mirror of
https://github.com/rn10950/RetroZilla.git
synced 2024-11-14 03:30:17 +01:00
51 lines
1.7 KiB
Plaintext
51 lines
1.7 KiB
Plaintext
Square Root
|
|
|
|
A simple iterative algorithm is used to compute the greatest integer
|
|
less than or equal to the square root. Essentially, this is Newton's
|
|
linear approximation, computed by finding successive values of the
|
|
equation:
|
|
|
|
x[k]^2 - V
|
|
x[k+1] = x[k] - ------------
|
|
2 x[k]
|
|
|
|
...where V is the value for which the square root is being sought. In
|
|
essence, what is happening here is that we guess a value for the
|
|
square root, then figure out how far off we were by squaring our guess
|
|
and subtracting the target. Using this value, we compute a linear
|
|
approximation for the error, and adjust the "guess". We keep doing
|
|
this until the precision gets low enough that the above equation
|
|
yields a quotient of zero. At this point, our last guess is one
|
|
greater than the square root we're seeking.
|
|
|
|
The initial guess is computed by dividing V by 4, which is a heuristic
|
|
I have found to be fairly good on average. This also has the
|
|
advantage of being very easy to compute efficiently, even for large
|
|
values.
|
|
|
|
So, the resulting algorithm works as follows:
|
|
|
|
x = V / 4 /* compute initial guess */
|
|
|
|
loop
|
|
t = (x * x) - V /* Compute absolute error */
|
|
u = 2 * x /* Adjust by tangent slope */
|
|
t = t / u
|
|
|
|
/* Loop is done if error is zero */
|
|
if(t == 0)
|
|
break
|
|
|
|
/* Adjust guess by error term */
|
|
x = x - t
|
|
end
|
|
|
|
x = x - 1
|
|
|
|
The result of the computation is the value of x.
|
|
|
|
------------------------------------------------------------------
|
|
This Source Code Form is subject to the terms of the Mozilla Public
|
|
# License, v. 2.0. If a copy of the MPL was not distributed with this
|
|
# file, You can obtain one at http://mozilla.org/MPL/2.0/.
|