EdddlmZddlmZddlmZddlmZmZm Z ddl m Z ddl m Z ddlmZmZddlmZdd lmZdd lmZdd lmZdd lmZdd lmZddlmZddlm 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-m.Z.m/Z/ddl0m1Z1ddl2m3Z3ddl4m5Z5ddl6m7Z7m8Z8ddl9m:Z:dZ;dZdZ?dZ@dZAdAdZBdAd ZCd!ZDdBd#ZEd$ZFdBd%ZGd&ZHdCd(ZId)ZJdBd*ZKd+ZLd,ZMd-ZNdBd.ZOdAd/ZPdAd0ZQd1ZRdAd2ZSd3ZTd4ZUe:rBeVeWe8e;eeNeOePeQeLeReSf\Z;ZZNZOZPZQZLZRZSeBe;feCe;fe7gZXeIeBe;feCe;fe;gfZYeNeEe;feNeEeHe;fe7gZZe@eHfe7gZ[e@e?e@eKe@eMe@e;fZ\e@e?eGe@e?eIfeBeDeIe@feZeXeFeYe@eFeFe[fe7gZ]d5fd6Z^dDd7Z_d8`ZaebeVeceaeVeWedjeeaZfd9Zgd'ahd@d:Zid;Zjd<Zkd=Zld>Zmd?Znd'S)E) defaultdictAdd)Expr)Factors gcd_terms factor_terms expand_mul)Mul)piI)Pow)S)orderedDummy)sympify bottom_up)binomial)coshsinhtanhcothsechcschHyperbolicFunction)cossintancotseccscsqrtTrigonometricFunction) perfect_power)factor)greedy)identitydebug) SYMPY_DEBUGcr|S)zSimplification of rational polynomials, trying to simplify the expression, e.g. combine things like 3*x + 2*x, etc.... )normalr(expandrvs 1lib/python3.11/site-packages/sympy/simplify/fu.pyTR0r3s* 99;;     & & ( ((c(d}t||S)zReplace sec, csc with 1/cos, 1/sin Examples ======== >>> from sympy.simplify.fu import TR1, sec, csc >>> from sympy.abc import x >>> TR1(2*csc(x) + sec(x)) 1/cos(x) + 2/sin(x) ct|tr)|jd}tjt |z St|t r)|jd}tjt|z S|SNr) isinstancer#argsrOnerr$r r1as r2fzTR1..f4se b#    A5Q<  C   A5Q<  r4rr1r=s r2TR1r?(s# R  r4c(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|Sr7)r8r!r9r rr"r;s r2r=zTR2..fRsm b#   ! Aq66#a&&= C  ! Aq66#a&&=  r4rr>s r2TR2rB@s#$ R  r4Fc.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)r8ris_Powbase).0ais r2 z8TR2i..f..ok...sR,,#2s++Kry/KRW^,,,,,,r4N)anyr make_argsrLr<s r2rNz.TR2i..f..ok..sf==01,,-**,,,,,======r4) is_integer is_positivefuncr ris_Addlenr9rO)kehalfs r2okzTR2i..f..oks.?3*$>*=*=AF q *===56V=====  @r4cbg|]+}||||f,Spop)rLrWnrZs r2 z#TR2i..f..:JJJ1bbAaDkkJ!QUU1XXJJJr4cbg|]+}||||f,Sr\r])rLrWdrZs r2r`z#TR2i..f..rar4cg}|D]^}|jrUt|jdkr=rt|nt |}||kr|||f_|r}t |D]\}\}}||=|||<t|}|D]<}||||z}||r|||<%|||f=~dSdSN) rUrVr9r(r append enumerater as_powers_dict) rcddonenewkrWknewivrYrZs r2 factorizez"TR2i..f..factorizes$D / /8/AF a/(,A6!999,q//Dqy/ QI... $-dOO##LAy4!"DGGDz0022--A!tAwAr!Qxx- ! aV,,,,DD  r4rFevaluaterfrFc"g|] \}}|||z Sr\r\rLbrXs r2r`z#TR2i..f..s%<<.f..s%666tq!A6ad666r4cg|] \}}||z Sr\r\rss r2r`z#TR2i..f..s ///A1///r4cg|] \}}||z Sr\r\rss r2r`z#TR2i..f..s 6N6N6N1q!t6N6N6Nr4)is_Mulas_numer_denomis_Atomrilistkeysr8r rr9rgr!rUrr:rRrSr items) r1ndonerjrotrWr<a1rcr_rZrYs @@@r2r=zTR2i..fzs(y I  ""1 9   I @ @ @ @ @     JJJJJQVVXXJJJ I    JJJJJQVVXXJJJ I      &  !U !U  ' 'A!S!! 'q E2226,adadl,HHS^^QqT1222"&&AaD1Q44,QBQw,1R5AaD=,#afQik"2"2QqT!9:::'++!quAs## 'q E2226'adadl'HHS^^adU2333"&&AaD1Q4 '!( 'qvayAE'9 'qvay#.. 'q q)E:::6'adadl'!' 'HHS1--!u4555"&&AaD1Q4  Pq<66qwwyy66678B #/////06N6N6N6N6N1OO OB r4r)r1rYr=s ` r2TR2ir^s48SSSSSj R  r4c:ddlmfd}t||S)aRInduced formula: example sin(-a) = -sin(a) Examples ======== >>> from sympy.simplify.fu import TR3 >>> from sympy.abc import x, y >>> from sympy import pi >>> from sympy import cos >>> TR3(cos(y - x*(y - x))) cos(x*(x - y) + y) >>> cos(pi/2 + x) -sin(x) >>> cos(30*pi/2 + x) -cos(x) rsignsimpc Tt|ts|S||jd}t|ts|S|jdtjdz z jtjdz |jdz jcxurdurnntttttttttttti}|t|tjdz |jdz }|S)NrrFT)r8r&rTr9rPirSrr r!r"r#r$type)r1fmaprs r2r=zTR3..fs"344 I WWXXbgaj)) * *"344 I GAJa  ,a"'!*1D0Q 5 5 5 5UY 5 5 5 5 5c3S#sCc3ODd2hhQ 344B r4)sympy.simplify.simplifyrr)r1r=rs @r2TR3rsB$100000      R  r4c|S)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 r\r0s r2TR4rs 0 Ir4c>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)NrTrfrFr)rJrKrTexpis_realr9r')r1rXpr=ghmaxpows r2_fz_TR56.._f1s   bgla/ Iv~ I FQJ4  I FSLT ! I 6Q; I 6Q; /1QQrw|A''*++ +vzQ FAIqa))!!AAbgl1o,>,>,A*B*BA*EEE1  6A:IFAI!"&))IFAI1QQrw|A''*++Q. .r4r)r1r=rrrrrs ````` r2_TR56rsG4!/!/!/!/!/!/!/!/!/F R  r4rcBt|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 Srer\xs r2zTR5..i Qr4rr)rr rr1rrs r2TR5rW!$ S#CS A A AAr4cBt|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 Srer\rs r2rzTR6..~rr4r)rrr rs r2TR6rlrr4c(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)NrFrfr)rJrKrTrrr9r0s r2r=zTR7..fsS  bglc1 bfk IC"',q/)***A--r4rr>s r2TR7rs# ... R  r4Tc.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 c^ |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%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}t1t|t|}t3|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 |tt7t!|S)Nc,g|]}t|Sr\r rLrms r2r`z"TR8..f..s???aJqMM???r4FfirstrrFrf)rxrJrKrTrr rrRrSryTR8rr9 is_RationalrVrUr as_coeff_MulrrPrrg is_Integerextendminranger^r ) r1r_rcnewnnewdr9r<csrmra2rs r2r=zTR8..fs9 I  I  GLS#J & V   #%'"5   I  ??2+<+<+>+>???DAqq&&&Dq&&&Dqy 1DAI 1tDy))91!71BG )1.0gaj.?1boo//0BIRb$+r**++ % %Av#s# %T!WW $$QVAY////( %qu/ %AEAI %FKC:- %T!&\\"))16;q>*:15*@AAAAT !!!$$$$ I I a 3q66A: Q! IDz AA  q 9 9ABB KKR"WBG 4a7 8 8 8 8!ffqj 9BB KKR"WBG 4a7 8 8 8!ffqj 9  & KKAEEGG % % %!ffqj :BB KK#b2g,,R"W5q8 9 9 9!ffqj :  & KKAEEGG % % %:c4j))***r4rr1rr=s ` r2rrs/"5+5+5+5+5+n R  r4c(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)NrFFrfTcg|]}||Sr\r\rLrs r2r`z.TR9..f..do..#777bB7r777r4r rUr{rr9rVrr trig_splitrr )r1rr9hitrmrMjajwasnewsplitgcdn1n2r<rtiscosdos r2rzTR9..f..dosn9  (())D4yyA~ s4yy))""AaB! "1q5#d))44 " "!!W%$ 2g bgg#:"&)DG&*DG"&C!E " $77D7778By$RVV %E  ', $CRAu ;8Br6!8CQ NN23Aqy>>AA6 aqA#vc1q5!)nn,S!a%^^;;8Br6!8CQ NN23Aqy>>AA6 aqAuS!a%^^+CQ NN::r4T)rUprocess_common_addends)r1rs @r2r=zTR9..fsCy I< ;< ;< ;< ;< ;< ;|&b"---r4rr>s r2TR9rs'.B.B.B.H R  r4c.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) rTrr r9rUr{rr^r _from_argsTR10)r1r=argr9r<rtrs r2r=zTR10..fTs 73* $ I Ggaj : 9 &GCH--..CH~~ At$$Ax 989q66$s1vvU";";";;AtCFF%888899q66$s1vvU";";";;AtCFF%88889989q66#a&&=3q66#a&&=88q66#a&&=3q66#a&&=88 r4rrs ` r2rrBs.$6 R  r4cRttd}t||S)aSum of products to function of sum. Examples ======== >>> from sympy.simplify.fu import TR10i >>> from sympy import cos, sin, sqrt >>> from sympy.abc import x >>> TR10i(cos(1)*cos(3) + sin(1)*sin(3)) cos(2) >>> TR10i(cos(1)*sin(3) + sin(1)*cos(3) + cos(3)) cos(3) + sin(4) >>> TR10i(sqrt(2)*cos(x)*x + sqrt(6)*sin(x)*x) 2*sqrt(2)*x*sin(x + pi/6) Nc  |js|Sd fd t| d}|jrtt}|jD]}d}|jrO|jD]G}|jr>|jtj ur+|j j r|| |d}nH|s%|tj  |g}|D]}t|ztfD]}||vrt!t#||D]}|||t!t#||D]y}|||t%||||||z}  | } | | kr-| | d|||<d|||<nz׌|r)t%|d|Dz}n  |}n|j|S)NTc|js|Stt|j}t |dkrd}t t |D]_}||}| t |dzt |D]1}||}| ||z}|} | |kr| ||<d||<d}n2`|r%t d|D}|jr |}|St|ddi} | s|S| \} } } }}}|r5| | z} | | kr| t||z zS| t||zzS| | z} | | kr| t||zzS| t||z zS)NrFFrfTcg|]}||Sr\r\rs r2r`z0TR10i..f..do..rr4twor)r1rr9rrmrMrrrrrrrrr<rtsamers r2rzTR10i..f..dos9  (())D4yyA~ s4yy))""AaB! "1q5#d))44 " "!!W%$ 2g bgg#:"&)DG&*DG"&C!E " $77D7778By$RVV /$//E  &+ #CRAt &f8*s1q5zz>)3q1u::~%f8*s1q5zz>)3q1u::~%r4cDtt|jSrI)tupler free_symbolsrs r2rz"TR10i..f..seGAN$;$;<<r4rrfc4g|]}td|DS)cg|]}||Sr\r\rs r2r`z/TR10i..f...s(>(>(>2(>(>(>(>r4r)rLrns r2r`z$TR10i..f..s<#-#-#-$'(>(>a(>(>(>#?#-#-#-r4r)rUrrr{r9rxrJrrHalfrKrrgr:_ROOT3 _invROOT3rrVrvalues) r1byradr<rrMr9rtrmrrrrs @r2r=zTR10i..fsy I6 &6 &6 &6 &6 &6 &p$ <<>> i& %%EW + +8"f""9"16)9" " 2"!"I,,Q///"#C!E+!%L''***D * * (I.**AEz *!&s58}}!5!5 * *A$Qx{) (%*3uQx==%9%9 * *#(8A;!-$,&)%(1+a *C&D&D&(bgg#&#:!*$(KK$4$4$426E!HQK26E!HQK$)E !** 4#-#-"\\^^#-#-#--/RVVMi& P r4)_ROOT2_rootsrr>s r2TR10irrs:&iiiV R  r4Nc.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)NrFrT)rational) rTrr rr:rxrr9 is_NumberrqTR11) r1r=rcorrmrrKs r2r=zTR11..f s 73* $ I  %A$q& ABx )((Avc3Z'  wqzQVAY& $IIII8$qD1a4K++Q3q58OI% %71:**D*99DAqsQw!| %c1fQhqslSNNSNN7c>%1QBBA1B r4r)r1rKr=s ` r2rrs0T!!!!!F R  r4c(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|Sr7) rsetr rP as_base_exprrTrr raddr9)flatr9firtrXs r2 sincos_argsz%_TR11..f..sincos_args_ss##DmD)) 5 5~~''1<5AE5v#s+5T!WW ))!&)444Kr4c|tD]a}|dz }||tvrt}n||tvrt}n6t||}|||b|SNrF)r rrremove)r1num_argsden_argsnargrYrTs r2 handle_matchz&_TR11..f..handle_matchks! , ,Av8C=(DDXc]*DD"d^^%%d++++Ir4)r8rmapry)r1rrrrs r2r=z_TR11..f[s"d## I   !b.?.?.A.ABB(    \"h 1 1 \"h 1 1 r4rr>s r2_TR11rFs$*###J R  r4c.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)NrFrrf) rTr!r9rUr{rr^rrTR12)r1rr9r<rttbrs r2r=zTR12..fsw#~ Igaj : 1 &GCH--..CH~~ At$$Ax #a&&...VVFFRK!c!ffRi-0 0 r4rrs ` r2rrs.& R  r4c(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)NrFc3@K|]}t|tVdSrI)r8r!)rLrs r2rNz/TR12i..f..ok..s,AABJr3//AAAAAAr4) as_f_sign_1r NegativeOnerxrVr9all)dirrr=rs r2rZzTR12i..f..oksBA 1a % !( s16{{a7G AA!&AAAAA a4K           r4c(g|]}|jdSrr9rL_s r2r`z$TR12i..f..s444!&)444r4c(g|]}|jdSrrrs r2r`z$TR12i..f..s888AafQi888r4c|jrPt|jdkr:|j\}}t|trt|tr ||fSdSdSdSdSr)rUrVr9r8r!)nir<rts r2rZzTR12i..f..oksy S\\Q. w1a%% *Q*<*< a4K        r4FTc(g|]}|jdSrrrs r2r`z$TR12i..f..s+++AafQi+++r4cPg|]#\}}td|jDdz |z$S)c,g|]}t|Sr\)r!rQs r2r`z/TR12i..f...s+8(8(8(A8(8(8(r4rf)rr9)rLrmrXs r2r`z$TR12i..f..se1J1J1J59Q368(8( !8(8(8(3)+,3-/0211J1J1Jr4)rUrxrJryr9r{r rPrhrrr:r(rrrRrKrSr rextract_additivelyr^r!r})r1r_rcdokrZd_argsrmrrrrrn_argsrredneweds r2r=zTR12i..fs  RY ") I  ""1v QV I   cmA&&''v&& * *EAr2A 144QV4445Aq y *BZZ9&MM"'*** !F1I * 1 *RW5H *BrwKK *DAq888889AVCF !26 F1IIBy* bg...$%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 r4rr>s r2TR12ir"s'*aaaF R  r4c(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 j|js|Stgtgdgi}tt j|D]f}|jttfvr4|t||j dK|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)NrrFrf) rxr!r"rr rPrTrrgr9rVr^)r1r9r<rrt1t2s r2r=zTR13..f/sLy IRb$+r**++ % %Av#s# %T!WW $$QVAY////T !!!$$$$ I I q66A: #a&&1* IDz!ffqj KBB KKSWWSb\\1CGGCRLL4HHI J J J!ffqj K  & KKAEEGG % % %!ffqj IBB KKCGGCRLL003r773rBw<<3GG H H H!ffqj I  & KKAEEGG % % %Dzr4rr>s r2TR13r'!s#< R  r4c0dfd t|S)aReturns cos(x)*cos(2*x)*...*cos(2**(k-1)*x) -> sin(2**k*x)/(2**k*sin(x)) Examples ======== >>> from sympy.simplify.fu import TRmorrie, TR8, TR3 >>> from sympy.abc import x >>> from sympy import Mul, cos, pi >>> TRmorrie(cos(x)*cos(2*x)) sin(4*x)/(4*sin(x)) >>> TRmorrie(7*Mul(*[cos(x) for x in range(10)])) 7*sin(12)*sin(16)*cos(5)*cos(7)*cos(9)/(64*sin(1)*sin(3)) Sometimes autosimplification will cause a power to be not recognized. e.g. in the following, cos(4*pi/7) automatically simplifies to -cos(3*pi/7) so only 2 of the 3 terms are recognized: >>> TRmorrie(cos(pi/7)*cos(2*pi/7)*cos(4*pi/7)) -sin(3*pi/7)*cos(3*pi/7)/(4*sin(pi/7)) A touch by TR8 resolves the expression to a Rational >>> TR8(_) -1/8 In this case, if eq is unsimplified, the answer is obtained directly: >>> eq = cos(pi/9)*cos(2*pi/9)*cos(3*pi/9)*cos(4*pi/9) >>> TRmorrie(eq) 1/16 But if angles are made canonical with TR3 then the answer is not simplified without further work: >>> TR3(eq) sin(pi/18)*cos(pi/9)*cos(2*pi/9)/2 >>> TRmorrie(_) sin(pi/18)*sin(4*pi/9)/(8*sin(pi/9)) >>> TR8(_) cos(7*pi/18)/(16*sin(pi/9)) >>> TR3(_) 1/16 The original expression would have resolve to 1/16 directly with TR8, however: >>> TR8(eq) 1/16 References ========== .. [1] https://en.wikipedia.org/wiki/Morrie%27s_law Tc`|js|S|r0|\}}|d|dz Stti}g}|jD]}|\}}|jrXt|trC|jd \} } |  | |||<x| |g} D]} | }| |r~d} |dx} }| |vr| dz } | dz} | |v| dkrtd| z|z| zd| zz t|| zz }d}g}t| D]N}| dz} t| | zd}| | t|||p||}Ot| D]W}|} t| | zd}||xx|zcc<||s|| X| ||znCt|d| z}| |||z|~| rt#| |zfdDz}|S)NrrfrFFrpcNg|]!}|D]}t||zd"S)Frp)r)rLr<rWr9s r2r`z'TRmorrie..f..sX&I&I&I-.Q&I&I;<AaC%(((&I&I&I&Ir4)rxryrr{r9rrr8rrrgsortr rrr^rr )r1rr_rccossotherrrtrXrr<rrWcccinewargtakeccsrmkeyr9r=s @r2r=zTRmorrie..fs y I  #$$&&DAq1Q7711Q77? "4    A==??DAq| 1c 2 2 q ..00AQr"""Q Q - -AQA FFHHH -A$RAgFA!GBAgq5- 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 r4rrr>s @r2TRmorrier4Ps5v777777r R  r4c.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) NFrrfrc g|] }| Sr\r\)rLrmBs r2r`z#TR14..f..&:&:&:qt&:&:&:r4c g|] }| Sr\r\)rLrmAs r2r`z#TR14..f..r:r4rF)rxryrr:TR14r9rJrrRrSrgr rTrr rr{rrVrr^rinsertr8r )r1r_rcrrr-processr<rtrXrrr=sinotherr|rr1remr<r9rs @@r2r=zTR14..fsty I  $$&&DAq~ AU+++AU+++19# #dB  : :Ax }}1  LLOOOEAA ! #s3 :'LLOOOOLLA&&&HAq" NNAq{Aq"a8 9 9 9 9ww''((U&*%((^^3"1aB, % AA' %AJQ4>$%adn$%tqt|%R5AbE>% ' AA#&qtQqT??D !tt|7&:&:&:&:T&:&:&: #A$ 'q# 6 6 6 6!"17&:&:&:&: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 r4rrs ` r2r=r=s4,^^^^^@ R  r4c2fd}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)NrFrfc d|zSrer\rs r2rz!TR15..f..Y !a%r4r)r8rrKr rTR15rr"r1rXiar<rrs r2r=zTR15..fP2s##  27C(@(@ I F q5A: 2!a%())"'1 1 rT "c3Sc B B B 7 B r4rr1rrr=s `` r2rGrG@4       R  r4c2fd}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)NrFrfc d|zSrer\rs r2rz!TR16..f..zrFr4r)r8rrKrrrGrr!rHs r2r=zTR16..fqrJr4rrKs `` r2TR16rParLr4c(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|Sr7)r8rrKrSrrR is_negativer!r"r9r r$rr#r0s r2r=zTR111..fs r3    W  $&F$5 :<&:L I bgs # # 1rw|A''"&0 0  % % 1rw|A''"&0 0  % % 1rw|A''"&0 0 r4rr>s r2TR111rTs#    R  r4c2fd}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 Srer\rs r2rz!TR22..f.. 1q5r4rc |dz Srer\rs r2rz!TR22..f..rXr4) r8rrKrTr"r!rr#r$rs r2r=zTR22..fsk2s##  c (B I 2sCcs C C C 2sCcs C C C r4rrKs `` r2TR22rZs4$ R  r4c(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)NrrFrfcbg|]+}t|td|zz zz,SrFrrrLrWr_rs r2r`z&TRpower..f..sJ$/$/$/%-QNN3AaC{3C3C$C$/$/$/r4cg|];}t|tj|zztd|zz zz.f..sc=Q=Q=Q:;>Fa^^M1$>%%(!ac'1%5%5>6=Q=Q=Qr4cbg|]+}t|td|zz zz,Sr^r_r`s r2r`z&TRpower..f..sJ$)$)$)%-QNN3AaC{3C3C$C$)$)$)r4cg|];}t|tj|zztd|zz zz.f..sc9K9K9K:;:B!QM1$:%%(!ac'1%5%5:69K9K9Kr4)r8rrKr rrr9rrSis_oddrrrris_evenr)r1rtr_rs @@r2r=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#.... r4rr>s r2TRpowerrfs#*, R  r4cPt|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&r0s r2Lris RXX+ , , - --r4cHt||fSrI)ri count_opsrs r2rr!saddAKKMM2r4ctt}tt}|}t|}t |t s|jfd|jDSt|}| ttrT||}||kr|}| ttrt|}| ttr<||}tt!|}t#||||g}t#t%||S)aHAttempt 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://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.657.2478&rep=rep1&type=pdf c2g|]}t|S))measure)fu)rLr<rns r2r`zfu..ms&AAAAAw///AAAr4)r3)r)RL1RL2rr8rrTr9r?hasr!r"rBr rrr4rr)r1rnfRL1fRL2rrv1rv2s ` r2roro!sJL #w  D #w  D C B b$  CrwAAAAAAABB RB vvc3d2hh GCLL772;; & B 66#s   RB vvc33d2hh(3--   #r3$' 2 2 2 tBxx ) ) ))r4ctt}|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. rrfzmust have at least one keyFrqT) rr{r9rrgrr: ValueErrorrVr)r1rkey2key1abscr<rr9rrWrnrrXrs r2rr|s t  D 7 8 8A>>##DAq1u BB !+TT!WWW!, - 4 4Q 7 7 7 7  8 7 - -A !%a! " ) )! , , , , -5666 D C     G1 q66A: Q'''A"Q%%Cax  KK!     KK!A$      $Z Ir4z~ TR0 TR1 TR2 TR3 TR4 TR5 TR6 TR7 TR8 TR9 TR10 TR10i TR11 TR12 TR13 L TR2i TRmorrie TR12i TR14 TR15 TR16 TR111 TR22cXtdtdcaadtz adS)NrFrf)r%rrrr\r4r2rrs&!WWd1ggNFF&IIIr4c ttd||fD\}}||\}}||}dx}}t j|jvr#|t j}| }n5t j|jvr"|t j}| }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 t j} | s t j} | | ur%|tz}||||j dt d z d fS| | z t"kr#|d | zz}||||j dt d z d fS| | z t$kr#|d | zz}||||j dt d 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) Nc,g|]}t|Sr\rrs r2r`ztrig_split.. ' ' '1GAJJ ' ' 'r4rfc6g|]}|Sr\as_exprrs r2r`ztrig_split.. * * *AAIIKK * * *r4c2dx}}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. NrFr)rr:rxrrVr9r{r^r8rr rJrr)r<rrrrr9rts r2 pow_cos_sinztrig_split..pow_cos_sins( A U 8% NN$$EB16{{Q c tx AF||s A!S!! As##  aeqvo at Ga%%  3'' X !%16/ !GBB415222dAq8 8 3   AA 3   A    FQU? ,RR1axr4rch|] }|j Sr\r)rLrs r2 ztrig_split..Ps1111111r4c3*K|] }|jvVdSrIr)rLrmr9s r2rNztrig_split..Qs)<<<<<<>> 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) rFNrc,g|]}t|Sr\rrs r2r`zas_f_sign_1..rr4rfc6g|]}|Sr\rrs r2r`zas_f_sign_1..rr4) rUrVr9rrr:rxrr.rrrr) rXr<rtrrrrrrs r2r r js* 8s16{{a' 6DAqQ]AE "" E 8 q + q A  2rqAA!Qw ' ' ' ' 'DAq XXa[[FB %%((    C} "  VVAM " "   "* $ VVAM " "   R * *"b * * *DAqAEz!1RB RxdSAEzArzr4c.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\)rLrmrcs r2r`z'_osborne..f..s4I4I4IQQqS4I4I4Ir4 unhandled %s)r8rr9rUrrrrr rrrr!rr"rr#rr$NotImplementedErrorrT)r1r<rcs r2r=z_osborne..fs4"011 I GAJx JAaCCS^4I4I4I4I!&4I4I4I%J%J b$   @SVV8O D ! ! @q66M D ! ! @SVV8O D ! ! @q66!8O D ! ! @q66M D ! ! @q66!8O%nrw&>?? ?r4rrXrcr=s ` r2_osborners1"@@@@@( Q??r4c.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$rt'|tzSt)d|jz)NrT)as_Addr)r8r&r9as_independentxreplacerr:rr rrrr!rr"rr#rr$rrrT)r1constrr<rcs r2r=z_osbornei..fs:"344 I71:,,Qt,<<q JJ15z " "U1W , b#   @7719  C @77N C @7719  C  @7719  C  @77N C  @7719 %nrw&>?? ?r4rrs ` r2 _osborneirs1 @@@@@( Q??r4cddlmddlm|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)rLrs r2r`z!hyper_as_trig..s ( ( (QQL ( ( (r4cg|] \}}||f Sr\r\)rLrWrns r2r`z!hyper_as_trig..s $ $ $tq!QF $ $ $r4c t|ttjSrI)rrdictr ImaginaryUnit)rrrcrepsrs r2rzhyper_as_trig..sG''((!Q  d,,3.3./0+@+@r4) rrsympy.simplify.radsimpratomsr&rrrr)r1trigsmaskedrrcrrs @@@@r2 hyper_as_trigrs4100000...... HH* + +E ( (% ( ( (D [[d $ $F % $t $ $ $D A FA  !@!@!@!@!@!@!@ @@r4c|tts|Stt t |S)aConvert products and powers of sin and cos to sums. Explanation =========== Applied power reduction TRpower first, then expands products, and converts products to sums with TR8. Examples ======== >>> from sympy.simplify.fu import sincos_to_sum >>> from sympy.abc import x >>> from sympy import cos, sin >>> sincos_to_sum(16*sin(x)**3*cos(2*x)**2) 7*sin(x) - 5*sin(3*x) + 3*sin(5*x) - sin(7*x) )rrrr rr rf)exprs r2 sincos_to_sumrs;& 88C  . :gdmm,,---r4)F)rFrrI)NT)o collectionsrsympy.core.addrsympy.core.exprrsympy.core.exprtoolsrrr sympy.core.functionr sympy.core.mulr sympy.core.numbersr rsympy.core.powerrsympy.core.singletonrsympy.core.sortingrsympy.core.symbolrsympy.core.sympifyrsympy.core.traversalr(sympy.functions.combinatorial.factorialsr%sympy.functions.elementary.hyperbolicrrrrrrr(sympy.functions.elementary.trigonometricrr r!r"r#r$r%r&sympy.ntheory.factor_r'sympy.polys.polytoolsr(sympy.strategies.treer)sympy.strategies.corer*r+sympyr,r3r?rBrrrrrrrrrrrrrrr"r'r4r=rGrPrTrZrfrir{rCTR1CTR2CTR3CTR4rprqrorrfufuncsrziplocalsgetFUrrrr rrrrr\r4r2rs###### AAAAAAAAAA******$$$$$$$$ """"""&&&&&&######&&&&&&******======<<<<<<<<<<<<<<<<<<????????????????????//////((((((((((((11111111 )))0 >?? s7s7s7s7l666r%%%P$$$N(@(@(@V.....r4