Updated: 11 December 1998

DIGITAL Portable Mathematics Library

Not all mathematical functions are capable of returning a meaningful result for all input argument values. Any argument value passed to a DPML routine that does not return a meaningful result, or is defined differently for different environments, is referred to as an exceptional argument. Exceptional arguments that result in an exception behavior are documented in the Exceptions section of each DPML routine in Chapter 2.

Exceptional arguments typically fall into one of two categories:

• Domain errors or invalid arguments. These are arguments for which a function is not defined. For example, the inverse sine function, asin, is defined only for arguments between -1 and +1 inclusive. Attempting to evaluate acos(-2) or acos(3) results in a domain error or invalid argument error.
• Range errors. These errors occur when a mathematically valid argument results in a function value that exceeds the range of representable values for the floating-point data type. Appendix A gives the approximate minimum and maximum values representable for each floating-point data type.

DPML routines are designed to provide predictable and platform-consistent exception conditions and behavior. When an exception is triggered in a DPML routine, two pieces of information can be generated and made available to the calling program for exception handling:

• A notification that an exception has occurred. The mechanics of exception notification vary from platform to platform (for example, signaling, trapping, set errno).
• A return value. If your environment allows your routine to continue after raising an exception condition (with an exception handler for example), then a return value is made available upon completion of the routine.

The exception condition-handling mechanisms on your platform dictate how you can recover from an exception condition, and whether you can expect to receive an exception notification, a return value, or both, from a DPML routine.

The Exceptions section of each DPML routine documents each exceptional argument that results in an exception behavior. In addition to the exceptional arguments, an indication of how the DPML routines treat each argument is given. Exceptional arguments are sometimes presented in terms of symbolic constants.

For example, the following table lists the exceptional arguments of the exponential routine, exp(x):

Exceptional Argument	Exception Condition/Routine Behavior
x > ln(max_float)	Overflow
x < ln(min_float)	Underflow

The exceptional arguments indicate that whenever x > ln(max_float)
or x < ln(min_float), DPML recognizes an overflow or underflow condition, respectively.

The symbolic constants ln(max_float) and ln(min_float) represent the natural log of the maximum and minimum representable values of the floating-point data type in question. The actual values of ln(max_float) and ln(min_float) are described in Appendix A.

DPML recognizes three predefined conditions: overflow, underflow, and invalid argument. Table 1-3 describes the default action and return value of each condition.

Table 1-3 Default Action and Return Values for Exception Conditions

Exception Condition	Default Action	Return Value
Overflow	Trap	HUGE_RESULT
Underflow	Continue Quietly	0
Invalid argument	Trap	INV_RESULT

The values HUGE_RESULT and INV_RESULT are data-type dependent.

For IEEE data types, HUGE_RESULT and INV_RESULT are the floating-point encodings for Infinity and NaN respectively.

For VAX data types, HUGE_RESULT and INV_RESULT are max_float and 0 respectively.

1.5 IEEE Std 754 Considerations

The Institute of Electrical and Electronics Engineers (IEEE) ANSI/IEEE Std 754-1985, IEEE Standard for Binary Floating-Point Arithmetic data types include denormalized numbers (very close to zero). The standard supports the concept of "Not-a-Number" or NaN to represent indeterminate quantities, and uses plus infinity or minus infinity (so that they behave in arithmetic) like the mathematical infinities. Whenever a DPML routine produces an overflow or indeterminate condition, it generates an infinity or NaN value.

All DPML routines, except one, return a NaN result when presented with a NaN input. The only exception is pow(NaN,0) = 1 in ANSI C.

1.6 X/Open Portability Guide Considerations

Table 1-4 lists the routines described in this manual that conform to the requirements of the X/Open Portability Guide, Version 4 (XPG4), or are implemented as UNIX extensions to the XPG4 standard (XPG4-UNIX). Descriptions of these routines appear in Chapter 2 under the generic function name listed in Table 1-4. Platform-specific entry-points are listed in Appendix B.

Table 1-4 XPG4 Conformant Routines

Routine	Conforms to Standard	Generic Function Name
acos	XPG4	acos
acosh	XPG4-UNIX	acosh
asin	XPG4	asin
asinh	XPG4-UNIX	asinh
atan	XPG4	atan
atan2	XPG4	atan
atanh	XPG4-UNIX	atanh
ceil	XPG4	ceil
cos	XPG4	cos
cosh	XPG4	cosh
cot	XPG4	cot
erf	XPG4	erf
erfc	XPG4	erf
exp	XPG4	exp
expm1	XPG4-UNIX	exp
fabs	XPG4	fabs
floor	XPG4	floor
fmod	XPG4	fmod
frexp	XPG4	frexp
gamma	XPG4	lgamma
hypot	XPG4	hypot
ilogb	XPG4-UNIX	ilogb
isnan	XPG4	isnan
j0	XPG4	bessel
j1	XPG4	bessel
jn	XPG4	bessel
ldexp	XPG4	ldexp
lgamma	XPG4	lgamma
log	XPG4	log
log10	XPG4	log
log1p	XPG4-UNIX	log
logb	XPG4-UNIX	logb
modf	XPG4	modf
nextafter	XPG4-UNIX	nextafter
pow	XPG4	pow
remainder	XPG4-UNIX	remainder
rint	XPG4-UNIX	rint
scalb	XPG4-UNIX	scalb
sin	XPG4	sin
sinh	XPG4	sinh
tan	XPG4	tan
tanh	XPG4	tanh
y0	XPG4	bessel
y1	XPG4	bessel
yn	XPG4	bessel

Each DPML routine documented in this chapter is presented in the following format:

• Routine name---A brief name to identify the function of the routine. A routine may contain more than one function.
• Interface---What the routine expects to receive and what it returns. See Section 2.1 for more information.
• Description---Additional information, including the permitted range of input values and generic calculations used to compute the results.
• Exceptions---A description of how the routine behaves when given a specific exceptional input argument.

The interface to each function is:

RETURN_TYPE generic_interface_name (INPUT_ARG_TYPE...)

Each of these is described below.

RETURN_TYPE

The data type of the value returned by the routine to your application program. Each routine returns a specific class of data type. For example, either F_TYPE or F_COMPLEX can appear in a DPML interface described in Chapter 2. The supported data types are described in Section 1.2.

generic_interface_name

The generic name. DPML routines in this chapter are listed in alphabetic order by their interface names. Some DPML routines may be available in the syntax of your high-level language. Fortran and C are examples. To maximize the portability of your application, use the corresponding mathematical routine described in your high-level language, and directly call only the routines documented in this manual that are not supported by your language. Refer to Appendix B for the specific entry-point names needed to directly call a DPML routine from your platform.

INPUT_ARG_TYPE...

The number and type of input arguments provided by your application. Some routines require more than one argument. Arguments must be coded in the order shown in the interface section of each routine described in this chapter. The supported data types for arguments are described in Section 1.2.

Note Unless otherwise noted, arguments are read-only and passed by value. Arguments passed by another mechanism are prefaced by an asterisk (*); for example, *n in the frexp() routine.

Each generic interface name documented in the interface section of a routine description corresponds to one or more specific entry-point names described in Appendix B. For example, on OpenVMS Alpha systems, the acosd function has five entry-point names; one for each available floating-point data type. The acosd entry-point names are math$acosd_f, math$acosd_s, math$acosd_x, math$acosd_g, and math$acosd_t. On DIGITAL UNIX Alpha systems, the acosd function has two entry-point names corresponding to their supported data types: S_FLOAT and T_FLOAT. The two entry-point names are acosdf for S_FLOAT input arguments and acosd for T_FLOAT arguments. Use the specific entry-point name that corresponds to the input argument data type.

2.3 Working with Exception Conditions

Each DPML routine description contains a table of exceptions. Each exception listed in the table represents an exceptional case that is handled in a platform-specific manner. For example, the atan2() exception table contains the following two entries:

Exceptional Argument	Routine Behavior
y = x = 0	Invalid argument
|y| = |x| = infinity	Invalid argument

The first entry describes an exception condition containing two input arguments with zero values. Upon detecting this error, the routine behavior signals the "invalid argument" condition. The second entry is applicable only to platforms supporting signed or unsigned infinity values. Here, if the absolute value of both input arguments is equal to infinity, an "invalid argument" condition is signaled.

The exact behavior of a routine that detects an exceptional argument varies from platform to platform and is sometimes dependent on the environment in which it is called. The behavior you see depends on the platform and language used. It also depends on how the routine was called and the interaction of the various layers of software through which the call to the routine was made. Remember, access to a DPML routine can be made through direct access (a CALL statement written by a programmer in a source code statement) or through indirect access (from compiler-implemented mathematical syntax).

The default behavior for detecting the x=y=0 arguments is to generate an exception trap when accessing atan2() indirectly through Fortran compiler syntax. C compiler syntax for the atan2() routine sets errno and returns a NaN when give the same input. In these cases, your compiler documentation provides you with information on how to work with exception conditions.

2.4 DPML Routine Interface Examples

This section discusses the atan2() and cdiv() interfaces and explains how to interpret them. The explanations given in this section apply to all DPML routines.

2.4.1 atan2() Interface

The interface to the atan2() routine is:

F_TYPE atan2 (F_TYPE y, F_TYPE x)

The routine name atan2() is the high-level language source-level name that gets mapped to a specific entry-point name documented in Appendix B. This is the name that appears in compiler documentation for this mathematical routine. The appropriate entry-point name is automatically selected when atan2() is called from high-level language syntax. This selection depends upon the data type of the input arguments. If you make direct calls to this routine, you must manually select the proper entry-point name documented in Appendix B for the data type of your input arguments.

The format of the atan2() routine shows that it expects to receive two input arguments by value. Both arguments must be the same F_TYPE. The returned value will also be the same F_TYPE as the input arguments.

For example, on OpenVMS Alpha systems, the G_FLOAT entry-point name is math$atan2_g(). It takes two G_FLOAT arguments by value and returns a G_FLOAT result.

For DIGITAL UNIX Alpha systems, the S_FLOAT entry-point name is atan2f(). The routine takes two S_FLOAT input arguments by value and returns an S_FLOAT result.

2.4.2 cdiv() Interface

The interface to the cdiv() routine is:

F_COMPLEX cdiv (F_TYPE a, F_TYPE b, F_TYPE c, F_TYPE d)

The routine name cdiv() is the generic name that gets mapped to a specific entry-point name documented in Appendix B. Selection of the appropriate entry-point name is done automatically when cdiv() is called from high-level language syntax. This selection depends upon the data type of the input arguments. Again, if you make direct calls to this routine, you must manually select the proper entry-point name documented in Appendix B for the data type of your input arguments.

The format of the cdiv() routine shows that it expects to receive four input arguments by value. All arguments must be the same F_TYPE. The returned value will be an F_COMPLEX data type and will be the same base data type as the input arguments.

For example, on OpenVMS Alpha systems, the F_FLOAT entry-point name is math$cdiv_f(). This routine takes four F_FLOAT input arguments by value and returns an F_FLOAT_COMPLEX result in an ordered pair of F_FLOAT quantities.

For DIGITAL UNIX Alpha systems, the S_FLOAT entry-point name is cdivf(). This routine takes four S_FLOAT input arguments by value and returns an S_FLOAT_COMPLEX result.

Interface

F_TYPE acos (F_TYPE x)

F_TYPE acosd (F_TYPE x)

Description

acos() computes the principal value of the arc cosine of x in the interval [0,pi] radians for x in the interval [-1,1].

acosd() computes the principal value of the arc cosine of x in the interval [0,180] degrees for x in the interval [-1,1].

Exceptions

Interface

F_TYPE acosh (F_TYPE x)

Description

acosh() returns the hyperbolic arc cosine of x for x in the interval [1,+infinity], where acosh(x) = ln(x + sqrt(x**2 - 1)). acosh() is the inverse function of cosh(), where acosh(cosh (x)) = x.

Exceptions

Interface

F_TYPE asin (F_TYPE x)

F_TYPE asind (F_TYPE x)

Description

asin() computes the principal value of the arc sine of x in the interval [-pi/2,pi/2] radians for x in the interval [-1,1].

asind() computes the principal value of the arc sine of x in the interval [-90,90] degrees for x in the interval [-1,1].

Exceptions

Interface

F_TYPE asinh (F_TYPE x)

Description

asinh() returns the hyperbolic arc sine of x for x in the interval [-infinity, +infinity], where asinh(x) = ln(x + sqrt(x**2 + 1)). asinh() is the inverse function of sinh(), where asinh(sinh (x)) = x.

Exceptions

Interface

F_TYPE atan (F_TYPE x)

F_TYPE atand (F_TYPE x)

Description

atan() computes the principal value of the arc tangent of x in the interval [-pi/2,pi/2] radians for x in the interval [-infinity, +infinity].

atand() computes the principal value of the arc tangent of x in the interval [-90,90] degrees for x in the interval [-infinity, +infinity].

Exceptions

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