A Pythonic way to take a n-th root without using math

log.py

#####################################################################################################
#  Copyright (c) 2023-2024 NWSOFT                                                                   #
#                                                                                                   #
#  Permission is hereby granted, free of charge, to any person obtaining a copy                     #
#  of this software and associated documentation files (the "Software"), to deal                    #
#  in the Software without restriction, including without limitation the rights                     #
#  to use, copy, modify, merge, publish, distribute, sublicense, and/or sell                        #
#  copies of the Software, and to permit persons to whom the Software is                            #
#  furnished to do so, subject to the following conditions:                                         #
#                                                                                                   #
#  The above copyright notice and this permission notice shall be included in all                   #
#  copies or substantial portions of the Software.                                                  #
#                                                                                                   #
#  THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR                       #
#  IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,                         #
#  FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE                      #
#  AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER                           #
#  LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,                    #
#  OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE                    #
#  SOFTWARE.                                                                                        #
#####################################################################################################


def log(number: float, base: int, decimals: int = 10) -> float:
    """
    Takes a logarithm.
    :param number: The number to take logarithm of.
    :param base: The base of the logarithm.
    :param decimals: The precision of the calculation.
    :returns: The result of n-th root.
    """
    raised_times = 0
    while number < 1:
        number *= base
        raised_times += 1

    x = number
    y = 0
    idx = 0

    while True:
        radicand = y + (x - y) / 2
        test = base**radicand
        if abs(test - number) < (1 / 10 ** (decimals + 1)):
            return radicand - raised_times
        if test > number:
            x = radicand
        elif test < number:
            y = radicand

        idx += 1
        print(f"{idx:10d} {x} {y} {abs(test - number)} {radicand:3.8f}")


if __name__ == "__main__":
    print(log(0.5, 5))

root.py

#####################################################################################################
#  Copyright (c) 2023-2024 NWSOFT                                                                   #
#                                                                                                   #
#  Permission is hereby granted, free of charge, to any person obtaining a copy                     #
#  of this software and associated documentation files (the "Software"), to deal                    #
#  in the Software without restriction, including without limitation the rights                     #
#  to use, copy, modify, merge, publish, distribute, sublicense, and/or sell                        #
#  copies of the Software, and to permit persons to whom the Software is                            #
#  furnished to do so, subject to the following conditions:                                         #
#                                                                                                   #
#  The above copyright notice and this permission notice shall be included in all                   #
#  copies or substantial portions of the Software.                                                  #
#                                                                                                   #
#  THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR                       #
#  IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,                         #
#  FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE                      #
#  AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER                           #
#  LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,                    #
#  OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE                    #
#  SOFTWARE.                                                                                        #
#####################################################################################################


def nthroot(number: float, base: int, decimals: int = 10) -> float:
    """
    Takes a n-th root.
    :param number: The number to take sqare root of.
    :param base: The base of the root.
    :param decimals: The precision of the calculation.
    :returns: The result of n-th root.
    """
    raised_times = 0
    while number < 1:
        number *= 10**base
        raised_times += 1

    x = number
    y = 0
    idx = 0

    while True:
        radicand = y + (x - y) / 2
        test = radicand**base
        if abs(test - number) < (1 / 10 ** (decimals + 1)):
            return radicand / (10**raised_times)
        if test > number:
            x = radicand
        elif test < number:
            y = radicand

        idx += 1
        print(f"{idx:10d} {x} {y} {abs(test - number)} {radicand:3.8f}")


if __name__ == "__main__":
    print(nthroot(0.5, 5))

Should support all types of numbers, and any base.

---

To-do list:

• Add tests
• Check for illegal inputs (0-base...)
• Faster return by logical guessing
• Support for numbers less than one