math.3m

来自「<B>Digital的Unix操作系统VAX 4.2源码</B>」· 3M 代码 · 共 550 行 · 第 1/2 页

.NE
.RE
\fBRounding:\fP
.RS
Every algebraic operation (+, \-, \(**, /,
.if n \
sqrt)
.if t \
\(sr)
is rounded by default to within half an \*(up, and
when the rounding error is exactly half an \*(up then
the rounded value's least significant bit is zero.
This kind of rounding is usually the best kind,
sometimes provably so; for instance, for every
x = 1.0, 2.0, 3.0, 4.0, ..., 2.0**52, we find
(x/3.0)\(**3.0 == x and (x/10.0)\(**10.0 == x and ...
despite that both the quotients and the products
have been rounded.  Only rounding like IEEE 754
can do that.  But no single kind of rounding can be
proved best for every circumstance, so IEEE 754
provides rounding towards zero or towards
.If +
or towards
.If \-
at the programmer's option.  And the
same kinds of rounding are specified for
Binary\-Decimal Conversions, at least for magnitudes
between roughly 1.0e\-10 and 1.0e37.
.RE
\fBExceptions:\fP
.RS
IEEE 754 recognizes five kinds of floating\-point exceptions,
listed below in declining order of probable importance.
.PP
.TS
tab(@);
lfHB lfHB
lfR  lfR .
_
.sp 4p
Exception@Default Result
.sp 4p
_
.sp 6p
Invalid Operation@\*(nn, or FALSE
.if n \{\
Overflow@\(+-Infinity
Divide by Zero@\(+-Infinity \}
.if t \{\
Overflow@\(+-\(if
Divide by Zero@\(+-\(if \}
Underflow@Gradual Underflow
Inexact@Rounded value
.sp 6p
_
.TE
.PP
.NT
An Exception is not an Error unless handled
badly.  What makes a class of exceptions exceptional
is that no single default response can be satisfactory
in every instance.  On the other hand, if a default
response will serve most instances satisfactorily,
the unsatisfactory instances cannot justify aborting
computation every time the exception occurs.
.NE
.RE
.PP
For each kind of floating\-point exception, IEEE 754
provides a Flag that is raised each time its exception
is signaled, and stays raised until the program resets
it.  Programs may also test, save and restore a flag.
Thus, IEEE 754 provides three ways by which programs
may cope with exceptions for which the default result
might be unsatisfactory:
.IP 1) \w'\0\0\0\0'u
Test for a condition that might cause an exception
later, and branch to avoid the exception.
.IP 2) \w'\0\0\0\0'u
Test a flag to see whether an exception has occurred
since the program last reset its flag.
.IP 3) \w'\0\0\0\0'u
Test a result to see whether it is a value that only
an exception could have produced.
.NT
The only reliable ways to discover
whether Underflow has occurred are to test whether
products or quotients lie closer to zero than the
underflow threshold, or to test the Underflow
flag.  (Sums and differences cannot underflow in
IEEE 754; if x
.if n \
!=
.if t \
\(!=
y then x\-y is correct to
full precision and certainly nonzero regardless of
how tiny it may be.)  Products and quotients that
underflow gradually can lose accuracy gradually
without vanishing, so comparing them with zero
(as one might on a VAX) will not reveal the loss.
Fortunately, if a gradually underflowed value is
destined to be added to something bigger than the
underflow threshold, as is almost always the case,
digits lost to gradual underflow will not be missed
because they would have been rounded off anyway.
So gradual underflows are usually \fIprovably\fR ignorable.
The same cannot be said of underflows flushed to 0.
.NE
.PP
At the option of an implementor conforming to IEEE 754,
other ways to cope with exceptions may be provided:
.IP 4) \w'\0\0\0\0'u
ABORT.  This mechanism classifies an exception in
advance as an incident to be handled by means
traditionally associated with error\-handling
statements like "ON ERROR GO TO ...".  Different
languages offer different forms of this statement,
but most share the following characteristics:
.IP \(em \w'\0\0\0\0'u
No means is provided to substitute a value for
the offending operation's result and resume
computation from what may be the middle of an
expression.  An exceptional result is abandoned.
.IP \(em \w'\0\0\0\0'u
In a subprogram that lacks an error\-handling
statement, an exception causes the subprogram to
abort within whatever program called it, and so
on back up the chain of calling subprograms until
an error\-handling statement is encountered or the
whole task is aborted and memory is dumped.
.IP 5) \w'\0\0\0\0'u
STOP.  This mechanism, requiring an interactive
debugging environment, is more for the programmer
than the program.  It classifies an exception in
advance as a symptom of a programmer's error; the
exception suspends execution as near as it can to
the offending operation so that the programmer can
look around to see how it happened.  Quite often
the first several exceptions turn out to be quite
unexceptionable, so the programmer ought ideally
to be able to resume execution after each one as if
execution had not been stopped.
.IP 6) \w'\0\0\0\0'u
\&... Other ways lie beyond the scope of this document.
.RE
.PP
The crucial problem for exception handling is the problem of
Scope, and the problem's solution is understood, but not
enough manpower was available to implement it fully in time
to be distributed in 4.3 BSD's \fIlibm\fR.  Ideally, each
elementary function should act as if it were indivisible, or
atomic, in the sense that ...
.IP i) \w'iii)'u+2n
No exception should be signaled that is not deserved by
the data supplied to that function.
.IP ii) \w'iii)'u+2n
Any exception signaled should be identified with that
function rather than with one of its subroutines.
.IP iii) \w'iii)'u+2n
The internal behavior of an atomic function should not
be disrupted when a calling program changes from
one to another of the five or so ways of handling
exceptions listed above, although the definition
of the function may be correlated intentionally
with exception handling.
.PP
Ideally, every programmer should be able \fIconveniently\fR to
turn a debugged subprogram into one that appears atomic to
its users.  But simulating all three characteristics of an
atomic function is still a tedious affair, entailing hosts
of tests and saves\-restores; work is under way to ameliorate
the inconvenience.
.PP
Meanwhile, the functions in \fIlibm\fR are only approximately
atomic.  They signal no inappropriate exception except
possibly ...
.RS
Over/Underflow
.RS
when a result, if properly computed, might have lain barely within range, and
.RE
Inexact in \fIcabs\fR, \fIcbrt\fR, \fIhypot\fR, \fIlog10\fR and \fIpow\fR
.RS
when it happens to be exact, thanks to fortuitous cancellation of errors.
.RE
.RE
Otherwise, ...
.RS
Invalid Operation is signaled only when
.RS
any result but \*(nn would probably be misleading.
.RE
Overflow is signaled only when
.RS
the exact result would be finite but beyond the overflow threshold.
.RE
Divide\-by\-Zero is signaled only when
.RS
a function takes exactly infinite values at finite operands.
.RE
Underflow is signaled only when
.RS
the exact result would be nonzero but tinier than the underflow threshold.
.RE
Inexact is signaled only when
.RS
greater range or precision would be needed to represent the exact result.
.RE
.RE
 
\fBExceptions on RISC machines:\fP
.RS
The exception enables and the flags that are raised when an exception
occurs (as well as the rounding mode) are in the floating\-point control and
status register.  This register can be read or written by the routines
described on the man page
.IR fpc (3).
This register's layout is described in the file
.RI < mips/fpu.h >
in UMIPS\-BSD releases and in
.RI < sys/fpu.h >
in UMIPS\-SYSV releases.
.sp +.5
What is currently available is only the raw interface which was only
intended to be used by the code to implement IEEE user trap handlers.
IEEE floating\-point exceptions are enabled by setting the enable bit
for that exception in the floating\-point control and status register.
If an exception then occurs the UNIX signal SIGFPE is sent to the
process.  It is up to the signal handler to determine the instruction
that caused the exception and to take the action specified by the
user.  The instruction that caused the exception is in one of two
places.  If the floating\-point board is used (the floating\-point
implementation revision register indicates this in it's implementation
field) then the instruction that caused the exception is in the
floating\-point exception instruction register.  In all other
implementations the instruction that caused the exception is at
the address of the program counter as modified by the branch delay
bit in the cause register.  Both the program counter and cause register
are in the sigcontext structure passed to the signal handler (see
.MS signal 3 ).
If the program is to be continued past the instruction that caused
the exception the program counter in the signal context must be
advanced.  If the instruction is in a branch delay slot then the
branch must be emulated to determine if the branch is taken and
then the resulting program counter can be calculated (see
.MS emulate_branch 3
and 
.MS signal 3 ).
.RE
.SH Restrictions
When signals are appropriate, they are emitted by certain
operations within the codes, so a subroutine\-trace may be
needed to identify the function with its signal in case
method 5) above is in use.  And the codes all take the
IEEE 754 defaults for granted; this means that a decision to
trap all divisions by zero could disrupt a code that would
otherwise get correct results despite division by zero.
.SH See Also
fpc(3), signal(3), emulate_branch(3)
.br
.I "R2010 Floating Point Coprocessor Architecture"
.br
.I "R2360 Floating Point Board Product Description"
.PP
An explanation of IEEE 754 and its proposed extension p854
was published in the IEEE magazine MICRO in August 1984 under
the title "A Proposed Radix\- and Word\-length\-independent
Standard for Floating\-point Arithmetic" by W. J. Cody et al.
.PP
Articles in the IEEE magazine COMPUTER vol. 14 no. 3 (Mar.
1981), and in the ACM SIGNUM Newsletter Special Issue of
Oct. 1979, may be helpful although they pertain to
superseded drafts of the standard.