Page 332 - Embedded Microprocessor Systems Real World Design

Now we can create a 16-bit floating-point number from our example value:
0 100000 010001 010
sign exponent mantissa, leading 1 implied

The general steps for converting a decimal number in the form xxx.yyy to floating-point
format are:
Convert xxx (digits to the left of the decimal point) to binary (call it aaaa). Convert yyy
(digits to right of decimal point) to fractional binary, call it bbb. Write as a fractional binary
number:

aaa.bbbb
Shift the number to the right, keeping track of the exponent, until there is a single 1 to
the left of the radix point:

a.aabbb (6bit example shown, works for any size number)
exponent = 2 (because we shifted two positions right)
Drop the leading 1 and calculate the exponent using the bias of the exponent field. If the
number is positive, make the sign bit 0. Otherwise the sign bit is 1.
The IEEE has developed a standard for representation of floating-point numbers. The
IEEE format defines single and double precision values. The IEEE single-precision format
uses 1 sign bit, 8 exponent bits, and 23 mantissa bits, for a total of 32 bits. The double
precision standard uses 64 bits: 1 sign bit, 11 exponent bits, and 52 fractional bits. The single
precision exponent can range from -127 to +127, and the double precision exponent can
range from -1023 to +1023.
Finally, what do we do to indicate zero? Zero can’t be represented by 1.xxxx. The IEEE
standard defines zero as being represented when the exponent and mantissa are both zero.
The sign bit can be either.
The IEEE standard also reserves the maximum exponent value (FF for single precision,
7FF for double precision) to indicate an overflow condition-numbers that are either too
small or too large to be represented.

Appendix B 313