MOTOROLA DSP96002 USER’S MANUAL D-3

Figure D-1. SP and DP Formats

31 30 23 22 0
S8-bit biased
exponent 23-bit fraction

Single Precision
Double Precision

S11-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

):