Re: ddfloat library: the behavior of >dd

From	Krishna Myneni <krishna.myneni@ccreweb.org>
Newsgroups	comp.lang.forth
Subject	Re: ddfloat library: the behavior of >dd
Date	2026-09-15 07:33 -0500
Organization	A noiseless patient Spider
Message-ID	<118bdvk$2e9ov$1@dont-email.me> (permalink)
References	(11 earlier) <6aa7c5cf$1@news.ausics.net> <1188htd$2vabu$1@dont-email.me> <2026Sep14.170043@mips.complang.tuwien.ac.at> <1189ler$1sfui$1@dont-email.me> <2026Sep14.225816@mips.complang.tuwien.ac.at>

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On 9/14/26 3:58 PM, Anton Ertl wrote:
> Krishna Myneni <krishna.myneni@ccreweb.org> writes:
>> On 9/14/26 10:00 AM, Anton Ertl wrote:
>> ...
>>>
>>> I am debating with myself whether one should use the closest FP number
>>> with round-to-nearest, or the closest with round-to-absolute-higher.
>>> Maybe depends on the purpose of this normalization.
>>>
>> Is your idea to stay within 1 ulp error for the conversion, but use a
>> simpler algorithm?
> 
> No.
> 
> The idea here is as follows:
> 
> You have your number X that you want to normalize to a power of 10.
> For that number there exist numbers Y, Y', Z, and Z', where
> 
> Y > X >= Z
> Y >= 10^(n+1) > prev(Y)
> Z >= 10^n     > prev(Z)
> 
> Where prev(V) produces the largest (in absolute terms) FP value <V.
> So Y or Z are either powers of 10 or the smallest FP numbers just
> above a power of 10, and they are those numbers which are the
> normalization boundaries of X.
> 
> For normalizing X, we would love to divide by 10^n, but if that's not
> representable as FP number, and if we want to use our FP words to do
> that, what do we use?
> 
> 1) The result that has the best chance of being close to the proper
> result would be to use the FP number Z' that's closest to 10^n (round
> to nearest).  But if Z'=prev(Z) (which will probably be the case half
> of the time when 10^n cannot be represented exactly), the result of
> X/Z' will be larger than it should be, and, in particular, even if
> X=Z, X/Z' might have a value >1.0, which is probably not desired.
> 
> 2) Instead, you could normalize as X/Z, i.e., the number that's
> closest to 10^n if you round away from 0 (what I described as
> round-to-absolute-higher above).  But Z can be up to 1ulp away from
> 10^n, so the result of X/Z will, on averge be further from the
> intended value than X/Z'.
> 
> Another option is to normalize in a more precise way than performing a
> double-double FP multiplication or division, but that's more costly.
> 

The present implementation of >DD decimal string to double-double 
conversion in kForth (dd_io.4th) is off by 14 ulp for the case of pi,
which is not good enough, so it will be useful to compare your proposed 
conversion procedure.

\ For 64-bit Forth with a separate fp stack

3.1415926535897931e0  1.2246467991473532e-16  ddconstant ddpi

32 set-precision
ddpi ddfs.
+3.1415926535897932384626433832795 dd 0 ok

ddvariable ref
ddvariable cnv

ddpi ref dd!  \ reference
\ convert decimal string to double double with >DD
S" 3.1415926535897932384626433832795dd0" >dd cnv dd!

cnv dd@ ddfs.
+3.1415926535897932384626433832791 dd 0 ok

17 set-precision
\ print high and low double precision fp numbers
ref dd@ fswap fs. 2 spaces fs.
3.1415926535897931e+00   1.2246467991473532e-16  ok
cnv dd@ fswap fs. 2 spaces fs.
3.1415926535897931e+00   1.2246467991473498e-16  ok

\ Note the loss of significant digits in the low double for cnv
\ What is the ulp error for cnv?
hex
ref 8 + @ u.  \ print the 64-bit hex value of ref, low double
3CA1A62633145C07  ok
cnv 8 + @ u.  \ print the 64-bit hex value of cnv, low double
3CA1A62633145BF9  ok

\ difference in ulp
decimal
cnv 8 + @ ref 8 + @ - .
-14  ok

--
Krishna

Thread

ddfloat library: the behavior of >dd marcel hendrix <mhx@iae.nl> - 2026-09-10 19:33 +0200
  Re: ddfloat library: the behavior of >dd Krishna Myneni <krishna.myneni@ccreweb.org> - 2026-09-10 13:35 -0500
    Re: ddfloat library: the behavior of >dd marcel hendrix <mhx@iae.nl> - 2026-09-10 22:10 +0200
      Re: ddfloat library: the behavior of >dd Krishna Myneni <krishna.myneni@ccreweb.org> - 2026-09-10 16:51 -0500
  Re: ddfloat library: the behavior of >dd dxf <dxforth@gmail.com> - 2026-09-11 12:16 +1000
    Re: ddfloat library: the behavior of >dd marcel hendrix <mhx@iae.nl> - 2026-09-11 12:05 +0200
      Re: ddfloat library: the behavior of >dd marcel hendrix <mhx@iae.nl> - 2026-09-12 00:27 +0200
        Re: ddfloat library: the behavior of >dd Krishna Myneni <krishna.myneni@ccreweb.org> - 2026-09-11 20:29 -0500
          Re: ddfloat library: the behavior of >dd dxf <dxforth@gmail.com> - 2026-09-12 12:02 +1000
            Re: ddfloat library: the behavior of >dd Krishna Myneni <krishna.myneni@ccreweb.org> - 2026-09-12 10:24 -0500
          Re: ddfloat library: the behavior of >dd marcel hendrix <mhx@iae.nl> - 2026-09-12 10:45 +0200
            Re: ddfloat library: the behavior of >dd dxf <dxforth@gmail.com> - 2026-09-12 19:51 +1000
            Re: ddfloat library: the behavior of >dd dxf <dxforth@gmail.com> - 2026-09-12 22:47 +1000
              Re: ddfloat library: the behavior of >dd dxf <dxforth@gmail.com> - 2026-09-13 03:14 +1000
                Re: ddfloat library: the behavior of >dd Krishna Myneni <krishna.myneni@ccreweb.org> - 2026-09-12 13:02 -0500
                Re: ddfloat library: the behavior of >dd dxf <dxforth@gmail.com> - 2026-09-13 15:44 +1000
                Re: ddfloat library: the behavior of >dd marcel hendrix <mhx@iae.nl> - 2026-09-13 09:00 +0200
                Re: ddfloat library: the behavior of >dd dxf <dxforth@gmail.com> - 2026-09-13 20:33 +1000
                Re: ddfloat library: the behavior of >dd dxf <dxforth@gmail.com> - 2026-09-14 13:28 +1000
                Re: ddfloat library: the behavior of >dd marcel hendrix <mhx@iae.nl> - 2026-09-14 10:09 +0200
                Re: ddfloat library: the behavior of >dd dxf <dxforth@gmail.com> - 2026-09-14 20:00 +1000
                Re: ddfloat library: the behavior of >dd marcel hendrix <mhx@iae.nl> - 2026-09-14 12:22 +0200
                Re: ddfloat library: the behavior of >dd dxf <dxforth@gmail.com> - 2026-09-14 21:59 +1000
                Re: ddfloat library: the behavior of >dd marcel hendrix <mhx@iae.nl> - 2026-09-14 14:26 +0200
                Re: ddfloat library: the behavior of >dd dxf <dxforth@gmail.com> - 2026-09-15 01:32 +1000
                Re: ddfloat library: the behavior of >dd marcel hendrix <mhx@iae.nl> - 2026-09-14 21:08 +0200
                Re: ddfloat library: the behavior of >dd Krishna Myneni <krishna.myneni@ccreweb.org> - 2026-09-14 15:25 -0500
                Re: ddfloat library: the behavior of >dd dxf <dxforth@gmail.com> - 2026-09-15 13:11 +1000
                Re: ddfloat library: the behavior of >dd marcel hendrix <mhx@iae.nl> - 2026-09-15 14:06 +0200
                Re: ddfloat library: the behavior of >dd dxf <dxforth@gmail.com> - 2026-09-16 00:05 +1000
                Re: ddfloat library: the behavior of >dd anton@mips.complang.tuwien.ac.at (Anton Ertl) - 2026-09-14 15:00 +0000
                Re: ddfloat library: the behavior of >dd marcel hendrix <mhx@iae.nl> - 2026-09-14 21:15 +0200
                Re: ddfloat library: the behavior of >dd Krishna Myneni <krishna.myneni@ccreweb.org> - 2026-09-14 15:29 -0500
                Re: ddfloat library: the behavior of >dd anton@mips.complang.tuwien.ac.at (Anton Ertl) - 2026-09-14 20:58 +0000
                Re: ddfloat library: the behavior of >dd Krishna Myneni <krishna.myneni@ccreweb.org> - 2026-09-15 07:33 -0500
                Re: ddfloat library: the behavior of >dd anton@mips.complang.tuwien.ac.at (Anton Ertl) - 2026-09-15 15:53 +0000
                Re: ddfloat library: the behavior of >dd Krishna Myneni <krishna.myneni@ccreweb.org> - 2026-09-15 15:23 -0500
                Re: ddfloat library: the behavior of >dd Krishna Myneni <krishna.myneni@ccreweb.org> - 2026-09-15 17:53 -0500

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