MATH(3) Library Functions Manual MATH(3)
NAME
math  introduction to mathematical library functions
LIBRARY
Math Library (libm, lm)
SYNOPSIS
#include <<math.h>>
DESCRIPTION
These functions constitute the C Math Library (libm, lm). Declarations
for these functions may be obtained from the include file <math.h>.
List of Functions
Name Man 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
cbrt sqrt(3) cube root 1
ceil ceil(3) integer no less than 0
copysign copysign(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 finite(3) test for finity 0
floor floor(3) integer no greater than 0
fmod fmod(3) remainder ???
hypot hypot(3) Euclidean distance 1
ilogb ilogb(3) exponent extraction 0
isinf isinf(3) test for infinity 0
isnan isnan(3) test for notanumber 0
j0 j0(3) Bessel function ???
j1 j0(3) Bessel function ???
jn j0(3) Bessel function ???
lgamma lgamma(3) log gamma function ???
log log(3) natural logarithm 1
log10 log(3) logarithm to base 10 3
log1p log(3) log(1+x) 1
nan nan(3) return quiet NaN 0
nextafter nextafter(3) next representable number 0
pow pow(3) exponential x**y 60500
remainder remainder(3) remainder 0
rint rint(3) round to nearest integer 0
scalbn scalbn(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
trunc trunc(3) nearest integral value 3
y0 j0(3) Bessel function ???
y1 j0(3) Bessel function ???
yn j0(3) Bessel function ???
List of Defined Values
Name Value Description
M_E 2.7182818284590452354 e
M_LOG2E 1.4426950408889634074 log 2e
M_LOG10E 0.43429448190325182765 log 10e
M_LN2 0.69314718055994530942 log e2
M_LN10 2.30258509299404568402 log e10
M_PI 3.14159265358979323846 pi
M_PI_2 1.57079632679489661923 pi/2
M_PI_4 0.78539816339744830962 pi/4
M_1_PI 0.31830988618379067154 1/pi
M_2_PI 0.63661977236758134308 2/pi
M_2_SQRTPI 1.12837916709551257390 2/sqrt(pi)
M_SQRT2 1.41421356237309504880 sqrt(2)
M_SQRT1_2 0.70710678118654752440 1/sqrt(2)
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 doubleprecision arithmetic conforming to the IEEE Standard
754 for Binary FloatingPoint 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 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 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 significant bits, roughly like 17 significant
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. 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 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 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 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 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 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 floatingpoint
arithmetic exceptions neatly.
How do the functions in 4.3 BSD's new math library 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_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
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 standard'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 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 Semiconductor 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(), 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 significant 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 DoublePrecision:
Wordsize: 64 bits, 8 bytes.
Radix: Binary.
Precision: 53 significant bits, roughly like 16 significant
decimals. If x and x' are consecutive positive
DoublePrecision 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 infinity. Underflow
is Gradual, rounding to the nearest integer multiple of
0.5**1074 = 4.9e324.
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 xx
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/(xy) != 1/(yx) = 
infinity .
infinity is signed: it persists when added to itself or to any
finite number. Its sign transforms correctly through
multiplication and division, and infinity (finite)/+ =
+0 (nonzero)/0 = + infinity. But oooo, oo*0 and oo/oo
are, like 0/0 and sqrt(3), invalid operations that produce
NaN.
Reserved operands: there are 2**532 of them, all called NaN (Not A
Number). Some, called Signaling NaNs, trap any
floatingpoint 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 (+, , *, /, \/) 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
BinaryDecimal Conversions, at least for magnitudes between
roughly 1.0e10 and 1.0e37.
Exceptions: IEEE 754 recognizes five kinds of floatingpoint
exceptions, listed below in declining order of probable
importance.
Exception Default Result
Invalid Operation NaN, or FALSE
Overflow +oo
Divide by Zero +oo
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 floatingpoint 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 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 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
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 abandoned.
 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 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 ...
1. No exception should be signaled that is not deserved by the data
supplied to that function.
2. Any exception signaled should be identified with that function
rather than with one of its subroutines.
3. 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 savesrestores; 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.
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.
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 Wordlengthindependent Standard for Floatingpoint
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.
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.
NetBSD 6.1.5 February 23, 2007 NetBSD 6.1.5
