4.4BSD/usr/share/man/cat3/math.0
MATH(3) BSD Programmer's Manual MATH(3)
NNAAMMEE
math - introduction to mathematical library functions
DDEESSCCRRIIPPTTIIOONN
These functions constitute the C math library, _l_i_b_m_. The
link editor searches this library under the "-lm" option.
Declarations for these functions may be obtained from the
include file <_m_a_t_h_._h>. The Fortran math library is
described in ``man 3f intro''.
LLIISSTT OOFF FFUUNNCCTTIIOONNSS
_N_a_m_e _A_p_p_e_a_r_s _o_n _P_a_g_e _D_e_s_c_r_i_p_t_i_o_n _E_r_r_o_r _B_o_u_n_d _(_U_L_P_s_)
acos sin.3m inverse trigonometric function 3
acosh asinh.3m inverse hyperbolic function 3
asin sin.3m inverse trigonometric function 3
asinh asinh.3m inverse hyperbolic function 3
atan sin.3m inverse trigonometric function 1
atanh asinh.3m inverse hyperbolic function 3
atan2 sin.3m inverse trigonometric function 2
cabs hypot.3m complex absolute value 1
cbrt sqrt.3m cube root 1
ceil floor.3m integer no less than 0
copysign ieee.3m copy sign bit 0
cos sin.3m trigonometric function 1
cosh sinh.3m hyperbolic function 3
drem ieee.3m remainder 0
erf erf.3m error function ???
erfc erf.3m complementary error function ???
exp exp.3m exponential 1
expm1 exp.3m exp(x)-1 1
fabs floor.3m absolute value 0
floor floor.3m integer no greater than 0
hypot hypot.3m Euclidean distance 1
infnan infnan.3m signals exceptions
j0 j0.3m bessel function ???
j1 j0.3m bessel function ???
jn j0.3m bessel function ???
lgamma lgamma.3m log gamma function; (formerly gamma.3m)
log exp.3m natural logarithm 1
logb ieee.3m exponent extraction 0
log10 exp.3m logarithm to base 10 3
log1p exp.3m log(1+x) 1
pow exp.3m exponential x**y 60-500
rint floor.3m round to nearest integer 0
scalb ieee.3m exponent adjustment 0
sin sin.3m trigonometric function 1
sinh sinh.3m hyperbolic function 3
sqrt sqrt.3m square root 1
tan sin.3m trigonometric function 3
tanh sinh.3m hyperbolic function 3
y0 j0.3m bessel function ???
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y1 j0.3m bessel function ???
yn j0.3m bessel function ???
NNOOTTEESS
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 dou-
ble-precision arithmetic conforming to the IEEE Standard
754 for Binary Floating-Point Arithmetic. The two ver-
sions 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 _u_l_ps tabulated above; an _u_l_p is one
_Unit in the _Last _Place. And the programs have been cured
of anomalies that afflicted the older math library _l_i_b_m 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.
DDEECC VVAAXX--1111 DD__ffllooaattiinngg--ppooiinntt::
This is the format for which the original math library
_l_i_b_m was developed, and to which this manual is still
principally dedicated. It is _t_h_e 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 sig. bits, roughly like 17 sig.
decimals.
If x and x' are consecutive positive
D_floating-point numbers (they differ by 1
_u_l_p), 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.
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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 hap-
pening 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 over-
flow 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 _u_l_p, and when
the rounding error is exactly half an _u_l_p
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 ele-
mentary 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 computa-
tion! The situation is described in more detail in manual
pages.
_T_h_i_s _r_e_s_p_o_n_s_e _s_e_e_m_s _e_x_c_e_s_s_i_v_e_l_y _p_u_n_i_t_i_v_e_, _s_o
_i_t _i_s _d_e_s_t_i_n_e_d _t_o _b_e _r_e_p_l_a_c_e_d _a_t _s_o_m_e _t_i_m_e _i_n
_t_h_e _f_o_r_e_s_e_e_a_b_l_e _f_u_t_u_r_e _b_y _a _m_o_r_e _f_l_e_x_i_b_l_e _b_u_t
_s_t_i_l_l _u_n_i_f_o_r_m _s_c_h_e_m_e _b_e_i_n_g _d_e_v_e_l_o_p_e_d _t_o _h_a_n_d_l_e
_a_l_l _f_l_o_a_t_i_n_g_-_p_o_i_n_t _a_r_i_t_h_m_e_t_i_c _e_x_c_e_p_t_i_o_n_s
_n_e_a_t_l_y_. _S_e_e _i_n_f_n_a_n_(_3_M_) _f_o_r _t_h_e _p_r_e_s_e_n_t _s_t_a_t_e
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_o_f _a_f_f_a_i_r_s_.
How do the functions in 4.3 BSD's new _l_i_b_m for UNIX com-
pare 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
_l_i_b_m. The VMS codes interpolate in large table to achieve
speed and accuracy; the _l_i_b_m 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 _l_i_b_m codes in 4.3 BSD are intended for the public
domain; they may be copied freely provided their prove-
nance 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 _l_i_b_m.
IIEEEEEE SSTTAANNDDAARRDD 775544 FFllooaattiinngg--PPooiinntt AArriitthhmmeettiicc::
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 stan-
dard'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 con-
verted to run on a MicroVAX, though nobody has volunteered
to do that yet.
The codes in 4.3 BSD's _l_i_b_m 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
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freely) by those companies and designed by some of the
authors of the codes above. Except for _a_t_a_n, _c_a_b_s, _c_b_r_t,
_e_r_f, _e_r_f_c, _h_y_p_o_t, _j_0_-_j_n, _l_g_a_m_m_a, _p_o_w and _y_0_-_y_n, the
Motorola 68881 has all the functions in _l_i_b_m 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 _l_i_b_m codes is that they are intended for the public
domain; they may be copied freely provided their prove-
nance 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 _l_i_b_m.
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 Dou-
ble-Precision numbers (they differ by 1
_u_l_p), 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 Infin-
ity.
Underflow is _G_r_a_d_u_a_l_, rounding to the near-
est integer multiple of 0.5**1074 =
4.9e-324.
Zero is represented ambiguously as +0 or -0.
Its sign transforms correctly through multi-
plication 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 divi-
sion by zero and copysign(x,+-0). In par-
ticular, 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 cor-
rectly through multiplication and division,
and (finite)/+-Infinity = +-0 (nonzero)/0 =
+-Infinity. But Infinity-Infinity, Infin-
ity*0 and Infinity/Infinity are, like 0/0
and sqrt(-3), invalid operations that pro-
duce _N_a_N. ...
Reserved operands:
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there are 2**53-2 of them, all called _N_a_N
(_Not _a _Number). Some, called Signaling
_N_a_Ns, trap any floating-point operation per-
formed upon them; they are used to mark
missing or uninitialized values, or nonexis-
tent elements of arrays. The rest are Quiet
_N_a_Ns; they are the default results of
Invalid Operations, and propagate through
subsequent arithmetic operations. If x != x
then x is _N_a_N; every other predicate (x > y,
x = y, x < y, ...) is FALSE if _N_a_N is
involved.
NOTE: Trichotomy is violated by _N_a_N.
Besides being FALSE, predicates that
entail ordered comparison, rather
than mere (in)equality, signal
Invalid Operation when _N_a_N is
involved.
Rounding:
Every algebraic operation (+, -, *, /, sqrt)
is rounded by default to within half an _u_l_p,
and when the rounding error is exactly half
an _u_l_p 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 rounding like IEEE
754 can do that. But no single kind of
rounding can be proved best for every cir-
cumstance, 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 float-
ing-point exceptions, listed below in
declining order of probable importance.
Exception Default Result
__________________________________________
Invalid Operation _N_a_N, or FALSE
Overflow +-Infinity
Divide by Zero +-Infinity
Underflow Gradual Underflow
Inexact Rounded value
NOTE: An Exception is not an Error unless
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handled badly. What makes a class of excep-
tions 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 satisfac-
torily, 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 excep-
tion is signaled, and stays raised until the pro-
gram 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 excep-
tion 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 can-
not 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 any-
way. So gradual underflows are usually _p_r_o_v_-
_a_b_l_y 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 pro-
vided:
4) ABORT. This mechanism classifies an exception
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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 aban-
doned.
-- 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 encoun-
tered or the whole task is aborted and memory
is dumped.
5) STOP. This mechanism, requiring an interactive
debugging environment, is more for the program-
mer than the program. It classifies an excep-
tion 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 excep-
tions 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 _l_i_b_m. 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 subrou-
tines.
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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 _c_o_n_v_e_n_i_e_n_t_l_y 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 _l_i_b_m 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 _c_a_b_s, _c_b_r_t, _h_y_p_o_t, _l_o_g_1_0 and _p_o_w
when it happens to be exact, thanks to for-
tuitous cancellation of errors.
Otherwise, ...
Invalid Operation is signaled only when
any result but _N_a_N would probably be mis-
leading.
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.
BBUUGGSS
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.
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SSEEEE AALLSSOO
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 Newslet-
ter Special Issue of Oct. 1979, may be helpful although
they pertain to superseded drafts of the standard.
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