Hardware and Software for Digital Signal Processors - Digital Signal Processing: Fundamentals and Applications - page 425

000000000 00000000000000000000001 ¼
ð 1 Þ

0

ð 0 : 00000000000000000000001 2 Þ 2 0 126 ¼ 2 149

ð smallest positive value Þ

Double Precision Format
The IEEE double precision format is described in Figure 9.12 .
The IEEE double precision floating-point standard representation requires a 64-bit word, which
may be numbered from 0 to 63, left to right. The first bit is the sign bit S , the next eleven bits are the
exponent bits E , and the final 52 bits are the fraction bits F . The IEEE floating-point format in double
precision significantly increases the dynamic range of number representation since there are eleven
exponent bits; the double-precision format also reduces the interval size in the mantissa normalized
range of þ 1to þ 2, since there are 52 mantissa bits as compare with the single precision case of 23 bits.
Applying the conversion formula shown in Figure 9.12 is similar to the single precision case.

31 30

20 19 0 31 0

s exponent fraction fraction

o r e r

11 2 1023

s E

x () (. F )

FIGURE 9.12
IEEE double precision floating-point format.

EXAMPLE 9.12
Convert the following number in IEEE double precision format to the decimal format:
001000.0:110.000

Solution:
Using the bit pattern in Figure 9.12 , we have
s ¼ 0; E ¼ 2 9 ¼ 512 and

1:F ¼ 1:11 2 ¼ð2Þ 0 þð2Þ 1 þð2Þ 2 ¼ 1:75

Then, applying the double precision formula yields
x ¼ð1Þ 0 ð1:75Þ2 5121023 ¼ 1:75 2 511 ¼ 2:6104 10 154

For the purpose of completeness, rules for determining the value x represented by the double-
precision word are listed as follows:

• f E ¼ 2 ; 047 and F is nonzero, then x ¼ NaN ("Not a number").
• f E ¼ 2 ; 047, F is zero, and S is 1, then x ¼ Inifinity.
• f E ¼ 2 ; 047, F is zero, and S is 0, then x ¼þ Inifinity.
• f0 < E < 2 ; 047, then x ¼ð 1 Þ s ð 1 :FÞ 2 E 1 ; 023 , where "1 :F " is intended to represent the
binary number created by prefixing F with an implicit leading 1 and a binary point.