ChorusOS man pages section 3M: Mathematical Libraries

Intro(3M)

NAME

Intro- Introduction to mathematical library functions

DESCRIPTION

These functions constitute the C mathematical library, libm. The link editor searches this library under the -lm option. Declarations for these functions are in the include file math.h. References of the form name(3M) refer to pages in this section of this document.

LIST OF FUNCTIONS

Name      Appears on Page       Description           Error Bound (ULPs)

acos         acos(3M)    inverse trigonometric function      3
acosh        acosh(3M)   inverse hyperbolic function         3
asin         asin(3M)    inverse trigonometric function      3
asinh        asinh(3M)   inverse hyperbolic function         3
atan         sin(3M)     inverse trigonometric function      1
atanh        atanh(3M)   inverse hyperbolic function         3
atan2        atan2(3M)   inverse trigonometric function      2
cabs         hypot(3M)   complex absolute value              1
cbrt         sqrt(3M)    cube root                           1
ceil         ceil(3M)    integer no less than                0
copysign     ieee(3M)    copy sign bit                       0
cos          cos(3M)     trigonometric function              1
cosh         cosh(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
finite       ieee(3M)    Is the value a valid finite number?
floor        floor(3M)   integer no greater than             0
fmod         fmod(3M)    floating-point remainder           ???
gamma        lgamma(3M)  gamma function                      4
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      2 for positive arguments
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         rint(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          tan(3M)     trigonometric function              3
tanh         tanh(3M)    hyperbolic function                 3
y0           j0(3M)      bessel function                    ???
y1           j0(3M)      bessel function                    ???
yn           j0(3M)      bessel function                    ???

A ulp is one Unit in the Last Place.

NOTES

The foregoing functions assume double-precision arithmetic conforming to the IEEE Standard 754 for Binary Floating-Point Arithmetic.

Properties of IEEE 754 Double-Precision

Wordsize:

64 bits, 8 bytes

Radix:

Binary

Precision:

53 significant bits, approximate to 16 significant decimals

If x and x' are consecutive positive Double-Precision numbers (that is, 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**102 = 1.8e308

Underflow threshold = 0.5**1022 [ap ] 2.2e-308

Overflow goes by default to a signed [infin ]

Underflow is Gradual, rounding to the nearest integer multiple of:

0.5**1074 [ap ] 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. However, 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) cannot be affected by the sign of zero. But, if finite x = y, then:

[infin ] = 1/(x-y) [ne ] -1/(y-x) = -[infin ]

[infin ] is signed:

It persists when added to itself or to any finite number. Its sign transforms correctly through multiplication and division:

(finite)/+-[infin ] = +-0

(nonzero)/0 =+-[infin ]

But, [infin ]-[infin ], [infin ]*0, and [infin ]/[infin ] are invalid operations that produce NaN (Not a Number). For example, 0/0 and sqrt(-3),

Reserved operands:

There are 2**53-2 of them, all called NaN. Some, called Signaling NaNs, trap any floating-point operation performed upon them. These are used to mark missing or uninitialized values, or nonexistent elements of arrays. The rest are Quiet NaNs, which are the default results of Invalid Operations and propagate through subsequent arithmetic operations. If x [ne ] x, then x is NaN and every other predicate (x > y, x = y, x < y, and so on) 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 off by default to within half a ulp, and when the rounding error is exactly half a ulp, the rounded value's least significant bit is zero. This kind of rounding is usually best. For instance, for every x = 1.0, 2.0, 3.0, 4.0, ..., 2.0**52, both (x/3.0)*3.0 == x and (x/10.0)*10.0 == x, despite both the quotients and the products having been rounded. However, no single kind of rounding is best in every circumstance. Therefore, IEEE 754 provides rounding towards zero, +[infin ], or -[infin ], as required. The same kinds of rounding are specified for Binary-Decimal Conversions, at least for magnitudes roughly between 1.0e-10 and 1.0e37.

Exceptions:

IEEE 754 recognizes five kinds of floating-point exceptions, listed below in declining order of importance.

Exception           Default Result
        
Invalid Operation   NaN, or FALSE
Overflow            +-[infin ]
Divide by Zero      +-[infin ] 
Underflow           Gradual Underflow
Inexact             Rounded value

Note -

An exception is not an error unless it is badly handled. A class of exceptions is exceptional because a default response is not 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 for programs to manage exceptions for which the default result might be unsatisfactory:

Test for a condition that might cause an exception later, and branch to avoid the exception.
Test a flag to see if an exception has occurred since the program last reset its flag.
Test a result to see if it is a value that only an exception could have produced.

Caution -

To discover Underflow, either test whether products or quotients lie closer to zero than the underflow threshold, or test the Underflow flag. Sums and differences cannot underflow in IEEE 754. Thus, if x [ne ] y, then x-y is correct to full precision and certainly nonzero regardless of how small it may be. Products and quotients that underflow gradually can lose accuracy gradually without vanishing, since comparing them with zero will not reveal the loss. If a gradually underflowed value is to be added to something bigger than the underflow threshold, then digits lost to gradual underflow will not be missed because they would have been rounded off anyway. So gradual underflows are usually, therefore provably, ignorable. This is not true for underflows flushed to 0.

With the option of an implementor conforming to IEEE 754, other ways to manage exceptions are:

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 the program that 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.
STOP--This mechanism requires an interactive debugging environment. It classifies an exception in advance as a symptom of a programing error. The exception suspends execution as near as it can to the offending operation, so that the programmer can check to see how it happened.

RESTRICTIONS

This library implements very little of the above-mentioned IEEE 754 signaling requirements. As it does not rely upon the floating-point unit/operating system cooperation to signal errors asynchronously, it catches most errors explicitly. However, some NaN or results may be issued by the floating-point unit and are returned to the application without any warning better other than the value of the result. Detected errors are reported by setting errno to either ERANGE or EDOM, performing a system-dependant notification, and returning either +, -, or NaN, depending on the type of error. The above system-dependent notification is non-operational in ChorusOS products for which this library is currently distributed.

ATTRIBUTES

See attributes(5) for descriptions of the following attributes:

ATTRIBUTE TYPE	ATTRIBUTE VALUE
Interface Stability	Evolving

ChorusOS 5.0 Last Revised December 2001