How can I compute the range of signed and unsigned types
James Cownie
jcownie at etnus.com
Thu Apr 19 05:01:34 EDT 2001
For floating point you may also want to look at
http://www.netlib.org/blas/machar.c
which appears to calculate most (all ?) of the interesting properties
of your floating point representation. (Of course it may get confused
by x86s working in 80 bit intermediates unless you're very careful how
you compile it, so, as ever, YMMV).
This subroutine is intended to determine the parameters of the
floating-point arithmetic system specified below. The
determination of the first three uses an extension of an algorithm
due to M. Malcolm, CACM 15 (1972), pp. 949-951, incorporating some,
but not all, of the improvements suggested by M. Gentleman and S.
Marovich, CACM 17 (1974), pp. 276-277. An earlier version of this
program was published in the book Software Manual for the
Elementary Functions by W. J. Cody and W. Waite, Prentice-Hall,
Englewood Cliffs, NJ, 1980. The present program is a
translation of the Fortran 77 program in W. J. Cody, "MACHAR:
A subroutine to dynamically determine machine parameters".
TOMS (14), 1988.
Parameter values reported are as follows:
ibeta - the radix for the floating-point representation
it - the number of base ibeta digits in the floating-point
significand
irnd - 0 if floating-point addition chops
1 if floating-point addition rounds, but not in the
IEEE style
2 if floating-point addition rounds in the IEEE style
3 if floating-point addition chops, and there is
partial underflow
4 if floating-point addition rounds, but not in the
IEEE style, and there is partial underflow
5 if floating-point addition rounds in the IEEE style,
and there is partial underflow
ngrd - the number of guard digits for multiplication with
truncating arithmetic. It is
0 if floating-point arithmetic rounds, or if it
truncates and only it base ibeta digits
participate in the post-normalization shift of the
floating-point significand in multiplication;
1 if floating-point arithmetic truncates and more
than it base ibeta digits participate in the
post-normalization shift of the floating-point
significand in multiplication.
machep - the largest negative integer such that
1.0+FLOAT(ibeta)**machep .NE. 1.0, except that
machep is bounded below by -(it+3)
negeps - the largest negative integer such that
1.0-FLOAT(ibeta)**negeps .NE. 1.0, except that
negeps is bounded below by -(it+3)
iexp - the number of bits (decimal places if ibeta = 10)
reserved for the representation of the exponent
(including the bias or sign) of a floating-point
number
minexp - the largest in magnitude negative integer such that
FLOAT(ibeta)**minexp is positive and normalized
maxexp - the smallest positive power of BETA that overflows
eps - the smallest positive floating-point number such
that 1.0+eps .NE. 1.0. In particular, if either
ibeta = 2 or IRND = 0, eps = FLOAT(ibeta)**machep.
Otherwise, eps = (FLOAT(ibeta)**machep)/2
epsneg - A small positive floating-point number such that
1.0-epsneg .NE. 1.0. In particular, if ibeta = 2
or IRND = 0, epsneg = FLOAT(ibeta)**negeps.
Otherwise, epsneg = (ibeta**negeps)/2. Because
negeps is bounded below by -(it+3), epsneg may not
be the smallest number that can alter 1.0 by
subtraction.
xmin - the smallest non-vanishing normalized floating-point
power of the radix, i.e., xmin = FLOAT(ibeta)**minexp
xmax - the largest finite floating-point number. In
particular xmax = (1.0-epsneg)*FLOAT(ibeta)**maxexp
Note - on some machines xmax will be only the
second, or perhaps third, largest number, being
too small by 1 or 2 units in the last digit of
the significand.
-- Jim
James Cownie <jcownie at etnus.com>
Etnus, LLC. +44 117 9071438
http://www.etnus.com
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