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## Fd05 - hsl mathematical software library

**1 SUMMARY**
This function

**supplies real-valued machine constants relating to the floating-point storage and arithmetic **of

A nonzero floating-point number is stored in the form ±

*m *β , where β is known as the base (or radix) of the
arithmetic,

*m *is the mantissa (or significand or fraction) and

*e *is the exponent (or characteristic). The mantissa isusually normalized so that any floating-point number has a unique representation. Individual machines differ in theway that the normalization is performed. The exponent is stored as a sequence of binary digits (bits); the sign of theexponent either occupies one of these digits, or, more commonly, the actual value of the exponent is obtained by
adding the stored binary representation to a fixed negative bias. The mantissa is represented as

*m *= ∑

*mi *β
0 ≤

*m *< β,

*j *is usually 0 or 1 and

*m *is usually normalized so that

*m *> 0. (Under special circumstances,

*m *may be
zero; such circumstances are typically associated with implementations of gradual underflow on a particularmachine.)

**ATTRIBUTES **—

**Version: **1.0.0.

**Types: **FD05A, FD05AD.

**Remark: **This package is also included in the HSL

Archive.

**Calls: **None.

**Original date: **April 1988.

**Origin: **N.I.M. Gould and S. Marlow, Harwell.

**Licence: **A

third-party licence for this package is available without charge.

**2 HOW TO USE THE PACKAGE**
Two versions of the function are available corresponding to the use of single and double precision arithmetic.

**2.1 Argument list**
INUM is an INTEGER variable which must be set by the user to select which one of the real machine constants is
required. Possible values of INUM and the corresponding constants are:
1. FD05A/FD05AD(1) is the smallest REAL/DOUBLE PRECISION number of the form β such that 1.0 + β
and 1.0 are different stored numbers. On most machines

*i *= 1 −

*n*. This value is that returned by theNumeric Inquiry Function EPSILON in the ANSI Fortran 90 standard and is commonly called themachine precision.

2. FD05A/FD05AD(2) is the smallest REAL/DOUBLE PRECISION number of the form β such that 1.0 − β
and 1.0 are different stored numbers. On most machines

*i *= −

*n*.

3. FD05A/FD05AD(3) is a close approximation to the smallest positive REAL/DOUBLE PRECISION
number which may be stored on the machine. This number is normally β
smallest allowable value of the exponent, but may be as small as β
4. FD05A/FD05AD(4) is a close approximation to the smallest positive REAL/DOUBLE PRECISION
number which may be stored on the machine to full precision, i.e., for which

*m *> 0. This number is
is the smallest allowable value of the exponent. The numbers
FD05A/FD05AD(3) and FD05A/FD05AD(4) will be identical on machines which have no provision for

**All use is subject to licence.**
**HSL ARCHIVE**
gradual underflow. This value is that returned by the Numeric Inquiry Function TINY in the ANSIFortran 90 standard.

5. FD05A/FD05AD(5) is a close approximation to the largest finite positive REAL/DOUBLE PRECISION
number which may be stored on the machine. This number is normally β
the largest allowable value of the exponent, and is the value returned by the Numeric Inquiry FunctionHUGE in the ANSI Fortran 90 standard.

INUM is not altered by the function.

**Restriction: **1 ≤ INUM ≤ 5. Note: FD05 does not indicate an error when the

value of INUM is out of range; a value of INUM ≤ 0 is treated as 1 and a value of INUM > 5 is treated as 5.

FD05A/FD05AD is a REAL (DOUBLE PRECISION in the D version) function whose value will be set to the required

**3 GENERAL INFORMATION**
**Use of common:**
**Other routines called directly:**
**Input/output:**
**Restrictions:**
The constants have been set by the original implementor of HSL on your machine. Further detail of the machine
representation of floating-point numbers may be found in, for instance, the Dictionary of Computing (OxfordUniversity Press, 1983).

**5 EXAMPLE OF USE**
As a very simple example, we wish to find the machine precision and the largest finite floating-point number which
may be stored on the machine in double precision; in this case an IBM 3084Q is used.

DOUBLE PRECISION DNUM, FD05ADDNUM = FD05AD( 1 )WRITE( 6, 2000 ) DNUMDNUM = FD05AD( 5 )WRITE( 6, 2010 ) DNUMSTOP
2000 FORMAT( ' MACHINE PRECISION (DOUBLE PRECISION) = ', 1P, D12.4 )2010 FORMAT( ' LARGEST FLOATING-POINT NUMBER (DOUBLE PRECISION) = ',
LARGEST FLOATING-POINT NUMBER (DOUBLE PRECISION) =

**All use is subject to licence.**
Source: http://www.hsl.rl.ac.uk/archive/specs/fd05.pdf

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