Jtj0ApdZddlZddlmZddlmZddlmZmZmZmZm Z m Z m Z dgdzZ e ddD]Zegdd ez zze dezddedzz<dAd Zd Zd ZedkrddlZejZejZdZedkrejdkrdZndZde dDZdZdZdZedkreZeZnedkr ejZeZeZneZeZedkrdeevrejZde dDZde dDZ dZ!dZ"de"fd Z#dde"fd!Z$dde"fd"Z%edkre%Z&ne$Z&dd#zZ'dd$zZ(dd%zZ)dd&zZ*d'Z+d(Z,d)Z-d*Z.d+Z/d,Z0d-Z1e1Z2edkr9ejdkrej3xZ4xZ5Z3ej6Z7n7ej8xZ4xZ5Z3ej7Z7n$edkre9ed.d/xZ4xZ5Z3d0Z7ne-Z4e.Z5e0Z3e/Z7ifd1Z:d2Z;ddd3fd4ZZZHd?ZId@ZJdS)Bzw Utility functions for integer math. TODO: rename, cleanup, perhaps move the gmpy wrapper code here from settings.py N)bisect)xrange)BACKENDgmpysage sage_utilsMPZMPZ_ONEMPZ_ZEROc~|g}|d||zkr!||d|zdzgz}|d||zk!|dddS)a Return a list of integers ~= [start, n*start, ..., target/n^2, target/n, target] but conservatively rounded so that the quotient between two successive elements is actually slightly less than n. With n = 2, this describes suitable precision steps for a quadratically convergent algorithm such as Newton's method; with n = 3 steps for cubic convergence (Halley's method), etc. >>> giant_steps(50,1000) [66, 128, 253, 502, 1000] >>> giant_steps(50,1000,4) [65, 252, 1000] rN)starttargetnLs U/var/www/html/CA-Chatbot/venv/lib/python3.11/site-packages/mpmath/libmp/libintmath.py giant_stepsrsV& A B%%'// 2A  B%%'// TTrT7Nc$|dkr||z S|| zS)zFor an integer x, calculate x >> n with the fastest (floor) rounding. Unlike the plain Python expression (x >> n), n is allowed to be negative, in which case a left shift is performed.rrxrs rrshiftr+! Avva1f}QBirc$|dkr||zS|| z S)zFor an integer x, calculate x << n. Unlike the plain Python expression (x << n), n is allowed to be negative, in which case a right shift with default (floor) rounding is performed.rrrs rlshiftr!2rrrc|sdS|dz}|r t|Sd}|dz}|dzs|dz}|dz }|dz|t|dzzS)z1Count the number of trailing zero bits in abs(n).rr)small_trailing)rlow_bytets rpython_trailingr'>s{ q4xH(h'' A!GA$h a Q$h ~a$h' ''rr2cL|r!t|SdSzr4Ys # # #1!Q$ # # #r,ctt|}|dkr|Sttj|ddz }|t ||z zS)0Calculate bit size of the nonnegative integer n.r5r)rpowersintmathlogbctable)rbcs rpython_bitcountr?[sN   B Syy TXa^^  q B 2 rcN|r"t|dSdS)r7rr)r numdigitsr,s r gmpy_bitcountrBcs(Q!!!$$ $rcDt|SN)r trailing_zero_bitsr,s r sage_trailingrFls q66 $ $ & &&r bit_lengthc,g|]}t|Sr)trailingr2rs rr4r4~s . . .ahqkk . . .rc,g|]}t|Sr)bitcountrJs rr4r4s , , ,18A;; , , ,ric2|t||zz|z S)zaChanges radix of a fixed-point number; i.e., converts x * 2**xbits to floor(x * 10**bdigits).r )rxbitsbasebdigitss r bin_to_radixrRs D 7" #u ,,r$0123456789abcdefghijklmnopqrstuvwxyz c|dkrt|Sg}|r0t||\}}||||0d|dddS)ziReturn the string numeral of a positive integer in an arbitrary base. Most efficient for small input.rTNr)strdivmodappendjoin)rrPdigitsdigsdigits r small_numeralr^st rzz1vv D #!T??5 F5M""" # 774":  rc,|dkr|sdSdt| |||zS|dkrt|||S|dz|dzz}t|||z\}}t||||}t|||||d}||zS)_Represent the integer n as a string of digits in the given base. Recursive division is used to make this function about 3x faster than Python's str() for converting integers to decimal strings. The 'size' parameters specifies the number of digits in n; this number is only used to determine splitting points and need not be exact.r0-rr)numeralr^rXrjust rrPsizer[halfABadbds rnumeral_pythonrms Avv 3WaRtV4444 czzQf--- AI$( #D !T4Z DAq D$ ' 'B D$ ' ' - -dC 8 8B 7NrcF|dkrdt| |||zS|dkrtj||S|dz|dzz}t|t ||z\}}t||||}t|||||d}||zS)r`rrbi`rrra)rdrr[rXr rerfs r numeral_gmpyros 1uuWaRtV4444 g~~{1d### AI$( #D !SYY_ % %DAq D$ ' 'B D$ ' ' - -dC 8 8B 7Nri iXill c*|s|S|tkr6|tkrt|dzSt|dzdzdz}n8t|}|dz}t|d|zdz z dzdz|dz z} |||zzdz }||kr|S|})zd Correctly (floor) rounded integer square root, using division. Fast up to ~200 digits. ?g-?rrd2)_1_800_1_50r:rL)rrr>rys risqrt_small_pythonrys 6zz u99q#v;;  3)) * *Q . a[[ E QqSW#A% & &2 . q!tVaK 66H  rc,|tkrVt|dz}|tkr7|||zzdz }|tkr!|||zzdz }|tkr |||zzdz }|St |}d}|d|zz}|d|zz }||dzz }|dz}t d|}tdd|zz||d|zz z dzz}|}t||D]1}||zd|z|z z } |||z z | z|z } |d|z| z z|dzz }|}2|||z z||zz S) a Fast approximate integer square root, computed using division-free Newton iteration for large x. For random integers the result is almost always correct (floor(sqrt(x))), but is 1 ulp too small with a roughly 0.1% probability. If x is very close to an exact square, the answer is 1 ulp wrong with high probability. With 0 guard bits, the largest error over a set of 10^5 random inputs of size 1-10^5 bits was 3 ulp. The use of 10 guard bits almost certainly guarantees a max 1 ulp error. rrrrTrrtg@g)rur:_1_100_1_200_1_400rLminr) rrxr> guard_bitshbc startprecrwpppr2xr2s risqrt_fast_pythonrso$ 6zz 3KK ;;QTaAF{{AX!O;;QTaA !BJ!J,A!J,B2a4LB a%CB I C!I+ !1Y;"7D!@ @AAA B C ( (cqtax bd r!a' 1a43, RT *  q#vJAjL ))rc|tkrt|}||||zz fSt|dz}|||zz }|dkr|dz}|dd|zzz }|dk|r(|dd|zzkr|dz }|dd|zzz}|dd|zzk||fS)z=Correctly rounded integer (floor) square root with remainder.rrr)_1_600ryr)rrxrems rsqrtrem_pythonrs 6zz q ! !!ac'z!q A ac'C '' Q !A# ''  1Q3--Q!A#1Q3-- c6Mrc,t|dS)z2Integer square root with correct (floor) rounding.r)r)rs r isqrt_pythonr+s !  Q rc&t||zSrD) isqrt_fast)rprecs r sqrt_fixedr/s ag  risqrtcDt|SrD)r rr,s rr=ss1vv||~~rcDt|SrD)r sqrtremr,s rrr>sA((rcD|dkrd| dzzt| zS||vr||S|}ttttf\}}}}|rE|dzr!||z}||z|z||zz||z|z}}|dz}n||z}||z|z|d|z|zz}}|dz}|E|dkr|||<|S)zCComputes the nth Fibonacci number as an integer, for integer n.rrrrrc)ifibr r ) r_cachemabrqaqqqs rrrFs 1uuqbd|dA2hh&&F{{ay A(Hg5JAq!Q  q5 1BQ3r6!A#:qs2vqA FAA1BQ3r62ac!e8qA !GA  3wwq Hri)rrc||}|r|St|}||dz }t}||kr||z}||kr|||<|dz }||k|S)z.Return n factorial (for integers n >= 0 only).r)getlenMAX_FACTORIAL_CACHE)rmemofkrMAXs rifacrasz  A D A QqS A C q&& Q 88DG Q q&& Hrc||dz}||}|r|St|}||}t}||kr|dz }||z}||kr|||<||k|S)z4Return n!! (double factorial), integers n >= 0 only.rr)rmaxr)r memo_pairrrrrrs rifac2rps QqS>D  A D A QA C a%% Q Q 88DG a%% HrcDttj|SrD)r:r factorialr,s rrrsS**++rc|dz}tt|}ddg|dd<tdt|dzdzD]&}||rt|dz||D]}d||<'d|DS)Nrrrrrcg|]}||Srr)r2rs rr4zlist_primes..s " " "! "A " " "r)listrr:)rsieveijs r list_primesrs AA OOEAE"1"I As1c6{{1} % % 8 AqD!Q''  a " "u " " ""rcDdtj|dzDS)Nc,g|]}t|Sr)r:r1s rr4zlist_primes..s1111A111rr)rprimesr,s rrrs$11 AaC 0 01111r)r{r %)+/c&tdzsdkSdkr tvStD] }|zsdS dz tz fd}dkrddg}ndkrgd }nt}|D]}||sdSd S) a& Determines whether n is a prime number. A probabilistic test is performed if n is very large. No special trick is used for detecting perfect powers. >>> sum(list_primes(100000)) 454396537 >>> sum(n*isprime(n) for n in range(100000)) 454396537 rrrtFct|}|dks|krdStdD]}|dzz}|krdSdS)NrTrF)powr)rrrwdrrss rtestzisprime..testsf !AJJ 66Q!VV4!  A1qAAvvtturir{lHe%Z )rr{rrrrrT)r:small_odd_primes_setsmall_odd_primesrI)rrr witnessesrrrrs` @@@risprimers  AA q5Av 2vv((( 1u 55  !A A QA 7{{qE _  &&& $ tAww 55  4rc tt|}|dkr|Sg}td|dzD]B|zs;|dzzsdStfd|Ds|Cdt |zS)z Evaluates the Moebius function which is `mu(n) = (-1)^k` if `n` is a product of `k` distinct primes and `mu(n) = 0` otherwise. TODO: speed up using factorization rrrc3"K|] }|zV dSrDr)r2rrs r zmoebius..s'..q1u......rr)absr:rsumrYr)rfactorsrs @rmoebiusrs CFF A1uuG Aqs^^""A "1H qq....g..... "q!!! W rc4d}|D]}|r |r |||z}}| |}|S)Nrr)argsrrs rgcdrsL A    !a%1 AA Hric|dzrtS||}|r|St}|}ddD}td|dzD]}t|dzddD](}|dz ||z|dz||dzzz||dz<)|dd}t|dzddD]*}|||dzz }||krd|dzz|d|zzz||<+||krd|dzz|zd|zzcSdS) a Computes the Euler numbers `E(n)`, which can be defined as coefficients of the Taylor expansion of `1/cosh x`: .. math :: \frac{1}{\cosh x} = \sum_{n=0}^\infty \frac{E_n}{n!} x^n Example:: >>> [int(eulernum(n)) for n in range(11)] [1, 0, -1, 0, 5, 0, -61, 0, 1385, 0, -50521] >>> [int(eulernum(n)) for n in range(11)] # test cache [1, 0, -1, 0, 5, 0, -61, 0, 1385, 0, -50521] rc,g|]}t|SrrNr1s rr4zeulernum..s'''AQ'''r)rrrrrrrrrN)r rMAX_EULER_CACHErangerY) rrrrrrrsumars reulernumrs^$ 1u 1 A C A'''''AAqsmm / /qsB## / /Ac1Q4Z1Q3!A#,.AacFF  qsB## : :A AacFNDCxx AqD\DAqDL9q 661a4L$&!Q$. . . .  / /rcp|dks|dkrt||krt||kS|dkrtStg|dzz}t|d<t d|dzD]A}t t ||ddD]}|dz ||z||dz z||< Bd||zz||zS)z, Stirling number of the first kind. rrrr) ValueErrorr r r rr)rrrrrs r stirling1r%s 1uuAAvv16{{1uu  acA AaD Aqs^^))Aq 1b)) ) )AaC1Q4rsJ LLLLLLLLLLLLLLLLLLs q66A&'SA!H%5N1a4>QqS>""0       fOOO _F _F ( ( ( ft|~~         $ #c # # #''' fHHH 'MHHHHH fT22H/ .55:: . . . , ,d , , ,--- 3 Y    Ai,!I, fGGG C C C C  4.*.*.*`(     ft|~~+/:5 5j5.+/94 4j5,  G%=%=>>?K?*u((GG$K"J EG    2     1!u~     f 8DD  + +D >D### f222<s+,,***X&   J'{$/$/$/$/L"r