[hݯ SrSrSSKrSSKJr SSKrSqSSKJrJrJ r J r J r J r J r JrJrJrJrJrJr SSKJrJrJrJrJrJrJrJrJrJrJrJ r J!r! \ S:XaS r"OS r"S r#"S S \$5r%\& \&"S5r(\&"S5r)\&"S5r*\&"S5r+\&"S5r,\,r-Sr.Sr/Sr0S\ SS4r1S\ SS4r2S\ SS4r3S\ SS4r4S\ SS4r5S\ SS4r6S\ SS4r7S\ SS4r8S\ SS4r9S\ SS4r:Sr;Sr<"S S!5r=S/\>"SS"5Vs/sHn\ US- -S- PM sn-r?\="5\?/r@\)S#\*S$\,S%\+S&0rAS'rBS(rC\D\E4rFS)rGS*rH\ S+:Xa$S,\I"\5;a\RrB\RrC\ S:Xa\R=rBrC\(a\GrK\HrLO\BrK\CrLS\-4S-jrM\N"S.\>"S/S0555rO\ S+:XaS1\I"\5;a \RrM\ S:Xa \RrMS\-4S2jrQS3rRShS4jrSS5rTS\-4S6jrUS\-4S7jrVS\-4S8jrWS\-4S9jrXS:\-4S;jrYS<\-4S=jrZS\-4S>jr[S?\-4S@jr\\-4SAjr]SBr^SCr_SDr`SEraSFrbSGrcSHrdSIreSJrfSKrgSLrhS\-4SMjriS\-4SNjrjS\-4SOjrkSPrlS\-S4SQjrmS\-4SRjrnS\-S?4SSjroS\-4STjrp\-4SUjrqS\-4SVjrr\-4SWjrs\ S+:Xa\prt\qruO\rrt\sruSXrvSYrw\-4SZjrx\-4S[jry\-4S\jrz\,\+\+\,\)\*\*\)\(\(0r{\,\,\+\+\)\*\*\)\(\(0r|\-4S]jr}S^r~S_rSiS`jrSjSajr\9\9\:\8Sb.r\-4ScjrSdrSer\-4Sfjr\-4Sgjr\ S:XaKSSKJs Js Jr \Rrm\Rrn\Rrt\Rrx\GR rgg!\'a Sr&GNf=fs snf!\'a \D4rFGNDf=f!\a gf=f)kzH Low-level functions for arbitrary-precision floating-point arithmetic. plaintextN)bisect) MPZMPZ_TYPEMPZ_ZEROMPZ_ONEMPZ_TWOMPZ_FIVEBACKENDSTRICT HASH_MODULUS HASH_BITSgmpysage sage_utils) giant_steps trailtablebctablelshiftrshiftbitcounttrailing sqrt_fixednumeralisqrt isqrt_fastsqrtrem bin_to_radixrc(Uupp4U[U5X44$Nhexxsignmanexpbcs K/var/www/auris/envauris/lib/python3.13/site-packages/mpmath/libmp/libmpf.py to_pickabler+s3SXs&&c.Uupp4U[U5SSX44$)Nr"r$s r*r+r+#s#3SXab\3**r,c*Uupp4U[US5X44$)N)rr$s r* from_pickabler1'sDs #c2, ((r,c\rSrSrSrg) ComplexResult+N)__name__ __module__ __qualname____firstlineno____static_attributes__r5r,r*r3r3+sr,r3cU$r!r5)r%s r*r<1sqr,nfcudc \[S[[[U5S- 5S- 55$)zVReturn number of accurate decimals that can be represented with a precision of n bits.rry O @maxintroundr=s r* prec_to_dpsrI;s* q#eCF#556q89 ::r,c \[S[[[U5S-S-555$)zFReturn the number of bits required to represent n decimals accurately.rrCrDrHs r* dps_to_precrK@s) q#eSVAX'99:; <*A     ax  c r,cU(d[$X4::aXX#4$X4- nU[:XaAXS- - nUS-(a*US-(dU[US:U-(a US- S-nO#US- nO[UU(aX-nOU*U- *nX&- nUnUS-(db[[ US-5nU(d:US-(dUS-nUS- nUS-nUS-(dM[[ US-5nX-nX'- nX7-nUS:XaSnXX#4$)zHsame as normalize, but with the added condition that man is odd or zero rr.rVrhrirjrms r* _normalize1rqs%   z#!! A m aCL q5q1u#qu a(8"8a4(CQ$C S $   lHC B 7 s39~ &Ci qaCii3sSy>*A     ax  c r,c[U5[:Xde[U5[;de[U5[;deU[U5:Xde[ XX#XE5$)sAdditional checks on the components of an mpf. Enable tests by setting the environment variable MPMATH_STRICT to Y.)typer _exp_typesrror&r'r(r)rnr^s r*strict_normalizerws[ 9  8z !! ! 9 "" " #   d$ 44r,c[U5[:Xde[U5[;de[U5[;deU[U5:XdeU(a US-(de[ XX#XE5$)rsr)rtrrurrqrvs r*strict_normalize1rysj 9  8z !! ! 9 "" " #  q! ! t#4 55r,r_mpmath_normalizec[U5nSnUS:aSnU*nUS:a[[U5nO [U5nU(dU(d[$US-(dzUS-(aX@S- US-US- 4$[ [US-5nU(d:US-(dUS-nUS- nUS-nUS-(dM[ [US-5nX-nX- nXV-nX@X4$[ X@XX#5$)z~Create raw mpf from (man, exp) pair. The mantissa may be signed. If no precision is specified, the mantissa is stored exactly.rrr.rhri)rrrFrrkr normalize)r'r(rnr^r&r)r_s r* from_man_expr~#s c(C D Qwd Tz SX  c] LQwQwQhaa883sSy>*A)AIC1HC!GB))s39~. IC HC GB3## T 33r,c#<# UHo[US54v M g7f)rN)r~).0r=s r* rBsB/Q\!Q'(/sii_mpmath_createcPU(dU[;a [U$[USX5$)z_Create a raw mpf from an integer. If no precision is specified, the mantissa is stored exactly.r) int_cacher~)r=rnr^s r*from_intrJs'  >Q<  1d ((r,cJUupp4U(dU(a[SU-5eX#4$)z:Return (man, exp) of a raw mpf. Raise an error if inf/nan.z*mantissa and exponent are undefined for %s ValueErrorsr&r'r(r)s r* to_man_exprRs)Ds SEKLL 8Or,cUup#pEU(dU(a [S5eUS:aU(aU*U-$X4-$U(dU(aX4*- *$X4*- $U(a[U*U*U5$[X4*U5$)zConvert a raw mpf to the nearest int. Rounding is done down by default (same as int(float) in Python), but can be changed. If the input is inf/nan, an exception is raised.z cannot convert inf or nan to intr)rrY)rr^r&r'r(r)s r*to_intrYsDs S;<< ax DS= z  T]# #4=  #tS))dC((r,cbUup#pEU(d U(aU$US:aU$XE-nUS:asU[:XaU(a[$[$U[:XaU(a[$[$U[ :Xa)US:d U[ :Xa[$U(a[$[$[e[U[XV5U5$Nrr) r[rkfonerZfnonerWr NotImplementedErrormpf_posmin)rr^r&r'r(r)mags r* mpf_round_introsDs S ax &C Qw - E\ [ K E\!\ M !Qw#.,e|"{% % 1c"lC ((r,cL[U[5nU(a [X1U5nU$r!)rrZrrrnr^vs r* mpf_floorrs"a%A AS ! Hr,cL[U[5nU(a [X1U5nU$r!)rr[rrs r*mpf_ceilr"a'A AS ! Hr,cL[U[5nU(a [X1U5nU$r!)rrWrrs r*mpf_nintrrr,c.[U[U5X5$r!)mpf_subr)rrnr^s r*mpf_fracrs 1ilD ..r,5c(X:wa[$[R"U5up4U[:Xa[$U[*:Xa[ $[ [US-5US- X5$! U[:Xa[s$U[*:Xa[ s$[s$=f)zCreate a raw mpf from a Python float, rounding if necessary. If prec >= 53, the result is guaranteed to represent exactly the same number as the input. If prec is not specified, use prec=53.lr)fnanmathfrexpmath_float_inffinffninfr~rF)r%rnr^mes r* from_floatrs  v zz!}  N4K^OE\ AuI"d 88   t   sA""B6B Bqct[U5nX:Xa [X1U5$SSKnURU5(aEUR U5upV[ [ URUS55[ US- 5X5$URU5(a[$URU5(a[$[$)zCreate a raw mpf from a numpy float, rounding if necessary. If prec >= 113, the result is guaranteed to represent exactly the same number as the input. If prec is not specified, use prec=113.rNr) floatrnumpyisfiniterr~rFldexpisposinfrisneginfrr)r%rnr^ynprrs r* from_npfloatrs aAv!3'' {{1~~xx{CC 013qu:tII {{1~~d{ {{1~~e| Kr,c$UR5(a[$UR5(a!UR5(a[$[ $Uc([ [UR5S5S-5n[[U5X5$)zCreate a raw mpf from a Decimal, rounding if necessary. If prec is not specified, use the equivalent bit precision of the number of significant digits in x.rrC) is_nanr is_infinite is_signedrrrFlenas_tuplefrom_strstr)r%rnr^s r* from_Decimalrsg xxzz$;}} u?4? |3qzz|A'(::; CFD &&r,FclUup4pVU(d9U[:XagU[:Xa[$U[:Xa[*$[[- $US:a[ X4XVSU5up4pVU(aU*n[ R "XE5$![a+ U(aeXV-S:aU(a [*s$[s$gf=f)a Convert a raw mpf to a Python float. The result is exact if the bitcount of s is <= 53 and no underflow/overflow occurs. If the number is too large or too small to represent as a regular float, it will be converted to inf or 0.0. Setting strict=True forces an OverflowError to be raised instead. Warning: with a directed rounding mode, the correct nearest representable floating-point number in the specified direction might not be computed in case of overflow or (gradual) underflow. grr)rkrrr normalize1rr OverflowError)rstrictr^r&r'r(r)s r*to_floatrsDs  :c 9^+ :~o-n,, Bw'3BD3 d zz###     8a<&&%% s(A>>(B3(B32B3c@[[U5[U5X#5$)z@Create a raw mpf from a rational number p/q, round if necessary.)mpf_divr)pqrnr^s r* from_rationalrs 8A; T 77r,cxUupp4U(aU*nUS:Xa[SU-5eUS:a USU--S4$USU*-4$)zYConvert a raw mpf to a rational number. Return integers (p, q) such that s = p/q exactly.rRz&cannot convert %s to a rational numberrrrrs r* to_rationalrsZDs d RxACGHH axaf~q  AI~r,cfUup#pEXA-nU(aUS:aU*U-$U*U*- $US:aX6-$X6*- $)z.Convert a raw mpf to a fixed-point big integerrr5)rrnr&r'r(r)offsets r*to_fixedrsQDs ZF Q;//!$6'22 Q;s},"w//r,cp[(dSSKnURq[[U5U*U[5$)zMReturn a raw mpf chosen randomly from [0, 1), with prec bits in the mantissa.rN) getrandbitsrandomr~rZ)rnrs r*mpf_randrs/ ;((  D)D5$ DDr,c\US(a US(dU[:Xd U[:XagX:H$)z~Test equality of two raw mpfs. This is simply tuple comparison unless either number is nan, in which case the result is False.rF)rrr_s r*mpf_eqrs( Q4qt 9T  6Mr,c([RS:aUupp4U(dmU[:Xa[RR$U[ :Xa[RR $U[:XaR *$U[-nUS:a U[-nO[S- SU- [-- nXS-[-nU(aU*nUS:XaSn[U5$[[USS95$![a [U5s$f=f)N)rOr.rrrRrS)r)sys version_infor hash_infonanrinfrrrrFhashrr)rssignsmansexpsbchs r*mpf_hashr!s 6!!"TDy!2!22Dy!2!22Ez3==#4#4"44 <  19)#Dq=R$Y)$; t. (Same convention as Python's cmp() function.)rrrR)rkmpf_signrrrrrZ) rr_rrrrtsigntmantexptbcabdeltas r*mpf_cmpr>sEE t :x{l* :hqk) 6! 9Q 9Q :a ~Q | < ;RQ A A 5 55 5 5 A![ )E Qx r,cHU[:Xd U[:Xag[X5S:$NFrrrrs r*mpf_ltrr!DyAI 1=1 r,cHU[:Xd U[:Xag[X5S:*$rrrs r*mpf_lerw!DyAI 1=A r,cHU[:Xd U[:Xag[X5S:$rrrs r*mpf_gtr|rr,cHU[:Xd U[:Xag[X5S:$rrrs r*mpf_gerrr,cxUS=pUSSH)n[X15(aUn[X25(dM'UnM+ X4$r)rr)seqrrEr%s r* mpf_min_maxrsAAC W !>>3 !>>3 8Or,cXU(a"Uup4pVU(d U(aU$[X4XVX5$U$)zLCalculate 0+s for a raw mpf (i.e., just round s to the specified precision).)rrrnr^r&r'r(r)s r*rrs/ 3H$Sd88 Hr,cUup4pVU(d)U(a U[:Xa[$U[:Xa[$U$U(dSU- XEU4$[SU- XEXaU5$)zNegate a raw mpf (return -s), rounding the result to the specified precision. The prec argument can be omitted to do the operation exactly.r)rrrrs r*mpf_negrsXDs  Dy,Ez$; $"%% afc# 66r,cUup4pVU(dU(aU[:Xa[$U$U(dU(aSXEU4$U$[SXEXaU5$)zReturn abs(s) of the raw mpf s, rounded to the specified precision. The prec argument can be omitted to generate an exact result.r)rrrrs r*mpf_absrsMDs S :K  s$ $ a2S 11r,cRUupp4U(dU[:XagU[:XaggSU-$)zReturn -1, 0, or 1 (as a Python int, not a raw mpf) depending on whether s is negative, zero, or positive. (Nan is taken to give 0.)rrRr)rrrs r*rrs0Ds  9Q :b 4<r,cUupVpxUuppX-n U(GaU (GaX{- n U (GaYU S:aU S:aKU(aDX-U - U - nXS-:a2US-n Xm-nX:XaUS- nOUS-n[XVX}- [U5X#5$XY:XaXU --nO$U(aXU -- nOXm-U - nUS:aSnOU*nSn[U5n[X_U UU=(d UU5$U S:aU S:aKU(aDX-U- U- nXS-:a2US-n X-n XY:XaU S- n OU S-n [XX- [U 5X#5$XY:Xa XjU *--nO&U (a XjU *-- nOX*-U- nUS:aSnOU*nSn[U5n[X_UUU=(d UU5$XY:XaX-nOU(aX- nOXj- nUS:aSnOU*nSn[U5n[X_U UU=(d UU5$U(a [U5nU(dBU(aX:XdU (dU (dU$[$U (a[XXU=(d U U5$U$U (aU$U(a[XVXxU=(d UU5$U$)z Add the two raw mpf values s and t. With prec=0, no rounding is performed. Note that this can produce a very large mantissa (potentially too large to fit in memory) if exponents are far apart. rdri)rrr}rr)rr_rnr^_subrrrrrrrrrrr'r)s r*mpf_addr sEE ME  zC4194+/194)%t{$TND 77>&.1CDFN$;c%)^t$;cax !"d !c]!%dB CHH!D=TJ,t3Eax'!% >4194+/194)%t{$TND 77>6'/2CDVGO$+CDKc Kcaxd c]T2tzr3?? AJ  vTK e4dkc3G G  %t$+#sCC Hr,c[XX#S5$)ztReturn the difference of two raw mpfs, s-t. This function is simply a wrapper of mpf_add that changes the sign of t.r)r )rr_rnr^s r*rrs 1A &&r,cSnSnUS-=(d SnSnUHnUuppU (asU (a U(dU *n X- n X:a6X:a(U(aU [[U55- U:aU nU nMVXJU -- nM_U *n X- U:a U(dXpTMuMwXM-U -nU nMU (dMU(a [U5n[U=(d [US5nM U(aU$[ XEX5$)a" Sum a list of mpf values efficiently and accurately (typically no temporary roundoff occurs). If prec=0, the final result will not be rounded either. There may be roundoff error or cancellation if extremely large exponent differences occur. With absolute=True, sums the absolute values. rr.i@BNr)rabsrr rkr~)xsrnr^absoluter'r(max_extra_precspecialr%xsignxmanxexpxbcrs r*mpf_sumr"s C C!V&wNG !"T XuJE{*%S(:":^"KCCEM*C9~-#'S<4/CC TAJg.15G7: $ ,,r,cUupEpgUuppXH- n XY-n U (a)[U 5nU(a[XXj-XU5$XXj-U4$U(+=(a UnU (+=(a U nU(d U(d[$[X4;a[$U (d U (aXpU[:Xa[$[[ S.[ U5[ U5-$)Multiply two raw mpfsrrR)rrrkrrrrrr_rnr^rrrrrrrrr&r'r) s_special t_specials r* gmpy_mpf_mulrRsEE =D )C c] dBcB Bty"- -#tI#tI Y  v~d{ d1qEz$;u hqkHQK7 88r,cUupEpgU(d[U[U5X#5$U(d[$US:aUS-nU*nXQ-n[XEU[ U5X#5$)Multiply by a Python integer.rr)mpf_mulrrkr}rrr=rnr^r&r'r(r)s r*gmpy_mpf_mul_intr"gs]Ds q(1+t11  1u   BHC TXc]D >>r,cUupEpgUuppXH- n XY-n U (a5X{-S- nU[X- 5- nU(a[XXj-XU5$XXj-U4$U(+=(a UnU (+=(a U nU(d U(d[$[X4;a[$U (d U (aXpU[:Xa[$[[ S.[ U5[ U5-$)rrr)rFrrkrrrrrs r*python_mpf_mulr$tsEE =D )C Y] c#'l dBcB Bty"- -#tI#tI Y  v~d{ d1qEz$;u hqkHQK7 88r,cUupEpgU(d[U[U5X#5$U(d[$US:aUS-nU*nXQ-nUS:aU[[ U5S- - nOU[ U5S- - nU[ XW- 5- n[ XEXgX#5$)rrrr|)r rrkrrFrr}r!s r*python_mpf_mul_intr&sDs q(1+t11  1u   BHC4x gc!fo!! hqkAo#cg,B T 33r,c,Uup#pEU(dU$X#XA-U4$)z8Quickly multiply the raw mpf s by 2**n without rounding.r5)rr=r&r'r(r)s r* mpf_shiftr(s#Ds  ceR r,cpUupp4U(dU[:Xa[S4$[e[X*U- 5XC-4$)z?Convert x = y*2**n to (y, n) with abs(y) in [0.5, 1) if nonzeror)rkrr(r$s r* mpf_frexpr*s>Ds  :1:   QC "& ((r,cUupEpgUuppU(aU (dU[:Xa&U[:Xa[eU[:Xa[$[$U[:Xa[eU(+=(a Un U (+=(a U n U (a U (a[$U[:Xd U[:Xa[$U (d5U[:Xa[$[[S.[ U5[ U5-$[$XH- nU S:Xa[ XXj- XrU5$X'- U -S-nUS:aSn[X_-U 5unnU(a*US-S-nUS- n[ UUXj- U- [U5X#5$[UUXj- U- [U5X#5$)zFloating-point divisionrrr) rkZeroDivisionErrorrrrrrdivmodrr})rr_rnr^rrrrrrrrrrr&extraquotrems r*rrsVEE t :Ez!22Dy+L :# #X'4 X'4 K 9T KEz u%hqkHQK&?@ @ =D qy$diC@@ J q E qyt{D)ID# a1}  $diox~tQQ T45(4.$ LLr,c2UupEpgU(aU(d[[U5XU5$US:aUS-nU*nX'-S-n[X-U5upU (a(U S-S-n US- n[XIU*U- [ U 5X#5$[ XIU*U- [ U 5X#5$)z>Floating-point division n/t with a Python integer as numeratorrrr)rrr-rrr}) r=r_rnr^r&r'r(r)r.r/r0s r* mpf_rdiv_intr2sDs Cx{AS111u   B IMEqx%ID a1}  $sd5j(4.$LL T#eXd^T GGr,c>UupEpgUuppU(dU(dU (d U (a[$XH:Xa XU-:aU$U S:XaXjU -:a[$[Xj5n SU-U-nSU-U -n XVU - -XU - --n U S:aSnOU *n Sn[XU [ U 5X#5$)NrrRr)rrkrr}r)rr_rnr^rrrrrrrrbaser'r&s r*mpf_modr5sEE td  ~$c/ qyTH_ t?D ; D ; D I 4I#6 7C axd ThsmT ??r,cUupEpgU(deU(a^U[:XaUS:aU$US:Xa[$[$U[:Xa*US:a[[/US-$US:Xa[$[$[$[ U5nUS:Xa[ $US:Xa [ XU5$US:XaWUuppgU(d[$XU-nUS:Xa S[Xf-S4$Xw-S- nU[[ XW- 5- n[SXVU-XrU5$US:Xa[[ XU5$US:a)[X*US-[U5n [[ XU5$XA-n US:Xa U [Xa-S4$Xq-S:aXQ-n[XXa-[U5X#5$U[:H=(d [UU n US[U5--S-n [ uppUS-(aYX-n X-nXS- - nU[[ X- 5-nX:a!U (aXU - - n O U *X- - *n XU - - nU nUS-nU(dOSXU-nXf-nXw-S- nU[[ XW- 5-nX|:a!U (aXWU - - nO U*X|- - *nXgU - - nU nUS-nM[!XXX#5$)z=Compute s**n, where s is a raw mpf and n is a Python integer.rrr.rRrir)rrrkrrFrrr rrr mpf_pow_intreciprocal_rndrrWrlr})rr=rnr^r&r'r(r)_inverse result_sign rounds_downworkprecpmpepbcs r*r7r7sDs S 91uQhAvd{L :1udE]1q511Avd{L  AAAvd{Avgas++AvLg !8w+ + Wq[ gc#'l##!Sc'2S99BwwtQc221uaT!V^C-@AtWC00(K axWceQ// td{ +CE8C=$LL-'&C% a m#a'HNA2 q5BB 6MCBI//C~H -BCS\23BHn$ FAgg Wq[ '#ci.) ) =k*2;/0 = CB F7 : [bt 99r,cU[:Xa [XU5$UupEpgU[Xg-U- S- S4nU(aU[[4;U- n OU[ [4;U- n U (a [ XX#5$[XU5$)z For nonzero x, calculate x + eps with directed rounding, where eps < prec relatively and eps has the given sign (0 for positive, 1 for negative). With rounding to nearest, this is taken to simply normalize x to the given precision. r)rWrr r\r[r]r ) r%eps_signrnr^r&r'r(r)epsaways r* mpf_perturbrEfs mq$$Ds WcfTk!mQ /C  M22h>-00H< qt))q$$r,cUS(aSn[U5nOSnUup4pVU(dg[U[R"SS5-5S-nXV-n[ U5S:a{SS KJn Jn [[ U55S -n [U5n [X"U 55n [X"U 5U 5n [U 5n [U[[X5U5nUup4pVU nOSn[Xu- U- S5n[U[R"SS5- S -5n[!X5n[#UUSU5n[%USUS 9nU['U5U- S- - nUUU4$) a Helper function for representing the floating-point number s as a decimal with dps digits. Returns (sign, string, exponent) where sign is '' or '-', string is the digit string, and exponent is the decimal exponent as an int. If inexact, the decimal representation is rounded toward zero.r-)rH0r r.i r)mpf_ln2mpf_ln10rg?)r4size)rrFrlogr  libelefunrKrLrrr rrr7ftenrErrrr)rrPr&_signr'r(r)bitprec exp_from_1rKrLexpprectmprexponentfixprecfixdpssfsddigitss r* to_digits_expr\sX t AJE #A&'",GJ :03s8$q(smc77+,c8G,g6 3K A{44g >C '-"$a(G 488Bq>)C/ 0F ! B b'2v .B Rbs +F F f$q((H  !!r,cUS(dKU[:XaU(aSnOSnU(aUS- nU$U[:XagU[:XagU[:Xag[eUc[ US -*S 5nUcUn[ XS -5upxn U(dUS S ;aU S- n SnGO[U5U:awXS ;aoUSUnUS- n U S :aXS :XaU S-n U S :a XS :XaMU S :a+USU [[X5S-5-SX- S- --nOSSUS- --nU S- n OUSUnX9s=:aU:a5O O2U S :aS[U *5-U-nSn OU S-n X:a USX- -- nS n OSn USU S-XS-nU(aURS5nUSS:XaUS- nU S :XaU(a U(dXx-$U S :aXx-S-[U 5-$U S :aXx-S-[U 5-$g)aK Convert a raw mpf to a decimal floating-point literal with at most `dps` decimal digits in the mantissa (not counting extra zeros that may be inserted for visual purposes). The number will be printed in fixed-point format if the position of the leading digit is strictly between min_fixed (default = min(-dps/3,-5)) and max_fixed (default = dps). To force fixed-point format always, set min_fixed = -inf, max_fixed = +inf. To force floating-point format, set min_fixed >= max_fixed. The literal is formatted so that it can be parsed back to a number by to_str, float() or Decimal(). rz0.0z.0ze+0+inf-infrNrOr567899rI1.rRze+r) rkrrrrrr\rrrFrstrip) rrP strip_zeros min_fixed max_fixedshow_zero_exponentr_r&r[rVisplits r*to_strrls;( Q4 :AA!U H 9V :f 9UcCF)R&8)c)+1!e4D(  !9  MH v; !7DS\FaAq&VY#-Qq&VY#-Avc#fi.1*<&==swQR{@SSscAg.A DS\F  +) +!|c8)n,6 1 ;c59o-FHE%.3&7 ]]3'FbzS # 1}%7 9M1}T]T1CMAA!|DMC/#h-??|r,cUR5RS5n[U5 URS5n[ U5S:XaSnOUSn[ US5nURS5n[ U5S:Xa*USUSRS5pTU[ U5-nXE-n[ [ X55nX4$)zHelper function for from_str.lrrrrdr.rI)lowerrerrkrrFr)r%r4partsr(rrs r*str_to_man_exprq s  A !H GGCLE 5zQ !H%(m GGCLE 5zQQxq-1 s1v  E CLA 6Mr,)rr^r_rc UR5R5nU[;a [U$SU;aSURS5up4UR S5UR S5pC[ [ U5[ U5X5$[USS9upV[U5S:a.[XQS-5n[U[[XaS-5X5nU$US:a[USU--X5nU$[ USU*-X5nU$)a>Create a raw mpf from a decimal literal, rounding in the specified direction if the input number cannot be represented exactly as a binary floating-point number with the given number of bits. The literal syntax accepted is the same as for Python floats. TODO: the rounding does not work properly for large exponents. /rnrJr4ir) rostrip special_strrkrerrFrqr rr r7rP)r%rnr^rrr'r(rs r*rr!s  AK1~ axwws|xx}ahhsm1SVSVT77ab)HC 3x#~ Sr' " A{42g6 B H !8r3w2A Hc2t8T7A Hr,c[USS9up[U5nSnUS:aU*nSn[U5n[X1X$U[5$)Nr.rtrr)rqrrr}rZ)r%r'r(r&r)s r* from_bstrrxDsKaa(HC c(C D Qwd #B T[ 99r,cPUupp4SS/U[U[U5SS9-SU--$)NrHrGr.)rMr4ze%i)rrr$s r*to_bstrrzNs6Ds s8D>GChsm!D DPS TTr,cnUup4pVU(a [S5eU(dU$US-(aUS-nUS-nUS- nOUS:Xa[X4US-XaU5$[SSU-U- S-5nXwS-- nUS;a[XG-5nO#[ XG-5upHU(a US-S-nUS- n[ XEU- S-X5$)zV Compute the square root of a nonnegative mpf value. The result is correctly rounded. z square root of a negative numberrr.rfd)r3rrErrr~) rrnr^r&r'r(r)shiftr0s r*mpf_sqrtr~Xs Ds >??  Qw q   a $S!VRs;; 1T6"9Q; E QYE d{CJ3:& 61*C QJE %i!^T 77r,cU[:Xa [XU5$U[:Xa [XU5$[[X5[X5US-5n[ XBU5$)zECompute the Euclidean norm sqrt(x**2 + y**2) of two raw mpfs x and y.r)rkrr r r~)r%rrnr^hypot2s r* mpf_hypotrtsQ Ez'!3//Ez'!3// WQ\71<a 8F F# &&r,r!)TNNF)rJ)__doc__ __docformat__rrrrbackendrrrr r r r r rrrrr libintmathrrrrrrrrrrrrrr+r1rr3intern NameErrorrWrZr[r]r\ round_fastrIrKrQrkfnzerorrftworPfhalfrrrrrYrarange h_mask_smallrXrlrorqrFlongrurwrydirrzr}rr~dictrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr rrrr"r$r&r  mpf_mul_intr(r*rr2r5r8 negative_rndr7rEr\rlrqrvrrxrzr~rsage.libs.mpmath.ext_libmplibsmpmath ext_libmpext_lib ImportError)r9s0r*rsS   FFFF f'+) J  s Sk s #; C[  ; =  Ha Xq!  7Aq GQ 7Aq 8Q GR 8T2 8T2 HdB2""s%3-@-Qg!na'-@@ , % 5-uhu& 5n'RtJ56 f,D 9''J((K f)333J  I"JIJ !%*4> B%S/B B  f)SY6&&L f**LJ)),),Z J J J/z9&* :'*#J#-8  0&E:2 h    : j 7j 2jq] ~j' J.-`:9*&0 ? Z9,(24( fG"KG$K )'#MJ",H '@0 x Z +-  z X +-  !+Q:h%62"h@DS@j*$uDA $ F:U%88)' f 44////////##[* FpAfJT$    s=-M:M%.M* A M: M"!M"* M76M7:NN