MATH(3) UNIX Programmer's Manual MATH(3)NAMEmath - 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.
LIST OF FUNCTIONS
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
nan nan.3 return quiet NaN 0
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
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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 ver-
sions, 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 fol-
lowing 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
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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 can-
not 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 computa-
tion 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 pro-
perties 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)MirOS BSD #10-current March 26, 2006 3
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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 fore-
seeable future by a more flexible but still uni-
form 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 pro-
duced 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 Dou-
ble format, so similar that the C programs for the IEEE ver-
sions 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
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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 pro-
vided 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 any-
thing 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 multiplica-
tion 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 mul-
tiplication 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
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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 +Infin-
ity or towards -Infinity at the programmer's
option. And the same kinds of rounding are speci-
fied 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 excep-
tional 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 can-
not 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
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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 under-
flowed 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 usu-
ally 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 trad-
itionally 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 state-
ment, 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
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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 unex-
ceptionable, so the programmer ought ideally to be
able to resume execution after each one as if exe-
cution had not been stopped.
6) ... Other ways lie beyond the scope of this docu-
ment.
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 elemen-
tary 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 excep-
tions listed above, although the definition of the
function may be correlated intentionally with excep-
tion 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 possi-
bly ...
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
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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 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 Newsletter Spe-
cial Issue of Oct. 1979, may be helpful although they per-
tain 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 other-
wise get correct results despite division by zero.
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