
    \hg                        S r SSKJr  SSKJr  SSKJr  SSKJr  SSK	J
r
  SSKJr  SSKJr  SS	KJr  SS
KJr  SS jrSS jrSS jr\SS j5       r\SS j5       r\SS j5       r\SS j5       rSS jrg)a  A module for special angle formulas for trigonometric functions

TODO
====

This module should be developed in the future to contain direct square root
representation of

.. math
    F(\frac{n}{m} \pi)

for every

- $m \in \{ 3, 5, 17, 257, 65537 \}$
- $n \in \mathbb{N}$, $0 \le n < m$
- $F \in \{\sin, \cos, \tan, \csc, \sec, \cot\}$

Without multi-step rewrites
(e.g. $\tan \to \cos/\sin \to \cos/\sqrt \to \ sqrt$)
or using chebyshev identities
(e.g. $\cos \to \cos + \cos^2 + \cdots \to \sqrt{} + \sqrt{}^2 + \cdots $),
which are trivial to implement in sympy,
and had used to give overly complicated expressions.

The reference can be found below, if anyone may need help implementing them.

References
==========

.. [*] Gottlieb, Christian. (1999). The Simple and straightforward construction
   of the regular 257-gon. The Mathematical Intelligencer. 21. 31-37.
   10.1007/BF03024829.
.. [*] https://resources.wolframcloud.com/FunctionRepository/resources/Cos2PiOverFermatPrime
    )annotations)Callable)reduce)Expr)S)igcdex)Integersqrt)cacheitc                    ^ U (       d  g[        U 5      S:X  a  SU S   4$ [        U 5      S:X  a  [        U S   U S   5      u  nmnUT4U4$ [        U SS 6 u  p4[        U S   U5      u  nmnU/U4S jU 5       Q7U4$ )a  Compute extended gcd for multiple integers.

Explanation
===========

Given the integers $x_1, \cdots, x_n$ and
an extended gcd for multiple arguments are defined as a solution
$(y_1, \cdots, y_n), g$ for the diophantine equation
$x_1 y_1 + \cdots + x_n y_n = g$ such that
$g = \gcd(x_1, \cdots, x_n)$.

Examples
========

>>> from sympy.functions.elementary._trigonometric_special import migcdex
>>> migcdex()
((), 0)
>>> migcdex(4)
((1,), 4)
>>> migcdex(4, 6)
((-1, 1), 2)
>>> migcdex(6, 10, 15)
((1, 1, -1), 1)
) r      )r   r      Nc              3  .   >#    U  H
  nTU-  v   M     g 7fNr   ).0ivs     i/var/www/auris/envauris/lib/python3.13/site-packages/sympy/functions/elementary/_trigonometric_special.py	<genexpr>migcdex.<locals>.<genexpr>S   s     "1Qs   )lenr   migcdex)xuhygr   s        @r   r   r   .   s    2 
1v{QqTz
1v{1qt$1a1vqyAabE?DAQqT1oGAq!#""#Q&&    c                 |    U (       d  gSS jn[        X5      nU  Vs/ s H  o2U-  PM	     nn[        U6 u  pVU$ s  snf )a*  Compute the partial fraction decomposition.

Explanation
===========

Given a rational number $\frac{1}{q_1 \cdots q_n}$ where all
$q_1, \cdots, q_n$ are pairwise coprime,

A partial fraction decomposition is defined as

.. math::
    \frac{1}{q_1 \cdots q_n} = \frac{p_1}{q_1} + \cdots + \frac{p_n}{q_n}

And it can be derived from solving the following diophantine equation for
the $p_1, \cdots, p_n$

.. math::
    1 = p_1 \prod_{i \ne 1}q_i + \cdots + p_n \prod_{i \ne n}q_i

Where $q_1, \cdots, q_n$ being pairwise coprime implies
$\gcd(\prod_{i \ne 1}q_i, \cdots, \prod_{i \ne n}q_i) = 1$,
which guarantees the existence of the solution.

It is sufficient to compute partial fraction decomposition only
for numerator $1$ because partial fraction decomposition for any
$\frac{n}{q_1 \cdots q_n}$ can be easily computed by multiplying
the result by $n$ afterwards.

Parameters
==========

denoms : int
    The pairwise coprime integer denominators $q_i$ which defines the
    rational number $\frac{1}{q_1 \cdots q_n}$

Returns
=======

tuple[int, ...]
    The list of numerators which semantically corresponds to $p_i$ of the
    partial fraction decomposition
    $\frac{1}{q_1 \cdots q_n} = \frac{p_1}{q_1} + \cdots + \frac{p_n}{q_n}$

Examples
========

>>> from sympy import Rational, Mul
>>> from sympy.functions.elementary._trigonometric_special import ipartfrac

>>> denoms = 2, 3, 5
>>> numers = ipartfrac(2, 3, 5)
>>> numers
(1, 7, -14)

>>> Rational(1, Mul(*denoms))
1/30
>>> out = 0
>>> for n, d in zip(numers, denoms):
...    out += Rational(n, d)
>>> out
1/30
r   c                
    X-  $ r   r   )r   r   s     r   mulipartfrac.<locals>.mul   s	    ur    )r   intr   r%   returnr%   )r   r   )denomsr#   denomr   ar   _s          r   	ipartfracr+   V   sG    ~  3E#$V!VA$A;DAH 	%s   9c                ~    / nS H5  n[        X5      u  p4US:X  d  M  Un UR                  U5        U S:X  d  M3  Us  $    g)zqIf n can be factored in terms of Fermat primes with
multiplicity of each being 1, return those primes, else
None
)           i  r   r   N)divmodappend)nprimespquotient	remainders        r   fermat_coordsr8      sH    
 F#$Ql>AMM!Av $ r    c                 "    [         R                  $ )z-Computes $\cos \frac{\pi}{3}$ in square roots)r   Halfr   r    r   cos_3r;      s     66Mr    c                 $    [        S5      S-   S-  $ )z-Computes $\cos \frac{\pi}{5}$ in square rootsr.   r      r
   r   r    r   cos_5r>      s     GaK1r    c                 V   [        S[        S5      -   S-  [        S5      [        S[        S5      -
  5      [        [        S5      S[        S[        S5      -   5      -  S[        S5      -
  [        S[        S5      -
  5      -  -
  -  S[        S5      -  -   S-   5      -   -  S-  -   5      $ )	z.Computes $\cos \frac{\pi}{17}$ in square roots   r/       r   ir      "   r
   r   r    r   cos_17rD      s     	d2h"tAw$rDH}*=T!WT"tBx-00ARL
rDH}
4  !"T"X.023 	4+4  579 : 	:; ;r    c                    SS jn SS jnU " [         R                  [        S5      5      u  p#U " U[        S5      5      u  pEU " U[        S5      5      u  pgU " USSU-   SU-  -   -  5      u  pU " USSU-   SU-  -   -  5      u  pU " USSU-   SU-  -   -  5      u  pU " USSU-   SU-  -   -  5      u  pU " USX(-   U-   SU
-  -   -  5      u  nnU " USX;-   U-   SU-  -   -  5      u  nnU " USX,-   U	-   SU-  -   -  5      u  nnU " USX?-   U
-   SU-  -   -  5      u  nnU " U	SX)-   U-   SU-  -   -  5      u  nnU " U
SX:-   U-   SU-  -   -  5      u  nnU " USX--   U-   SU-  -   -  5      u  nnU " USX>-   U-   SU	-  -   -  5      u  nnU" USUU-   U-   U-   -  5      n U" USUU-   U-   U-   -  5      n!U" USUU-   U-   U-   -  5      n"U" USUU-   U-   U-   -  5      n#U" USUU-   U-   U-   -  5      n$U" USUU-   U-   U-   -  5      n%U" U * SU!U"-   -  5      * n&U" U#* SU$U%-   -  5      * n'S	U" U&* SU'-  5      -  n([        [        S5      [        U(S-   5      -  S
-  [         R                  -   5      $ )zComputes $\cos \frac{\pi}{257}$ in square roots

References
==========

.. [*] https://math.stackexchange.com/questions/516142/how-does-cos2-pi-257-look-like-in-real-radicals
.. [*] https://r-knott.surrey.ac.uk/Fibonacci/simpleTrig.html
c                ^    U [        U S-  U-   5      -   S-  U [        U S-  U-   5      -
  S-  4$ Nr   r
   r)   bs     r   f1cos_257.<locals>.f1   s9    DAN"a'!d1a4!8n*<)AAAr    c                0    U [        U S-  U-   5      -
  S-  $ rG   r
   rH   s     r   f2cos_257.<locals>.f2   s    DAN"A%%r       @   r=   r.   r      )r)   r   rI   r   r&   ztuple[Expr, Expr])r)   r   rI   r   r&   r   )r   NegativeOner	   r   r:   ))rJ   rM   t1t2z1z3z2z4y1y5y6y2y3y7y8y4x1x9x2x10x3x11x4x12x5x13x6x14x15x7x8x16v1v2v3v4v5v6u1u2w1s)                                            r   cos_257r|      s   B& ws|,FBGBK FBGBK FBAq2v"}%&FBAq2v"}%&FBAq2v"}%&FBAq2v"}%&FBB"qt+,-FBR2",-.GBR2",-.GBR2",-.GBR2",-.GBR2",-.GBR2",-.GCR2",-.GB	BBGbL2%&	'B	BBGbL2%&	'B	BBGcMC'(	)B	BBHsNS()	*B	CS3Y_s*+	,B	CS2X]R'(	)B
bS"b2g,
	B
bS"b2g,
	B	BsBrEN	BQR!V$Q&/00r    c                 0    [         [        [        [        S.$ )aC  Lazily evaluated table for $\cos \frac{\pi}{n}$ in square roots for
$n \in \{3, 5, 17, 257, 65537\}$.

Notes
=====

65537 is the only other known Fermat prime and it is nearly impossible to
build in the current SymPy due to performance issues.

References
==========

https://r-knott.surrey.ac.uk/Fibonacci/simpleTrig.html
)r-   r.   r/   r0   )r;   r>   rD   r|   r   r    r   	cos_tabler~      s      	 r    N)r   r%   r&   ztuple[tuple[int, ...], int])r'   r%   r&   ztuple[int, ...])r3   r%   r&   zlist[int] | None)r&   r   )r&   zdict[int, Callable[[], Expr]])__doc__
__future__r   typingr   	functoolsr   sympy.core.exprr   sympy.core.singletonr   sympy.core.intfuncr   sympy.core.numbersr	   (sympy.functions.elementary.miscellaneousr   sympy.core.cacher   r   r+   r8   r;   r>   rD   r|   r~   r   r    r   <module>r      s   !D #     " % & 9 $%'PHV  	 	
 	 	
 	; 	; 	'1 	'1Tr    