
    "`Er                        d Z ddlmZ ddlmZmZ d Z G d de      ZdId
Z	dJdZ
d Zd Zd Zd Zd ddfd ddfd ddfd ddfd ddfd ddfd ddfd d dfd! d"dfd# d$dfd% d&dfd' d(dfd) d*dfd+ d,dfd- d.dfd/ d0dfd1 d2dfd3 d4dfd5 d6dfd7 d8dfd9 d:dfd; d<dfd= d>dfd? d@dfdA dBdfdC dDdfdE dFdfgZg ddd	d	fdGZe	e_	        e
e_
        ee_        edHk(  rddlZ ej&                          yy)Kzs
Implements the PSLQ algorithm for integer relation detection,
and derivative algorithms for constant recognition.
   )xrange)	int_types
sqrt_fixedc                 $    | d|dz
  z  z   |z	  |z  S Nr    )xprecs     7/usr/lib/python3/dist-packages/mpmath/identification.pyround_fixedr   
   s    !d1f+4'D00    c                       e Zd Zy)IdentificationMethodsN)__name__
__module____qualname__r   r   r   r   r      s    r   r   N  Fc                    t        |      }|dk  rt        d      | j                  }|dk  rt        d      |r|t        d|      z  dk  rt	        d       t        |dz        }|| j                  d      | z  }n| j                  |      }d	}	||	z  }|rt	        d
|| j                  |      fz         | j                  ||      }|sJ dg|D 
cg c]#  }
| j                  | j                  |
      |      % c}
z   }t        d |dd D              }|st        d      ||dz  k  r|rt	        d       yt        d|z  dz  |      }i }i }i }t        d|dz         D ]1  }t        d|dz         D ]  }||k(  |z  x|||f<   |||f<   d|||f<    3 dgdg|z  z   }t        d|dz         D ]5  }d}t        ||dz         D ]  }|||   dz  |z	  z  } t        ||      ||<   7 |d   }|dd }t        d|dz         D ]  }||   |z  |z  ||<   ||   |z  |z  ||<     t        d|dz         D ]  }t        |dz   |      D ]	  }d|||f<    ||dz
  k  r#||   r||dz      |z  ||   z  |||f<   nd|||f<   t        d|      D ]1  }||   ||dz      z  }|r||    ||   z  |z  |z  |||f<   +d|||f<   3  t        d|dz         D ]  }t        |dz
  dd      D ]  }|||f   rt        |||f   |z  |||f   z  |      }n(||   |||   z  |z	  z   ||<   t        d|dz         D ]  }|||f   ||||f   z  |z	  z
  |||f<    t        d|dz         D ]6  }|||f   ||||f   z  |z	  z
  |||f<   |||f   ||||f   z  |z	  z   |||f<   8   t        |      D ]  }d}d}t        d|      D ]-  }|||f   }||z  t        |      z  ||dz
  z  z	  }||kD  s*|}|}/ ||dz      ||   c||<   ||dz   <   t        d|dz         D ]!  }||dz   |f   |||f   c|||f<   ||dz   |f<   # t        d|dz         D ]!  }||dz   |f   |||f   c|||f<   ||dz   |f<   # t        d|dz         D ]!  }|||dz   f   |||f   c|||f<   |||dz   f<   # ||dz
  k  rt        |||f   dz  |||dz   f   dz  z   |z	  |      }|s n|||f   |z  |z  }|||dz   f   |z  |z  }t        ||dz         D ]=  }|||f   }|||dz   f   } ||z  || z  z   |z	  |||f<   | |z  || z  z   |z	  |||dz   f<   ? t        |dz   |dz         D ]  }t        t        |dz
  |dz         dd      D ]  }	 t        |||f   |z  |||f   z  |      }||   |||   z  |z	  z   ||<   t        d|dz         D ]  }|||f   ||||f   z  |z	  z
  |||f<    t        d|dz         D ]6  }|||f   ||||f   z  |z	  z
  |||f<   |||f   ||||f   z  |z	  z   |||f<   8   ||z  }!t        d|dz         D ]  }t        ||         }"|"|k  rt        d|dz         D cg c]  }t        t        |||f   |      |z	        ! }#}t        d |#D              |k  r>|r6t	        d||| j                  |"| j                  d      |z  z  d      fz         |#c c S t        |"|!      }! t        d |j#                         D              }$|$rdd|z  z  |$z  |z	  }%|%dz  }%n| j$                  }%|r7t	        d||| j                  |!| j                  d      |z  z  d      |%fz         |%|k\  s n |rt	        d|fz         t	        d%z         yc c}
w # t         $ r Y  Hw xY wc c}w )a  
    Given a vector of real numbers `x = [x_0, x_1, ..., x_n]`, ``pslq(x)``
    uses the PSLQ algorithm to find a list of integers
    `[c_0, c_1, ..., c_n]` such that

    .. math ::

        |c_1 x_1 + c_2 x_2 + ... + c_n x_n| < \mathrm{tol}

    and such that `\max |c_k| < \mathrm{maxcoeff}`. If no such vector
    exists, :func:`~mpmath.pslq` returns ``None``. The tolerance defaults to
    3/4 of the working precision.

    **Examples**

    Find rational approximations for `\pi`::

        >>> from mpmath import *
        >>> mp.dps = 15; mp.pretty = True
        >>> pslq([-1, pi], tol=0.01)
        [22, 7]
        >>> pslq([-1, pi], tol=0.001)
        [355, 113]
        >>> mpf(22)/7; mpf(355)/113; +pi
        3.14285714285714
        3.14159292035398
        3.14159265358979

    Pi is not a rational number with denominator less than 1000::

        >>> pslq([-1, pi])
        >>>

    To within the standard precision, it can however be approximated
    by at least one rational number with denominator less than `10^{12}`::

        >>> p, q = pslq([-1, pi], maxcoeff=10**12)
        >>> print(p); print(q)
        238410049439
        75888275702
        >>> mpf(p)/q
        3.14159265358979

    The PSLQ algorithm can be applied to long vectors. For example,
    we can investigate the rational (in)dependence of integer square
    roots::

        >>> mp.dps = 30
        >>> pslq([sqrt(n) for n in range(2, 5+1)])
        >>>
        >>> pslq([sqrt(n) for n in range(2, 6+1)])
        >>>
        >>> pslq([sqrt(n) for n in range(2, 8+1)])
        [2, 0, 0, 0, 0, 0, -1]

    **Machin formulas**

    A famous formula for `\pi` is Machin's,

    .. math ::

        \frac{\pi}{4} = 4 \operatorname{acot} 5 - \operatorname{acot} 239

    There are actually infinitely many formulas of this type. Two
    others are

    .. math ::

        \frac{\pi}{4} = \operatorname{acot} 1

        \frac{\pi}{4} = 12 \operatorname{acot} 49 + 32 \operatorname{acot} 57
            + 5 \operatorname{acot} 239 + 12 \operatorname{acot} 110443

    We can easily verify the formulas using the PSLQ algorithm::

        >>> mp.dps = 30
        >>> pslq([pi/4, acot(1)])
        [1, -1]
        >>> pslq([pi/4, acot(5), acot(239)])
        [1, -4, 1]
        >>> pslq([pi/4, acot(49), acot(57), acot(239), acot(110443)])
        [1, -12, -32, 5, -12]

    We could try to generate a custom Machin-like formula by running
    the PSLQ algorithm with a few inverse cotangent values, for example
    acot(2), acot(3) ... acot(10). Unfortunately, there is a linear
    dependence among these values, resulting in only that dependence
    being detected, with a zero coefficient for `\pi`::

        >>> pslq([pi] + [acot(n) for n in range(2,11)])
        [0, 1, -1, 0, 0, 0, -1, 0, 0, 0]

    We get better luck by removing linearly dependent terms::

        >>> pslq([pi] + [acot(n) for n in range(2,11) if n not in (3, 5)])
        [1, -8, 0, 0, 4, 0, 0, 0]

    In other words, we found the following formula::

        >>> 8*acot(2) - 4*acot(7)
        3.14159265358979323846264338328
        >>> +pi
        3.14159265358979323846264338328

    **Algorithm**

    This is a fairly direct translation to Python of the pseudocode given by
    David Bailey, "The PSLQ Integer Relation Algorithm":
    http://www.cecm.sfu.ca/organics/papers/bailey/paper/html/node3.html

    The present implementation uses fixed-point instead of floating-point
    arithmetic, since this is significantly (about 7x) faster.
       zn cannot be less than 25   zprec cannot be less than 53   z*Warning: precision for PSLQ may be too lowg      ?N<   zPSLQ using prec %i and tol %sc              3   2   K   | ]  }t        |        y wNabs).0xxs     r   	<genexpr>zpslq.<locals>.<genexpr>   s     '2s2w'   r   z)PSLQ requires a vector of nonzero numbersd   z#STOPPING: (one number is too small)          c              3   2   K   | ]  }t        |        y wr   r   )r   vs     r   r   zpslq.<locals>.<genexpr>  s     +!s1v+r    z'FOUND relation at iter %i/%i, error: %sc              3   2   K   | ]  }t        |        y wr   r   )r   hs     r   r   zpslq.<locals>.<genexpr>'  s     1c!f1r    z%i/%i:  Error: %8s   Norm: %szCANCELLING after step %i/%i.z2Could not find an integer relation. Norm bound: %s)len
ValueErrorr
   maxprintintmpfconvertnstrto_fixedminr   r   ranger   r   ZeroDivisionErrorvaluesinf)&ctxr	   tolmaxcoeffmaxstepsverbosenr
   targetextraxkminxgABHijsktysjj1REPmszmaxr)   szt0t1t2t3t4best_errerrvecrecnormnorms&                                         r   pslqr[      s	   f 	AA1u233 88Dby67743q8#a':;F
{ggajF7#kk#EEMD-sxx}0EEF
,,sD
!CJ3 
A>b#,,swwr{D1>>A '12''DDEEc3h78AtGa<&A
A
A
A Aqs^ 1Q3 	A !tn,AacFQqsVAacF	
 
!qAAqs^ #1Q3 	#A!A$'T/"A	#!T"!	#
 	
!A	!AAqs^ #!"!!"!# Aqs^ !Q 	AAacF	!8tAaC&D.QqT1!A#!A#q! 	AQ4!A#;DaD51:,t3!A#!A#	 Aqs^ 5!Q# 	5A1v1Q34!AaC& 8$? Q41QqT6T>*AaDAqs^ 51Q31QqsV8t#34!A#5Aqs^ 51Q31QqsV8t#34!A#1Q31QqsV8t#34!A#5	55 X Mq! 	A!A#AQ$Q-T1Q3Z0BEz	 1vqt!a!f!A#CA1QqSU8QqsV 0!A#!A#a%C!A#CA1QqSU8QqsV 0!A#!A#a%C!A#CA1QqsU8QqsV 0!A#!AaC%CA:QqsVQY1QqS514t;TBB AaC&D.R'BAacE(d"r)BAqs^ 2qsVq1uXR%2+$.!A#CF2b5LT1!AaC%	2 !QqS! 	9AC!QqSM1b1 9#QqsVt^a!f$<dCA t!A$4/0!1Q3 9AqsVq1Q3x4'78AacF91Q3 9AqsVq1Q3x4'78AacFqsVq1Q3x4'78AacF99	9& T>1Q3 	*Aad)CSy a!!s;q1vt4<=  +s++h6G (CHHS3771:t;K5KQ,OPQ RJ3)H	*  1ahhj111T6]w.47DSLD77D1hCGGAJ4D)Da H$OP Q8[M\ ,X>?BTIJc ?H ) (s   (]4"]9$^
9	^	^	c                 
   | j                  |      }|dk  rt        d      |dk(  rddgS | j                  d      g}t        d|dz         D ]5  }|j                  ||z          | j                  |fi |}|-|ddd   c S  y)a  
    ``findpoly(x, n)`` returns the coefficients of an integer
    polynomial `P` of degree at most `n` such that `P(x) \approx 0`.
    If no polynomial having `x` as a root can be found,
    :func:`~mpmath.findpoly` returns ``None``.

    :func:`~mpmath.findpoly` works by successively calling :func:`~mpmath.pslq` with
    the vectors `[1, x]`, `[1, x, x^2]`, `[1, x, x^2, x^3]`, ...,
    `[1, x, x^2, .., x^n]` as input. Keyword arguments given to
    :func:`~mpmath.findpoly` are forwarded verbatim to :func:`~mpmath.pslq`. In
    particular, you can specify a tolerance for `P(x)` with ``tol``
    and a maximum permitted coefficient size with ``maxcoeff``.

    For large values of `n`, it is recommended to run :func:`~mpmath.findpoly`
    at high precision; preferably 50 digits or more.

    **Examples**

    By default (degree `n = 1`), :func:`~mpmath.findpoly` simply finds a linear
    polynomial with a rational root::

        >>> from mpmath import *
        >>> mp.dps = 15; mp.pretty = True
        >>> findpoly(0.7)
        [-10, 7]

    The generated coefficient list is valid input to ``polyval`` and
    ``polyroots``::

        >>> nprint(polyval(findpoly(phi, 2), phi), 1)
        -2.0e-16
        >>> for r in polyroots(findpoly(phi, 2)):
        ...     print(r)
        ...
        -0.618033988749895
        1.61803398874989

    Numbers of the form `m + n \sqrt p` for integers `(m, n, p)` are
    solutions to quadratic equations. As we find here, `1+\sqrt 2`
    is a root of the polynomial `x^2 - 2x - 1`::

        >>> findpoly(1+sqrt(2), 2)
        [1, -2, -1]
        >>> findroot(lambda x: x**2 - 2*x - 1, 1)
        2.4142135623731

    Despite only containing square roots, the following number results
    in a polynomial of degree 4::

        >>> findpoly(sqrt(2)+sqrt(3), 4)
        [1, 0, -10, 0, 1]

    In fact, `x^4 - 10x^2 + 1` is the *minimal polynomial* of
    `r = \sqrt 2 + \sqrt 3`, meaning that a rational polynomial of
    lower degree having `r` as a root does not exist. Given sufficient
    precision, :func:`~mpmath.findpoly` will usually find the correct
    minimal polynomial of a given algebraic number.

    **Non-algebraic numbers**

    If :func:`~mpmath.findpoly` fails to find a polynomial with given
    coefficient size and tolerance constraints, that means no such
    polynomial exists.

    We can verify that `\pi` is not an algebraic number of degree 3 with
    coefficients less than 1000::

        >>> mp.dps = 15
        >>> findpoly(pi, 3)
        >>>

    It is always possible to find an algebraic approximation of a number
    using one (or several) of the following methods:

        1. Increasing the permitted degree
        2. Allowing larger coefficients
        3. Reducing the tolerance

    One example of each method is shown below::

        >>> mp.dps = 15
        >>> findpoly(pi, 4)
        [95, -545, 863, -183, -298]
        >>> findpoly(pi, 3, maxcoeff=10000)
        [836, -1734, -2658, -457]
        >>> findpoly(pi, 3, tol=1e-7)
        [-4, 22, -29, -2]

    It is unknown whether Euler's constant is transcendental (or even
    irrational). We can use :func:`~mpmath.findpoly` to check that if is
    an algebraic number, its minimal polynomial must have degree
    at least 7 and a coefficient of magnitude at least 1000000::

        >>> mp.dps = 200
        >>> findpoly(euler, 6, maxcoeff=10**6, tol=1e-100, maxsteps=1000)
        >>>

    Note that the high precision and strict tolerance is necessary
    for such high-degree runs, since otherwise unwanted low-accuracy
    approximations will be detected. It may also be necessary to set
    maxsteps high to prevent a premature exit (before the coefficient
    bound has been reached). Running with ``verbose=True`` to get an
    idea what is happening can be useful.
    r   zn cannot be less than 1r$   Nr%   )r/   r+   r4   appendr[   )r8   r	   r=   kwargsxsrF   as          r   findpolyra   7  s    R 	
A1u233Av1v
''!*B1QqS\ 
		!Q$CHHR"6"=TrT7N	r   c                 V    | |}}|r
|||z  }}|r
|dk7  r
| |z  } ||z  }|dk(  r| S | |fS r   r   )pqr	   rK   s       r   fracgcdre     sN    aqA
!a%1 Av	a	aAva4Kr   c                 z   | d   }| dd  } g }t        t        |             D ]r  }| |   }|st        | |      }||   d   }|dk(  rd}nd|z   }t        |t              r|dkD  rt        |      |z   }nd|z  |z   }nd|z  |z   }|j                  |       t dj                  |      }d	|v sd|v rd
|z   dz   }|xs dS )Nr$   r   1 *z(%s)z(%s/%s)z + +()0)r4   r*   re   
isinstancer   strr]   join)	r	constantsrd   rH   rF   rc   zcsterms	            r   
pslqstringrv     s    	!A	!"A
A3q6] aD1A1aBSy2X!Y'q5Q"$"(1*!2$!A+HHTN 	

1A
ax3!8!GcM8Or   c                    | d   }| dd  } g }g }t        t        |             D ]  }| |   }|st        | |      }||   d   }t        |t              r;t        |      dk(  r|}	n|dt        |      }	||g|dk     j                  |	       k|dt        |d         d|d   d}	||g|d   dk     j                  |	        dj                  |      }dj                  |      }|r|r	d|d	|dS |r|S |rd
|z  S y )Nr$   r   z**z**(/rl   ri   rk   z)/(z1/(%s))r4   r*   re   rn   r   r   r]   rp   )
rq   rr   rd   numdenrF   rc   rs   rt   rJ   s
             r   
prodstringr{     s   	!A	!"A
C
C3q6] .aD1A1aB!Y'q6Q;B02CF$;c1Q3''*%'QqTAaD9c1Q46"**1-. ((3-C
((3-C
s#s33
3J
8c>!sr   c                    |dk  r	| | | }}}| | j                  |dz  d|z  |z  z
        z   d|z  z  }| | j                  |dz  d|z  |z  z
        z
  d|z  z  }t        ||z
        t        ||z
        k  r4|rd| d|dz  d|z  |z  z
  dd|z  d}|S dd	|z  |z  d
d|z  d}|S |rd| d|dz  d|z  |z  z
  dd|z  d}|S dd	|z  |z  d
d|z  d}|S )Nr$   r   r"   z((z+sqrt(z))/rl   z(sqrt(z)/z-sqrt(z(-sqrt()sqrtr   )r8   rJ   r`   bcu1u2rH   s           r   quadraticstringr     s(   1uA2qbA!"SXXad1Q3q5j!
!AaC	(B"SXXad1Q3q5j!
!AaC	(B
2a4y3r!t9A2ad1Q3q5j1=q
 H &(d1fQqS1q H A2ad1Q3q5j1=qH ')!tAvac2qHr   c                     ||z  S r   r   r8   r	   r   s      r   <lambda>r     
    1Q3 r   z$y/$cr$   c                     ||z  S r   r   r   s      r   r   r     r   r   z$c*$yc                     ||z  S r   r   r   s      r   r   r     r   r   z$c/$yc                     ||z  dz  S Nr   r   r   s      r   r   r         AaC!8 r   zsqrt($y)/$cc                     ||z  dz  S r   r   r   s      r   r   r     r   r   z$c*sqrt($y)c                     ||z  dz  S r   r   r   s      r   r   r     r   r   z$c/sqrt($y)c                     ||dz  z  S r   r   r   s      r   r   r         1QT6 r   zsqrt($y)/sqrt($c)c                     |dz  |z  S r   r   r   s      r   r   r     s    1a46 r   zsqrt($c)*sqrt($y)c                     ||dz  z  S r   r   r   s      r   r   r     r   r   zsqrt($c)/sqrt($y)c                 *    | j                  ||z        S r   r~   r   s      r   r   r          388AaC= r   z$y**2/$cc                 *    | j                  ||z        S r   r   r   s      r   r   r     r   r   z$c*$y**2c                 *    | j                  ||z        S r   r   r   s      r   r   r     r   r   z$c/$y**2c                 *    || j                  |      z  S r   r   r   s      r   r   r         1SXXa[= r   z$y**2/$c**2c                 *    | j                  |      |z  S r   r   r   s      r   r   r     s    388A;q= r   z$c**2*$y**2c                 *    || j                  |      z  S r   r   r   s      r   r   r     r   r   z$c**2/$y**2c                 *    | j                  ||z        S r   expr   s      r   r   r         3771Q3< r   z
log($y)/$cc                 *    | j                  ||z        S r   r   r   s      r   r   r     r   r   z
$c*log($y)c                 *    | j                  ||z        S r   r   r   s      r   r   r     r   r   z
$c/log($y)c                 *    || j                  |      z  S r   r   r   s      r   r   r   	      1SWWQZ< r   z
log($y/$c)c                 *    | j                  |      |z  S r   r   r   s      r   r   r   
  s    3771:a< r   z
log($c*$y)c                 *    || j                  |      z  S r   r   r   s      r   r   r     r   r   z
log($c/$y)c                 *    | j                  ||z        S r   lnr   s      r   r   r         366!A#; r   z
exp($y)/$cc                 *    | j                  ||z        S r   r   r   s      r   r   r     r   r   z
$c*exp($y)c                 *    | j                  ||z        S r   r   r   s      r   r   r     r   r   z
$c/exp($y)c                 *    || j                  |      z  S r   r   r   s      r   r   r         1SVVAY; r   z
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k  r|d   r |t/        ||             |sd   S |rt        t              S yc c}	w c c}}w c c}w c c}}w c c}w c c}w c c}w )an  
    Given a real number `x`, ``identify(x)`` attempts to find an exact
    formula for `x`. This formula is returned as a string. If no match
    is found, ``None`` is returned. With ``full=True``, a list of
    matching formulas is returned.

    As a simple example, :func:`~mpmath.identify` will find an algebraic
    formula for the golden ratio::

        >>> from mpmath import *
        >>> mp.dps = 15; mp.pretty = True
        >>> identify(phi)
        '((1+sqrt(5))/2)'

    :func:`~mpmath.identify` can identify simple algebraic numbers and simple
    combinations of given base constants, as well as certain basic
    transformations thereof. More specifically, :func:`~mpmath.identify`
    looks for the following:

        1. Fractions
        2. Quadratic algebraic numbers
        3. Rational linear combinations of the base constants
        4. Any of the above after first transforming `x` into `f(x)` where
           `f(x)` is `1/x`, `\sqrt x`, `x^2`, `\log x` or `\exp x`, either
           directly or with `x` or `f(x)` multiplied or divided by one of
           the base constants
        5. Products of fractional powers of the base constants and
           small integers

    Base constants can be given as a list of strings representing mpmath
    expressions (:func:`~mpmath.identify` will ``eval`` the strings to numerical
    values and use the original strings for the output), or as a dict of
    formula:value pairs.

    In order not to produce spurious results, :func:`~mpmath.identify` should
    be used with high precision; preferably 50 digits or more.

    **Examples**

    Simple identifications can be performed safely at standard
    precision. Here the default recognition of rational, algebraic,
    and exp/log of algebraic numbers is demonstrated::

        >>> mp.dps = 15
        >>> identify(0.22222222222222222)
        '(2/9)'
        >>> identify(1.9662210973805663)
        'sqrt(((24+sqrt(48))/8))'
        >>> identify(4.1132503787829275)
        'exp((sqrt(8)/2))'
        >>> identify(0.881373587019543)
        'log(((2+sqrt(8))/2))'

    By default, :func:`~mpmath.identify` does not recognize `\pi`. At standard
    precision it finds a not too useful approximation. At slightly
    increased precision, this approximation is no longer accurate
    enough and :func:`~mpmath.identify` more correctly returns ``None``::

        >>> identify(pi)
        '(2**(176/117)*3**(20/117)*5**(35/39))/(7**(92/117))'
        >>> mp.dps = 30
        >>> identify(pi)
        >>>

    Numbers such as `\pi`, and simple combinations of user-defined
    constants, can be identified if they are provided explicitly::

        >>> identify(3*pi-2*e, ['pi', 'e'])
        '(3*pi + (-2)*e)'

    Here is an example using a dict of constants. Note that the
    constants need not be "atomic"; :func:`~mpmath.identify` can just
    as well express the given number in terms of expressions
    given by formulas::

        >>> identify(pi+e, {'a':pi+2, 'b':2*e})
        '((-2) + 1*a + (1/2)*b)'

    Next, we attempt some identifications with a set of base constants.
    It is necessary to increase the precision a bit.

        >>> mp.dps = 50
        >>> base = ['sqrt(2)','pi','log(2)']
        >>> identify(0.25, base)
        '(1/4)'
        >>> identify(3*pi + 2*sqrt(2) + 5*log(2)/7, base)
        '(2*sqrt(2) + 3*pi + (5/7)*log(2))'
        >>> identify(exp(pi+2), base)
        'exp((2 + 1*pi))'
        >>> identify(1/(3+sqrt(2)), base)
        '((3/7) + (-1/7)*sqrt(2))'
        >>> identify(sqrt(2)/(3*pi+4), base)
        'sqrt(2)/(4 + 3*pi)'
        >>> identify(5**(mpf(1)/3)*pi*log(2)**2, base)
        '5**(1/3)*pi*log(2)**2'

    An example of an erroneous solution being found when too low
    precision is used::

        >>> mp.dps = 15
        >>> identify(1/(3*pi-4*e+sqrt(8)), ['pi', 'e', 'sqrt(2)'])
        '((11/25) + (-158/75)*pi + (76/75)*e + (44/15)*sqrt(2))'
        >>> mp.dps = 50
        >>> identify(1/(3*pi-4*e+sqrt(8)), ['pi', 'e', 'sqrt(2)'])
        '1/(3*pi + (-4)*e + 2*sqrt(2))'

    **Finding approximate solutions**

    The tolerance ``tol`` defaults to 3/4 of the working precision.
    Lowering the tolerance is useful for finding approximate matches.
    We can for example try to generate approximations for pi::

        >>> mp.dps = 15
        >>> identify(pi, tol=1e-2)
        '(22/7)'
        >>> identify(pi, tol=1e-3)
        '(355/113)'
        >>> identify(pi, tol=1e-10)
        '(5**(339/269))/(2**(64/269)*3**(13/269)*7**(92/269))'

    With ``full=True``, and by supplying a few base constants,
    ``identify`` can generate almost endless lists of approximations
    for any number (the output below has been truncated to show only
    the first few)::

        >>> for p in identify(pi, ['e', 'catalan'], tol=1e-5, full=True):
        ...     print(p)
        ...  # doctest: +ELLIPSIS
        e/log((6 + (-4/3)*e))
        (3**3*5*e*catalan**2)/(2*7**2)
        sqrt(((-13) + 1*e + 22*catalan))
        log(((-6) + 24*e + 4*catalan)/e)
        exp(catalan*((-1/5) + (8/15)*e))
        catalan*(6 + (-6)*e + 15*catalan)
        sqrt((5 + 26*e + (-3)*catalan))/e
        e*sqrt(((-27) + 2*e + 25*catalan))
        log(((-1) + (-11)*e + 59*catalan))
        ((3/20) + (21/20)*e + (3/20)*catalan)
        ...

    The numerical values are roughly as close to `\pi` as permitted by the
    specified tolerance:

        >>> e/log(6-4*e/3)
        3.14157719846001
        >>> 135*e*catalan**2/98
        3.14166950419369
        >>> sqrt(e-13+22*catalan)
        3.14158000062992
        >>> log(24*e-6+4*catalan)-1
        3.14158791577159

    **Symbolic processing**

    The output formula can be evaluated as a Python expression.
    Note however that if fractions (like '2/3') are present in
    the formula, Python's :func:`~mpmath.eval()` may erroneously perform
    integer division. Note also that the output is not necessarily
    in the algebraically simplest form::

        >>> identify(sqrt(2))
        '(sqrt(8)/2)'

    As a solution to both problems, consider using SymPy's
    :func:`~mpmath.sympify` to convert the formula into a symbolic expression.
    SymPy can be used to pretty-print or further simplify the formula
    symbolically::

        >>> from sympy import sympify # doctest: +SKIP
        >>> sympify(identify(sqrt(2))) # doctest: +SKIP
        2**(1/2)

    Sometimes :func:`~mpmath.identify` can simplify an expression further than
    a symbolic algorithm::

        >>> from sympy import simplify # doctest: +SKIP
        >>> x = sympify('-1/(-3/2+(1/2)*5**(1/2))*(3/2-1/2*5**(1/2))**(1/2)') # doctest: +SKIP
        >>> x # doctest: +SKIP
        (3/2 - 5**(1/2)/2)**(-1/2)
        >>> x = simplify(x) # doctest: +SKIP
        >>> x # doctest: +SKIP
        2/(6 - 2*5**(1/2))**(1/2)
        >>> mp.dps = 30 # doctest: +SKIP
        >>> x = sympify(identify(x.evalf(30))) # doctest: +SKIP
        >>> x # doctest: +SKIP
        1/2 + 5**(1/2)/2

    (In fact, this functionality is available directly in SymPy as the
    function :func:`~mpmath.nsimplify`, which is essentially a wrapper for
    :func:`~mpmath.identify`.)

    **Miscellaneous issues and limitations**

    The input `x` must be a real number. All base constants must be
    positive real numbers and must not be rationals or rational linear
    combinations of each other.

    The worst-case computation time grows quickly with the number of
    base constants. Already with 3 or 4 base constants,
    :func:`~mpmath.identify` may require several seconds to finish. To search
    for relations among a large number of constants, you should
    consider using :func:`~mpmath.pslq` directly.

    The extended transformations are applied to x, not the constants
    separately. As a result, ``identify`` will for example be able to
    recognize ``exp(2*pi+3)`` with ``pi`` given as a base constant, but
    not ``2*exp(pi)+3``. It will be able to recognize the latter if
    ``exp(pi)`` is given explicitly as a base constant.

    c                 D    rt        d|        j                  |        y )NzFound: )r-   r]   )rH   	solutionsr<   s    r   addsolutionzidentify.<locals>.addsolution  s    E)Q'r   r$   rm   Nz-(%s)gffffff?c              3   :   K   | ]  }|t        |      f  y wr   )getattr)r   namer8   s     r   r   zidentify.<locals>.<genexpr>  s     L4dGC$56Ls   r   rg   r   c              3   2   K   | ]  }t        |        y wr   r   r   uws     r   r   zidentify.<locals>.<genexpr>  s     $9SW$9r    r#   z/$cz$yrh   z$c.)r   r#   r      c           	   3      K   | ]@  }t        j                  j                        j                  |      z  d              B yw)r   N)boolra   r   )r   rF   r`   r8   s     r   r   zidentify.<locals>.<genexpr>8  s6     PQtCLL366!9)<Q?@Ps   AA	c              3   2   K   | ]  }t        |        y wr   r   r   s     r   r   zidentify.<locals>.<genexpr><  s      5RR 5r    )key)r/   identifyepsrn   dictsorteditemsdireval
transformsr   r[   r,   rv   oner*   r   replacer-   sumr]   r   ro   r{   ) r8   r	   rr   r9   r:   fullr<   r   solrH   Mr   r'   	namespacerc   valueftftnredr   cnrJ   r`   rq   rd   aabbccilogslogsrF   r   s    `     `               `        @r   r   r     s   j I 	
A 	Av1ullA2y#xwG;J'*+!GAI++S= 
ggclggslAi&=CIOODU=VW	q#''!*d+WIWL3s8LLI:CDQ$q),a0DID	 	I6=D%66ggaj#&')3	 # C 	EArrSy3qA1v1}A!i8!88#qAAA}$9q$9!9Q!>1Q4q), HHcggq!Q$/a8=SVq[QqT!"JBB3r73r73r73q8+C"R;9%3,D!,44UB?AD!,44T2>AAIaL0c
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 7  9> 3#7s*   O6O8OOO#O!&O&
__main__)Nr   r!   F)r   )__doc__libmp.backendr   libmpr   r   r   objectr   r[   ra   re   rv   r{   r   r   r   r   doctesttestmodr   r   r   <module>r      s  
 " (1	F 	dL	sj	0"." ###]A.]A.]A..2.2.2 *a0 *a0 *a0 -3 -3 -3q1q1q1q1q1q1a0a0a0a0a0a07
<  "tdob	 "  !)  !)   zGOO r   