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>.
Each of the following double functions has a float counterpart with the
name ending in f, as an example the float counterpart of double
acos(double x) is float acosf(float x).
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
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
ilogb ieee.3m exponent extraction 0
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
log10 exp.3m logarithm to base 10 3
log1p exp.3m log(1+x) 1
pow exp.3m exponential x**y 60-500
remainder ieee.3m remainder 0
rint floor.3m round to nearest integer 0
scalbn 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 ???
y1 j0.3m bessel function ???
yn j0.3m 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.
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 (AT&T) 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 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 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.
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.
fpgetround(3), fpsetround(3), fpgetprec(3), fpsetprec(3), fpgetmask(3),
fpsetmask(3), fpgetsticky(3), fpresetsticky(3) - IEEE floating point
interface
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.
4th Berkeley Distribution May 6, 1991 MATH(3M)
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