Date: Thu, 9 Mar 2017 12:14:09 -0800 From: Steve Kargl <sgk@troutmask.apl.washington.edu> To: Bruce Evans <brde@optusnet.com.au> Cc: freebsd-numerics@freebsd.org Subject: Re: Bit twiddling question Message-ID: <20170309201409.GA37219@troutmask.apl.washington.edu> In-Reply-To: <20170309195134.GA36213@troutmask.apl.washington.edu> References: <20170308202417.GA23103@troutmask.apl.washington.edu> <20170309173152.F1269@besplex.bde.org> <20170309075236.GA30199@troutmask.apl.washington.edu> <20170309152307.GA32781@troutmask.apl.washington.edu> <20170310025417.U3723@besplex.bde.org> <20170309195134.GA36213@troutmask.apl.washington.edu>
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On Thu, Mar 09, 2017 at 11:51:34AM -0800, Steve Kargl wrote: > On Fri, Mar 10, 2017 at 05:02:13AM +1100, Bruce Evans wrote: > > > > > > Something similar can be applied to ld128, but reduction may take > > > two rounds (ie., a comparison with UINT64_MAX and then UINT32_MAX). > > > > I don't like that much. The 0x1p52 magic is tricky and probably buggy. > > s_rint.c uses the same magic but needs many complications to avoid > > double rounding. > > Yep. Seems to have some issue. From my working copy of s_sinpif(x), > I tried > > #if 0 > ix = (uint32_t)ax; > a = ax == ix ? zero : __compute_sinpif(ax - ix); > if (ix & 1) ax = -ax; > return ((hx & 0x80000000) ? -ax : ax); > #else > volatile float vx; > float y; > vx = ax + 0x1p23F; > y = vx - 0x1p23F; > ix = (uint32_t)y; > ax = ax == y ? zero : __compute_sinpif(ax - ix); > if (ix & 1) ax = -ax; > return ((hx & 0x80000000) ? -ax : ax); > #endif > > My max ULP went from 0.5 to 0.50398505 for exhaustive testing > in the interval [1,100]. If I grab the worse case x and use the > '#if 1' code, I see > > ./testf -S -a 1.50121641e+00f > a = 1.50121641e+00f, /* 0x3fc027dc */ > sinpif(a) = -9.99992669e-01f, /* 0xbf7fff85 */ > sin(pi*a) = -9.99992669e-01f, /* 0xbf7fff85 */ MPFR 96-bits rounded 24-bits. > ULP: 0.49601495 > > The '#if 0' code gives > > ./testf -S -a 1.50121641e+00f > a = 1.50121641e+00f, /* 0x3fc027dc */ > sinpif(a) = -9.99992728e-01f, /* 0xbf7fff86 */ > sin(pi*a) = -9.99992669e-01f, /* 0xbf7fff85 */ MPFR 96-bits rounded 24-bits. > ULP: 0.50398505 > Certainly looks like a double rounding issue. An instrumented copy of the code shows that y = 2 instead of 1 in the above. a = 1.50121641e+00f, /* 0x3fc027dc */ 1.50121641e+00 2.00000000e+00 sinpif(a) = -9.99992728e-01f, /* 0xbf7fff86 */ sin(pi*a) = -9.99992669e-01f, /* 0xbf7fff85 */ ULP: 0.50398505 -- Steve 20161221 https://www.youtube.com/watch?v=IbCHE-hONow
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