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160 lines
5.1 KiB
160 lines
5.1 KiB
'''
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This implementation is a heavily modified fixed point implementation of
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BBP_formula for calculating the nth position of pi. The original hosted
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at: https://web.archive.org/web/20151116045029/http://en.literateprograms.org/Pi_with_the_BBP_formula_(Python)
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# Permission is hereby granted, free of charge, to any person obtaining
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# a copy of this software and associated documentation files (the
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# "Software"), to deal in the Software without restriction, including
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# without limitation the rights to use, copy, modify, merge, publish,
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# distribute, sub-license, and/or sell copies of the Software, and to
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# permit persons to whom the Software is furnished to do so, subject to
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# the following conditions:
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#
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# The above copyright notice and this permission notice shall be
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# included in all copies or substantial portions of the Software.
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#
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# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
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# EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
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# MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.
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# IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY
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# CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT,
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# TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE
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# SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
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Modifications:
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1.Once the nth digit and desired number of digits is selected, the
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number of digits of working precision is calculated to ensure that
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the hexadecimal digits returned are accurate. This is calculated as
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int(math.log(start + prec)/math.log(16) + prec + 3)
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--------------------------------------- --------
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/ /
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number of hex digits additional digits
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This was checked by the following code which completed without
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errors (and dig are the digits included in the test_bbp.py file):
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for i in range(0,1000):
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for j in range(1,1000):
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a, b = pi_hex_digits(i, j), dig[i:i+j]
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if a != b:
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print('%s\n%s'%(a,b))
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Deceasing the additional digits by 1 generated errors, so '3' is
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the smallest additional precision needed to calculate the above
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loop without errors. The following trailing 10 digits were also
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checked to be accurate (and the times were slightly faster with
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some of the constant modifications that were made):
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>> from time import time
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>> t=time();pi_hex_digits(10**2-10 + 1, 10), time()-t
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('e90c6cc0ac', 0.0)
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>> t=time();pi_hex_digits(10**4-10 + 1, 10), time()-t
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('26aab49ec6', 0.17100000381469727)
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>> t=time();pi_hex_digits(10**5-10 + 1, 10), time()-t
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('a22673c1a5', 4.7109999656677246)
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>> t=time();pi_hex_digits(10**6-10 + 1, 10), time()-t
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('9ffd342362', 59.985999822616577)
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>> t=time();pi_hex_digits(10**7-10 + 1, 10), time()-t
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('c1a42e06a1', 689.51800012588501)
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2. The while loop to evaluate whether the series has converged quits
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when the addition amount `dt` has dropped to zero.
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3. the formatting string to convert the decimal to hexadecimal is
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calculated for the given precision.
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4. pi_hex_digits(n) changed to have coefficient to the formula in an
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array (perhaps just a matter of preference).
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'''
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import math
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from sympy.utilities.misc import as_int
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def _series(j, n, prec=14):
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# Left sum from the bbp algorithm
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s = 0
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D = _dn(n, prec)
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D4 = 4 * D
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k = 0
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d = 8 * k + j
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for k in range(n + 1):
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s += (pow(16, n - k, d) << D4) // d
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d += 8
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# Right sum iterates to infinity for full precision, but we
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# stop at the point where one iteration is beyond the precision
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# specified.
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t = 0
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k = n + 1
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e = 4*(D + n - k)
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d = 8 * k + j
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while True:
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dt = (1 << e) // d
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if not dt:
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break
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t += dt
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# k += 1
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e -= 4
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d += 8
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total = s + t
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return total
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def pi_hex_digits(n, prec=14):
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"""Returns a string containing ``prec`` (default 14) digits
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starting at the nth digit of pi in hex. Counting of digits
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starts at 0 and the decimal is not counted, so for n = 0 the
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returned value starts with 3; n = 1 corresponds to the first
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digit past the decimal point (which in hex is 2).
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Examples
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========
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>>> from sympy.ntheory.bbp_pi import pi_hex_digits
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>>> pi_hex_digits(0)
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'3243f6a8885a30'
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>>> pi_hex_digits(0, 3)
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'324'
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References
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==========
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.. [1] http://www.numberworld.org/digits/Pi/
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"""
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n, prec = as_int(n), as_int(prec)
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if n < 0:
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raise ValueError('n cannot be negative')
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if prec == 0:
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return ''
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# main of implementation arrays holding formulae coefficients
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n -= 1
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a = [4, 2, 1, 1]
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j = [1, 4, 5, 6]
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#formulae
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D = _dn(n, prec)
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x = + (a[0]*_series(j[0], n, prec)
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- a[1]*_series(j[1], n, prec)
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- a[2]*_series(j[2], n, prec)
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- a[3]*_series(j[3], n, prec)) & (16**D - 1)
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s = ("%0" + "%ix" % prec) % (x // 16**(D - prec))
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return s
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def _dn(n, prec):
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# controller for n dependence on precision
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# n = starting digit index
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# prec = the number of total digits to compute
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n += 1 # because we subtract 1 for _series
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return int(math.log(n + prec)/math.log(16) + prec + 3)
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