Inside Macintosh: PowerPC Numerics / Part 1 - The PowerPC Numerics Environment

Chapter 2 - Floating-Point Data Formats

Range and Precision of Data Formats

Table 2-7 shows the precision, range, and memory usage for each numeric data format. You can use this table to compare the data formats and choose which one is needed for your application. Typically, choosing a data format requires that you determine the tradeoffs between

• fixed-point or floating-point form
• precision
• range
• memory usage
• speed

In the table, decimal ranges are expressed as rounded, two-digit decimal representations of the exact binary values. The speed of a given data format varies depending on the particular implementation of PowerPC Numerics. (See Chapter 5, "Conversions," for information on aspects of conversion relating to precision.)

Table 2-7 Summary of PowerPC Numerics data formats

	Single	Double	Double-double
Size (bytes:bits)	4:32	8:64	16:128
Range of binary exponents
Minimum	$-126$	$-1022$	$-1022$
Maximum	127	1023	1023
Significand precision
Bits	24	53	107
Decimal digits	7-8	15-16	32
Decimal range (approximate)
Maximum positive	$3.4\times 10+38$	$1.8\times 10+308$	$1.8\times 10+308$
Minimum positive norm	$1.2\times 10-38$	$2.2\times 10-308$	$2.2\times 10-308$
Minimum positive denorm	$1.4\times 10-45$	$4.9\times 10-324$	$4.9\times 10-324$
Maximum negative denorm	$-1.4\times 10-45$	$-4.9\times 10-324$	$-4.9\times 10-324$
Maximum negative norm	$-1.2\times 10-38$	$-2.2\times 10-308$	$-2.2\times 10-308$
Minimum negative	$-3.4\times 10+38$	$-1.8\times 10+308$	$-1.8\times 10+308$

For example, in single format, the largest representable number is composed as follows:

significand

$=(2+-2-23)$

$=1$ .11111111111111111111111₂

exponent

$=127$

value

$=(2+-2-23)\times 2127$

\xC5 $3.403\times 1038$

The smallest positive normalized number representable in single format is made up as follows:

significand

$=1$

$=1$ .00000000000000000000000₂

exponent

$=-126$

value

$=1\times 2-126$

\xC5 $1.175\times 10-38$

For denormalized numbers, the smallest positive value representable in the single format is made up as follows:

significand

$=2-23$

$=0$ .00000000000000000000001₂

exponent

$=-126$

value

$=2-23\times 2-126$

\xC5 $1.401\times 10-45$