rhbartels@watcgl.UUCP (Richard Bartels) (12/20/84)
The following may need some explanation. A few days ago, Paul Schinder
broadcast a general request to the net about raising negative reals and
negative doubles to integer powers. I forwarded the request to
Brian Smith of Argonne National Laboratory, who is the ACM-SIGNUM official
representative on the IEEE X3J3 Committee (FORTRAN Standards).
The following is his response, which he tried to forward directly to
Paul, with a "destination unknown" result. The answer may have general
interest.
Furthermore: You should be aware that FORTRAN-77, and its standards,
were set up by X3J3, whose members welcome general feedback. The
committee is currently debating "FORTRAN 8x", proposals for a successor
to FORTRAN-77. For details about the proposals as they now stand,
the fall SIGNUM Newsletter can be consulted. (I could get a limited
number of copies if you can't find them elsewhere. Oh, and memberships
are welcomed.)
Richard Bartels
SIGNUM Chair
Computer Science Department
University of Waterloo
Waterloo, Ontario N2L 3G1
CANADA
---------------------------------------------------------------
Date: 19-Dec-84 11:28:38-CST (Wed)
From: "Brian T. Smith" <watmath!ANL-MCS!smith>
Subject: Interpretation of x**i, i<0
Return-Path: <smith@ANL-MCS.ARPA>
To: paul@ANL-MCS@oddjob.UChicago
Cc: SU-SCORE!na.bartels
To: Paul Schinder
Richard Bartels forwarded your message to me at Argonne. We are having
problems with our net mail to certain locations, and so will mail this
to Richard as well as you. Richard, please send a copy of this to Paul
Schinder.
Although the standard does not say so explicitly, I believe that it does
require the interpretation of x**i where i is negative. The argument
to support this view is that the standard appeals to the mathematical
definition of raising x to the power y (as with other intrinsic
operations such as sin, cos, etc.) to define the Fortran operation
x**y. This appears in section 6.1.2 of the Fortran 77 standard. It
goes on to say that once this interpretation in section 6.1.2 is
established, then the processor may evaluate it in any mathematically
equivalent fashion. (For example, x**i where i is negative and x is
non-zero can be evaluated as 1.0/x**(-i).) It then goes on to say that
wherever there is no mathematical definition, the operation is
prohibited. These latter two statements appear in section 6.6 of the
Fortran 77 standard. Therefore, given these statements, x**i where i
is negative is mathematically defined (provided x is non-zero) and
therefore must be evaluated (it does not say how accurately, though).
As with all standards, the document is written in a concise but
intended rigorous manner whose primary purpose is to define the
language. Therefore, it does not always define constructs directly as
in the above case. It appeals to more general definitions
(mathematical concepts and models) that apply to many contexts (all
intrinsic operations and intrinsic functions).
By the way, I am a member of the current Fortran standards committee.
The above comments are my best effort in trying to help you interpret
the standard and what it says. This is therefore not an opinion from
the Fortran committee. If you want, you can present this as a point of
clarification for the committee, for which you will get an official
response.
Needless to say, as one of the contributors to the written document for
the next standard, I will try to make sure that this is clarified in
the next standard. The difficulty with trying to make a clear
statement such as "Intrinsic operations (and intrinsic functions) must
be evaluated for all operands (and arguments) for which they are
mathematically defined" is that it does not address finite exponent
ranges or asking for the sine of an argument that is extremely large
and therefore the result has no meaning. If you want to discuss these
remarks by phone, my number is 312-972-7232.
Brian T. Smith
Mathematics and Computer Science Division
Argonne National Laboratorygwyn@brl-tgr.ARPA (Doug Gwyn <gwyn>) (12/26/84)
So how about an answer to the original question: x ** i where x is negative?
rhbartels@watcgl.UUCP (Richard Bartels) (12/28/84)
Sorry, Brian mixed i<0 for x<0, but the same answer should apply: "...the [FORTRAN 77] standard appeals to the mathematical definition of raising x to the power y..." since x**i is mathematically x*x*...*x (i times) for integer i>0 and (1/x)*(1/x)*...*(1/x) (-i times) for integer i<0 and x!=0 and is 1 for x!=0 and i=0, the standard in section 6.1.2 is implying that FORTRAN 77 should work for x<0 and integer i. If some particular compiler blows it, then it's the compiler which is at fault, and not the standard.