a iZ?dZ@iZAdZBdZCiZDdZEdZFdZGiZHddgZIedeeBdD]*ZJeIeKdeJeBdgdeJd7ZIq~ddZLddZMddZNddZOdd d!ZPeLd"d#ZQeLd$d%ZRed&ZSed'ZTed(ZUed)ZVd*d+ZWeLdd,d-ZXd.d/ZYd0d1ZZeLd2d3Z[eLd4d5Z\eMe\Z]eMeXZ^eMe[Z_eMeYZ`eMeQZaeMeRZbeLd6d7ZceLd8d9ZdeMedZeeMecZfefd:d;Zgdd?Ziefd@dAZjefdBdCZkddDdEZldFdGZmdHdIZnddJdKZodLdMZpefdNdOZqdPdQZrdRdSZsdTdUZtdVdWZudXdYZvefdZd[Zwefd\d]Zxefd^d_Zyefd`daZzefdbdcZ{efdddeZ|efdfdgZ}efdhdiZ~ddjdkZdldmZdndoZdpdqZefdrdsZedfdtduZdvdwZeddfdxdyZefdzd{Zefd|d}Zefd~dZefddZefddZefddZefddZefddZefddZdddZdddZe dkrzRddlmmmZej4Z4ejZejqZqejZejZejgZgejZejZejlZlWn eefyedYn0dS)a( This module implements computation of elementary transcendental functions (powers, logarithms, trigonometric and hyperbolic functions, inverse trigonometric and hyperbolic) for real floating-point numbers. For complex and interval implementations of the same functions, see libmpc and libmpi. N)bisect)xrange)MPZMPZ_ZEROMPZ_ONEMPZ_TWOMPZ_FIVEBACKEND)- round_floor round_ceiling round_downround_up round_nearest round_fast ComplexResultbitcountbctablelshiftrshift giant_steps sqrt_fixedfrom_intto_int from_man_expto_fixedto_float from_float from_rational normalizefzerofonefnonefhalffinffninffnanmpf_cmpmpf_signmpf_absmpf_posmpf_negmpf_addmpf_submpf_mulmpf_div mpf_shift mpf_rdiv_int mpf_pow_intmpf_sqrtreciprocal_rnd negative_rnd mpf_perturb isqrt_fast)ifibpythonXiiii i cs,d_d_fdd}j|_j|_|S)z Decorator for caching computed values of mathematical constants. This decorator should be applied to a function taking a single argument prec as input and returning a fixed-point value with the given precision. NcsRj}||krj||?St|dd}|fi|_|_j||?S)Ng? ) memo_precmemo_valint)preckwargsrDZnewprecff/Users/vegardjervell/Documents/master/model/venv/lib/python3.9/site-packages/mpmath/libmp/libelefun.pyg^szconstant_memo..g)rDrE__name____doc__)rJrMrKrIrL constant_memoUs  rPcstffdd }j|_|S)z Create a function that computes the mpf value for a mathematical constant, given a function that computes the fixed-point value. Assumptions: the constant is positive and has magnitude ~= 1; the fixed-point function rounds to floor. cs<|d}|}|ttfvr$|d7}td|| t|||S)Nr>rr)rr rr)rGrndwpvfixedrKrLrJrs  zdef_mpf_constant..f)rrO)rUrJrKrTrLdef_mpf_constantjsrVc Cs||dkrNtd|d}|s(|d@r:t||d|fSt ||d|fS||d}t||||\}}}t||||\} } } | ||| || || fS)NrrA)rrbsp_acot) qab hyperbolicZa1mp1q1Zr1p2q2r2rKrKrLrX{s   rXcCsBtd|t|d}t|d||\}}}|||>||S)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)rFmathlogrX)rZrGr\NprYrrKrKrL acot_fixedsriFcCs>d}t}|D](\}}|t|tt||||7}q ||?S)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) + ... rC)rrri)ZcoefsrGr\Z extraprecsrZr[rKrKrLmachins  "rkcCstgd|dS)zz Computes ln(2). This is done with a hyperbolic Machin-type formula, with binary splitting at high precision. )))i)r<i-"TrkrGrKrKrL ln2_fixedsrqcCstgd|dS)zN Computes ln(10). This is done with a hyperbolic Machin-type formula. )).)"1)r>TrorprKrKrL ln10_fixedsrwiqci-~ i@ cCs||dkrbtd|dd|dd|d}|dtdd}d||tt|}nt|rz|dkrztd ||||d}t|||d|\}} } t|||d|\} } } | | }|| }| | | |}|||fS) 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. rrArWrBz binary splitting)rCHUD_CCHUD_ACHUD_Bprint bs_chudnovsky)rZr[levelverboserMrgrYZmidZg1r^r_Zg2r`rarKrKrLrs (   rc Csft|ddd}|r"td|td|d|\}}}ttd|>}|t||t|t}|S)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],@rAzbinary splitting with N =r)rFrrr7r}r~CHUD_D) rGrZ verbose_baserfrMrgrYZsqrtCrSrKrKrLpi_fixeds  rcCs t|dS)N)rrprKrKrL degree_fixedsrcCsT||dkrtt|fS||d}t||\}}t||\}}|||||fS)ze Sum series for exp(1)-1 between a, b, returning the result as an exact fraction (p, q). rrA)rrbspe)rZr[r]r^r_r`rarKrKrLrs    rcCs8td|t|d}td|\}}|||>|S)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)rFrdrer)rGrfrgrYrKrKrLe_fixed s rcCs(|d7}ttd|>t|>}|d?S)z2 Computes the golden ratio, (1+sqrt(5))/2 rCrA )r7r r)rGrZrKrKrL phi_fixedsrcCs&|d}tttt|d||dS)NrCr)rmpf_logr0mpf_pi)rGrRrKrKrLln_sqrt2pi_fixed*srcCstt||S)N)rrrprKrKrL sqrtpi_fixed0src Cs|\}}}}|\}} } } |r,| dkr,td| dkrNt|d|| | >||S| dkr| dkr|rttt||dt|||St|||S|rtt||dt|| ||Stt||d|| ||St||d|} tt|| ||S)zV Compute s**t. Raises ComplexResult if s is negative and t is fractional. rz,negative number raised to a fractional powerrBrrC) rr2r/r!r3r4rmpf_expr.) rjtrGrQZssignZsmanZsexpZsbctsigntmanZtexptbccrKrKrLmpf_pow>s0     rc Cs|dkr||dfSt|}d}d|dt|d}t\}}}} |d@r||}||}| |d7} | tt|| ?} | |kr|| |?}|| |7}|} |d8}|sq||}||}||d}|tt||?}||kr|||?}|||7}|}|d}qD||fS)zn-th power of a fixed point number with precision prec Returns the power in the form man, exp, man * 2**exp ~= y**n rArr|r)rr!rrF) ynrGbcexpZworkprec_pmpeZpbcrKrKrL int_pow_fixedZs8          rcCsd}z*t||||}tt|d|}WnDtyrt||}t|}td||}t|||}t|}Yn0d}|} |} t|||D]r} t ||d| \} } t| |d| | | | }t |d| || |}||dt || | |}| } q|S)N2g?rrCrA) rrrF OverflowErrorrr1rrrrr)rrrGexp1starty1rhfnextraZextra1prevprgrrrbBrKrKrL nthroot_fixeds(    rcCs||\}}}}|rtd|sd|tkr(tS|tkrL|dkri NgL<@gףp= ?rCrWurdrJN)rr&r r!r$r*r/r4rFrr1rrrrr)rjrrGrQsignmanrrZ flag_inverseZ extra_inverseprec2rZnthrhshiftZsign1esrrZ rnd_shiftrKrKrL mpf_nthroots  , "      rcCst|d||S)zcubic root of a positive numberrW)r)rjrGrQrKrKrLmpf_cbrtsrc Cs|tvr(t|\}}||kr(|||?S|d}|tkrp|durHt|}t|}|||>}t||||}nttt||d|}|tkr||ft|<|||?S)z` Fast computation of log(n), caching the value for small n, intended for zeta sums. rCNrz) log_int_cacheLOG_TAYLOR_SHIFTrqrlog_taylor_cachedrrrMAX_LOG_INT_CACHE) rrGln2valueZvprecrRrhxrSrKrKrL log_int_fixeds    rcCsJd}||d?}|dkr,t||dkr,|St||}|}|d7}q|S)z^ Fixed-point computation of agm(a,b), assuming a, b both close to unit magnitude. rrr|r<)absr7)rZr[rGiZanewrKrKrL agm_fixeds   rcCs|||?}|}}}|r>|||?}|||?}||7}q|t|>7}|||d?}|t||>|?}|}}}|r|||?}|||?}||7}qzt|>|d>}|||?}t|||}t||>|S)a* Fixed-point computation of -log(x) = log(1/x), suitable for large precision. It is required that 0 < x < 1. The algorithm used is the Sasaki-Kanada formula -log(x) = pi/agm(theta2(x)^2,theta3(x)^2). [1] For faster convergence in the theta functions, x should be chosen closer to 0. Guard bits must be added by the caller. HYPOTHESIS: if x = 2^(-n), n bits need to be added to account for the truncation to a fixed-point number, and this is the only significant cancellation error. The number of bits lost to roundoff is small and can be considered constant. [1] Richard P. Brent, "Fast Algorithms for High-Precision Computation of Elementary Functions (extended abstract)", http://wwwmaths.anu.edu.au/~brent/pd/RNC7-Brent.pdf rAr)rr7rr)rrGx2rjrZr[rrgrKrKrLlog_agm)s$            rc Cst|D]}t||>}qt|>}|||>||}|dk}|rH| }|||?}|||?}|} |d} |||?}d} |r| || 7} | d7} | || 7} |||?}| d7} q|| ||?} | | d|>} |r| S| S)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. rrWrzrAr)rr7r) rrGrhronerSrv2v4s0s1krjrKrKrL log_taylorXs0         rcCs&||t?}t|}||}||ftvr:t||f\}}n(||t>}t||d}||ft||f<||L}||L}|||>|}||>t|>|}|||?} | | |?} |} |d} || |?}d} |r| || 7} | d7} | || 7} || |?}| d7} q| | |?} | | d>}||S)zd Fixed-point computation of log(x), assuming x in (0.5, 2) and prec <= LOG_TAYLOR_PREC. r<rWrzrAr)rcache_prec_stepslog_taylor_cacherr)rrGrZ cached_precdprecrZZlog_arrSrrrrrrjrKrKrLrzs6             rcCs|\}}}}|s4|tkrtS|tkr(tS|tkr4tS|r@td|d}|dkrp|sXtSt|t|| ||S||}t|} | dkrd| } | rt|>|} n|t|d>} t | } || } | |krt | | | || | d}t || ||S|| 7}| dkr&t | |kr&t|t|| ||S|t kr\t t||||}|r||t|7}nH| t}||}t||}|| 7}tt||| }||t|8}t|| ||S)zj 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$r&rrrqrrrrr6LOG_TAYLOR_PRECrrLOG_AGM_MAG_PREC_RATIOr0rr)rrGrQrrrrrRmagZabs_magrrrZ cancellationrr]Z optimal_magrrKrKrLrsN         rc Cs|ds||}}|dsz|dsR||kr6tkr>nntSt||fvrNtStS|tkrjtt|||S|tkrvtStSt||}t||}d}t||||}t|td}|d|d} |tks| | dkrt||||t |d|d}t t|||dS)z1 Computes log(sqrt(a^2+b^2)) accurately. rr>rCrArWrB) r r%r&r$rr)r.r,r"minr0) rZr[rGrQZa2b2rh2 cancelledZ mag_cancelledrKrKrL mpf_log_hypots.     "rc Cs|dkr$tt||d?d}ntt|d|}d}tt|dd|?}d}t||D]h}||7}|||>}t||\}}||>|}|t||||>t|>|d|?} || }|}qdt|||S)Nd5g@Cg@rrA)rdatanrFrr cos_sin_fixedrr) rrGrhrZextra_prRcossintanrZrKrKrL atan_newtons  *rcCspdt|d>d}||}||ftvr:t||f\}}n&||t>}t||}||ft||f<||?||?fS)Nrr>)ratan_taylor_cacheATAN_TAYLOR_SHIFTr)rrGrrrZatan_arKrKrLatan_taylor_get_cached"s   rc Cs||t?}t||\}}||}||>|d|?|||?t|>}}|d|?}|||?} |d} || |?}d} |r||| 7}| d7} | || 7} || |?}| d7} q~| ||?} || } || S)NrArWrz)rrr) rrGrrZrrrrSrrrrrjrKrKrL atan_taylor1s$ ,        rcCs,|stt||dSttt|t|dS)NrB)r0rr+r5)rrGrQrKrKrLatan_infEsrc Cs|\}}}}|sH|tkrtS|tkr0td||S|tkrDtd||StS||}||dkrht|||S| |dkrt|d|||S|dt|}|dkrtd||}d} nd} t||} |r| } |t krt | |} n t | |} | rt |d?d| } |r| } t | | ||S)Nrrr>rATF)r r$rr%r&r6rr1rATAN_TAYLOR_PRECrrrr) rrGrQrrrrrrRZ reciprocalrrZrKrKrLmpf_atanJs6        rc CsD|\}}}}|\}} } } | s|tkrF|tkrFt|dkr|ttfvr,td|tkr>t||S|d}t||}ttt|||} t| tt|||} t t | ||dS)Nrz%acos(x) is real only for -1 <= x <= 1rr) r!r"rrr.r3r-r/r,r0rrrKrKrLmpf_acoss     rc Cs|d}|\}}}}||}|dkrJ|| kr@t|d|||S|| 7}ttt||t||} tt|| |} |rtt| |t|St| ||SdS)Nr>r) r6r3r,r.r!r)r+rr5) rrGrQrRrrrrrrYrKrKrL mpf_asinhs   rcCsJ|d}t|tdkrtdttt||t||}tt|||||S)NrrBz acosh(x) is real only for x >= 1)r'r!rr3r,r.r"r)rrGrQrRrYrKrKrL mpf_acoshs rc Cs|\}}}}|s,|r,|ttfvr$|Std||}|dkr`|dkrX|dkrXttg|Std|d}|dkr|| krt||||S|| 7}t|t|} tt||} t t t | | |||dS)Nz&atanh(x) is real only for -1 <= x <= 1rrrrrB) r r&rr$r%r6r,r!r-r0rr/) rrGrQrrrrrrRrZr[rKrKrL mpf_atanhs$       rc Cs|\}}}}|s |tkrtS|St||}|dkr\|dksH|t|kr\ttt|||S||d}t|} tt | dt |} t | ||} t ||} t | | |} t| | |} t | | ||} | S)NrrCr>r)r%r&rrrr8rmpf_phir,r0r"r mpf_cos_pir/r-) rrGrQrrrrsizerRrZr[rrSrKrKrL mpf_fibonaccis$       rcCs|dkr| }d}nd}td|d}t||}td||}ddt|| }||}|||K}t|>}|dk} |tkr.|||?} } | | |?} t} }d}| r| |d|} | | 7} |d7}| |d|} || 7}|d7}| | |?} q| ||?}| r|| |}n || |}ntd|d}|||?} } || g}td|D]}||d| |?q`tg|}d}| rt|D]P}| |d|} | r|d@r||| 8<n||| 7<|d7}q| |d|?} qtd|D]}|||||?||<q t||}|dkrt ||||>}|rd||}n||}t|D]}|||?}qt||?S|d}t|D]}|||?|}qt t ||>||}|r| }||?||?fSd S) 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?rCrAg333333?rcrBN) rFrmaxrEXP_SERIES_U_CUTOFFrrappendsumr7r)rrGtyperrhZxmagrrRrZaltrrZZx4rrrrrZxpowersrZsumsrjrSZpshiftrKrKrLexponential_seriessr         "      rc Cs|tkrt||dSt|d}||7}t|>}}d}|||?}}|r||}||7}|d7}||}||7}|d7}|||?}qH|||?}||}|} |r|||?}|d8}q|| ?S)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)). rrrAr)EXP_COSH_CUTOFFrrFr) rrGrhrrrrZrrjrrKrKrL exp_basecase>s$      rcCsJ|tkr(t||d\}}||||fSt||}t||>|}||fS)z( Computation of exp(x), exp(-x) r)rrrr)rrGcoshsinhrZr[rKrKrLexp_expneg_basecaseWs  rc Cs4|tkrt||dS|t}||?}t|}|tvrl|dtt>}t|dtd\}}|d?|d?ft|<t|\}}t|}||L}||L}|||>8}t|>} |} d} |||? } | r | | } | | 7} | d7} | ||?} | | } | | 7} | d7} | ||? } q| || ||?| || ||?fS)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. rArCr)COS_SIN_CACHE_PRECrCOS_SIN_CACHE_STEPrF cos_sin_cacher) rrGZprecsrrwZcos_tZsin_toffsetrrrrZrKrKrLcos_sin_basecasebs,   $(rcCs6|\}}}}|r||}|d}|r,| }|dkrb|dkrbt|td|} t| ||>||S|| krztt|||S|dkr||} || } | dkr|| >} n || ?} t| } t| | \}} t|}| |L} n(||} | dkr|| >} n || ?} d}t| |}t|||||S|s$tS|t kr2t S|S)Nr:rg333333?r) mpf_erFr2r6r!rqdivmodrrr%r )rrGrQrrrrrrRewpmodrrlg2rrKrKrLrsB         rcCsP|\}}}}|s\|r\|r4|tkr$tS|tkr0tStS|tkrDttfS|tkrTttfSttfS||}|d} |dkr|| kr|rt|d|||Sttd||} t||||} | | fS| | 7} |dkr2dd|d>| kr2|rtttg|d|||Sttt|||d} } |r*t | } | | fS|dkr| |}||}|dkr`||>}n || ?}t |}t ||\}}t |}||L}n*|| }|dkr||>}n || ?}d}t || \}}||d|?} ||d|?} |r| } |r| | >| }t|| ||St| || d||} t| || d||} | | fSd S) z4Simultaneously compute (cosh(x), sinh(x)) for real xrrrrCrWrBrAN)r$r!r%r"r&r6r0rr)r+rqrrFrr)rrGrQtanhrrrrrrRrrrrjrrrrrrZr[rKrKrL mpf_cosh_sinhsj               r cCs|dkrd}d|>}|||}t|d}|d?}||} | dkrN|| >} n || ?} t| |\} } | |krx|| } n| } | ||d?rt| } | |?} ||}q|d7}q n2|| 7}||} | dkr|| >} n || ?} d} | | |fS)Nrr>rrC)rrrF)rrrrRrZcancellation_precrpi2Zpi4rrrrsmallrKrKrLmod_pi2s6         r cCs|\}}}}|s^|r tt} } n tt} } |dkr:| | fS|dkrF| S|dkrR| S|dkr^| S||} |d} | dkr| | kr|rt|t| }ttd||} t|d|||} |dkr| | fS|dkr| S|dkr| S|dkrt||||S|r|dkr|dkr(t} ttft|d@|A} n |dkr>tt} } n tt} } |dkrZ| | fS|dkrh| S|dkrv| S|dkrt| | ||S|| d?dd?} || | d>}t ||}|d|} || }|dkr||>}n || ?}|t | | ?}nt ||| | \}} } t || \} } | d@}|dkrH| | } } n0|dkrb| | } } n|dkrx| | } } |r| } |dkrt | | ||} t | | ||} | | fS|dkrt | | ||S|dkrt | | ||S|dkrt| | ||SdS)z 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 rrrArWrCrBN)r&r!r r.rr6r"boolr/rrr rrr)rrGrQwhichpirrrrrrjrrRrZmag2rrr]rKrKrL mpf_cos_sinsx                       rcCst|||dSNrrrrGrQrKrKrLmpf_cosbrcCst|||dS)NrArrrKrKrLmpf_sincrrcCst|||dS)NrWrrrKrKrLmpf_tandrrcCst|||ddS)NrrrrrKrKrLmpf_cos_sin_pierrcCst|||ddSrrrrKrKrLrfrrcCst|||ddS)NrArrrrKrKrL mpf_sin_pigrrcCst|||dSNrr rrKrKrLmpf_coshhrrcCst|||dSrrrrKrKrLmpf_sinhirrcCst|||ddS)Nr)r rrrKrKrLmpf_tanhjrrcCs|durt|d}t||\}}t|}t||\}}|d@}|dkrP||fS|dkrb| |fS|dkrv| | fS|dkr|| fSdS)NrrWrrA)rrrFr)rrGr rrrrjr]rKrKrLros rcCsN|durt|}t||\}}t|}t||}|dkr@||>S|| ?SdSr)rqrrFr)rrGrrrrSrKrKrL exp_fixed{s r Zsagez)Warning: Sage imports in libelefun failed)F)FN)N)r)r)N)N)rOrdrbackendrrrrrr r Zlibmpfr r r rrrrrrrrrrrrrrrrrrr r!r"r#r$r%r&r'r(r)r*r+r,r-r.r/r0r1r2r3r4r5r6r7Z libintmathr8rrrrrrrrrrrrrrrrrrPrVrXrirkrqrwr~rr}rrrrrrrrrrZ mpf_degreeZmpf_ln2Zmpf_ln10rrZ mpf_sqrtpiZmpf_ln_sqrt2pirrrrrrrrrrrrrrrrrrrrrrrrrrrrrr r rrrrrrrrrrrr Zsage.libs.mpmath.ext_libmpZlibsZmpmathZ ext_libmpZ_lbmp ImportErrorAttributeErrorrrKrKrKrLs      (    #       6 S / "" H, # #      K  -C$O