
    "`0A                        d Z ddlZddlmZ ddlmZ ddlmZmZmZmZm	Z	m
Z
mZ dgdz  Z edd      D ]  Zegdd	ez
  z  z  edez  ddedz   z  <    d=d
Zd Zd Zedk(  rddlZej                   Zej"                  Zd Zedk(  r ej(                         dk\  rd Znd Z ed      D  cg c]  } d| z  	 c} Zd Zd Zd Zedk(  reZeZnedk(  rej4                  ZeZeZneZeZedk(  rd ee      v rej<                  Z ed      D cg c]
  } e|       c}Z ed      D cg c]
  } e|       c}Z d Z!dZ"de"fdZ#dde"fdZ$dde"fdZ%edk(  re%Z&ne$Z&ddz  Z'dd z  Z(dd!z  Z)dd"z  Z*d#Z+d$Z,d% Z-d& Z.d' Z/d( Z0d) Z1e1Z2edk(  rN ej(                         dk\  rejf                  xZ4xZ5Z3ejl                  Z7n=ejp                  xZ4xZ5Z3ejn                  Z7n edk(  r e9ed*d+       xZ4xZ5Z3d, Z7ne-Z4e.Z5e0Z3e/Z7i fd-Z:d.Z;ddd/fd0Z<ddiddigfd1Z=edk(  rej|                  Z<nedk(  rd2 Z<ej~                  Z:d3 Z@edk(  rd4 Z@d5ZA eBeA      ZCd6 ZDd7 ZEd8 ZFd9ZGde
ifd:ZHd; ZId< ZJyc c} w c c}w c c}w )>zw
Utility functions for integer math.

TODO: rename, cleanup, perhaps move the gmpy wrapper code
here from settings.py

    N)bisect   )xrange)BACKENDgmpysage
sage_utilsMPZMPZ_ONEMPZ_ZERO         c                 d    |g}|d   | |z  kD  r||d   |z  dz   gz   }|d   | |z  kD  r|ddd   S )a  
    Return a list of integers ~=

    [start, n*start, ..., target/n^2, target/n, target]

    but conservatively rounded so that the quotient between two
    successive elements is actually slightly less than n.

    With n = 2, this describes suitable precision steps for a
    quadratically convergent algorithm such as Newton's method;
    with n = 3 steps for cubic convergence (Halley's method), etc.

        >>> giant_steps(50,1000)
        [66, 128, 253, 502, 1000]
        >>> giant_steps(50,1000,4)
        [65, 252, 1000]

       N )starttargetnLs       9/usr/lib/python3/dist-packages/mpmath/libmp/libintmath.pygiant_stepsr      sP    & 
A
B%%'/2A B%%'/TrT7N    c                 "    |dk\  r| |z	  S | | z  S )zFor an integer x, calculate x >> n with the fastest (floor)
    rounding. Unlike the plain Python expression (x >> n), n is
    allowed to be negative, in which case a left shift is performed.r   r   xr   s     r   rshiftr   +        	Ava1f}QBir   c                 "    |dk\  r| |z  S | | z	  S )zFor an integer x, calculate x << n. Unlike the plain Python
    expression (x << n), n is allowed to be negative, in which case a
    right shift with default (floor) rounding is performed.r   r   r   s     r   lshiftr!   2   r   r   r   c                 ~    | sy| dz  }|r	t         |   S d}| dz  } | dz  s| dz  } |dz  }| dz  s|t         | dz     z   S )z1Count the number of trailing zero bits in abs(n).r      r   )small_trailing)r   low_bytets      r   python_trailingr'   >   se    4xHh''	A!GA$h	a	Q $h ~a$h'''r   r   2c                 :    | rt        |       j                         S yz<Count the number of trailing zero bits in abs(n) using gmpy.r   )r
   	bit_scan1r   s    r   gmpy_trailingr-   N   s    Q))++r   c                 :    | rt        |       j                         S yr*   )r
   scan1r,   s    r   r-   r-   S   s    Q'r   ,  c                     t        t        |       }|dk7  r|S t        t        j                  | d            dz
  }|t
        | |z	     z   S )0Calculate bit size of the nonnegative integer n.r0   r      )r   powersintmathlogbctable)r   bcs     r   python_bitcountr:   [   sF    		B	Sy		TXXa^	q	 B2r   c                 <    | rt        |       j                  d      S y)r2   r   r   )r
   	numdigitsr,   s    r   gmpy_bitcountr=   c   s    Q!!!$
$r   c                 4    t        |       j                         S N)r
   trailing_zero_bitsr,   s    r   sage_trailingrA   l   s    q6$$&&r   
bit_lengthi   c                 *    | t        |      |z  z  |z	  S )zaChanges radix of a fixed-point number; i.e., converts
    x * 2**xbits to floor(x * 10**bdigits).)r
   )r   xbitsbasebdigitss       r   bin_to_radixrG      s     D	7"#u,,r   $0123456789abcdefghijklmnopqrstuvwxyz
   c                     |dk(  rt        |       S g }| r&t        | |      \  } }|j                  ||          | r&dj                  |ddd         S )ziReturn the string numeral of a positive integer in an arbitrary
    base. Most efficient for small input.rI    Nr   )strdivmodappendjoin)r   rE   digitsdigsdigits        r   small_numeralrS      sY     rz1vD
!T?5F5M"  774":r   c                     | dk  r| sydt        |  |||      z   S |dk  rt        | ||      S |dz  |dz  z   }t        | ||z        \  }}t        ||||      }t        ||||      j                  |d      }||z   S )_  Represent the integer n as a string of digits in the given base.
    Recursive division is used to make this function about 3x faster
    than Python's str() for converting integers to decimal strings.

    The 'size' parameters specifies the number of digits in n; this
    number is only used to determine splitting points and need not be
    exact.r   0-   r   r   )numeralrS   rM   rjust	r   rE   sizerP   halfABadbds	            r   numeral_pythonrb      s     	AvWaRtV444czQf--AI$(#D!T4Z DAq	D$	'B	D$	'	-	-dC	8B7Nr   c                    | dk  rdt        |  |||      z   S |dk  rt        j                  | |      S |dz  |dz  z   }t        | t	        |      |z        \  }}t        ||||      }t        ||||      j                  |d      }||z   S )rU   r   rW   i` r   r   rV   )rY   r   rP   rM   r
   rZ   r[   s	            r   numeral_gmpyrd      s     	1uWaRtV444 g~{{1d##AI$(#D!SY_%DAq	D$	'B	D$	'	-	-dC	8B7Nr   i   iX  i     l                l           c                     | s| S | t         k  r,| t        k  rt        | dz        S t        | dz  dz        dz   }n0t        |       }|dz  }t        | d|z  dz
  z	  dz  dz         |dz
  z  }	 || |z  z   dz	  }||k\  r|S |})zd
    Correctly (floor) rounded integer square root, using
    division. Fast up to ~200 digits.
          ?g-     ?r   r   d   2   )_1_800_1_50r5   bitcount)r   rr9   r   ys        r   isqrt_small_pythonro      s    
 6zu9q#v;3))*Q.a[EQqSW#A%&2. q!tVaK6H	 r   c                    | t         k  rLt        | dz        }| t        k\  r3|| |z  z   dz	  }| t        k\  r|| |z  z   dz	  }| t        k\  r|| |z  z   dz	  }|S t        |       }d}| d|z  z  } |d|z  z  }||dz  z  }|dz  }t        d|      }t        dd|z  z  | |d|z  z
  z	  dz  z        }|}t        ||      D ]1  }||z  d|z  |z
  z	  }	| ||z
  z	  |	z  |z	  }
|d|z  |
z
  z  |dz   z	  }|}3 || |z	  z  |z   z	  S )	a  
    Fast approximate integer square root, computed using division-free
    Newton iteration for large x. For random integers the result is almost
    always correct (floor(sqrt(x))), but is 1 ulp too small with a roughly
    0.1% probability. If x is very close to an exact square, the answer is
    1 ulp wrong with high probability.

    With 0 guard bits, the largest error over a set of 10^5 random
    inputs of size 1-10^5 bits was 3 ulp. The use of 10 guard bits
    almost certainly guarantees a max 1 ulp error.
    rg   r   rI   r   ri   g       @g         )rj   r5   _1_100_1_200_1_400rl   minr   )r   rn   r9   
guard_bitshbc	startprecrm   pppr2xr2s              r   isqrt_fast_pythonr}      s[   $ 	6z3K;QTaAF{AX!O;QTaA	!BJ!J,A!J,B2a4LB
a%CBIC!I+!1Y;"7D!@@AA	BC( cqtax bdr!a'1a43,RT* q#vJAjL))r   c                     | t         k  rt        |       }|| ||z  z
  fS t        |       dz   }| ||z  z
  }|dk  r|dz  }|dd|z  z   z  }|dk  r|r'|dd|z   z  kD  r|dz  }|dd|z  z   z  }|dd|z   z  kD  r||fS )z=Correctly rounded integer (floor) square root with remainder.r   r   r   )_1_600ro   r}   )r   rn   rems      r   sqrtrem_pythonr     s     	6zq!!ac'z!q A
ac'C
'	Q!A# ' 1Q3-Q!A# 1Q3- c6Mr   c                     t        |       d   S )z2Integer square root with correct (floor) rounding.r   )r   )r   s    r   isqrt_pythonr   +  s    !Qr   c                     t        | |z        S r?   )
isqrt_fast)r   precs     r   
sqrt_fixedr   /  s    agr   isqrtc                 4    t        |       j                         S r?   )r
   r   r,   s    r   <lambda>r   =  s    s1v||~ r   c                 4    t        |       j                         S r?   )r
   sqrtremr,   s    r   r   r   >  s    A( r   c                 ,   | dk  rd|  dz   z  t        |        z  S | |v r||    S | }t        t        t        t        f\  }}}}| rF| dz  r!||z  }||z  |z   ||z  z   ||z  |z   }}| dz  } n||z  }||z  |z   |d|z  |z  z   }}| dz  } | rF|dk  r|||<   |S )zCComputes the nth Fibonacci number as an integer, for
    integer n.r   r   r   r   rX   )ifibr   r   )	r   _cachemabrz   qaqqqs	            r   r   r   F  s     	1uqbd|dA2h&&F{ay	A (Hg5JAq!Q
q51BQ3r6!A#:qs2vqAFA1BQ3r62ac!e8qA!GA  	3wq	Hr   i  )r   r   c                     |j                  |       }|r|S t        |      }||dz
     }t        }|| k  r||z  }||k  r|||<   |dz  }|| k  r|S )z.Return n factorial (for integers n >= 0 only).r   )getlenMAX_FACTORIAL_CACHE)r   memofkrz   MAXs         r   ifacr   a  sk    AD	AQqS	A
C
q&	Q8DG	Q	 q&
 Hr   c                     || dz     }|j                  |       }|r|S t        |      }||   }t        }|| k  r|dz  }||z  }||k  r|||<   || k  r|S )z4Return n!! (double factorial), integers n >= 0 only.r   r   )r   maxr   )r   	memo_pairr   r   r   rz   r   s          r   ifac2r   p  st    QqS>DAD	AQA
C
a%	Q	Q8DG	 a%
 Hr   c                 >    t        t        j                  |             S r?   )r5   r   	factorialr,   s    r   r   r     s    S*+ r   c                     | dz   } t        t        |             }ddg|d d t        dt        | dz        dz         D ]"  }||   s	t        |dz  | |      D ]  }d||<   	 $ |D cg c]  }|s|	 c}S c c}w )Nr   r   r   rg   )listr   r5   )r   sieveijrz   s        r   list_primesr     s    	AAOEAE"1IAs1c6{1}% 8AqD!Q' a "!A"""s   %A4-A4c                 j    t        j                  | dz         D cg c]  }t        |       c}S c c}w )Nr   )r   primesr5   )r   _s     r   r   r     s'     $AaC 011A111s   0)rq      r                        %   )   +   /   c                     t                 dz  s dk(  S  dk  r t        v S t        D ]	  } |z  r	 y  dz
  t              z	   fd} dk  rddg}n dk  rg d	}nt        }|D ]  } ||      r y y
)a&  
    Determines whether n is a prime number. A probabilistic test is
    performed if n is very large. No special trick is used for detecting
    perfect powers.

        >>> sum(list_primes(100000))
        454396537
        >>> sum(n*isprime(n) for n in range(100000))
        454396537

    r   r   ri   Fc                 v    t        |       }|dk(  s|k(  ryt        d      D ]  }|dz  z  }|k(  s y y)Nr   Tr   F)powr   )r   r   rm   dr   r   ss      r   testzisprime.<locals>.test  sQ    !AJ6Q!V! 	A1qAAv	 r   i rq   l   He%Z	 )r   rq   r   r   r   r   r   T)r5   small_odd_primes_setsmall_odd_primestrailing)r   rz   r   	witnessesr   r   r   r   s   `    @@@r   isprimer     s     	AAq5Av2v((( 1u 	
!AA	QA 	7{qE		
_	&	$	 Aw r   c                     t        t        |             } | dk  r| S g }t        d| dz         D ]8  | z  r	| dz  z  s yt        fd|D              r(|j	                         : dt        |      z  S )z
    Evaluates the Moebius function which is `mu(n) = (-1)^k` if `n`
    is a product of `k` distinct primes and `mu(n) = 0` otherwise.

    TODO: speed up using factorization
    r   r   r   c              3   (   K   | ]	  }|z    y wr?   r   ).0r   rz   s     r   	<genexpr>zmoebius.<locals>.<genexpr>  s     .q1u.s   r   )absr5   r   sumrN   r   )r   factorsrz   s     @r   moebiusr     s|     	CFA1uGAqs^ "A1H.g..q!" Wr   c                  <    d}| D ]  }|r|s|||z  }}|r
|} |S )Nr   r   )argsr   r   s      r   gcdr     s<    	A !a%1  A Hr   i  c                    | dz  rt         S |j                  |       }|r|S t        }| }dD cg c]  }t        |       }}t	        d| dz         D ]  }t	        |dz   dd      D ]"  }|dz
  ||   z  |dz   ||dz      z  z   ||dz   <   $ |j                  d       d}t	        |dz   dd      D ]'  }	|||	dz      z  }||k  sd|dz  z  |d|z  z  z  ||<   ) || k(  sd|dz  z  |z  d|z  z  c S  yc c}w )a  
    Computes the Euler numbers `E(n)`, which can be defined as
    coefficients of the Taylor expansion of `1/cosh x`:

    .. math ::

        \frac{1}{\cosh x} = \sum_{n=0}^\infty \frac{E_n}{n!} x^n

    Example::

        >>> [int(eulernum(n)) for n in range(11)]
        [1, 0, -1, 0, 5, 0, -61, 0, 1385, 0, -50521]
        >>> [int(eulernum(n)) for n in range(11)]   # test cache
        [1, 0, -1, 0, 5, 0, -61, 0, 1385, 0, -50521]

    r   )r   r   r   r   r   r   r   r   r   N)r   r   MAX_EULER_CACHEr
   rangerN   )
r   r   r   r   r   r   r   r   sumar   s
             r   eulernumr     s5   $ 	1u

1A
C	A&'AQ'A'Aqsm 
/qsB# 	/Ac1Q4Z1Q3!A#,.AacF	/	qsB# 	:AAacFNDCx AqD\DAqDL9q		: 61a4L$&!Q$..
/ 	(s   C0c                 4   | dk  s|dk  rt         || k\  rt        | |k(        S |dk  rt        S t        g|dz   z  }t        |d<   t	        d| dz         D ]5  }t	        t        ||      dd      D ]  }|dz
  ||   z  ||dz
     z   ||<    7 d| |z   z  ||   z  S )z,
    Stirling number of the first kind.
    r   r   r   r   )
ValueErrorr
   r   r   r   ru   )r   r   r   r   r   s        r   	stirling1r   %  s     	1uAAv16{1u	
acAAaDAqs^ )Aq	1b) 	)AaC1Q4<!AaC&(AaD	)) !A#;1r   c                 F   | dk  s|dk  rt         || k\  rt        | |k(        S |dk  rt        |dk(        S t        }t        }t	        |dz         D ]A  }||z   dz  r||t        |      | z  z  z  }n||t        |      | z  z  z  }|||z
  z  |dz   z  }C |t        |      z  S )z-
    Stirling number of the second kind.
    r   r   )r   r
   r   r   r   r   )r   r   r   r&   r   s        r   	stirling2r   6  s     	1uAAv16{Av16{AAAaC[ #EQ;SVQYASVQYAQKAE"# Q<r   )r   )K__doc__r6   r   backendr   r   r   r   r	   r
   r   r   r$   r   r   r   r   r!   operatorr'   versionr-   r4   r:   r=   rA   rl   r   sage_bitcountdirrB   
trailtabler8   rG   	stddigitsrS   rb   rd   rY   rj   r   rt   rs   rr   rk   ro   r}   r   r   r   sqrt_fixed2r   isqrt_smallr   	isqrt_remr   sqrtgetattrr   r   r   r   fac	fibonaccir   r   setr   r   r   r   r   r   r   r   )r   r   s   00r   <module>r      sX      L L Ls	q 6A&'SA!H%5N1a4>QqS>"60   f__F__F( ft||~	
	 c
	#1!Q$	#' fHH''MHHHH
fT2H $):.ahqk.
 %d
,18A;
,-
 3	Y 	 Ai , !I , fGG	
C	
C	
C	
C	4.*`(  
ft||~+/::55j5..+/9944j5,,
G%=>?K ?*u(G$K"JEG  2    1!u~   f88D+D>>D# f2 < +, *X&J '{ $/L"{ 
$J /
,s   ,I(I-)I2