Date: Sun, 23 Apr 2017 11:15:39 -0400 From: Rich Felker <dalias@...c.org> To: musl@...ts.openwall.com Subject: Re: [PATCH] math: rewrite fma with mostly int arithmetics 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]: > > A few thoughts, inline below. I'm not entirely opposed to this, if it > > turns out to be better than the alternatives, but I would like to > > understand whether it really is... > > > > On Wed, Apr 19, 2017 at 12:41:40AM +0200, Szabolcs Nagy wrote: > > > the freebsd fma code failed to raise underflow exception in some > > > cases in nearest rounding mode (affects fmal too) e.g. > > > > > > fma(-0x1p-1000, 0x1.000001p-74, 0x1p-1022) > > > > > > and the inexact exception may be raised spuriously since the fenv > > > is not saved/restored around the exact multiplication algorithm > > > (affects x86 fma too). > > > > 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. > > > depends on the a_clz_64 patch and previous scalbn fix. > > > > > > fmal should be possible to do in a similar way. > > > > > > i expect it to be faster than the previous code on most targets as > > > the rounding mode is not changed and has less multiplications > > > (it is faster on x86_64 and i386), the code size is a bit bigger > > > though. > > > > Kinda surprising on i386 -- I'd expect the 64x64 multiplications to be > > costly compared to float ones. > > > > i implement 64x64 int mul by four 32x32->64 mul, > i386 has 32x32->64 mul op so that works out well. Most archs should have a 32x32->64; if not this is an ISA quality issue and not one I really want to focus on remedying. > the float code has to do dekker's exact multiplication > which uses six(!) fp mul (each of which is probably > slower than an int mul) and a lot more fp add. OK, this makes your approach make a lot more sense! Thanks for sharing that info. > > > /* normalize so top 10bits and last bit are 0 */ > > > struct num nx, ny, nz; > > > nx = normalize(ix); > > > ny = normalize(iy); > > > nz = normalize(iz); > > > > 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(). Rich
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