
    gV`                        d dl mZmZmZ d dlmZ d dlmZmZm	Z	m
Z
mZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZm Z m!Z!m"Z"m#Z#m$Z$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/m0Z0m1Z1m2Z2m3Z3m4Z4m5Z5m6Z6 d dl7m8Z8 d dl9m:Z:m;Z; d	 Z<d
 Z=d Z>d Z?d Z@d ZAd ZBd ZCd ZDd ZEd ZFd ZGd ZHd ZId ZJd ZKd ZLd ZMd ZNd ZOd ZPd ZQd ZRd  ZSd! ZTd" ZUd# ZVd$ ZWd% ZXd& ZYd' ZZd( Z[d) Z\e)d*             Z]d+S ),    )QQEXRR)ring) _invert_monomsrs_integraters_truncrs_mul	rs_squarers_pow_has_constant_termrs_hadamard_exprs_series_from_listrs_exprs_log	rs_newtonrs_series_inversionrs_compose_addrs_asinrs_atanrs_atanhrs_tanrs_cotrs_sinrs_cos
rs_cos_sinrs_sinhrs_coshrs_tanh_tan1rs_funrs_nth_rootrs_LambertWrs_series_reversionrs_is_puiseux	rs_series)raisesslow)symbols)
sincosexptancotatanatanhtanhlogsqrt)Rational)expandSc                 :    d}t          | |z
            |k     sJ d S )Ng|=)abs)abtols      ^/var/www/html/ai-engine/env/lib/python3.11/site-packages/sympy/polys/tests/test_ring_series.pyis_closer=      s(    
Cq1u::    c                     t          dt                    \  } }|dz  d|dz  z  z   d|z  z   dz   }t          |          d|dz  z  d|dz  z  z   d|z  z   dz   k    sJ t          |          |dz  dz  |dz  dz  z   d|z  z   dz   k    sJ t          dt                    \  } }|dz  d|dz  z  z   d|z  z   dz   }t	          ||          |dz  dz  |dz  dz  z   d|dz  z  dz  z   d|z  z   k    sJ t          dt                    \  } }}|dz  |dz  z  |z   dz   }t	          ||          |dz  |dz  z  dz  |dz  dz  z   |z   k    sJ t	          ||          |dz  |dz  z  dz  ||z  z   |z   k    sJ d S )	Nx                  x, y)r   r   r   r   r   )Rr@   pys       r<   test_ring_series1rK      s   R==DAq	1qAv!aA!!Q$1a4!A# 5 999991Ab1a46!1AaC!7!!;;;;;R==DAq	1qAv!aA1Aa!Q$q&1QT6!8!;ac!AAAAA62GAq!	1QT	AA1Aad1q!tAv!5!999991Aad1qs!2Q!6666666r>   c                      t          dt                    \  } }}}|||z  z   dz  }t          ||d          }||dz  d|dz  z  |z  |z  z   d|dz  z  |dz  z  |dz  z  z   k    sJ d S )Nx, y, trA   rC      rB   )r   r   r	   )rH   r@   rJ   trI   p1s         r<   
test_truncrQ   !   s    i$$JAq!Q	
QqS1A	!Q		BA!Q$q
"Qq!tVAqD[A%55555555r>   c                    	 t          dt                    \  } }}d|z  z   ||z  z   t          d          D ]}t          |d          ddz  z  |dz  z  dz  |z  |dz  z  z   d|dz  z  |dz  z  z   dz  |z  z   d|z  |z  z   dz   k    sJ d|z  z   ||z  z   |dz  z  |z  z   t          |d          }|dd|z  z  z   d|z  |z  z   k    sJ t          dt                    \  }	t	          t
          	fd	           ddz  z   ddz  z  z   }ddz  z   }t          ||d          ddz  z  d
dz  z  z   dz  z   dz   k    sJ d S )NrM   rD   rB   rC   rN      rA   zc                  (    t           d          S )NrB   )r
   )rI   r@   rT   s   r<   <lambda>z test_mul_trunc.<locals>.<lambda>2   s    vaAq11 r>      )r   r   ranger
   r'   
ValueError)
rH   rJ   rO   irP   R1p2rI   r@   rT   s
          @@@r<   test_mul_truncr]   '   s   i$$JAq!Q	AaC!A#A1XX  1aA!Q$q!tbd1fQTk)AadF1a4K7!A#a%?!A#a%G!KKKKK	AaC!A#1Qq A	1a		BQqSUQqSU"""""bMMEB
:111111222	
QqS1QT6	B	
QTB"b!Q1QT6Bq!tG#3ac#9A#=======r>   c                  2   t          dt                    \  } }}}d||z  z   ||z  z   dz  }t          |||d          }t          ||d          }||k    sJ d|z   |dz  z   |dz  z   }t          ||d          d|dz  z  d|dz  z  z   d|z  z   dz   k    sJ d S )NrM   rD   rB   rC   rA   )r   r   r
   r   )rH   r@   rJ   rO   rI   rP   r\   s          r<   test_square_truncr_   8   s    i$$JAq!Q	
QqS1Q3A	1a		B	1a		B8888	A1q!tAQ11a4!AqD&1Q3!6!:::::::r>   c                     t          dt                    \  } }}}|||z  z   }|dz  }|||fD ]-}t          ||d          }t          |d|d          }||k    sJ .d|z   }t          |d|d          }|dd|z  z   k    sJ t          |d|d          dk    sJ t          |d|d          dd|z  z
  k    sJ ||z   }t          |d|d          |dz  d|dz  z  |z  z   d|z  |dz  z  z   k    sJ t          d|z   t	          dd          |d	          d	|dz  z  d
z  |dz  dz  z
  |t	          dd          z  z   dz   k    sJ d S )Nx, y, z      rD   rC   rB   r   rA   Q   	   )r   r   r	   r   r4   )	rH   r@   rJ   rT   p0rI   xxrP   r\   s	            r<   test_pow_truncri   A   s   i$$JAq!Q	
QqSB
BA!Qi  aQBB""Rxxxxx	AA	1a		BQqS====!Q1""""!RA!ac'))))	AA!Q1A!Q$q1Q3q!t8!;;;;;!a%!QA..!AqD&)ad1f2DqRSUVGW2WZ[2[[[[[[[r>   c                      t          dt                    \  } }}}|||z  z   }t          ||          sJ ||dz  z   }t          ||          rJ d|z   |dz  z   }t          ||          sJ ||z   ||z  z   }d S )Nra   rA   rD   )r   r   r   )rH   r@   rJ   rT   rI   s        r<   test_has_constant_termrk   S   s    i$$JAq!Q	AaCAa#####	AqDA!!Q'''''	A1Aa#####	A!AAAr>   c                  &   t          dt                    \  } dz   ddz  z  z   d}t          |          }t          |z  |          dk    sJ t          dt                    \  } }dz   ddz  z  z   |z  z   dz  |z  z   t          |          }t          |z  |          dk    sJ t          dt                    \  } }dz   |z   t	          t
          fd           | j        t	          t          fd           d S )Nr@   rB   rF   rD   rG   c                  &    t           d          S )NrA   r   rI   r@   s   r<   rV   z test_inversion.<locals>.<lambda>j   s    (;Aq!(D(D r>   c                  &    t           d          S )NrC   rn   ro   s   r<   rV   z test_inversion.<locals>.<lambda>l   s    &9!Q&B&B r>   )r   r   r   r	   r'   NotImplementedErrorzeroZeroDivisionError)rH   nrP   rJ   rI   r@   s       @@r<   test_inversionru   ]   s>   R==DAq	A!Q$A	A	Q1	%	%BAbD!Q1$$$$62GAq!	A!Q$1q!tAv%A	Q1	%	%BAbD!Q1$$$$62GAq!	A	A
 D D D D DEEE	A
BBBBBCCCCCr>   c                  <   t          dt                    \  } }}t          ||d          }t          ||d|          t	          ||d          k    sJ t          ||d          }t          ||d|          d|dz  z  dz  d|dz  z  dz  z   |dz  d	z  z   |z   k    sJ d S )
NrG   
   rc   rF      p   rC   (   rN   )r   r   r   r$   r   r   rH   r@   rJ   rI   s       r<   test_series_reversionr|   o   s    62GAq!q!RAq!Q**gaA.>.>>>>>q!RAq!Q**a1fSj1QT6"9.D	1Q//      r>   c            	         t          dt                    \  } }dd|z  z   |dz  z   d|dz  z  z   }g d}t          |||d          }|                     t	          t          |                              }t          |                    ||          |d          }||k    sJ t          dt                    \  } }}g d}t          ||z   ||dd	
          }t          dd||z   z  z   d||z   dz  z  z   d||z   dz  z  z   |d          }	||	k    sJ t          dt                    \  } }d}
t          |||
          dz
  }t          ||||
          }d	}	t          |          D ]\  }}|	|t          ||||
          z  z  }	||	k    sJ d S )Nr@   rD   rB   rC   )rD   rB   r   rA   rA   rF   rG   )rD   rC   rF   rx   r   )concurrx      )r   r   r   	from_listlistreversedr	   composer   	enumerater   )rH   r@   rI   crpcr1rJ   rP   r\   hrZ   cxs                r<   test_series_from_listr   y   s   R==DAq	AaC!Q$1a4AAAq!Q''A	
T(1++&&	'	'B	"**Q""Aq	)	)B777762GAq!A	QUAq!A	6	6	6B	1q!A#w;AaC!8+a1qj81a	@	@B8888R==DAq
Aq!Q!A	Q1a	(	(B	
B1 $ $2
b1a####888888r>   c                  f   t          dt                    \  } }d|z   }t          ||d          |dz  z  }|t          dd          |z  t          j        z
  |dz  z   k    sJ d|z   d|dz  z  dz  z   }t          ||d          }|d|d	z  z  d
z  d|dz  z  dz  z   d|dz  z  dz  z
  |dz  dz  z
  d|dz  z  dz  z   |dz  dz  z
  |dz  dz  z   |z   k    sJ t          ||d          }t          ||d          }|| k    sJ t          dt                    \  } }}d|z   d|z  |dz  z  z   }t          ||d          }|d|dz  z  |dz  z  d|dz  z  |z  z
  d|dz  z  |dz  z  z
  |dz  dz  z   d|dz  z  |z  z   |dz  dz  z
  d|dz  z  |z  z
  |dz  dz  z   d|dz  z  |z  z   |dz  dz  z
  |z   k    sJ t          d          }t          dt                    \  } }}t          ||z   |d          t          dd|dz  z  z             |dz  z  t          dd|dz  z  z            |dz  z  z   t          dd|dz  z  z            |dz  z  z
  t          d|z            |z  z   t          t          |                    z   k    sJ t          ||dz  |z  z   |z   |d          t          |dz             |dz  z  |z  t          dd|dz  z  z            |dz  z  z   t          d|z            |dz  z  |z  z   t          dd|dz  z  z            |dz  z  z
  t          d|z            |z  z   t          t          |                    z   k    sJ ||dz  z   dz   }t          ||d          
                    |d          t          t          d          t          dd          z             k    sJ d S )Nr@   rD   rA   rB   rC   rf   rc   i     rx      rW   rN      rF   -   $   rG   r9   rd   rw   l   K)z i^ )r   r   r   r4   r6   Halfr   r)   r   r2   r   )rH   r@   rI   rP   r\   p3rJ   r9   s           r<   test_logr      sr   R==DAq	AA	1aA	B!Q!AF*QW44444	AqAvaxA	1aBQT#1a4+bAgck9AqDGC1fRiQ$q&a46"$%& & & & &	Q1	%	%B	Aq		B"999962GAq!	A!AqDA	1aB!AqD&A+!Q$q(1QT6!Q$;6Aa?!AqD&(JQ$q&QT6!8$&'d1f-/0Avax8:;Q$q&ACDE F F F F 	A62GAq!!a%A2a1a4j>>/!Q$"6AqAvJ19L"L
Q!Q$Z..A
# "1Q3	#*,.s1vvJJ#7 7 7 7 7!ad1f*q.!Q''Bq2wKK<1+<Q+>
1a1f:q!t, 1ggadl1n,-/1!Qq!tV*~~ad/B,C
1Q3	,s1vvJJ,     	
AqD1A!Q##Aq))RA+w9W9W0W-X-XXXXXXXr>   c                  	   t          dt                    \  } }||dz  z   }dD ]C}t          d|z   ||          dz
  }t          |||          }t	          |||          }||k    sJ Dt          ||d          }|                    |dz            t          dd          k    sJ d	}t	          d|z   ||          }t          |||          }||dz   k    sJ t          d
          }t          dt          t          |          |f                   \  } }}	t          ||z   |d          t          |          |dz  z  dz  t          |          |dz  z  dz  z   t          |          |dz  z  dz  z   t          |          |z  z   t          |          z   k    sJ t          ||dz  |	z  z   |z   |d          t          |          |dz  z  |	dz  z  dz  t          |          |dz  z  |	z  dz  z   t          |          |dz  z  dz  z   t          |          |dz  z  |	z  z   t          |          |dz  z  dz  z   t          |          |dz  z  |	z  z   t          |          |dz  z  dz  z   t          |          |z  z   t          |          z   k    sJ t          dt                    \  } }}	t          ||z   |d          t          t          |          dz            |dz  z  t          t          |          dz            |dz  z  z   t          t          |          dz            |dz  z  z   t          t          |                    |z  z   t          t          |                    z   k    sJ t          ||dz  |	z  z   |z   |d          t          t          |          dz            |dz  z  |	dz  z  t          t          |          dz            |dz  z  |	z  z   t          t          |          dz            |dz  z  z   t          t          |                    |dz  z  |	z  z   t          t          |          dz            |dz  z  z   t          t          |                    |dz  z  |	z  z   t          t          |          dz            |dz  z  z   t          t          |                    |z  z   t          t          |                    z   k    sJ d S )Nr@   rA   )rw      rD   r      l   O$obl      LeFSX   r9   rG   rF   rE   rC   rN   rB   )	r   r   r   r   r   coeffr)   r,   r   )
rH   r@   rI   r   qrP   q1precr9   rJ   s
             r<   test_expr      s   R==DAq	AqDA  Aq!,,q0Aq!__B1Qwwwww	1b		B88ArE??b!;=[\\\\\\Dq1uaA	1d		BQ;;;; 	A62c!ffai=))GAq!!a%A#a&&A+b.3q66!Q$;q="@Aq!tA#Aq#!#&q66#* * * * *!ad1f*q.!Q''3q66!Q$;q!t+;A+=FF1a4KM!O,!!ffQTk"n,-/21vvad{1},=FF1a4KM,FF1a4KM,*,/FF1a4KM,:<?FF1H,EGJ1vv,N N N N N 62GAq!!a%A2c!ffRi==A#53q66!8QT8I#I
3q66!8QT$s1vvJJqL$)+-c!ff::$6 6 6 6 6!ad1f*q.!Q''2c!ffQh<<1+<QT+A
3q66!8QT!, QmmAqD0,135c!ff::ad?13D,E
3q66!8QT,s1vvJJq!tOA-,.023q66!8QT0A,B 	3q66

1, #a&&zz," " " " " " "r>   c                      t          dt                    \  } }|dz  dz
  }t          ||d          }|d|dz  z  d|dz  z  z   dz   k    sJ d S )Nr@   rB   rA   rc   )r   r   r   )rH   r@   rI   r   s       r<   test_newtonr      sa    R==DAq	1qA!QA!Q$1a4!#######r>   c                      t          dt                    \  } }|dz  dz
  }|dz  dz
  }t          ||          |dz  d|dz  z  z
  d|dz  z  z
  d|dz  z  z   d|z  z
  dz
  k    sJ d S )	Nr@   rC   rD   rB   rN   rA   rS   rx   )r   r   r   )rH   r@   rP   r\   s       r<   test_compose_addr      s    R==DAq	
AB	
AB"b!!QTAadF]Qq!tV%;bAg%E1%Lq%PPPPPPPr>   c                     t          dt                    \  } }}||z  |dz  |dz  z  z   |dz  |z  z   }t          |t          |d          t          ||d          k    sJ t          |t          |d          t	          ||d          k    sJ d S )NrG   rB   rC   rF   rw   )r   r   r!   r   r    r{   s       r<   test_funr      s    62GAq!	!ad1a4i!Q$q& A!VQ##vaB'7'77777!UAr""eAq"oo555555r>   c                  	   t          dt                    \  } }}t          d|dz  |z  z   d|d          d|dz  z  |dz  z  dz  d	|d
z  z  |dz  z  dz  z   d|dz  z  |dz  z  dz  z
  |dz  |z  dz  z   dz   k    sJ t          d||z  z   |dz  |dz  z  z   d|d          |dz   |d
z  z  dz  d|dz  z  |dz  z  dz  z   d|dz  z  |dz  z  dz  z
  d|dz  z  |dz  z  dz  z
  d|dz  z  |dz  z  dz  z   |dz  |dz  z  dz  z   |dz  |dz  z  dz  z
  ||z  dz  z   dz   k    sJ t          d|z  d|d          d|t          dd          z  z  k    sJ t          d|z  |dz  z   |dz  z   d|d          |t          dd          z  dz  d|t          dd          z  z  z   k    sJ t          d|z  |dz  |z  z   |dz  z   d|d          }||t          d	d          z   |dz  z  dz  |t          d	d          z  dz  z   |t          dd          z  |z  dz  z   d|t          dd          z  z  z   k    sJ t          d          }t          dt                    \  } }}t          ||z   d|d          t	          dd|t          dd          z  z  z            |dz  z  t	          dd|t          dd          z  z  z            |dz  z  z
  t	          dd|t          dd          z  z  z            |z  z   t	          |t          dd          z            z   k    sJ t          |t          dd          z  |dz  |z  z   dz   d|d          t	          t          d          dz             |t          dd          z  z  |z  t	          t          d          dz            |t          dd          z  z  z
  t	          t          d          dz            |dz  z  |z  z   t	          t          d          dz            |dz  z  z   t	          t          d          dz            |t          dd          z  z  z
  t	          t          d          dz            |t          dd          z  z  z   t	          t          d                    z   k    sJ d S )NrG   rD   rB   rA   rw   irc   i   rx   rN   rC          rF   rf         re   rS   i   r9   d   i>  i     )r   r   r"   r)   r   r3   )rH   r@   rJ   r   r9   s        r<   test_nth_rootr      s3   62GAq!q1a46z1a,,AqDAd0B	!Q$q!tC1AqD&A+b.1)+,a46!8146718 8 8 8 8q1Q3wAad*Aq!44Aad
1	!Q$q!tB9AqDAc)9*,-adF1a4KM9:<=adF1a4KN9K	1QT	!9d1a4ik9"$%aCE9*,-9. . . . . qsAq!$$!R1XX+5555qsQTzAqD(!Q22aAajma2aPQ77
l6RRRRRAaC!Q$q&L1a4'Aq11ABqGGAqD $q"Qq''z"}4q"Qq''z!|BF1bQRSTgg:UUUUU 	A62GAq!q1uaA&&"Q1bAhh;-?*@*@A*E
1a2a88mad"+#%'1Q1a[=(9%:%:1%<+=?A!R1XX++O O O O Oq"Q(({QT!V+a/Aq99baoo=M	2a88>>4775=))!R1XX+5>68:4772:q!t8KA8M>N
4774<A> "4773;2a88 ;>< 	4772:q"Q(({"># &(Q[[>1 1 1 1 1 1 1r>   c            	      v   t          dt                    \  } }}t          ||d          |dz   dz  |dz  dz  z   |dz  dz  z
  |z   k    sJ t          ||z  |dz  |dz  z  z   |d          d|dz  z  |dz  z  |dz  |dz  z  z
  d|dz  z  |dz  z  z   |dz  |dz  z  dz  z
  |d	z  |dz  z  dz  z
  |d	z  |dz  z  z   |dz  |dz  z  z
  |dz  |dz  z  dz  z   |d
z  |dz  z  z
  |dz  |dz  z  dz  z
  |dz  |dz  z  z   ||z  z   k    sJ t          d          }t          dt                    \  } }}t          ||z   |d          t	          |dz  |z
  |dz  d
|d	z  z  z   d	|d
z  z  z   d
|dz  z  z   dz   z             |d
z  z  t	          d|dz  z  dz
  d|d	z  z  d|d
z  z  z   d|dz  z  z   dz   z            |dz  z  z   t	          ||d
z  d|dz  z  z   dz   z            |dz  z  z
  t	          d|dz  dz   z            |z  z   t	          t          |                    z   k    sJ t          ||dz  |z  z   |z   |d
          t	          d|z  |d
z  d|dz  z  z   dz   z             |dz  z  |z  t	          d|dz  z  dz
  d|d	z  z  d|d
z  z  z   d|dz  z  z   dz   z            |dz  z  z   t	          d|dz  dz   z            |dz  z  |z  z   t	          ||d
z  d|dz  z  z   dz   z            |dz  z  z
  t	          d|dz  dz   z            |z  z   t	          t          |                    z   k    sJ d S )NrG   rf   rx   rF   rC   rB   rc   rW   rN   rA   r9   rD   )r   r   r   r)   r   r/   rH   r@   rJ   r9   s       r<   	test_atanr      s   62GAq!1a1uQwAa/!Q$q&81<<<<<1Q3Aad?Aq))Qq!tVArE\AqDAI-E	!Q$q!t.d1a4ik."$%qDAIaK.023Q$q!t).<>?d1a4i.H	1QT	!.d1a4i. "#Q$q!t)A+..011QT	.:<=aC.@ @ @ @ @
 	A62GAq!1q5!QB1q1a4!AqD&=1QT63I	!Q$44 ( % % $T$!Q$
Qq!tVa1f_	!Q$..    T$q!Q$1a4-!"3455ad:$; 	1adQh<$  Q[[$) ) ) ) ) 1q!tAv:>1a((R!QTAadF]Q5F0G-H-H,H	
A--a1fqj1QT6AadF?Qq!tV#;a#?@AA!Q$F-G
1adQh<Aa- "$Q1qAv(9%:";";AqD"@-ACEaA
J G D DD- T!WW++-      r>   c                     t          dt                    \  } }}t          |||z  z   |d          |dz  |dz  z  dz  |dz  |dz  z  dz  z   |dz  |z  dz  z   |dz  dz  z   ||z  z   |z   k    sJ t          ||z  |dz  |dz  z  z   |d          |dz  |dz  z  dz  d|dz  z  |dz  z  dz  z   |dz  |dz  z  dz  z   |dz  |dz  z  dz  z   |dz  |dz  z  z   ||z  z   k    sJ d S )	NrG   rF   rC   rN   rB   rx   rz   rA   )r   r   r   rH   r@   rJ   s      r<   	test_asinr   
  sZ   62GAq!1qs7Aq!!QT!Q$Yq[1a419Q;%>Aa%I	1Q&1&&    1Q3Aad?Aq))QT!Q$Yq[1QT6!Q$;r>-I	1QT	!.d1a4ik."$%qDAI..01!.4 4 4 4 4 4 4r>   c            	      :   t          dt                    \  } }}t          ||d          |dz  z  t          dd          |dz  z  t          dd          z   t          dd	          |d
z  z  z   |dz  z   k    sJ t          ||z  |dz  |d	z  z  z   |d          d|dz  z  |dz  z  d	z  d|dz  z  |dz  z  dz  z   d|dz  z  |dz  z  d	z  z   d|dz  z  |dz  z  dz  z   |dz  |dz  z  d	z  z   d|dz  z  |dz  z  d	z  z   |dz  |dz  z  z   d|dz  z  |dz  z  dz  z   |dz  |dz  z  z   |d	z  |d	z  z  d	z  z   |dz  |d	z  z  z   ||z  z   k    sJ t	          d          }t          dt          t          |          |f                   \  } }}t          ||z   |d          t          |          dz  dt          |          d	z  z  d	z  z   dt          |          z  d	z  z   |dz  z  t          |          dz  dt          |          dz  z  d	z  z   t          dd	          z   |d	z  z  z   t          |          d	z  t          |          z   |dz  z  z   t          |          dz  dz   |z  z   t          |          z   k    sJ t          ||dz  |z  z   |z   |d          dt          |          d	z  z  dt          |          z  z   |d	z  z  |z  t          |          dz  t          dd	          t          |          dz  z  z   t          dd	          z   |d	z  z  z   t          |          dz  dz   |dz  z  |z  z   t          |          d	z  t          |          z   |dz  z  z   t          |          dz  dz   |z  z   t          |          z   k    sJ t          dt                    \  } }}t          ||z   |d          t          t          |          dz  dt          |          d	z  z  d	z  z   dt          |          z  d	z  z             |dz  z  t          t          |          dz  dt          |          dz  z  d	z  z   t          d          d	z  z             |d	z  z  z   t          t          |          d	z  t          |          z             |dz  z  z   t          t          |          dz  dz             |z  z   t          t          |                    z   k    sJ t          ||dz  |z  z   |z   |d          t          dt          |          d	z  z  dt          |          z  z             |d	z  z  |z  t          t          |          dz  dt          |          dz  z  d	z  z   t          d          d	z  z             |d	z  z  z   t          t          |          dz  dz             |dz  z  |z  z   t          t          |          d	z  t          |          z             |dz  z  z   t          t          |          dz  dz             |z  z   t          t          |                    z   k    sJ ||dz  z   dz   }t          ||d                              |d          t          t          d          t          d          dz  z             k    sJ d S )NrG   rf   rF      ;  rB      rD   rC   rd   rA   rc   rW   r   rx   rN   r9   rw   l   PH]l   n)r   r   r   r4   r)   r-   r   r   r   r/   r6   rH   r@   rJ   r9   rI   s        r<   test_tanr     s?   62GAq!!Q??1a4S!Q$!R08Aq>>!b'3IIAPRGST T T T!A#1QT	/1a((AadF1b5LNR1WQT\"_,L	!Q$q!tA-1a41S(-)+,a419Q;-79:1a41Q-G	1QT	-adF1a4KN-#%&T!Q$Y-/12Aad1-=?@!tAqDy-IKLQ3-O O O O O
 	A62c!ffai=))GAq!!a%A3q6619qQ{1}#<	#a&&
$qD#FFAI#a&&!)A5AF1L#M	QSVV	QT!#"%(VVQY]A$5#68;A#? ? ? ? ? !ad1f*q.!Q''Ac!ffaiK!CFF(,BAqD+H+J	QXa^^CFFAI-	-A	>1D,EHKAPQ	TUWXZ[W[G[\]G],^	QSVV	QT!,"%(VVQY]A$5,68;A,? ? ? ? ? 62GAq!!a%A"SVVQY3q6619Q%>	#a&&
& # #qD#c!ffai!CFFAI+a-7"Q%%'ABB1a4G#H
3q6619s1vvq!t##$&(QQ&7&7&9#:<>s1vvJJ#G G G G G !ad1f*q.!Q''2aA	k	#a&&/ , ,Q$,,c!ffai!CFFAI+a-7"Q%%'ABB1a4G,H
3q6619q=!Q$q ,!#%c!ffai#a&&&8#9#9!Q$#>,? 	3q6619q=!, !Qjj,) ) ) ) )
 	
AqD1A1a$$Q++r$q''A>O<P<P= 3 0 0      r>   c                     t          dt                    \  } }}t          |dz  |dz  z   |d          |dz  |dz  z
  |dz  z   |dz  z
  |d	z  z   |d
z  z
  dz   |z
  |dz  z   |dz  z
  |dz  z   |dz  z
  d|dz  z  dz  z   d|dz  z  dz  z
  k    sJ t          ||dz  |z  z   |d          |dz   |dz  z  |dz  |z  dz  z
  |dz  |dz  z  z   |dz  dz  z
  |dz  |dz  z  z
  |dz  |z  dz  z
  ||dz  z  z   |dz  z
  |z
  |d
z  z   k    sJ d S )NrG   rN   rx   rc   ir   rd   r   rD   rB   rC   rA   rF   r   r   )r   r   r   r   s      r<   test_cotr   0  s   62GAq!!Q$A+q!$$B!b'(9AG(C	B)b')G)$&')(*+),./d)356T):<=qD)ACDa4)H	!Q$q)QT6!8)     !ad1f*a##1uQTzAqDF2I'=1QT	'I	1R(Q$q!t)(d1fQh(')*1a4(023A#(689(:<=G(D D D D D D Dr>   c                  
   t          dt                    \  } }}t          ||d          |dz  z  t          dd          |dz  z  t          dd          z   t          dd	          |d
z  z  z
  |dz  z   k    sJ t          ||z  |dz  |dz  z  z   |d          |dz  |dz  z  dz  |dz  |dz  z  dz  z
  |dz  |dz  z  dz  z   |dz  |dz  z  dz  z
  |d	z  |dz  z  d	z  z
  |d	z  |dz  z  dz  z   |dz  |dz  z  dz  z
  |dz  |dz  z  dz  z   |dz  |dz  z  dz  z
  |dz  |dz  z  d	z  z
  |dz  |dz  z  z   ||z  z   k    sJ t	          d          }t          dt          t          |          t          |          |f                   \  } }}t          ||z   |d          t          |          |dz  z  dz  t          |          |dz  z  d	z  z
  t          |          |dz  z  dz  z
  t          |          |z  z   t          |          z   k    sJ t          ||dz  |z  z   |z   |d          t          |           |dz  z  |dz  z  dz  t          |          |dz  z  |z  dz  z
  t          |          |dz  z  dz  z   t          |          |dz  z  |z  z
  t          |          |dz  z  d	z  z
  t          |          |dz  z  |z  z   t          |          |dz  z  dz  z
  t          |          |z  z   t          |          z   k    sJ t          dt                    \  } }}t          ||z   |d          t          t          |          dz            |dz  z  t          t          |          d	z            |dz  z  z
  t          t          |          dz            |dz  z  z
  t          t          |                    |z  z   t          t          |                    z   k    sJ t          ||dz  |z  z   |z   |d          t          t          |          dz             |dz  z  |dz  z  t          t          |          dz            |dz  z  |z  z
  t          t          |          dz            |dz  z  z   t          t          |                    |dz  z  |z  z
  t          t          |          d	z            |dz  z  z
  t          t          |                    |dz  z  |z  z   t          t          |          dz            |dz  z  z
  t          t          |                    |z  z   t          t          |                    z   k    sJ d S )NrG   rf   rF   r     rB   rD   x   rN   rd   r   rC   rc   rW   rS     rx   rE   rA   r9   )r   r   r   r4   r)   r*   r+   r   r   s       r<   test_sinr   8  s   62GAq!!Q??1a4T1a4(1c"2"22Xa^^AG5KKaRTgUV V V V!A#1QT	/1a((AqDBJrM	1QT	#-1QT	"-%'(!tAqDy~-6891QT	!-D	1QT	"-!tAqDy{-#%&T!Q$Ys]-356T!Q$Yq[-A 	
1QT	!- d1a4i-  #$A#-& & & & & 	A62c!ffc!ffa/011GAq!!a%A#a&&A+b.3q66!Q$;q="@Aq!tA#Aq#!#&q66#* * * * *!ad1f*q.!Q''CFF71a4<1+<Q+>Aq!tAa,a&&A+b.,)+.q66!Q$;q=,9;>q66!Q$;q=,IAq!tA,Aq!tA,&(+Aq,136q66,: : : : : 62GAq!!a%A"SVVBY--1"4r#a&&(||AqD7H"H
3q66!8QT#s1vvJJqL#)+-c!ff::#6 6 6 6 6!ad1f*q.!Q''Bs1vvaxLL=A+=ad+B
3q66!8QT!, QmmAqD0,135c!ff::ad?13D,E
3q66!8QT,s1vvJJq!tOA-,.023q66!8QT0A,B 	3q66

1, #a&&zz," " " " " " "r>   c                     t          dt                    \  } }}t          ||d          |dz  z  t          dd          |dz  z  t          dd          |z  z
  t          dd          |d	z  z  z   t          j        |d
z  z  z
  |dz  z   k    sJ t          ||z  |dz  |dz  z  z   |d          |dz  |dz  z  dz  |dz  |dz  z  dz  z
  |dz  |dz  z  dz  z   |dz  |dz  z  dz  z   |dz  |dz  z  dz  z
  |dz  |dz  z  dz  z   |dz  |dz  z  dz  z
  |dz  |dz  z  dz  z   |dz  |dz  z  dz  z
  |dz  |dz  z  dz  z   |dz  |dz  z  z
  |dz  |dz  z  dz  z
  dz   k    sJ t          d          }t          dt          t          |          t          |          |f                   \  } }}t          ||z   |d          t          |          |dz  z  dz  t          |          |dz  z  dz  z   t          |          |dz  z  dz  z
  t          |          |z  z
  t          |          z   k    sJ t          ||dz  |z  z   |z   |d          t          |           |dz  z  |dz  z  dz  t          |          |dz  z  |z  dz  z   t          |          |dz  z  dz  z   t          |          |dz  z  |z  z
  t          |          |dz  z  dz  z   t          |          |dz  z  |z  z
  t          |          |dz  z  dz  z
  t          |          |z  z
  t          |          z   k    sJ t          dt                    \  } }}t          ||z   |d          t          t          |          dz            |dz  z  t          t          |          dz            |dz  z  z   t          t          |          dz            |dz  z  z
  t          t          |                    |z  z
  t          t          |                    z   k    sJ t          ||dz  |z  z   |z   |d          t          t          |          dz             |dz  z  |dz  z  t          t          |          dz            |dz  z  |z  z   t          t          |          dz            |dz  z  z   t          t          |                    |dz  z  |z  z
  t          t          |          dz            |dz  z  z   t          t          |                    |dz  z  |z  z
  t          t          |          dz            |dz  z  z
  t          t          |                    |z  z
  t          t          |                    z   k    sJ d S )NrG   rf   rF   rD   逝  rC   r   rE   r   r   r   rB   rc   rS   rw   0   rx   rN   r   rA   r9   )
r   r   r   r4   r6   r   r)   r*   r+   r   r   s       r<   test_cosr   R  s   62GAq!!Q??1a4E1a4(1c"2"21"44x2q2w7NNQRQWXY\^X_Q__bcfhbiij j j j!A#1QT	/1a((AqDBJrM	1QU
2-1QT	%-(*+Q$q"u*Q,-7	1QT	#-1QT	!-$&'d1a4im-467d1a4ik-B 	
1QT	!- d1a4il-# &'T!Q$Y-/ 23Aad1-= @A-A A A A A 	A62c!ffc!ffa/011GAq!!a%A#a&&A+b.3q66!Q$;q="@Aq!tA#Aq#!#&q66#* * * * *!ad1f*q.!Q''CFF71a4<1+<Q+>Aq!tAa,a&&A+b.,)+.q66!Q$;q=,9;>q66!Q$;q=,IAq!tA,Aq!tA,&(+Aq,136q66,: : : : : 62GAq!!a%A"SVVBY--1"4r#a&&(||AqD7H"H
3q66!8QT#s1vvJJqL#)+-c!ff::#6 6 6 6 6!ad1f*q.!Q''Bs1vvaxLL=A+=ad+B
3q66!8QT!, QmmAqD0,135c!ff::ad?13D,E
3q66!8QT,s1vvJJq!tOA-,.023q66!8QT0A,B 	3q66

1, #a&&zz," " " " " " "r>   c                  d   t          dt                    \  } }}t          ||d          \  }}|t          ||d          k    sJ |t	          ||d          k    sJ t          |||z  z   |d          \  }}|t          |||z  z   |d          k    sJ |t	          |||z  z   |d          k    sJ d S )NrG   rf   rF   )r   r   r   r   r   )rH   r@   rJ   r+   r*   s        r<   test_cos_sinr   l  s    62GAq!!Q""HC&Aq//!!!!&Aq//!!!!!ac'1a((HC&QqS!Q''''''&QqS!Q''''''''r>   c            	         t          dt                    \  } }}t          ||d          |dz  z  t          dd          |dz  z  t          dd          z   t          dd          |dz  z  z   |d	z  z   k    sJ t          ||z  |dz  |dz  z  z   |d          d|d
z  z  |dz  z  |d
z  |dz  z  z   d|dz  z  |dz  z  z   |dz  |dz  z  dz  z   |dz  |dz  z  dz  z   |dz  |dz  z  z   |dz  |dz  z  z   |dz  |dz  z  dz  z   |dz  |dz  z  z   |dz  |dz  z  dz  z   |dz  |dz  z  z   ||z  z   k    sJ t	          d          }t          dt
                    \  } }}t          ||z   |d          t          |dz  |z   |d
z  d|dz  z  z
  d|dz  z  z   d|dz  z  z
  dz   z            |dz  z  t          d|dz  z  dz   d|dz  z  d|dz  z  z
  d|dz  z  z   dz
  z            |dz  z  z
  t          ||dz  d|dz  z  z
  dz   z            |dz  z  z   t          d|dz  dz
  z            |z  z
  t          t          |                    z   k    sJ t          ||dz  |z  z   |z   |d          t          d|z  |dz  d|dz  z  z
  dz   z            |dz  z  |z  t          d|dz  z  dz   d|dz  z  d|dz  z  z
  d|dz  z  z   dz
  z            |dz  z  z
  t          d|dz  dz
  z            |dz  z  |z  z
  t          ||dz  d|dz  z  z
  dz   z            |dz  z  z   t          d|dz  dz
  z            |z  z
  t          t          |                    z   k    sJ ||dz  z   dz   }t          ||d                              |d          t          t          dd          t          d          z             k    sJ d S )NrG   rf   rF   rD   rx   rB   rC   rd   r   rc   rW   rN   rA   r9   rw   lyY?l l    P{] )r   r   r   r4   r)   r   r0   r   r   s        r<   
test_atanhr   u  s   62GAq!Aq!QT!Xa^^AqD%88Aq>>%IHUVXYNN[\_a[bLb%befikel%lllllAaC!Q$q!t)OQ**a1fQUlQT!Q$Y.F	!Q$q!t/d1a4ik/"$%qDAIaK/023Q$q!t)/<>?d1a4i/H	1QT	!/d1a4i/ "#Q$q!t)A+/.011QT	/:<=aC/@ @ @ @ @
 	A62GAq!AE1a  B1q1a4!AqD&=1QT63I	!Q$44 ( % %T%!Q$
Qq!tVa1f_	!Q$..    T%q!Q$1a4-!"3455ad:%;=?1a4	D
 A > >>% 588%     A1QJNAq))R!QTAadF]	6
 1 . .qD..1a4!a1fqAvo!Q$&>&BCDDQTI.J
1adQh<Aa. "$Q1qAv(9%:";";AqD"@.A 	1adQh<.  a\\.* * * * *
 	
AqD1AAq"%%a,,8<Lj3Y3Y
((4 1 1      r>   c                     t          dt                    \  } }}t          ||d          |dz  z  t          dd          |dz  z  t          dd          z   t          dd          |d	z  z  z   |d
z  z   k    sJ t          ||z  |dz  |dz  z  z   |d          |dz  |dz  z  dz  |dz  |dz  z  dz  z   |dz  |dz  z  dz  z   |dz  |dz  z  dz  z   |dz  |dz  z  dz  z   |dz  |dz  z  dz  z   |dz  |dz  z  dz  z   |dz  |dz  z  dz  z   |dz  |dz  z  dz  z   |dz  |dz  z  dz  z   |dz  |dz  z  z   ||z  z   k    sJ d S )NrG   rf   rF   rD   r   rB   r   rN   rd   r   rC   rc   rW   rS   r   rx   rE   rA   )r   r   r   r4   r   s      r<   	test_sinhr     s   62GAq!1aAqD HQ$5$5ad$:Xa=M=M$MPXYZ\]P^P^_`ce_fPf$fijmoip$ppppp1Q3Aad?Aq))QT!R%Z]	1QT	#.1QT	".%'(!tAqDy~.6891QT	!.D	1QT	".!tAqDy{.#%&T!Q$Ys].356T!Q$Yq[.A 	
1QT	!. d1a4i.  #$A#.& & & & & & &r>   c                     t          dt                    \  } }}t          ||d          |dz  z  t          dd          |dz  z  t          dd          |z  z   t          dd          |d	z  z  z   t          j        |d
z  z  z   |dz  z   k    sJ t          ||z  |dz  |dz  z  z   |d          |dz  |dz  z  dz  |dz  |dz  z  dz  z   |dz  |dz  z  dz  z   |dz  |dz  z  dz  z   |dz  |dz  z  dz  z   |dz  |dz  z  dz  z   |dz  |dz  z  dz  z   |dz  |dz  z  dz  z   |dz  |dz  z  dz  z   |dz  |dz  z  dz  z   |dz  |dz  z  z   |dz  |dz  z  dz  z   dz   k    sJ d S )NrG   rf   rF   rD   r   rC   r   rE   r   r   r   rB   rc   rS   rw   r   rx   rN   r   rA   )r   r   r   r4   r6   r   r   s      r<   	test_coshr     s   62GAq!1aAqD HQ$6$6q!t$;hq#>N>Nq>P$PS[\]_aSbSbcdgicjSj$j	q2w%R%! ! ! ! !1Q3Aad?Aq))QT!R%Z]	1QU
2.1QT	%.(*+Q$q"u*Q,.7	1QT	#.1QT	!.$&'d1a4im.467d1a4ik.B 	
1QT	!. d1a4il.# &'T!Q$Y./ 23Aad1.= @A.A A A A A A Ar>   c            	         t          dt                    \  } }}t          ||d          |dz  z  t          dd          |dz  z  t          dd          z   t          dd	          |d
z  z  z
  |dz  z   k    sJ t          ||z  |dz  |d	z  z  z   |d          d|dz  z  |dz  z  d	z  d|dz  z  |dz  z  dz  z
  d|dz  z  |dz  z  d	z  z   d|dz  z  |dz  z  dz  z
  |dz  |dz  z  d	z  z
  d|dz  z  |dz  z  d	z  z   |dz  |dz  z  z
  d|dz  z  |dz  z  dz  z   |dz  |dz  z  z
  |d	z  |d	z  z  d	z  z
  |dz  |d	z  z  z   ||z  z   k    sJ t	          d          }t          dt
                    \  } }}t          ||z   |d          t          t          |          dz  dt          |          d	z  z  d	z  z
  dt          |          z  d	z  z             |dz  z  t          t          |          dz   dt          |          dz  z  d	z  z   t          dd	          z
            |d	z  z  z   t          t          |          d	z  t          |          z
            |dz  z  z   t          t          |          dz   dz             |z  z   t          t          |                    z   k    sJ t          ||dz  |z  z   |z   |d          }|                    |d                              |d          t          dt          |          dz  z  dt          |          d	z  z  z   dt          |          dz  z  d	z  z   dt          |          z  z
  t          dd	          z             k    sJ d S )NrG   rf   rF   r   r   rB   r   rD   rC   rd   r   rA   rc   rW   r   r   rx   rN   r9   rw   iit'  i	  is'  i  )r   r   r   r4   r)   r   r1   r   r   s        r<   	test_tanhr     s   62GAq!1aAqD HS#$6$6q!t$;hq"oo$MPXYZ\]P^P^_`ce_fPf$fijmoip$ppppp1Q3Aad?Aq))Qq!tVArE\!^
1a41R.AqD&A+a-.(*,QT'!Q$,s*:.;=>T!Q$Yq[.I	!Q$q!tA.1QT	."$%adF1a4KN.356T!Q$Y.? 	
1QT	!. d1a4i.  #$A#.& & & & & 	A62GAq!1q5!Q2d1ggqj1T!WWaZ<>&A	$q''	!' $ $T$Q
{QtAwwz\!^;bAhhFGG1L$M
477A:Q  A%$&(*DGGQJ;?(;(;A(=$>@B477$L L L L L 	AqDF
Q1%%AIIa%%a++r%Q
2Bd1ggqj3Q
?1,3-/4T!WW}3=?A#qzz3J 0K 0K K K K K K Kr>   c                     t           t          t          t          t          t
          g} t          t          t          t          t          t          g}t          dt                    \  }}}t          d          }t          | |          D ]\  }} |d|z   |d                              |d          } |d|z                                 |dd                                          }	t'          |                                |	                    |d                                                     t/          d|z   d|d                              |d          }d|z   t1          dd          z                      |dd                                          }	t'          |                                |	                    |d                                                     d S )NrG   r9   rB   rF   r   rD   )r   r   r   r   r   r   r*   r+   r-   r.   r/   r1   r   r   r)   zipr   seriesremoveOr=   as_exprsubsrt   r"   r   )
rs_funcssympy_funcsrH   r@   rJ   r9   rs_func
sympy_funcrI   r   s
             r<   test_RRr     s|   AHS#tT2K62GAq!A"8[99 0 0GAE1a  ((A..Jq1u$$Q1--5577affQllnn..////AE1a##++Aq11A
a%"Q((	""1a++3355AQYY[[!&&A,,..**+++++r>   c                  f   t          dt                    \  } }}dd|z  z   |dz  z   d|dz  z  z   }t          ||          rJ ||t          dd          z  |z  z   }t          ||          sJ t          ||          rJ ||dz  |t          dd          z  z  |z  z   }t          ||          rJ d S )NrG   rD   rB   rC   rF   )r   r   r%   r{   s       r<   test_is_regularr     s    62GAq!	AaC!Q$1a4AQ"""""	Ar!AwwJqLAAQ"""""	AqDBqGGOAAQ"""""""r>   c                     t          dt                    \  } }}|t          dd          z  |t          dd          z  z   |z   }t          ||d          }|t          dd          z   |t          dd          z  z   d|t          d	d          z  z  z
  |t          dd          z  z   d|t          d
d          z  z  z   |t          dd          z  z
  |t          dd          z  z
  |t          dd          z  z   |t          dd          z  z
  |t          dd          z  z   }||k    sJ t          d|z   d|d          }||t          dd          z   dz  |t          dd          z  dz  z   |t          dd          z  dz  z   dz   k    sJ t	          d|z   |d          }||t          dd          z   dz  |t          dd          z  z   |t          dd          z  z   k    sJ t          ||d          }||t          dd          z   |t          dd          z  z   |t          dd          z  z   k    sJ ||t          dd          z  |z  z   }t          ||d          }||t          dd          z  |dz  z  dz  |t          dd          z  |dz  z  dz  z   |t          dd          z  |dz  z  dz  z   |t          dd          z  |z  z   dz   k    sJ t          ||d          }||t          dd          z   |t          dd          z  z
  |t          dd          z  z
  |t          dd          z  dz  z
  |z   |t          dd          z  z   |t          dd          z  z   k    sJ t          ||d          }||t          dd          z  |dz  z  dz  |t          dd          z  |dz  z  z   |t          d
d          z  |d
z  z  d
z  z
  |t          d
d          z  |dz  z  z
  ||dz  z  dz  z   |z   |t          dd          z  |dz  z  dz  z
  |t          dd          z  |z  z   k    sJ t          ||d          }||t          dd          z  dz  |t          dd          z  dz  z   |t          dd          z  dz  z   |t          dd          z  dz  z   |z   |t          dd          z  z   |t          dd          z  z   k    sJ t          ||d          }||t          dd          z   |t          dd          z  z   d|t          d	d          z  z  z
  d|t          dd          z  z  dz  z   d|t          d
d          z  z  z   d|t          dd          z  z  dz  z
  |t          dd          z  z
  |t          dd          z  z   |t          dd          z  z
  |t          dd          z  z   k    sJ t          ||d          }|d         |t          dd          z  dz  |t          dd          z  z
  |t          dd          z  dz  z   |t          d
d          z  z
  |t          dd          z  dz  z
  |t          dd          z  z
  |t          dd          z  dz  z
  dz   k    sJ |d         |t          dd          z   dz  |t          dd          z  dz  z
  |t          dd          z  dz  z
  |t          dd          z  dz  z
  |z   |t          dd          z  z   |t          dd          z  z   k    sJ t          ||d          }||t          dd          z  |t          dd          z  z   |t          dd          z  z   |t          dd          z  dz  z   |z   |t          dd          z  z   |t          dd          z  z   k    sJ t          ||d          }||t          dd          z  dz  |t          dd          z  dz  z   |t          dd          z  dz  z   |t          dd          z  dz  z   |z   |t          dd          z  z   |t          dd          z  z   k    sJ t          ||d          }||t          dd          z  dz  |t          dd          z  z   |t          dd          z  dz  z   |t          d
d          z  z   |t          dd          z  dz  z   |t          dd          z  z   |t          dd          z  dz  z   dz   k    sJ t          ||d          }||t          dd          z   |t          dd          z  z
  |t          dd          z  z
  |t          dd          z  dz  z
  |z   |t          dd          z  z   |t          dd          z  z   k    sJ d S )NrG   rB   rF   rC   rD      r   rA   rW   rx   rd   rf   rE   rN         r      rc   rb   )r   r   r   r"   r   r#   r   r   r   r   r   r   r   r   r   )rH   r@   rJ   rI   r   r   rP   s          r<   test_puiseuxr     sK
   62GAq!	2a77
Q1QZ!#AAq!$$A
R2YY,BqGG	#a2b99n	4q"Qq''z	A	!R"XX+
2a77

#%&1QZ
023R"XX+
>@A2b99
M
R1XX+
B 7777AE1a##ABqGGA2a77
1,q"Qq''z!|;a?????q1uaABqGGA2a77
*Q1QZ77777Aq!ABqGGaAaj(1b1gg:55555	
Q1QZ\	Br1aA2a77
1a4"Q1QZ1_Q%66BqGGAqD9JJ	2a77
1     	1aA!R!WW*q"R))|+aBrl:Q1QZ\I	
r!AwwJBqGG$ $ $ $ $ 	AqABqGGAqD"Q1QZ1_4q"Qq''z!Q$q7HH	2a77
1a4AqD&(#%&')*BqGGAqD):;=>1QZ\J J J J J 	1aA2a77
1q"R))|A~-2b99Q>	2a77
1b1gg:&()2a77
3 3 3 3 3 	q!QABr"II2a77
*Qq"R))|^;	!R!WW*Q1b2hh;')*1b1gg:a8:;R!WW*E	2a88Brl#%&2a[1 1 1 1 1 	1aAQ41bBii<>Ar!AwwJ.BqGGB>BqGGK	2a77
1"R))|$&'Aajl3567 7 7 7 7Q4Ar!AwwJ;q=1bBii<>1Ar"RyyLNB	2a77
1b1gg:&()2a77
3 3 3 3 3 	AqA2a77
Q2b		\)Ar"RyyL81b1gg:a<G!K	2a77
1QZ          	1aA2a77
1q"R))|A~-2b99Q>	2a77
1b1gg:&()2a77
3 3 3 3 3 	1aA2b99QBqGG+aAajm;aAajH	2a77
1"R))|$&'Aajl3567 7 7 7 7 	1aABqGGaBrl*Q2b		\9Ar!AwwJqLH	
r!AwwJBqGG$ $ $ $ $ $ $r>   c                     t          j        t          d                    } |                     t          d                    }t	          d          \  }}t          ||g|           \  }}}d|z   |t          dd          z  z  d|z
  |t          dd          z  z  z   }t          |          t          dd          t          d          fd|z   t          d          t          dd          fd|z
  ik    sJ |                                dt          d          z   |t          d          dz  z  z  dt          d          z
  |t          d          dz  z  z  z   k    sJ d S )NrB   rG   rD   rC   r   )	r   algebraic_fieldr3   
from_sympyr)   r   dictr   r6   )Ksqrt2r@   rJ   rH   xryrrI   s           r<   test_puiseux_algebraicr   	  s3   
477##ALLa!!E6??DAqaVQIAr2	
5"b1gg+5"b1gg+ 55A771Q1qwAr!Aww%HHHHH99;;1tAww;AaDDF3q477{A!QK6OOOOOOOOr>   c                  H   t          dt                    \  } }t          ||d          |dz  z  }||dz  dz  |dz  dz  z
  |dz  d	z  z   |d
z  dz  z
  t          dd          z   |dz  dz  z
  |dz  z   k    sJ t          ||d          }t          |d
|d          }|d|t          dd
          z  z  dz  |t          dd
          z  dz  z
  |t          dd
          z  dz  z   |t          dd
          z  dz  z
  |t          dd
          z  z   k    sJ t          ||d          }t          |d|d          }t	          |d|d          }|d|t          dd          z  z  dz  |t          dd          z  dz  z
  d |t          d!d          z  z  d"z  z   d|t          d#d          z  z  d$z  z
  |t          dd          z  z   k    sJ t          |t          dd
          z  |d          }||t          d#d
          z  d%z  |dz  d&z  z   |t          dd
          z  d'z  z   |dz  d(z  z   |t          dd
          z  d)z  z   |dz  d*z  z   |t          dd
          z  dz  z   |dz  d+z  z   |t          d d
          z  dz  z   |dz  d,z  z   |t          dd
          z  d	z  z   |dz  d-z  z   |t          dd
          z  dz  z   |d.z  d/z  z   |t          dd
          z  dz  z   |d
z  d0z  z   |t          d.d
          z  dz  z   |d
z  z   |t          dd
          z  z   dz   k    sJ d S )1Nr@   r   r   rc   l    LQf rN   i arA   i rB   r   rD   r   rd   r   rw   ir   i r   i^  rf   i  rF   rS   rx   i=   i:k/   iP@  rW   !   i     *   l     L.l     { l     ]
 l     nLl    Xnl    (vQ i i _7 r   rC   r   rE   )r   r   r   r"   r   r   )rH   r@   r   rI   s       r<   test1r     s   R==DAqq!RR A1Z!Q$x-/!Q$v+=1T	I
1S		rE!Ge$ $ $ $ $ 	q!RAAq!R  AQ2a[)Ar"QxxK,==	2a77
4R!WW*R-(*+R!WW*5 5 5 5 5 	q!RAAq!R  Aq!QAAr"QxxK	)Ar"QxxK,==
1bAhh;t2b88mB./12BqGG< < < < < 	q"Qq''z1b!!A2b88..A6F1FF	2b88O#$&'d>&9:	2b88M!"$%qD$45 	
2b88J "#Ai0 34R1XX+h2FG 	
1W	 "Qq''z&(	) ,-a4:	6 9:2a77
4	H
 	
1S
 b1gg:c>"
 %&qDG,
 /0Aajl;
 >?qSA 	
2a77
       r>   c                  p   t          dt                    \  } }t          d|           \  }}||t          dd          z  |z  z   }t          ||d          }||dz  dz  |dz  z   d|dz  z  z   |t          dd          z  z  |d	z  d	z  |d
z  z   |z   |t          d	d          z  z  z
  |dz  dz  |dz  z   |t          dd          z  z  z   |dz  dz  |dz  z   |t          dd          z  z  z
  |dz  dz  dz   |z  z   |dz  |t          dd          z  z  dz  z
  ||t          dd          z  z  z   k    sJ d S )NrJ   r@   rD   rF   rC   r   rc   rB   rW   rN   rf   rA   rx   )r   r   r   )rH   rJ   r6   r@   rI   r   s         r<   test_puiseux2r   /  s   R==DAqQ<<DAq	Ar!AwwJqLA1aABrAqD1QT6)1bAhh;6!R%(QT/	;
bAhh;: Q$q&1a4-BqGG3478!tAv	18Aaj7qDFQJ>*,-qDBqGGOA,=>@A!R!WW*M M M M M M Mr>   c                     t          d          \  } }}}t          ||d                                          |k    sJ t          t          |          |d                                          t          |                              |dd                                          k    sJ t          t          |          t          |          z   |d                                          t          |          t          |          z                       |dd                                          k    sJ t          t          |          t          |          z  |d                                          t          |          t          |          z                      |dd                                          k    sJ t          |          |z
  t          |dz            |dz  dz  z   z  }t          t          ||d                                                    t          |                    |dd                                                    k    sJ t          |dz  dz  |dz  z             t          |dz            t          |dz            dz  z  z   }t          t          ||d                                                    t          |                    |dd                                                    k    sJ t          | dz  |z             t          | dz  dz
            |z
  |dz  z
  z  }t          t          ||d                                                    t          |                    |dd                                                    k    sJ t          |dz  |dz  z
  dz             dz  t          |dz  |dz  z
            z  }t          t          ||d                                                    t          |                    |dd                                                    k    sJ t          ||z   |z             }t          t          ||d                                                    t          |                    |dd                                                    k    sJ t          t          |dz  dz             |z   |z             }t          t          ||d	                                                    t          |                    |dd	                                                    k    sJ |t          dd          z  |t          dd          z  z   |z   }t          t          |          |d          }|                                |t          d
d          z  |t          dd          z  z   |t          dd          z  z   |t          d	d          z  dz  z   |z   |t          dd          z  z   |t          dd          z  z   k    sJ t          t          |          |d          }|                                |t          dd          z  dz  |t          dd          z  z   |t          dd          z  z   dz   k    sJ t          t          |          |d          }|                                |t          d
d          z   dz  |t          dd          z  dz  z
  |t          dd          z  dz  z
  |t          d	d          z  d	z  z
  |z   |t          dd          z  z   |t          dd          z  z   k    sJ t          t          |          |d          }|                                |t          dd          z  d	z  |t          dd          z  z
  |t          dd          z  dz  z   |t          dd          z  z
  |t          dd          z  dz  z
  |t          dd          z  z
  |t          dd          z  dz  z
  dz   k    sJ t          t          |          dz  |d                                          t          |          dz                      |dd                                          k    sJ t          t          d| z             | d                                          | dz   dz  | dz  dz  z   | dz  dz  z
  | z   k    sJ t          t          dd| z  z             | d                                          d| dz  z  d| dz  z  dz  z   d| dz  z  z
  d| z  z   k    sJ t          t          d| z   | dz  z             | d                                          d| d
z  z  d
z  | dz  dz  z   | dz  dz  z   | d	z  dz  z
  | dz  dz  z   | dz  dz  z   d| dz  z  dz  z
  | dz  dz  z   | z   k    sJ t          t          d| |dz  z  z             | d                                          | d	z   |dz  z  d	z  | dz  |dz  z  dz  z   | dz  |dz  z  dz  z
  | dz  |d	z  z  dz  z   | dz  |dz  z  dz  z
  | |dz  z  z   k    sJ d S )Nz
x, a, b, crF   r   rB   rA   rw   rC   rD   rN   rf   r   r   r   r   rc   rE   rx   rb   i@   rd   rS   )r)   r&   r   r*   r   r   r+   r5   r,   r-   r   r2   )r@   r9   r:   r   rI   r   s         r<   test_rs_seriesr   :  s
   &&JAq!QQ1%%''1,,,,SVVQ""**,,Qq!	2 2GII   SVVc!ff_a++33553q66A<q!Q+ + + +SVVCFF]Aq))1133QA:q!Q+ + + + 
Q!c!Q$ii!Q$q&()A)Aq"%%--//00F188Aq
< <GII5 5     	AqDFQqSLC!HHS1XXq[00A)Aq!$$,,..//6!((1a	; ;7994 4     	AqD1Hs1a4!8}}q(1a4/0A)Aq!$$,,..//6!((1a	; ;7994 4     	AqD1Q3JNQs1a4!A#:.A)Aq"%%--//00F188Aq
< <GII5 5     	AEAIA)Aq!$$,,..//6!((1a	; ;7994 4     	C1qMMA!""A)Aq!$$,,..//6!((1a	; ;7994 4     	
2a77
Q1QZ!#A#a&&!QA99;;!R!WW*q"R))|3aBrlBQ1QZPQ\Q	
r!AwwJBqGG$ $ $ $ $ 	#a&&!QA99;;!R!WW*Q,BqGG3aAaj@1DDDDD#a&&!QA99;;1b1gg:+a-!R2YY,q.81bBii<>I	2a77
1b1gg:&()2a77
3 3 3 3 3 	#a&&!QA99;;!R2YY,q.1b1gg:52a77
2E2aPQ77
R	2a77
1"R))|$&'Aajl3567 7 7 7 7 SVVAXq!$$,,..3q66!82C2CAq3 3wyy    SQZZA&&..00QTE!Gad1f4DqDF55    SQqS\\1a((0022c!Q$hAqD6JadF7qS7! ! ! ! !SQA&&2..6688Bq!tGAIqDF=T!V=$&'d1f=-/0!tAv=6891Q=?AB1a4=JqDF==     SQq!tV__a++3355!Q$q"uQqDBJqL:!#$a419Q;:/12Aad1:=qDAIaK: "#AqD&:) ) ) ) ) ) )r>   N)^sympy.polys.domainsr   r   r   sympy.polys.ringsr   sympy.polys.ring_seriesr   r   r	   r
   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r    r!   r"   r#   r$   r%   r&   sympy.testing.pytestr'   r(   sympy.core.symbolr)   sympy.functionsr*   r+   r,   r-   r.   r/   r0   r1   r2   r3   sympy.core.numbersr4   
sympy.corer5   r6   r=   rK   rQ   r]   r_   ri   rk   ru   r|   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r    r>   r<   <module>r     s   * * * * * * * * * * " " " " " "@ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ . - - - - - - - % % % % % %                        ' ' ' ' ' '                  7 7 76 6 6> > >"; ; ;\ \ \$  D D D$    .Y Y Y>" " "@$ $ $Q Q Q6 6 61 1 1,  &4 4 4  >D D D" " "4" " "4( ( (  .& & &A A AK K K&, , ,
# 
# 
#=$ =$ =$~	P 	P 	P  4M M M @) @) @) @) @)r>   