1. COMS 361 Computer Organization
Title: Real Numbers
Date: 11/16/2004
Lecture Number: 20

2. Announcements Homework 9 Exam 2 Due Thursday, 11/18/2004
Tuesday, 11/23/2004

3. Review Memory elements Multiplication SR flip-flop (unclocked)
D Latch (clocked)
D flip-flop (clocked)
Master-Slave flop-flops
Clock
Multiplication

4. Outline Multiplication Real numbers IEEE standard 754 (floating point)
Single precision
Double precision
Decimal to floating point conversion
Limitations

5. Multiplication (version 1)

6. Multiplication (version 1)
Left shifted multiplicand
Multiplicand register size must be at least
Multiplicand bits + multiplier bits long
ALU size same as the multiplicand register

7. Multiplication (version 1)
x 1011
x 0001
x 0101
x 0010

8. Multiplication (version 2)
At least half the multiplicand bits are zero
Waste to make a 64 bit adder
Left shift inserts zeros on the right end of multiplicand
Right part of the product will not change
Shift the product register right instead
Fixes the multiplicand
32 bits
Multiplicand register
ALU

9. Multiplication (version 2)

10. Multiplication (version 2)
0010
0010
x 0010
x 1011
0010
0010
x 0001
x 0101

11. Multiplication (version 3)
Product register has unused space
Large enough to hold the multiplier
As unused bits are filled
Multiplier requires less bits
Initialize product register
Zeros in the left half
Multiplier in the right half

12. Multiplication (version 3)

13. Real Numbers
Are these real numbers or are they Sears numbers?

14. Real Numbers Scientific Notation Normalized scientific notation
Number times 10 to a power
= * 103
Normalized scientific notation
Zero to the left of the decimal point
= * 106
Advantageous to store real numbers in 32 bits (word size)

15. Real Numbers 0.1012 = 0 * 20 + 1 * 2-1 + 0 * 2-2 + 1 * 2-3
/ / / /16
= 0 * * * * 2-3
= =

16. Real Numbers Decimal to binary conversion algorithm
1210 = 0 * * 23
1210 – 810 = 410 2 1 1
410 = 1 * 22
410 – 410 = 010

17. Real Numbers
= 0.?????2
20 = 1
2 -1 = 0.5 1
= 1 * 2 -1
2 -2 = 0.25
– 0.5 =
2 -3 = 0.125
2 -4 =
= 1 * 2 -2 1
2 -5 =
– 0.25 =
2 -6 =
= 1 * 2 -3 1
2 -7 =
– = 010
2 -8 = 0. 2

18. Real Number Storage
Real numbers are stored in floating point representation
IEEE Standard 754
Issued 1985
Allows using data on different machines
A sign
An exponent
A mantissa also called a significand (normalized decimal fraction)
Single digit to the left of the decimal point

19. IEEE Standard 754 Provides two floating point types Single precision
24-bits of significand precision
Double precision
53-bits of significand precision

20. IEEE Single Precision IEEE standard 754
Floating point number representation
32-bit
s eeeeeeee fffffff ffffffffffffffff
s: (1) sign bit
0 means positive, 1 means negative s
exponent
significand 31 30 23 22

21. IEEE Single Precision s eeeeeeee fffffff ffffffffffffffff
e: (8) exponent bits [-126 … 127]
A bias of 127 is added to the exponent
Exponent of 0 is stored as 127, stored exponent of 200 means actual exponent is (200 – 127) = 73
Stored exponent of all zeros and ones are reserved for special numbers
f: (24) fractional part [23 bits + 1 implied bit]
Since number to the left of the decimal point is not zero, its binary representation will have a leading one
Saves a bit, a one is implied and does not need to be explicitly stored

22. Special Single Cases Two zeros Signed zero
e = 0, f = 0 (exponent and fractional bits are all 0)
(-1)s x 0.0
0x (+0)
0x (-0)

23. Special Single Cases Positive infinity Negative infinity +INF
s = 0, e = 255, f = 0 (all fractional bits are all 0)
0x7f
Negative infinity
-INF
s = 1, e = 255, f = 0 (all fractional bits are all 0)
0xff

24. Special Single Cases Not-A-Number (NaN)
s = 0 | 1, e = 255, f != 0 (at least one fractional bit is NOT 0)
There are many representations for NaN
Here is one example
0x7fc0 0000

25. Special Single Cases
The smallest incremental value in IEEE single precision representation
A single one bit in the least significant bit of the significand
23-bits, smallest represents
Small but a finite number
Gives rise to the accuracy number can be represented

26. Special Single Cases Example:
No digits to the left of the decimal point
1.3 rounded to 1.0, 1.7 rounded to 2.0
Error is no more than 1/2
23 digits to the left of the binary point
Error is no more than
Accurate to 7 decimal places
Compute 1

27. IEEE Floating Point Numbers
Minimum representable number is:
Minimum exponent (-126)
Minimum significand
Minimum single precision floating point number 0x

28. IEEE Floating Point Numbers
Maximum representable number is:
Maximum exponent (127)
Maximum significand
Maximum single precision floating point number
0x7f7f ffff

29. IEEE Double Precision IEEE standard 754
Floating point number representation
64-bit
s eeeeeeeeeeeffffffffffffffffffffffffffffffffffffffffffffffffff
s: (1) sign bit
0 means positive, 1 means negative s
exponent
significand 63 62 52 51 32
significand 31

30. Double Precision
s eeeeeeeeeeeffffffffffffffffffffffffffffffffffffffffffffffffff
e: (11) exponent bits [-1022 … 1023]
A bias of 1023 is added to the exponent
Exponent of 0 is stored as 1023, stored exponent of 2000 means actual exponent is (2000 – 1023) = 977
Stored exponent of all zeros and ones are reserved for special numbers
f: (53) fractional part [52 bits + 1 implied bit]
Since number to the left of the decimal point is not zero, its binary representation will have a leading one
Saves a bit, a one is implied and does not need to be explicitly stored

31. Real (Decimal) Number Storage
Double precision floating point numbers
s: (1) sign bit
e: (11) exponent bits [-1022 … 1023]
f: (53) fractional part [52 bits + 1 implied bit]
seeeeeee
eee f f f f
f f f f f f f f
Byte
Byte

32. Special Double Cases Two zeros Signed zero
e = 0, f = 0 (exponent and fractional bits are all 0)
(-1)s x 0.0
64 bits
… 0000
0x (+0)
… 0000
0x (-0)

33. Special Double Cases Positive infinity Negative infinity +INF
s = 0, e = 2047, f = 0 (all fractional bits are all 0)
… 0000
0x7ff
Negative infinity
-INF
s = 1, e = 2047, f = 0 (all fractional bits are all 0)
… 0000
0xfff

34. Special Double Cases Not-A-Number (NaN)
s = 0 | 1, e = 2047, f != 0 (at least one fractional bit is NOT 0)
There are many representations for NaN
Here is one example
… 0000
0x7ff

35. Special Double Cases Maximum double number
… 1111
0x7fef ffff ffff ffff
x 10308
Minimum positive single number
… 0000 0x x

36. Decimal to Float Conversion
Show – in IEEE single precision format
First, save sign (negative so 1) and convert to binary…
= x 20
Normalize…
= x 24
Strip 1 off the mantissa and extend to form significand =
Bias the exponent…
Exp + Bias = = 131 =

37. Real Number Storage Hex value : 0xC1C10000 Link me baby
Hex value : 0xC1C10000
Link me baby

38. Real Number Storage
Numbers have limited precision
Compute 1

39. Real Number Storage #include <iostream.h> void main() {
cout << "precision example" << endl;
cout << "Number of bytes in a float: " << sizeof(float) << endl;
float epsilon = 1.0f, value;
int iteration = 0;
int maxIteration = 100;
while(iteration < maxIteration) {
epsilon /= 2.0;
value = 1.0f + epsilon;
if (value == 1) break;
iteration++;
} // end while(...)
cout << "Iteration: " << iteration << " Epsilon: " << epsilon
<< " Value: " << value << endl << endl;
iteration = 0;
double epsilonD = 1.0, valueD;
cout << "Number of bytes in a double: " << sizeof(double) << endl;
epsilonD /= 2.0;
valueD = epsilonD;
if (valueD == 1) break;
cout << "Iteration: " << iteration << " Epsilon: " << epsilonD
<< " Value: " << valueD << endl; }

40. Real Number Storage Numbers have limited precision
Most real numbers have an infinite decimal expansion

41. Real Number Storage Limited Range and Precision
There are three categories of numbers left out when floating point representation is used
Numbers out of range because their absolute value is too large (similar to integer overflow)
Numbers out of range because their absolute value is too small (numbers too near zero to be stored given the precision available
Numbers whose binary representations require either an infinite number of binary digits or more binary digits than the bits available

42. Real Number Storage Limited Range and Precision Illustrated
With one bit to the right of the decimal point, only the real number 0.5 can be represented.

43. Real Number Storage Limited Range and Precision Illustrated
real numbers that can be represented with two bits
0.25, 0.5, 0.75
real numbers that can be represented with three bits
0.125, 0.25, 0.375, 0.5, 0.625, 0.75, 0.875
The holes correspond to all the unrepresented numbers: 0.126, 0.255, 0.3, …