MATH(3) BSD Programmer's Manual MATH(3)
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 <math.h>. The Fortran math library is
described in ``man 3f intro''.
LIST OF FUNCTIONS
Name Appears on Page Description Error Bound (ULPs)
acos sin.3m inverse trigonometric function 3
acosh asinh.3m inverse hyperbolic function 3
asin sin.3m inverse trigonometric function 3
asinh asinh.3m inverse hyperbolic function 3
atan sin.3m inverse trigonometric function 1
atanh asinh.3m inverse hyperbolic function 3
atan2 sin.3m inverse trigonometric function 2
cabs hypot.3m complex absolute value 1
cbrt sqrt.3m cube root 1
ceil floor.3m integer no less than 0
copysign ieee.3m copy sign bit 0
cos sin.3m trigonometric function 1
cosh sinh.3m hyperbolic function 3
drem ieee.3m remainder 0
erf erf.3m error function ???
erfc erf.3m complementary error function ???
exp exp.3m exponential 1
expm1 exp.3m exp(x)1 1
fabs floor.3m absolute value 0
floor floor.3m integer no greater than 0
hypot hypot.3m Euclidean distance 1
infnan infnan.3m signals exceptions
j0 j0.3m bessel function ???
j1 j0.3m bessel function ???
jn j0.3m bessel function ???
lgamma lgamma.3m log gamma function; (formerly gamma.3m)
log exp.3m natural logarithm 1
logb ieee.3m exponent extraction 0
log10 exp.3m logarithm to base 10 3
log1p exp.3m log(1+x) 1
pow exp.3m exponential x**y 60500
rint floor.3m round to nearest integer 0
scalb ieee.3m exponent adjustment 0
sin sin.3m trigonometric function 1
sinh sinh.3m hyperbolic function 3
sqrt sqrt.3m square root 1
tan sin.3m trigonometric function 3
tanh sinh.3m hyperbolic function 3
y0 j0.3m bessel function ???
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y1 j0.3m bessel function ???
yn j0.3m bessel function ???
NOTES
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 doubleprecision "D" format in the
DEC VAX11 family of computers, another for dou
bleprecision arithmetic conforming to the IEEE Standard
754 for Binary FloatingPoint Arithmetic. The two ver
sions 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(1.0e11) > 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(1.0e30) 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 VAX11 D_floatingpoint:
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 doubleprecision format
for the PDP11 and the earlier VAX11 machines; VAX11s
after 1983 were provided with an optional "G" format
closer to the IEEE doubleprecision format. The earlier
DEC MicroVAXs have no D format, only G doubleprecision.
(Why? Why not?)
Properties of D_floatingpoint:
Wordsize: 64 bits, 8 bytes. Radix: Binary.
Precision: 56 sig. bits, roughly like 17 sig.
decimals.
If x and x' are consecutive positive
D_floatingpoint numbers (they differ by 1
ulp), then
1.3e17 < 0.5**56 < (x'x)/x <= 0.5**55 <
2.8e17.
Range: Overflow threshold = 2.0**127 = 1.7e38.
Underflow threshold = 0.5**128 = 2.9e39.
NOTE: THIS RANGE IS COMPARATIVELY NARROW.
Overflow customarily stops computation.
Underflow is customarily flushed quietly to
zero.
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CAUTION:
It is possible to have x != y and
yet xy = 0 because of underflow.
Similarly x > y > 0 cannot prevent
either x*y = 0 or y/x = 0 from hap
pening 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
floatingpoint 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 over
flow 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 PDP11), 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_floatingpoint is one of
the better computer arithmetics designed in the 1960's.
Its properties are reflected fairly faithfully in the ele
mentary 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 computa
tion! 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 floatingpoint arithmetic exceptions
neatly. See infnan(3M) for the present state
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of affairs.
How do the functions in 4.3 BSD's new libm for UNIX com
pare 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 prove
nance 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_floatingpoint need use
anything worse than the new libm.
IEEE STANDARD 754 FloatingPoint 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 WTL1032, ... , 1165
Zilog Z8070 Western Electric (AT&T) WE32106.
Other implementations range from software, done thoroughly
in the Apple Macintosh, through VLSI in the
HewlettPackard 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 stan
dard's way of handling rounding and exceptions like
over/underflow. The DEC VAX G_floatingpoint 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 con
verted 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
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freely) by those companies and designed by some of the
authors of the codes above. Except for atan, cabs, cbrt,
erf, erfc, hypot, j0jn, lgamma, pow and y0yn, 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 prove
nance 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 DoublePrecision:
Wordsize: 64 bits, 8 bytes. Radix: Binary.
Precision: 53 sig. bits, roughly like 16 sig.
decimals.
If x and x' are consecutive positive Dou
blePrecision numbers (they differ by 1
ulp), then
1.1e16 < 0.5**53 < (x'x)/x <= 0.5**52 <
2.3e16.
Range: Overflow threshold = 2.0**1024 = 1.8e308
Underflow threshold = 0.5**1022 = 2.2e308
Overflow goes by default to a signed Infin
ity.
Underflow is Gradual, rounding to the near
est integer multiple of 0.5**1074 =
4.9e324.
Zero is represented ambiguously as +0 or 0.
Its sign transforms correctly through multi
plication or division, and is preserved by
addition of zeros with like signs; but xx
yields +0 for every finite x. The only
operations that reveal zero's sign are divi
sion by zero and copysign(x,+0). In par
ticular, comparison (x > y, x >= y, etc.)
cannot be affected by the sign of zero; but
if finite x = y then Infinity = 1/(xy) !=
1/(yx) = Infinity.
Infinity is signed.
it persists when added to itself or to any
finite number. Its sign transforms cor
rectly through multiplication and division,
and (finite)/+Infinity = +0 (nonzero)/0 =
+Infinity. But InfinityInfinity, Infin
ity*0 and Infinity/Infinity are, like 0/0
and sqrt(3), invalid operations that pro
duce NaN. ...
Reserved operands:
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there are 2**532 of them, all called NaN
(Not a Number). Some, called Signaling
NaNs, trap any floatingpoint operation per
formed upon them; they are used to mark
missing or uninitialized values, or nonexis
tent 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 sig
nificant 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 cir
cumstance, 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
BinaryDecimal Conversions, at least for
magnitudes between roughly 1.0e10 and
1.0e37.
Exceptions:
IEEE 754 recognizes five kinds of float
ingpoint 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
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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 floatingpoint exception, IEEE 754
provides a Flag that is raised each time its excep
tion is signaled, and stays raised until the pro
gram 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 excep
tion 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 can
not underflow in IEEE 754; if x != y then xy
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 any
way. 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 pro
vided:
4) ABORT. This mechanism classifies an exception
in advance as an incident to be handled by
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means traditionally associated with
errorhandling 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 aban
doned.
 In a subprogram that lacks an errorhandling
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 errorhandling statement is encoun
tered or the whole task is aborted and memory
is dumped.
5) STOP. This mechanism, requiring an interactive
debugging environment, is more for the program
mer than the program. It classifies an excep
tion 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 excep
tions 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 subrou
tines.
iii) The internal behavior of an atomic function should
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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 savesrestores; work is under way to amelio
rate 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 for
tuitous cancellation of errors.
Otherwise, ...
Invalid Operation is signaled only when
any result but NaN would probably be mis
leading.
Overflow is signaled only when
the exact result would be finite but beyond
the overflow threshold.
DividebyZero 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 subroutinetrace 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
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was published in the IEEE magazine MICRO in August 1984
under the title "A Proposed Radix and
Wordlengthindependent Standard for Floatingpoint Arith
metic" 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 Newslet
ter Special Issue of Oct. 1979, may be helpful although
they pertain to superseded drafts of the standard.
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