Date: Tue, 14 Aug 2012 07:14:28 +1000 (EST) From: Bruce Evans <brde@optusnet.com.au> To: Stephen Montgomery-Smith <stephen@missouri.edu> Cc: freebsd-numerics@FreeBSD.org, Bruce Evans <brde@optusnet.com.au> Subject: Re: Complex arg-trig functions Message-ID: <20120814055931.Q4897@besplex.bde.org> In-Reply-To: <50295887.2010608@missouri.edu> References: <5017111E.6060003@missouri.edu> <501C361D.4010807@missouri.edu> <20120804165555.X1231@besplex.bde.org> <501D51D7.1020101@missouri.edu> <20120805030609.R3101@besplex.bde.org> <501D9C36.2040207@missouri.edu> <20120805175106.X3574@besplex.bde.org> <501EC015.3000808@missouri.edu> <20120805191954.GA50379@troutmask.apl.washington.edu> <20120807205725.GA10572@server.rulingia.com> <20120809025220.N4114@besplex.bde.org> <5027F07E.9060409@missouri.edu> <20120814003614.H3692@besplex.bde.org> <50295887.2010608@missouri.edu>
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On Mon, 13 Aug 2012, Stephen Montgomery-Smith wrote:
> On 08/13/2012 11:57 AM, Bruce Evans wrote:
>
>> I finally tested a version of this. I only did simple comparisons (float
>> vs double and double vs long double). The results look promising after
>> fixing a few bugs:
>
> Thank you very much for doing the testing, and for fixing the bugs.
>
>>
>> % amd64 float prec, on 2**12 * 2**12 args:
>
>> % icacosh:max_er = 0x3690000000 436.5000, avg_er = 0.317, #>=1:0.5 =
>> 29104:255732
>
>> There are negative reasons to have the float versions unless they are not
>> wrappers. The reasons to have non-wrappers are to test the algorithm and
>> run faster.
>
> That large max-err for the imaginary part of icacosh for float bothers me.
> It means that I haven't thought it through properly. Could you send me the
> input values that created this error(s)?
It was just a bug in my test program, for scaling denormals. Some cases
have an infinite-precision result that is very close to the smallest
denormal. When this is rounded imperfectly to 0, implementation details
give a special case. I had just fixed this for +0, but forget to mask
out the sign bit for testing -0. Somehow this didn't turn up for other
functions.
> The float versions really are much harder than the double and long-double
> versions. And it just doesn't seem worth the effort, because who uses them
> when the double versions are available?
I doubt that they are really harder.
> In my case, not for speed. Because on my machine the float versions are
> slightly slower than the double version.
The double constants might do that on amd64. 'r *= 0.5F;' would compile
to a single multiplication, where r and 0.5F are already in registers.
'r *= 0.5;' would compile to: convert r to double; multiply; convert
back to float. Conversions are slower than multiplications. On Phenom,
double <-> float conversion has a latency of 7 (?) cycles, while
multiplication has a latency of 4 cycles; both have a throughput of
1 instruction/cycle. Sometimes latencies can be hidden, but here there
are 3 dependent operations.
> Also, you made the comment that in the float version, all the 0.5 should
> become 0.5F. Two questions:
> 1. Doesn't the compiler do this conversion for me?
> 2. What is wrong with using x/2 instead of 0.5*x? You told me in a far
> earlier email to use 0.5*x. (Similarly in one place I have a 0.25.)
1. It can't, since x * 0.5F is quite different from x * 0.5 when x is
float and float expressions are evaluated in float precision. The
former is a float expression, and for example overflows when x =
FLT_MAX. The latter is a double expression, so it doesn't overflow
when x = FLT_MAX. Maybe the compiler could optimize it to a float
expression if its result is immediately assigned to a float variable
or cast to float, but this is not a very easy optimization -- the
compiler would have to prove that it has the right side effects for
all possible values of x. The side effect of overflow happens on
assignment.
2. Division tends to be slow, and is slow on x86. On Phenom, division
has a latency of 20 cycles and a throughput of 1 per 15 cycles in
scalar double precision (17 in vector[2] double precision). Maybe
the compiler can optimize division by a power of 2 to a multiplication
though.
In fact, gcc on x86 does both of these optimizations. With float x,
and double y, in a function returning float, 'return x/2.0;' gets
turned into float multiplication by 0.5F, but 'y = x/2.0; return y;'
only have the division optimization (it converts x to double, multiplies
doubles, stores to y, and converts to float).
So all of those '* 0.5[F]'s can be turned back into '/ 2's (no need to
depend on the optimization by writing 2.0).
Optimization of [long] double constants to float (or double) constants
is more routine. When you write 'static const long double one = 1;',
gcc normally puts a long double at the address of `one' (this is visible
to debuggers), but never actually uses it it uses either fld1 on i387
or an unnamed float constant with the same value. At least on i387;
loading a float constant automatically extends it and is much faster
than loading an (already-extended) long double constant; however, on
amd64 a float constant would have to be converted to double for use
in double expressions.
I added old complex functions to the simple test run. See the attachment
for the full list.
% amd64 float prec, on 2**12 x 2**12 args:
% % ...
% rctan: max_er = 0x9708bb4b 4.7198, avg_er = 0.104, #>=1:0.5 = 177976:980356
% rctanh:max_er = 0x9c7f786c 4.8906, avg_er = 0.159, #>=1:0.5 = 635992:2189748
% ...
% ictan: max_er = 0x9c7f786c 4.8906, avg_er = 0.159, #>=1:0.5 = 635992:2189748
% ictanh:max_er = 0x9708bb4b 4.7198, avg_er = 0.104, #>=1:0.5 = 177976:980356
All (amd64 float prec) below 4 ulps except these. Have to check my
denormal scaling again.
% amd64 double prec, on 2**12 x 2**12 args:
% ...
% rcatan:max_er = 0x8000000000000000 4503599627370496.0000, avg_er = 268435456.212, #>=1:0.5 = 4681:81365
% rcatanh:max_er = 0x8000000000000000 4503599627370496.0000, avg_er = 268435456.047, #>=1:0.5 = 428997:691341
% ...
% icatan:max_er = 0x8000000000000000 4503599627370496.0000, avg_er = 268435456.047, #>=1:0.5 = 428997:691341
% icatanh:max_er = 0x8000000000000000 4503599627370496.0000, avg_er = 268435456.212, #>=1:0.5 = 4681:81365
% ...
% icsqrt:max_er = 0x8000000000000000 4503599627370496.0000, avg_er = 268435456.476, #>=1:0.5 = 14279:7662849
This was tracked to a known annoying difference between SSE and i387
on NaNs. x+y clears the sign buit on i387 but not on SSE. SSE is
correct.
% i386 float prec, on 2**12 x 2**12 args:
% ...
% rcatan:max_er = 0x3c4078ec 1.8829, avg_er = 0.284, #>=1:0.5 = 13260:190836
% rcatanh:max_er = 0x2cef3171 1.4042, avg_er = 0.167, #>=1:0.5 = 4096:420916
i386 is generally more accurate. *atan* now below 4 ulps.
% rccos: max_er = 0xb7f10439f3111686 24688206287.5958, avg_er = 1415.000, #>=1:0.5 = 3881364:4046644
% rccosh:max_er = 0xb7f10439f3111686 24688206287.5958, avg_er = 1415.000, #>=1:0.5 = 3881364:4046644
% rcexp: max_er = 0xb7e10439f3111686 24679817679.5958, avg_er = 312.908, #>=1:0.5 = 3884684:4078258
% rcsin: max_er = 0xb80bf11b402ee934 24702322906.0057, avg_er = 807.518, #>=1:0.5 = 3841652:4115524
% rcsinh:max_er = 0xb7f10439f3111686 24688206287.5958, avg_er = 1693.508, #>=1:0.5 = 3880876:4368000
% rctan: max_er = 0xb68d1b1c5151fcc0 24501606626.5413, avg_er = 1627.536, #>=1:0.5 = 3890932:4214696
% rctanh:max_er = 0x125ead9c2ea875d 154097358.0911, avg_er = 1416.635, #>=1:0.5 = 2897540:3523200
% ...
% iccos: max_er = 0xb80bf11b402ee934 24702322906.0057, avg_er = 1477.124, #>=1:0.5 = 3824832:4420052
% iccosh:max_er = 0xb80bf11b402ee934 24702322906.0057, avg_er = 1477.124, #>=1:0.5 = 3824832:4420052
% icexp: max_er = 0xb7fbf11b402ee934 24693934298.0057, avg_er = 1736.298, #>=1:0.5 = 3839256:4219864
% icsin: max_er = 0xb7f10439f3111686 24688206287.5958, avg_er = 1693.508, #>=1:0.5 = 3880876:4368000
% icsinh:max_er = 0xb80bf11b402ee934 24702322906.0057, avg_er = 807.518, #>=1:0.5 = 3841652:4115524
% ictan: max_er = 0x125ead9c2ea875d 154097358.0911, avg_er = 1416.635, #>=1:0.5 = 2897540:3523200
% ictanh:max_er = 0xb68d1b1c5151fcc0 24501606626.5413, avg_er = 1632.536, #>=1:0.5 = 3890988:4214752
i387 hardware trig functions are very inaccurate.
% sparc64 double prec, on 2**12 x 2**12 args:
% rcacos:max_er = 0x36b5 3.4192, avg_er = 0.228, #>=1:0.5 = 2394:125984
% rcacosh:max_er = 0x1f15 1.9426, avg_er = 0.258, #>=1:0.5 = 8464:2737832
% rcasin:max_er = 0x2b8c 2.7217, avg_er = 0.113, #>=1:0.5 = 33296:99148
% rcasinh:max_er = 0x1f15 1.9426, avg_er = 0.258, #>=1:0.5 = 8464:2737800
% rcatan:max_er = 0x2c46 2.7671, avg_er = 0.212, #>=1:0.5 = 4680:81364
% rcatanh:max_er = 0x2970 2.5898, avg_er = 0.047, #>=1:0.5 = 129452:691356
% rclog: max_er = 0xe09 0.8772, avg_er = 0.250, #>=1:0.5 = 0:18348
Everything seemed to be passing on sparc64, but it is 300 to 1000 times
slower on long doubles, so not many tests completed.
Bruce
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