I need to perform a multiplication operation on a fixed-point variable x
(unsigned 16-bit integer [U16] type with binary point 6 [BP6]) with a coefficient A
, which I know will always be between 0 and 1. Code is being written in C for a 32-bit embedded platform.
I know that if I were to also make this coefficient a U16 BP6, then I would end up with a U32 BP12 from the multiplication. I want to rescale this result back down to U16 BP6, so I just lop off the first 10 bits and the last 6.
However, since the coefficient is limited in precision by the number of fractional bits, and I do not necessarily need the full 10 bits of integer, I was thinking that I could just make the coefficient variable A
a U16 BP15 to yield a more precise result.
I have worked out the following example (bear with me):
Let's say that x = 172.0
(decimal) and I want to use a coefficient A = 0.82
(decimal). The ideal decimal result would be 172.0 * 0.82 = 141.04.
In binary, x = 0010101100.000000
.
If I am using BP6 for A, the binary representation will be either
A_1 = 0000000000.110100 = 0.8125 or
A_2 = 0000000000.110101 = 0.828125
(depending on whether value is based on floor or ceiling).
Performing the binary multiplication between x and either value of A yields (leaving out leading zeroes):
A_1 * x = 10001011.110000000000 = 139.75
A_2 * x = 10001110.011100000000 = 142.4375
In both cases, triming down the last 6 bits would not affect the result.
Now, if I expanded A to have BP15, then
A_3 = 0.110100011110110 = 0.82000732421875
and the resulting multiplication yields
A_3 * x = 10001101.000010101001000000000 = 141.041259765625
When trimming the extra 15 fractional bits, the result is
A_3 * x = 10001101.000010 = 141.03125
So it's pretty clear here that by expanding the coefficient to have more fractional bits yields a more precise result (at least in my example). Is this something which will hold true in general? Is this good/bad to use in practice? Am I missing or misunderstanding something?
EDIT: I should have said "accuracy" in place of "precision" here. I am looking for a result which is closer to my expected value rather than a result which contains more fractional bits.