math(3)
NAME
math - introduction to mathematical library functions
DESCRIPTION
These functions constitute the C math library, libm. This library is
based Sun Microsystems FDLIBM (Freely Distributable LIBM), a C math
library for machines that support IEEE 754 floating-point arithmetic.
Declarations for the functions in this library may be obtained from the
include file <math.h>.
LIST OF FUNCTIONS
Name Appears on Page Description
acos sin(3) inverse trigonometric function
acosh asinh(3) inverse hyperbolic function
asin sin(3) inverse trigonometric function
asinh asinh(3) inverse hyperbolic function
atan sin(3) inverse trigonometric function
atanh asinh(3) inverse hyperbolic function
atan2 sin(3) inverse trigonometric function
cbrt sqrt(3) cube root
ceil floor(3) integer no less than
copysign ieee(3) copy sign bit
cos sin(3) trigonometric function
cosh sinh(3) hyperbolic function
remainder ieee(3) remainder
erf erf(3) error function
erfc erf(3) complementary error function
exp exp(3) exponential
expm1 exp(3) exp(x)-1
fabs floor(3) absolute value
floor floor(3) integer no greater than
hypot hypot(3) Euclidean distance
ilogb ieee(3) exponent extraction
j0 j0(3) bessel function
j1 j0(3) bessel function
jn j0(3) bessel function
log exp(3) natural logarithm
logb ieee(3) exponent extraction
log10 exp(3) logarithm to base 10
log1p exp(3) log(1+x)
pow exp(3) exponential x**y
rint floor(3) round to nearest integer
scalb ieee(3) exponent adjustment
scalbn ieee(3) exponent adjustment
sin sin(3) trigonometric function
sinh sinh(3) hyperbolic function
sqrt sqrt(3) square root
tan sin(3) trigonometric function
tanh sinh(3) hyperbolic function
y0 j0(3) bessel function
y1 j0(3) bessel function
yn j0(3) bessel function
NOTES
From the FDLIBM README
FDLIBM (double precision version) assumes:
a. IEEE 754 style (if not precise compliance) arithmetic;
b. 32 bit 2's complement integer arithmetic;
c. Each double precision floating-point number must be in IEEE 754
double format, and that each number can be retrieved as two 32-bit
integers;
Example: let y = 2.0
double fp number y: 2.0
IEEE double format: 0x4000000000000000
Referencing y as two integers:
*(int*)&y,*(1+(int*)&y) = {0x40000000,0x0} (on sparc)
{0x0,0x40000000} (on 386)
Note: FDLIBM will detect, at run time, the correct ordering of
the high and low part of a floating-point number.
d. IEEE exceptions may trigger "signals" as is common in Unix
implementations.
-------------------
2. EXCEPTION CASES
-------------------
All exception cases in the FDLIBM functions will be mapped
to one of the following four exceptions:
+huge*huge, +tiny*tiny, +1.0/0.0, +0.0/0.0
(overflow) (underflow) (divided-by-zero) (invalid)
For example, log(0) is a singularity and is thus mapped to
-1.0/0.0 = -infinity.
That is, FDLIBM's log will compute -one/zero and return the
computed value. On an IEEE machine, this will trigger the
divided-by-zero exception and a negative infinity is returned by
default.
Similarly, exp(-huge) will be mapped to tiny*tiny to generate
an underflow signal.
IEEE STANDARD 754 Floating-Point Arithmetic:
Properties of IEEE 754 Double-Precision:
Wordsize: 64 bits, 8 bytes. Radix: Binary.
Precision: 53 significant bits, roughly 16 significant decimals.
If x and x' are consecutive positive Double-Precision numbers
(they differ by 1 ulp), then
1.1e-16 < 0.5**53 < (x'-x)/x < 0.5**52 < 2.3e-16.
Range: Overflow threshold = 2.0**1024 = 1.8e308
Underflow threshold = 0.5**1022 = 2.2e-308
Overflow goes by default to a signed Inf.
Underflow is Gradual, rounding to the nearest integer multiple
of 0.5**1074 = 4.9e-324.
Zero is represented ambiguously as +0 or -0.
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.)
cannot be affected by the sign of zero; but if finite x = y
then Inf = 1/(x-y) != -1/(y-x) = -Inf.
Inf is signed.
it persists when added to itself or to any finite number. Its
sign transforms correctly through multiplication and division,
and (finite)/+Inf = +0 (nonzero)/0 = +Inf. But Inf-Inf, Inf*0
and Inf/Inf are, like 0/0 and sqrt(-3), invalid operations that
produce NaN. ...
Reserved operands:
there are 2**53-2 of them, all called NaN (Not a Number).
Some, called Signaling NaNs, 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 NaNs; they are the default results of Invalid
Operations, and propagate through subsequent arithmetic
operations. If x != x then x is NaN; every other predicate (x
> y, x = y, x < y, ...) is FALSE if NaN is involved.
NOTE: Trichotomy is violated by NaN.
Besides being FALSE, predicates that entail ordered
comparison, rather than mere (in)equality, signal Invalid
Operation when NaN is involved.
Rounding:
Every algebraic operation (+, -, *, /, sqrt) is rounded by
default to within half an ulp, and when the rounding error is
exactly half an 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 +Inf or towards -Inf 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.
Exceptions:
IEEE 754 recognizes five kinds of floating-point exceptions,
listed below in declining order of probable importance.
Exception Default Result
Invalid Operation NaN, or FALSE
Overflow +Inf
Divide by Zero +Inf
Underflow Gradual Underflow
Inexact Rounded value
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.
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:
1) Test for a condition that might cause an exception later, and
branch to avoid the exception.
2) Test a flag to see whether an exception has occurred since the
program last reset its flag.
3) Test a result to see whether it is a value that only an exception
could have produced.
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
provably ignorable. The same cannot be said of underflows
flushed to 0.
At the option of an implementor conforming to IEEE 754, other ways to
cope with exceptions may be provided:
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:
-- 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.
-- 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.
5) 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.
6) ... Other ways lie beyond the scope of this document.
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
libm. Ideally, each elementary function should act as if it were
indivisible, or atomic, in the sense that ...
i) No exception should be signaled that is not deserved by the data
supplied to that function.
ii) Any exception signaled should be identified with that function
rather than with one of its subroutines.
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.
Ideally, every programmer should be able conveniently 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.
Meanwhile, the functions in libm are only approximately atomic. They
signal no inappropriate exception except possibly ...
Over/Underflow
when a result, if properly computed, might have lain barely
within range, and
Inexact in cbrt, hypot, log10 and pow
when it happens to be exact, thanks to fortuitous cancellation
of errors.
Otherwise, ...
Invalid Operation is signaled only when
any result but NaN would probably be misleading.
Overflow is signaled only when
the exact result would be finite but beyond the overflow
threshold.
Divide-by-Zero is signaled only when
a function takes exactly infinite values at finite operands.
Underflow is signaled only when
the exact result would be nonzero but tinier than the underflow
threshold.
Inexact is signaled only when
greater range or precision would be needed to represent the
exact result.
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.
The math manual pages have been adapted from the 4.3BSD 3M manual pages
for FDLIBM by Kees J. Bot <kjb@cs.vu.nl> who normally avoids floating
point like the plague. Some text may not apply to FDLIBM, but KJB didn't
know whether to remove it or not. Don't blame the original authors
mentioned on these pages for inaccuracies introduced.
SEE ALSO
asinh(3), erf(3), exp(3), floor(3), hypot(3), ieee(3), j0(3), sin(3),
sinh(3), sqrt(3).
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.
AUTHOR
W. Kahan, with the help of Z-S. Alex Liu, Stuart I. McDonald, Dr.
Kwok-Choi Ng, Peter Tang.