Page 232 - Introduction to Microcontrollers Architecture, Programming, and Interfacing of The Motorola 68HC12

7.5 Floating-Point Arithmetic and Conversion 209

1. Attach a zero round byte to each significand and unnormalize the number with the
smaller exponent.
2, Add significands of operands (including the round byte).
3, If an overflow occurs in the significand bits, shift the bits for the magnitude and
round byte right one bit and increment the exponent.
4. If all bits of the unrounded result are zero, put the sign of the result equal to + and
the exponent of the result to the most negative value; otherwise, renormalize the
result, if necessary, by shifting the bits of the magnitude and round byte left and
decrementing the exponent for each shift.
5. If underflow occurs, flush the result to zero, and set the underflow flag; otherwise,
round the result.
6, If overflow occurs, put the magnitude equal to the maximum value, and set the
overflow flag.

Figure 7.20. Procedure for Floating-Point Addition

Converting the decimal fraction into a binary floating-point number can be carried
out in two steps.
1. Convert the decimal fraction to a binary fraction.
2, Convert the binary fraction to a binary floating-point number.
Step 2 is straightforward, so we concentrate our discussion on Step 1, converting a
decimal fraction to a binary fraction.
Converting fractions between different bases presents a difficulty not found when
integers are converted between different bases. For example, if

is a base-r fraction, then it can happen that when f is converted to a base-s fraction.

that is, bi is not equal to 0 for infinitely many values of i. (As an example of this,
expand the decimal fraction 0.1 into a binary one.) Rather than trying to draw analogies
to the conversion of integer representations, it is simpler to notice that multiplying the
right-hand side of (18) by s yields bi as the integer part of the result, multiplying the
resulting fractional part by s yields b 2 as the integer part, and so forth. We illustrate the
technique with an example.
Suppose that we want to convert the decimal fraction .345786 into a binary fraction
so that

l 2
.345786 = b x * 2~ + b 2 * 2" + ...
Then bi is the integer part of 2 * (.345786), b2 is the integer part of 2 times the
fractional part of the first multiplication, and so on for the remaining binary digits. More
often than not, this conversion process from a decimal fraction to a binary one does not