MATH(3) BSD Library Functions Manual MATH(3)NAMEmath — 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 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 dou‐
ble-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 simi‐
larly, 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 dou‐
ble-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 deci‐
mals. If x and x' are consecutive positive D_float‐
ing-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 cus‐
tomarily flushed quietly to zero. CAUTION: It is possible
to have x ≠ y and yet x-y = 0 because of underflow. Simi‐
larly 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 represen‐
tations of zero are accepted by the hardware, only the
obvious representation is ever produced. There is no -0 on
a VAX.
∞ 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 round‐
ing is away from 0.
Except for its narrow range, D_floating-point is one of the better com‐
puter 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 arith‐
metic operation that over/underflowed would behave. Similarly, expres‐
sions 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 computa‐
tion! 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 puri‐
tanical 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 elemen‐
tary 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 Mac‐
intosh or ELXSI 6400, is to forego the use of better codes provided (per‐
haps 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 acknowl‐
edged, 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 deci‐
mals. If x and x' are consecutive positive Double-Preci‐
sion 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 ∞. 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 pre‐
served 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 ∞ =
1/(x-y) ≠ -1/(y-x) = - ∞ .
∞ is signed: it persists when added to itself or to any finite num‐
ber. Its sign transforms correctly through multiplication
and division, and ∞ (finite)/± = ±0 (nonzero)/0 = ± ∞.
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 float‐
ing-point operation performed upon them; they are used to
mark missing or uninitialized values, or nonexistent ele‐
ments 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 sig‐
nificant 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 round‐
ing 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 +∞ or towards
-∞ 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 excep‐
tions, listed below in declining order of probable impor‐
tance.
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 excep‐
tions 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 pro‐
gram 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 quo‐
tients 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 vanish‐
ing, 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 thresh‐
old, 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 can‐
not 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 lan‐
guages 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 environ‐
ment, 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 hap‐
pened. 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 avail‐
able 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 han‐
dling.
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 amelio‐
rate 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 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 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 deci‐
sion to trap all divisions by zero could disrupt a code that would other‐
wise get correct results despite division by zero.
BSD February 23, 2007 BSD
