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Table d-1. parameters for numerical formats – Motorola DSP96002 User Manual

Page 755

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MOTOROLA

DSP96002 USER’S MANUAL

D-3

Figure D-1. SP and DP Formats

31 30

23 22

0

S

8-bit biased
exponent

23-bit fraction

Single Precision

Double Precision

S

11-bit biased
exponent

52-bit fraction

63 62

52 51

0

p-1

bias

e

min

e

max

SP

23

127

+1

+254

DP

52

1023

+1

+2046

Table D-1. Parameters for Numerical Formats

f = •b

1

b

2

•••b

p-1

There are 23 fractional bits (p=24) (bits 0 through 22) in the SP format, and 52 fractional bits

(p=53) (bits 0 through 51) in the DP format.

The sign bit, exponent, and fraction fields encode the numerical values of floating-point numbers, as well as

±

0,

±

, and NaNs as follows:

1.

Normalized Numerical Values (

E

min

E

E

max

): For numerical values, the biased exponent

e

lies between

e

min

and

e

max

, inclusive. Equivalently, the exponent

E

takes on values between

E

min

and

E

max

inclusive. Table D-1 summarizes these values for SP and DP. If the biased ex-

ponent

e

is larger than

e

min

(

E

is larger than

E

min

), the number in question is called normalized,

i.e. the implicit integer value b0 is equal to one. Note that this integer value is not stored in mem-

ory. Normalized numbers x are equal in value to:

x = (-1)

s

• 2

e - bias

1.f

where

1.f

is a binary, fixed point number, i.e.:

1.f = 1+(o.5) • b

1

+ (0.25) • b

2

+...+ (–

1

2

)

p-1

• b

p-1

Therefore, the smallest magnitude of any normalized number is equal to (

e=e

min

, f=0):

x

min,n

= 1 • 2

emin - bias

Using the value from Table D-1, this equals approximately 1.18 • 10

-38

for SP numbers.

The largest (normalized) numerical value that can be represented equals (all

b

i

=1, e=e

max

):