math - introduction to mathematical library functions
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 math.h.
Name Appears on Page Description Error Bound (ULPs)
acos acos.3 inverse trigonometric function 3
acosh acosh.3 inverse hyperbolic function 3
asin asin.3 inverse trigonometric function 3
asinh asinh.3 inverse hyperbolic function 3
atan atan.3 inverse trigonometric function 1
atanh atanh.3 inverse hyperbolic function 3
atan2 atan2.3 inverse trigonometric function 2
cabs hypot.3 complex absolute value 1
cbrt sqrt.3 cube root 1
ceil ceil.3 integer no less than 0
copysign ieee.3 copy sign bit 0
cos cos.3 trigonometric function 1
cosh cosh.3 hyperbolic function 3
erf erf.3 error function ???
erfc erf.3 complementary error function ???
exp exp.3 exponential 1
expm1 exp.3 exp(x)-1 1
fabs fabs.3 absolute value 0
finite ieee.3 test for finity 0
floor floor.3 integer no greater than 0
fmod fmod.3 remainder ???
hypot hypot.3 Euclidean distance 1
ilogb ieee.3 exponent extraction 0
isinf isinf.3 test for infinity 0
isnan isnan.3 test for not-a-number 0
j0 j0.3 bessel function ???
j1 j0.3 bessel function ???
jn j0.3 bessel function ???
lgamma lgamma.3 log gamma function ???
log exp.3 natural logarithm 1
log10 exp.3 logarithm to base 10 3
log1p exp.3 log(1+x) 1
nextafter ieee.3 next representable number 0
pow exp.3 exponential x**y 60-500
remainder ieee.3 remainder 0
rint rint.3 round to nearest integer 0
scalbn ieee.3 exponent adjustment 0
sin sin.3 trigonometric function 1
sinh sinh.3 hyperbolic function 3
sqrt sqrt.3 square root 1
tan tan.3 trigonometric function 3
tanh tanh.3 hyperbolic function 3
y0 j0.3 bessel function ???
y1 j0.3 bessel function ???
yn j0.3 bessel function ???
In 4.3 BSD, distributed from the University of California
in late 1985, most of the foregoing functions come in two
versions, one for the double-precision "D" format in the
DEC VAX-11 family of computers, another for
double-precision arithmetic conforming to the IEEE Standard
754 for Binary Floating-Point Arithmetic. The two
versions behave very similarly, as should be expected from
programs more accurate and robust than was the norm when
UNIX was born. For instance, the programs are accurate to
within the numbers of ulps tabulated above; an ulp is one
Unit in the Last Place. And the programs have been cured
of anomalies that afflicted the older math library libm in
which incidents like the following had been reported:
sqrt(-1.0) = 0.0 and log(-1.0) = -1.7e38.
cos(1.0e-11) cos(0.0) 1.0.
pow(x,1.0) != x when x = 2.0, 3.0, 4.0, ..., 9.0.
pow(-1.0,1.0e10) trapped on Integer Overflow.
sqrt(1.0e30) and sqrt(1.0e-30) were very slow.
However the two versions do differ in ways that have to be
explained, to which end the following notes are provided.
DEC VAX-11 D_floating-point:
This is the format for which the original math library
libm was developed, and to which this manual is still
principally dedicated. It is the double-precision format
for the PDP-11 and the earlier VAX-11 machines; VAX-11s
after 1983 were provided with an optional "G" format
closer to the IEEE double-precision format. The earlier
DEC MicroVAXs have no D format, only G double-precision.
(Why? Why not?)
Properties of D_floating-point:
Wordsize: 64 bits, 8 bytes. Radix: Binary.
Precision: 56 sig. bits, roughly like 17 sig.
decimals.
If x and x' are consecutive positive
D_floating-point numbers (they differ by 1
ulp), then
1.3e-17 0.5**56 (x'-x)/x 0.5**55
2.8e-17.
Range: Overflow threshold = 2.0**127 = 1.7e38.
Underflow threshold = 0.5**128 = 2.9e-39.
NOTE: THIS RANGE IS COMPARATIVELY NARROW.
Overflow customarily stops computation.
Underflow is customarily flushed quietly to
zero.
CAUTION:
It is possible to have x != y and
yet x-y = 0 because of underflow.
Similarly x y 0 cannot prevent
either x*y = 0 or y/x = 0 from happening
without warning.
Zero is represented ambiguously.
Although 2**55 different representations of
zero are accepted by the hardware, only the
obvious representation is ever produced.
There is no -0 on a VAX.
Infinity is not part of the VAX architecture.
Reserved operands:
of the 2**55 that the hardware recognizes,
only one of them is ever produced. Any
floating-point operation upon a reserved
operand, even a MOVF or MOVD, customarily
stops computation, so they are not much
used.
Exceptions:
Divisions by zero and operations that overflow
are invalid operations that customarily
stop computation or, in earlier machines,
produce reserved operands that will stop
computation.
Rounding:
Every rational operation (+, -, *, /) on a
VAX (but not necessarily on a PDP-11), if
not an over/underflow nor division by zero,
is rounded to within half an ulp, and when
the rounding error is exactly half an ulp
then rounding is away from 0.
Except for its narrow range, D_floating-point is one of
the better computer arithmetics designed in the 1960's.
Its properties are reflected fairly faithfully in the elementary
functions for a VAX distributed in 4.3 BSD. They
over/underflow only if their results have to lie out of
range or very nearly so, and then they behave much as any
rational arithmetic operation that over/underflowed would
behave. Similarly, expressions like log(0) and atanh(1)
behave like 1/0; and sqrt(-3) and acos(3) behave like 0/0;
they all produce reserved operands and/or stop computation!
The situation is described in more detail in manual
pages.
This response seems excessively punitive, so
it is destined to be replaced at some time in
the foreseeable future by a more flexible but
still uniform scheme being developed to handle
all floating-point arithmetic exceptions
neatly. See infnan(3) for the present state
of affairs.
How do the functions in 4.3 BSD's new libm for UNIX compare
with their counterparts in DEC's VAX/VMS library?
Some of the VMS functions are a little faster, some are a
little more accurate, some are more puritanical about
exceptions (like pow(0.0,0.0) and atan2(0.0,0.0)), and
most occupy much more memory than their counterparts in
libm. The VMS codes interpolate in large table to achieve
speed and accuracy; the libm codes use tricky formulas
compact enough that all of them may some day fit into a
ROM.
More important, DEC regards the VMS codes as proprietary
and guards them zealously against unauthorized use. But
the libm codes in 4.3 BSD are intended for the public
domain; they may be copied freely provided their provenance
is always acknowledged, and provided users assist
the authors in their researches by reporting experience
with the codes. Therefore no user of UNIX on a machine
whose arithmetic resembles VAX D_floating-point need use
anything worse than the new libm.
IEEE STANDARD 754 Floating-Point Arithmetic:
This standard is on its way to becoming more widely
adopted than any other design for computer arithmetic.
VLSI chips that conform to some version of that standard
have been produced by a host of manufacturers, among them
...
Intel i8087, i80287 National Semiconductor 32081
Motorola 68881 Weitek WTL-1032, ... , -1165
Zilog Z8070 Western Electric (ATT) WE32106.
Other implementations range from software, done thoroughly
in the Apple Macintosh, through VLSI in the
Hewlett-Packard 9000 series, to the ELXSI 6400 running ECL
at 3 Megaflops. Several other companies have adopted the
formats of IEEE 754 without, alas, adhering to the
standard's way of handling rounding and exceptions like
over/underflow. The DEC VAX G_floating-point format is
very similar to the IEEE 754 Double format, so similar
that the C programs for the IEEE versions of most of the
elementary functions listed above could easily be converted
to run on a MicroVAX, though nobody has volunteered
to do that yet.
The codes in 4.3 BSD's libm for machines that conform to
IEEE 754 are intended primarily for the National Semi.
32081 and WTL 1164/65. To use these codes with the Intel
or Zilog chips, or with the Apple Macintosh or ELXSI 6400,
is to forego the use of better codes provided (perhaps
freely) by those companies and designed by some of the
authors of the codes above. Except for atan, cabs, cbrt,
erf, erfc, hypot, j0-jn, lgamma, pow and y0-yn, the
Motorola 68881 has all the functions in libm on chip, and
faster and more accurate; it, Apple, the i8087, Z8070 and
WE32106 all use 64 sig. bits. The main virtue of 4.3
BSD's libm codes is that they are intended for the public
domain; they may be copied freely provided their provenance
is always acknowledged, and provided users assist
the authors in their researches by reporting experience
with the codes. Therefore no user of UNIX on a machine
that conforms to IEEE 754 need use anything worse than the
new libm.
Properties of IEEE 754 Double-Precision:
Wordsize: 64 bits, 8 bytes. Radix: Binary.
Precision: 53 sig. bits, roughly like 16 sig.
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, 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 (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 prov-
ably 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 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.
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.
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.
May 6, 1991 MATH(3)
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