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)

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.

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 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 significant bits, roughly like 17 significant 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 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 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.

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 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
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 Semiconductor 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(), 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 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 significant bits, roughly like 16 significant 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 infinity (finite)/±  = ±0 (nonzero)/0 = ± infinity. 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 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.

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 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.

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	±∞
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 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 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. 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 ...

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 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 ...

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.