🚀 go-pugleaf

         I said:
 >                                             I believe the papers to
 > which you refer basically require a way to obtain the double precis-
 > ion product of two single precision numbers.  I consider this a some-
 > what different issue.
         Dik Winter said:
It would be a different issue if it where true.  But it is not true, the
papers I refer to are about the accumulation of an inner product in
double (extra) precision.  See for instance: J.H.Wilkinson, Rounding Errors
in Numeric Processes, Notes on Applied Science No. 32, HMSO, London;
Prentice Hall, New Jersey, 1963.   (Yes fairly old; it was already known
in that time!)

         I have checked this reference (actually "Rounding Errors in
Algebraic Processes").  I believe it is consistent with what I said.
For example p. 10 "Nearly all digital computers produce in the first
instance, the 2t-digit product of two t-digit numbers", p. 14 "The
subroutine accepts single-precision factors and gives a double preci-
sion product", p. 32 "If we are working on a computer on which the
inner products can be accumulated, then there may be stages in the
work when we obtain results which are correct to 2t binary figures".
Throughout this section it is clear what is being discussed is a way
to obtain inner products to twice working precision, not slightly
more than working precision as in the IEEE extended formats.
         I said:
 >           I agree that binary formats are superior to hex in that they
 > provide 2 extra bits of precision.
         Dik Winter said:
A nit; they provide 3 more; nearly a digit.

         How do you get 3?  Suppose the leading hex digit is 1, then we
lose 3 bits from the leading zeros, 1 bit from the hidden 1, but gain 2
bits from the smaller exponent field for a net loss of 2 bits.
         I said
 >                                     I don't agree (particually for 64-bit
 > formats) that this 2 bits makes it significantly easier to avoid numer-
 > ical problems.
         Dik Winter said:
Oh, but they are, see the paper I mentioned above.

         I found nothing in the above reference to justify this state-
ment.
                       James B. Shearer