+ i^RIHt^RIHtHtHt^RIHt^RIH t H t H t H t H t ^RIHtHt^RIHtHt^RIHtHtRtR R R R /R ltR#))combinations_with_replacement)symbolsAddDummy)Rational)cancelComputationFailedparallel_poly_from_exprreducedPoly)Monomial monomial_div) DomainErrorPolificationFailed)debugdebugfc\V4P4wr\W.RRR7wr4\ T!\YB, 4,# \d Y, u#i;i)z Put an expression over a common denominator, cancel and reduce. Examples ======== >>> from sympy import ratsimp >>> from sympy.abc import x, y >>> ratsimp(1/x + 1/y) (x + y)/(x*y) TF)fieldexpand)ras_numer_denomr r r)exprfgQrs& v/Users/tonyclaw/.openclaw/workspace/skills/math-calculator/venv/lib/python3.14/site-packages/sympy/simplify/ratsimp.pyratsimpr s^ $< & & (DAq#T%8 7VAC[   s sA A#"A#quickT polynomialFc aaaaaaaa^RIHo\RV4\V4P 4wrg\ Wg.S,.VO5/VBwpoSPp T P'dT P4SnM\RT ,4hTR,U u.uFqPSP4NK up o\4oTTT3RloRTTTTTTT3Rllo\TSSPSPR7^,p\TSSPSPR7^,pT'dYg, P4#S!\!TSPSPR7\!TSPSPR7.4wrp S'g{T 'ds\#R \%T 44.pT FEwppppS!TTR R R 7pTP'TP)T4TP)T434KG \+TR R7wrT P,'g8T P/R R7wpp T P/R R7wpp \1TT4pM \1^4pT TP2,T TP4,, # \ dTu#i;iuup i)a Simplifies a rational expression ``expr`` modulo the prime ideal generated by ``G``. ``G`` should be a Groebner basis of the ideal. Examples ======== >>> from sympy.simplify.ratsimp import ratsimpmodprime >>> from sympy.abc import x, y >>> eq = (x + y**5 + y)/(x - y) >>> ratsimpmodprime(eq, [x*y**5 - x - y], x, y, order='lex') (-x**2 - x*y - x - y)/(-x**2 + x*y) If ``polynomial`` is ``False``, the algorithm computes a rational simplification which minimizes the sum of the total degrees of the numerator and the denominator. If ``polynomial`` is ``True``, this function just brings numerator and denominator into a canonical form. This is much faster, but has potentially worse results. References ========== .. [1] M. Monagan, R. Pearce, Rational Simplification Modulo a Polynomial Ideal, https://dl.acm.org/doi/pdf/10.1145/1145768.1145809 (specifically, the second algorithm) )solveratsimpmodprimez.Cannot compute rational simplification over %s:NNc0<aV^8Xd^.#.p\\\SP44V4Fp^.\SP4,oVFpSV;;,^, uu&K \;QJdV3RlS4F 'dK RM RM!V3RlS44'gKVP S4K VUu.uF&p\ V4P!SP!NK( upS!V^, 4,#uupi)zs Compute all monomials with degree less than ``n`` that are not divisible by any element of ``leading_monomials``. c3@<"TFp\SV4RJxK R#5iN)r).0lmgms& r 5ratsimpmodprime..staircase..bs$&$58<3'4/$sFT)rrangelengensallappendr as_expr) nSmiisr)leading_monomialsopt staircases & @rr9"ratsimpmodprime..staircaseVs 63J /c#((m0DaHBCM!A! s&$&sss&$&&& I9::1 ##SXX.:Yq1u=MMM:s,Dc<aaaaYre^pVP4VP4,pS'd V^, p MTp W4,V 8:EdEW43S9dEM:SPW434S!V4oS!V4o\RW4SS34\R\ S4,\ R7o\R\ S4,\ R7oSS,p \ \VV3Rl\\ S4444SPV ,4p \ \VV3Rl\\ S4444SPV ,4p \W ,W,, SSPV ,SPRR7^,p \ V SPR 7P4pS!VSS,RRR 7pV'Ed\;QJd*R VP44F 'dK R M RM!R VP444'Eg>V PV4pV PV4pVP\!\#\%SS,^.\ S4\ S4,,4444pVP\!\#\%SS,^.\ S4\ S4,,4444p\ VSP4p\ VSP4pV^8Xd \'R 4hVP)WVSS,34W4,V8wd VR,.pMV^, pV^, pV^, pEKRV^8d(S!WVW#WG, 4wrVpS!WVW#V, V4wrVpWVV3#)a Computes a rational simplification of ``a/b`` which minimizes the sum of the total degrees of the numerator and the denominator. Explanation =========== The algorithm proceeds by looking at ``a * d - b * c`` modulo the ideal generated by ``G`` for some ``c`` and ``d`` with degree less than ``a`` and ``b`` respectively. The coefficients of ``c`` and ``d`` are indeterminates and thus the coefficients of the normalform of ``a * d - b * c`` are linear polynomials in these indeterminates. If these linear polynomials, considered as system of equations, have a nontrivial solution, then `\frac{a}{b} \equiv \frac{c}{d}` modulo the ideal generated by ``G``. So, by construction, the degree of ``c`` and ``d`` is less than the degree of ``a`` and ``b``, so a simpler representation has been found. After a simpler representation has been found, the algorithm tries to reduce the degree of the numerator and denominator and returns the result afterwards. As an extension, if quick=False, we look at all possible degrees such that the total degree is less than *or equal to* the best current solution. We retain a list of all solutions of minimal degree, and try to find the best one at the end. z%s / %s: %s, %szc:%d)clszd:%dc3R<"TFpSV,SV,,xK R#5ir&)r'r5CsM1s& rr*._ratsimpmodprime..:>aBqEBqEMM>$'c3R<"TFpSV,SV,,xK R#5ir&r>)r'r5DsM2s& rr*rArBrCT)orderpolys)r. particularrc3*"TF q^8HxK R#5i)rNr>)r'r6s& rr*rAs<|!Av|sFzIdeal not prime?) total_degreeaddrrr-rr sumr,r.r rGcoeffsr/valuessubsdictlistzip ValueErrorr0)aballsolNDcdstepsmaxdegboundngc_hatd_hatrr3solr?rEr@rFG_ratsimpmodprimer8rr!r9testeds&&&&& @@@@rrf)ratsimpmodprime.._ratsimpmodprimehs:1!ANN$44 QJEEeunv JJv 1B1B $qRn 5#b')u5B#b')u5BbB:5R>::CHHrMKE:5R>::CHHrMKE AI-q#((R-!iit5568AQSXX&--/A27t4@Cs33!ratsimpmodprime..s'QqTZZ\):S1=N)Nri)key)convert)rr)sympy.solvers.solversr!rrrr rrjhas_assoc_Field get_fieldrLMrGsetr r.r rr-r0rRminis_Field clear_denomsrqp)rrerrr.argsnumdenomrHrjrr\r]rYnewsolrbrcr3rardcndnrrfr7r8r!r9rgs&fd$*, @@@@@@rr"r"s<<, T",,.JC,c\A-=MMM sZZF %%'  rirrs;3**'YY8B.!,$5ri