#### Bailey Borwein Plouffe

```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.
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 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
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()
```  