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E-4 Series 90-30/20/Micro Programmable Controllers Reference Manual – September 1998 GFK-0467K
E
Values of Floating-Point Numbers
Use the following table to calculate the value of a floating-point number from the binary number
stored in two registers.
Exponent (e) Mantissa (f) Value of Floating Point Number
255 Non-zero Not a valid number (NaN).
255 0
–1
s
* ∞
0 < e < 255 Any value
–1
s
* 2
e–127
* 1.f
0 Non-zero
–1
s
* 2
–126
* 0.f
0 0 0
f = the mantissa. The mantissa is a binary fraction.
e = the exponent. The exponent is an integer E such that E+127 is the power of 2 by which the mantissa
must be multiplied to yield the floating-point value.
s = the sign bit.
* = the multiplication operator.
For example, consider the floating-point number 12.5. The IEEE floating-point binary
representation of the number is:
01000001 01001000 00000000 00000000
or 41480000 hex in hexadecimal form. The most significant bit (the sign bit) is zero (s=0). The
next eight most significant bits are 10000010, or 130 decimal (e=130).
The mantissa is stored as a decimal binary number with the decimal point preceding the most
significant of the 23 bits. Thus, the most significant bit in the mantissa is a multiple of 2
–1
, the next
most significant bit is a multiple of 2
–2
, and so on to the least significant bit, which is a multiple of
2
–23
. The final 23 bits (the mantissa) are:
1001000 00000000 00000000
The value of the mantissa, then, is .5625 (that is, 2
–1
+ 2
–4
).
Since e > 0 and e < 255, we use the third formula in the table above:
number = –1
s
* 2
e–127
* 1.f
= –1
0
* 2
130–127
* 1.5625
= 1 * 2
3
* 1.5625
= 8 * 1.5625
= 12.5
Thus, you can see that the above binary representation is correct.
The range of numbers that can be stored in this format is from ± 1.401298E–45 to
± 3.402823E+38 and the number zero.