mirror of
https://github.com/rn10950/RetroZilla.git
synced 2024-11-14 03:30:17 +01:00
73 lines
2.3 KiB
Plaintext
73 lines
2.3 KiB
Plaintext
Squaring Algorithm
|
|
|
|
When you are squaring a value, you can take advantage of the fact that
|
|
half the multiplications performed by the more general multiplication
|
|
algorithm (see 'mul.txt' for a description) are redundant when the
|
|
multiplicand equals the multiplier.
|
|
|
|
In particular, the modified algorithm is:
|
|
|
|
k = 0
|
|
for j <- 0 to (#a - 1)
|
|
w = c[2*j] + (a[j] ^ 2);
|
|
k = w div R
|
|
|
|
for i <- j+1 to (#a - 1)
|
|
w = (2 * a[j] * a[i]) + k + c[i+j]
|
|
c[i+j] = w mod R
|
|
k = w div R
|
|
endfor
|
|
c[i+j] = k;
|
|
k = 0;
|
|
endfor
|
|
|
|
On the surface, this looks identical to the multiplication algorithm;
|
|
however, note the following differences:
|
|
|
|
- precomputation of the leading term in the outer loop
|
|
|
|
- i runs from j+1 instead of from zero
|
|
|
|
- doubling of a[i] * a[j] in the inner product
|
|
|
|
Unfortunately, the construction of the inner product is such that we
|
|
need more than two digits to represent the inner product, in some
|
|
cases. In a C implementation, this means that some gymnastics must be
|
|
performed in order to handle overflow, for which C has no direct
|
|
abstraction. We do this by observing the following:
|
|
|
|
If we have multiplied a[i] and a[j], and the product is more than half
|
|
the maximum value expressible in two digits, then doubling this result
|
|
will overflow into a third digit. If this occurs, we take note of the
|
|
overflow, and double it anyway -- C integer arithmetic ignores
|
|
overflow, so the two digits we get back should still be valid, modulo
|
|
the overflow.
|
|
|
|
Having doubled this value, we now have to add in the remainders and
|
|
the digits already computed by earlier steps. If we did not overflow
|
|
in the previous step, we might still cause an overflow here. That
|
|
will happen whenever the maximum value expressible in two digits, less
|
|
the amount we have to add, is greater than the result of the previous
|
|
step. Thus, the overflow computation is:
|
|
|
|
|
|
u = 0
|
|
w = a[i] * a[j]
|
|
|
|
if(w > (R - 1)/ 2)
|
|
u = 1;
|
|
|
|
w = w * 2
|
|
v = c[i + j] + k
|
|
|
|
if(u == 0 && (R - 1 - v) < w)
|
|
u = 1
|
|
|
|
If there is an overflow, u will be 1, otherwise u will be 0. The rest
|
|
of the parameters are the same as they are in the above description.
|
|
|
|
------------------------------------------------------------------
|
|
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/.
|