date and time created 85/09/06 17:52:49 by zliu

[unix-history] / usr / src / lib / libm / common_source / math.3

.TH INTRO 3M "19 August 1985"
.UC 4
.ds up \fIulp\fR
.ds nn \fINaN\fR
.de If
.if n \\
\\$1Infinity\\$2
.if t \\
\\$1\\(if\\$2
..
.SH NAME
intro \- introduction to mathematical library functions
.SH DESCRIPTION
These functions constitute the math library,
.I libm.
They are automatically loaded as needed by the Fortran compiler
.IR f77 (1).
The link editor searches this library under the \*(lq\-lm\*(rq option.
Declarations for these functions may be obtained from the include file
.RI < math.h >.
.SH "LIST OF FUNCTIONS"
.sp 2
.nf
.ta \w'copysign'u+2n +\w'infnan.3m'u+10n +\w'inverse trigonometric func'u
\fIName\fP	\fIAppears on Page\fP	\fIDescription\fP	\fIError Bound (ULPs)\fP
.ta \w'copysign'u+4n +\w'infnan.3m'u+4n +\w'inverse trigonometric function'u+6nC
.sp 5p
acos	sin.3m	inverse trigonometric function	3
acosh	asinh.3m	inverse hyperbolic function	3
asin	sin.3m	inverse trigonometric function	3
asinh	asinh.3m	inverse hyperbolic function	3
atan	sin.3m	inverse trigonometric function	1
atanh	asinh.3m	inverse hyperbolic function	3
atan2	sin.3m	inverse trigonometric function	2
cabs	hypot.3m	complex absolute value	1
cbrt	sqrt.3m	cube root	1
ceil	floor.3m	integer no less than	0
copysign	ieee.3m	copy sign bit	0
cos	sin.3m	trigonometric function	1
cosh	sinh.3m	hyperbolic function	3
drem	ieee.3m	remainder	0
erf	erf.3m	error function	???
erfc	erf.3m	complementary error function	???
exp	exp.3m	exponential	1
expm1	exp.3m	exp(x)\-1	1
fabs	floor.3m	absolute value	0
floor	floor.3m	integer no greater than	0
gamma	gamma.3m	log gamma function
hypot	hypot.3m	Euclidean distance	1
infnan	infnan.3m	signals exceptions
j0	j0.3m	bessel function	???
j1	j0.3m	bessel function	???
jn	j0.3m	bessel function	???
log	exp.3m	natural logarithm	1
logb	ieee.3m	exponent extraction	0
log10	exp.3m	logarithm to base 10	3
log1p	exp.3m	log(1+x)	1
pow	exp.3m	exponential x**y	60\-500
scalb	ieee.3m	exponent adjustment	0
sin	sin.3m	trigonometric function	1
sinh	sinh.3m	hyperbolic function	3
sqrt	sqrt.3m	square root	1
tan	sin.3m	trigonometric function	3
tanh	sinh.3m	hyperbolic function	3
y0	j0.3m	bessel function	???
y1	j0.3m	bessel function	???
yn	j0.3m	bessel function	???
.ta
.fi
.SH NOTES
In 4.3 BSD, distributed from the University of California
in late 1985, most of the foregoing functions come in two
versions, one for the double\-precision "D" format in the
DEC VAX\-11 family of computers, another for double\-precision
arithmetic conforming to the IEEE Standard 754 for Binary
Floating\-Point Arithmetic.  The two versions behave very
similarly, as should be expected from programs more accurate
and robust than was the norm when UNIX was born.  For
instance, the programs are accurate to within the numbers
of \*(ups tabulated above; an \*(up is one \fIU\fRnit in the \fIL\fRast
\fIP\fRlace.  And the programs have been cured of anomalies that
afflicted the older math library \fIlibm\fR in which incidents like
the following had been reported:
.RS
sqrt(\-1.0) = 0.0 and log(\-1.0) = \-1.7e38.
.br
cos(1.0e\-11) > cos(0.0) > 1.0.
.br
pow(x,1.0)
.if n \
!=
.if t \
\(!=
x when x = 2.0, 3.0, 4.0, ..., 9.0.
.br
pow(\-1.0,1.0e10) trapped on Integer Overflow.
.br
sqrt(1.0e30) and sqrt(1.0e\-30) were very slow.
.RE
However the two versions do differ in ways that have to be
explained, to which end the following notes are provided.
.PP
\fBDEC VAX\-11 D_floating\-point:\fR
.PP
This is the format for which the original math library \fIlibm\fR
was developed, and to which this manual is still principally
dedicated.  It is \fIthe\fR double\-precision format for the PDP\-11
and the earlier VAX\-11 machines; VAX\-11s after 1983 were
provided with an optional "G" format closer to the IEEE
double\-precision format.  The earlier DEC MicroVAXs have no
D format, only G double\-precision. (Why?  Why not?)
.PP
Properties of D_floating\-point:
.RS
Wordsize: 64 bits, 8 bytes.  Radix: Binary.
.br
Precision: 56 bits; equivalent to about 17 significant decimals.
.RS
If x and x' are consecutive positive D_floating\-point
numbers (they differ by 1 \*(up), then
.br
1.3e\-17 < 0.5**56 < (x'\-x)/x \(<= 0.5**55 < 2.8e\-17.
.RE
.nf
.ta \w'Range:'u+1n +\w'Underflow threshold'u+1n +\w'= 2.0**127'u+1n
Range:	Overflow threshold	= 2.0**127	= 1.7e38.
	Underflow threshold	= 0.5**128	= 2.9e\-39.
	NOTE:  THIS RANGE IS COMPARATIVELY NARROW.
.ta
.fi
.RS
Overflow customarily stops computation.
.br
Underflow is customarily flushed quietly to zero.
.br
CAUTION:
.RS
It is possible to have x
.if n \
!=
.if t \
\(!=
y and yet
x\-y = 0 because of underflow.  Similarly
x > y > 0 cannot prevent either x\(**y = 0
or  y/x = 0 from happening without warning.
.RE
.RE
Zero is represented ambiguously.
.RS
Although 2**55 different representations of zero are accepted by
the hardware, only the obvious representation is ever produced.
There is no \-0 on a VAX.
.RE
.If
is not part of the VAX architecture.
.br
Reserved operands:
.RS
of the 2**55 that the hardware
recognizes, only one of them is ever produced.
Any floating\-point operation upon a reserved
operand, even a MOVF or MOVD, customarily stops
computation, so they are not much used.
.RE
Exceptions:
.RS
Divisions by zero and operations that
overflow are invalid operations that customarily
stop computation or, in earlier machines, produce
reserved operands that will stop computation.
.RE
Rounding:
.RS
Every rational operation  (+, \-, \(**, /) on a
VAX (but not necessarily on a PDP\-11), if not an
over/underflow nor division by zero, is rounded to
within half an \*(up, and when the rounding error is
exactly half an \*(up then rounding is away from 0.
.RE
.RE
.PP
Except for its narrow range, D_floating\-point is one of the
better computer arithmetics designed in the 1960's.
Its properties are reflected fairly faithfully in the elementary
functions for a VAX distributed in 4.3 BSD.
They over/underflow only if their results have to lie out of range
or very nearly so, and then they behave much as any rational
arithmetic operation that over/underflowed would behave.
Similarly, expressions like log(0) and atanh(1) behave
like 1/0; and sqrt(\-3) and acos(3) behave like 0/0;
they all produce reserved operands and/or stop computation!
The situation is described in more detail in manual pages.
.RS
.ll -0.5i
\fIThis response seems excessively punitive, so it is destined
to be replaced at some time in the foreseeable future by a
more flexible but still uniform scheme being developed to
handle all floating\-point arithmetic exceptions neatly.
See infnan(3M) for the present state of affairs.\fR
.ll +0.5i
.RE
.PP
How do the functions in 4.3 BSD's new \fIlibm\fR for UNIX
compare with their counterparts in DEC's VAX/VMS library?
Some of the VMS functions are a little faster, some are
a little more accurate, some are more puritanical about
exceptions (like pow(0.0,0.0) and atan2(0.0,0.0)),
and most occupy much more memory than their counterparts in
\fIlibm\fR.
The VMS codes interpolate in large table to achieve
speed and accuracy; the \fIlibm\fR codes use tricky formulas
compact enough that all of them may some day fit into a ROM.
.PP
More important, DEC regards the VMS codes as proprietary
and guards them zealously against unauthorized use.  But the
\fIlibm\fR codes in 4.3 BSD are intended for the public domain;
they may be copied freely provided their provenance is always
acknowledged, and provided users assist the authors in their
researches by reporting experience with the codes.
Therefore no user of UNIX on a machine whose arithmetic resembles
VAX D_floating\-point need use anything worse than the new \fIlibm\fR.
.PP
\fBIEEE STANDARD 754 Floating\-Point Arithmetic:\fR
.PP
This standard is on its way to becoming more widely adopted
than any other design for computer arithmetic.
VLSI chips that conform to some version of that standard have been
produced by a host of manufacturers, among them ...
.nf
.ta 0.5i +\w'Intel i8070, i80287'u+6n
	Intel i8087, i80287	National Semiconductor  32081
	Motorola 68881	Weitek WTL-1032, ... , -1065
	Zilog Z8070	Western Electric (AT&T) 32106.
.ta
.fi
Other implementations range from software, done thoroughly
in the Apple Macintosh, through VLSI in the Hewlett\-Packard
9000 series, to the ELXSI 6400 running at 3 Megaflops.
Several other companies have adopted the formats
of IEEE 754 without, alas, adhering to the standard's way
of handling rounding and exceptions like over/underflow.
The DEC VAX G_floating\-point format is very similar to the IEEE
754 Double format, so similar that the C programs for the
IEEE versions of most of the elementary functions listed
above could easily be converted to run on a MicroVAX, though
nobody has volunteered to do that yet.
.PP
The codes in 4.3 BSD's \fIlibm\fR for machines that conform to
IEEE 754 are intended primarily for the National Semi. 32081
and Weitek 1065.  To use these codes with the Intel or Zilog
chips, or with the Apple Macintosh or ELXSI 6400, is to
forego the use of better codes available (for a price) from
those companies and designed by some of the authors of the
codes above.  The Motorola 68881 has all the \fIelementary\fR
functions above except \fIpow\fR implemented on chip already,
and faster and more accurate.  The main virtue of 4.3 BSD's
\fIlibm\fR codes is that they are intended for the public domain;
they may be copied freely provided their provenance is always
acknowledged, and provided users assist the authors in their
researches by reporting experience with the codes.
Therefore no user of UNIX on a machine that conforms to
IEEE 754 need use anything worse than the new \fIlibm\fR.
.PP
Properties of IEEE 754 Double\-Precision:
.RS
Wordsize: 64 bits, 8 bytes.  Radix: Binary.
.br
Precision: 53 bits; equivalent to about 16 significant decimals.
.RS
If x and x' are consecutive positive Double\-Precision
numbers (they differ by 1 \*(up), then
.br
1.1e\-16 < 0.5**53 < (x'\-x)/x \(<= 0.5**52 < 2.3e\-16.
.RE
.nf
.ta \w'Range:'u+1n +\w'Underflow threshold'u+1n +\w'= 2.0**1024'u+1n
Range:	Overflow threshold	= 2.0**1024	= 1.8e308
	Underflow threshold	= 0.5**1022	= 2.2e\-308
.ta
.fi
.RS
Overflow goes by default to a signed
.If "" .
.br
Underflow is \fIGradual,\fR rounding to the nearest
integer multiple of 0.5**1074 = 4.9e\-324.
.RE
Zero is represented ambiguously as +0 or \-0.
.RS
Its sign transforms correctly through multiplication or
division, and is preserved by addition of zeros
with like signs; but x\-x yields +0 for every
finite x.  The only operations that reveal zero's
sign are division by zero and copysign(x,\(+-0).
In particular, comparison (x > y, x \(>= y, etc.)
cannot be affected by the sign of zero; but if
finite x = y then
.If
\&= 1/(x\-y)
.if n \
!=
.if t \
\(!=
\-1/(y\-x) =
.If \- .
.RE
.If
is signed.
.RS
it persists after addition of itself
or of any finite number.  Its sign transforms
correctly through multiplication and division, and
.If (finite)/\(+- \0=\0\(+-0
(nonzero)/0 =
.If \(+- .
But 
.if n \
Infinity\-Infinity, Infinity\(**0 and Infinity/Infinity
.if t \
\(if\-\(if, \(if\(**0 and \(if/\(if
are, like 0/0 and sqrt(\-3),
invalid operations that produce \*(nn. ...
.RE
Reserved operands:
.RS
there are 2**53\-2 of them, all
called \*(nn (\fIN\fRot \fIa N\fRumber).
Some, called Signaling \*(nns, trap any floating\-point operation
performed upon them; they are used to mark missing
or uninitialized values, or nonexistent elements
of arrays.  The rest are Quiet \*(nns; they are
the default results of Invalid Operations, and
propagate through subsequent arithmetic operations.
If x
.if n \
!=
.if t \
\(!=
x then x is \*(nn; every other predicate
(x > y, x = y, x < y, ...) is FALSE if \*(nn is involved.
.br
NOTE: Trichotomy is violated by \*(nn.
.RS
Besides being FALSE, predicates that entail ordered
comparison, rather than mere (in)equality,
signal Invalid Operation when \*(nn is involved.
.RE
.RE
Rounding:
.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
Exceptions:
.RS
IEEE 754 recognizes five kinds of floating\-point exceptions,
listed below in declining order of probable importance.
.RS
.nf
.ta \w'Invalid Operation'u+6n +\w'Gradual Underflow'u+2n
Exception	Default Result
.tc \(ru
		
.tc
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
.ta
.fi
.RE
NOTE:  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.
.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.
.RS
CAUTION: 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.
.RE
.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 cannot 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 pow(x,y)
.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
.SH BUGS
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 6) 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
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.
The manuals for Pascal, C and BASIC on the Apple Macintosh
document the features of IEEE 754 pretty well.
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.
.SH AUTHOR
W. Kahan, with the help of Alex Z\-S. Liu, Stuart I. McDonald,
Dr. Kwok\-Choi Ng, Peter Tang.