sin(y), 1, 1, x, y, True) >>> trig_split(cos(x), -sqrt(3)*sin(x), two=True) (2, 1, -1, x, pi/6, False) >>> trig_split(cos(x), sin(x), two=True) (sqrt(2), 1, 1, x, pi/4, False) >>> trig_split(cos(x), -sin(x), two=True) (sqrt(2), 1, -1, x, pi/4, False) >>> trig_split(sqrt(2)*cos(x), -sqrt(6)*sin(x), two=True) (2*sqrt(2), 1, -1, x, pi/6, False) >>> trig_split(-sqrt(6)*cos(x), -sqrt(2)*sin(x), two=True) (-2*sqrt(2), 1, 1, x, pi/3, False) >>> trig_split(cos(x)/sqrt(6), sin(x)/sqrt(2), two=True) (sqrt(6)/3, 1, 1, x, pi/6, False) >>> trig_split(-sqrt(6)*cos(x)*sin(y), -sqrt(2)*sin(x)*sin(y), two=True) (-2*sqrt(2)*sin(y), 1, 1, x, pi/3, False) >>> trig_split(cos(x), sin(x)) >>> trig_split(cos(x), sin(z)) >>> trig_split(2*cos(x), -sin(x)) >>> trig_split(cos(x), -sqrt(3)*sin(x)) >>> trig_split(cos(x)*cos(y), sin(x)*sin(z)) >>> trig_split(cos(x)*cos(y), sin(x)*sin(y)) >>> trig_split(-sqrt(6)*cos(x), sqrt(2)*sin(x)*sin(y), two=True) r_cS=p#[RnUR(GaqUR5up@[ UR 5S:dU(dgUR(a[ UR 5nOU/nURS5n[U[5(aUnOL[U[5(aUnO4UR(a"UR[RLaX@-nOgU(a}USn[U[5(a U(aUnOYUnOV[U[5(a U(aUnO7UnO4UR(a"UR[RLaXF-nOgU[RLaXBU4$SX#4$[U[5(aUnO[U[5(aUnUcUcgU[RLaUOSnXBU4$)aEReturn ``a`` as a tuple (r, c, s) such that ``a = (r or 1)*(c or 1)*(s or 1)``. Three arguments are returned (radical, c-factor, s-factor) as long as the conditions set by ``two`` are met; otherwise None is returned. If ``two`` is True there will be one or two non-None values in the tuple: c and s or c and r or s and r or s or c with c being a cosine function (if possible) else a sine, and s being a sine function (if possible) else oosine. If ``two`` is False then there will only be a c or s term in the tuple. ``two`` also require that either two cos and/or sin be present (with the condition that if the functions are the same the arguments are different or vice versa) or that a single cosine or a single sine be present with an optional radical. If the above conditions dictated by ``two`` are not met then None is returned. NrFr)rr8rmrrWr7rprrr6rrrJrr)r:rrrrr7rxs r0 pow_cos_sintrig_split..pow_cos_sinsr(  UU 888NN$EB166{QcxxAFF|s A!S!!As##aeeqvvoGa%%3''XX!%%166/GB1552q8 8dA8 8 3  A 3  A 9 QUU?Raxr2Nrc3@># UHoRT;v M g7frI)r7)rLrgr7s r0rNtrig_split..^s<8a66T>8srFrFrr)rr,ras_exprrr!factorsquor6rrUr7rr"r8rr rr)r:rxrrguaubrrrrrcoacasacobcbsbrrrr7s @r0rrsd"# '1GAJ 'DA XXa[FB %%(   CKB}} " VVAMM "S "** $ VVAMM "S"$ *AIIK *DA?D AAyKCRAAyKCR B"B!4!4#&B#; "bB  H" H"!QVV$$AFF1Iqvvay*Q2DDD3rbRb''**R2II)+111>> from sympy.simplify.fu import as_f_sign_1 >>> from sympy.abc import x >>> as_f_sign_1(x + 1) (1, x, 1) >>> as_f_sign_1(x - 1) (1, x, -1) >>> as_f_sign_1(-x + 1) (-1, x, -1) >>> as_f_sign_1(-x - 1) (-1, x, 1) >>> as_f_sign_1(2*x + 2) (2, x, 1) rFNrr_)rVrWr7rr!r8rmrrr,rrrr) rYr:rxrrgrrrrrs r0r r ws* 88s166{a' 66DAQ]]AEE "" EE 88q ++q A 2rqAQw ! '1GAJ 'DA XXa[FB %%(   C}} " VVAMM "   "** $ VVAMM "   "$ *AIIK *DAAEEz1B RxdSAEEzrz+ ( +s G;"Hc&^U4Sjn[X5$)aReplace all hyperbolic functions with trig functions using the Osborne rule. Notes ===== ``d`` is a dummy variable to prevent automatic evaluation of trigonometric/hyperbolic functions. References ========== .. [1] https://en.wikipedia.org/wiki/Hyperbolic_function c>[U[5(dU$URSnUR(dUT-O2[R "URVs/sHo"T-PM sn5n[U[ 5(a[[U5-$[U[5(a [U5$[U[5(a[[U5-$[U[5(a[U5[- $[U[5(a [!U5$[U["5(a[%U5[- $['SUR(-5es snf)Nr unhandled %s)r6rr7rVrrrr rrrrrrr rr!rr"NotImplementedErrorrU)r/r:rgrcs r0r;_osborne..fs"011I GGAJxxAaCS^^!&&4I&QqS&4I%J b$  SV8O D ! !q6M D ! !SV8O D ! !q6!8O D ! !q6M D ! !q6!8O%nrww&>? ?5JsE%rrYrcr;s ` r0_osborners"@( Q?r2c&^U4Sjn[X5$)a Replace all trig functions with hyperbolic functions using the Osborne rule. Notes ===== ``d`` is a dummy variable to prevent automatic evaluation of trigonometric/hyperbolic functions. References ========== .. [1] https://en.wikipedia.org/wiki/Hyperbolic_function c>[U[5(dU$URSRTSS9upUR T[ R 05U[--n[U[5(a[U5[- $[U[5(a [U5$[U[5(a[U5[- $[U[5(a[U5[-$[U[ 5(a [#U5$[U[$5(a['U5[-$[)SUR*-5e)NrT)as_Addr)r6r$r7as_independentxreplacerr8r rrrrrrr rr!rr"rrrU)r/constrr:rcs r0r;_osbornei..fs"344I771:,,Qt,< JJ155z "U1W , b#  719  C 7N C 719  C 719  C 7N C 719 %nrww&>? ?r2rrs ` r0 _osborneirs @( Q?r2c<^^^^ SSKJm SSKJm UR [ 5nUVs/sHo"[ 54PM snmUR[T55nTVVs/sHupEXT4PM snnm[ 5m[UT5UUUU 4Sj4$s snfs snnf)aReturn an expression containing hyperbolic functions in terms of trigonometric functions. Any trigonometric functions initially present are replaced with Dummy symbols and the function to undo the masking and the conversion back to hyperbolics is also returned. It should always be true that:: t, f = hyper_as_trig(expr) expr == f(t) Examples ======== >>> from sympy.simplify.fu import hyper_as_trig, fu >>> from sympy.abc import x >>> from sympy import cosh, sinh >>> eq = sinh(x)**2 + cosh(x)**2 >>> t, f = hyper_as_trig(eq) >>> f(fu(t)) cosh(2*x) References ========== .. [1] https://en.wikipedia.org/wiki/Hyperbolic_function rr|)collectc >T"T"[UT5R[T555[R5$rI)rrdictr ImaginaryUnit)rrrcrepsr}s r0rhyper_as_trig..&s1'(!Q  d,3./0+@r2) rr}sympy.simplify.radsimpratomsr$rrrr) r/trigsrvmaskedrXrhrrcrr}s @@@@r0 hyper_as_trigrs41. HH* +E"' (%QL% (D [[d $F $ $ttqQFt $D A FA !@ @@ ) %s B!Bc~UR[[5(dU$[[ [ U555$)aConvert products and powers of sin and cos to sums. Explanation =========== Applied power reduction TRpower first, then expands products, and converts products to sums with TR8. Examples ======== >>> from sympy.simplify.fu import sincos_to_sum >>> from sympy.abc import x >>> from sympy import cos, sin >>> sincos_to_sum(16*sin(x)**3*cos(2*x)**2) 7*sin(x) - 5*sin(3*x) + 3*sin(5*x) - sin(7*x) )r~rrrr rs)exprs r0 sincos_to_sumr*s.& 88C   :gdm,--r2)F)rFrrI)NT)r collectionsrsympy.core.addrsympy.core.cachersympy.core.exprrsympy.core.exprtoolsrrr sympy.core.functionr sympy.core.mulr sympy.core.numbersr r sympy.core.powerrsympy.core.singletonrsympy.core.sortingrsympy.core.symbolrsympy.core.sympifyrsympy.core.traversalr(sympy.functions.combinatorial.factorialsr%sympy.functions.elementary.hyperbolicrrrrrrr(sympy.functions.elementary.trigonometricrrrr r!r"r#r$sympy.ntheory.factor_r%sympy.polys.polytoolsr&sympy.strategies.treer'sympy.strategies.corer(r)sympyr*r1r>rBrzrrrrrrrrrrrrrr/r6rCrFrTr`rermrsrvrprCTR1CTR2CTR3CTR4r{r|r}rrfufuncsrziplocalsgetFUrrrrr rrrrrr2r0rs#$ AA*$ "&#&*=<<<???/((1 )0?      o7d6r%P$N(@V.r2