1. COMP201 Computer Systems Floating Point Numbers

2. Floating Point Numbers  Representations considered so far have a limited range dependent on the number of bits: 16 bits0 to 65535 for unsigned -32768 to 32767 for 2's complement. 32 bits0 to 4294967295 for unsigned -2147483648 to 2147483647 2's Complement  How do we represent the following numbers: mass of an electron = 0.00000000000000000000000000000910956Kg Mass of the earth = 5975000000000000000000000Kg  Both these numbers have very few significant digits but are well beyond the range above.

3. Floating Point Numbers  We normally use scientific notation: Mass of electron = 9.10956 x 10-31 Kg Mass of earth = 5.975 x 1024 Kg These numbers can be split into two components:  Mantissa  Exponent e.g. 9.10956 x 10-31 Kg mantissa = 9.10956exponent = -31

4. Floating Point Numbers  Also need to be able to represent negative numbers  Same approach can be taken with binary numbers i.e. a x re where a is the mantissa, r is the base( Radix) and e is the exponent  Notice, we are trading off precision and possibly making computation times longer, in exchange for being able to represent a larger range of numbers.

5. Defining a floating point number Several things must be defined, in order to define a floating point number:  Size of mantissa  Sign of mantissa  Size of exponent  Sign of exponent  Number base in use  9.10956  Positive  31  Negative  10

6. Floating Point Numbers  Consider the following using eight digits and radix 10: MSDLSD Exponent (2 digits) Mantissa (5 digits) The high order bit is the sign bit (0 for + and 1 for - )

7. Floating Point Numbers But this has serious limitations! Range of numbers which can be represented is 10 0 – 10 99 since only two digits are devoted to exponent. Precision is only 5 digits No provision for negative exponents

8. Floating Point Numbers (Continued) One solution for the limitations of range and negative exponents, is to decrease the positive range to 49, by applying an offset of 50 (excess-50 notation). Then, the range of numbers which can be expressed becomes: 0.00001 x 10 -50 to 0.99999 x 10 49 Often implementations require that the most significant digit not be zero so restrict the most negative value to 0.10000 x 10 -50 and the all-0’s represents 0.

9. From the text: Excess-50 notation Range of represented numbers

10. Examples (from textbook) Problem: Convert 246.8035 to the standard format Write as 0.2468035 x 10 3 Truncate at 5 digits: 0.24680 x 10 3 Final number: 05324680 Problem: Convert - 0.00000075 to FP format. Write as -0.75 x 10 -6 Zero fill to 5 digits Final number:  54475000 3 + 50 (the offset) NOTE: Plus is represented by 0; minus by 5 in this format.

11. Floating point in the computer Within the computer, we work in base 2, and use a format such as: NOTE: Bit order depends upon the particular computer

12. Numbers represented Using 32 bits to represent a number, with 1 sign bit, 8 bits for exponent, and the remaining (23 bits) for mantissa. In order to represent negative exponents, use excess-128 notation. Range of numbers represented is approximately 10 -38 to 10 +38 (in decimal terms)

13. But this is not the end! The precision of the mantissa can be improved from 23 bits to 24 bits by noticing that the MSB of the mantissa in a normalized binary number is always “1” And so, it can be implied, rather than expressed directly. The added complications (see below) are felt to be a good tradeoff for the added precision. This complication is the fact that certain numbers are too small to be normalized, and the number 0.0 cannot be represented at all! The solution is to utilize certain codes for these special cases.

14. Floating point numbers– the IEEE standard IEEE Standard 754  Most (but not all) computer manufactures use IEEE-754 format  Number represented: (-1) S * (1.M)*2 (E - Bias) 2 main formats: single and double SExponentMantissa 1823 Single Bias = 127 S ExponentMantissa 11152 Double Bias = 1023

15. And there are special cases to be considered: Zero, represented by +/- 2 -126 x 0.M 2E -127 x 1.M +/- infinity Special condition Exponent Mantissa 0+/- 0 0not 0 1 -254 any 255+/- 0 255not 0

16. Special cases (continued)  The thing to remember about the special cases, is that they are SPECIAL, that is, not expected to occur.  They cover numbers outside the range expected, by using some unlikely-to-be used codes: Special code for 0.0 Special code for numbers too small to be normalized… Special code for infinity  What you are expected to remember is that these special codes exist, and they limit the range of numbers that can be represented by the standard.

17. Final ranges represented 2 -126 to 2 127 Approx. 10 -38 to 10 38 Similarly, the double-precision used sixty-four bits, and represents a range of approximately 10 -300 to 10 300

18. Floating Point Numbers IEEE Standard 754  Example: What Decimal number does the following IEEE floating point number represent? 0 10000001 10110000000000000000000 Sign = 0 (positive) Exponent = 2 (0x81 or 129 – 127 = 2) Mantissa = 1.10110000000000000000000 implied Final answer: 110.11 2 or 6.75 10 The bias

19. More…  What is the IEEE f.p. number 01000001111111100000000000000000 Converted to decimal? Begin by dividing up the number into the fields of sign, exponent and mantissa: 0 10000011 11111100000000000000000 Then, convert the exponent ( 10000011 = 131) and subtract the bias (127) to get the shift (4) Add in the “implied one” to the mantissa 1.11111100000000000000000 And shift 4 places, to get the final number 11111.11, or 31.75 in decimal. Mantissa (23 bits)

20. And more…  What is the decimal number 32.125 converted to IEEE fp format? First, convert the number to binary: 100000.001 Then, normalize for 1.xxxx format: 1.00000001 shift = 5 Add in the bias (127) and convert the total (132) to binary (10000100 or 0x84) Assemble final number: 0 1000010 00000001000000000000000 I’ve added spaces for emphasis

21. Floating Point Application Floating point arithmetic is very important in certain types of application  Primarily scientific computation Many applications use no floating point arithmetic at all  Communications  Operating systems  Most graphics

22. Floating point implementation Floating point arithmetic may be implemented in software or hardware. It is much more complex than integer arithmetic.  Hardware instruction implementations require more gates than equivalent integer instructions and are often slower  Software implementations are generally very slow CPU Floating performance can be very different from Integer performance