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
(ULP s) |
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 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 |
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 |
60-500 |
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 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 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 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 significant bits, roughly like 17 significant 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 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_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 |
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 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(),
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 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 Double-Precision:
- Wordsize:
- 64 bits, 8 bytes.
- Radix:
- Binary.
- Precision:
- 53 significant bits, roughly like 16 significant 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
infinity (finite)/± = ±0 (nonzero)/0 = ±
infinity. But ∞-∞, ∞∗0 and ∞/∞ 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 (+, -, ∗, /, √) 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 |
±∞ |
Divide by Zero |
±∞ |
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:
- Test for a condition that might cause an exception later,
and branch to avoid the exception.
- Test a flag to see whether an exception has occurred since
the program last reset its flag.
- 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:
- 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.
- 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.
- ... 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 ...
- 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 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.
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.
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.