Sanitize the markup, as prompted.

1 files changed, 173 insertions, 182 deletions

diff --git a/lib/msun/man/math.3 b/lib/msun/man/math.3
index 52c50a8..9297982 100644
--- a/lib/msun/man/math.3
+++ b/lib/msun/man/math.3
@@ -35,15 +35,13 @@
 .Dd June 11, 2004
 .Dt MATH 3
 .Os
-.ds up \fIulp\fR
-.de If
-.if n \\
-\\$1Infinity\\$2
-.if t \\
-\\$1\\(if\\$2
-..
+.if n \{\
+.char \[if] "Infinity
+.char \[sr] "sqrt
+.\}
 .Sh NAME
-math \- floating-point mathematical library
+.Nm math
+.Nd "floating-point mathematical library"
 .Sh LIBRARY
 .Lb libm
 .Sh SYNOPSIS
@@ -58,14 +56,14 @@ functions has a
 counterpart with an
 .Ql f
 appended to the name and a
-.Vt long double
+.Vt "long double"
 counterpart with an
 .Ql l
 appended.
 As an example, the
 .Vt float
 and
-.Vt long double
+.Vt "long double"
 counterparts of
 .Ft double
 .Fn acos "double x"
@@ -73,19 +71,22 @@ are
 .Ft float
 .Fn acosf "float x"
 and
-.Ft long double
+.Ft "long double"
 .Fn acosl "long double x" ,
 respectively.
 .Pp
 The programs are accurate to within the numbers
-of \*(ups tabulated below; an \*(up is one \fIU\fRnit in the \fIL\fRast
-\fIP\fRlace.
-.sp 2
-.nf
-.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
+of
+.Em ulp Ns s
+tabulated below; an
+.Em ulp
+is one
+.Em U Ns nit
+in the
+.Em L Ns ast
+.Em P Ns lace .
+.Bl -column "nexttoward" "remainder with partial quotient"
+.Em "Name	Description	Error Bound (ULPs)"
 .\" XXX Many of these error bounds are wrong for the current implementation!
 acos	inverse trigonometric function	???
 acosh	inverse hyperbolic function	???
@@ -133,7 +134,7 @@ modf	extract fractional part	0
 nearbyint	round to integer	0
 nextafter	next representable value	0
 .\" nexttoward	next representable value	0
-pow	exponential x**y	60\-500
+pow	exponential x**y	60-500
 remainder	remainder	0
 .\" remquo	remainder with partial quotient	???
 rint	round to nearest integer	0
@@ -150,8 +151,7 @@ trunc	round towards zero	0
 y0	bessel function	???
 y1	bessel function	???
 yn	bessel function	???
-.ta
-.fi
+.El
 .Sh NOTES
 Virtually all modern floating-point units attempt to support
 IEEE Standard 754 for Binary Floating-Point Arithmetic.
@@ -161,168 +161,144 @@ except for the few documented in
 it primarily defines representations of numbers and abstract
 properties of arithmetic operations relating to precision, rounding,
 and exceptional cases, as described below.
+.Ss IEEE STANDARD 754 Floating-Point Arithmetic
+.\" XXX mention single- and extended-/quad- precisions
+Properties of IEEE 754 Double-Precision:
+.Bd -ragged -offset indent -compact
+Wordsize: 64 bits, 8 bytes.
 .Pp
-\fBIEEE STANDARD 754 Floating\-Point Arithmetic:\fR
+Radix: Binary.
 .Pp
-.\" XXX mention single- and extended-/quad- precisions
-Properties of IEEE 754 Double\-Precision:
-.Bd -filled -offset indent
-Wordsize: 64 bits, 8 bytes.  Radix: Binary.
-.br
-Precision: 53
-.if n \
-sig.
-.if t \
-significant
-bits, roughly like 16
-.if n \
-sig.
-.if t \
-significant
-decimals.
-.Bd -filled -offset indent -compact
-If x and x' are consecutive positive Double\-Precision
-numbers (they differ by 1 \*(up), then
-.br
+Precision: 53 significant bits,
+roughly like 16 significant decimals.
+.Bd -ragged -offset indent -compact
+If x and x' are consecutive positive Double-Precision
+numbers (they differ by 1
+.Em ulp ) ,
+then
+.Bd -ragged -compact
 1.1e\-16 < 0.5**53 < (x'\-x)/x \(<= 0.5**52 < 2.3e\-16.
 .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
-.Bd -filled -offset indent -compact
-Overflow goes by default to a signed
-.If "" .
-.br
-Underflow is \fIGradual,\fR rounding to the nearest
+.Ed
+.Pp
+.Bl -column "XXX" -compact
+Range:	Overflow threshold  = 2.0**1024 = 1.8e308
+	Underflow threshold = 0.5**1022 = 2.2e\-308
+.El
+.Bd -ragged -offset indent -compact
+Overflow goes by default to a signed \(if.
+Underflow is
+.Em Gradual ,
+rounding to the nearest
 integer multiple of 0.5**1074 = 4.9e\-324.
 .Ed
+.Pp
 Zero is represented ambiguously as +0 or \-0.
-.Bd -filled -offset indent -compact
+.Bd -ragged -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
-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.)
+finite x.
+The only operations that reveal zero's
+sign are division by zero and
+.Fn 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 \- .
+finite x = y then \(if = 1/(x\-y) \(!= \-1/(y\-x) = \-\(if.
 .Ed
-.If
-is signed.
-.Bd -filled -offset indent -compact
-it persists when added to itself
-or to any finite number.  Its sign transforms
+.Pp
+\(if is signed.
+.Bd -ragged -offset indent -compact
+It persists when added to itself
+or to any finite number.
+Its sign transforms
 correctly through multiplication and division, and
-.If (finite)/\(+- \0=\0\(+-0
-(nonzero)/0 =
-.If \(+- .
+(finite)/\(+-\(if\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 \*(Na. ...
 .Ed
+.Pp
 Reserved operands:
-.Bd -filled -offset indent -compact
+.Bd -ragged -offset indent -compact
 there are 2**53\-2 of them, all
-called \*(Na (\fIN\fRot \fIa N\fRumber).
-Some, called Signaling \*(Nas, trap any floating\-point operation
+called \*(Na
+.Em ( N Ns ot Em a N Ns umber ) .
+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 \*(Nas; 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
-.if n \
-!=
-.if t \
-\(!=
-x then x is \*(Na; every other predicate
+If x \(!= x then x is \*(Na; every other predicate
 (x > y, x = y, x < y, ...) is FALSE if \*(Na is involved.
-.br
+.Pp
 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 \*(Na is involved.
 .Ed
-.Ed
+.Pp
 Rounding:
-.Bd -filled -offset indent -compact
+.Bd -ragged -offset indent -compact
 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
+is rounded by default to within half an
+.Em ulp ,
+and when the rounding error is exactly half an
+.Em ulp
+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
+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
++\(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
+Binary-Decimal Conversions, at least for magnitudes
 between roughly 1.0e\-10 and 1.0e37.
 .Ed
+.Pp
 Exceptions:
-.Bd -filled -offset indent -compact
-IEEE 754 recognizes five kinds of floating\-point exceptions,
+.Bd -ragged -offset indent -compact
+IEEE 754 recognizes five kinds of floating-point exceptions,
 listed below in declining order of probable importance.
-.Bd -filled -offset indent -compact
-.nf
-.ta \w'Invalid Operation'u+6n +\w'Gradual Underflow'u+2n
-Exception	Default Result
-.tc \(ru
-
-.tc
+.Bl -column -offset indent "Invalid Operation" "Gradual Underflow"
+.Em "Exception	Default Result"
 Invalid Operation	\*(Na, or FALSE
-.if n \{\
-Overflow	\(+-Infinity
-Divide by Zero	\(+-Infinity \}
-.if t \{\
 Overflow	\(+-\(if
-Divide by Zero	\(+-\(if \}
+Divide by Zero	\(+-\(if
 Underflow	Gradual Underflow
 Inexact	Rounded value
-.ta
-.fi
-.Ed
-NOTE:  An Exception is not an Error unless handled
-badly.  What makes a class of exceptions exceptional
+.El
+.Pp
+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
+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.
 .Ed
 .Pp
-For each kind of floating\-point exception, IEEE 754
+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.
+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:
@@ -336,20 +312,17 @@ since the program last reset its flag.
 .It
 Test a result to see whether it is a value that only
 an exception could have produced.
-.RS
+.Pp
 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
+flag.
+(Sums and differences cannot underflow in
+IEEE 754; if x \(!= y then x\-y is correct to
 full precision and certainly nonzero regardless of
-how tiny it may be.)  Products and quotients that
+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.
@@ -358,19 +331,22 @@ 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.
+So gradual underflows are usually
+.Em provably
+ignorable.
 The same cannot be said of underflows flushed to 0.
-.RE
 .El
 .Pp
 At the option of an implementor conforming to IEEE 754,
 other ways to cope with exceptions may be provided:
-.Bl -hang -width 3n
-.It 4.
-ABORT.  This mechanism classifies an exception in
+.Bl -enum
+.It
+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
+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:
 .Bl -dash
@@ -378,28 +354,32 @@ but most share the following characteristics:
 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.
+expression.
+An exceptional result is abandoned.
 .It
-In a subprogram that lacks an error\-handling
+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
+an error-handling statement is encountered or the
 whole task is aborted and memory is dumped.
 .El
-.It 5.
-STOP.  This mechanism, requiring an interactive
+.It
+STOP.
+This mechanism, requiring an interactive
 debugging environment, is more for the programmer
-than the program.  It classifies an exception in
+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
+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.
-.It 6.
+.It
 \&... Other ways lie beyond the scope of this document.
 .El
 .Ed
@@ -407,14 +387,14 @@ execution had not been stopped.
 Ideally, each
 elementary function should act as if it were indivisible, or
 atomic, in the sense that ...
-.Bl -tag -width "iii)"
-.It i)
+.Bl -enum
+.It
 No exception should be signaled that is not deserved by
 the data supplied to that function.
-.It ii)
+.It
 Any exception signaled should be identified with that
 function rather than with one of its subroutines.
-.It iii)
+.It
 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
@@ -423,51 +403,60 @@ of the function may be correlated intentionally
 with exception handling.
 .El
 .Pp
-The functions in \fIlibm\fR are only approximately atomic.
+The functions in
+.Nm libm
+are only approximately atomic.
 They signal no inappropriate exception except possibly ...
-.Bd -filled -offset indent -compact
+.Bl -tag -width indent -offset indent -compact
+.It Xo
 Over/Underflow
-.Bd -filled -offset indent -compact
+.Xc
 when a result, if properly computed, might have lain barely within range, and
-.Ed
-Inexact in \fIcabs\fR, \fIcbrt\fR, \fIhypot\fR, \fIlog10\fR and \fIpow\fR
-.Bd -filled -offset indent -compact
+.It Xo
+Inexact in
+.Fn cabs ,
+.Fn cbrt ,
+.Fn hypot ,
+.Fn log10
+and
+.Fn pow
+.Xc
 when it happens to be exact, thanks to fortuitous cancellation of errors.
-.Ed
-.Ed
+.El
 Otherwise, ...
-.Bd -filled -offset indent -compact
+.Bl -tag -width indent -offset indent -compact
+.It Xo
 Invalid Operation is signaled only when
-.Bd -filled -offset indent -compact
+.Xc
 any result but \*(Na would probably be misleading.
-.Ed
+.It Xo
 Overflow is signaled only when
-.Bd -filled -offset indent -compact
+.Xc
 the exact result would be finite but beyond the overflow threshold.
-.Ed
-Divide\-by\-Zero is signaled only when
-.Bd -filled -offset indent -compact
+.It Xo
+Divide-by-Zero is signaled only when
+.Xc
 a function takes exactly infinite values at finite operands.
-.Ed
+.It Xo
 Underflow is signaled only when
-.Bd -filled -offset indent -compact
+.Xc
 the exact result would be nonzero but tinier than the underflow threshold.
-.Ed
+.It Xo
 Inexact is signaled only when
-.Bd -filled -offset indent -compact
+.Xc
 greater range or precision would be needed to represent the exact result.
-.Ed
-.Ed
+.El
 .Sh BUGS
 Several functions required by
 .St -isoC-99
 are missing, and many functions are not available in their
-.Vt long double
+.Vt "long double"
 variants.
 .Pp
 On some architectures, trigonometric argument reduction is not
-performed accurately, resulting in errors greater than 1 ulp for large
-arguments to
+performed accurately, resulting in errors greater than 1
+.Em ulp
+for large arguments to
 .Fn cos ,
 .Fn sin ,
 and
@@ -478,8 +467,8 @@ and
 .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 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.\&
@@ -488,8 +477,10 @@ 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
+.At v7 .
+The library was substantially rewritten for
+.Bx 4.3
+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