1. Integers & Floating Point Numbers: Limits of Representation
CSE 351 Autumn 2016
Section 3

2. Key Points Remember that there are limitations! Design Decisions
Memory is finite, numbers/data are not finite
We can only represent so much
We have 𝟐 𝒘 distinct bit patterns with w bits
Design Decisions
Efficient/Fast and Easy to Implement
Accuracy
Range
Precision

3. Unsigned Integers Unsigned values follow base 2 system
Example of converting from base 2 to base 10
b 7 b 6 b 5 b 4 b 3 b 2 b 1 b 0 = b b …+ b b
Benefit:
Add and subtract using the normal “carry” and “borrow” rules, just in binary

4. Signed Integers: Two’s Complement
𝐛 𝐰−𝟏 has weight − 𝟐 𝐰−𝟏 , other bits have usual weights + 𝟐 𝐢
. . . b0
bw-1
bw-2
TMax
TMin –1 –2
UMax
UMax – 1
TMax + 1
2’s Complement Range
Unsigned
Range
Benefits:
Roughly same number of (+) and (–) numbers
Positive number encodings match unsigned
Single representation of zero
All zeros encoding (000000…) = 0
Negation is easy: ~x + 1 == -x

5. Values To Remember! Two’s Complement Values
Unsigned Values UMin = 0 000…0 UMax = 2w – 1 111…1
Two’s Complement Values
TMin = –2w–1
100…0
TMax = 2w–1 – 1
011…1
Negative one
111… xF...F
Values for W = 32
Decimal
Hex
Binary
UMax
4,294,967,296
FF FF FF FF
TMax
2,147,483,647
7F FF FF FF
TMin
-2,147,483,648 -1
LONG_MIN =
Values for W = 64
LONG_MAX =
ULONG_MAX =

6. Floating Point Numbers: The Vision
What do we want?
Large range of values
Large numbers and very small numbers
Precise values
Reflect real arithmetic
Support values such as  +∞, ‐∞, Not‐A‐Number (NaN)
Similar encoding to Two’s Complement

7. Floating Point Numbers
V = (–1)s * M * 2E s
exp
frac
Numerical Form
Sign bit s determines whether number is negative or positive
Significand (mantissa) M normally a fractional value in range [1.0, 2.0)
Exponent E weights value by a (possibly negative) power of two
Representation in Memory
MSB s is sign bit s
exp field encodes E (but is not equal to E) – remember the bias
frac field encodes M (but is not equal to M)

8. Floating Point Numbers
Value: ±1 × Mantissa × 2Exponent
Bit Fields: (‐1)S × 1.M × 2(E+bias)
Bias
Read exponent as unsigned, but with bias of –(2w‐1‐1) = –127
Representable exponents roughly ½ positive and ½ negative
Exponent 0 (Exp = 0) is represented as E = 0b 0111 1111
Why?
Floating point arithmetic = easier
Somewhat compatible with 2’s complement

9. Floating Point Numbers: Denormalized
No leading 1
Remember! Implicit exponent is –126
(not –127) even though  E = 0x00
Why?
To represent really smaller numbers that are close to 0

10. Floating Point Representation Summary
Exponent
Mantissa
Meaning
0x00
± 0
Non-zero
± denorm num
0x01 – 0xFE
Anything
± norm num
0xFF
± ∞
NaN

11. Floating Point Limitations: Math Properties
Exponent overflow yields +∞ or -∞
Floats with value +∞, -∞, and NaN can be used in operations
Result usually still +∞, -∞, or NaN; sometimes intuitive, sometimes not
Floating point ops do not work like real math, due to rounding!
Not associative:
( e100) – 1e100 != (1e100 – 1e100)
Not distributive:
100 * ( ) != 100 * * 0.2
Not cumulative
Repeatedly adding a very small number to a large one may do nothing

12. Distribution of Values
What can’t we get?
Between largest norm and infinity: Overflow
Between zero and smallest denorm: Underflow
Between norm numbers?: Rounding

13. Problems Problems Problems!
Consider the decimal number Give the IEEE-754 representation of this number as a 32-bit floating-point number.
Convert 1.1 x 2-128 to IEEE 754 single precision a
b. (note, 1.1x2^-128 is in base 2)
Shift the radix left by 2 to get 0.011x2^-126 which is denormalized

14. Problems Problems Problems!
If x and y have type float, give two different reasons that (x+2*y)-y == x+y might evaluate to 0 (i.e., false).
Refer to midterm 2016 winter
Overflow: If x and y are certain (very large) values, then x+2*y might produce the special value “infinity” (or negative infinity is possible too) and then subtracting y still produces infinity while x+y is still small enough not to overflow.
Rounding error: If x and y are the right number of orders of magnitude apart, we might due to rounding get that x+y is still x while x+2*y-y is slightly more than x.

15. Problems Problems Problems!
What is the largest positive number we can represent with a 10-bit signed two’s complement integer?
Bit pattern?
Decimal value?
2^8 + … + 2^0 = 2^(9 – 1) = = 511

16. Problems Problems Problems!
Assuming unsigned integers, what is the result when you compute UMAX+1?
Assuming two’s complement signed representation, what is the result when you compute TMAX+1?
TMIN (0x )

17. Problems Problems Problems!
Is the ‘==’ operator a good test of equality for floating point values? Why or why not?
No, since floating point suffers from rounding issues. Instead, use the <= or >= operator to test a range once you take a difference.

18. Problems Problems Problems!
Give an example of three floating-point numbers x, y, and z, such that the distributive property x (y + z) = x y + x z does not hold.
X = large number
Y = small number
Z = large number
We might lose y when we add it to z

19. How to use GDB Download calculator.c from class webpage
For debugging, we need to compile the file with debugging symbols. This an be done using –g flag in GCC.
gcc -Wall -std=gnu99 -g calculator.c -o calculator
To load binary into GDB, use following command:
gdb calculator
You should see bunch of information including version and license information
To run binary in GDB, use run command (type run or just r). This will start executing your program till any error occurs in your program.
If you want to start stepping through main(), use start command.
Passing command line arguments in GDB
run calculator 3 4 +
View source code while debugging
Use list command. For example,
if you want to look at the main function, type list main()
If you want to list a content around line 45, then type list 45
If you want to display a range of line numbers such as lines 10-15, then use list 10,15

20. How to use GDB (continued)
Setting Breakpoints
break command creates break point (example: break main).
Each break point is associated with a number. To enable/disable breakpoint, use enable or disable command.
TO see summary of all breakpoints, use info command (example: info break)
To continue execution after breakpoint, use continue or c command
Stepping through source code in GDB
To step one line of source code at a time, use next or n command.
To step through functions, use step or s command.
To step out of the function, use finish command.
Printing values while debugging
Use print command.
Exiting GDB
Press Ctrl-D or type quit or type q