+ jUpRt^RIt^RIHt^RIHt^RIHtHtHtHtH t H t ^RI H t H t HtHtHtHtHtHtHtHtHtHtHtHtHtHtHtHtHtHtH t H!t!H"t"H#t#H$t$H%t%H&t&H't'H(t(H)t)H*t*H+t+H,t,H-t-H.t.H/t/H0t0H1t1H2t2H3t3H4t4H5t5H6t6H7t7H8t8^RI9H:t:] R8XdRt;MR t;R t<] R8XdR t=M^t=^t>/t?R t@/tAR tB^ tC/tD^tER tF^tG/tH^^.tI]!^]!]B4^,4F7tJ]I]K!^]J,]B4^,.^]J^, ,,, tIK9 RtLRtMRtNRtORNRltP]LR4tQ]LR4tR]!R4tS]!R4tT]!R4tU]!^ 4tVRtW]LRORl4tXRtYRtZ]LR4t[]LR4t\]M!]\4t]]M!]X4t^]M!][4t_]M!]Y4t`]M!]Q4ta]M!]R4tb]LR4tc]LR 4td]M!]d4te]M!]c4tf]3R!ltgR"thR#ti]3R$ltj]3R%ltkRPR&ltlR'tmR(tnRQR)ltoR*tp]3R+ltqR,trR-tsR.ttR/tuR0tv]3R1ltw]3R2ltx]3R3lty]3R4ltz]3R5lt{]3R6lt|]3R7lt}]3R8lt~RQR9ltR:tR;tR<t]3R=lt]^3R>ltR?t]^R3R@lt]3RAlt]3RBlt]3RClt]3RDlt]3RElt]3RFlt]3RGlt]3RHlt]3RIltRPRJltRPRKlt] RL8Xd^RIHuHuHt]Pht4]EPt]Ptq]EPt]EPt]Ptg]EP"t]EP t]PtlR#R# ]]3d ]!RM4R#i;i)Ra( 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 ctaRSnRSnV3RlpSPVnSPVnV#)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<SPpW8:dSPW , , #\VR,^ ,4pS!V3/VBSnVSnSPW0, , #)g?) memo_precmemo_valint)preckwargsr<newprecfs&, W/home/luca/screen-checking/.venv/lib/python3.14/site-packages/mpmath/libmp/libelefun.pygconstant_memo..g^s^KK  ::).1 1d4il#w)&)  zzgl++)r<r=__name____doc__)rBrDsf rC constant_memorJUs5AKAJ,AJ AI HrFcDa\3V3RllpSPVnV#)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. c<V^,pS!V4pV\\39d V^, p\^W2)\V4W4#))r r rr)r?rndwpvfixeds&& rCrBdef_mpf_constant..frsB BY "I 8]+ + FAAsHQK;;rF)rrI)rQrBsf rCdef_mpf_constantrSjs<  AI HrFcxW!, ^8Xd^\^V,^,4pV'gV^,'d\W@^,,V3#\)W@^,,V3#W,^,p\WWS4wrgp\WW#4wrp W,W,,Wz,W,3#)rrbsp_acot) qab hyperbolica1mp1q1r1p2q2r2s &&&& rCrWrW{suz 1q\ 1BAIr) )8RQ$Y* * qA!.JBB!.JBB 525="% &&rFc\RV,\P!V4, ^,4p\V^W24wrEpWE,V,WP,,#)z Compute acot(a) or acoth(a) for an integer a with binary splitting; see http://numbers.computation.free.fr/Constants/Algorithms/splitting.html ffffff?)r>mathlogrW)rYr?r[NprXrs&&& rC acot_fixedrksI D4K #b ()Aq!A*GA! S4K13 rFFc^ p\pVF8wrVV\V4\\V4W,V4,, pK: WC, #)z Evaluate a Machin-like formula, i.e., a linear combination of acot(n) or acoth(n) for specific integer values of n, using fixed- point arithmetic. The input should be a list [(c, n), ...], giving c*acot[h](n) + ... )rrrk)coefsr?r[ extraprecsrYrZs&&& rCmachinrpsDIA SVjQD DD NrFc \.ROVR4#)zn Computes ln(2). This is done with a hyperbolic Machin-type formula, with binary splitting at high precision. T)))i)i-"rpr?s&rC ln2_fixedrxs 3T4 @@rFc \.ROVR4#)zF Computes ln(10). This is done with a hyperbolic Machin-type formula. T)).)"1)rMrvrws&rC ln10_fixedrs 14 >>rFiqci-~ i@ cDW, ^8Xd\^V,^, ^V,^, ,^V,^, ,4pV^,\^,,^,pRV,V,\\V,,,pM}V'dV^8d \ RW4W,^,p\ WV^,V4wrp \ WqV^,V4wrp W,pW,pW,W,,pWEV3#)z Computes the sum from a to b of the series in the Chudnovsky formula. Returns g, p, q where p/q is the sum as an exact fraction and g is a temporary value used to save work for recursive calls. z binary splittingrG)rCHUD_CCHUD_ACHUD_Bprint bs_chudnovsky)rYrZlevelverboserDrirXmidg1r^r_g2rarbs&&&& rCrrs sax 1Q1Q1Q' ( qD619  " !GaK6&(? + uqy & -sQh"157G< "357G<  E E EBEM 7NrFc:\VR, R, ^,4pV'd \RV4\^V^V4wrEp\\^V,,4pV\,V,V\ V,,\ ,,pV#)z Compute floor(pi * 2**prec) as a big integer. This is done using Chudnovsky's series (see comments in libelefun.py for details). gv O @gbi],@zbinary splitting with N =)r>rrr6rrCHUD_D) r?r verbose_baserhrDrirXsqrtCrPs &&& rCpi_fixedrsv D l *Q ./A )1-Aq!W-GA! v$' (E &!F1H*f,-A HrFc&\V4^,#))rrws&rC degree_fixedrs D>3 rFcW, ^8Xd\\V43#W,^,p\W4wr4\W!4wrVW6,V,WF,3#)zY Sum series for exp(1)-1 between a, b, returning the result as an exact fraction (p, q). )rrbspe)rYrZr]r^r_rarbs&& rCrrsN  saxA qA !ZFB !ZFB 58RU?rFc\RV,\P!V4, ^,4p\^V4wr#W#,V,V,#)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>rfrgr)r?rhrirXs& rCe_fixedr sB CHTXXd^ #b ()A !9DA S4K! rFcV^ , p\\^V,,4\V,,pV^ , #)z* Computes the golden ratio, (1+sqrt(5))/2 )r6rr)r?rYs& rC phi_fixedrs2  BJD8af%&'T/:A 7NrFc tV^ ,p\\\\V4^4V4V^, 4#) )rmpf_logr/mpf_pi)r?rOs& rCln_sqrt2pi_fixedr*s. B GIfRj!4b946 BBrFc,\\V4V4#N)rrrws&rC sqrtpi_fixedr0s htnd ++rFc (VwrErgVwrrV'dV ^8d \R4hV ^8d"\VRV,W,,W#4#V R8XdV ^8XdBV'd.\\\ W^ ,\ V,4W#4#\ WV4#V'd+\\ W^ ,\ V,4V )W#4#\\ W^ ,V4WV4#\ W^ ,V4p \\W4W#4#)zJ Compute s**t. Raises ComplexResult if s is negative and t is fractional. z,negative number raised to a fractional powerrG) rr1r.r r2r3rmpf_expr-) rotr?rNssignsmansexpsbctsigntmantexptbccs &&&& rCmpf_powr>s EE JKK qy1rEkTZ8$DD rz 19tXab"3'&)*.55AS) )"8ABw"3'$)+/%<<x7C8$cJ J 7C A 71=$ ,,rFcV^8Xd W,^3#\V4p^p^V^\V4,,^,,p\wrgrV^,'d~Wp,pW,pW^, , p V \\Wy, 4,,p W8d!WyV, , pWV, , pTp V^,pV'gWx3#W,pWD,pW3,^, pV\\W, 4,,pW58d!WV, , pWCV, , pTpV^,pEK)zun-th power of a fixed point number with precision prec Returns the power in the form man, exp, man * 2**exp ~= y**n )rr rr>) ynr?bcexpworkprec_pmpepbcs &&& rC int_pow_fixedrZs  Avax !B CD1Xa[=(1,-HNA2 q55BB 6MCBI//C~L)Hn$ FA 6M Cg Wq[ '#ag,' ' =k"A = CB FrFc^2p\WW,, 4p\\VRV, ,44p^ pTp Tp \WBV,4Fp \Wa^, V 4wr\W^, V ,V , V , V , 4p\V^V ,V, V ,4V,pW^, \WkV , 4,,V,pT p K V# \d?\ XT4p\ T4p\ ^Yt4p\ YWT4p\T4pELi;i)2g?) rrr> OverflowErrorrr0rrrrr)rrr?exp1starty1rjfnextraextra1prevprirrrcBs&&&& rC nthroot_fixedrs E Aag~ & BQK ! E F E U +qA#u- B1e a",v5 6 1ac$hvo & * A#U7++ +a / , H  b%  a[ !R ' BE " 1I s:C99AEEclVwrErgV'd \R4hV'g^V\8Xd\#V\8Xd!V^8d\#V^8Xd\#\#V'g\#V^8d\#\#RpV^8dUV^8Xd\#V^8Xd \ WV4#VR 8Xd\ \WV4#\V,pRp^p W), pV)pV^8dVR8g&V\^RVR,,,48dzV^ ,p \V4p \^W4p \W W4p \V ^,V ^,V ^,V ^,W#4pV'd\ \WX , V4#V#V^V,,W!,, p V^ 8dW^ ,, p WV,, p Wz, p^pWn,pV^8d^pV)pV'dVVV,, pMVVV,,p\W^4p^ pWn,V^, V ,, V,V, p^pV'dVR8XgVR8Xd^pMVR 8XgVR 8Xd^p\VV,WV4p\VVW#4pV'd\ \WX , V4#V#) zYnth-root of a positive number Use the Newton method when faster, otherwise use x**(1/n) znth root of a negative numberFTi NgL<@gףp= ?urdrBrG)rr%rr r#r)r.r3r>rr0rrrrr)rorr?rNsignmanrr flag_inverse extra_inverseprec2rnthrjshiftsign1esrr rnd_shifts&&&& rC mpf_nthrootrsi Ds ;<<  9K :1u Av KK q5L L1u 6K 61C( ( 74#. .S!    B2v1:C$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$*AtQ] 2C88rFc\V^W4#)zcubic root of a positive number)r)ror?rNs&&&rCmpf_cbrtrs q!T ''rFcV\9d%\V,wr4WA8dW4V, , #V^ ,pV\8:dCVf \V4p\V4pWV, ,p\ Wu4Wb,,pM&\ \ \V4V^,4V4pV\8d W3\V&WV, , #)zT Fast computation of log(n), caching the value for small n, intended for zeta sums. ) log_int_cacheLOG_TAYLOR_SHIFTrxrlog_taylor_cachedrrrMAX_LOG_INT_CACHE) rr?ln2valuevprecrOrjxrPs &&& rC log_int_fixedrs  M$Q'  =T\* * B  ;B-C QK Q$K a $qu , WXa["Q$/ 4 7 a D>rFc^pW,^, pV^8d\W, 4^8dV#\W,4pTpV^, pKM)zR Fixed-point computation of agm(a,b), assuming a, b both close to unit magnitude. )absr6)rYrZr?ianews&&& rC agm_fixedrsH A ax q5S[1_H qsO  QrFc$W,V, pT;p;rEV'd)WR,V, pWE,V, pW4, pK0V\V,, pW3,V^, , pV\W,4,V, pT;p;rEV'd)WR,V, pWE,V, pWd, pK0\V,V^,,pWf,V, p\W6V4p\V4V,V,#)a Fixed-point computation of -log(x) = log(1/x), suitable for large precision. It is required that 0 < x < 1. The algorithm used is the Sasaki-Kanada formula -log(x) = pi/agm(theta2(x)^2,theta3(x)^2). [1] For faster convergence in the theta functions, x should be chosen closer to 0. Guard bits must be added by the caller. HYPOTHESIS: if x = 2^(-n), n bits need to be added to account for the truncation to a fixed-point number, and this is the only significant cancellation error. The number of bits lost to roundoff is small and can be considered constant. [1] Richard P. Brent, "Fast Algorithms for High-Precision Computation of Elementary Functions (extended abstract)", http://wwwmaths.anu.edu.au/~brent/pd/RNC7-Brent.pdf )rr6rr)rr?x2rorYrZrris&& rClog_agmr)s2 #$BNAN TdN STM '4-A QA :ag  %AMAM TdN STM  $1a4 A t A!A TNd "q ((rFc$\V4Fp\W,4pK \V,pW, V,W,,pV^8pV'dV)pWU,V, pWw,V, pTp V^,p WX,V, p^p V'dBWV ,, p V ^, p WV ,, p WX,V, pV ^, p KIW,V, p W,^V,,p V'dV )#V #)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. )rr6r) rr?rjronerPrv2v4s0s1kros &&& rC log_taylorrXsAY qw  T/C %$!% A q5D B #$B %DB B AB $A A  1f  Q 1f  TdN Q %DB AaCA r HrFcW\, , p\V,pW1, pW#3\9d\W#3,wrVM*W#\, ,p\WS^4pWV3\W#3&WT,pWd,pW, V,V,pWq,\V,V,,pW,V, p W,V, p Tp V^,p W,V, p^p V'dBWV ,, p V ^, p WV ,, p W,V, pV ^, p KIW,V, p W,^,pWn,#)zX Fixed-point computation of log(x), assuming x in (0.5, 2) and prec <= LOG_TAYLOR_PREC. )rcache_prec_stepslog_taylor_cacherr)rr?r cached_precdprecrYlog_arrPrrrrrros&& rCrrzs$ ##$A"4(K  E ++#AN35 00 111-,-:(KA OE %DQA DA-.A #$B %DB B AB $A A  d  Q d  TdN Q %DB 1 A 9rFc$Vwr4rVV'g4V\8Xd\#V\8Xd\#V\8Xd\#V'd \ R4hV^,pV^8Xd-V'g\#\ V\ V4,V)W4#WV,p\V4p V ^8:d^V , p V 'd\V,V, p MV\V^, ,, p \V 4p Wl, p W8d!\WW, WR4p\WW4#W}, pV R8d/\V 4V8d\ V\ V4,V)W4#V\8:d>\\WGV, 4V4pV'dW\ V4,, pM\V)\,pVV, p\!VV4pVV), p\#\%W4V4)pVV\ V4,,p\ W)W4#)z^ Compute the natural logarithm of the mpf value x. If x is negative, ComplexResult is raised. zlogarithm of a negative numberri')rr$r#r%rrrxrrrrr5LOG_TAYLOR_PRECrrLOG_AGM_MAG_PREC_RATIOr/rr)rr?rNrrrrrOmagabs_magrrr cancellationrr] optimal_magrs&&& rCrrs Ds  :e| 9Tk 9Tk <== B axLC " -sD>> &C#hG !|'  RK3&D'BqD/*Dtnx  %wz3SAAq3 3  B G r !IbM 1B3B B _ fSR%0" 5  Yr]" "Ac11 #  aO | Xa_b ) ) Qy}_ 3 **rFc V^,'gYrV^,'gyV^,'g3Yu;8Xd\8Xd \#\W39d\#\#V\8Xd\ \ V4W#4#V\8Xd\#\#\ W4p\ W4p^p\WEW&,4p\V\^ 4pV^,V^,,p V\8XgW)^,8d2\WEW&,\V^,V^,4, 4p\\ WrV4R4#)z) Computes log(sqrt(a^2+b^2)) accurately. rG) rr$r%r#rr(r-r+r!minr/) rYrZr?rNa2b2rh2 cancelled mag_cancelleds &&&& rC mpf_log_hypotrs Q441 Q44tt v~ K :71:t1 1 9K B B E  $BE2&IaL1-ME]VQY6 RTZBqE"Q%(88 9 WRs+R 00rFc|V^d8d5\P!\W^5, , 4R, 4pM-\P!\V4RV,, 4p^2p\\VR,4^5V, , 4p^2p\ W14FpWT, pW%V, ,p\ W%4wrgWu,V,pV\ WV, 4, V,\V,V^,V, ,,p W), pTpK \ W#V, 4#)dg@g@C)rfatanr>rr cos_sin_fixedrr) rr?rjrextra_prOcossintanrYs && rC atan_newtonr s s{ IIc1Bw<)'1 2 IIc!fS$Y& ' E CG E *+AG%&  U(O 'yS &G$$ +'2+36B,1O P E' !4Z  rFc^\V^, 4,^,pW!, pW3\9d\W3,wrEM)W\, ,p\WB4pWE3\W3&WC, WS, 3#rU)ratan_taylor_cacheATAN_TAYLOR_SHIFTr )rr?rrrYatan_as&& rCatan_taylor_get_cachedr"st $q&! "b (E LE z&&%ah/ 6 ++ ,Q&'(k!(# J&/ **rFcW\, , p\W!4wr4W, pWQ,V^,V, W5,V, ,\V,,,;rgV^,V, pW,V, p V^,p Wy,V, p^p V'dBWgV ,, pV ^, p WV ,, p Wy,V, pV ^, p KIW,V, p Wj, p WL,#)rrr) rr?rrYrrrrPrrrrros && rC atan_taylorr1s %% &A&q/IA Aiaddlqsd{;w$O PPB Q$$,B 'd B AB DA A  1f  Q 1f  V  Q 'd B A :rFc V'g\\W4R4#\\\V\V,4R44#)rVrG)r/rr*r4)rr?rNs&&&rCatan_infrEs5 *B// 9VD,s*;= 1rG)r&r rr2r+r-r!r)rr?rNrOrXs&&& rC mpf_acoshr2sS Bq$2>??ub126A 71$d 00rFc Vwr4rVV'g'V'dV\\39dV#\R4hWe,pV^8d-V^8XdV^8Xd\\.V,#\R4hV^,pVR8dWx)8d \ WW4#W), p\ V\V4p \\W4p \\\WV4W4R4#)z&atanh(x) is real only for -1 <= x <= 1r.rG) rr%rr#r$r5r+r r,r/rr.) rr?rNrrrrrrOrYrZs &&& rC mpf_atanhr4sDs S  HDEE (C Qw !8q%=& &DEE B Rx 9q2 2 t 4AaA WWQ2.:B ??rFcVwr4rVV'gV\8Xd\#V#\WV,4pV^8d6V^ 8gV\V48:d\ \ \ V44W4#W,^,p\V4p \\V ^4\V4p \WV4p \W4p \WV4p \WV4p \WW4p V #r)r$r%rrrr7rmpf_phir+r/r!r mpf_cos_pir.r,) rr?rNrrrrsizerOrYrZrrPs &&& rC mpf_fibonaccir:sDs  :K sv;D ax "9.DOT7 7 r B A !Q+AbA1AbAbAd A HrFc&V^8dV)p^pM^p\RVR,,4p\V4V, p\^WT,4p^ ^\WE)4,,pW,pWV, ,p\V,pV^8Hp V\8dW,V, ;rW,V, p \ ;r^pV 'd^W^, V,,qV , q^, pW^, V,,qV , q^, pW,V, p KeW,V, pV 'dW, V,pEM`W,V,pEMO\RVR,,4pW,V, ;rW.p\ ^V4F)pVPVR,V ,V, 4K+ \ .V,p^pV 'd\ V4FdpW^, V,,p V 'd&V^,'dVV;;,V ,uu&MVV;;,V , uu&V^, pKf V VR,,V, p K\ ^V4F$pVV,VV,,V, VV&K& \V4V,pV^8Xdf\VV,W,, 4pV'd VV, pM VV,p\ V4FpVV,V, pK VV, #V^, p\ V4FpVV,V, V, pK \\W,VV,, 44pV'dV)pVV, VV, 3#)z Taylor series for cosh/sinh or cos/sin. type = 0 -- returns exp(x) (slightly faster than cosh+sinh) type = 1 -- returns (cosh(x), sinh(x)) type = 2 -- returns (cos(x), sin(x)) ?g333333?rerG) r>rmaxrEXP_SERIES_U_CUTOFFrrappendsumr6r)rr?typerrjxmagrrOraltrrYx4rrrrrxpowersrsumsrorPpshifts&&& rCexponential_seriesrHs 1u B Cc MA A; D AtxA 3q< E B19A R-C 19C !!#"e]  Q3'MA72FA Q3'MA72FA" Ae] # A# A D$J #")1A NNGBKNR/ 0zA~ AYsAg 1q55$q'Q,'"&q'Q,'Q  72;2%A1AAwwqz)b0DG IO qy qscg ' AAAAA1 AEz AAA#&C'A sCGqs?+ , A5AuH%%rFcV\8d \W^4#\VR,4pW, p\V,;r4^pW,V, ;rgV'dBWe,qcV, q5^, pWe,qdV, qE^, pWg,V, pKIW@,V, pW4,pTp V'dW,V, pV^,pK"W, #)z Compute exp(x) as a fixed-point number. Works for any x, but for speed should have |x| < 1. For an arbitrary number, use exp(x) = exp(x-m*log(2)) * 2^m where m = floor(x/log(2)). r<)EXP_COSH_CUTOFFrHr>r) rr?rjrrrrYrrors && rC exp_basecaserK>s  o!!1-- D#IAID$B Acd]A  qq&! qq&! TdN $4B A A STM Q 6MrFcV\8d\W^4wr#W#,W#, 3#\W4p\W,,V,pWE3#)z Computation of exp(x), exp(-x) )rJrHrKr)rr?coshsinhrYrZs&& rCexp_expneg_basecaserOWsK o'3 y$)##QA TY A%A 4KrFcV\8d \W^4#V\, pW, p\V4pV\9dSV^ \,\, ,p\V^ \,^4wrgV^ , V^ , 3\V&\V,wrg\V, pWh,pWx,pWV,,p\ V,p Tp ^p W,V, )p V 'dRW,qV , q^, qV,V, p W,qV , q^, qV,V, )p KYW,W,, V, W,W,,V, 3#)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. )COS_SIN_CACHE_PRECrHCOS_SIN_CACHE_STEPr> cos_sin_cacher) rr?precsrrwcos_tsin_toffsetr r rrYs && rCcos_sin_basecaserYbs9    !!1-- % %E A AA  %%&88 9)!R0B-BAF !2I3 a #LE $ &F E EeOA T/C C A 34-A  61!}1 61A#$'71 Ysy T )ci .Ad-J KKrFcVwr4rVV'EdWe,pV^,pV'dV)pVR8d=V^8d6\V\RV,4,4p \WV,W4#Wx)8d\\W1V4#V^8dVW,p WZ,p V ^8d WK,p M WK), p \ V 4p \ W4wr\V4pW,p M#WX,p V ^8d WK,p M WK), p ^p\W4p\WNV, W4#V'g\#V\8Xd\#V#)r9g333333?) mpf_er>r1r5r rxdivmodrKrr$r)rr?rNrrrrrrOewpmodrXrlg2rs&&& rCrrs"Ds sh BY $C #:#(bT#X&'Aqs(D6 6 9tT5 5 7HE[F{MG$E"C!>DAAA IAXF{MG$A1!C2t11  Ez HrFc"VwrErgV'gsV'dkV'd)V\8Xd\#V\8Xd\#\#V\8Xd \\3#V\8Xd \\3#\\3#Wg,pV^,p VR8dMW)8d=V'd\ V^V, W4#\ \^W4p \ WW4p W3#W), p V^ 8d^^V^, ,,V 8ddV'd&\ \\.V,^V, W4#\ \\V4W4R4;rV'd \V 4p W3#V^8dYW,pWn,pV^8d W_,pM W_), p\V4p\VV4wpp\V4pVV,pM#Wi,pV^8d W_,pM W_), p^p\VV 4wppVV^V,, ,p VV^V,, , p V'dV )p V'dW,V ,p\WY)W4#\V VV , ^, W4p \V VV , ^, W4p W3#)z4Simultaneously compute (cosh(x), sinh(x)) for real xrG)r#r r$r!r%r5r/rr(r*rxr]r>rOr)rr?rNtanhrrrrrrOrMrNrror_rXrr`rrYrZs&&&& rC mpf_cosh_sinhrds)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&$4zrFcV^8d^p^V,pW2,V,p\V^, 4pV^, pWa,p V ^8d W ,p M W ), p \W4wrW8d W|, p MT p WV,^ , , 'd\V 4p W, p Wb, pM7V^, pKW2), pW,p V ^8d W ,p M W ), p ^p WV3#r6)rr]r>)rrrrOrcancellation_precr_pi2pi4rXrrrsmalls&&&& rCmod_pi2rjs Qw  "a H00E57#C(C[F{MG$!>DAwC##FH[ FA t  Q; A A  8OrFc|VwrVrxV'gEV'd \\rM \\rV^8XdW3#V^8XdV #V^8XdV #V^8XdV #W,p V^ ,p V ^8dyW)8drV'd\V\ V 44p\ \^W4p \ V^V, W4p V^8XdW3#V^8XdV #V^8XdV #V^8Xd \ WW4#V'EdVR8dVR8Xd2\p \\ 3\V^,4V, ,p MV^8Xd \ \rM \\rV^8XdW3#V^8XdV #V^8XdV #V^8Xd \WW4#Wg)^, , ^,^, p WmV)^, ,, p\V4V,pV^ ,V, p W|,pV^8d Wo,pM Wo), pV\V 4,V , pM\WgW4wpr\VV 4wrV ^,pV^8XdV )T rMV^8XdV )V )rM V^8XdY)rV'dV )p V^8Xd\W)W4p \W)W4p W3#V^8Xd\W)W4#V^8Xd\W)W4#V^8Xd \WW4#R#)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 NrG)r%r rr-rr5r!boolr.rrrjrYrr)rr?rNwhichpirrrrrrorrOrmag2rXrr]s&&&&& rC mpf_cos_sinrpsDs  qq A:ad{ A:ax A:ax A:ax (C B Qw 9Avbz*D!T/AAqvt1Az!$;z!8z!8z+at"AA r "9by5M$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--rFc\WV^4#rUrprr?rNs&&&rCmpf_cosrtb[#q-I&IrFc\WV^4#rrrrss&&&rCmpf_sinrwcrurFc\WV^4#)rrrss&&&rCmpf_tanrzdrurFc\WV^^4#r6rrrss&&&rCmpf_cos_sin_pir|esKaQR4S-SrFc\WV^^4#rUrrrss&&&rCr8r8f AS!Q0O)OrFc\WV^^4#rrrrss&&&rC mpf_sin_pirgr~rFc(\WV4^,#r6rdrss&&&rCmpf_coshrhmAS.I!.L'LrFc(\WV4^,#rUrrss&&&rCmpf_sinhrirrFc\WV^R7#)rV)rcrrss&&&rCmpf_tanhrjsmASq.Q'QrFcVf\V^, 4p\W4wr4\V4p\WA4wrVV^,pV^8XdWV3#V^8XdV)V3#V^8XdV)V)3#V^8XdWe)3#R#r)rr]r>rY)rr?rgrrrror]s&&& rCrros {tAv !>DA AA A $DA AAAvad{Avqb!e|Avqb1"f}Avae|vrFcVf \V4p\W4wr4\V4p\WA4pV^8d WS,#WS), #r)rxr]r>rK)rr?rrrrPs&&& rC exp_fixedr{sF {o !>DA AAQAAvv RyrFsagez)Warning: Sage imports in libelefun failed)F)FNrr6)rIrfrbackendrrrrrrr libmpfr r r r rrrrrrrrrrrrrrrrrrr r!r"r#r$r%r&r'r(r)r*r+r,r-r.r/r0r1r2r3r4r5r6 libintmathr7rJr>rQrRrSrrrrrrrrrrrrrJrSrWrkrprxrrrrrrrrrrrr7rr\ mpf_degreempf_ln2mpf_ln10rr mpf_sqrtpimpf_ln_sqrt2pirrrrrrrrrrrrr rrrrrr*r,r/r2r4r:rHrKrOrYrrdrjrprtrwrzr|r8rrrrrrsage.libs.mpmath.ext_libmplibsmpmath ext_libmp_lbmp ImportErrorAttributeErrorrrFrCrs.  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=L L54L5