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>.
Name Manual Description
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 tan(3) inverse trigonometric function 1
atanh atanh(3) inverse hyperbolic function 3
atan2 tan2(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 sin(3) trigonometric function 1
cosh sinh(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
floor floor(3) integer no greater than 0
hypot hypot(3) Euclidean distance 1
ilogb ieee(3) exponent extraction 0
isinf isinf(3) check exceptions
isnan isnan(3) check exceptions
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
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
tan tan(3) trigonometric function 3
tanh tanh(3) hyperbolic function 3
y0 j0(3) Bessel function
???
y1 j0(3) Bessel function
???
yn j0(3) Bessel function
???
In 4.3BSD, 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 VAX-11 family of computers, another for
double-precision arithmetic conforming to IEEE Std 754-1985.
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 number of ulps tabulated above; a
ulp is one Unit in the Last Place. The functions 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 doubleprecision
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 17 sig. decimal digits. If x and
x' are consecutive positive D_floatingpoint 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 a ulp, and
when the rounding error is exactly half a
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
4.3BSD. 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 4.3BSD'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
implementations
interpolate in large table to achieve speed and accuracy; the libm
implementations use tricky formulas compact enough that all
of them may
some day fit into a ROM.
More importantly, DEC considers the VMS implementation proprietary and
guards it zealously against unauthorized use. In contrast,
the libm included
in 4.3BSD is freely distributable; it may be copied
freely provided
their provenance is always acknowledged. 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 is the most widely adopted standard 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
for 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 Std 754-1985 without, alas, adhering to
the standard's
method of handling rounding and exceptions such as
over/underflow.
The DEC VAX G_floating-point format is very similar to IEEE
Std 754-1985
Double format. It is 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 code in 4.3BSD's libm for machines that conform to IEEE
Std 754-1985
is intended primarily for the National Semi. 32081 and WTL
1164/65. To
use this code with the Intel or Zilog chips, or with the Apple Macintosh
or ELXSI 6400, is to forego the use of better code provided
(perhaps for
free) by those companies and designed by some of the authors
of the code
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 is faster and more accurate
to boot; it,
Apple, the i8087, Z8070 and WE32106 all use 64 sig. bits.
The main
virtue of 4.3BSD's libm is that it is freely distributable;
it may be
copied freely provided its provenance is always acknowledged. Therefore
no user of UNIX on a machine that conforms to IEEE Std
754-1985 need use
anything worse than the new libm.
Properties of IEEE Std 754-1985 Double-Precision:
Wordsize: 64 bits, 8 bytes.
Radix: Binary.
Precision: 53 sig. bits, roughly equivalent to 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 InfinityInfinity, 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 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 (+, -, *, /,
sqrt) is rounded
by default to within half a ulp, and when
the rounding
error is exactly half a ulp then the
rounded value's
least sig. 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 Std
754-1985 can
do that. But no single kind of rounding
can be proved
best for every circumstance, so IEEE Std
754-1985 provides
rounding towards zero or towards
+Infinity or towards
-Infinity at the programmer's discretion. 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 Std 754-1985 recognizes five kinds of
floatingpoint
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 Std 754-1985
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 Std 754-1985 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 Std
754-1985; 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 Std
754-1985, 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 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.
Often times 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
4.3BSD'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 to conveniently
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.
Properties of IEEE Std 754-1985 Single-Precision:
Wordsize: 32 bits, 4 bytes.
Radix: Binary.
Precision: 24 sig. bits, roughly equivalent to 7 sig.
decimals.
If x and x' are consecutive positive Double-Precision
numbers (they differ by 1 ulp, then
6.0e-8 < 0.5**24 < (x'-x)/x <= 0.5**23 <
1.2e-7.
Range: Overflow threshold = 2.0**128 = 3.4e38.
Underflow threshold = 0.5**126 = 1.2e-38
Overflow goes by default to a signed Infinity.
Underflow is Gradual, rounding to the
nearest integer
multiple of 0.5**149 = 1.4e-45.
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 InfinityInfinity, Infinity*0
and Infinity/Infinity are, like
0/0 and
sqrt(-3), invalid operations that produce
NaN.
Reserved operands:
There are 2**24-2 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 (+, -, *, /,
sqrt) is rounded
by default to within half a ulp, and when
the rounding
error is exactly half a ulp then the
rounded value's
least sig. 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 Std
754-1985 can
do that. But no single kind of rounding
can be proved
best for every circumstance, so IEEE Std
754-1985 provides
rounding towards zero or towards
+Infinity or towards
-Infinity at the programmer's discretion. 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 Std 754-1985 recognizes five kinds of
floatingpoint
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.
An explanation of IEEE Std 754-1985 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 Std
754-1985 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.
When signals are appropriate, they are emitted by certain
operations
within libm, so a subroutine-trace may be needed to identify
the function
with its signal in case method 5) above is in use. All the
code in libm
takes the IEEE Std 754-1985 defaults for granted; this means
that a decision
to trap all divisions by zero could disrupt a function
that would
otherwise get a correct result despite division by zero.
May 6, 1991
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