
    gW                     \   d dl mZ d dlmZ d dlmZmZ d dlmZ d dl	m
Z
mZmZmZ d dlmZ d dlmZ d dlmZ d d	lmZmZ d d
lmZmZ d dlmZ d dlmZmZmZm Z m!Z! d dl"m#Z# d dl$m%Z% d dl&m'Z' d dl(m)Z) d dl*m+Z+ d dl,m-Z- d dl*m.Z.m/Z/m0Z0m1Z1m2Z2m3Z3m4Z4m5Z5m6Z6 d dl7m8Z8m9Z9m:Z:  ed          \  Z;Z<Z= edd          \  Z>Z?Z@ ed           ed          cZAZBd ZCd ZDd ZEd ZFd ZGd  ZHe:d!             ZIe:d"             ZJd# ZKd$ ZLd% ZMd& ZNd' ZOe9d(             ZPd) ZQd* ZRd+ ZSd, ZTd- ZUd. ZVd/S )0    )Sum)Add)
DerivativeFunction)Mul)IRationaloopi)S)symbols)	factorial)explog)acoshasech)sqrt)acosasinatancossin)airyai)erf)gamma)	integrate)fps)O)	rational_algorithmFormalPowerSeriesFormalPowerSeriesProductFormalPowerSeriesComposeFormalPowerSeriesInversesimpleDErational_independentexp_rehyper_re)raisesXFAILslowzx y zzn m kT)integerfrc                     dt           dz
  dz  t           dz
  z  z  } t          | t           t                    dt           dz
  z   dz   t          t          dz             t          t                    z  z
  ddfk    sJ dt           z   t           dz  z   t           dz  z   t           dz
  t           dz
  z  z  } t          | t           t                    ddt           dz
  z  z  dz   t           dz   dfk    sJ t          t
          t          z  t          t           z  z
  t
          t           z  z
  t           dz  z   z  } t          | t           t                    t
          t           dz
  z   t          z  t
          t          z
  z  t          t           dz
  z  t          z  t
          t          z
  z  z   ddfk    sJ t           dt           z
  t           dz  z
  z  } t          | t           t                    J t          | t           t          d          t          d	d          t          d
          dz  z   t           dz
  z  t          d
           dz  t          j
        z   z  t          d
           dz  t          j
        z
  t           dz
  z  t          d
          dz  t          j
        z   z  z   ddfk    sJ dt           dz  dt           z  z   dz   z  } t          | t           t                    J t          | t           t          d          t          d	t          z   t           dz
  z  z  dz  t          d	t          z
  t           dz
  z  z  dz  z
  ddfk    sJ t          dt           z             } t          | t           t                    d	t           z   t          z  ddfk    sJ t          t                     } t          | t           t                    J t          | t           t          d          t          t          t           z  z  dz  t          t           t           z  z  dz  z
  t          z  ddfk    sJ t           t          t                     z  t          dt           dz  z             dz  z
  } t          | t           t                    J t          | t           t          d          t          t          t           dz   z  z  dz  t          t           t           dz   z  z  dz  z
  t          t          dz
  z  z  ddfk    sJ t          dt           z   dt           z
  z            dz  t          t                     z
  } t          | t           t                    J t          | t           t          d          d	t           z   dz  t          t          t           z  z  dz  z
  t          t           t           z  z  dz  z   t          j
        z   t          z  ddfk    sJ t          t          t                     t           t                    J d S )N      r      i   Tfull   
   )xr   kr   zymr	   r   r   Halfr   r   r   r   r,   s    Z/var/www/html/ai-engine/env/lib/python3.11/site-packages/sympy/series/tests/test_formal.pytest_rational_algorithmr@      sd   	a!eaZ1q5!"AaA&&
qb1f+	Yq1u--	!<	=q!DE E E E 
QA1	!a%AE!23AaA&&	Q!a[1	a!eQ'( ( ( ( 	
QqS1Q3Y1_q!t#$AaA&&QK<>a!e
$!qb1f+a-AE)B	CQJK K K K 	
QUQT\AaA&&...aAD111B??T!WWQY&1"q&
177(2+
 GG8A:1"q&
)
q''"*qv

  "#A	'' ' ' ' 	
QTAaCZ!^AaA&&...aAD111
R!VrAv
!	#q"q&QBF);';q&@	@!QGH H H H 	AE

AaA&&!*q!Q        	QAaA&&...aAD111Q!W9/Qqbz\Q.
.!	3Q:; ; ; ; 	
$q''	CAqDMMA%%AaA&&...aAD111Q!a[=A
QB1"q&>!1Q 6
6
QU)
	    	QUq1u"T!WW,AaA&&...aAD1111":+/QqA2wY!O
+q1"|q.@
@
&
!	    c!ffa++33333    c            
      d   t           }  | g t                    g k    sJ  | t          t                    t          t                    gt                    t          t                    t          t                    gk    sJ  | t          dz  t          t                    t          t          t                    z  t          dz  gt                    t          dz  t          dz  z   t          t          t                    z  t          t                    z   gk    sJ  | t          j        t          t          t                    z  t          t                    t          t                    t          z  t          t                    t          t                    t          gt                    t          dz   t          t          t                    z  t          t                    z   t          t                    t          z  t          t                    z   t          t                    gk    sJ d S )Nr0   r1   r/   )r%   r8   r   r   r   Oner   )ris    r?   test_rational_independentrE   P   sn   	B2b!99????2s1vvs1vv""s1vvs1vv&666662q!tSVVQs1vvXq!t,a00	
A1aAhQ'() ) ) )2quaAhAAq#a&&#a&&!DaHH	
Q#a&&3q66!3q66!8c!ff#4c!ff=> > > > > >rA   c            	         t          t          t                    t          t                    D ]G} | t          t                     t	          t          t                    t                    z   dfk    sJ  t          t          t                    t          t                    D ]L} | t          t                    t	          t          t                    t          t                    z   dfk    sJ  t          t          dt          z             t          t                    D ]e} | t          dz   t	          t          t                    t          d          z  t	          t          t                    t                    z   dfk    sJ  t          t          t                    t          t                    D ]u} | t          t	          t          t                    t                    z  t          dz  dz
  t	          t          t                    t          t                    z  z   dfk    sJ  t          t          t                    t          t                    z  t          t                    D ]t} | dt          t                    z  dt	          t          t                              z  z
  t	          t          t                    t          t                    z   dfk    sJ  t          dt          z   dt          z
  z  t          z  t          t                    D ]_} | dt          z  t          t                    z  t          dz  dz
  t	          t          t                    t                    z  z   dfk    sJ  t          t          t                    t          t                    D ]W} | t           t          t                    z  t	          t          t                    t          t                    z   dfk    sJ  d S d S )Nr/   r0   )
r$   r   r8   r,   r   r   r   r   nr   )DEs    r?   test_simpleDErI   Z   s   s1vvq!$$  qttej1q111155555s1vvq!$$  addZ!a333Q77777s1q5zz1a((  q1uj1q!444z!A$$7J7JJANNNNNtAww1%%  a
1Q44+++q!taxAaDD!Q9O9O.OO    s1vvc!ff}a++  a!fqAaDD!1!111JqttQ4J4JJANNNNNQQ!+Q22  ac!A$$h!Q$(JqttQ,?,?!??CCCCCvayy!Q''  qb1g
1Q44A 6 66::::: rA   c                     t          t                     t          t          t                    t                    z   } t          | t          t
                    t	          t
                     t	          t
          dz             z   k    sJ t          t                    t          t          t                    t          t                    z   } t          | t          t
                    t	          t
                    t	          t
          dz             z   k    sJ t          t                    t          t          t                    t                    z   t          t          t                    t          t                    z   } t          | t          t
                    t	          t
                    t	          t
          dz             z   t	          t
          dz             z   k    sJ t          t          t                    t                    t          t          t                    t          t                    z   } t          | t          t
                    t	          t
                    t	          t
          dz             z   k    sJ t          t          t                    t          d          t          t          t                    t          d          z   t          t          t                              z   } t          | t          t
                    t	          t
                    t	          t
          dz             z   t	          t
          dz             z   k    sJ d S )Nr/   r0   r1   r2   )r,   r8   r   r&   r-   r9   ds    r?   test_exp_rerM   t   s   	
1
1Q44###A!Q??qtteaAhh.....	!z!A$$1%%%A!Q??addQq1uXXo----	!z!A$$"""Z!a%;%;;A!Q??addQq1uXXo!a%888881Q44j1q!444A!Q??addQq1uXXo----1Q44AAaDD!Q!7!77*QqTT:J:JJA!Q??addQq1uXXo!a%8888888rA   c                  
   t          t                    t          t          t                    t          t                    z   } t          | t          t
                    t	          t
                    t
          dz   t
          dz   z  t	          t
          dz             z  z   k    sJ t           t          t                    z  t          t          t                    t          t                    z   } t          | t          t
                    t
          dz   t
          dz   z  t	          t
          dz             z  t	          t
                    z
  k    sJ dt          t                    z  dt          t          t                    t                    z  z
  t          t          t                    t          t                    z   } t          | t          t
                    dt
          z  dz
  t	          t
          dz             z  t
          dz   t
          dz   z  t	          t
          dz             z  z   dt	          t
                    z  z   k    sJ dt          z  t          t                    z  t          dz  dz
  t          t          t                    t                    z  z   } t          | t          t
                    t
          t	          t
                    z  dt          z  t	          t
          dz             z  z   t
           dz
  t	          t
          dz             z  z   k    sJ t          dz  dz   t          t          t                    t                    z  t          t          dz  dz
  z  t          t          t                    t          t                    z  z   } t          | t          t
                    t
          t
          dz
  z  t
          z   t	          t
                    z  dt
          z  t
          dz   t
          dz   z  z
  dz   t	          t
          dz             z  z   k    sJ t          dz  dz
  t          t          t                    t          d          z  dt          z  t          t          t                    t          t                    z  z   t          t          t                    t                    z   } t          | t          t
                    t
          t
          dz
  z  t
          dz
  z  dt
          z  t
          dz
  z  z   t
          z   t	          t
                    z  t
           t
          dz   z  t
          dz   z  t	          t
          dz             z  z   k    sJ d S )	Nr/   r0   r1   r7   r2   	   (   )r,   r8   r   r'   r-   r9   rG   rK   s    r?   test_hyper_rerR      s   	!z!A$$1%%%AAq!!!ac{1QU88'; ;;;;;	
1Q44*QqTT1a(((AAq!QQ!a% 81Q44 ?????	!A$$:addA&&&&AaDD!Q)?)??AAq!	A1QU88q1uq1uoaAhh661Q44?@ @ @ @ 	
!AaDDAqD1Hj1q1111AAq!	!A$$1Qq1uXX!a1q5 112 2 2 2 
BJqttQ'''!QUQY-
1Q44A8N8N*NNAAq!	
AEQ!!q1uq2v&6 6 ;Qq2vYYFFG G G G Q$(JqttQ**	*QqSAaDD!Q1G1G-G	G	AaDD!		
AAq!
QU)QU
ac1q5k
)A
-qtt	3"a!e*a!e
aAhh	&
'( ( ( ( ( (rA   c                     t          d          dk    sJ t          dt                    dk    sJ t          dt          d          dk    sJ t          dt          d          dk    sJ t          dt          z  dt          dz  z  z             dt          z  dt          dz  z  z   k    sJ t          t          dt          z             dd          t          dt          z             k    sJ t          t          dz  t          z   dz             } t          | t                    sJ | j        t          dz  t          z   dz   k    sJ | d         dk    sJ | d         t          dz  k    sJ |                     d	          t          dz  t          z   dz   t          t          d	z            z   k    sJ |                                 t          dz  t          z   dz   k    sJ t          t          dt          z                       } t          | t                    sJ | j        t          dt          z             k    sJ | 	                    t          t                    | k    sJ | d d
         dt          t          dz   dz  t          dz  dz  t          d	z   d	z  gk    sJ |                     t                    t          k    sJ |                     d          t          t          dz  dz  z
  t          dz  dz  z   t          d	z  d	z  z
  t          d
z  d
z  z   k    sJ | j        j        d         }| j        t          d| z   t          |z  z  |z  |dt           f          k    sJ |                     d           | d d
         }}t#          |          D ]\  }}|d
k    r n||         |k    sJ t%          t                                         t                    } t          | t                    sJ |                                 t          t          dz  dz  z
  t          d
z  dz  z   t          t          dz            z   k    sJ t'          t(          d            t'          t*          d            d S )Nr/   r0   +dir-F)hyperrationalr   r2   r6   r1      r5   rG   x   c                  :    t          t          t          z            S N)r   r;   r8    rA   r?   <lambda>ztest_fps.<locals>.<lambda>   s    AaC rA   c                  .    t          t          d          S )Nr   rU   )r   r8   r_   rA   r?   r`   ztest_fps.<locals>.<lambda>   s    s1!}}} rA   )r   r8   r   
isinstancer    functiontruncater   
polynomialsubsr;   as_leading_termak	variablesinfiniter   r
   	enumerater   r(   NotImplementedError
ValueError)r,   r9   ftsits         r?   test_fpsrr      s   q66Q;;;;q!99>>>>q!""""q!""""qsQq!tV|!a1f,,,,s1q5zz7773q1u::EEEEAqD1HqLAa*+++++:AA%%%%Q419999Q41a4<<<<::a==AqD1HqL1QT7722222<<>>QTAX\))))CAJJAa*+++++:QU####66!Q<<1RaR5QAqD57AqDFQTE!G44444Q1$$$$<<??a!Q$q&j1a461AqDF:QT!VCCCCC	qA:qbzk!Q$.1Aq":>>>>>>JJJ"1"B"  166EtqyyyyyA

1Aa*+++++::<<1q!tAv:1S01QT77:::::
 0 0111
:,,-----rA   c                  
   t          dt          z            dt          z  k    sJ t          t          dz  t          z   dz   t          dz  z  d          t          dz  t          z   dz   t          dz  z  k    sJ dt          dz
  dz  t          dz
  z  z  } t          | t                                                    t          dd          t          t          dd          z  z
  dt          dz  z  d	z  z
  d
t          dz  z  dz  z
  dt          dz  z  dz  z
  dt          dz  z  dz  z
  t	          t          dz            z   k    sJ dt          z   t          dz  z   t          dz  z   t          dz
  t          dz
  z  z  } t          | t                                                    t
          j        t          t          dd          z  z   dt          dz  z  d	z  z   d
t          dz  z  dz  z   dt          dz  z  dz  z   dt          dz  z  dz  z   t	          t          dz            z   k    sJ t          dt          z
  t          dz  z
  z  } t          | t          d                                          t          t          dz  z   dt          dz  z  z   dt          dz  z  z   dt          dz  z  z   t	          t          dz            z   k    sJ dt          dz  dt          z  z   dz   z  } t          | t          d                                          t
          j        t          dz  z
  t          dz  dz  z   t          dz  d	z  z
  t          dz  d	z  z   t	          t          dz            z   k    sJ t          dt          z             } t          | t                                                    t          t          dz  dz  z
  t          dz  dz  z   t          dz  dz  z
  t          dz  dz  z   t	          t          dz            z   k    sJ t          | t          d                                          t          | t          d                                          k    sJ t          | t          d                                          t          d          t          dd          z
  t          dz
  dz  dz  z
  t          dz
  dz  dz  z   t          dz
  dz  dz  z
  t          dz
  dz  dz  z   t          dz  z   t	          t          dz
  dz  t          df          z   k    sJ t          | t          dd                                          t          d          t          dd          z
  t           dz   dz  dz  z
  t           dz   dz  dz  z
  t           dz   dz  dz  z
  t           dz   dz  dz  z
  t          dz  z   t	          t          dz
  dz  t          df          z   k    sJ t          t                    } t          | t          d                                          t          t          dz  dz  z
  t          dz  dz  z   t	          t          dz            z   k    sJ t          | t          dd                                          t          | t          dd                                          k    sJ t          | t          dd                                          t          d          t          dd          z
  dt          dz
  dz  z  dz  z
  dt          dz
  dz  z  dz  z   dt          dz
  dz  z  dz  z
  dt          dz
  dz  z  dz  z   t          dz  z   t	          t          dz
  dz  t          df          z   k    sJ t          | t          ddd                                          t          d          t          dd          z
  dt           dz   dz  z  dz  z
  dt           dz   dz  z  dz  z
  dt           dz   dz  z  dz  z
  dt           dz   dz  z  dz  z
  t          dz  z   t	          t          dz
  dz  t          df          z   k    sJ t          t          t                    z  t          dt          dz  z             dz  z
  } t          | t          d                                          t          dz  dz  t          dz  d z  z
  t	          t          dz            z   k    sJ t          dt          z   dt          z
  z            dz  t          t                    z
  } t          | t          d                              d!"          dt          dz  z  dz  dt          d#z  z  d#z  z   t	          t          d!z            z   k    sJ d S )$Nr/   r0   r1   r5   rU   r6   r2         1             iA  @   rZ   q      Tr3      Q   D    )r4   rV         iw  iq  )   i	=     r7   r[      )	r   r8   rd   r	   r   r   r=   r   r   r>   s    r?   test_fps__rationalr      s   qs88!1q11$"---!Q$(Q,!Q$1FFFFF	a!eaZ1q5!"Aq!99	"a1Xa^^+	+bAgai	7"QT'"*	Ds1a4xPR{	R	QT"
Aww
        
QA1	!a%AE!23Aq!99	
!HQNN"	"R1WQY	.AqD	;c!Q$hrk	I	QT"
Aww
        	
QUQT\Aq!$((**	AqD1QT6AadF"Qq!tV+a1gg56 6 6 6 	
QTAaCZ!^Aq!$((**	1q!tAv1Q&Aa/!AqD''9: : : : 	AE

Aq!99	AqDF
QT!Vad1f$q!tAv-!Q$78 8 8 8q!$$&&#a*;*;*;*D*D*F*FFFFFq!Q<<  ""	Q(1a..	 AEA:b=	0AEA:b=	@
a%!C
q51*T/
*,-aC
023QUQJA2G2G
HI I I I q!QB((**	Q(1a..	 QBFQ;r>	1aR!VaKN	B"q&1S
BFQ;t+
,./c
245q1uqj1a&4I4I
JK K K K 	QAq!$((**a!Q$q&j1a46.AAadGG.KKKKKq!$A&&&//11Aqt$$$--//0 0 0 0q!QT"""++--	a8Aq>>	!Aq1uqjLO	3b!a%!mC6G	G	
AEA:c	
AzM%/
023A#
6891q51*q!f8M8M
NO O O O q!QTr***3355	a8Aq>>	!ArAvkM"$4	4rA26A+~c7I	I	
QBFQ;s	
!a!^E1
245aC
8:;QUQJA:O:O
PQ Q Q Q 	
$q''	CAqDMMA%%Aq!$((**ad1fq!tBw.>1a4.HHHHHQUq1u"T!WW,Aq!$((2(..!AqD&(Qq!tVAX2E!R%2PPPPPPPrA   c            
         t          t                    } t          | t                                                    t          t          dz  dz  z
  t          dz  dz  z   t	          t          dz            z   k    sJ t          t                    } t          | t                                                    dt          dz  dz  z
  t          dz  dz  z   t	          t          dz            z   k    sJ t          t                    } t          | t                                                    dt          z   t          dz  dz  z   t          dz  dz  z   t          dz  dz  z   t          dz  dz  z   t	          t          dz            z   k    sJ t          t                    } t          | t                                                    t          t          dz  dz  z
  t          dz  dz  z   t	          t          dz            z   k    sJ t          t          t                              } t          | t                                                    t          t          dz            t          t          t          dz            z  z
  t          dz  t          t          dz            z  dz  z   t          dz  t          t          dz            z  dz  z
  dt          dz  z  t          t          dz            z  dz  z   t          dz  t          t          dz            z  dz  z
  t	          t          dz            z   k    sJ t          t          t                              } t          | t                                                    t          t          z   t          t          dz  z  dz  z
  t          t          dz  z  d	z  z
  t	          t          dz            z   k    sJ t          dt          z            } t          | t                                                    t          dz  t          z
  t          dz  dz  z   t          dz  dz  z
  t	          t          dz            z   k    sJ t          t          t                    z  t          dt          dz  z             dz  z
  } t          | t          d
                                          t          dz  dz  t          dz  dz  z
  t	          t          dz            z   k    sJ t          dt          z             } t          | t          d
                                          t          t          dz  dz  z
  t          dz  dz  z   t          dz  dz  z
  t          dz  dz  z   t	          t          dz            z   k    sJ t          t          dz            } t          | t                                                    dt          dd          z  t          t          dd                    z  dt          z  z  dt          dd          z  t          dz  z  dt          t          dd                    z  z  z
  t	          t          dz            z   k    sJ t          t                    t          t                    z  } t          | t                                                    t          t          dz  z   t          dz  dz  z   t          dz  dz  z
  t	          t          dz            z   k    sJ t          t                    t          t                    z  t          z  } t          | t                                                    dt          z   t          dz  dz  z   t          dz  dz  z
  t          dz  dz  z
  t	          t          dz            z   k    sJ t          t                    t          t                    z  } t          | t                                                    t          dt          dz  z  dz  z
  dt          dz  z  dz  z   t	          t          dz            z   k    sJ d S )Nr1   rZ   r6   r\   r/   r0   r2      ru   FrY   r      Z      )r   r8   r   rd   r   r   r   r   r   r   r   r   r   r   r	   r   r>   s    r?   test_fps__hyperr      sP   AAq!991q!tAv:1S#81QT77#BBBBBAAq!991q!tAv:1R#7!AqD''#AAAAAAAq!99	A1QAa!Q$r')AqDH4qAww>? ? ? ? 	QAq!991q!tAv:1Q#61a4#@@@@@DGGAq!99	RTQs2a4yy[	 1a4BqD		>!#3	3ad3r!t99nQ6F	F	
1a4BqD			"	
 !tC1II~a/
023AqD''
:; ; ; ; 	E!HHAq!991q51QT6!8#3a1fQh#>1a4#HHHHHQqS		Aq!992a4!8ad1f#4q!tAv#=!Q$#GGGGG	$q''	CAqDMMA%%Aq!e$$$--//1a46AqDG3Ca1gg3MMMMMAE

Aq!e$$$--//	AqDF
QT!Vad1f$q!tAv-!Q$78 8 8 8 	q!tAq!99	
HQNN	5!Q00	0!B$	7	
HQNN	1a4	5!Q#8#8!8	9
:<=adGG
DE E E E 	As1vvAq!991q!t8ad1f#4q!tBw#>1a4#HHHHHAs1vvaAq!991q51a46>AqDG#;ad2g#E!Q$#OOOOOAQAq!991qAvax<!AqD&)#;a1gg#EEEEEEErA   c                     t           dz  t          t                     z  } t          | t                                                     dt           dz  z  ddt           dz  z  z  z
  t	          dd          z   t           dz  dz  z
  t           dz  dz  z   t          t           dz            z   k    sJ t           dz  t          t                     z  } t          | t           d	
                                          t           dz  t           dz  dz  z
  t          t           dz            z   k    sJ t          t          t                               t           z  } t          | t                                                     t           t           dz  dz  z
  t           dz  dz  z   t           dz  dz  z
  t           dz  dz  z   t          t           dz            z   k    sJ t           dz  t          t          t                               z  } t          | t                                                     t           dz  t           dz  dz  z
  t           dz  dz  z   t           dz  dz  z
  t          t           dz            z   k    sJ d S )Nr/   r2   rZ   r0   r\     选 Fr   r1   r6   r     逝  )	r8   r   r   rd   r	   r   r   r   r   r>   s    r?   test_fps_shiftr   ,  s   	2c!ffAq!99	!Q$AadFhq#...Ad:QT&[H1QPQT77RS S S S 	
1T!WWAq!e$$$--//	1q!tAv!Q$        	DGGQAq!99	AqDF
QT"Wq!tCx'!Q$u*4qAww>? ? ? ? 	
1Sa\\Aq!99	1q!tAv1R!Q$s(*Qq!tWW45 5 5 5 5 5rA   c                  j   t           t          t                     z  t          dt           dz  z             dz  z
  } t          | t                                                     t           dz  dz  t           dz  dz  z
  t          t           dz            z   k    sJ t          t                     t          t                     z   t          t                     z
  t          dt           z             z   } t          | t                                                     t           dt           dz  z  dz  z
  t           dz  dz  z
  t           dz  dz  z   t          t           dz            z   k    sJ dt           z  t          t                     z   } t          | t                                                     dt           z  t           z   t           dz  dz  z
  t           dz  dz  z   t          t           dz            z   k    sJ t          t                     t          t                     z
  dt           dz
  z  z   } t          | t                                                     d	t           dz  dz  z
  d
t           dz  z  dz  z
  dt           dz  z  dz  z
  dt           dz  z  dz  z
  t          t           dz            z   k    sJ d S )Nr/   r0   r2   r   rZ   r1   r6   r\   rO   r   r   r   w   )	r8   r   r   r   rd   r   r   r   r   r>   s    r?   test_fps__Add_exprr   >  s   	$q''	CAqDMMA%%Aq!991a46AqDG#3a1gg#=====AQ#a&& 3q1u::-Aq!991qAvax<!Q$q&#81a46#AAadGG#KKKKK	!c!ffAq!991Q37QT!V#3ad3h#>1a4#HHHHHAQ!QU)#Aq!99
QT!Va1fQhAqD+c!Q$hsl:Qq!tWWDE E E E E ErA   c                  .   t          t                    } t          | t          t                    | k    sJ t          | t          t                                                     t          dt          dz  z  t          t          f          k    sJ t          t                    } t          | t          t                                                    dt          dt          dz  z  t          t          f          z   k    sJ t          | t          t                                                     dt          dt          dz  z  t          t          f          z   k    sJ t          t                    } t          | t          t          d                                          ddt          dz  z  z  ddt          dz  z  z  z   dt          z  z
  t          dz  z   t          dt          dz  z  t          t          f          z   k    sJ t          | t          t           d                                          ddt          dz  z  z  ddt          dz  z  z  z   dt          z  z
  t          dz  z
  t          dt          dz  z  t          t          f          z   k    sJ t          dt          z             } t          | t          t                    ddt          dz  z  z  dd	t          d	z  z  z  z
  ddt          dz  z  z  z   ddt          dz  z  z  z
  dt          z  z   t          dt          z            z
  t          dt          dz  z  t          t          f          z   k    sJ t          | t          t                     ddt          dz  z  z  dd	t          d	z  z  z  z
  ddt          dz  z  z  z   ddt          dz  z  z  z
  dt          z  z   t          t          z  z   t          dt          z            z
  t          dt          dz  z  t          t          f          z   k    sJ d S )
Nr/   rZ   r5   Tr3   r6   r1   r0   r2   )r   r8   r   r
   rd   r   r   r   r   r   r   r>   s    r?   test_fps__asymptoticr   M  sl   AAq!R==Aq!bS>>""$$!AqD&1b'(:(:::::AAq!R==!!##q1Qq!tVaW+=+='=====q!bS>>""$$QqAv2w-?-?(?????QAq!Rd###,,..
AadFa1a4j 1Q3&A-!AqD&1b'0B0BBC C C Cq!bSt$$$--//
AadFa1a4j 1Q3&A-!AqD&1b'0B0BBC C C C 	AE

Aq!R==	Qq!tVq!AqD&z	!AqAvJ	.AadF	;ac	AC!HH	L	
1QT6Ar7		
    q!bS>>	Qq!tVq!AqD&z	!AqAvJ	.AadF	;ac	AAbD	H	RT
qAv2w''
() ) ) ) ) )rA   c                     t          t          t                              t          z  } t          | t                                                    dt          t                    z  t          t                    dz  z
  t          t          dd          z  dz  z   t          t          dd          z  dz  z
  t          t          dd          z  d	z  z   t          t          d
d          z  dz  z
  t          t          dd          z  dz  z   t          t          dz            z   k    sJ t          t          t                              t          z  } t          | t                                                    t          t          dd          z  t          t          dd          z  dz  z
  t          t          dd          z  dz  z   t          t          d
d          z  dz  z
  t          t          dd          z  d	z  z   t          t          dz            z   k    sJ t          t          t                              t          dz  z  } t          | t                                                    t          t          dd          z  t          t          dd          z  dz  z
  t          t          j	        z  dz  z   t          t          dd          z  dz  z
  t          t          dd          z  d
z  z   t          t          dd          z  dz  z
  t          t          d
d          z  dz  z   t          t          dd          z  dz  z
  t          t          dz            z   k    sJ t          t          t                              } t          | t                                                                                    dt          dz  z   t          dz  dz  z   t          dz  dz  z   t          dz  dz  z   t          dz  dz  z   t          t                    z   t          t          dd          z  dz  z   t          t          dd          z  dz  z   t          t          dd          z  dz  z   t          t          d
d          z  d	z  z   t          t          dd          z  dz  z   t          t          dz            z   k    sJ t          t          t                              t          z  } t          | t                                                                                    t          t          dz  dz  z   t          dz  dz  z   t          dz  dz  z   t          dz  dz  z   t          t          dd          z  z   t          t          dd          z  dz  z   t          t          dd          z  dz  z   t          t          d
d          z  dz  z   t          t          dd          z  d	z  z   t          t          dz            z   k    sJ d S )Nr/   rZ   r1   r0   r\   r6   r   r   r   rP   i ar   l    LQf r5      r   r   r   r2   r   i _7 )r   r   r8   r   rd   r	   r   r   r   r=   r   expandr>   s    r?   test_fps__fractionalr   e  s#   DGGqAq!99	
477T!WWQY	HQNN!23!6	6	
HQNN	4	
 "#Xa^^"3F":
;	
HQNN	8	#
$&'"a&8&C
DFG1gg
NO O O O
 	DGGqAq!99	
HQNN	QA.q0	01hq!nn3DS3H	H	
HQNN	4	
 "#Xb!__"4V";
<>?1gg
FG G G G 	T!WW1Aq!99	
HROO	a"a02	2QYq[	@	
HQNN	1	
 (1a..02
356A5Fr5I
J	
HQNN	2	
 !8B?? 22 5
689!Q$
@A A A A
 	DGGAq!99&&((	
QqS1a47	QT#X	%1U
	2QT'\	ADGG	K	
HQNN	1	
 (1a..04
578(1a..7H7M
N	
HQNN	6	!
"$%xA$6x$?
@BCAqD''
JK K K K
 	DGGQAq!99&&((	
QT!Vad2g	1S	(1a4:	58Aq>>8I	I	
HQNN	1	
 (1a..04
578(1a..7H7M
N	
HROO	F	"
#%&q!tWW
-. . . . . .rA   c                  V   t          ddt          z  z             } t          | t                    t          t                     t          z   t          dz  dz  z
  t          dz  dz  z   t          dz  dz  z
  t          dz  dz  z   t          t          dz            z   k    sJ t          | t          d          t          t                     t          z   t          dz  dz  z
  t          dz  dz  z   t          dz  dz  z
  t          dz  dz  z   t          t          dz            z   k    sJ d S )	Nr/   r0   r1   r2   r6   rZ   Fr   )r   r8   r   r   r>   s    r?   !test_fps__logarithmic_singularityr     s   A!GAq!99	Q!ad1fq!tAv%1Q.Aa7!AqD''AB B B Bq!e$$$	Q!ad1fq!tAv%1Q.Aa7!AqD''AB B B B B BrA   c                     t          t                    } t          | t                    t          d          t          t                    z
  t          dz  dz  z
  dt          dz  z  dz  z
  t	          t          dz            z   k    sJ d S )Nr0   r2   r1   rz   rZ   )r   r8   r   r   r   r>   s    r?   &test_fps__logarithmic_singularity_failr     si    aAq!99AQ!Q$q&01QT6"9<qAwwFFFFFFFrA   c                  "   t           t          z  t          t           dz            z  } t          | t                                         d          t           t          dz   z  t           t          dz   z  dz  z
  t          t           t          dz   z  t                     z   k    sJ t           t          z  t          dt           z             z  } t          | t                     }|j        j        d         }|j	        t          d| z   t           |t          z   z  z  |z  |dt          f          k    sJ t           dz
  t          z  t          dt           z             z  } t          | t           d                                          t           dz
  t          z  t          d          z  t           dz
  t          dz   z  dz  z   t           dz
  t          dz   z  dz  z
  t           dz
  t          dz   z  d	z  z   t           dz
  t          d
z   z  dz  z
  t           dz
  t          dz   z  dz  z   t          t           dz
  t          dz   z  t           df          z   k    sJ t           t          dz
  z  t          t                     z  } t          | t                                                     t           t          dz
  z  t           t          z  dz  z
  t           t          dz   z  dz  z   t          t           t          d
z   z  t                     z   k    sJ t           t          dz
  z  t          t                     z  t           t          z  t          t                     z  z   } t          | t                                                     t           t          dz
  z  t           t          dz   z  z   t           t          dz   z  dz  z   t           t          z  z   t           t          d
z   z  dz  z   t           t          dz   z  dz  z   t          t           t          dz   z  t                     z   k    sJ t           t          z  t          t                     z  } t          | t           t                                                    t           t          dz
  z   dz  t           t          dz
  z  dz  z   t           t          z  t          dz  dt           z  z
  z  z   t          dt           z  t           z  t           dz  z  t           t          f          z   k    sJ t           t          dz  z  t          t                     z  } t          | t                                                     t           t          dz  z  t           t          dz  dz   z  dz  z
  t           t          dz  d
z   z  dz  z   t          t           t          dz  dz   z  t                     z   k    sJ t           t          t           z   z  t          t                     z  } t          | t                                                     t           t           t          z   dz   z  t           t           t          z   dz   z  dz  z
  t           t           t          z   dz   z  dz  z   t          t           t           t          z   dz   z  t                     z   k    sJ d S )Nr0   ru   rZ   r/   r   r5   r1   r}   r~   r2   r   r6   r   r   <   r\   )r8   rG   r   r   rd   r   r   rh   ri   rj   r   r
   r   r   r   r   r<   )r,   fpr9   s      r?   test_fps_symbolicr     s/   	1SAYYAq!99a  AAJQUA$=!a!e*a@P@P$PPPPP	1SQZZA	QB
A;raRj[QU#Q&Ar
334 4 4 4 
Q
3q1u::Aq!Q<<  ""
a%!CFF	a!eq1u-a/	/1q5AE2B22E	EQRSVWRWHXY[H[	[
a%1q5	#	
!"Q!a% 0 5
6891q5AE:JQPQF8S8S
TU U U U 	
AE
3q66Aq!99	
QUad1f	q1q5z"}	,qQUQ/?/?	?A A A A 	
AE
3q66AqDQK'Aq!99	
QUa!a%j	 1q1u:a<	/!Q$	6	
QUB
QUB
')*1q1u:q)9)9
:; ; ; ; 	
1T!WWAq!R==!!##
a!e*QQUA	%1bdQqSj(9	9	
AaCA2;q!taW	%	%
&' ' ' ' 	
AaCQAq!99	AaC1qsQw<>!A!aLO3aAaC!Ga6H6HHI I I I 	
AE
3q66Aq!99	AEAIQUQY))AA	N3,>>1q1uqy>STAUAUUV V V V V VrA   c                  \   t           t          t                     z  t          dt           z            z  } t          | t                                                     dt           dz  z  dt           dz  z  z   t           dz  dz  z
  t           dz  z
  t          t           dz            z   k    sJ d S )Nr0   r1   r2   r6   rZ   )r8   r   r   r   rd   r   r>   s    r?   test_fps__slowr     s    	#a&&QqSAq!991QT6AadF?QT!V#;ad#BQq!tWW#LLLLLLLrA   c                     t          t          t                              t          t          t                              c} | z   }|j        t          t                    t          t                    z   k    sJ |                                dt          z   t          dz  dz  z
  t          dz  dz  z
  t          dz  dz  z   t          dz  dz  z   t          t          dz            z   k    sJ dz   }|j        t          t                    dz   k    sJ |                                dt          z   t          dz  dz  z
  t          dz  dz  z   t          t          dz            z   k    sJ d| z   }|j        t          t                    dz   k    sJ |                                dt          dz  dz  z
  t          dz  dz  z   t          t          dz            z   k    sJ t          z   t          t                    k    sJ |                                  d	t          dz  dz  z   t          dz  dz  z
  t          t          dz            z   k    sJ z
  t          j	        u sJ | z
  }|j        t          t                    t          t                    z
  k    sJ |                                d	t          z   t          dz  dz  z   t          dz  dz  z
  t          dz  dz  z
  t          dz  dz  z   t          t          dz            z   k    sJ dz
  }|j        t          t                    dz
  k    sJ |                                d	t          z   t          dz  dz  z
  t          dz  dz  z   t          t          dz            z   k    sJ d| z
  }|j        t          t                     dz   k    sJ |                                t          dz  dz  t          dz  dz  z
  t          t          dz            z   k    sJ t          t          fd
           t          t          fd           dz  }|j        dt          t                    z  k    sJ |                                dt          z  t          dz  dz  z
  t          dz  dz  z   t          t          dz            z   k    sJ d| z  }|j        dt          t                    z  k    sJ |                                ddt          dz  z  dz  z
  t          dz  dz  z   t          t          dz            z   k    sJ | z  t          |           k    sJ t          z  t          t                    k    sJ                                 }|j        t          t                    k    sJ |                                dt          dz  dz  z
  t          dz  dz  z   t          t          dz            z   k    sJ |                                 }|j        t          t                     k    sJ |                                t           t          dz  dz  z   t          dz  dz  z
  t          t          dz            z   k    sJ |                                                                 }|j        t          t                     k    sJ |                                d	t          dz  dz  z   t          dz  dz  z
  t          t          dz            z   k    sJ t          t          t          t                                        }|                                }|                                                                ddt          t                    z  z  t          j        z   t          dz  z   t          dz  dz  z   t          dz  dz  z   t          dz  dz  z   t          dz  dz  z   t          t                    dz  z   t          t%          dd          z  dz  z   t          t%          dd          z  dz  z   t          t%          dd          z  dz  z   t          t%          dd          z  dz  z   t          t%          dd          z  dz  z   t          t          dz            z   k    sJ                     t          ddf          t          d           dz   k    sJ t'          t          ddf          t          d           dz   k    sJ t'          t                    }|j        t          t                     k    sJ |                                d	t          dz  dz  z   t          dz  dz  z
  t          t          dz            z   k    sJ |                     t                    }|j        t          t                    k    sJ |                                t          t          dz  dz  z
  t          dz  dz  z   t          t          dz            z   k    sJ d S )Nr/   r0   r1   rZ   r2   r   r6   r\   r5   c                  P     t          t          t                    d          z   S )Nr5   rU   r   r   r8   f1s   r?   r`   z&test_fps__operations.<locals>.<lambda>  s    rCAB$7$7$77 rA   c                  P     t          t          t                    d          z   S )Nr/   x0r   r   s   r?   r`   z&test_fps__operations.<locals>.<lambda>  s    rCA1$5$5$55 rA   rQ   ru   r      i`'  i  i *0   i  r   i ; rP   i n r   i 9r   )r   r   r8   r   rc   rd   r   r   r   Zeror(   rm   r   diffr   r   r   r=   r	   r   )f2fsumfsubfmfdf3fir   s          @r?   test_fps__operationsr     s   Q[[#c!ff++FB7D=CFFSVVO++++==??	A1QAa!Q$r')AqDH4qAww>? ? ? ? 6D=CFFQJ&&&&==??a!ead1fnq!tCx7!AqD''AAAAAr6D=CFFQJ&&&&==??a!Q$q&j1a472Qq!tWW<<<<<Fs2qzz!!!!KKMM>R!Q$q&[1a472Qq!tWW<<<<<G7D=CFFSVVO++++==??
QAa!Q$q& 1a47*QT#X5!Q$?@ @ @ @ 6D=CFFQJ&&&&==??b1fq!tAvo1S81QT77BBBBBr6D=SVVGaK''''==??ad1fq!tBw.1a488888
:7777888
:5555666	aB;!CFF("""";;==AaC!Q$q&L1a472Qq!tWW<<<<<	
RB;!CFF("""";;==A!Q$qL1a461AadGG;;;;;GB####Fs2qzz!!!!	B;#a&&    ;;==A1QJAb01QT77:::::	B;3q66'!!!!;;==QBAaK!Q$s(2Qq!tWW<<<<<			B;3q66'!!!!;;==BAaK!Q$r'1AadGG;;;;;	Sa\\		B	B;;==!!	
Ad1ggI	!B$	&Ac	1AqDJ	>Af	L	
Ah
a
#%&A%6r%9
:<=x1~~<Md<R
S	
HQNN	5	 
!#$hq!nn#4W#<
=?@(2q//?QR[?[
\ 
1a4
    <<Aq	""s1vvgk1111R!Q##Aw{2222	2q		B;3q66'!!!!;;==BAaK!Q$r'1AadGG;;;;;	aB;#a&&    ;;==A1QJAc1AadGG;;;;;;;rA   c                     t          t          t                              t          t          t                              t          t	          t                              c} }t          t          fd           t          t          fd           t          t          fd           t          t          fd                               | t                    t          t                    sJ t          j
        t                    sJ t          j        t                    sJ j        t          t                    k    sJ j        t          t                    k    sJ j        t          t                    t          t                    z  k    sJ                     d          t          t          dz  z   t          dz  dz  z   k    sJ                     d          t          t          dz  z   t          dz  dz  z   t%          t          dz            z   k    sJ                     d          t          t          dz  z   t          dz  dz  z   k    sJ t          t(          fd           t          t(          fd	           t          t(          fd
           t          t(          fd                               |t                                        d          t          dt          dz  z  dz  z
  k    sJ                     |t                                        d          t          dt          dz  z  dz  z
  t%          t          dz            z   k    sJ d S )Nc                  ^                          t          t                    t                    S r^   )productr   r8   r   s   r?   r`   z#test_fps__product.<locals>.<lambda>      rzz#a&&!44 rA   c                  ~                          t          t          t                    d          t          d          S Nr5   rU   r2   r   r   r   r8   r   s   r?   r`   z#test_fps__product.<locals>.<lambda>  *    rzz#c!ff"*=*=*=q!DD rA   c                  ~                          t          t          t                    d          t          d          S Nr/   r   r2   r   r   s   r?   r`   z#test_fps__product.<locals>.<lambda>  *    rzz#c!ff*;*;*;QBB rA   c                  z                          t          t          t                              t          d          S Nr2   )r   r   r   r;   r8   r   s   r?   r`   z#test_fps__product.<locals>.<lambda>  #    rzz#c!ff++q!<< rA   r2   r0   r1   c                  .                          d          S Nr6   
_eval_termfprods   r?   r`   z#test_fps__product.<locals>.<lambda>!      (8(8(;(; rA   c                       j         S r^   rj   r   s   r?   r`   z#test_fps__product.<locals>.<lambda>"  	     rA   c                  8                          t                    S r^   _eval_derivativer8   r   s   r?   r`   z#test_fps__product.<locals>.<lambda>#      (>(>q(A(A rA   c                  8                          t                    S r^   r   r8   r   s   r?   r`   z#test_fps__product.<locals>.<lambda>$      (:(: rA   )r   r   r8   r   r   r(   rm   r   rb   r!   ffpsr    gfpsr,   grc   _eval_termsrd   r   re   rl   )r   r   r   r   s     @@r?   test_fps__productr     s   SVVc#a&&kk3s1vv;;JBB
:4444555
:DDDDEEE
:BBBBCCC
:<<<<===JJr1Ee566666ej"344444ej"3444447c!ff7c!ff>SVVc!ff_,,,,Q1q!t8ad1f#44444>>!AqD1a46 1AadGG ;;;;;A!ad(QT!V"33333
 ; ; ; ;<<<
 6 6 6 6777
 A A A ABBB
 : : : :;;;::b!((++q1QT6!8|;;;;::b!%%a((A!Q$qL1QT77,BBBBBBBrA   c                      t          t          t                              t          t          t                              t          t	          t                              c t          t           fd           t          t           fd           t          t           fd           t          t           fd           t          t           fd           t          t          fd                                t                    t          t                    sJ t          j
        t                    sJ t          j        t                    sJ j        t          t                    k    sJ j        t          t                    k    sJ j        t          t          t                              k    sJ                     d          dt          z   t          d	z  d	z  z   t          d
z  dz  z
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  k    sJ                                 dt          z   t          d	z  d	z  z   t          d
z  dz  z
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  t%          t          dz            z   k    sJ                     d          dt          z   t          d	z  d	z  z   t          d
z  dz  z
  t%          t          dz            z   k    sJ t          t&          fd           t          t&          fd           t          t&          fd           t          t&          fd                                t                                        d
          dt          z   t          d	z  d	z  z   t%          t          d
z            z   k    sJ                      t                                        d          dt          z   t          d	z  d	z  z   t          d
z  dz  z
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  t          dz  dz  z   t%          t          dz            z   k    sJ                      t                                        d          dt          z   t          d	z  d	z  z   t          d
z  dz  z
  t          dz  dz  z
  t%          t          dz            z   k    sJ                     t                                        d
          t          t          dz  dz  z
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z            z   k    sJ                     t                                        d          t          t          dz  dz  z
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  t%          t          dz            z   k    sJ                     t                                        d          t          t          dz  dz  z
  t          dz  dz  z   t%          t          dz            z   k    sJ d S )Nc                  ^                          t          t                    t                    S r^   )composer   r8   r   s   r?   r`   z#test_fps__compose.<locals>.<lambda>-  r   rA   c                  ~                          t          t          t                    d          t          d          S r   r   r   r   r8   r   s   r?   r`   z#test_fps__compose.<locals>.<lambda>.  r   rA   c                  ~                          t          t          t                    d          t          d          S r   r   r   s   r?   r`   z#test_fps__compose.<locals>.<lambda>/  r   rA   c                  z                          t          t          t                              t          d          S r   )r   r   r   r;   r8   r   s   r?   r`   z#test_fps__compose.<locals>.<lambda>0  r   rA   c                  :                          t                    S r^   r   r8   )r   r   s   r?   r`   z#test_fps__compose.<locals>.<lambda>2      rzz"a00 rA   c                  :                          t                    S r^   r   )r   r   s   r?   r`   z#test_fps__compose.<locals>.<lambda>3  r   rA   rZ   r/   r0   r2   ru   r6   r   c                  .                          d          S r   r   fcomps   r?   r`   z#test_fps__compose.<locals>.<lambda>@  r   rA   c                       j         S r^   r   r   s   r?   r`   z#test_fps__compose.<locals>.<lambda>A  r   rA   c                  8                          t                    S r^   r   r   s   r?   r`   z#test_fps__compose.<locals>.<lambda>B  r   rA   c                  8                          t                    S r^   r   r   s   r?   r`   z#test_fps__compose.<locals>.<lambda>C  r   rA   r   r   r   r1   r7   i;  )r   r   r8   r   r   r(   rm   r   rb   r"   r   r    r   r,   r   rc   r   rd   r   rl   )r   r   r   r   s   @@@@r?   test_fps__composer   *  s3   SVVc#a&&kk3s1vv;;JBB
:4444555
:DDDDEEE
:BBBBCCC
:<<<<===
:00000111
:00000111JJr1Ee566666ej"344444ej"3444447c!ff7c!ff>SQ[[((((Q1q51a46>AqDF#:QT"W#DDDDD>>q1uq!tAv~1Q6Ab@1QT77JJJJJ>>!A1QAa 7!AqD'' AAAAA
 ; ; ; ;<<<
 6 6 6 6777
 A A A ABBB
 : : : :;;;::b!%%a((AEAqDFNQq!tWW,DDDDD::b!%%a((	A1QAa!Q$r')AqDH4q!tBw>1a4HI I I I::b!%%a((	A1QAa!Q$r')AadGG34 4 4 4 ::b!%%a((A1QJ1a4,@@@@@::b!%%a((A1QJAb,@1QT6#:,MPQRSUVRVPWPW,WWWWW::b!%%a((A1QJAb,@1QT77,JJJJJJJrA   c                     t          t          t                              t          t          t                              t          t	          t                              c} }t          t          fd           |                     t                    t          t                    sJ t          j
        t                    sJ t          t          fd           j        t          t                    k    sJ j        t          t                     k    sJ                     d          dt          z
  t          dz  dz  z   t          dz  dz  z
  t          dz  d	z  z   k    sJ                                 dt          z
  t          dz  dz  z   t          dz  dz  z
  t          dz  d	z  z   t          dz  d
z  z
  t!          t          dz            z   k    sJ                     d          dt          z
  t          dz  dz  z   t          dz  dz  z
  t          dz  d	z  z   t!          t          dz            z   k    sJ t          t"          fd           t          t          fd           t          t"          fd           t          t"          fd           t          t"          fd           |                     t                                        d          dt          z
  t          dz  dz  z   t          dz  dz  z
  t          dz  d	z  z   t          dz  d
z  z
  t          dz  dz  z   t          dz  dz  z
  t!          t          dz            z   k    sJ |                    t                                                    dt          dz  dz  z   dt          dz  z  d	z  z   t!          t          dz            z   k    sJ |                    t                                        d          dt          dz  dz  z   dt          dz  z  d	z  z   dt          dz  z  dz  z   t!          t          dz            z   k    sJ d S )Nc                  8                          t                    S r^   )inverser8   r   s   r?   r`   z#test_fps__inverse.<locals>.<lambda>S  s    rzz!}} rA   c                       j         S r^   )r   finvs   r?   r`   z#test_fps__inverse.<locals>.<lambda>X  s	    ty rA   r6   r/   r0   r1   rZ   r2   r   r\   c                  .                          d          S r   r   r   s   r?   r`   z#test_fps__inverse.<locals>.<lambda>`  s    (:(: rA   c                       j         S r^   )r   r   s   r?   r`   z#test_fps__inverse.<locals>.<lambda>a  s	    tv rA   c                       j         S r^   r   r   s   r?   r`   z#test_fps__inverse.<locals>.<lambda>b  s	     rA   c                  8                          t                    S r^   r   r   s   r?   r`   z#test_fps__inverse.<locals>.<lambda>c  s    (=(=a(@(@ rA   c                  8                          t                    S r^   r   r   s   r?   r`   z#test_fps__inverse.<locals>.<lambda>d  s    q(9(9 rA   ru   r   r   r   =   )r   r   r8   r   r   r(   rm   r   rb   r#   r   r    r,   rc   r   rd   r   rl   )r   r   r   r   s     @@r?   test_fps__inverser   P  s   SVVc#a&&kk3s1vv;;JBB
:,,,,---::a==Dd455555di!233333
:(((()))6SVV=CGG####A!a%!Q$q&.1a46"9AqDG"CCCCC==??a!ead1fnq!tAv51R?!Q$s(JQqRStWWTTTTT==q1uq!tAv~1Q6Ab@1QT77JJJJJ
 : : : :;;;
:~~~~&&&
 5 5 5 5666
 @ @ @ @AAA
 9 9 9 9:::::a==!!!$$	A1QAa!Q$r')AqDH4q!tCx?!Q$t)KaPQSTPTggUV V V V ::a==!!##q1a46zAadF2I'=!Q$'GGGGG::a==!!!$$AqDF
Qq!tVBY(>AqD(LqQRTUQUww(VVVVVVVrA   N)Wsympy.concrete.summationsr   sympy.core.addr   sympy.core.functionr   r   sympy.core.mulr   sympy.core.numbersr   r	   r
   r   sympy.core.singletonr   sympy.core.symbolr   (sympy.functions.combinatorial.factorialsr   &sympy.functions.elementary.exponentialr   r   %sympy.functions.elementary.hyperbolicr   r   (sympy.functions.elementary.miscellaneousr   (sympy.functions.elementary.trigonometricr   r   r   r   r   sympy.functions.special.besselr   'sympy.functions.special.error_functionsr   'sympy.functions.special.gamma_functionsr   sympy.integrals.integralsr   sympy.series.formalr   sympy.series.orderr   r   r    r!   r"   r#   r$   r%   r&   r'   sympy.testing.pytestr(   r)   r*   r8   r;   r:   rG   r<   r9   r,   r-   r@   rE   rI   rM   rR   rr   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r_   rA   r?   <module>r     s   ) ) ) ) ) )       6 6 6 6 6 6 6 6       4 4 4 4 4 4 4 4 4 4 4 4 " " " " " " % % % % % % > > > > > > = = = = = = = = @ @ @ @ @ @ @ @ 9 9 9 9 9 9 Q Q Q Q Q Q Q Q Q Q Q Q Q Q 1 1 1 1 1 1 7 7 7 7 7 7 9 9 9 9 9 9 / / / / / / # # # # # #            I I I I I I I I I I I I I I I I I I I I I I 5 4 4 4 4 4 4 4 4 4
''

1a
''4
(
(
(1ax}}hhsmm1/4 /4 /4d> > >  49 9 9"( ( (4&. &. &.R 0Q 0Q 0Qf ,F ,F ,F^5 5 5$E E E) ) )0. . .>B B B G G G
#V #V #VLM M M
O< O< O<bC C C8#K #K #KLW W W W WrA   