Motorola DSP96002 - Manual (Page 755)

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
):