Lecture 6. Fixed and Floating Point Numbers

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1 Lecture 6. Fixed and Floating Point Numbers
COMP211 Computer Logic Design Lecture 6. Fixed and Floating Point Numbers Prof. Taeweon Suh Computer Science Education Korea University

2 What Would You Get? #include <stdio.h> int main() {
signed int sa = 7; signed int sb = -7; unsigned int ua = *((unsigned int *) &sa); unsigned int ub = *((unsigned int *) &sb); printf("sa = %d : ua = 0x%x\n", sa, ua); printf("sb = %d : ub = 0x%x\n", sb, ub); return 0; }

3 What Would You Get? #include <stdio.h> int main() {
float f1 = -58.0; unsigned int u1 = *((unsigned int *) &f1); printf("f1 = %f\n", f1); printf("f1 = %3.20f\n", f1); printf("u1 = 0x%X\n", u1); return 0; } What is this?

4 What Would You Get? #include <stdio.h> int main() {
double d1 = -58.0; unsigned long long u1 = *((unsigned long long *) &d1); printf("d1 = %lf\n", d1); printf("d1 = %3.20lf\n", d1); printf("u1 = 0x%llX\n", u1); return 0; } What is this?

5 What Would You Get? #include <stdio.h> int main() {
float f2 = -0.1; unsigned int u2 = *((unsigned int *) &f2); printf("f2 = %f\n", f2); printf("f2 = %3.20f\n", f2); printf("u2 = 0x%X\n", u2); return 0; } Why are these different? And What is this?

6 What Would You Get? #include <stdio.h> int main() {
float f3 = 0.7; unsigned int u3 = *((unsigned int *) &f3); printf("f3 = %f\n", f3); printf("f3 = %3.20f\n", f3); printf("u3 = 0x%X\n", u3); return 0; } Why are these different? What is this?

7 0.07 != 0.07 ?

8 a x b x c != b x c x a ? We are going to answer to these questions!

9 Number Systems So far we have studied the following integer number systems in computer Unsigned numbers Sign/magnitude numbers Two’s complement numbers What about rational numbers? For example, 2.5, , 0.75 etc Two common notations to represent rational numbers in computer Fixed-point numbers Floating-point numbers

10 Fixed-Point Numbers Fixed point notation has an implied binary point between the integer and fraction bits The binary point is not a part of the representation but is implied Example: Fixed-point representation of 6.75 using 4 integer bits and 4 fraction bits: The number of integer and fraction bits must be agreed upon by those generating and those reading the number There is no way of knowing the existence of the binary point except through agreement of those people interpreting the number

11 Signed Fixed-Point Numbers
As with whole numbers, negative fractional numbers can be represented in two ways Sign/magnitude notation Two’s complement notation Example: using 8 bits (4 bits each to represent integer and fractional parts) 2.375 = Sign/magnitude notation: Two’s complement notation: 1. flip all the bits: 2. add 1: Addition and subtraction works easily in computer with 2’s complement notation like integer addition and subtraction

12 Example Suppose that we have 8 bits to represent a number
4 bits for integer and 4 bits for fraction Compute (-0.625) = 0.625 = in 2’s complement form: 0.125

13 Fixed-Point Number Systems
Fixed-point number systems have a limitation of having a constant number of integer and fractional bits What are the largest and the smallest rational numbers you can represent with 32 bits, assuming 16 bits each for integer and fractional parts? Some low-end digital signal processors support fixed-point numbers Example: TMS320C550x TI (Texas Instruments) DSPs:

14 Floating-Point Numbers
Floating-point number systems circumvent the limitation of having a constant number of integer and fractional bits They allow the representation of very large and very small numbers The binary point floats to the right of the most significant 1 Similar to decimal scientific notation For example, write in scientific notation: Move the decimal point to the right of the most significant digit and increase the exponent: 273 = 2.73 × 102 In general, a number is written in scientific notation as: ± M × BE Where, M = mantissa B = base E = exponent In the example, M = 2.73, B = 10, and E = 2 (that is, × 102)

15 Floating-Point Numbers
Floating-point number representation using 32 bits 1 sign bit 8 exponent bits 23 bits for the mantissa. The following slides show three versions of floating-point representation with using a 32-bit The final version is called the IEEE 754 floating-point standard

16 Floating-Point Representation #1
First, convert the decimal number to binary 22810 = = × 27 Next, fill in each field in the 32-bit: The sign bit (1 bit) is positive, so 0 The exponent (8 bits) is 7 (111) The mantissa (23 bits) is

17 Floating-Point Representation #2
You may have noticed that the first bit of the mantissa is always 1, since the binary point floats to the right of the most significant 1 Example: = = × 27 Thus, storing the most significant 1 (also called the implicit leading 1) is redundant information We can store just the fraction parts in the 23-bit field Now, the leading 1 is implied

18 Floating-Point Representation #3
The exponent needs to represent both positive and negative The final change is to use a biased exponent The IEEE 754 standard uses a bias of 127 Biased exponent = bias + exponent For example, an exponent of 7 is stored as = 134 = Thus , using the IEEE bit floating-point standard is

19 Example Represent -5810 using the IEEE 754 floating-point standard
First, convert the decimal number to binary 5810 = = × 25 Next, fill in each field in the 32-bit number The sign bit is negative (1) The 8 exponent bits are ( ) = 132 = (2) The remaining 23 bits are the fraction bits: (2) It is 0xC in the hexadecimal form Check this out with the result of the sample program in the slide# 3

20 Floating-Point Numbers: Special Cases
The IEEE 754 standard includes special cases for numbers that are difficult to represent, such as 0 because it lacks an implicit leading 1 Number Sign Exponent Fraction X ∞ - ∞ 1 NaN non-zero NaN is used for numbers that don’t exist, such as √-1 or log(-5)

21 Floating-Point Number Precision
The IEEE 754 standard also defines 64-bit double-precision that provides greater precision and greater range Single-Precision (use the float declaration in C language) 32-bit notation 1 sign bit, 8 exponent bits, 23 fraction bits bias = 127 It spans a range from ± X to ± X 1038 Double-Precision (use the double declaration in C language) 64-bit notation 1 sign bit, 11 exponent bits, 52 fraction bits bias = 1023 It spans a range from ± X to ± X 10308 Most general purpose processors (including Intel and AMD processors) provide hardware support for double-precision floating-point numbers and operations

22 Double Precision Example
Represent using the IEEE 754 double precision First, convert the decimal number to binary 5810 = = × 25 Next, fill in each field in the 64-bit number The sign bit is negative (1) The 11 exponent bits are ( ) = 1028 = (2) The remaining 52 bits are the fraction bits: (2) It is 0xC04D0000_ in the hexadecimal form Check this out with the result of the sample program in the slide# 4

23 Represent 0.7 Represent 0.7 in IEEE 754 single precision form
½ = = 0.1(2) // = 0.2 1/8 = = 0.001(2) // =0.075 1/16 = = (2) // =0.0125 1/128 = = (2) // = 1/256 = = (2) // = …… Thus, 0.7 = …(2) = …(2) X 2-1 In IEEE754 single precision, 0.7 = 0x3F333333 Check it out with the slide#6 IEEE754 floating-point standard can’t represent some numbers exactly

24 Binary Coded Decimal (BCD)
Since floating-point number systems can’t represent some numbers exactly such as 0.1 and 0.7, some application (calculators) use BCD (Binary coded decimal) BCD numbers encode each decimal digit using 4 bits with a range of 0 to 9 BCD fixed-point notation examples 1.7 = 4.9 = BCD is very common in electronic systems where a numeric value is to be displayed, especially, in systems consisting solely of digital logic (not containing a microprocessor) - Wiki Decimal BCD Digit 0000 1 0001 2 0010 3 0011 4 0100 5 0101 6 0110 7 0111 8 1000 9 1001

25 Backup Slides

26 Floating-Point Numbers: Rounding
Arithmetic results that fall outside of the available precision must round to a neighboring number Rounding modes Round down Round up Round toward zero Round to nearest Example Round ( ) so that it uses only 3 fraction bits Round down: Round up: Round toward zero: 1.100 Round to nearest: 1.625 is closer to than 1.5 is

27 Floating-Point Addition with the Same Sign
Addition with floating-point numbers is not as simple as addition with 2’s complement numbers The steps for adding floating-point numbers with the same sign are as follows Extract exponent and fraction bits Prepend leading 1 to form mantissa Compare exponents Shift smaller mantissa if necessary Add mantissas Normalize mantissa and adjust exponent if necessary Round result Assemble exponent and fraction back into floating-point format

28 Floating-Point Addition Example
Add the following floating-point numbers: 1.5(10) = 1.1(2) x 20 3.25(10) = 11.01(2) = 1.101(2) x 21 1.1(10) = 0x3FC00000 in IEEE 754 single precision 3.25(10) = 0x in IEEE 754 single precision

29 Floating-Point Addition Example
1. Extract exponent and fraction bits For first number (N1): S = 0, E = 127, F = .1 For second number (N2): S = 0, E = 128, F = .101 2. Prepend leading 1 to form mantissa N1: 1.1 N2:

30 Floating-Point Addition Example
3. Compare exponents 127 – 128 = -1, so shift N1 right by 1 bit 4. Shift smaller mantissa if necessary shift N1’s mantissa: 1.1 >> 1 = (× 21) 5. Add mantissas × 21 × 21 × 21

31 Floating-Point Addition Example
6. Normalize mantissa and adjust exponent if necessary × 21 = × 22 7. Round result No need (fits in 23 bits) 8. Assemble exponent and fraction back into floating-point format S = 0, E = = 129 = , F = 4.75(10) = 0x in the hexadecimal form