MZd\-ddlmZddlmZddlmZmZddl m Z ddl m Z ddl mZddlmZdd lmZdd lmZdd lmZmZmZmZmZdd lmZmZmZmZdd l m!Z!ddl"m#Z#m$Z$ddl%m&Z&m'Z'ddl(m)Z)m*Z*ddl+m,Z,ddl-m.Z.ddl/m0Z0ddl1m2Z2ddl3m4Z5dZ6dZ7dZ8dZ9GddeZ:e2dZ;e;jye=e=fe:ddlZ>ddl?m@Z@ddlAmBZBmCZCdd lDmEZEmFZFmGZGy!)") annotations)Callable)logsqrt)product)_sympify)cacheit)S)Expr)PrecisionExhausted)expand_complexexpand_multinomial expand_mul_mexpand PoleError) fuzzy_bool fuzzy_not fuzzy_andfuzzy_or)global_parameters)is_gtis_lt) NumberKind UndefinedKind)HAS_GMPYgmpy)sift)sympy_deprecation_warning)as_int) Dispatcher)sqrtremc|dkr tdt|}|dkr+tt|}d|||zz cxkr d|zkr|St|ddS)z9Return the largest integer less than or equal to sqrt(n).rzn must be nonnegativel) ValueErrorint_sqrtinteger_nthroot)nss 2/usr/lib/python3/dist-packages/sympy/core/power.pyisqrtr,sm1u011 AA  aM AaC 1Q3 H  1a  ##c@t|t|}}|dkr td|dkr tdtrW|dkrRtdk\rtj||\}}ntj ||\}}t|t |fSt||S)a@ Return a tuple containing x = floor(y**(1/n)) and a boolean indicating whether the result is exact (that is, whether x**n == y). Examples ======== >>> from sympy import integer_nthroot >>> integer_nthroot(16, 2) (4, True) >>> integer_nthroot(26, 2) (5, False) To simply determine if a number is a perfect square, the is_square function should be used: >>> from sympy.ntheory.primetest import is_square >>> is_square(26) False See Also ======== sympy.ntheory.primetest.is_square integer_log rzy must be nonnegativerzn must be positivelr$)r r%rrirootrootbool_integer_nthroot_python)yr)xts r+r(r(.s6 !9fQiqA1u0111u-..AI q=::a#DAq99Q?DAqay$q'!! "1a ((r-cX|dvr|dfS|dk(r|dfS|dk(rt|\}}t|| fS||jk\ry t|d|z zdz}|d kDr3d |}} ||dz z}||dz |z||zz|z}}t ||z dkrn.|}||z}||kr|dz }||z}||kr||kDr|dz}||z}||kDrt|||k(fS#t$rKt |d|z }|dkDr&t|dz }td ||z zdz|z}ntd |z}YwxYw) N)rrTrr$)rFg?g?5g@l )mpmath_sqrtremr& bit_length OverflowError_logabs) r3r)r4remguessexpshiftxprevr5s r+r2r2YsF{$wAv$wAv"31v3wALLN"A1IO$ u}uqAE AAE19q!t+a/1E1u9~!   1A a% Q qD a% a% Q qD a% q616>3 "1ajl 8bMEcEk*Q./58ESME "sCAD)(D)c|dk(r td|dk(r td|dvr3t|}t|}|jdz }|||z|k(fS|dkr7t |dkDr|n| | \}}||xrt |dkr|dzn|dz fSt|}t|}dx}}||k\r>|}d}||k\r/t ||\}}|xs|}||z }||kDr ||z}|dz}||k\r/||k\r>||dk(xr|dk(fS)a  Returns ``(e, bool)`` where e is the largest nonnegative integer such that :math:`|y| \geq |x^e|` and ``bool`` is True if $y = x^e$. Examples ======== >>> from sympy import integer_log >>> integer_log(125, 5) (3, True) >>> integer_log(17, 9) (1, False) >>> integer_log(4, -2) (2, True) >>> integer_log(-125,-5) (3, True) See Also ======== integer_nthroot sympy.ntheory.primetest.is_square sympy.ntheory.factor_.multiplicity sympy.ntheory.factor_.perfect_power rzx cannot take value as 1rzy cannot take value as 0)r$r$)r%r&r r: integer_logr1divmod) r3r4er)brdmr>s r+rErEsH2 Av344Av344G| F 1I LLNQ !Q$!)|1uA1A2r21!;a!eQUQU;;;q Aq A IA q&  1fAq\FAsSA FA1uQQ 1f q& a1fa r-ceZdZUdZdZdZded<ded<ed=dZd>dZ e d?d Z e d?d Z e d Z ed Zd ZdZdZdZdZdZdZdZdZdZdZdZdZdZdZdZdZ dZ!dZ"d Z#d!Z$d"Z%d#Z&d$Z'd%Z(d@d&Z)d'Z*d(Z+d)Z,d*Z-d+Z.d,Z/d-Z0d.Z1d/Z2d0Z3dAd1Z4dBd2Z5dCd3Z6ed4Z7fd5Z8d6Z9d7Z:d8Z;d9Zd<Z?xZ@S)EPowa% Defines the expression x**y as "x raised to a power y" .. deprecated:: 1.7 Using arguments that aren't subclasses of :class:`~.Expr` in core operators (:class:`~.Mul`, :class:`~.Add`, and :class:`~.Pow`) is deprecated. See :ref:`non-expr-args-deprecated` for details. Singleton definitions involving (0, 1, -1, oo, -oo, I, -I): +--------------+---------+-----------------------------------------------+ | expr | value | reason | +==============+=========+===============================================+ | z**0 | 1 | Although arguments over 0**0 exist, see [2]. | +--------------+---------+-----------------------------------------------+ | z**1 | z | | +--------------+---------+-----------------------------------------------+ | (-oo)**(-1) | 0 | | +--------------+---------+-----------------------------------------------+ | (-1)**-1 | -1 | | +--------------+---------+-----------------------------------------------+ | S.Zero**-1 | zoo | This is not strictly true, as 0**-1 may be | | | | undefined, but is convenient in some contexts | | | | where the base is assumed to be positive. | +--------------+---------+-----------------------------------------------+ | 1**-1 | 1 | | +--------------+---------+-----------------------------------------------+ | oo**-1 | 0 | | +--------------+---------+-----------------------------------------------+ | 0**oo | 0 | Because for all complex numbers z near | | | | 0, z**oo -> 0. | +--------------+---------+-----------------------------------------------+ | 0**-oo | zoo | This is not strictly true, as 0**oo may be | | | | oscillating between positive and negative | | | | values or rotating in the complex plane. | | | | It is convenient, however, when the base | | | | is positive. | +--------------+---------+-----------------------------------------------+ | 1**oo | nan | Because there are various cases where | | 1**-oo | | lim(x(t),t)=1, lim(y(t),t)=oo (or -oo), | | | | but lim( x(t)**y(t), t) != 1. See [3]. | +--------------+---------+-----------------------------------------------+ | b**zoo | nan | Because b**z has no limit as z -> zoo | +--------------+---------+-----------------------------------------------+ | (-1)**oo | nan | Because of oscillations in the limit. | | (-1)**(-oo) | | | +--------------+---------+-----------------------------------------------+ | oo**oo | oo | | +--------------+---------+-----------------------------------------------+ | oo**-oo | 0 | | +--------------+---------+-----------------------------------------------+ | (-oo)**oo | nan | | | (-oo)**-oo | | | +--------------+---------+-----------------------------------------------+ | oo**I | nan | oo**e could probably be best thought of as | | (-oo)**I | | the limit of x**e for real x as x tends to | | | | oo. If e is I, then the limit does not exist | | | | and nan is used to indicate that. | +--------------+---------+-----------------------------------------------+ | oo**(1+I) | zoo | If the real part of e is positive, then the | | (-oo)**(1+I) | | limit of abs(x**e) is oo. So the limit value | | | | is zoo. | +--------------+---------+-----------------------------------------------+ | oo**(-1+I) | 0 | If the real part of e is negative, then the | | -oo**(-1+I) | | limit is 0. | +--------------+---------+-----------------------------------------------+ Because symbolic computations are more flexible than floating point calculations and we prefer to never return an incorrect answer, we choose not to conform to all IEEE 754 conventions. This helps us avoid extra test-case code in the calculation of limits. See Also ======== sympy.core.numbers.Infinity sympy.core.numbers.NegativeInfinity sympy.core.numbers.NaN References ========== .. [1] https://en.wikipedia.org/wiki/Exponentiation .. [2] https://en.wikipedia.org/wiki/Zero_to_the_power_of_zero .. [3] https://en.wikipedia.org/wiki/Indeterminate_forms Tis_commutativeztuple[Expr, Expr]args_argsch |tj}t|}t|}ddlm}t ||s t ||r t d||fD]9}t |trtdt|jdddd ;|r|tjurtjS|tjurt|tj rtjSt|tj"r*t%|tj rtj&St%|tj"r:|j(rtjS|j(d urtjS|tj&urtj S|tj ur|S|d k(r|stjS|j*jd k(rJ|tj,k(rd dlm}|t3||j4t3||j6S|j8r |j:s |j<r^|j>r |j@s |jBr:|jEr*|jFr| }n|jHrt3| | Stj||fvrtjS|tj ur5tK|jLrtjStj Sd dl'm(}|jRs |tj,urt ||sddl*m+}d dl'm,} d dl-m.} ||d j_\} } | | \} }t || r(|j`d |k(rtj,| | zzS|jbrsd dl2m3}m4}|||}|jBrQ|rO|| ||d  |tjjztjlzzk(rtj,| | zzS|jo|}||Stjp|||}|js|}t |t2s|S|jtxr |jt|_:|S)Nr) Relationalz Relational cannot be used in Powzf Using non-Expr arguments in Pow is deprecated (in this case, one of the arguments is of type zf). 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Notes ===== Algorithms: 1. For unevaluated integer power, use built-in ``pow`` function with 3 arguments, if powers are not too large wrt base. 2. For very large powers, use totient reduction if $e \ge \log(m)$. Bound on m, is for safe factorization memory wise i.e. $m^{1/4}$. For pollard-rho to be faster than built-in pow $\log(e) > m^{1/4}$ check is added. 3. For any unevaluated power found in `b` or `e`, the step 2 will be recursed down to the base and the exponent such that the $b \bmod q$ becomes the new base and $\phi(q) + e \bmod \phi(q)$ becomes the new exponent, and then the computation for the reduced expression can be done. r)totientPrTr)ModFraN)rr@rt is_positiver rlsympy.ntheory.factor_rrur&r:IntegerpowmodrrcrMrv) rrrr@rrHrGrKmbphirr:s r+ _eval_Modz Pow._eval_Mods0IItxxc >>coo||qA vv 53>>alld)SXs1va1\\^8RALLNA,=,Bgaj/C"3q##+q#9::s1a|,, $$T^^4|3tS591==#s#3== V..0 #!!*CC -Cs4u=qAA$ r-c|jjr-|jjr|jjSyyr)r@rtrrrzrs r+ _eval_is_evenzPow._eval_is_even+s3 88  488#7#799$$ $$8 r-cPtj|}|dur |jS|SNT)rM_eval_is_extended_negativerm)rext_negs r+_eval_is_negativezPow._eval_is_negative/s(006 d?>> !r-cf|j|jk(r|jjryy|jjr|jjryy|jj r/|jj ry|jjryy|jjr-|jjr|jjSy|jjr|jjryy|jjr|jjr7|jdz}|jry|jr|jdury|jjr"ddl m}||jjSyy)NTFrTrr])rr@rris_realis_extended_negativerzr{is_zeroris_extended_nonpositiverrtr}r)rrKrs r+_eval_is_extended_positivezPow._eval_is_extended_positive5s< 99 yy001 YY " "xx YY + +xxxx YY  xx((xx''') YY . .xx YY # #xx""HHqL99<>88888?+D% YY TXX%9%9&: r-c|j\}}|jr|jdur |jry|jr8|jr,|tj ury|j s |jry|jrU|jrI|js |jr1t|dz jrt|dzjry|jr1|jr%|j|j}|jS|jr|jr|dz jry|jr|jr|dzjryyyy)NFTr)rP is_rationalrtrr rkrrrmrrrxfuncru)rrHrGchecks r+_eval_is_integerzPow._eval_is_integersyy1 ==||u$ <yy~~$)C)Cxx,,,yy~~$166)AdiimmF`F`xx,,, ** >  fyy--22txx7W7W$$)F)F$$)@)@//88'' dhh33 8I8IU8Rtyy488),== =yy%%xx$$ xx""88##XX__ #dii.55xx,,.1 1 diilUDDTDTU/AAHHQJ**e3 dyyAMM)q/A99((Q]]yy++Q0J0Jq||$DII&qtt+77>I U?v @DIItxx',A||||# &?r-c|jtjk(r5t|jj |jj gStd|jDr|jryyy)Nc34K|]}|jywr)r).0rs r+ z'Pow._eval_is_complex..s/q||/T) rr rorr@rrallrP_eval_is_finiters r+_eval_is_complexzPow._eval_is_complexs_ 99 TXX00$((2O2OPQ Q /TYY/ /D4H4H4J5K /r-c|jjdury|jjr1|jjr|jj }||Sy|jt jk(rLd|jzt jt jzz }|jry|j ryy|jjr%ddl m }||jj}|y|jjr|jjr{|jjry|jj}|s|S|jjryd|jzj}|r|jj S|S|jjdurJddlm}||j|jzt jz }d|zj } | | Syy)NFr$Trr]r)rrOrr@rtr{r rorrrzr}rrrrrrr) rrfrimlograthalfrrisodds r+_eval_is_imaginaryzPow._eval_is_imaginarys 99 # #u , 99 ! !xx""hhoo?J 99 DHH Q__ 45Ayyxx 88 B N//E  99 % %$((*C*Cyy$$hh**J88&& dhhJ22D#yy444K 99 % % . @DIItxx',AqSLLE  ! /r-c|jjrw|jjr|jjS|jj r|jjry|jt juryyyr)r@rtrrr{rr rkrs r+ _eval_is_oddzPow._eval_is_oddse 88  xx##yy'''((TYY-=-=amm+, r-c|jjrD|jjry|jjs|jj ry|jj }|y|jj }|y|r:|r7|jjst|jjryyyyr) r@rrrr|rrmrr)rc1c2s r+rzPow._eval_is_finites 88  yy  yy$$ (<(< YY  :  XX   :  "xx&&)DII4E4E*F+G2r-c|jjr2|jjr|jdz jryyyy)zM An integer raised to the n(>=2)-th power cannot be a prime. rFN)rrtr@rrs r+_eval_is_primezPow._eval_is_prime.s< 99  DHH$7$7TXX\syy!!!r-cddlm}t|j|rg|jj ||}|jj ||}t||r|j |S|j||Sddlm}m }d}||jk(s"||k(rz|jtjk(r]|jr2t|tr"||jj||S||jj||zSt||jrN|j|jk(r5||j|j} | jr t!|| St||jr|j|jk(r|jj"dur|jj%t&d} |jj%t&d} || | |\} } }| r@|j|| }| t)|t!|j|}|S|j}g}g}|j+} |jj,D]}|j||}|j+} || | |\} } }| r(|j/|| z||j/|]|j0s|j2sy|j/||rHt5|}|j/|dk7rt!|j|dn |jt)|St||s(|j6r|jtjur|jj8r|jj:r|jj%t&d} |j||jzj%t&d} || | |\} } }| r6|j|| }| t)|t!|j|}|Syyyyy) NrrYrc8|\}}|\}}||k(r|jr||z } t|dd}||dfSt|ts|f}td|Dsy tt|t|\}} |dkr| dk7r|dz }| t|z} | dk(rd} n t| g|} d|| fSy#t$rL|j\} } | jxr | j xs| j xr | j }YwxYw#t$rYywxYw) a*Return (bool, pow, remainder_pow) where, if bool is True, then the exponent of Pow `old` will combine with `pow` so the substitution is valid, otherwise bool will be False. For noncommutative objects, `pow` will be an integer, and a factor `Pow(old.base, remainder_pow)` needs to be included. If there is no such factor, None is returned. For commutative objects, remainder_pow is always None. cti are the coefficient and terms of an exponent of self or old In this _eval_subs routine a change like (b**(2*x)).subs(b**x, y) will give y**2 since (b**x)**2 == b**(2*x); if that equality does not hold then the substitution should not occur so `bool` will be False. FrTNc34K|]}|jywr)rt)rterms r+rz1Pow._eval_subs.._check..qsB4tBr)FNNrr) rOr r%rrrrrctuplerrFr) ct1ct2oldcoeff1terms1coeff2terms2rcombinesrHrG remainder remainder_pows r+_checkzPow._eval_subs.._checkMsH"!NFF NFF%% -Chs51#' $S$..&fe4"(B6BB0)/vv)OY7yA~1HC%7I$>,0M,/ ,CF,CM#S-77 %=&h"01#$==#>QYY#g!BRBRBgWXWgWgh4&$ s%B5AD 5AD  D  DDF)as_Addrr)rprZrcr@rsubs__rpow__rr}rr ro is_Functionr_subsrxrMras_independentSymbolr as_coeff_mulrPappendrOrtAddis_Powrr)rrnewrZrHrGr@rr%lrrrrr$resultoargnew_lo_alrnewaexpos r+ _eval_subszPow._eval_subsAsA dhh , sC(A c3'A![)zz!}$99Q? "C8 %t $)) s tyyAFF/B:c8#<488>>#s344DHHNN3444 c499 %$((cgg*=DIIsxx(A{{3{" c499 %$))sxx*?xx%'hh--fU-Cgg,,VE,B)/S#)>&C!YYsC0F$0!$VS=-I!J!Mww'') &A773,D++-C-3Cc-B*B] S#X.(4 KK 6  //KK% &:DLLQRTYYu!EX\XaXab;& sC SZZCHH4FTXXMfMfkoktktlAlA''(((>C88C N*::u;&C%+Cc%: "B]3, , SXX})EFF  lAMf4FZr-c|j\}}|jr0|j|jkr|jdkDrd|z | fS||fS)aReturn base and exp of self. Explanation =========== If base a Rational less than 1, then return 1/Rational, -exp. If this extra processing is not needed, the base and exp properties will give the raw arguments. 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