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```Date: Mon, 24 Apr 2017 00:34:48 +0200
From: Szabolcs Nagy <nsz@...t70.net>
To: musl@...ts.openwall.com
Subject: Re: [PATCH] math: rewrite fma with mostly int arithmetics

* 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.

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
```

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