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from: @(#)math.3 6.10 (Berkeley) 5/6/91
$FreeBSD: head/lib/msun/man/math.3 140355 2005-01-16 16:49:10Z ru $
.Dd January 11, 2005 .Dt MATH 3 .Os .char \[sr] "sqrt .\} .Sh NAME .Nm math .Nd "floating-point mathematical library" .Sh LIBRARY .Lb libm .Sh SYNOPSIS n 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.
p
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 .
l -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:
d -ragged -offset indent -compact Wordsize: 64 bits, 8 bytes.
p Radix: Binary.
p Precision: 53 significant bits, roughly like 16 significant decimals. d -ragged -offset indent -compact If x and x' are consecutive positive Double-Precision numbers (they differ by 1 .Em ulp ) , then d -ragged -compact 1.1e-16 < 0.5**53 < (x'-x)/x \(<= 0.5**52 < 2.3e-16. .Ed .Ed
p l -column "XXX" -compact Range: Overflow threshold = 2.0**1024 = 1.8e308 Underflow threshold = 0.5**1022 = 2.2e-308 .El d -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
p Zero is represented ambiguously as +0 or -0. d -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
p \*(If is signed. d -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
p Reserved operands: d -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.
p 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
p Rounding: d -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
p Exceptions: d -ragged -offset indent -compact IEEE 754 recognizes five kinds of floating-point exceptions, listed below in declining order of probable importance. l -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
p 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
p 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: l -enum t Test for a condition that might cause an exception later, and branch to avoid the exception. t Test a flag to see whether an exception has occurred since the program last reset its flag. t Test a result to see whether it is a value that only an exception could have produced.
p 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
p At the option of an implementor conforming to IEEE 754, other ways to cope with exceptions may be provided: l -enum t 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: l -dash t 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. t 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 t 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. t ... Other ways lie beyond the scope of this document. .El .Ed
p Ideally, each elementary function should act as if it were indivisible, or atomic, in the sense that ... l -enum t No exception should be signaled that is not deserved by the data supplied to that function. t Any exception signaled should be identified with that function rather than with one of its subroutines. t 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
p The functions in .Nm libm are only approximately atomic. They signal no inappropriate exception except possibly ... l -tag -width indent -offset indent -compact t Xo Over/Underflow .Xc when a result, if properly computed, might have lain barely within range, and t 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, ... l -tag -width indent -offset indent -compact t Xo Invalid Operation is signaled only when .Xc any result but \*(Na would probably be misleading. t Xo Overflow is signaled only when .Xc the exact result would be finite but beyond the overflow threshold. t Xo Divide-by-Zero is signaled only when .Xc a function takes exactly infinite values at finite operands. t Xo Underflow is signaled only when .Xc the exact result would be nonzero but tinier than the underflow threshold. t Xo Inexact is signaled only when .Xc greater range or precision would be needed to represent the exact result. .El .Sh BUGS Several functions required by .St -isoC-99 are missing, and many functions are not available in their .Vt "long double" variants.
p 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 . .Sh SEE ALSO .Xr fenv 3 , .Xr ieee 3
p 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 x 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 .