[hhSrSSKrSSKJrJrJrJrJr SSKJ r J r J r J r J r JrJrJrJrJrJrJrJrJrJrJrJrJrJrJrJrJrJrJ r J!r!J"r"J#r#J$r$J%r%J&r&J'r'J(r(J)r)J*r*J+r+J,r,J-r-J.r.J/r/J0r0J1r1J2r2J3r3J4r4J5r5J6r6J7r7J8r8J9r9 SSK:J;r;Jr>J?r?J@r@JArAJBrBJCrCJDrDJErEJFrFJGrGJHrHJIrIJJrJJKrKJLrLJMrMJNrNJOrOJPrPJQrQJRrR \ \4rS\\4rT\!\4rU\"\4rV\#\$4rW\#\$\%4rXSrYSrZS r[S \4S jr\S r]\4S jr^Sr_\4Sjr`\4SjraS\4SjrbS\4Sjrc\4SjrdS\4SjreSrf\4Sjrg\4Sjrh\4Sjri\4Sjrj\4Sjrk\4Sjrl\4Sjrm\4Sjrn\4Sjro\4Sjrp\4S jrq\4S!jrr\4S"jrs\4S#jrt\4S$jruS%rv\4S&jrw\4S'jrx\4S(jry\4S)jrzS*r{\4S+jr|\4S,jr}\4S-jr~\4S.jr\4S/jr\4S0jr\4S1jr\4S2jr\4S3jr\4S4jr\4S5jr\4S6jr\4S7jr\4S8jr\4S9jr\"S:5r\"S;5rS<r\4S=jr\4S>jr\4S?jr\4S@jr\4SAjr\4SBjrSISCjrSISDjrSISEjrSISFjr\SG:Xa&SSKJs Js Jr \Rr~\Rrzgg!\\4a \"SH5 gf=f)Jz- Low-level functions for complex arithmetic. N)MPZMPZ_ZEROMPZ_ONEMPZ_TWOBACKEND)1 round_floor round_ceiling round_downround_up round_nearest round_fastbitcountbctable normalize normalize1reciprocal_rndrshiftlshift giant_steps negative_rndto_strto_fixed from_man_exp from_floatto_floatfrom_intto_intfzerofoneftwofhalffinffninffnanfnonempf_absmpf_posmpf_negmpf_addmpf_submpf_mulmpf_div mpf_mul_int mpf_shiftmpf_sqrt mpf_hypot mpf_rdiv_int mpf_floormpf_ceilmpf_nintmpf_fracmpf_signmpf_hash ComplexResult)mpf_pimpf_expmpf_log mpf_cos_sin mpf_cosh_sinhmpf_tan mpf_pow_int mpf_log_hypotmpf_cos_sin_pimpf_phimpf_cosmpf_sin mpf_cos_pi mpf_sin_pimpf_atan mpf_atan2mpf_coshmpf_sinhmpf_tanhmpf_asinmpf_acos mpf_acosh mpf_nthroot mpf_fibonaccic8UupU[;agU[;agg)z2Check if either real or imaginary part is infiniteTF)_infszreims K/var/www/auris/envauris/lib/python3.13/site-packages/mpmath/libmp/libmpc.py mpc_is_infrY)s FB U{4 U{4 c8UupU[;agU[;agg)z9Check if either real or imaginary part is infinite or nanTF) _infs_nanrTs rX mpc_is_infnanr]0s FB Yt Yt rZc Uup4[X15nUS(aUS-[[U5U40UD6-S-$US-[XA40UD6-S-$)Nrz - jz + )rr))rUdpskwargsrVrWrss rX mpc_to_strrc7s[ FB B !uEzF72;>v>>DDEzF25f55;;rZFcJUup4[[X1U5[XAU55$N)complexr)rUstrictrndrVrWs rXmpc_to_complexri?s$ FB 8B,hr3.G HHrZcJ[RS:aaUup[U5[RR[U5--nUS[RR --n[ U5$[[USS95$![a [U5s$f=f)N)rlT)rg) sys version_infor8 hash_infoimagwidthinthashri OverflowError)rUrVrWhs rXmpc_hashrvCs 6! RL3==-- < < CMM''' (1v  q67 7 7N s7B B"!B"c&Uup4U[XAU54$rer)rUprecrhrVrWs rX mpc_conjugater{Ps FB wr% %%rZcU[:g$re)mpc_zero)rUs rXmpc_is_nonzeror~Ts =rZcBUupEUupg[XFX#5[XWX#54$rer*rUwrzrhabcds rXmpc_addrW) DA DA 1 #WQ4%= ==rZc&UupE[XAX#5U4$rer)rUxrzrhrrs rX mpc_add_mpfr\ DA 1 #Q &&rZcBUupEUupg[XFX#5[XWX#54$rer+rs rXmpc_subr`rrZc&UupE[XAX#5U4$rer)rUprzrhrrs rX mpc_sub_mpfrerrZc:Uup4[X1U5[XAU54$re)r(rUrzrhrrs rXmpc_posri" DA 1C '!3"7 77rZc:Uup4[X1U5[XAU54$rerxrs rXmpc_negrmrrZc6Uup#[X!5[X154$re)r/)rUnrrs rX mpc_shiftrqs DA Q?IaO ++rZc"Uup4[X4X5$)zAAbsolute value of a complex number, |a+bi|. Returns an mpf value.)r1rs rXmpc_absrus DA Q4 %%rZc"Uup4[XCX5$)z3Argument of a complex number. Returns an mpf value.)rIrs rXmpc_argr{s DA Q4 %%rZc:Uup4[X1U5[XAU54$re)r3rs rX mpc_floorrs" DA Qc "Ias$; ;;rZc:Uup4[X1U5[XAU54$re)r4rs rXmpc_ceilr" DA AS !8AS#9 99rZc:Uup4[X1U5[XAU54$re)r5rs rXmpc_nintrrrZc:Uup4[X1U5[XAU54$re)r6rs rXmpc_fracrrrZcUupEUupg[XF5n[XW5n [XG5n [XV5n [XX#5n [XX#5n X4$)z Complex multiplication. Returns the real and imaginary part of (a+bi)*(c+di), rounded to the specified precision. The rounding mode applies to the real and imaginary parts separately. )r,r+r*)rUrrzrhrrrrrqrsrVrWs rXmpc_mulrsU DA DA A A A A t !B t !B 6MrZcUup4[X35n[XD5n[X4X5n[XVX5n[US5n X4$Nr)r,r+r/) rUrzrhrrrrrrVrWs rX mpc_squarersE DA A ATA t !B 1aB 6MrZc@UupE[XAX#5n[XQX#5nXg4$re)r,rUrrzrhrrrVrWs rX mpc_mul_mpfrs( DA t !B t !B 6MrZcRUupE[[XQX#55n[XAX#5nXg4$)z: Multiply the mpc value z by I*x where x is an mpf value. )r)r,)rUrrzrhrrrVrWs rXmpc_mul_imag_mpfrs/ DA t) *B t !B 6MrZc@UupE[XAX#5n[XQX#5nXg4$re)r.)rUrrzrhrrrVrWs rX mpc_mul_intrs( DA Q4 %B Q4 %B 6MrZcUupEUupgUS-n[[Xf5[Xw5U5n [[XF5[XW5U5n [[XV5[XG5U5n [XX#5[XX#54$N )r*r,r+r-) rUrrzrhrrrrwpmagtus rXmpc_divrst DA DA B '!- 3C galB/A galB/A 1 "GA$$; ;;rZc@UupE[XAX#5n[XQX#5nXg4$)zCalculate z/p where p is real)r-rs rX mpc_div_mpfrs( DA t !B t !B 6MrZcUup4[[X35[XD5US-5n[X5X5n[[XEX55nXg4$)zCalculate 1/z efficientlyrr*r,r-r))rUrzrhrrmrVrWs rXmpc_reciprocalrsG DA WQ\$r'2A t !B t) *B 6MrZcUupE[[XD5[XU5US-5n[[X@5XbU5n[[[XP55XbU5nXx4$)z)Calculate p/z where p is real efficientlyrr) rrUrzrhrrrrVrWs rX mpc_mpf_divrsS DA WQ\473A q ,B & 5B 6MrZcSnSnU(a>US-(aX0-XA-- X@-X1--pCUS-nX-X-- SU-U-pUS-nU(aM>X44$)zcComplex integer power: computes (a+b*I)**n exactly for nonnegative n (a and b must be Python ints).rrrl)rrrwrewims rXcomplex_int_powrsh C C q5usu}cecem FAsQSy!A#a%1 a ! 8OrZc US[:Xa[XSX#5$[[[ XS-5XS-5X#5$)Nrrr)r mpc_pow_mpfmpc_exprmpc_log)rUrrzrhs rXmpc_powrs?tu}1dD.. 7712g.7;T GGrZc UupEpgUS:a[USU-XV--X#5$US:Xa![XS-5n[USU-U-X#5$[[[ XS-5XS-5X#5$)Nrr) mpc_pow_intmpc_sqrtrrr) rUrrzrhpsignpmanpexppbcsqrtzs rXrrs{E qy1rEkTZ8$DD rzG$52+"4d@@ ;wqr'2ABw? KKrZc PUupEU[:Xa[XAX#5[4$U[:Xa[[XQX#5nUS-nUS:XaU[4$US:Xa[U4$US:Xa[U5[4$US:Xa[[U54$US:Xa[$US:Xa [ XU5$US:Xa [ XU5$US:Xa [ XU5$US:a[ [X*US-5X#5$UupxpUuppU(aU*nU (aU *n X- n[U5nUU[X5--nUS:aWUS:aX-nU n OX*-n U n [XU5unn[U[X-5X#5n[U[X-5X#5nUU4$[[[XS-5XS-5X#5$) Nrrrlrkri'r)rr@r)mpc_onerrrrabsmaxrrrrrrr)rUrrzrhrrvasignamanaexpabcbsignbmanbexpbbcdeabs_de exact_sizerVrWs rXrrs DAEz1+U22Ez d ( Q 6e8O !V!8O !V1:u$ $ !V'!*$ $Avg~Avgas++Avj#..Bw~as331u^K2tAv$>JJEE dUd dUd B WFFS]*+JE 6 KDD cNDD Q/B "c!&k4 5 "c!&k4 52v ;wqr'2ABw? KKrZc`Uup4U[:XaHU[:XaX44$US(a[[U5X5n[U4$[X1U5nU[4$US-nUS(dU[[ X44U5X75n[ US5n [XU5n[ US5n [X5n [ XKX5nXe4$[[ X44U5X75n[ US5n [XU5n[ US5n [X5n [ XKX5nUS(a[U5n[U5nXe4$)zComplex square root (principal branch). We have sqrt(a+bi) = sqrt((r+a)/2) + b/sqrt(2*(r+a))*i where r = abs(a+bi), when a+bi is not a negative real number.rrr)rr0r)r*rr/r-r+) rUrzrhrrrWrVrrrrrs rXrr's, DAEz :6M Q4'!*d0B2; !3'B;  bB Q4 WaVR(! 0 a  as # aO a_ Q4 % 6M GQFB' / a  as # aO a_ Q4 % Q4BB 6MrZcSn[[XX$-- 55n[[XX$-- 55nUSU--SU- -nURnURn [ [U55n[ [U 55n Sn Un Un[XCU -5Hn[XUS- 5unn[UUS- U -U- U- 5n[UUS- U -U- U- 5nUU-UU--X-- n[XU- 5n[XU- 5nUU-UU--U- nU*U-UU--U- nUU-U-nUU-U-nUUS- [XU - 5--U-nUUS- [XU - 5--U-n Un M X4$![ a] [ XT5n[ Xd5n[ U5n [SX5n [XV4U [4U5up[U5n[U 5n GNUf=f)N2y?g?rr)rrrrealrprrtrr2rrrrrr)rrrrzstarta1b1rrVrWfnnthextraprevpextra1rre2im2r4apbprecimcrebimbs rXmpc_nthroot_fixedrKs E VAag~ & 'B VAag~ & 'B  "r'\SU # VV VV R\ R\ E E F U +"21Q3/SS1Q3+/F23S1Q3+/F23#gCQZ 0 Aax  Aax Cx"s("q(sSy28#)axBaxBQqS&uW--- 1QqS&uW--- 1, 6M5  b  b  a[1b("C<7 BZ BZsAE33A#GGc>UupEUSS:XaU[:Xa[XAX#5nU[4$US:abUS:Xa[$US:Xa [XE4X#5$US:Xa[ [XE4X#5$[ XE4U*US-[ U5n[ [XrU5$US::a[SUS--5nUuppUupnn[XE4U5nUS US-S :a\US US-U:aM[XH5n[XX5n[UUX5unnSn[Xh*U- X5n[UU*U- X5nUU4$[U5nUS-S-n[SUU5n[XE4U[4X5unn[USUSUSUS X#5n[USUSUSUS X#5nUU4$) ze Complex n-th root. Use Newton method as in the real case when it is faster, otherwise use z**(1/n) rrlrrrg333333?rirk)rrPrrr mpc_nthrootrrrrrrrrr2rr)rUrrzrhrrrVinverseprec2rrrrrrrrpfafbfrWrrrs rXrrrs DAtqyQ%Z t )E{1u 6N 6A64- - 77QFD6 6qfqb$q&.2EFws33BwC4"9%&!"T!"T3 aUD ! b6BrF?S bfr"vo&<!#B!#B&r2q8FBEb&,;Bb5&,;Br6M !B GbLE q"e $C aVc5\5 6FB 2a5"Q%A1t 9B 2a5"Q%A1t 9B r6MrZc[USX5$)z Complex cubic root. rk)rrUrzrhs rXmpc_cbrtrs q!T ''rZcUup4U[:Xa [XAU5$U[:Xa[X1U5[4$[X1S-U5n[XAS-U5upg[XVX5n[XWX5n X4$)aJ Complex exponential function. We use the direct formula exp(a+bi) = exp(a) * (cos(b) + sin(b)*i) for the computation. This formula is very nice because it is pefectly stable; since we just do real multiplications, the only numerical errors that can creep in are single-ulp rounding errors. The formula is efficient since mpmath's real exp is quite fast and since we can compute cos and sin simultaneously. It is no problem if a and b are large; if the implementations of exp/cos/sin are accurate and efficient for all real numbers, then so is this function for all complex numbers. r)rr=r;r,) rUrzrhrrrrrrVrWs rXrrsy DAEz1C((Ezq$e++ !!VS !C qq&# &DA  #B  #B 6MrZcF[USUSX5n[XU5nX44$)Nrr)rArrys rXrrs+ qtQqT4 -B # B 6MrZcUup4U[:Xa[X1U5[4$U[:Xa[XAU5[4$US-n[X55upg[ XE5up[ XhX5n [ XyX5n U [ U 54$)a?Complex cosine. The formula used is cos(a+bi) = cos(a)*cosh(b) - sin(a)*sinh(b)*i. The same comments apply as for the complex exp: only real multiplications are pewrormed, so no cancellation errors are possible. The formula is also efficient since we can compute both pairs (cos, sin) and (cosh, sinh) in single stwps.)rrDrJr=r>r,r) rUrzrhrrrrrchshrVrWs rXmpc_cosrs DAEzq$e++Ez%u,, B q DA 1 !FB  "B  "B wr{?rZcUup4U[:Xa[X1U5[4$U[:Xa[[XAU54$US-n[X55upg[ XE5up[ XxX5n [ XiX5n X4$)z{Complex sine. We have sin(a+bi) = sin(a)*cosh(b) + cos(a)*sinh(b)*i. See the docstring for mpc_cos for additional comments.r)rrErKr=r>r,rs rXmpc_sinrs} DAEzq$e++Ezhq,,, B q DA 1 !FB  "B  "B 6MrZcTUup4UupVpxUuppU[:Xa[X1U5[4$U[:Xa[[XAU54$US-n [US5n[US5n[ X=5up[ XM5unn[ UUU 5n[UUX5n[UUX5nUU4$)z_Complex tangent. Computed as tan(a+bi) = sin(2a)/M + sinh(2b)/M*i where M = cos(2a) + cosh(2b).r)rr?rLr/r=r>r*r-)rUrzrhrrrrrrrrrrrrrrrrrVrWs rXmpc_tanrs DAEEEz'!3/66Ez%!3!777 B!QA!QA q DA 1 !FB !R C C #B S$ $B r6MrZc:Uup4U[:Xa[X1U5[4$[U[US-5US-5nU[:Xa[ XAU5[4$US-n[ X55upg[ XE5up[XhX5n [XyX5n U [U 54$Nrr)rrFr,r:rJrBr>r)rs rX mpc_cos_pirs DAEz!3'..6$q&>46*AEz%u,, B ! DA 1 !FB  "B  "B wr{?rZc&Uup4U[:Xa[X1U5[4$[U[US-5US-5nU[:Xa[[ XAU54$US-n[ X55upg[ XE5up[XxX5n [XiX5n X4$r)rrGr,r:rKrBr>rs rX mpc_sin_pir s DAEz!3'..6$q&>46*AEzhq,,, B ! DA 1 !FB  "B  "B 6MrZchUup4U[:Xa[XAU5upVU[4[U44$U[:Xa[X1U5upxU[4U[44$US-n [X95upx[XI5upV[XuX5n [XX5n [XX5n [XvX5n U [ U 54X44$)Nr)rr>r=r,r))rUrzrhrrrrrrrcrecimsresims rX mpc_cos_sinr&s DAEzq,E{UBK''Ez1C(5zAu:%% B q DA 1 !FB ! #C ! #C ! #C ! #C   **rZcUup4U[:Xa[X1U5upVU[4U[44$[U[US-5US-5nU[:Xa[ XAU5upxU[4[U44$US-n [X95upV[ XI5upx[XWX5n [XhX5n [XgX5n [XXX5n U [ U 54X44$r)rrBr,r:r>r))rUrzrhrrrrrrrr"r#r$r%s rXmpc_cos_sin_pir(%s DAEzas+5zAu:%%6$q&>46*AEzq,E{UBK'' B ! DA 1 !FB ! #C ! #C ! #C ! #C   **rZc8Uup4[U[U54X5$)z:Complex hyperbolic cosine. Computed as cosh(z) = cos(z*i).)rr)rs rXmpc_coshr*7s DA Awqz?D ..rZc.Uup4[XC4X5upCX44$)z;Complex hyperbolic sine. Computed as sinh(z) = -i*sin(z*i).)rrs rXmpc_sinhr,< DA A64 %DA 4KrZc.Uup4[XC4X5upCX44$)z>Complex hyperbolic tangent. Computed as tanh(z) = -i*tan(z*i).)rrs rXmpc_tanhr/Br-rZcLUup4US-n[[XE5[U54n[[XE5U4n[ Xe5n[ Xu5n [ XX5up4[[ US55[ US54n U S[:Xa[U5(a U S[4n U $)Nrrrr) r*r r)r+rrr/r%rYr) rUrzrhrrrryl1l2rs rXmpc_atanr4Is DA Bagaj(AaaA B B 24 %DA !B )Ab/1A tt| 1 qT5M HrZg:pΈ?g?c ` UupEUS-nU[:Xa[[[U5U5nUS(d*US:Xa[ XAU5[4$[ XAU5[4$US(aJ[ X5n[[U5X5n US:Xa U[U 54$[[US55U 4$[XAU5n US:Xa[U 4$[ X5n[US5[U 54$S=pUS(a [U5nSn US(a [U5nSn [[XF5n[[XF5n [XU5n [XuU5n[[XU5S5n[XOU5n[XUU5n[[UU5S(d"US:Xa[ UU5nGOB[ UU5nGO4[XU5nUS(d[U[XU5U5n [XU5n[[U[U UU5U5S5nUS:Xa![[[!UU5XF5U5nO[[U[!UU5U5U5nO[U[XU5U5n [U[XU5U5n[[U UU5S5n[U[!UU5U5nUS:Xa[[UXF5U5nO[[UUU5U5n[["X5S(d[U[XU5U5n[U5S(a1[XU5n[UUU5n[[UUU5S5nO#[XU5n[[UUU5S5n[U[U[U5U5n[%[[[U[!UU5U5U5U5nO<[![[XU5[U5U5n[%[UUU5U5nU (a(US:Xa[[ U5UU5nO [U5nU (dUS:Xa [U5nU (aUS:Xa [U5n['USUSUSUSX5n['USUSUSUSX5nUU4$)acomplex acos for n = 0, asin for n = 1 The algorithm is described in T.E. Hull, T.F. Fairgrieve and P.T.P. Tang 'Implementing the Complex Arcsine and Arcosine Functions using Exception Handling', ACM Trans. on Math. 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