libm - mathematical library functions

cc [ flags ] files -lm [ libraries ]

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>.

/usr/lib/libm.a -lm

Name Appears on Page Description Error Bound

acos trig(3) inverse trigonometric function 3
acosh hyperbolic(3) inverse hyperbolic function 3
asin trig(3) inverse trigonometric function 3
asinh hyperbolic(3) inverse hyperbolic function 3
atan trig(3) inverse trigonometric function 1
atanh hyperbolic(3) inverse hyperbolic function 3
atan2 trig(3) inverse trigonometric function 2
cabs hypot(3) complex absolute value 1
cbrt sqrt(3) cube root 1
ceil floor(3) integer no less than 0
copysign ieee(3) copy sign bit 0
cos trig(3) trigonometric function 1
cosh hyperbolic(3) hyperbolic function 3
drem ieee(3) remainder 0
erf erf(3) error function ???
erfc erf(3) complementary error function ???
exp exp(3) exponential 1
exp10 exp(3) base-10 exponential 1
exp2 exp(3) base-2 exponential 1
expm1 exp(3) exp(x)-1 1
fabs floor(3) absolute value 0
finite ieee(3) finite number check -
floor floor(3) integer no greater than 0
hypot hypot(3) Euclidean distance 1
j0 bessel(3) bessel function ???
j1 bessel(3) bessel function ???
jn bessel(3) bessel function ???
lgamma lgamma(3) log gamma function; (formerly gamma.3)
log exp(3) natural logarithm 1
logb ieee(3) exponent extraction 0
log10 exp(3) logarithm to base 10 3
log1p exp(3) log(1+x) 1
log2 exp(3) logarithm to base 2 1
pow exp(3) exponential x**y 60-500
rint floor(3) round to nearest integer 0
scalb ieee(3) exponent adjustment 0
sin trig(3) trigonometric function 1
sinh hyperbolic(3) hyperbolic function 3
sqrt sqrt(3) square root 1
tan trig(3) trigonometric function 3
tanh hyperbolic(3) hyperbolic function 3
y0 bessel(3) bessel function ???
y1 bessel(3) bessel function ???
yn bessel(3) bessel function ???

MachTen uses the Standard Apple Numerics Environment (SANE) package
on the Macintosh to perform basic floating point arithmetic
operations (addition, subtraction, multiplication, division,
comparison, and conversion), and many auxillary and elementary
functions. The balance of libm routines are as coded in 4.3 BSD,
distributed from the University of California in late 1985. SANE
fully supports the IEEE Standard 754 for binary floating-point
arithmetic. SANE assumes responsibility for using the Motorola
MC68881 Floating-Point Coprocessor, if it is available.

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

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.
Overflow customarily stops computation.
Underflow is customarily flushed quietly to zero.
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.
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.
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. See infnan(3) for the present state
of affairs.

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.) 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 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.
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.
IEEE 754 recognizes five kinds of floating-point
exceptions, listed below in declining order of probable
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

- 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

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 ...
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
Divide-by-Zero is signaled only when
a function takes exactly infinite values at finite
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.

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.

Apple Numerics Manual, Second Edition; Addison-Wesley Publishing
Company, Inc.; February 1990; ISBN 0-201-17738-2.

W. Kahan, with the help of Z-S. Alex Liu, Stuart I. McDonald, Dr.
Kwok-Choi Ng, Peter Tang.