mirror of
https://github.com/TheAlgorithms/Python.git
synced 2024-11-27 15:01:08 +00:00
9316e7c014
* flake8 --max-line-length=88
* fixup! Format Python code with psf/black push
Co-authored-by: github-actions <${GITHUB_ACTOR}@users.noreply.github.com>
91 lines
3.5 KiB
Python
91 lines
3.5 KiB
Python
def bailey_borwein_plouffe(digit_position: int, precision: int = 1000) -> str:
|
|
"""
|
|
Implement a popular pi-digit-extraction algorithm known as the
|
|
Bailey-Borwein-Plouffe (BBP) formula to calculate the nth hex digit of pi.
|
|
Wikipedia page:
|
|
https://en.wikipedia.org/wiki/Bailey%E2%80%93Borwein%E2%80%93Plouffe_formula
|
|
@param digit_position: a positive integer representing the position of the digit to
|
|
extract.
|
|
The digit immediately after the decimal point is located at position 1.
|
|
@param precision: number of terms in the second summation to calculate.
|
|
A higher number reduces the chance of an error but increases the runtime.
|
|
@return: a hexadecimal digit representing the digit at the nth position
|
|
in pi's decimal expansion.
|
|
|
|
>>> "".join(bailey_borwein_plouffe(i) for i in range(1, 11))
|
|
'243f6a8885'
|
|
>>> bailey_borwein_plouffe(5, 10000)
|
|
'6'
|
|
>>> bailey_borwein_plouffe(-10)
|
|
Traceback (most recent call last):
|
|
...
|
|
ValueError: Digit position must be a positive integer
|
|
>>> bailey_borwein_plouffe(0)
|
|
Traceback (most recent call last):
|
|
...
|
|
ValueError: Digit position must be a positive integer
|
|
>>> bailey_borwein_plouffe(1.7)
|
|
Traceback (most recent call last):
|
|
...
|
|
ValueError: Digit position must be a positive integer
|
|
>>> bailey_borwein_plouffe(2, -10)
|
|
Traceback (most recent call last):
|
|
...
|
|
ValueError: Precision must be a nonnegative integer
|
|
>>> bailey_borwein_plouffe(2, 1.6)
|
|
Traceback (most recent call last):
|
|
...
|
|
ValueError: Precision must be a nonnegative integer
|
|
"""
|
|
if (not isinstance(digit_position, int)) or (digit_position <= 0):
|
|
raise ValueError("Digit position must be a positive integer")
|
|
elif (not isinstance(precision, int)) or (precision < 0):
|
|
raise ValueError("Precision must be a nonnegative integer")
|
|
|
|
# compute an approximation of (16 ** (n - 1)) * pi whose fractional part is mostly
|
|
# accurate
|
|
sum_result = (
|
|
4 * _subsum(digit_position, 1, precision)
|
|
- 2 * _subsum(digit_position, 4, precision)
|
|
- _subsum(digit_position, 5, precision)
|
|
- _subsum(digit_position, 6, precision)
|
|
)
|
|
|
|
# return the first hex digit of the fractional part of the result
|
|
return hex(int((sum_result % 1) * 16))[2:]
|
|
|
|
|
|
def _subsum(
|
|
digit_pos_to_extract: int, denominator_addend: int, precision: int
|
|
) -> float:
|
|
# only care about first digit of fractional part; don't need decimal
|
|
"""
|
|
Private helper function to implement the summation
|
|
functionality.
|
|
@param digit_pos_to_extract: digit position to extract
|
|
@param denominator_addend: added to denominator of fractions in the formula
|
|
@param precision: same as precision in main function
|
|
@return: floating-point number whose integer part is not important
|
|
"""
|
|
sum = 0.0
|
|
for sum_index in range(digit_pos_to_extract + precision):
|
|
denominator = 8 * sum_index + denominator_addend
|
|
exponential_term = 0.0
|
|
if sum_index < digit_pos_to_extract:
|
|
# if the exponential term is an integer and we mod it by the denominator
|
|
# before dividing, only the integer part of the sum will change;
|
|
# the fractional part will not
|
|
exponential_term = pow(
|
|
16, digit_pos_to_extract - 1 - sum_index, denominator
|
|
)
|
|
else:
|
|
exponential_term = pow(16, digit_pos_to_extract - 1 - sum_index)
|
|
sum += exponential_term / denominator
|
|
return sum
|
|
|
|
|
|
if __name__ == "__main__":
|
|
import doctest
|
|
|
|
doctest.testmod()
|