1. Floating Point Numbers
Prof. Giancarlo Succi, Ph.D., P.Eng.
Prof. G. Succi, Ph.D., P.Eng.

2. Content Introduction Scientific Notation IEEE 754 Scientific Notation
Converting numbers
Special numbers
Prof. G. Succi, Ph.D., P.Eng.

3. Numbers since now What can we represent in N bits? Unsigned integers:
0 to 2N – 1
Signed Integers (One’s Complement)
-2(N-1) + 1 to 2(N-1) - 1
Signed Integers (Two’s Complement)
-2(N-1) to 2(N-1) - 1 …
Prof. G. Succi, Ph.D., P.Eng.

4. Numbers since now What can we represent in 32 bits? Unsigned integers:
0 to 4,294,967,295
Signed Integers (One’s Complement)
- 2,147,483,647 to 2,147,483,647
Signed Integers (Two’s Complement)
- 2,147,483,648 to 2,147,483,647
Prof. G. Succi, Ph.D., P.Eng.

5. Regions not covered since now
Other numbers
Regions covered:
Unsigned integers
Signed integers -1 1 2
-2,147,483,648
2,147,483,647
4,294,967,295
Regions not covered since now
Prof. G. Succi, Ph.D., P.Eng.

6. Other numbers What about other numbers?
Very large numbers? (seconds/century) 3,155,760,00010 ( x 109)
Very small numbers? (atomic diameter) (1.010 x 10-8)
Rationals (repeating pattern) 2/3 ( )
Irrationals /2 ( )
Transcendentals e ( ),  ( )
All represented in scientific notation
Prof. G. Succi, Ph.D., P.Eng.

7. Content Introduction Scientific Notation IEEE 754 Scientific Notation
Converting numbers
Special numbers
Prof. G. Succi, Ph.D., P.Eng.

8. Scientific Notation exponent mantissa 6.02 x 1023 radix (base)
decimal point
radix (base)
Normalized form: no leadings 0s (exactly one digit to left of decimal point)
Alternatives to representing 1/1,000,000,000
Normalized: 1.0 x 10-9
Not normalized: 0.1 x 10-8,10.0 x 10-10
Prof. G. Succi, Ph.D., P.Eng.

9. Converting base to 2 exponent mantissa 1.0two x 2-1 radix (base)
binary point
radix (base)
Computer arithmetic that supports it called floating point, because it represents numbers where binary point is not fixed, as it is for integers
Declare such variables in Java as float
Prof. G. Succi, Ph.D., P.Eng.

10. Binary Scientific Notation in detail31 S
Exponent 30 23 22
Significand
1 bit
8 bits
23 bits
Normal format: +1.xxxxxxxxxxtwo*2yyyytwo
S represents Sign
Exponent represents y’s
Significand represents x’s
Prof. G. Succi, Ph.D., P.Eng.

11. Covered Range Regions covered: Largest Exponent: 2128 => 1038
-2.0 x 1038
-2.0 x 10-38
2.0 x 10-38
2.0 x 1038
Underflow
Overflow
Prof. G. Succi, Ph.D., P.Eng.

12. Extend covered range 31 30 20 22 1 bit 11 bits 20 bits 32 bits31 S
Exponent 30 20 22
Significand
1 bit
11 bits
20 bits
Significand
32 bits
Double Precision (vs. Single Precision)
C variable declared as double
Represent numbers almost as small as 2.0 x to almost as large as 2.0 x 10308
But primary advantage is greater accuracy due to larger significand
Prof. G. Succi, Ph.D., P.Eng.

13. Content Introduction Scientific Notation IEEE 754 Scientific Notation
Converting numbers
Special numbers
Prof. G. Succi, Ph.D., P.Eng.

14. IEEE 754 Floating Point Standard
Single Precision, DP similar
Sign bit: 1 means negative means positive
Significand:
To pack more bits, the leading 1 is implicit for normalized numbers
bits single, bits double
always true: Significand < 1 (for normalized numbers)
Note: 0 has no leading 1, so reserve exponent value 0 just for number 0
Prof. G. Succi, Ph.D., P.Eng.

15. IEEE 754: Usability
Introduced by Prof. Kahan: to use FP numbers even if no FP hardware is available; e.g., sort records with FP numbers using integer compares
It break FP number comparisons into 3 parts: compare signs, then exponents, then significands
To improve speed, try to have a single comparison, especially if a number is positive
Therefore, …
The highest bit is the sign ( negative < positive)
Exponent next, so larger exponent  larger #
Significand last
Prof. G. Succi, Ph.D., P.Eng.

16. IEEE 754: Signed Exponent
Using 2’s comp? 1.0 x 2-1 v. 1.0 x2+1 (1/2 v. 2)
1/2 2
This notation using integer compare of 1/2 v. 2 makes 1/2 > 2!
Instead, pick notation is smallest negative, and is the largest positive
1.0 x 2-1 v. 1.0 x2+1 (1/2 v. 2)
1/2 2
Prof. G. Succi, Ph.D., P.Eng.

17. IEEE 754: Summary
Called Biased Notation, where bias is number subtract to get real number
IEEE 754 uses bias of 127 for single prec.
Subtract 127 from Exponent field to get actual value for exponent
1023 is bias for double precision
(-1)S x (1 + Significand) x 2(Exponent-127)
Double precision identical, except with exponent bias of 1023
Prof. G. Succi, Ph.D., P.Eng.

18. Content Introduction Scientific Notation IEEE 754 Scientific Notation
Converting numbers
Special numbers
Prof. G. Succi, Ph.D., P.Eng.

19. Understanding significand: fractions
In decimal: => 34010/ => 3410/10010
Binary: => 1102/10002 = 610/810
=> 112/1002 = 310/410
Advantage: less purely numerical, more thought oriented; this method usually helps people understand the meaning of the significand better
Prof. G. Succi, Ph.D., P.Eng.

20. Understanding significand: positional values
Convert from scientific notation
In decimal: = (1x100) + (6x10-1) + (7x10-2) + (3x10-3) + (2x10-4)
In binary: = (1x20) + (1x2-1) + (0x2-2) + (0x2-3) + (1x2-4)
Interpretation of value in each position extends beyond the decimal/binary point
Advantage: good for quickly calculating significand value; use this method for translating FP numbers
Prof. G. Succi, Ph.D., P.Eng.

21. Converting Binary FP to Decimal
Sign: 0 => positive
Exponent:
two = 104ten
Bias adjustment: = -23
Significand:
1 + 1x2-1+ 0x x x x = =
Represents: ten*2-23 ~ 1.986*10-7
(about 2/10,000,000)
Prof. G. Succi, Ph.D., P.Eng.

22. Attention to non-typed data!!!
0011
0100
What does bit pattern mean:
1.986 *10-7? 878,003,010? “4UCB”? ori $s5, $v0, 17218?
Data can be anything; operation of instruction that accesses operand determines its type!
Side-effect of stored program concept: instructions stored as numbers
Power/danger of unrestricted addresses/ pointers: use ASCII as Fl. Pt., instructions as data, integers as instructions, ... (Leads to security holes in programs)
Prof. G. Succi, Ph.D., P.Eng.

23. Converting Decimal to FP
Simple Case: If denominator is an exponent of 2 (2, 4, 8, 16, etc.), then it’s easy.
Show MIPS representation of -0.75
-0.75 = -3/4
-11two/100two = -0.11two
Normalized to -1.1two x 2-1
(-1)S x (1 + Significand) x 2(Exponent-127)
(-1)1 x ( ) x 2( ) 1
Prof. G. Succi, Ph.D., P.Eng.

24. Converting Decimal to FP
If denominator is not an exponent of 2…
Then we cannot represent the number precisely, but that’s why we have so many bits in significand: for precision
Once we have the significand, normalizing a number to get the exponent is easy
So how do we get the significand?
Prof. G. Succi, Ph.D., P.Eng.

25. Converting Decimal to FP: Rationale
Fact: All rational numbers have a repeating pattern when written out in decimal.
Fact: This still applies in binary.
To finish conversion:
Write out binary number with repeating pattern.
Cut it off after correct number of bits (different for single v. double precision).
Derive Sign, Exponent and Significand fields.
Prof. G. Succi, Ph.D., P.Eng.

26. Conversion: Example How to represent 1/3 in MIPS? 1/3 = 0.33333…10
= …
= 1/4 + 1/16 + 1/64 + 1/ / …
= …
= … 2 * 20
= … 2 * 2-2
Prof. G. Succi, Ph.D., P.Eng.

27. Conversion: Example Sign: 0 Exponent = -2 + 127 = 12510=011111012
Significand = …
Prof. G. Succi, Ph.D., P.Eng.

28. Content Introduction Scientific Notation IEEE 754 Scientific Notation
Converting numbers
Special numbers
Prof. G. Succi, Ph.D., P.Eng.

29. Representation for +/- Infinity
In FP, divide by zero should produce +/- infinity, not overflow.
Why?
OK to do further computations with infinity e.g., X/0 > Y may be a valid comparison
IEEE 754 represents +/- infinity
Highest positive exponent reserved for infinity
Significand: all zeroes
Prof. G. Succi, Ph.D., P.Eng.

30. Representation for 0 Represent 0? Why two zeros? exponent all zeros
significand all zeros too
What about sign?
+0:
-0:
Why two zeros?
Helps in some comparisons
Prof. G. Succi, Ph.D., P.Eng.

31. Special numbers: summary
What have we defined so far? (Single Precision)
Exponent Significand Object
0 nonzero ???
anything +/- fl. pt. #
/- infinity
255 nonzero ???
Prof. G. Succi, Ph.D., P.Eng.

32. Proposed Example What value is represented by: 1 1000 0001
Prof. G. Succi, Ph.D., P.Eng.

33. Proposed Example: solution
What value is represented by: 1 S
Exponent
Significand
(-1)S x (1 + Significand) x 2(Exponent-127)
(-1)1 x ( ) x 2( )
-1 x (1.111) x 2(2) = = -7. 5
Prof. G. Succi, Ph.D., P.Eng.

34. Things to Remember
Floating Point numbers approximate the values that we want to use.
IEEE 754 Floating Point Standard is the most widely accepted attempt to standardize interpretation of such numbers
Every desktop computer sold since ~1997 follows these conventions
Type is not associated with data, bits have no meaning unless given in context
Prof. G. Succi, Ph.D., P.Eng.