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Scholar
4,946 Views
Registered: ‎01-28-2014

## Reciprocal - Divider Sizing

I need to calculate the reciprocal of a number and the fixed point functions in the HLS math library doesn't have supported for that directly. Instead I have a relatively large divide to calculate my reciprocal. The denominator has 13 integer bits and 16 fractional bits. The first problem is that the fractional precision of the numerator dictates the fractional precision of my output. I need 32 fractional bits so my 1 is represented with 1 integer bit and 32 fractional bits.  At the same time the fractional precision of my divider adds to the numerator's integer bits. As a result HLS ends up generating a relatively larger divider that is consuming a large number of resources. This would be necessary if the denominator could be less than one, but I know from context won't happen. Is there anyway I can get a smaller divider? Perhaps's there's an alternative to calculate the reciprocal that is smaller?

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Highlighted
8,714 Views
Registered: ‎04-26-2015

Straight long division is what I normally use.

Depending on the resource and timing requirements (and bit-width) you might be able to get away with just using a LUT (having only a single input makes reciprocals far more practical than division). For 12-bit input, 12-bit output you'd get away with three 18K block RAMs, (for 4096*12-bit total) which is pretty affordable in most systems - and because they're dual-ported you can potentially do two divides per clock cycle (although persuading HLS to do that requires care). However, RAM requirements go up very quickly as the bit width rises; 16-bit input/output will take 64 18K RAMs.

For the larger widths, there are methods where you use a low-resolution LUT to get an approximation, then refine it with Newton's Method or something similar. Or you can have a shallower, wider LUT that gives you a couple of Taylor series terms for each point to allow accurate interpolation. These approaches make sense where you're dealing with large-ish numbers and you don't have enough clock cycles to do the usual 1-bit-per-cycle process, but they tend to be fairly DSP and RAM-intensive.

Edit: I see that your number won't be suitable for a pure lookup table. Realistically, if you can afford the 32 cycles it'll take to do it with long division, that'll be by far the easiest approach.

5 Replies
Highlighted
4,934 Views
Registered: ‎04-26-2015

I've found that you can often write your own division algorithm in C that works as well as - or better than - the built-in one (generally "better than" occurs when you can make assumptions that the tool cannot). That may well be the best way to do this one; a basic reciprocal function is essentially a division where lots of assumptions are available.

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Scholar
4,910 Views
Registered: ‎01-28-2014

Yea this is the best I can think of as well. I think you can implement binary long division but skip many of the bits since you know the remainder won't be greater than the denominator for some number of iterations.Are you aware of a more efficient algorithm?

Highlighted
8,715 Views
Registered: ‎04-26-2015

Straight long division is what I normally use.

Depending on the resource and timing requirements (and bit-width) you might be able to get away with just using a LUT (having only a single input makes reciprocals far more practical than division). For 12-bit input, 12-bit output you'd get away with three 18K block RAMs, (for 4096*12-bit total) which is pretty affordable in most systems - and because they're dual-ported you can potentially do two divides per clock cycle (although persuading HLS to do that requires care). However, RAM requirements go up very quickly as the bit width rises; 16-bit input/output will take 64 18K RAMs.

For the larger widths, there are methods where you use a low-resolution LUT to get an approximation, then refine it with Newton's Method or something similar. Or you can have a shallower, wider LUT that gives you a couple of Taylor series terms for each point to allow accurate interpolation. These approaches make sense where you're dealing with large-ish numbers and you don't have enough clock cycles to do the usual 1-bit-per-cycle process, but they tend to be fairly DSP and RAM-intensive.

Edit: I see that your number won't be suitable for a pure lookup table. Realistically, if you can afford the 32 cycles it'll take to do it with long division, that'll be by far the easiest approach.

Highlighted
Teacher
4,889 Views
Registered: ‎03-31-2012

you can implement SRT as described here: http://users.minet.uni-jena.de/~nez/rechnerarithmetik_5/fdiv_bug/intel_white11.pdf

Just make sure to include all the lut entries properly ;-)

Also these guys know something about division too: http://i.stanford.edu/pub/cstr/reports/csl/tr/95/675/CSL-TR-95-675.pdf

-K