bignum_root.erl

%% Power function (X ^ Y) and root function (X ^ (1/Y)) for
%% integers in Erlang
%% by Kenji Rikitake <kenji.rikitake@acm.org> 26-JAN-2010
%% modified by Hynek Vychodil <vychodil.hynek@gmail.com> 2-FEB-2010
%% Distributed under MIT license at the end of the source code.

-module(bignum_root).

-export([power/2, root/2, sqrt/1]).

%% computing X ^ Y

power(X, Y) when is_integer(X), is_integer(Y), Y >= 0 ->
    power(X, Y, 1).

power(_, 0, Acc) ->
    Acc;
power(X, Y, Acc) when Y rem 2 =:= 1 ->
    power(X, Y - 1, Acc * X);
power(X, Y, Acc) ->
    power(X * X, Y div 2, Acc).

sqrt(X) -> root(X, 2).

%% computing X ^ (1/Y)
%% with Newton-Raphson method

root(X, 1) when is_integer(X) -> X;
root(X, 2) when is_integer(X), X > 0 -> sqrt(X, trunc(math:sqrt(X)) + 1);
root(X, Y) when is_integer(X), X > 0, is_integer(Y), Y >= 2 ->
    % estimation of the root by the float calc
	E = trunc(math:exp(math:log(X) / Y)),
    root(X, Y, E + 1).

root(X, Y, E) ->
    % Newton-Raphson method of solving (Y)th-root
	EP = power(E, Y - 1),
	Err = E * EP - X,
    E2 = E - Err div (Y * EP),
    if
		E2 =/= E -> root(X, Y, E2);
		Err > 0 -> E - 1;	% Solve rounding error
		true -> E
    end.

sqrt(X, E) ->
    % Newton-Raphson method of solving square root
	Err = E * E - X,
    E2 = E - Err div (2 * E),
    if
		E2 =/= E -> sqrt(X, E2);
		Err > 0 -> E - 1;	% Solve rounding error
		true -> E
    end.

%%  MIT License:
%%  Copyright (c) 2010 by Kenji Rikitake.
%%  Copyright (c) 2010 by Hynek Vychodil.
%%
%%  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.

https://gist.github.com/jj1bdx/293169 has improved root estimation for numbers bigger than 10^300.