633 lines
22 KiB
Groff
633 lines
22 KiB
Groff
.\" Copyright (c) 1985, 1993
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.\" The Regents of the University of California. All rights reserved.
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.\"
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.\" Redistribution and use in source and binary forms, with or without
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.\" modification, are permitted provided that the following conditions
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.\" are met:
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.\" 1. Redistributions of source code must retain the above copyright
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.\" notice, this list of conditions and the following disclaimer.
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.\" 2. Redistributions in binary form must reproduce the above copyright
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.\" notice, this list of conditions and the following disclaimer in the
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.\" documentation and/or other materials provided with the distribution.
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.\" 3. All advertising materials mentioning features or use of this software
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.\" must display the following acknowledgement:
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.\" This product includes software developed by the University of
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.\" California, Berkeley and its contributors.
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.\" 4. Neither the name of the University nor the names of its contributors
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.\" may be used to endorse or promote products derived from this software
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.\" without specific prior written permission.
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.\"
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.\" THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
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.\" ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
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.\" IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
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.\" ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
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.\" FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
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.\" DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
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.\" OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
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.\" HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
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.\" LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
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.\" OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
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.\" SUCH DAMAGE.
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.\"
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.\" @(#)math.3 8.2 (Berkeley) 5/5/94
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.\" $FreeBSD$
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.\"
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.TH MATH 3 "May 5, 1994"
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.UC 4
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.ds up \fIulp\fR
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.ds nn \fINaN\fR
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.de If
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.if n \\
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\\$1Infinity\\$2
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.if t \\
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\\$1\\(if\\$2
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..
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.SH NAME
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math \- introduction to mathematical library functions
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.SH DESCRIPTION
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These functions constitute the C math library,
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.I libm.
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The link editor searches this library under the \*(lq\-lm\*(rq option.
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Declarations for these functions may be obtained from the include file
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.RI < math.h >.
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The Fortran math library is described in ``man 3f intro''.
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.SH "LIST OF FUNCTIONS"
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.sp 2
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.nf
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.ta \w'copysign'u+2n +\w'infnan.3m'u+10n +\w'inverse trigonometric func'u
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\fIName\fP \fIAppears on Page\fP \fIDescription\fP \fIError Bound (ULPs)\fP
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.ta \w'copysign'u+4n +\w'infnan.3m'u+4n +\w'inverse trigonometric function'u+6nC
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.sp 5p
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acos sin.3m inverse trigonometric function 3
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acosh asinh.3m inverse hyperbolic function 3
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asin sin.3m inverse trigonometric function 3
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asinh asinh.3m inverse hyperbolic function 3
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atan sin.3m inverse trigonometric function 1
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atanh asinh.3m inverse hyperbolic function 3
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atan2 sin.3m inverse trigonometric function 2
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cabs hypot.3m complex absolute value 1
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cbrt sqrt.3m cube root 1
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ceil floor.3m integer no less than 0
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copysign ieee.3m copy sign bit 0
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cos sin.3m trigonometric function 1
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cosh sinh.3m hyperbolic function 3
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drem ieee.3m remainder 0
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erf erf.3m error function ???
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erfc erf.3m complementary error function ???
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exp exp.3m exponential 1
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expm1 exp.3m exp(x)\-1 1
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fabs floor.3m absolute value 0
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floor floor.3m integer no greater than 0
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hypot hypot.3m Euclidean distance 1
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infnan infnan.3m signals exceptions
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j0 j0.3m bessel function ???
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j1 j0.3m bessel function ???
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jn j0.3m bessel function ???
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lgamma lgamma.3m log gamma function; (formerly gamma.3m)
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log exp.3m natural logarithm 1
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logb ieee.3m exponent extraction 0
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log10 exp.3m logarithm to base 10 3
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log1p exp.3m log(1+x) 1
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pow exp.3m exponential x**y 60\-500
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rint floor.3m round to nearest integer 0
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scalb ieee.3m exponent adjustment 0
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sin sin.3m trigonometric function 1
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sinh sinh.3m hyperbolic function 3
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sqrt sqrt.3m square root 1
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tan sin.3m trigonometric function 3
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tanh sinh.3m hyperbolic function 3
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y0 j0.3m bessel function ???
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y1 j0.3m bessel function ???
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yn j0.3m bessel function ???
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.ta
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.fi
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.SH NOTES
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In 4.3 BSD, distributed from the University of California
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in late 1985, most of the foregoing functions come in two
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versions, one for the double\-precision "D" format in the
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DEC VAX\-11 family of computers, another for double\-precision
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arithmetic conforming to the IEEE Standard 754 for Binary
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Floating\-Point Arithmetic. The two versions behave very
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similarly, as should be expected from programs more accurate
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and robust than was the norm when UNIX was born. For
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instance, the programs are accurate to within the numbers
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of \*(ups tabulated above; an \*(up is one \fIU\fRnit in the \fIL\fRast
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\fIP\fRlace. And the programs have been cured of anomalies that
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afflicted the older math library \fIlibm\fR in which incidents like
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the following had been reported:
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.RS
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sqrt(\-1.0) = 0.0 and log(\-1.0) = \-1.7e38.
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.br
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cos(1.0e\-11) > cos(0.0) > 1.0.
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.br
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pow(x,1.0)
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.if n \
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!=
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.if t \
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\(!=
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x when x = 2.0, 3.0, 4.0, ..., 9.0.
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.br
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pow(\-1.0,1.0e10) trapped on Integer Overflow.
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.br
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sqrt(1.0e30) and sqrt(1.0e\-30) were very slow.
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.RE
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However the two versions do differ in ways that have to be
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explained, to which end the following notes are provided.
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.PP
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\fBDEC VAX\-11 D_floating\-point:\fR
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.PP
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This is the format for which the original math library \fIlibm\fR
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was developed, and to which this manual is still principally
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dedicated. It is \fIthe\fR double\-precision format for the PDP\-11
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and the earlier VAX\-11 machines; VAX\-11s after 1983 were
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provided with an optional "G" format closer to the IEEE
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double\-precision format. The earlier DEC MicroVAXs have no
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D format, only G double\-precision.
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(Why? Why not?)
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.PP
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Properties of D_floating\-point:
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.RS
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Wordsize: 64 bits, 8 bytes. Radix: Binary.
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.br
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Precision: 56
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.if n \
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sig.
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.if t \
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significant
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bits, roughly like 17
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.if n \
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sig.
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.if t \
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significant
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decimals.
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.RS
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If x and x' are consecutive positive D_floating\-point
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numbers (they differ by 1 \*(up), then
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.br
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1.3e\-17 < 0.5**56 < (x'\-x)/x \(<= 0.5**55 < 2.8e\-17.
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.RE
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.nf
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.ta \w'Range:'u+1n +\w'Underflow threshold'u+1n +\w'= 2.0**127'u+1n
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Range: Overflow threshold = 2.0**127 = 1.7e38.
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Underflow threshold = 0.5**128 = 2.9e\-39.
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NOTE: THIS RANGE IS COMPARATIVELY NARROW.
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.ta
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.fi
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.RS
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Overflow customarily stops computation.
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.br
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Underflow is customarily flushed quietly to zero.
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.br
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CAUTION:
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.RS
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It is possible to have x
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.if n \
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!=
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.if t \
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\(!=
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y and yet
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x\-y = 0 because of underflow. Similarly
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x > y > 0 cannot prevent either x\(**y = 0
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or y/x = 0 from happening without warning.
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.RE
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.RE
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Zero is represented ambiguously.
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.RS
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Although 2**55 different representations of zero are accepted by
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the hardware, only the obvious representation is ever produced.
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There is no \-0 on a VAX.
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.RE
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.If
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is not part of the VAX architecture.
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.br
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Reserved operands:
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.RS
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of the 2**55 that the hardware
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recognizes, only one of them is ever produced.
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Any floating\-point operation upon a reserved
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operand, even a MOVF or MOVD, customarily stops
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computation, so they are not much used.
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.RE
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Exceptions:
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.RS
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Divisions by zero and operations that
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overflow are invalid operations that customarily
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stop computation or, in earlier machines, produce
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reserved operands that will stop computation.
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.RE
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Rounding:
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.RS
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Every rational operation (+, \-, \(**, /) on a
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VAX (but not necessarily on a PDP\-11), if not an
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over/underflow nor division by zero, is rounded to
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within half an \*(up, and when the rounding error is
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exactly half an \*(up then rounding is away from 0.
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.RE
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.RE
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.PP
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Except for its narrow range, D_floating\-point is one of the
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better computer arithmetics designed in the 1960's.
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Its properties are reflected fairly faithfully in the elementary
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functions for a VAX distributed in 4.3 BSD.
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They over/underflow only if their results have to lie out of range
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or very nearly so, and then they behave much as any rational
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arithmetic operation that over/underflowed would behave.
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Similarly, expressions like log(0) and atanh(1) behave
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like 1/0; and sqrt(\-3) and acos(3) behave like 0/0;
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they all produce reserved operands and/or stop computation!
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The situation is described in more detail in manual pages.
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.RS
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.ll -0.5i
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\fIThis response seems excessively punitive, so it is destined
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to be replaced at some time in the foreseeable future by a
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more flexible but still uniform scheme being developed to
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handle all floating\-point arithmetic exceptions neatly.
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See infnan(3M) for the present state of affairs.\fR
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.ll +0.5i
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.RE
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.PP
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How do the functions in 4.3 BSD's new \fIlibm\fR for UNIX
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compare with their counterparts in DEC's VAX/VMS library?
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Some of the VMS functions are a little faster, some are
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a little more accurate, some are more puritanical about
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exceptions (like pow(0.0,0.0) and atan2(0.0,0.0)),
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and most occupy much more memory than their counterparts in
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\fIlibm\fR.
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The VMS codes interpolate in large table to achieve
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speed and accuracy; the \fIlibm\fR codes use tricky formulas
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compact enough that all of them may some day fit into a ROM.
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.PP
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More important, DEC regards the VMS codes as proprietary
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and guards them zealously against unauthorized use. But the
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\fIlibm\fR codes in 4.3 BSD are intended for the public domain;
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they may be copied freely provided their provenance is always
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acknowledged, and provided users assist the authors in their
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researches by reporting experience with the codes.
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Therefore no user of UNIX on a machine whose arithmetic resembles
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VAX D_floating\-point need use anything worse than the new \fIlibm\fR.
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.PP
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\fBIEEE STANDARD 754 Floating\-Point Arithmetic:\fR
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.PP
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This standard is on its way to becoming more widely adopted
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than any other design for computer arithmetic.
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VLSI chips that conform to some version of that standard have been
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produced by a host of manufacturers, among them ...
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.nf
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.ta 0.5i +\w'Intel i8070, i80287'u+6n
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Intel i8087, i80287 National Semiconductor 32081
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Motorola 68881 Weitek WTL-1032, ... , -1165
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Zilog Z8070 Western Electric (AT&T) WE32106.
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.ta
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.fi
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Other implementations range from software, done thoroughly
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in the Apple Macintosh, through VLSI in the Hewlett\-Packard
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9000 series, to the ELXSI 6400 running ECL at 3 Megaflops.
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Several other companies have adopted the formats
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of IEEE 754 without, alas, adhering to the standard's way
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of handling rounding and exceptions like over/underflow.
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The DEC VAX G_floating\-point format is very similar to the IEEE
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754 Double format, so similar that the C programs for the
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IEEE versions of most of the elementary functions listed
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above could easily be converted to run on a MicroVAX, though
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nobody has volunteered to do that yet.
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.PP
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The codes in 4.3 BSD's \fIlibm\fR for machines that conform to
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IEEE 754 are intended primarily for the National Semi. 32081
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and WTL 1164/65. To use these codes with the Intel or Zilog
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chips, or with the Apple Macintosh or ELXSI 6400, is to
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forego the use of better codes provided (perhaps freely) by
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those companies and designed by some of the authors of the
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codes above.
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Except for \fIatan\fR, \fIcabs\fR, \fIcbrt\fR, \fIerf\fR,
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\fIerfc\fR, \fIhypot\fR, \fIj0\-jn\fR, \fIlgamma\fR, \fIpow\fR
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and \fIy0\-yn\fR,
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the Motorola 68881 has all the functions in \fIlibm\fR on chip,
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and faster and more accurate;
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it, Apple, the i8087, Z8070 and WE32106 all use 64
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.if n \
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sig.
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.if t \
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significant
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bits.
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The main virtue of 4.3 BSD's
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\fIlibm\fR codes is that they are intended for the public domain;
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they may be copied freely provided their provenance is always
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acknowledged, and provided users assist the authors in their
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researches by reporting experience with the codes.
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Therefore no user of UNIX on a machine that conforms to
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IEEE 754 need use anything worse than the new \fIlibm\fR.
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.PP
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Properties of IEEE 754 Double\-Precision:
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.RS
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Wordsize: 64 bits, 8 bytes. Radix: Binary.
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.br
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Precision: 53
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.if n \
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sig.
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.if t \
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significant
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bits, roughly like 16
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.if n \
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sig.
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.if t \
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significant
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decimals.
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.RS
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If x and x' are consecutive positive Double\-Precision
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numbers (they differ by 1 \*(up), then
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.br
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1.1e\-16 < 0.5**53 < (x'\-x)/x \(<= 0.5**52 < 2.3e\-16.
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.RE
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.nf
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.ta \w'Range:'u+1n +\w'Underflow threshold'u+1n +\w'= 2.0**1024'u+1n
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Range: Overflow threshold = 2.0**1024 = 1.8e308
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Underflow threshold = 0.5**1022 = 2.2e\-308
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.ta
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.fi
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.RS
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Overflow goes by default to a signed
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.If "" .
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.br
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Underflow is \fIGradual,\fR rounding to the nearest
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integer multiple of 0.5**1074 = 4.9e\-324.
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.RE
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Zero is represented ambiguously as +0 or \-0.
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.RS
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Its sign transforms correctly through multiplication or
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division, and is preserved by addition of zeros
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with like signs; but x\-x yields +0 for every
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finite x. The only operations that reveal zero's
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sign are division by zero and copysign(x,\(+-0).
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In particular, comparison (x > y, x \(>= y, etc.)
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cannot be affected by the sign of zero; but if
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finite x = y then
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.If
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\&= 1/(x\-y)
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.if n \
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!=
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.if t \
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\(!=
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\-1/(y\-x) =
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.If \- .
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.RE
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.If
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is signed.
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.RS
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it persists when added to itself
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or to any finite number. Its sign transforms
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correctly through multiplication and division, and
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.If (finite)/\(+- \0=\0\(+-0
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(nonzero)/0 =
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.If \(+- .
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But
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.if n \
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Infinity\-Infinity, Infinity\(**0 and Infinity/Infinity
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.if t \
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\(if\-\(if, \(if\(**0 and \(if/\(if
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are, like 0/0 and sqrt(\-3),
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invalid operations that produce \*(nn. ...
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.RE
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Reserved operands:
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.RS
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there are 2**53\-2 of them, all
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called \*(nn (\fIN\fRot \fIa N\fRumber).
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Some, called Signaling \*(nns, trap any floating\-point operation
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performed upon them; they are used to mark missing
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or uninitialized values, or nonexistent elements
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of arrays. The rest are Quiet \*(nns; they are
|
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the default results of Invalid Operations, and
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propagate through subsequent arithmetic operations.
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If x
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.if n \
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!=
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.if t \
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\(!=
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x then x is \*(nn; every other predicate
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(x > y, x = y, x < y, ...) is FALSE if \*(nn is involved.
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.br
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NOTE: Trichotomy is violated by \*(nn.
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.RS
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Besides being FALSE, predicates that entail ordered
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comparison, rather than mere (in)equality,
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signal Invalid Operation when \*(nn is involved.
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.RE
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.RE
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Rounding:
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.RS
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Every algebraic operation (+, \-, \(**, /,
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.if n \
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sqrt)
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.if t \
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\(sr)
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is rounded by default to within half an \*(up, and
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when the rounding error is exactly half an \*(up then
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the rounded value's least significant bit is zero.
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This kind of rounding is usually the best kind,
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|
sometimes provably so; for instance, for every
|
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x = 1.0, 2.0, 3.0, 4.0, ..., 2.0**52, we find
|
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(x/3.0)\(**3.0 == x and (x/10.0)\(**10.0 == x and ...
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|
despite that both the quotients and the products
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|
have been rounded. Only rounding like IEEE 754
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|
can do that. But no single kind of rounding can be
|
|
proved best for every circumstance, so IEEE 754
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|
provides rounding towards zero or towards
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|
.If +
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or towards
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.If \-
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at the programmer's option. And the
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|
same kinds of rounding are specified for
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|
Binary\-Decimal Conversions, at least for magnitudes
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|
between roughly 1.0e\-10 and 1.0e37.
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.RE
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|
Exceptions:
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.RS
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|
IEEE 754 recognizes five kinds of floating\-point exceptions,
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listed below in declining order of probable importance.
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.RS
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.nf
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|
.ta \w'Invalid Operation'u+6n +\w'Gradual Underflow'u+2n
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Exception Default Result
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.sp 0.5
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Invalid Operation \*(nn, or FALSE
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.if n \{\
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Overflow \(+-Infinity
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Divide by Zero \(+-Infinity \}
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.if t \{\
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Overflow \(+-\(if
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Divide by Zero \(+-\(if \}
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Underflow Gradual Underflow
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Inexact Rounded value
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.ta
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.fi
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.RE
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|
NOTE: An Exception is not an Error unless handled
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|
badly. What makes a class of exceptions exceptional
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|
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,
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|
the unsatisfactory instances cannot justify aborting
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computation every time the exception occurs.
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.RE
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.PP
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|
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 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
|
|
.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 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
|
|
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.
|