math.3

早期freebsd实现

.\" Copyright (c) 1985, 1993
.\"	The Regents of the University of California.  All rights reserved.
.\"
.\" Redistribution and use in source and binary forms, with or without
.\" modification, are permitted provided that the following conditions
.\" are met:
.\" 1. Redistributions of source code must retain the above copyright
.\"    notice, this list of conditions and the following disclaimer.
.\" 2. Redistributions in binary form must reproduce the above copyright
.\"    notice, this list of conditions and the following disclaimer in the
.\"    documentation and/or other materials provided with the distribution.
.\" 3. All advertising materials mentioning features or use of this software
.\"    must display the following acknowledgement:
.\"	This product includes software developed by the University of
.\"	California, Berkeley and its contributors.
.\" 4. Neither the name of the University nor the names of its contributors
.\"    may be used to endorse or promote products derived from this software
.\"    without specific prior written permission.
.\"
.\" THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
.\" ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
.\" IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
.\" ARE DISCLAIMED.  IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
.\" FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
.\" DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
.\" OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
.\" HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
.\" LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
.\" OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
.\" SUCH DAMAGE.
.\"
.\"	@(#)math.3	8.1 (Berkeley) 6/4/93
.\"
.TH MATH 3 "June 4, 1993"
.UC 4
.ds up \fIulp\fR
.ds nn \fINaN\fR
.de If
.if n \\
\\$1Infinity\\$2
.if t \\
\\$1\\(if\\$2
..
.SH NAME
math \- introduction to mathematical library functions
.SH DESCRIPTION
These functions constitute the C math library,
.I libm.
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 >.
The Fortran math library is described in ``man 3f intro''.
.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
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	???
lgamma	lgamma.3m	log gamma function; (formerly gamma.3m)
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
rint	floor.3m	round to nearest integer	0
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
.if n \
sig.
.if t \
significant
bits, roughly like 17
.if n \
sig.
.if t \
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, ... , -1165
	Zilog Z8070	Western Electric (AT&T) WE32106.
.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 ECL 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 WTL 1164/65.  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 provided (perhaps freely) by
those companies and designed by some of the authors of the
codes above.
Except for \fIatan\fR, \fIcabs\fR, \fIcbrt\fR, \fIerf\fR,
\fIerfc\fR, \fIhypot\fR, \fIj0\-jn\fR, \fIlgamma\fR, \fIpow\fR
and \fIy0\-yn\fR,
the Motorola 68881 has all the functions in \fIlibm\fR on chip,
and faster and more accurate;
it, Apple, the i8087, Z8070 and WE32106 all use 64
.if n \
sig.
.if t \
significant
bits.
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