Memory's free, but all the world ain't a VAX anymore. Bring math.3

kicking and screaming into the 1980's.  This change converts most of
the markup from man(7) to mdoc(7) format, and I believe it removes or
updates everything that was flat out wrong.  However, much work is
still needed to sanitize the markup, improve coverage, and reduce
overlap with other manpages.  Some of the sections would better belong
in a philosophy_of_w_kahan.3 manpage, but they are informative and
remain at least as reminders of topics to cover.

Reviewed by:	doc@, trhodes@

lib/msun/man/math.3

@@ -32,295 +32,139 @@
 .\"	from: @(#)math.3	6.10 (Berkeley) 5/6/91
 .\" $FreeBSD$
 .\"
-.TH MATH 3M "May 6, 1991"
-.UC 4
+.Dd June 11, 2004
+.Dt MATH 3
+.Os
 .ds up \fIulp\fR
-.ds nn \fINaN\fR
 .de If
 .if n \\
 \\$1Infinity\\$2
 .if t \\
 \\$1\\(if\\$2
 ..
-.SH NAME
+.Sh NAME
 math \- introduction to mathematical library functions
-.SH DESCRIPTION
+.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"
-Each of the following double functions has a float counterpart with the
-name ending in f, as an example the float counterpart of double acos(double
-x) is float acosf(float x).
+.In math.h .
+.Sh "LIST OF FUNCTIONS"
+Each of the following
+.Vt double
+functions has a
+.Vt float
+counterpart with an
+.Ql f
+appended to the name and a
+.Vt long double
+counterpart with an
+.Ql l
+appended.
+As an example, the
+.Vt float
+and
+.Vt long double
+counterparts of
+.Ft double
+.Fn acos "double x"
+are
+.Ft float
+.Fn acosf "float x"
+and
+.Ft long double
+.Fn acosl "long double x" ,
+respectively.
 .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
+.ta \w'nexttoward'u+10n +\w'remainder with partial quot'u
+\fIName\fP	\fIDescription\fP	\fIError Bound (ULPs)\fP
+.ta \w'nexttoward'u+4n +\w'remainder with partial quotient'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
-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
-ilogb	ieee.3m	exponent extraction	0
-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
-log10	exp.3m	logarithm to base 10	3
-log1p	exp.3m	log(1+x)	1
-pow	exp.3m	exponential x**y	60\-500
-remainder	ieee.3m	remainder	0
-rint	floor.3m	round to nearest integer	0
-scalbn	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	???
+.\" XXX Many of these error bounds are wrong for the current implementation!
+acos	inverse trigonometric function	3
+acosh	inverse hyperbolic function	3
+asin	inverse trigonometric function	3
+asinh	inverse hyperbolic function	3
+atan	inverse trigonometric function	1
+atanh	inverse hyperbolic function	3
+atan2	inverse trigonometric function	2
+cbrt	cube root	1
+ceil	integer no less than	0
+copysign	copy sign bit	0
+cos	trigonometric function	1
+cosh	hyperbolic function	3
+erf	error function	???
+erfc	complementary error function	???
+exp	exponential base e	1
+.\" exp2	exponential base 2	???
+expm1	exp(x)\-1	1
+fabs	absolute value	0
+.\" fdim	positive difference	???
+floor	integer no greater than	0
+.\" fma	multiply-add	???
+.\" fmax	maximum function	0
+.\" fmin	minimum function	0
+fmod	remainder function	???
+frexp	extract mantissa and exponent	0
+hypot	Euclidean distance	1
+ilogb	exponent extraction	0
+j0	bessel function	???
+j1	bessel function	???
+jn	bessel function	???
+ldexp	multiply by power of 2	0
+lgamma	log gamma function	???
+.\" llrint	round to integer	0
+.\" llround	round to nearest integer	0
+log	natural logarithm	1
+log10	logarithm to base 10	3
+log1p	log(1+x)	1
+.\" log2	base 2 logarithm	0
+logb	exponent extraction	0
+.\" lrint	round to integer	0
+.\" lround	round to nearest integer	0
+modf	extract fractional part	???
+.\" nan	return quiet \*(Na)	0
+.\" nearbyint	round to integer	0
+.\" nextafter	next representable value	0
+.\" nexttoward	next representable value	0
+pow	exponential x**y	60\-500
+remainder	remainder	0
+.\" remquo	remainder with partial quotient	???
+rint	round to nearest integer	0
+round	round to nearest integer	0
+.\" scalbln	exponent adjustment	0
+scalbn	exponent adjustment	0
+sin	trigonometric function	1
+sinh	hyperbolic function	3
+sqrt	square root	1
+tan	trigonometric function	3
+tanh	hyperbolic function	3
+tgamma	gamma function	???
+.\" trunc	round towards zero	0
+y0	bessel function	???
+y1	bessel function	???
+yn	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
+.Sh NOTES
+Virtually all modern floating-point units attempt to support
+IEEE Standard 754 for Binary Floating-Point Arithmetic.
+This standard does not cover particular routines in the math library
+except for the few documented in
+.Xr ieee 3 ;
+it primarily defines representations of numbers and abstract
+properties of arithmetic operations relating to precision, rounding,
+and exceptional cases, as described below.
+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
+\fIP\fRlace.
+.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
+.Pp
 Properties of IEEE 754 Double\-Precision:
-.RS
+.Bd -filled -offset indent
 Wordsize: 64 bits, 8 bytes.  Radix: Binary.
 .br
 Precision: 53
@@ -334,27 +178,27 @@ sig.
 .if t \
 significant
 decimals.
-.RS
+.Bd -filled -offset indent -compact
 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
+.Ed
 .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
+.Bd -filled -offset indent -compact
 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
+.Ed
 Zero is represented ambiguously as +0 or \-0.
-.RS
+.Bd -filled -offset indent -compact
 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
@@ -371,10 +215,10 @@ finite x = y then
 \(!=
 \-1/(y\-x) =
 .If \- .
-.RE
+.Ed
 .If
 is signed.
-.RS
+.Bd -filled -offset indent -compact
 it persists when added to itself
 or to any finite number.  Its sign transforms
 correctly through multiplication and division, and
@@ -387,16 +231,16 @@ 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
+invalid operations that produce \*(Na. ...
+.Ed
 Reserved operands:
-.RS
+.Bd -filled -offset indent -compact
 there are 2**53\-2 of them, all
-called \*(nn (\fIN\fRot \fIa N\fRumber).
-Some, called Signaling \*(nns, trap any floating\-point operation
+called \*(Na (\fIN\fRot \fIa N\fRumber).
+Some, called Signaling \*(Nas, 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
+of arrays.  The rest are Quiet \*(Nas; they are
 the default results of Invalid Operations, and
 propagate through subsequent arithmetic operations.
 If x
@@ -404,18 +248,18 @@ If x
 !=
 .if t \
 \(!=
-x then x is \*(nn; every other predicate
-(x > y, x = y, x < y, ...) is FALSE if \*(nn is involved.
+x then x is \*(Na; every other predicate
+(x > y, x = y, x < y, ...) is FALSE if \*(Na is involved.
 .br
-NOTE: Trichotomy is violated by \*(nn.
-.RS
+NOTE: Trichotomy is violated by \*(Na.
+.Bd -filled -offset indent -compact
 Besides being FALSE, predicates that entail ordered
 comparison, rather than mere (in)equality,
-signal Invalid Operation when \*(nn is involved.
-.RE
-.RE
+signal Invalid Operation when \*(Na is involved.
+.Ed
+.Ed
 Rounding:
-.RS
+.Bd -filled -offset indent -compact
 Every algebraic operation (+, \-, \(**, /,
 .if n \
 sqrt)
@@ -440,19 +284,19 @@ 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
+.Ed
 Exceptions:
-.RS
+.Bd -filled -offset indent -compact
 IEEE 754 recognizes five kinds of floating\-point exceptions,
 listed below in declining order of probable importance.
-.RS
+.Bd -filled -offset indent -compact
 .nf
 .ta \w'Invalid Operation'u+6n +\w'Gradual Underflow'u+2n
 Exception	Default Result
 .tc \(ru
 
 .tc
-Invalid Operation	\*(nn, or FALSE
+Invalid Operation	\*(Na, or FALSE
 .if n \{\
 Overflow	\(+-Infinity
 Divide by Zero	\(+-Infinity \}
@@ -463,7 +307,7 @@ Underflow	Gradual Underflow
 Inexact	Rounded value
 .ta
 .fi
-.RE
+.Ed
 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
@@ -471,8 +315,8 @@ 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
+.Ed
+.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
@@ -480,13 +324,14 @@ 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
+.Bl -enum
+.It
 Test for a condition that might cause an exception
 later, and branch to avoid the exception.
-.IP 2) \w'\0\0\0\0'u
+.It
 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
+.It
 Test a result to see whether it is a value that only
 an exception could have produced.
 .RS
@@ -514,29 +359,33 @@ 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
+.El
+.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
+.Bl -hang -width 3n
+.It 4.
 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
+.Bl -dash
+.It
 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
+.It
 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
+.El
+.It 5.
 STOP.  This mechanism, requiring an interactive
 debugging environment, is more for the programmer
 than the program.  It classifies an exception in
@@ -548,98 +397,91 @@ 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
+.It 6.
 \&... 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
+.El
+.Ed
+.Pp
+Ideally, each
 elementary function should act as if it were indivisible, or
 atomic, in the sense that ...
-.IP i) \w'iii)'u+2n
+.Bl -tag -width "iii)"
+.It i)
 No exception should be signaled that is not deserved by
 the data supplied to that function.
-.IP ii) \w'iii)'u+2n
+.It ii)
 Any exception signaled should be identified with that
 function rather than with one of its subroutines.
-.IP iii) \w'iii)'u+2n
+.It iii)
 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
+.El
+.Pp
+The functions in \fIlibm\fR are only approximately atomic.
+They signal no inappropriate exception except possibly ...
+.Bd -filled -offset indent -compact
 Over/Underflow
-.RS
+.Bd -filled -offset indent -compact
 when a result, if properly computed, might have lain barely within range, and
-.RE
+.Ed
 Inexact in \fIcabs\fR, \fIcbrt\fR, \fIhypot\fR, \fIlog10\fR and \fIpow\fR
-.RS
+.Bd -filled -offset indent -compact
 when it happens to be exact, thanks to fortuitous cancellation of errors.
-.RE
-.RE
+.Ed
+.Ed
 Otherwise, ...
-.RS
+.Bd -filled -offset indent -compact
 Invalid Operation is signaled only when
-.RS
-any result but \*(nn would probably be misleading.
-.RE
+.Bd -filled -offset indent -compact
+any result but \*(Na would probably be misleading.
+.Ed
 Overflow is signaled only when
-.RS
+.Bd -filled -offset indent -compact
 the exact result would be finite but beyond the overflow threshold.
-.RE
+.Ed
 Divide\-by\-Zero is signaled only when
-.RS
+.Bd -filled -offset indent -compact
 a function takes exactly infinite values at finite operands.
-.RE
+.Ed
 Underflow is signaled only when
-.RS
+.Bd -filled -offset indent -compact
 the exact result would be nonzero but tinier than the underflow threshold.
-.RE
+.Ed
 Inexact is signaled only when
-.RS
+.Bd -filled -offset indent -compact
 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
-\fBfpgetround\fR(3),
-\fBfpsetround\fR(3),
-\fBfpgetprec\fR(3),
-\fBfpsetprec\fR(3),
-\fBfpgetmask\fR(3),
-\fBfpsetmask\fR(3),
-\fBfpgetsticky\fR(3),
-\fBfpresetsticky\fR(3) - IEEE floating point interface
-.SH NOTES
+.Ed
+.Ed
+.Sh BUGS
+Several functions required by
+.St -isoC-99
+are missing, and many functions are not available in their
+.Vt long double
+variants.
+.Sh SEE ALSO
+.Xr fenv 3 ,
+.Xr ieee 3
+.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.
 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.
+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 HISTORY
+A math library with many of the present functions appeared in
+Version 7 AT&T UNIX.
+The library was substantially rewritten for 4.3BSD to provide
+better accuracy and speed on machines supporting either VAX
+or IEEE 754 floating-point.
+Most of this library was replaced with FDLIBM, developed at Sun
+Microsystems, in
+.Fx 1.1.5 .