PtjBddlmZddlmZddlmZddlmZddlm Z m Z m Z ddl m Z ddlmZddlmZmZdd lmZdd lmZdd lmZdd lmZdd lmZddlmZddlm Z ddl!m"Z"m#Z#m$Z$m%Z%m&Z&m'Z'm(Z(ddl)m*Z*m+Z+m,Z,m-Z-m.Z.m/Z/m0Z0m1Z1ddl2m3Z3ddl4m5Z5ddl6m7Z7ddl8m9Z9m:Z:ddl;mdZ?dCdZ@dZAdZBdZCdDd ZDdDd!ZEd"ZFdEd$ZGd%ZHdEd&ZId'ZJdFd)ZKd*ZLdEd+ZMd,ZNd-ZOd.ZPdEd/ZQdDd0ZRdDd1ZSd2ZTdDd3ZUd4ZVd5ZWee?eAeBeDeEeFeGeHeIeKeMeOe@ePeQeReSeNeTeUf\Z=Z>Z?ZAZBZDZEZFZGZHZIZKZMZOZ@ZPZQZRZSZNZTZUeDe=feEe=fe9gZZeKeDe=feEe=fe=gfZ[ePeGe=fePeGeJe=fe9gZ\eBeJfe9gZ]eBeAeBeMeBeOeBe=fZ^eBeAeIeBeAeKfeDeFeKeBfe\eZeHe[eBeHeHe]fe9gZ_d6fd7Z`dGd8Zad9bZcedeXeeeceXeYefjgecZhed:Zied;Zjed<ZkdCd=Zld>Zmd?Znd@ZodAZpdBZqd(S)H) defaultdictAdd)cacheit)Expr)Factors gcd_terms factor_terms expand_mul)Mul)piI)Pow)S)orderedDummy)sympify bottom_up)binomial)coshsinhtanhcothsechcschHyperbolicFunction)cossintancotseccscsqrtTrigonometricFunction) perfect_power)factor)greedy)identitydebug) SYMPY_DEBUGcr|S)zSimplification of rational polynomials, trying to simplify the expression, e.g. combine things like 3*x + 2*x, etc.... )normalr)expandrvs O/var/www/html/CA-Chatbot/venv/lib/python3.11/site-packages/sympy/simplify/fu.pyTR0r4 s* 99;;     & & ( ((c(d}t||S)zReplace sec, csc with 1/cos, 1/sin Examples ======== >>> from sympy.simplify.fu import TR1, sec, csc >>> from sympy.abc import x >>> TR1(2*csc(x) + sec(x)) 1/cos(x) + 2/sin(x) ct|tr)|jd}tjt |z St|t r)|jd}tjt|z S|SNr) isinstancer$argsrOner r%r!r2as r3fzTR1..f5se b#    A5Q<  C   A5Q<  r5rr2r>s r3TR1r@)s# R  r5c(d}t||S)a@Replace 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 t|tr,|jd}t|t |z St|t r,|jd}t |t|z S|Sr8)r9r"r:r!r r#r<s r3r>zTR2..fSsm b#   ! Aq66#a&&= C  ! Aq66#a&&=  r5rr?s r3TR2rCAs#$ R  r5Fc.fd}t||S)aConverts 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 cf |js|S|\  jsjr|S fd  fdt D} s|S fdt D}s|S fd}| |||g} D]M}t |trt|j dd}|vrS| |krA| t|j d |zdx |<|< r^d|z}|vrU| |krC| t|j dd z  |zdx |<|<t |trut|j dd}|vrS| |krA| t|j d | zdx |<|<v r|j r|j dtjurt |j dtrt|j dj dd}|vrj| |krX|js|jrD| t|j dd z  | zdx |<|<O|rxt#|d Dzt#d Dz }|t#d |Dt#d |Dz z}|S)Nc|js|joS|jttfvp>o<|jo5t |jdkotd|jDS)Nc3lK|]/}tdtj|DV0dS)c3jK|].}t|tp|jo |jtuV/dSN)r9r is_Powbase).0ais r3 z8TR2i..f..ok...sR,,#2s++Kry/KRW^,,,,,,r5N)anyr make_argsrMr=s r3rOz.TR2i..f..ok..sf==01,,-**,,,,,======r5) is_integer is_positivefuncr!r is_Addlenr:rP)kehalfs r3okzTR2i..f..oks.?3*$>*=*=AF q *===56V=====  @r5cbg|]+}||||f,Spop)rMrXnr[s r3 z#TR2i..f..:JJJ1bbAaDkkJ!QUU1XXJJJr5cbg|]+}||||f,Sr]r^)rMrXdr[s r3raz#TR2i..f..rbr5cg}|D]^}|jrUt|jdkr=rt|nt |}||kr|||f_|r}t |D]\}\}}||=|||<t|}|D]<}||||z}||r|||<%|||f=~dSdSN) rVrWr:r)r append enumerater as_powers_dict) rdddonenewkrXknewivrZr[s r3 factorizez"TR2i..f..factorizesD / /8/AF a(,A6!999,q//Dqyy QI... $-dOO##LAy4!"DGGDz0022--A!tAwAr!Qxx- ! aV,,,,DD  r5rFevaluatergrGc"g|] \}}|||z Sr]r]rMbrYs r3raz#TR2i..f..s%<<.f..s%666tq!A6ad666r5cg|] \}}||z Sr]r]rts r3raz#TR2i..f..s ///A1///r5cg|] \}}||z Sr]r]rts r3raz#TR2i..f..s 6N6N6N1q!t6N6N6Nr5)is_Mulas_numer_denomis_Atomrjlistkeysr9r!r r:rhr"rVrr;rSrTr items) r2ndonerkrptrXr=a1rdr`r[rZs @@@r3r>zTR2i..f{s y I  ""1 9   I @ @ @ @ @     JJJJJQVVXXJJJ I    JJJJJQVVXXJJJ I      &  !U !U  ' 'A!S!! 'q E22266adadllHHS^^QqT1222"&&AaD1Q44,QBQww1R5AaD==#afQik"2"2QqT!9:::'++!quAs## 'q E22266adadllHHS^^adU2333"&&AaD1Q4 '!( 'qvayAE'9'9qvay#..(:q q)E:::66adadll!l +HHS1--!u4555"&&AaD1Q4  Pq<66qwwyy66678B #/////06N6N6N6N6N1OO OB r5r)r2rZr>s ` r3TR2ir_s48SSSSSj R  r5cjddlmfd}|dd}t||S)aRInduced formula: example sin(-a) = -sin(a) Examples ======== >>> from sympy.simplify.fu import TR3 >>> from sympy.abc import x, y >>> from sympy import pi >>> from sympy import cos >>> TR3(cos(y - x*(y - x))) cos(x*(x - y) + y) >>> cos(pi/2 + x) -sin(x) >>> cos(30*pi/2 + x) -cos(x) rsignsimpc Tt|ts|S||jd}t|ts|S|jdtjdz z jtjdz |jdz jcxurdurnntttttttttttti}|t|tjdz |jdz }|S)NrrGT)r9r'rUr:rPirTr r!r"r#r$r%type)r2fmaprs r3r>zTR3..fs"344 I WWXXbgaj)) * *"344 I GAJa  ,a"'!*1D0Q Y Y Y YUY Y Y Y Y Yc3S#sCc3ODd2hhQ 344B r5c,t|tSrJ)r9r'xs r3zTR3..s*Q 566r5c2|ddS)Nc|jo|jSrJ) is_numberryr`s r3rz'TR3....sak.ahr5c |j|jSrJrUr:rs r3rz'TR3....sfafafor5replacers r3rzTR3..s !)) . . % %''r5)sympy.simplify.simplifyrrr)r2r>rs @r3TR3rsg$100000      66 ' ' ( (B R  r5c2|ddS)aIdentify values of special angles. a= 0 pi/6 pi/4 pi/3 pi/2 ---------------------------------------------------- sin(a) 0 1/2 sqrt(2)/2 sqrt(3)/2 1 cos(a) 1 sqrt(3)/2 sqrt(2)/2 1/2 0 tan(a) 0 sqt(3)/3 1 sqrt(3) -- Examples ======== >>> from sympy import pi >>> from sympy import cos, sin, tan, cot >>> for s in (0, pi/6, pi/4, pi/3, pi/2): ... print('%s %s %s %s' % (cos(s), sin(s), tan(s), cot(s))) ... 1 0 0 zoo sqrt(3)/2 1/2 sqrt(3)/3 sqrt(3) sqrt(2)/2 sqrt(2)/2 1 1 1/2 sqrt(3)/2 sqrt(3) sqrt(3)/3 0 1 zoo 0 cvt|to$|jdtz x}jo|jdvS)Nr)rgrGr)r9r'r:r is_Rationalq)rrs r3rzTR4..s? q/ 0 0 Eq " _Q ) E./c_.Dr5cr||jdj|jdjSr8rrs r3rzTR4.. s- FF>16!9>16!9>2 3 3r5rr1s r3TR4rs00 :: E E 4 4  5 55r5c>fd}t||S)a+Helper 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|jr|jjks|S|jjs|S|jdkdkr|S|jkdkr|S|jdkr|S|jdkr'|jjddzS|jdzdkrP|jdz}|jjd|jjddz|zzS|jdkrd}n;s|jdzr|S|jdz}n"t |j}|s|S|jdz}|jjddz|zS)NrTrgrGr)rKrLrUexpis_realr:r()r2rYpr>ghmaxpows r3_fz_TR56.._f>s   bgla//Iv~ I FQJ4  I FSLT ! !I 6Q;;I 6Q;;1QQrw|A''*++ +vzQFAIqa))!!AAbgl1o,>,>,A*B*BA*EEE1 6A:IFAI!"&))IFAI1QQrw|A''*++Q. .r5r)r2r>rrrrrs ````` r3_TR56r$sG4!/!/!/!/!/!/!/!/!/F R  r5rcBt|ttd||S)aReplacement of sin**2 with 1 - cos(x)**2. See _TR56 docstring for advanced use of ``max`` and ``pow``. Examples ======== >>> from sympy.simplify.fu import TR5 >>> from sympy.abc import x >>> from sympy import sin >>> TR5(sin(x)**2) 1 - cos(x)**2 >>> TR5(sin(x)**-2) # unchanged sin(x)**(-2) >>> TR5(sin(x)**4) (1 - cos(x)**2)**2 c d|z Srfr]rs r3rzTR5..v Qr5rr)rr!r r2rrs r3TR5rd!$ S#CS A A AAr5cBt|ttd||S)aReplacement 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 d|z Srfr]rs r3rzTR6..rr5r)rr r!rs r3TR6ryrr5c(d}t||S)aLowering 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 c|jr |jjtkr |jdks|Sdtd|jjdzzdz S)NrGrgr)rKrLrUr rr:r1s r3r>zTR7..fsN  bglc11bfkkIC"',q/)***A--r5rr?s r3TR7rs# ... R  r5Tc.fd}t||S)aqConverting 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 cD |js;|jr2|jjtt fvr|jjs|jjs|S rd| D\}}t|d}t|d}||ks||krpt||z }|jrW|j dj rEt|j dkr-|j djrt!|}|Stgt gdgi}t!j|D]}|jtt fvr4|t'||j dK|jrx|jjrl|jdkra|jjtt fvrG|t'|j|jj dg|jz|d||t}|t }|r|s(t|dkst|dks|S|d}t/t|t|}t1|D]e} |} |} |t | | zt | | z zdz ft|dkrv|} |} |t | | zt | | z zdz t|dkv|r4|t |t|dkrw|} |} |t | | z t | | z zdz t|dkw|r4|t |tt5t!|S)Nc,g|]}t|Sr]r rMrns r3raz"TR8..f..s???aJqMM???r5FfirstrrGrg)ryrKrLrUr r!rrSrTrzTR8r r:rrWrVr as_coeff_MulrQrrh is_Integerextendminranger_r ) r2r`rdnewnnewdr:r=csrnra2rs r3r>zTR8..fs I  I  GLS#J & & V  '"$'"5 'I  ??2+<+<+>+>???DAqq&&&Dq&&&DqyyDAIItDy))91!71BG ))bgaj.?)boo//0BIRb$+r"" % %Av#s##T!WW $$QVAY////( %qu/ %AEAIIFKC:--T!&\\"))16;q>*:15*@AAAAT !!!$$$$ I I a 3q66A::Q!IDz AA  q 9 9ABB KKR"WBG 4a7 8 8 8 8!ffqjjBB KKR"WBG 4a7 8 8 8!ffqjj  & KKAEEGG % % %!ffqjjBB KK#b2g,,R"W5q8 9 9 9!ffqjj  & KKAEEGG % % %:c4j))***r5rr2rr>s ` r3rrs/"5+5+5+5+5+n R  r5c(d}t||S)acSum 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) cB|js|Sdfd t|S)NTc|js|Stt|j}t |dkrd}t t |D]_}||}| t |dzt |D]1}||}| ||z}|} | |kr| ||<d||<d}n2`|r%t d|D}|jr |}|St|} | s|S| \} } } }}}|ru| | kr4| | zdzt||zdz zt||z dz zS| dkr||}}d| zt||zdz zt||z dz zS| | kr4| | zdzt||zdz zt||z dz zS| dkr||}}d| zt||zdz zt||z dz zS)NrGFrgTcg|]}||Sr]r]rMrs r3raz.TR9..f..do..0777bB7r777r5r rVr|rr:rWrr trig_splitr r!)r2rr:hitrnrNjajwasnewsplitgcdn1n2r=ruiscosdos r3rzTR9..f..do sT9  (())D4yyA~~s4yy))""AaBz "1q5#d))44 " "!!W:$ 2g bgg#::&)DG&*DG"&C!E & $77D7778By$RVV %E  ', $CRAu ;88r6!8CQ NN23Aqy>>AA66aqA#vc1q5!)nn,S!a%^^;;88r6!8CQ NN23Aqy>>AA66aqAuS!a%^^+CQ NN::r5T)rVprocess_common_addends)r2rs @r3r>zTR9..fsCy I< ;< ;< ;< ;< ;< ;|&b"---r5rr?s r3TR9rs'.B.B.B.H R  r5c.fd}t||S)aSeparate 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|jttfvr|S|j}|jd}|jrr"t t |j}nt |j}|}tj |}|jr|tkr]t|tt|dzt|tt|dzzSt|tt|dzt|tt|dzz S|tkr?t|t|zt|t|zzSt|t|zt|t|zz S|S)NrFr) rUr r!r:rVr|rr_r _from_argsTR10)r2r>argr:r=rurs r3r>zTR10..fas 73* $ $I Ggaj : 9 &GCH--..CH~~ At$$Ax 988q66$s1vvU";";";;AtCFF%888899q66$s1vvU";";";;AtCFF%88889988q66#a&&=3q66#a&&=88q66#a&&=3q66#a&&=88 r5rrs ` r3rrOs.$6 R  r5c(d}t||S)aSum of products to function of sum. Examples ======== >>> from sympy.simplify.fu import TR10i >>> from sympy import cos, sin, sqrt >>> from sympy.abc import x >>> TR10i(cos(1)*cos(3) + sin(1)*sin(3)) cos(2) >>> TR10i(cos(1)*sin(3) + sin(1)*cos(3) + cos(3)) cos(3) + sin(4) >>> TR10i(sqrt(2)*cos(x)*x + sqrt(6)*sin(x)*x) 2*sqrt(2)*x*sin(x + pi/6) c   |js|Sd fd t| d}|jrtt}|jD]}d}|jrO|jD]G}|jr>|jtj ur+|j j r|| |d}nH|s%|tj  |g}|D]}t|ztfD]}||vrt!t#||D]}|||t!t#||D]y}|||t%||||||z}  | } | | kr-| | d|||<d|||<nz׌|r)t%|d|Dz}n  |}n|j|S)NTc|js|Stt|j}t |dkrd}t t |D]_}||}| t |dzt |D]1}||}| ||z}|} | |kr| ||<d||<d}n2`|r%t d|D}|jr |}|St|ddi} | s|S| \} } } }}}|r5| | z} | | kr| t||z zS| t||zzS| | z} | | kr| t||zzS| t||z zS)NrGFrgTcg|]}||Sr]r]rs r3raz0TR10i..f..do..rr5twor)r2rr:rrnrNrrrrrrrrr=rusamers r3rzTR10i..f..dos9  (())D4yyA~~s4yy))""AaBz "1q5#d))44 " "!!W:$ 2g bgg#::&)DG&*DG"&C!E & $77D7778By$RVV /$//E  &+ #CRAt &f88s1q5zz>)3q1u::~%f88s1q5zz>)3q1u::~%r5cDtt|jSrJ)tupler free_symbolsrs r3rz"TR10i..f..seGAN$;$;<<r5rrgc4g|]}td|DS)cg|]}||Sr]r]rs r3raz/TR10i..f...s(>(>(>2(>(>(>(>r5r)rMros r3raz$TR10i..f..s<#-#-#-$'(>(>a(>(>(>#?#-#-#-r5r)rVrrr|r:ryrKrrHalfrLrrhr;_ROOT3 _invROOT3rrWrvalues) r2byradr=rrNr:rurnrrrrs @r3r>zTR10i..fsy I6 &6 &6 &6 &6 &6 &p$ <<>> i& %%EW + +8"f""9"16)9)9 " 2*:!"I,,Q///"#C!E+!%L''***D * * ((1*ikk2**AEzz!&s58}}!5!5 * *A$Qx{2 (%*3uQx==%9%9 * *#(8A;#6$,&)%(1+a *C&D&D&(bgg#&#::$(KK$4$4$426E!HQK26E!HQK$)E $.* 4#-#-"\\^^#-#-#--/RVVMi& P r5rr?s r3TR10irs'$iiiV R  r5Nc.fd}t||S)anFunction 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) c |jttfvr|Sr|j}|dz}tj}|jr|\}}|jttfvr|S|jd|jdkr@t}t}|tur|dz|dzz |z Sd|z|z|z S|S|jdjs|jdd\}}|j dzdkrq|j dz|z|j z }tt|}tt|}|jtkr d|z|z}n |dz|dzz }|S)NrGrT)rational) rUr r!rr;ryrr: is_NumberrrTR11) r2r>rcorrmrrLs r3r>zTR11..f)s 73* $ $I  %A$q& ABx )((Avc3Z'' wqzQVAY&&IIII88qD1a4K++Q3q58OI% %71:**D*99DAqsQw!||c1fQhqslSNNSNN7c>>1QBBA1B r5r)r2rLr>s ` r3rrs0T!!!!!F R  r5c(d}t||S)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)) ct|ts|Sd}t||\}}d}||||}||||}|S)Nc4tt}tj|D]n}|\}}|jrN|dkrH|jttfvr3|t| |j do|Sr8) rsetr rQ as_base_exprrUr r!raddr:)flatr:firurYs r3 sincos_argsz%_TR11..f..sincos_argshss##DmD)) 5 5~~''1<5AEEv#s++T!WW ))!&)444Kr5c|tD]a}|dz }||tvrt}n||tvrt}n6t||}|||b|SNrG)r!r rremove)r2num_argsden_argsnargrZrUs r3 handle_matchz&_TR11..f..handle_matchts|! , ,Av8C=((DDXc]**DD"d^^%%d++++Ir5)r9rmaprz)r2rr r rs r3r>z_TR11..fds"d## I   !b.?.?.A.ABB(    \"h 1 1 \"h 1 1 r5rr?s r3_TR11rOs$*###J R  r5c.fd}t||S)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|jtks|S|jd}|jrr"t t |j}nt |j}|}tj|}|jrtt|d}nt|}t||zdt||zz z S|S)NrFrrg) rUr"r:rVr|rr_rrTR12)r2rr:r=rutbrs r3r>zTR12..fsw#~~Igaj : 1 &GCH--..CH~~ At$$Ax #a&&...VVFFRK!c!ffRi-0 0 r5rrs ` r3rrs.& R  r5c(d}t||S)aKCombine 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)) c\|js|js |js|S|\}}|jr|js|Si}d}t t j|}t|D]1\}}||}|r2|\} } td| jD} tj || <| ||<E|jr@t|}|jr)| |jtj ||<|jr|jjs |jjr||j}|r5|\} } td| jD} |j|| <| |jz||<t|}|jr)| |jtj ||<3|s|Sd}t t jt%|} d} t| D]\}}||}|s|| }|rtj| |<n|jr@t|}|jr)| |jtj | |<x|jrz|jjs |jjrb||j}|rtj | |<nPt|}|jr)| |jtj | |<tj | |<d} td|D} || }|tj }||r||| <n|| | |xxt-|  zcc<| r9t | t |z t d|Dz }|S) Nct|}|rU|\}}}|tjurC|jr>t |jdkr(t d|jDr ||fSdSdSdSdSdS)NrGc3@K|]}t|tVdSrJ)r9r")rMrs r3rOz/TR12i..f..ok..s,AABJr3//AAAAAAr5) as_f_sign_1r NegativeOneryrWr:all)dirrr>rs r3r[zTR12i..f..oksBA 1a %%!(%s16{{a7G7GAA!&AAAAA8Ha4K   %%%%7G7G7G7Gr5c(g|]}|jdSrr:rM_s r3raz$TR12i..f..s444!&)444r5c(g|]}|jdSrr r!s r3raz$TR12i..f..s888AafQi888r5c|jrPt|jdkr:|j\}}t|trt|tr ||fSdSdSdSdSr )rVrWr:r9r")nir=rus r3r[zTR12i..f..oksyy S\\Q..w1a%% *Q*<*< a4K  ..    r5FTc(g|]}|jdSrr r!s r3raz$TR12i..f..s+++AafQi+++r5cPg|]#\}}td|jDdz |z$S)c,g|]}t|Sr])r"rRs r3raz/TR12i..f..."s+8(8(8(A8(8(8(r5rg)rr:)rMrnrYs r3raz$TR12i..f.."se1J1J1J59Q368(8( !8(8(8(3)+,3-/0211J1J1Jr5)rVryrKrzr:r|r rQrirrr;r)rrrSrLrTr rextract_additivelyr_r"r~)r2r`rddokr[d_argsrnrrrrrn_argsrr%edneweds r3r>zTR12i..fs  RY ") I  ""1v QV I   cmA&&''v&& * *EAr2A 144QV4445Aq y *BZZ9&MM"'*** !F1I * 1 *RW5H *BrwKK *DAq888889AVCF !26 F1IIBy* bg...$%Eq  I   cmLOO4455v&&% !% !EAr2A "BsGG! ! F1IIy!#BZZ9."MM"'222()F1I  !F- !131D !BrwKK%()F1II!'B!y2 & bg 6 6 6,-Eq $ Eq C+++++,AQB))!%00E "CFFGGAJJJ 1III#a&& IIII  Kfc6l*31J1J=@YY[[1J1J1J,KKB r5rr?s r3TR12ir/s'*aaaF R  r5c(d}t||S)aChange 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 P|js|Stgtgdgi}tj|D]f}|jttfvr4|t ||jdK|d|g|t}|t}t|dkrt|dkr|S|d}t|dkr| }| }|dt|t||zz t|t||zz zz t|dk|r4|t| t|dkr| }| }|dt|t||zzzt|t||zzzt|dk|r4|t| t|S)NrrGrg) ryr"r#r rQrUrrhr:rWr_)r2r:r=rrt1t2s r3r>zTR13..f8s,y IRb$+r"" % %Av#s##T!WW $$QVAY////T !!!$$$$ I I q66A::#a&&1**IDz!ffqjjBB KKSWWSb\\1CGGCRLL4HHI J J J!ffqjj  & KKAEEGG % % %!ffqjjBB KKCGGCRLL003r773rBw<<3GG H H H!ffqjj  & KKAEEGG % % %Dzr5rr?s r3TR13r4*s#< R  r5c0dfd t|S)aReturns cos(x)*cos(2*x)*...*cos(2**(k-1)*x) -> sin(2**k*x)/(2**k*sin(x)) Examples ======== >>> from sympy.simplify.fu import TRmorrie, TR8, TR3 >>> from sympy.abc import x >>> from sympy import Mul, cos, pi >>> TRmorrie(cos(x)*cos(2*x)) sin(4*x)/(4*sin(x)) >>> TRmorrie(7*Mul(*[cos(x) for x in range(10)])) 7*sin(12)*sin(16)*cos(5)*cos(7)*cos(9)/(64*sin(1)*sin(3)) Sometimes autosimplification will cause a power to be not recognized. e.g. in the following, cos(4*pi/7) automatically simplifies to -cos(3*pi/7) so only 2 of the 3 terms are recognized: >>> TRmorrie(cos(pi/7)*cos(2*pi/7)*cos(4*pi/7)) -sin(3*pi/7)*cos(3*pi/7)/(4*sin(pi/7)) A touch by TR8 resolves the expression to a Rational >>> TR8(_) -1/8 In this case, if eq is unsimplified, the answer is obtained directly: >>> eq = cos(pi/9)*cos(2*pi/9)*cos(3*pi/9)*cos(4*pi/9) >>> TRmorrie(eq) 1/16 But if angles are made canonical with TR3 then the answer is not simplified without further work: >>> TR3(eq) sin(pi/18)*cos(pi/9)*cos(2*pi/9)/2 >>> TRmorrie(_) sin(pi/18)*sin(4*pi/9)/(8*sin(pi/9)) >>> TR8(_) cos(7*pi/18)/(16*sin(pi/9)) >>> TR3(_) 1/16 The original expression would have resolve to 1/16 directly with TR8, however: >>> TR8(eq) 1/16 References ========== .. [1] https://en.wikipedia.org/wiki/Morrie%27s_law Tc`|js|S|r0|\}}|d|dz Stti}g}|jD]}|\}}|jrXt|trC|jd \} } |  | |||<x| |g} D]} | }| |r~d} |dx} }| |vr| dz } | dz} | |v| dkrtd| z|z| zd| zz t|| zz }d}g}t| D]N}| dz} t| | zd}| | t|||p||}Ot| D]W}|} t| | zd}||xx|zcc<||s|| X| ||znCt|d| z}| |||z|~| rt#| |zfdDz}|S)NrrgrGFrqcNg|]!}|D]}t||zd"S)Frq)r )rMr=rXr:s r3raz'TRmorrie..f..sX&I&I&I-.Q&I&I;<AaC%(((&I&I&I&Ir5)ryrzrr|r:rrr9r rrhsortr!rrr_r r )r2rr`rdcossotherrrurYrr=rrXcccinewargtakeccsrnkeyr:r>s @r3r>zTRmorrie..fsy I  #$$&&DAq1Q7711Q77? "4    A==??DAq| 1c 2 2 q ..00AQr"""Q Q - -AQA FFHHH -A$RAggFA!GBAggq55 Ab^^AqD0RT:FDC"1XXAAa!!B$777 2"49d.?d3i@@"1XX)) WWYY!!B$777S T) #Cy)HHRLLLJJvt|,,,,AEE!HHQJALLDG,,,5 -8  KsU{&I&I&I&I26&I&I&IIKB r5rrr?s @r3TRmorrierAYs5v777777r R  r5c.fd}t||S)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 ~|js|SrZ|\}}|tjur5t |d}t |d}||ks||kr||z }|Sg}g}|jD]}|jr>|\}} | js|j s| |D|}n tj} t|} | r| dj ttfvr=| tjur| |n| || z| \} } } | | | j| | | |ft!t#|}t%|}t!t'dx}\} }} } } }|rw|d|r8|d| jr[| jrM| | kr9| | kr&|dt+| | }| |kr5fd|D}|| xx|zcc<|d|n@| |kr4fd|D}|| xx|zcc<|d|t/| trt}nt}| |  | z|| jddzz|znlj| | kr| | kr| | kr|d| }t/| trt}nt}| |  | z|| jddzz|zS| || z|wt%||kr t1|}|S) NFrrgrrc g|] }| Sr]r])rMrnBs r3raz#TR14..f.."&:&:&:qt&:&:&:r5c g|] }| Sr]r])rMrnAs r3raz#TR14..f..&rFr5rG)ryrzrr;TR14r:rKrrSrTrhrrUr r!rr|rrWrr_rinsertr9r )r2r`rdrrr:processr=rurYrrr>sinotherr}rr>remrHrErs @@r3r>zTR14..fsOy I  $$&&DAq~~AU+++AU+++199 dB  : :Ax }}1  LLOOOEAA ! #s33::LLOOOOLLA&&&HAq" NNAq{Aq"a8 9 9 9 9ww''((U&*%((^^3"1aB, % AA' %AJQ4>$%adn$%tqt||R5AbE>> ' AA#&qtQqT??D !tt||&:&:&:&:T&:&:&: #A$ 'q# 6 6 6 6!"1&:&:&:&:T&:&:&: #A$ 'q# 6 6 6)!A$44($'$'!LL1Q4%!*QQqty|__a5G*G$)NOOO$qTQqT\\tqt||R5AbE>> ' AA#$Q4D)!A$44($'$'!LL1Q4%!*QQqty|__a5G*G$)NOOO$ LL1qt $ $ $Y, %\ u::  eB r5rrs ` r3rIrIs4,^^^^^@ R  r5c2fd}t||S)aConvert 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(t|trt|jts|S|j}|dzdkr"t |j|dzz|jz Sd|z }t |ttd}||kr|}|S)NrGrgc d|zSrfr]rs r3rz!TR15..f..b !a%r5r)r9rrLr!rTR15rr#r2rYiar=rrs r3r>zTR15..fY2s##  27C(@(@ I F q5A::!a%())"'1 1 rT "c3Sc B B B 77B r5rr2rrr>s `` r3rSrSI4       R  r5c2fd}t||S)aConvert 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(t|trt|jts|S|j}|dzdkr"t |j|dzz|jz Sd|z }t |ttd}||kr|}|S)NrGrgc d|zSrfr]rs r3rz!TR16..f..rRr5r)r9rrLr rrSrr"rTs r3r>zTR16..fzrVr5rrWs `` r3TR16r\jrXr5c(d}t||S)aDConvert 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 ct|tr$|jjs|jjr |jjs|St|jtr(t|jj d|j zSt|jtr(t|jj d|j zSt|jtr(t|jj d|j zS|Sr8)r9rrLrTrrS is_negativer"r#r:r!r%r r$r1s r3r>zTR111..fs r3    W  $&F$5 :<&:L I bgs # # 1rw|A''"&0 0  % % 1rw|A''"&0 0  % % 1rw|A''"&0 0 r5rr?s r3TR111r`s#    R  r5c2fd}t||S)ahConvert 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 ct|tr|jjtt fvs|St |t td}t |ttd}|S)Nc |dz Srfr]rs r3rz!TR22..f.. 1q5r5rc |dz Srfr]rs r3rz!TR22..f..rdr5) r9rrLrUr#r"rr$r%rs r3r>zTR22..fsi2s##  c (B(BI 2sCcs C C C 2sCcs C C C r5rrWs `` r3TR22rfs4$ R  r5c(d}t||S)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 ct|tr!t|jttfs|S|\}|jdjrjrj rIt|tr4ddz ztfdtdzdz Dz}nj r^t|trIddz ztj dz dz zztfdtdzdz Dz}njrEt|tr0ddz ztfdtdz Dz}n^jrWt|trBddz ztj dz zztfdtdz Dz}jr|d ztdz zz }|S)NrrGrgcbg|]+}t|td|zz zz,SrGrr rMrXr`rs r3raz&TRpower..f..sJ$/$/$/%-QNN3AaC{3C3C$C$/$/$/r5cg|];}t|tj|zztd|zz zz.f..sc=Q=Q=Q:;>Fa^^M1$>%%(!ac'1%5%5>6=Q=Q=Qr5cbg|]+}t|td|zz zz,Srjrkrls r3raz&TRpower..f..sJ$)$)$)%-QNN3AaC{3C3C$C$)$)$)r5cg|];}t|tj|zztd|zz zz.f..sc9K9K9K:;:B!QM1$:%%(!ac'1%5%5:69K9K9Kr5)r9rrLr!r rr:rrTis_oddrrrris_evenr)r2rur`rs @@r3r>zTRpower..fs2s##  27S#J(G(G I~~1 F1I < /AM /x LJq#.. L1Xc$/$/$/$/$/"AE19--$/$/$/00 LjC00 L1XamqsAg66s=Q=Q=Q=Q=Q?Da!eQY?O?O=Q=Q=Q8RR Lz!S11 L1Xc$)$)$)$)$)"1Q3ZZ$)$)$)** Lz!S11 L1Xamac2239K9K9K9K9K?DQqSzz9K9K9K4LLy /a1"ghq!A#.... r5rr?s r3TRpowerrrs#*, R  r5cPt|tS)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'r1s r3Lrus RXX+ , , - --r5cHt||fSrJ)ru count_opsrs r3rr*saddAKKMM2r5ctt}tt}|}t|}t |t s|jfd|jDSt|}| ttrT||}||kr|}| ttrt|}| ttr<||}tt!|}t#||||g}t#t%||S)a7Attempt 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 c2g|]}t|S))measure)fu)rMr=rzs r3razfu..vs&AAAAAw///AAAr5)r@)r*RL1RL2rr9rrUr:r@hasr"r#rCr!r rrArr)r2rzfRL1fRL2rrv1rv2s ` r3r{r{*sHL #w  D #w  D C B b$  CrwAAAAAAABB RB vvc3d2hh GCLL772;; & &B 66#s   RB vvc33d2hh(3--   #r3$' 2 2 2 tBxx ) ) ))r5ctt}|rX|jD]O}|\}}|dkr| }| }|||r ||ndf|PnL|r;|jD]2}|t j||f|3ntdg}d}|D]z} || } | \}} t| dkr:t| ddi} || } | | kr| } d}||| z\||| dz{|r t|}|S)aApply ``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. rrgzmust have at least one keyFrrT) rr|r:rrhrr; ValueErrorrWr)r2rkey2key1abscr=rr:rrXror"rYrs r3rrs t  D 7 8 8A>>##DAq1uuBB !+TT!WWW!, - 4 4Q 7 7 7 7  8 7 - -A !%a! " ) )! , , , , -5666 D C     G1 q66A::Q'''A"Q%%Caxx KK!     KK!A$      $Z Ir5z~ TR0 TR1 TR2 TR3 TR4 TR5 TR6 TR7 TR8 TR9 TR10 TR10i TR11 TR12 TR13 L TR2i TRmorrie TR12i TR14 TR15 TR16 TR111 TR22c tdSr r&r]r5r3_ROOT2r 77Nr5c tdS)Nrrr]r5r3rrrr5c&dtdz S)Nrgrrr]r5r3rrs T!WW9r5c d||fD\}}||\}}||}dx}}tj|jvr#|tj}| }n5tj|jvr"|tj}| }d||fD\}}d}|||} | dS| \} } } |||} | dS| \} }}| s|s| r(t| tr| ||| | | f\} } } } }}||}}|sP| p| }|p|}t||j sdS||||j d|j dt|tfS| s| s| r|r| r|rt| | j t||j urdSd| | fDtfd||fDsdS|||| j d| j dt| | j fS| r| s|r|s |r | | ||dS| p| }|p|}|j |j krdS| s tj } | s tj } | | ur,|tz}||||j dtd z d fS| | z t!kr#|d | zz}||||j dtd z d fS| | z t#kr#|d | zz}||||j dtd z d fSdS)a)Return 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) c,g|]}t|Sr]rrs r3raztrig_split.. ' ' '1GAJJ ' ' 'r5rgc6g|]}|Sr]as_exprrs r3raztrig_split.. * * *AAIIKK * * *r5c2dx}}tj}|jr4|\}}t |jdks|sdS|jrt |j}n|g}|d}t|tr|}n:t|tr|}n"|j r|j tj ur||z}ndS|rd|d}t|tr|r|}nB|}n?t|tr|r|}n%|}n"|j r|j tj ur||z}ndS|tjur|nd||fSt|tr|}nt|tr|}||dS|tjur|nd}|||fS)aReturn ``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. NrGr)rr;ryrrWr:r|r_r9r r!rKrr)r=rrrrr:rus r3 pow_cos_sinztrig_split..pow_cos_sins( A U 8% NN$$EB16{{Qctx AF||s A!S!! As##  aeqvooat Ga%%  3'' X !%16//!GBB41522dAq8 8 3   AA 3   A 9 FQU??RR1axr5Nrch|] }|j Sr]r )rMrs r3 ztrig_split..]s1111111r5c3*K|] }|jvVdSrJr )rMrnr:s r3rOztrig_split..^s)<<<<<<>> 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) rGNrc,g|]}t|Sr]rrs r3razas_f_sign_1..rr5rgc6g|]}|Sr]rrs r3razas_f_sign_1..rr5) rVrWr:rrr;ryrr/rrrr) rYr=rurrrrrrs r3rrws* 8s16{{a'' 6DAqQ]AE """ E 8 q + q A 2rqAA!Qw ' ' ' ' 'DAq XXa[[FB %%((    C} "" VVAM " "   "* $ $ VVAM " "   R * *"b * * *DAqAEzz!1RB RxxdSAEzzArzzr5c.fd}t||S)a2Replace 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 ct|ts|S|jd}|js|zn$t jfd|jD}t|t rtt|zSt|trt|St|trtt|zSt|trt|tz St|trt!|St|t"rt%|tz St'd|jz)Nrcg|]}|zSr]r])rMrnrds r3raz'_osborne..f..s4I4I4IQQqS4I4I4Ir5 unhandled %s)r9rr:rVrrrrr!rr rr"rr#rr$rr%NotImplementedErrorrU)r2r=rds r3r>z_osborne..fs4"011 I GAJx JAaCCS^4I4I4I4I!&4I4I4I%J%J b$   @SVV8O D ! ! @q66M D ! ! @SVV8O D ! ! @q66!8O D ! ! @q66M D ! ! @q66!8O%nrw&>?? ?r5rrYrdr>s ` r3_osborners1"@@@@@( Q??r5c.fd}t||S)a1Replace 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 ct|ts|S|jdd\}}|t ji|tzz}t|trt|tz St|trt|St|trt|tz St|trt|tzSt|t rt#|St|t$rt'|tzSt)d|jz)NrT)as_Addr)r9r'r:as_independentxreplacerr;rr!rr rr"rr#rr$rr%rrrU)r2constrr=rds r3r>z_osbornei..fs:"344 I71:,,Qt,<<q JJ15z " "U1W , b#   @7719  C @77N C @7719  C  @7719  C  @77N C  @7719 %nrw&>?? ?r5rrs ` r3 _osborneirs1 @@@@@( Q??r5cddlmddlm|t }d|D|t}dDtt|fdfS)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.g|]}|tfSr]r)rMrs r3raz!hyper_as_trig..s ( ( (QQL ( ( (r5cg|] \}}||f Sr]r])rMrXros r3raz!hyper_as_trig.."s $ $ $tq!QF $ $ $r5c t|ttjSrJ)rrdictr ImaginaryUnit)rrrdrepsrs r3rzhyper_as_trig..&sG''((!Q  d,,3.3./0+@+@r5) rrsympy.simplify.radsimpratomsr'rrrr)r2trigsmaskedrrdrrs @@@@r3 hyper_as_trigrs4100000...... HH* + +E ( (% ( ( (D [[d $ $F % $t $ $ $D A FA  !@!@!@!@!@!@!@ @@r5c|tts|Stt t |S)aConvert products and powers of sin and cos to sums. Explanation =========== Applied power reduction TRpower first, then expands products, and converts products to sums with TR8. Examples ======== >>> from sympy.simplify.fu import sincos_to_sum >>> from sympy.abc import x >>> from sympy import cos, sin >>> sincos_to_sum(16*sin(x)**3*cos(2*x)**2) 7*sin(x) - 5*sin(3*x) + 3*sin(5*x) - sin(7*x) )r~r r!rr rr)exprs r3 sincos_to_sumr*s;& 88C  . :gdmm,,---r5)F)rFrrJ)NT)r collectionsrsympy.core.addrsympy.core.cachersympy.core.exprrsympy.core.exprtoolsrr r sympy.core.functionr sympy.core.mulr sympy.core.numbersrrsympy.core.powerrsympy.core.singletonrsympy.core.sortingrsympy.core.symbolrsympy.core.sympifyrsympy.core.traversalr(sympy.functions.combinatorial.factorialsr%sympy.functions.elementary.hyperbolicrrrrrrr(sympy.functions.elementary.trigonometricr r!r"r#r$r%r&r'sympy.ntheory.factor_r(sympy.polys.polytoolsr)sympy.strategies.treer*sympy.strategies.corer+r,sympyr-r4r@rCrrrrrrrrrrrrrrr/r4rArIrSr\r`rfrrrur|rCTR1CTR2CTR3CTR4r|r}r{rrfufuncsrziplocalsgetFUrrrrrrrrrr]r5r3rs######$$$$$$ AAAAAAAAAA******$$$$$$$$ """"""&&&&&&######&&&&&&******======<<<<<<<<<<<<<<<<<<????????????????????//////((((((((((((11111111 )))0 >??      o7o7o7o7d666r%%%P$$$N(@(@(@V.....r5