🚀 go-pugleaf

In article 11460@news.lrz-muenchen.de, HUSFELD@usmv01.usm.uni-muenchen.de (Husfeld, Dirk) writes:
>In <1993Jan22.182648.7390@zia.aoc.nrao.edu> cflatter@nrao.edu writes:
>
>>>   <some cited lines about numerical precision deleted>
>> Unfortunately his discussion of numerical precision is very outdated.  Almost
>> anything you are likely to have on your desk these days will maintain much
>> better precision than Meeus would lead you to believe (most machines will
>> only lose one bit of precision in subtraction no matter how close the numbers
>> being subtracted for example).
>> 
>> 	Chris Flatters
>> 	cflatter@nrao.edu
>>
>Hmmm.  If you subtract two numbers that differ by only the last bit in their
>binary representation, then your result only has one bit of precision.
>So I think your above words were not very carefully chosen.  Could you
>elaborate on what you mean?

Most current floating-point implementations use an extra bit (called a
guard digit) during addition and subtraction.  It can be proved
(meaning I'm not going to do it here --- its awkward to reproduce it
text --- see "What Every Computer Scientist Should Know About
Floating-Point Arithmetic" by David Goldberg, ACM Computing Surveys,
23:5 (1991) if you really want to study it) that the use of a single
guard digit will restrict the maximum relative error in the subtraction
of two numbers to be less than 2e where e is the upper bound on the
relative error caused by rounding by half a unit in the least
significant digit (this is a general theorem and applies to both binary
and non-binary floating-point representations).

Floating-point implementations that conform to IEEE standard 754 (these
include IBM PC's (Intel 80x87), and most workstations (including SPARC,
MIPS, PA-RISC and IBM RS/6000) use a guard digit.  VAX floating-point
also uses a guard digit.  IBM added a guard digit to the
double-precision format of its system/360 architecture in the late
1960s (single-precision already used a guard digit) and retrofitted
every machine in the field.  Cray supercomputers do not use a guard
digit.

	Chris Flatters
	cflatter@nrao.edu