[hU*SrSSKrSSKJr SSKJr SSKJrJrJrJrJ r J r SSK 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/J0r0J1r1J2r2J3r3J4r4J5r5J6r6J7r7J8r8 SSK9J:r: \ S :XaS r;OS r;S r<\ S :XaS r=OS r=Sr>0r?Sr@0rASrBSrC0rDSrESrFSrG0rHSS/rI\"S\"\B5S-5HrJ\I\K"S\J-\B5S-/S\JS- --- rIM! SrLSrMSrNSrOSXSjrP\LS5rQ\LS5rR\"S5rS\"S 5rT\"S!5rU\"S"5rVS#rW\LSYS$j5rXS%rYS&rZ\LS'5r[\LS(5r\\M"\\5r]\M"\X5r^\M"\[5r_\M"\Y5r`\M"\Q5ra\M"\R5rb\LS)5rc\LS*5rd\M"\d5re\M"\c5rf\4S+jrgS,rhS-ri\4S.jrj\4S/jrkSZS0jrlS1rmS2rnS[S3jroS4rp\4S5jrqS6rrS7rsS8rtS9ruS:rv\4S;jrw\4S<jrx\4S=jry\4S>jrz\4S?jr{\4S@jr|\4SAjr}\4SBjr~S[SCjrSDrSErSFr\4SGjr\S4SHjrSIr\SS4SJjr\4SKjr\4SLjr\4SMjr\4SNjr\4SOjr\4SPjr\4SQjr\4SRjr\4SSjrSZSTjrSZSUjr\ SV:XaSSKJs Js Jr \Rhr4\GRr\Rrq\GRr\GRr\Rrg\GR"r\GR r\Rrlgg!\\4a \"SW5 gf=f)\a( This module implements computation of elementary transcendental functions (powers, logarithms, trigonometric and hyperbolic functions, inverse trigonometric and hyperbolic) for real floating-point numbers. For complex and interval implementations of the same functions, see libmpc and libmpi. N)bisect)xrange)MPZMPZ_ZEROMPZ_ONEMPZ_TWOMPZ_FIVEBACKEND)- round_floor round_ceiling round_downround_up round_nearest round_fast ComplexResultbitcountbctablelshiftrshift giant_steps sqrt_fixedfrom_intto_int from_man_expto_fixedto_float from_float from_rational normalizefzerofonefnonefhalffinffninffnanmpf_cmpmpf_signmpf_absmpf_posmpf_negmpf_addmpf_submpf_mulmpf_div mpf_shift mpf_rdiv_int mpf_pow_intmpf_sqrtreciprocal_rnd negative_rnd mpf_perturb isqrt_fast)ifibpythonXiiii i ct^STlSTlU4SjnTRUlTRUlU$)z Decorator for caching computed values of mathematical constants. This decorator should be applied to a function taking a single argument prec as input and returning a fixed-point value with the given precision. Nc>TRnX::aTRX - - $[US-S-5nT"U40UD6TlUTlTRX0- - $)Ng? ) memo_precmemo_valint)preckwargsrGnewprecfs N/var/www/auris/envauris/lib/python3.13/site-packages/mpmath/libmp/libelefun.pygconstant_memo..g^s^KK  ::).1 1d4il#w)&)  zzgl++)rGrH__name____doc__)rMrOs` rN constant_memorTUs5AKAJ,AJ AI HrQcD^[4U4SjjnTRUlU$)z Create a function that computes the mpf value for a mathematical constant, given a function that computes the fixed-point value. Assumptions: the constant is positive and has magnitude ~= 1; the fixed-point function rounds to floor. cx>US-nT"U5nU[[4;aUS- n[SX2*[U5X5$)Nr?rr)rr r r)rJrndwpvfixeds rNrMdef_mpf_constant..frsB BY "I 8]+ + FAAsHQK;;rQ)rrS)rZrMs` rNdef_mpf_constantr\js<  AI HrQcX!- S:Xa?[SU-S-5nU(d US-(a[X@S--U4$[*X@S--U4$X-S-n[XXS5upgn[XX#5upn X-X--Xz-X-4$)NrrB)rrbsp_acot) qab hyperbolica1mp1q1r1p2q2r2s rNr_r_{suz 1q\ 1BAIr) )8RQ$Y* * qA!.JBB!.JBB 525="% &&rQc[SU-[R"U5- S-5n[USX25upEnXE-U-XP--$)z Compute acot(a) or acoth(a) for an integer a with binary splitting; see http://numbers.computation.free.fr/Constants/Algorithms/splitting.html ffffff?r?r)rImathlogr_)rarJrcNpr`rs rN acot_fixedrssI D4K #b ()Aq!A*GA! S4K13 rQFc~Sn[nUH,upVU[U5[[U5X-U5-- nM. XC- $)z Evaluate a Machin-like formula, i.e., a linear combination of acot(n) or acoth(n) for specific integer values of n, using fixed- point arithmetic. The input should be a list [(c, n), ...], giving c*acot[h](n) + ... rF)rrrs)coefsrJrc extraprecsrarbs rNmachinrxsDIA SVjQD DD NrQc [/SQUS5$)zn Computes ln(2). This is done with a hyperbolic Machin-type formula, with binary splitting at high precision. )))i)r=i-"TrxrJs rN ln2_fixedrs 3T4 @@rQc [/SQUS5$)zF Computes ln(10). This is done with a hyperbolic Machin-type formula. )).)"1)r?Tr}r~s rN ln10_fixedrs 14 >>rQiqci-~ i@ cfX- S:XaO[SU-S- SU-S- -SU-S- -5nUS-[S--S-nSU-U-[[U---nOWU(aUS:a [ S X5 X-S-n[ XUS-U5upn [ XqUS-U5upn X-nX-nX-X--nXEU4$) z Computes the sum from a to b of the series in the Chudnovsky formula. Returns g, p, q where p/q is the sum as an exact fraction and g is a temporary value used to save work for recursive calls. rrBr^rDz binary splitting)rCHUD_CCHUD_ACHUD_Bprint bs_chudnovsky)rarblevelverboserOrqr`midg1rfrgg2rirjs rNrrs sax 1Q1Q1Q' ( qD619  " !GaK6&(? + uqy & -sQh"157G< "357G<  E E EBEM 7NrQc[US- S- S-5nU(a [SU5 [SUSU5upEn[[SU--5nU[-U-U[ U--[ --nU$)z Compute floor(pi * 2**prec) as a big integer. This is done using Chudnovsky's series (see comments in libelefun.py for details). gv O @gbi],@rBzbinary splitting with N =r)rIrrr8rrCHUD_D) rJr verbose_baserprOrqr`sqrtCrYs rNpi_fixedrsv D l *Q ./A )1-Aq!W-GA! v$' (E &!F1H*f,-A HrQc[U5S-$)N)rr~s rN degree_fixedrs D>3 rQcX- S:Xa[[U54$X-S-n[X5up4[X!5upVX6-U-XF-4$)zY Sum series for exp(1)-1 between a, b, returning the result as an exact fraction (p, q). rrB)rrbspe)rarbrerfrgrirjs rNrrsN  saxA qA !ZFB !ZFB 58RU?rQc[SU-[R"U5- S-5n[SU5up#X#-U-U-$)z Computes exp(1). This is done using the ordinary Taylor series for exp, with binary splitting. For a description of the algorithm, see: http://numbers.computation.free.fr/Constants/ Algorithms/splitting.html g?r?r)rIrnror)rJrprqr`s rNe_fixedr sB CHTXXd^ #b ()A !9DA S4K! rQcTUS- n[[SU--5[U--nUS- $)z* Computes the golden ratio, (1+sqrt(5))/2 rFrB )r8r r)rJras rN phi_fixedrs2  BJD8af%&'T/:A 7NrQc dUS-n[[[[U5S5U5US- 5$)NrFr)rmpf_logr1mpf_pi)rJrXs rNln_sqrt2pi_fixedr*s. B GIfRj!4b946 BBrQc,[[U5U5$N)rrr~s rN sqrtpi_fixedr0s htnd ++rQc UupEpgUuppU(aU S:a [S5eU S:a[USU-X--X#5$U S:XaU S:Xa8U(a%[[[ XS-[ U5X#5$[ XU5$U(a"[[ XS-[ U5U *X#5$[[ XS-U5XU5$[ XS-U5n [[X5X#5$)zJ Compute s**t. Raises ComplexResult if s is negative and t is fractional. rz,negative number raised to a fractional powerrDrrF) rr3r0r"r4r5rmpf_expr/) rwtrJrWssignsmansexpsbctsigntmantexptbccs rNmpf_powr>s EE JKK qy1rEkTZ8$DD rz 19tXab"3'&)*.55AS) )"8ABw"3'$)+/%<<x7C8$cJ J 7C A 71=$ ,,rQcUS:XaX-S4$[U5nSnSUS[U5--S--n[upgpUS-(aKXp-nX-nXS- - n U [[Xy- 5-n X:aXyU- - nXU- - nUn US-nU(dXx4$X-nXD-nX3-S- nU[[X- 5-nX5:aXU- - nXCU- - nUnUS-nM)zun-th power of a fixed point number with precision prec Returns the power in the form man, exp, man * 2**exp ~= y**n rBrrr)rr"rrI) ynrJbcexpworkprec_pmpepbcs rN int_pow_fixedrZs  Avax !B CD1Xa[=(1,-HNA2 q5BB 6MCBI//C~L)Hn$ FA 6M Cg Wq[ '#ag,' ' =k"A = CB F+ rQcSn[XX-- 5n[[USU- -55nSnUn Un [XBU-5Han [XaS- U 5up[XS- U -U - U - U - 5n[USU -U- U -5U-nXS- [XkU - 5--U-nU n Mc U$![a= [ WU5n[ U5n[ SXt5n[ XWU5n[U5nNf=f)N2g?rrFrB) rrrI OverflowErrorrr2rrrrr)rrrJexp1starty1rrfnextraextra1prevprqrrrkBs rN nthroot_fixedrs E Aag~ & BQK ! E F E U +qA#u- B1e a",v5 6 1ac$hvo & * A#U7++ +a / , H  b%  a[ !R ' BE " 1I s*B))AC0/C0c2UupEpgU(a [S5eU(dWU[:Xa[$U[:XaUS:a[$US:Xa[$[$U(d[$US:a[$[$SnUS:aIUS:Xa[$US:Xa [ XU5$US:Xa[ [XU5$[UnSnSn X)- nU*nUS :a~US :dU[S S US ---5:a`US-n [U5n [SX5n [X X5n [U SU SU SU SX#5nU(a[ [XW - U5$U$USU--X!-- n US:aXS-- n XU-- n Xz- nSnXn-nUS:aSnU*nU(a UUU-- nOUUU--n[X^5nSnXn-US- U -- U-U- nSnU(aUS:XdUS:XaSnOUS:XdUS:XaSn[UU-XU5n[UUX#5nU(a[ [XW - U5$U$)zYnth-root of a positive number Use the Newton method when faster, otherwise use x**(1/n) znth root of a negative numberrFrBrrDTrr?i NgL<@gףp= ?rFr^urdrM)rr'r!r"r%r+r0r5rIrr2rr rrr)rwrrJrWsignmanrr flag_inverse extra_inverseprec2rnthrrshiftsign1esrr rnd_shifts rN mpf_nthrootrsi Ds ;<<  9K :1u Av KK q5L L1u 6K 61C( ( 74#. .S!    B2v1:C$D.,@(A!Ar  a[1b( AE ' adAaD!A$!d 8 4$6< <H 1Q3J$& !E 2v a JE E B AvS  A  A   C E Y!U{ "Q &% /DI #:I #:I I q 6CS$*AtQ] 2C88rQc[USX5$)zcubic root of a positive numberr^)r)rwrJrWs rNmpf_cbrtrs q!T ''rQc>U[;a[Uup4XA:aX4U- - $US-nU[::a1Uc [U5n[U5nXU- -n[ Xu5Xb--nO"[ [ [U5US-5U5nU[:a X4[U'XU- - $)zT Fast computation of log(n), caching the value for small n, intended for zeta sums. rFr) log_int_cacheLOG_TAYLOR_SHIFTrrlog_taylor_cachedrrrMAX_LOG_INT_CACHE) rrJln2valuevprecrXrrxrYs rN log_int_fixedrs  M$Q'  =T\* * B  ;B-C QK Q$K a $qu , WXa["Q$/ 4 7 a D>rQctSnX-S- nUS:a[X- 5S:aU$[X-5nUnUS- nM6)zR Fixed-point computation of agm(a,b), assuming a, b both close to unit magnitude. rrrr=)absr8)rarbrJianews rN agm_fixedrsM A ax q5S[1_H qsO  Q rQcdX-U- nU=n=pEU(aXR-U- nXE-U- nX4- nU(aMU[U-- nX3-US- - nU[X-5-U- nU=n=pEU(aXR-U- nXE-U- nXd- nU(aM-US--nXf-U- n[X6U5n[U5U-U-$)a Fixed-point computation of -log(x) = log(1/x), suitable for large precision. It is required that 0 < x < 1. The algorithm used is the Sasaki-Kanada formula -log(x) = pi/agm(theta2(x)^2,theta3(x)^2). [1] For faster convergence in the theta functions, x should be chosen closer to 0. Guard bits must be added by the caller. HYPOTHESIS: if x = 2^(-n), n bits need to be added to account for the truncation to a fixed-point number, and this is the only significant cancellation error. The number of bits lost to roundoff is small and can be considered constant. [1] Richard P. Brent, "Fast Algorithms for High-Precision Computation of Elementary Functions (extended abstract)", http://wwwmaths.anu.edu.au/~brent/pd/RNC7-Brent.pdf rBr)rr8rr)rrJx2rwrarbrrqs rNlog_agmr)s2 #$BNAN TdN STM  !'4-A QA :ag  %AMAM TdN STM  ! $1a4 A t A!A TNd "q ((rQc\[U5Hn[X-5nM [U-nX- U-X--nUS:nU(aU*nXU-U- nXw-U- nUn US-n XX-U- nSn U(a(XU -- n U S- n XU -- n XX-U- nU S- n U(aM(X-U- n X-SU--n U(aU *$U $)a" Fixed-point calculation of log(x). It is assumed that x is close enough to 1 for the Taylor series to converge quickly. Convergence can be improved by specifying r > 0 to compute log(x^(1/2^r))*2^r, at the cost of performing r square roots. The caller must provide sufficient guard bits. rr^rrBr)rr8r) rrJrrronerYrv2v4s0s1krws rN log_taylorrXsAY qw  T/C %$!% A q5D B #$B %DB B AB $A A  1f  Q 1f  TdN Q ! %DB AaCA r HrQcX[- - n[UnX1- nX#4[;a [X#4upVO"X#[- -n[XSS5nXV4[X#4'XT-nXd-nX- U-U-nXq-[U-U--nX-U- n X-U- n Un US-n X-U- nSn U(a(XU -- n U S- n XU -- n X-U- nU S- n U(aM(X-U- n X-S-nXn-$)zX Fixed-point computation of log(x), assuming x in (0.5, 2) and prec <= LOG_TAYLOR_PREC. r=r^rrBr)rcache_prec_stepslog_taylor_cacherr )rrJr cached_precdprecralog_arrYrrrrrrws rNrrzs- ##$A"4(K  E ++#AN35 00 111-,-:(KA OE %DQA DA-.A #$B %DB B AB $A A  d  Q d  TdN Q ! %DB 1 A 9rQc`Uup4pVU(d0U[:Xa[$U[:Xa[$U[:Xa[$U(a [ S5eUS-nUS:Xa'U(d[$[ U[ U5-U*X5$XV-n[U5n U S::a\SU - n U (a [U-U- n OU[US- -- n [U 5n Xl- n X:a[XX- XS5n[XX5$X}- nU S:a)[U 5U:a[ U[ U5-U*X5$U[::a0[[XGU- 5U5nU(aX[ U5-- nOHU*[-nUU- n[!UU5nUU*- n[#[%X5U5*nUU[ U5--n[ X*X5$)z^ Compute the natural logarithm of the mpf value x. If x is negative, ComplexResult is raised. zlogarithm of a negative numberr?rri')r!r&r%r'rrrrrrr r7LOG_TAYLOR_PRECrrLOG_AGM_MAG_PREC_RATIOr1rr)rrJrWrrrrrXmagabs_magrrr cancellationrre optimal_magrs rNrrs Ds  :e| 9Tk 9Tk <== B axLC " -sD>> &C#hG !|'  RK3&D'BqD/*Dtnx  %wz3SAAq3 3  B G r !IbM 1B3B B _ fSR%0" 5  Yr]" "Ac11 #  aO | Xa_b ) ) Qy}_ 3 **rQc "US(dXpUS(dnUS(d/Xs=:Xa[:Xa[$ [X4;a[$[$U[:Xa[ [ U5X#5$U[:Xa[$[$[ X5n[ X5nSn[XEX&-5n[U[S5nUSUS-n U[:Xd X*S-:a![XEX&-[USUS5- 5n[[ XrU5S5$)z) Computes log(sqrt(a^2+b^2)) accurately. rr?rFrBr^rD) r!r&r'r%rr*r/r-r#minr1) rarbrJrWa2b2rh2 cancelled mag_cancelleds rN mpf_log_hypotrs Q41 Q4t v~ K :71:t1 1 9K B B E  $BE2&IaL1-ME]VQY6 RTZBqE"Q%(88 9 WRs+R 00rQcUS:a([R"[XS- - 5S- 5nO%[R"[U5SU-- 5nSn[[US-5SU- - 5nSn[ X15HOnXT- nX%U- -n[ X%5upgXu-U-nU[ XU- 5- U-[U-US-U- --n X)- nUnMQ [ X#U- 5$)Nd5g@Cg@rrB)rnatanrIrr cos_sin_fixedrr) rrJrrrextra_prXcossintanras rN atan_newtonrs s{ IIc1Bw<)'1 2 IIc!fS$Y& ' E CG E *+AG%&  U(O 'yS &G$$ +'2+36B,1O P E' !4Z  rQcS[US- 5-S-nX!- nX4[;a [X4upEO!X[- -n[XB5nXE4[X4'XC- XS- 4$)Nrr?)ratan_taylor_cacheATAN_TAYLOR_SHIFTr)rrJrrraatan_as rNatan_taylor_get_cachedr"st $q&! "b (E LE z&&%ah/ 6 ++ ,Q&'(k!(# J&/ **rQc0X[- - n[X!5up4X- nXQ-US-U- X5-U- -[U---=pgUS-U- nX-U- n US-n Xy-U- nSn U(a(XgU -- nU S- n XU -- n Xy-U- nU S- n U(aM(X-U- n Xj- n XL-$)NrBr^r)rrr) rrJrrarrrrYrrrrrws rN atan_taylorr!1s %% &A&q/IA Aiaddlqsd{;w$O PPB Q$$,B 'd B AB DA A  1f  Q 1f  V  Q ! 'd B A :rQc U(d[[X5S5$[[[U[U5S55$)NrD)r1rr,r6)rrJrWs rNatan_infr#Es5 *B// 9VD,s*;= 1)r(r"rr4r-r/r#r)rrJrWrXr`s rN mpf_acoshr>sS Bq$2>??ub126A 71$d 00rQc Uup4pVU(d$U(aU[[4;aU$[S5eXe-nUS:a&US:XaUS:Xa[[/U$[S5eUS-nUS:aXx*:a [ XX5$X*- n[ U[U5n [[X5n [[[XU5X5S5$)Nz&atanh(x) is real only for -1 <= x <= 1rrr5r;rD) r!r'rr%r&r7r-r"r.r1rr0) rrJrWrrrrrrXrarbs rN mpf_atanhr@sDs S  HDEE (C Qw !8q%=& &DEE B Rx 9q2 2 t 4AaA WWQ2.:B ??rQcUup4pVU(dU[:Xa[$U$[XV-5nUS:a3US:dU[U5::a[ [ [ U55X5$X-S-n[U5n [[U S5[U5n [XU5n [X5n [XU5n [XU5n [XX5n U $)NrrFr?r)r&r'rrrr9rmpf_phir-r1r#r mpf_cos_pir0r.) rrJrWrrrrsizerXrarbrrYs rN mpf_fibonaccirEsDs  :K sv;D ax "9.DOT7 7 r B A !Q+AbA1AbAbAd A HrQcUS:aU*nSnOSn[SUS--5n[U5U- n[SXT-5nSS[XE*5--nX-nXU- -n[U-nUS:Hn U[:aqX-U- =pX-U- n [ =pSnU (a4XS- U--oU - oS- nXS- U--oU - oS- nX-U- n U (aM4X-U- nU (aX- U-nOX-U-nO[SUS--5nX-U- =pX/n[ SU5HnURUSU -U- 5 M [ /U-nSnU (aa[ U5H>nXS- U--n U (aUS-(aUU==U -ss'O UU==U - ss'US- nM@ U US-U- n U (aMa[ SU5HnUUUU-U- UU'M [U5U-nUS:XaD[UU-X-- 5nU(aUU- nOUU-n[ U5H nUU-U- nM UU- $US- n[ U5HnUU-U- U- nM [[X-UU-- 55nU(aU*nUU- UU- 4$) z Taylor series for cosh/sinh or cos/sin. type = 0 -- returns exp(x) (slightly faster than cosh+sinh) type = 1 -- returns (cosh(x), sinh(x)) type = 2 -- returns (cos(x), sin(x)) rr?rFrBg333333?rmrD) rIrmaxrEXP_SERIES_U_CUTOFFrrappendsumr8r)rrJtyperrrxmagrrXraltrrax4rrrrrxpowersrsumsrwrYpshifts rNexponential_seriesrSs 1u B Cc MA A; D AtxA 3q< E B19A R-C 19C !!#"e]  Q3'MA72FA Q3'MA72FA" Aae] # A# A D$J #")1A NNGBKNR/ 0zA~ AYsAg 1q5$q'Q,'"&q'Q,'Q  72;2%A a1AAwwqz)b0DG IO qy qscg ' AAAAA1 AEz AAA#&C'A sCGqs?+ , A5AuH%%rQc2U[:a [XS5$[US-5nX- n[U-=p4SnX-U- =pgU(a(Xe-ocU- o5S- nXe-odU- oES- nXg-U- nU(aM(X@-U- nX4-nUn U(aX-U- nUS-nU(aMX- $)z Compute exp(x) as a fixed-point number. Works for any x, but for speed should have |x| < 1. For an arbitrary number, use exp(x) = exp(x-m*log(2)) * 2^m where m = floor(x/log(2)). rrGrBr)EXP_COSH_CUTOFFrSrIr) rrJrrrrrrarrwrs rN exp_basecaserV>s  o!!1-- D#IAID$B Acd]A  qq&! qq&! TdN ! $4B A A STM Q ! 6MrQczU[:a[XS5up#X#-X#- 4$[X5n[X--U-nXE4$)z Computation of exp(x), exp(-x) r)rUrSrVr)rrJcoshsinhrarbs rNexp_expneg_basecaserZWsK o'3 y$)##QA TY A%A 4KrQcU[:a [XS5$U[- nX- n[U5nU[;a:US[-[- -n[US[-S5upgUS- US- 4[U'[Uupg[U- nXh-nXx-nXU--n[ U-n Un Sn X-U- *n U (a0X-oU - oS- oU-U- n X-oU - oS- oU-U- *n U (aM0X-X-- U- X-X--U- 4$)z Compute cos(x), sin(x) as fixed-point numbers, assuming x in [0, pi/2). For an arbitrary number, use x' = x - m*(pi/2) where m = floor(x/(pi/2)) along with quarter-period symmetries. rBrFr)COS_SIN_CACHE_PRECrSCOS_SIN_CACHE_STEPrI cos_sin_cacher) rrJprecsrrwcos_tsin_toffsetrrrras rNcos_sin_basecaserdbsB    !!1-- % %E A AA  %%&88 9)!R0B-BAF !2I3 a #LE $ &F E EeOA T/C C A 34-A  61!}1 61A#$'71 !Ysy T )ci .Ad-J KKrQcUup4pVU(aXe-nUS-nU(aU*nUS:a/US:a)[U[SU-5-5n [XU-X5$Xx*:a[[X1U5$US:a@X-n XZ-n U S:aXK-n OXK*- n [ U 5n [ X5up[U5nX-n OXX-n U S:aXK-n OXK*- n Sn[X5n[XNU- X5$U(d[$U[:Xa[$U$)Nr;rg333333?r) mpf_erIr3r7r"rdivmodrVrr&r!)rrJrWrrrrrrXewpmodrcrlg2rs rNrrs Ds h BY $C #:#(bT#X&'Aqs(D6 6 9tT5 5 7HE[F{MG$E"C!>DAAA IAXF{MG$A1!C2t11  Ez HrQcUupEpgU(dlU(aeU(a&U[:Xa[$U[:Xa[$[$U[:Xa [[4$U[:Xa [[4$[[4$Xg-nUS-n US:aBX*:a7U(a[ USU- X5$[ [SX5n [ XX5n X4$X*- n US:ahSSUS- --U :aYU(a[ [[/USU- X5$[ [[U5X5S5=pU(a [U 5n X4$US:aCX-nXn-nUS:aX_-nOX_*- n[U5n[UU5unn[U5nUU-nOXi-nUS:aX_-nOX_*- nSn[UU 5unnUUSU-- -n UUSU-- - n U(aU *n U(aX-U -n[XY*X5$[U UU - S- X5n [U UU - S- X5n X4$) z4Simultaneously compute (cosh(x), sinh(x)) for real xrfrrrFr^rDrB)r%r"r&r#r'r7r1rr*r,rrhrIrZr)rrJrWtanhrrrrrrXrXrYrrwrjrcrrkrrarbs rN mpf_cosh_sinhros)Ds S Dy+Ez%<K 9dD\) :tUm+Tz &C bB Rx 9"1afd88tQ2Dq2D:  t  Rx a#a%j>B "D<#5qvtIIggaj$D AaC>D u zd"Cd00D!B$q&$4D!B$q&$4zrQc>US:aySnSU-nX2-U-n[US- 5nUS- nXa-n U S:aX -n OX *- n [X5upX:aX|- n OU n XU-S- - (a[U 5n X- n Xb- nO"US- nMvX2*- nX-n U S:aX -n OX *- n Sn XU4$)Nrrr?rF)rrhrI)rrrrXrcancellation_precrjpi2pi4rcrrrsmalls rNmod_pi2rus Qw  "a H00E57#C(C[F{MG$!>DAwC#FH[ FA), t  Q; A A  8OrQcUupVpxU(d?U(a [[pO [[pUS:XaX4$US:XaU $US:XaU $US:XaU $X-n US-n U S:anX*:ahU(a[U[ U 55n[ [SX5n [ USU- X5n US:XaX4$US:XaU $US:XaU $US:Xa [ XX5$U(aUS:asUS:Xa%[n [[ 4[US-5U- n OUS:Xa [ [pO [[pUS:XaX4$US:XaU $US:XaU $US:Xa [XX5$Xg*S- - S-S- n XmU*S- -- n[U5U-nUS-U- n X|-nUS:aXo-nOXo*- nU[U 5-U - nO[XgX5unp[UU 5upU S-nUS:XaU *U pOUS:XaU *U *pO US:XaX*pU(aU *n US:Xa[X*X5n [X*X5n X4$US:Xa [X*X5$US:Xa [X*X5$US:Xa [XX5$g)zz which: 0 -- return cos(x), sin(x) 1 -- return cos(x) 2 -- return sin(x) 3 -- return tan(x) if pi=True, compute for pi*x rrrBr^rFrDN)r'r"r!r/rr7r#boolr0rrrurdrr)rrJrWwhichpirrrrrrwrrXrmag2rcrres rN mpf_cos_sinr{sDs  qq A:ad{ A:ax A:ax A:ax (C B Qw 9Avbz*D!T/AAqvt1Az!$;z!8z!8z+at"AA "9by5M$sQw-$"67u1e1z!$;z!8z!8z'!"::d1fo "q (C46]#}s" BY  Q; A A x|^ "3S-1 Ar "DA AA aAA aQBA a2A B z C + C +t  zAsD.. zAsD.. zQ4--rQc[XUS5$Nrr{rrJrWs rNmpf_cosrb[#q-I&IrQc[XUS5$)NrBr~rs rNmpf_sinrcrrQc[XUS5$)Nr^r~rs rNmpf_tanrdrrQc[XUSS5$)Nrrr~rs rNmpf_cos_sin_piresKaQR4S-SrQc[XUSS5$r}r~rs rNrCrCf AS!Q0O)OrQc[XUSS5$)NrBrr~rs rN mpf_sin_pirgrrQc [XU5S$Nrrors rNmpf_coshrhmAS.I!.L'LrQc [XU5S$r}rrs rNmpf_sinhrirrQc[XUSS9$)Nr)rnrrs rNmpf_tanhrjsmASq.Q'QrQcUc[US- 5n[X5up4[U5n[XA5upVUS-nUS:XaXV4$US:XaU*U4$US:XaU*U*4$US:XaXe*4$g)Nrr^rrB)rrhrIrd)rrJrrrrrrwres rNrros {tAv !>DA AA A $DA AAAvad{Avqb!e|Avqb1"f}Avae|vrQcUc [U5n[X5up4[U5n[XA5nUS:aXS-$XS*- $r)rrhrIrV)rrJrrrrYs rN exp_fixedr{sF {o !>DA AAQAAvv RyrQsagez)Warning: Sage imports in libelefun failed)F)FNr)r)rSrnrbackendrrrrr r r libmpfr r rrrrrrrrrrrrrrrrrrr r!r"r#r$r%r&r'r(r)r*r+r,r-r.r/r0r1r2r3r4r5r6r7r8 libintmathr9rUrIr\r]r^rrrrrrr&rrrrr rTr\r_rsrxrrrrrrrrrrrrrBrrg mpf_degreempf_ln2mpf_ln10rr mpf_sqrtpimpf_ln_sqrt2pirrrrrrrrrrrrrrr!r#r(r*r7r9r<r>r@rErSrVrZrdrrorur{rrrrrCrrrrrrsage.libs.mpmath.ext_libmplibsmpmath ext_libmp_lbmp ImportErrorAttributeErrorrrQrNrs.  GG             hOO h  r7 8O,Q. /AQT/2256QqSAA 0 * " '   AA??8 X Y V R,     i ( h ' g & l + i ( j )CC ,, - #$45'-8"l ,!+Qf%(,  -)^  D D$F+P%1X!$ +(C % +F)!(F% 0% 0&%"&1&@* * 8I&V2 L:$* Z *@F!H(qUM.^$I#I#I *S&O&O$L$L$Q   f ;22>>----------OO ++ ++   (; 9:;sA=K??LL