Index of Section 3 Manual Pages
| Interix / SUA | libm.3 | Interix / SUA |
libm(3) libm(3)
math()
NAME
math - introduction to mathematical library functions
DESCRIPTION
These functions constitute the C math library, libm. The link editor
searches this library under the -lm option. Declarations for these
functions may be obtained from the include file .
LIST OF FUNCTIONS
Name Description Error Bound (ULPs)
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
cabs complex absolute value 1
cbrt cube root 1
ceil integer no less than 0
copysign copy sign bit 0
cos trigonometric function 1
cosh hyperbolic function 3
drem remainder 0
erf error function
erfc complementary error function
exp exponential 1
expm1 exp(x)-1 1
fabs absolute value 0
floor integer no greater than 0
hypot Euclidean distance 1
infnan signals exceptions
j0 bessel function
j1 bessel function
jn bessel function
lgamma log gamma function; (formerly
gamma.3)
log natural logarithm 1
logb exponent extraction 0
log10 logarithm to base 10 3
log1p log(1+x) 1
pow exponential x**y 60-500
rint round to nearest integer 0
scalb exponent adjustment 0
sin trigonometric function 1
sinh hyperbolic function 3
sqrt square root 1
tan trigonometric function 3
tanh hyperbolic function 3
y0 bessel function
y1 bessel function
yn bessel function
NOTES
This system provides for double-precision arithmetic conforming to the
IEEE Standard 754 for Binary Floating-Point Arithmetic.
Properties of IEEE 754 Double-Precision:
Wordsize: 64 bits, 8 bytes. Radix: Binary.
Precision: 53 bits, roughly like 16 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 Infinity.
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 Infinity = 1/(x-y) != -1/(y-x) = -Infinity.
Infinity is signed.
It persists when added to itself or to any finite number. Its sign
transforms correctly through multiplication and division, and
(finite)/+-Infinity = +-0 (nonzero)/0 = +-Infinity. But -Infinity,
Infinity*0 and Infinity/Infinity 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 +Infinity or towards -Infinity 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 +-Infinity
Divide by Zero +-Infinity
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 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:
1. 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.
2. 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.
3. 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, upon which much of the Interix math library is based. Ideally, each
elementary function should act as if it were indivisible, or atomic, in
the sense that:
* No exception should be signaled that is not deserved by the data
supplied to that function.
* Any exception signaled should be identified with that function
rather than with one of its subroutines.
* 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 cabs, 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.
SEE ALSO
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.
USAGE NOTES
None of these functions are thread safe.
None of these functions are async-signal safe.