advertisement

FLOATING POINT REPRESENTATION IN BINARY Parts of a Floating point number Floating point representation in Decimal Consider an example 123.34 It is represented as Or +123.34 x 10+0 +0.123 x 10+3 ………… Example 1 Excess- 50 representations (Exponent) (Symbolic representation of Decimal number as Binary digit) Assume our exponent is maximum two digits So the maximum positive value would be 99 (positive number only) In Binary number system, Positive and negative numbers represented by “bits” (Sign bits) For explanation we are using decimal numbers for Binary representation Ie Without sign symbol(+.-) we have to utilize digits or numbers for sign For that we are dividing the total possible quantity that to be represented in two digit into two parts 00 to 49 and 50 to 99 (for negative as well as positive) -50 to -01 and 0 to 49. Number above 50 represent positive exponent and number below 50 represents negative exponent So the Example 1 can be represented as +0.123 x 10+53 Consider the example: +.003456 x 10-5 After normalization (eliminating leading zeros) we get +0.3456 x 10-7 In excess 50 representation we represent as +0.3456 x 1043 Binary representation • The mantissa should be normalized • Has an implied “1” on left of the decimal place Note that After normalization in binary, the first bit must be 1 and that will not store in register 37.2510 = 100101.012 = 1.0010101 x 25 7.62510 = 111.1012 = 1.11101 x 22 0.312510 = 0.01012 = 1.01 x 2-2 Binary Scientific Notation The first, or leftmost, field of our floating point representation will be the sign bit: 0 for a positive number, 1 for a negative number. The second field of the floating point number will be the exponent. Since we must be able to represent both positive and negative exponents, we will use a convention which uses a value known as a bias of 127 (Excess 127) to determine the representation of the exponent. Excess -127 Representation (Exponent) As Excess 50 in decimal, binary numbers can represent in Excess 127 format (For single precession) and excess-1023 ( for double precession ) The mantissa is the set of 0’s and 1’s to the left of the radix point of the normalized (when the digit to the left of the radix point is 1) binary number. Ex:1.00101 X 23 The mantissa is stored in a 23 bit field. Ie with 23 bit storage, we can represent 24 bit mantissa with implied 1 bit on the left of decimal point