xref: /freebsd/lib/msun/man/math.3 (revision 24a0682c6465290759ed0b09ea16e40e7cd47053)

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.\"	from: @(#)math.3	6.10 (Berkeley) 5/6/91
.\" $FreeBSD$
.\"
.Dd January 11, 2005
.Dt MATH 3
.Os
.if n \{\
.char \[sr] "sqrt
.\}
.Sh NAME
.Nm math
.Nd "floating-point mathematical library"
.Sh LIBRARY
.Lb libm
.Sh SYNOPSIS
.In math.h
.Sh DESCRIPTION
These functions constitute the C math library.
.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.
.Pp
The programs are accurate to within the numbers
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	???
asin	inverse trigonometric function	???
asinh	inverse hyperbolic function	???
atan	inverse trigonometric function	???
atanh	inverse hyperbolic function	???
atan2	inverse trigonometric function	???
cbrt	cube root	1
ceil	integer no less than	0
copysign	copy sign bit	0
cos	trigonometric function	1
cosh	hyperbolic function	???
erf	error function	1
erfc	complementary error function	1
exp	exponential base e	1
.\" exp2	exponential base 2	???
expm1	exp(x)\-1	1
fabs	absolute value	0
fdim	positive difference	1
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	1
llrint	round to integer	0
llround	round to nearest integer	0
log	natural logarithm	1
log10	logarithm to base 10	1
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	0
.\" 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	???
sqrt	square root	1
tan	trigonometric function	1
tanh	hyperbolic function	???
tgamma	gamma function	1
trunc	round towards zero	0
y0	bessel function	???
y1	bessel function	???
yn	bessel function	???
.El
.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.
.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
Radix: Binary.
.Pp
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
.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 -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
.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) \(!= \-1/(y\-x) = \-\*(If.
.Ed
.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
(finite)/\(+-\*(If\0=\0\(+-0
(nonzero)/0 = \(+-\*(If.
But
\*(If\-\*(If, \*(If\(**0 and \*(If/\*(If
are, like 0/0 and sqrt(\-3),
invalid operations that produce \*(Na. ...
.Ed
.Pp
Reserved operands:
.Bd -ragged -offset indent -compact
there are 2**53\-2 of them, all
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
the default results of Invalid Operations, and
propagate through subsequent arithmetic operations.
If x \(!= x then x is \*(Na; every other predicate
(x > y, x = y, x < y, ...) is FALSE if \*(Na is involved.
.Pp
NOTE: Trichotomy is violated by \*(Na.
Besides being FALSE, predicates that entail ordered
comparison, rather than mere (in)equality,
signal Invalid Operation when \*(Na is involved.
.Ed
.Pp
Rounding:
.Bd -ragged -offset indent -compact
Every algebraic operation (+, \-, \(**, /,
\(sr)
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
proved best for every circumstance, so IEEE 754
provides rounding towards zero or towards
+\*(If or towards \-\*(If
at the programmer's option.
And the
same kinds of rounding are specified for
Binary-Decimal Conversions, at least for magnitudes
between roughly 1.0e\-10 and 1.0e37.
.Ed
.Pp
Exceptions:
.Bd -ragged -offset indent -compact
IEEE 754 recognizes five kinds of floating-point exceptions,
listed below in declining order of probable importance.
.Bl -column -offset indent "Invalid Operation" "Gradual Underflow"
.Em "Exception	Default Result"
Invalid Operation	\*(Na, or FALSE
Overflow	\(+-\*(If
Divide by Zero	\(+-\*(If
Underflow	Gradual Underflow
Inexact	Rounded value
.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
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
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:
.Bl -enum
.It
Test for a condition that might cause an exception
later, and branch to avoid the exception.
.It
Test a flag to see whether an exception has occurred
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.
.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 \(!= 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
.Em provably
ignorable.
The same cannot be said of underflows flushed to 0.
.El
.Pp
At the option of an implementor conforming to IEEE 754,
other ways to cope with exceptions may be provided:
.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
languages offer different forms of this statement,
but most share the following characteristics:
.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.
.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.
.El
.It
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.
.It
\&... Other ways lie beyond the scope of this document.
.El
.Ed
.Pp
Ideally, each
elementary function should act as if it were indivisible, or
atomic, in the sense that ...
.Bl -enum
.It
No exception should be signaled that is not deserved by
the data supplied to that function.
.It
Any exception signaled should be identified with that
function rather than with one of its subroutines.
.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
exceptions listed above, although the definition
of the function may be correlated intentionally
with exception handling.
.El
.Pp
The functions in
.Nm libm
are only approximately atomic.
They signal no inappropriate exception except possibly ...
.Bl -tag -width indent -offset indent -compact
.It Xo
Over/Underflow
.Xc
when a result, if properly computed, might have lain barely within range, and
.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.
.El
Otherwise, ...
.Bl -tag -width indent -offset indent -compact
.It Xo
Invalid Operation is signaled only when
.Xc
any result but \*(Na would probably be misleading.
.It Xo
Overflow is signaled only when
.Xc
the exact result would be finite but beyond the overflow threshold.
.It Xo
Divide-by-Zero is signaled only when
.Xc
a function takes exactly infinite values at finite operands.
.It Xo
Underflow is signaled only when
.Xc
the exact result would be nonzero but tinier than the underflow threshold.
.It Xo
Inexact is signaled only when
.Xc
greater range or precision would be needed to represent the exact result.
.El
.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
.An "W. J. Cody"
et al.
The manuals for Pascal, C and BASIC on the Apple Macintosh
document the features of IEEE 754 pretty well.
Articles in the IEEE magazine COMPUTER vol.\& 14 no.\& 3 (Mar.\&
1981), and in the ACM SIGNUM Newsletter Special Issue of
Oct.\& 1979, may be helpful although they pertain to
superseded drafts of the standard.
.Sh HISTORY
A math library with many of the present functions appeared in
.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
Microsystems, in
.Fx 1.1.5 .
.Sh BUGS
Several functions required by
.St -isoC-99
are missing, and many functions are not available in their
.Vt "long double"
variants.
.Pp
On some architectures, trigonometric argument reduction is not
performed accurately, resulting in errors greater than 1
.Em ulp
for large arguments to
.Fn cos ,
.Fn sin ,
and
.Fn tan .