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Date: Mon, 24 Apr 2017 00:35:33 +0200
From: Szabolcs Nagy <nsz@...t70.net>
To: musl@...ts.openwall.com
Subject: Re: [PATCH] math: rewrite fma with mostly int arithmetics
* Szabolcs Nagy <nsz@...t70.net> [2017-04-24 00:34:48 +0200]:
> * Rich Felker <dalias@...c.org> [2017-04-23 11:15:39 -0400]:
> > On Sun, Apr 23, 2017 at 01:00:52PM +0200, Szabolcs Nagy wrote:
> > > * Rich Felker <dalias@...c.org> [2017-04-22 18:24:25 -0400]:
> > > > Is it difficult to determine when the multiplication part of an fma is
> > > > exact? If you can determine this quickly, you can just return x*y+z in
> > > > this special case and avoid all the costly operations. For normal
> > > > range, I think it's roughly just using ctz to count mantissa bits of x
> > > > and y, and checking whether the sum is <= 53. Some additional handling
> > > > for denormals is needed of course.
> > >
> > > it is a bit more difficult than that:
> > >
> > > bits(a) + bits(b) < 54 || (bits(a) + bits(b) == 54 && a*b < 2)
> > >
> > > this is probably possible to handle when i do the int mul.
> > >
> > > however the rounding mode special cases don't get simpler
> > > and inexact flag still may be raised incorrectly when tail
> > > bits of x*y beyond 53 bits are eliminated when z is added
> > > (the result is exact but the dekker algorithm raises inexact).
> >
> > One thing to note: even if it's not a replacement for the whole
> > algorithm, this seems like a very useful optimization for a case
> > that's easy to test. "return x*y+z;" is going to be a lot faster than
> > anything else you can do. But maybe it's rare to hit cases where the
> > optimization works; it certainly "should" be rare if people are using
> > fma for the semantics rather than as a misguided optimization.
>
> i didn't see a simple way to check for exact x*y result
> (if it were easy then that could capture the exact 0 result
> case which means one less special case later, but this is
> not easy if x*y is in the subnormal range or overflows)
>
> > > > If the only constraint here is that top 10 bits and last bit are 0, I
> > > > don't see why clz is even needed. You can meet this constraint for
> > > > denormals by always multiplying by 2 and using a fixed exponent value.
> > >
> > > yeah that should work, but i also use clz later
> >
> > Ah, I missed that. Still it might be a worthwhile optimization here; I
> > think it shaves off a few ops in normalize().
>
> attached a new version with updated normalize.
>
now really
> on my laptop latency and code size:
>
> old x86_64: 67 ns/call 893 bytes
> new x86_64: 20 ns/call 960 bytes
> old i386: 80 ns/call 942 bytes
> new i386: 75 ns/call 1871 bytes
> old arm: - 960 bytes
> new arm: - 1200 bytes
View attachment "fma.c" of type "text/x-csrc" (3625 bytes)
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