NAME

math − introduction to mathematical library functions

DESCRIPTION

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>. The Fortran math library is described in “man 3f intro”.

LIST OF FUNCTIONS

NameAppears on PageDescriptionError Bound (ULPs)
acossin.3minverse trigonometric function3
acoshasinh.3minverse hyperbolic function3
asinsin.3minverse trigonometric function3
asinhasinh.3minverse hyperbolic function3
atansin.3minverse trigonometric function1
atanhasinh.3minverse hyperbolic function3
atan2sin.3minverse trigonometric function2
cabshypot.3mcomplex absolute value1
cbrtsqrt.3mcube root1
ceilfloor.3minteger no less than0
copysignieee.3mcopy sign bit0
cossin.3mtrigonometric function1
coshsinh.3mhyperbolic function3
dremieee.3mremainder0
erferf.3merror function???
erfcerf.3mcomplementary error function???
expexp.3mexponential1
expm1exp.3mexp(x)−11
fabsfloor.3mabsolute value0
floorfloor.3minteger no greater than0
hypothypot.3mEuclidean distance1
j0j0.3mbessel function???
j1j0.3mbessel function???
jnj0.3mbessel function???
lgammalgamma.3mlog gamma function; (formerly gamma.3m)
logexp.3mnatural logarithm1
logbieee.3mexponent extraction0
log10exp.3mlogarithm to base 103
log1pexp.3mlog(1+x)1
mathtrapmathtrap.3menable/disable floating−point exception trap
powexp.3mexponential x∗∗y60−500
rintfloor.3mround to nearest integer0
scalbieee.3mexponent adjustment0
sinsin.3mtrigonometric function1
sinhsinh.3mhyperbolic function3
sqrtsqrt.3msquare root1
tansin.3mtrigonometric function3
tanhsinh.3mhyperbolic function3
y0j0.3mbessel function???
y1j0.3mbessel function???
ynj0.3mbessel function???

NOTES

On NEWS, arithmetic functions are conformed to the IEEE Standard 754 for Binary Floating-Point Arithmetic. 

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 ... 

Motorola 68881/68882National Semiconductor  32081
Intel i8087, i80287Weitek WTL-1032, ... , −1165
Zilog Z8070Western Electric (AT&T) WE32106.

Except for atan, cabs, cbrt, erf, erfc, hypot, j0−jn, lgamma, pow and y0−yn, the Motorola 68881/68882 has all the functions in libm on chip, and faster and more accurate;

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 ∞. 
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 ∞ = 1/(x−y) ≠ −1/(y−x) = −∞. 

∞ is signed. 

it persists when added to itself or to any finite number.  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 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 +∞ 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 exceptions, listed below in declining order of probable importance. 

ExceptionDefault Result

Invalid OperationNaN, or FALSE
Divide by Zero±Infinity
Overflow±∞
Divide by Zero±∞
UnderflowGradual Underflow
InexactRounded 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 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. 

Under NEWS-OS, floating-point exceptions can be handled by signal(3C) and mathtrap(3M) functions. By default, all exceptions are ignored. However, using functions described in mathtrap(3M), floating-point exception trap will occur with signal SIGFPE.  Signal(3C) specifies trap handling routines. For more details, see signal(3C) and mathtrap(3).

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 Arithmetic" by W. J. Cody et al. 
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.

AUTHOR

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

NEWS-OS Release 4.1C