FLOATING-POINT NUMBERS - Structured Computer Organization - page 687

Item Single precision Double precision

Bits in sign 1 1

Bits in exponent 8 11

Bits in fraction 23 52

Bits, total 32 64

Exponent system Excess 127 Excess 1023

Exponent range − 126 to +127 − 1022 to +1023

2 − 126 2 − 1022 Smallest normalized number

approx. 2 128 approx. 2 1024 Largest normalized number

approx. 10 − 38 to 10 38 approx. 10 − 308 to 10 308

Decimal range

approx. 10 − 45 approx. 10 − 324 Smallest denormalized number

Figure B-5. Characteristics of IEEE floating-point numbers.

Normalized ± 0 < Exp < Max Any bit pattern

Denormalized ± 0 Any nonzero bit pattern

Zero ± 0 0

Infinity ± 1 1 1…1 0

Not a number ± 1 1 1…1 Any nonzero bit pattern

Sign bit

Figure B-6. IEEE numerical types.

really satisfactory, so IEEE invented denormalized numbers . These numbers
have an exponent of 0 and a fraction given by the following 23 or 52 bits. The
implicit 1 bit to the left of the binary point now becomes a 0. Denormalized num-
bers can be distinguished from normalized ones because the latter are not permit-
ted to have an exponent of 0.
The smallest normalized single precision number hasa1asexponent and 0 as
fraction, and represents 1. 0 2 −126 . The largest denormalized number hasa0as

exponent and all 1s in the fraction, and represents about 0. 9999999

×

2 −126 , which
is almost the same thing. One thing to note however, is that this number has only
23 bits of significance, versus 24 for all normalized numbers.
As calculations further decrease this result, the exponent stays put at 0, but the
first few bits of the fraction become zeros, reducing both the value and the number
of significant bits in the fraction. The smallest nonzero denormalized number con-
sists ofa1intherightmost bit, with the rest being 0. The exponent represents

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