Fortran 77 Language Reference

Appendix C

Data Representations

Whatever the size of the data element in question, the most significant bit of the data element is always stored in the lowest-numbered byte of the byte sequence required to represent that object.

This appendix is a brief introduction to data representation. For more in-depth explanations, see the Sun WorkShop Fortran Programming Guide and Numerical Computation Guide.

Real, Double, and Quadruple Precision

Real, double precision, and quadruple precision number data elements are represented according to the IEEE standard by the following form, where f is the bits in the fraction. Quad is SPARC only.

(-1)^sign * 2^{exponent-bias} *1.f

TABLE C-1   Floating-point Representation

	Single	Double	Quadruple
Sign	Bit 31	Bit 63	Bit 127
Exponent	Bits 30-23 Bias 127	Bits 62-52 Bias 1023	Bits 126-112 Bias 16583
Fraction	Bits 22-0	Bits 51-0	Bits 111-0
Range approx.	3.402823e+38 1.175494e-38	1.797693e+308 2.225074e-308	3.362E-4932 1.20E+4932

Extreme Exponents

The representations of extreme exponents are as follows.

Zero (signed)

Zero (signed) is represented by an exponent of zero and a fraction of zero.

Subnormal Number

The form of a subnormal number is:

(-1) ^sign * 2 ^1-bias *0.f

where f is the bits in the significand.

Signed Infinity

Signed infinity--that is, affine infinity--is represented by the largest value that the exponent can assume (all ones), and a zero fraction.

Not a Number (NaN)

Not a Number (NaN) is represented by the largest value that the exponent can assume (all ones), and a nonzero fraction.

Normalized REAL and DOUBLE PRECISION numbers have an implicit leading bit that provides one more bit of precision than is stored in memory. For example, IEEE double precision provides 53 bits of precision: 52 bits stored in the fraction, plus the implicit leading 1.

IEEE Representation of Selected Numbers

The values here are as shown by dbx, in hexadecimal.

TABLE C-2   IEEE Representation of Selected Numbers

Value	Single-Precision	Double-Precision
+0	00000000	0000000000000000
-0	80000000	8000000000000000
+1.0	3F800000	3FF0000000000000
-1.0	BF800000	BFF0000000000000
+2.0	40000000	4000000000000000
+3.0	40400000	4008000000000000
+Infinity	7F800000	7FF0000000000000
-Infinity	FF800000	FFF0000000000000
NaN	7Fxxxxxx	7FFxxxxxxxxxxxxx

Arithmetic Operations on Extreme Values

This section describes the results of basic arithmetic operations with extreme and ordinary values. We assume all inputs are positive, and no traps, overflow, underflow, or other exceptions happen.

TABLE C-3   Extreme Value Abbreviations  

Abbreviation	Meaning
Sub	Subnormal number
Num	Normalized number
Inf	Infinity (positive or negative)
NaN	Not a Number
Uno	Unordered

TABLE C-4   Extreme Values: Addition and Subtraction

Left Operand		Right Operand
		0	Sub	Num	Inf	NaN
0		0	Sub	Num	Inf	NaN
Sub		Sub	Sub	Num	Inf	NaN
Num		Num	Num	Num	Inf	NaN
Inf		Inf	Inf	Inf	Note	NaN
NaN		NaN	NaN	NaN	NaN	NaN

Note: Inf ± Inf and Inf + Inf = Inf ; Inf - Inf = NaN.

TABLE C-5   Extreme Values: Multiplication

Left Operand		Right Operand
		0	Sub	Num	Inf	NaN
0		0	0	0	NaN	NaN
Sub		0	0	NS	Inf	NaN
Num		0	NS	Num	Inf	NaN
Inf		NaN	Inf	Inf	Inf	NaN
NaN		NaN	NaN	NaN	NaN	NaN

In the above table, NS means either Num or Sub result possible.

TABLE C-6   Extreme Values: Division

Left Operand		Right Operand
		0	Sub	Num	Inf	NaN
0		NaN	0	0	0	NaN
Sub		Inf	Num	Num	0	NaN
Num		Inf	Num	Num	0	NaN
Inf		Inf	Inf	Inf	NaN	NaN
NaN		NaN	NaN	NaN	NaN	NaN

TABLE C-7   Extreme Values: Comparison

Left Operand		Right Operand
		0	Sub	Num	Inf	NaN
0		=	<	<	<	Uno
Sub		>		<	<	Uno
Num		>	>		<	Uno
Inf		>	>	>	=	Uno
NaN		Uno	Uno	Uno	Uno	Uno

Notes:

• If either X or Y is NaN, then X.NE.Y is .TRUE., and the others (.EQ., .GT., .GE., .LT., .LE.) are .FALSE.
• +0 compares equal to -0.
• If any argument is NaN, then the results of MAX or MIN are undefined.

Bits and Bytes by Architecture

The order in which the data--the bits and bytes--are arranged differs between VAX computers on the one hand, and SPARC computers on the other.

The bytes in a 32-bit integer, when read from address n, end up in the register as shown in the following tables.

TABLE C-8   Bits and Bytes for Intel and VAX Computers

Byte n+3	Byte n+2	Byte n+1	Byte n
31 30 29 28 27 26 25 24	23 22 21 20 19 18 17 16	15 14 13 12 11 10 09 08	07 06 05 04 03 02 01 00
Most Significant			Least significant

TABLE C-9   Bits and Bytes for 680x0 and SPARC Computers

Byte n	Byte n+1	Byte n+2	Byte n+3
31 30 29 28 27 26 25 24	23 22 21 20 19 18 17 16	15 14 13 12 11 10 09 08	07 06 05 04 03 02 01 00
Most Significant			Least significant

The bits are numbered the same on these systems, even though the bytes are numbered differently.

Following are some possible problem areas:

• Passing binary data over the network. Use External Data Representation (XDR) format or another standard network format to avoid problems.
• Porting raster graphics images between architectures. If your program uses graphics images in binary form, and they have byte ordering that is not the same as for images produced by SPARC system routines, you must convert them.
• If you convert character-to-integer or integer-to-character between architectures, you should use XDR.
• If you read binary data created on an architecture with a different byte order, then you must filter it to correct the byte order.

See also the xdr(3N) man page.