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Representation of Numbers 265
IEEE 754 standard In the early days of the digital computer revolution,
each processor design had its own internal representation for floatingpoint
numbers. Since floating-point arithmetic is more complicated to
implement, some of these designs did incorrect arithmetic. Therefore, in
1985 IEEE issued a standard (IEEE standard 754-1985 or IEEE-754 for
short) to allow floating-point data exchange among different computers
and to provide hardware designers with a model known to be correct. Currently,
almost all manufacturers design main processors or a dedicated
coprocessor for floating-point operations using the IEEE-754 standard
representation.
The IEEE 754 standard defines three formats for binary numbers: a
32-bit single precision format, a 64-bit double precision format, and an
80-bit temporary format (which is used internally by the processors or
arithmetic coprocessors to minimize rounding errors).
We will briefly describe the 32-bit single precision standard. This
standard has many similarities with the floating-point representation discussed
above, but there are also differences. Remember, this is another
model advocated by IEEE. The form of this model is
ˆx =
sign of M
↓
± xx ···x
} {{ }
8−bit E
xx ···x
} {{ }
23−bit M
(6.44)
The mantissa’s value is called the significand in this standard. Features
of this model are as follows:
• If the sign bit is 0, the number is positive; if the sign bit is 1, the
number is negative.
• The exponent is coded in 8-bit excess-127 (and not 128) format. Hence
the uncoded exponents are between −127 and 128.
• The mantissa is in 23-bit binary. A normalized mantissa always starts
with a bit 1, followed by the binary point, followed by the rest of the
23-bit mantissa. However, the leading bit 1, which is always present in a
normalized mantissa, is hidden (not stored) and needs to be restored for
computation. Again, note that this is different from the usual definition
of the normalized mantissa. If all the 23 bits representing the mantissa
are set to 0, the significand is 1 (remember the implicit leading 1). If
all 23 bits are set to 1, the significand is almost 2 (in fact 2 − 2 −23 ). All
IEEE 754 normalized numbers have a significand that is in the interval
1 ≤ M