1. CDA 3101 Fall 2013 Introduction to Computer Organization
Floating Point
Theory, Notation, MIPS
30 September, 2 October 2013

2. Overview Floating point numbers Scientific notation
Decimal scientific notation
Binary scientific notation
IEEE 754 FP Standard
Floating point representation inside a computer
Greater range vs. precision
Decimal to Floating Point conversion
Type is not associated with data
MIPS floating point instructions, registers

3. Computer Numbers Computers are made to deal with numbers
What can we represent in n bits?
Unsigned integers: to 2n - 1
Signed integers: (n-1) to 2(n-1) - 1
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: 21/2 ( )
Transcendentals : e ( ),  ( )

4. 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

5. Binary Scientific Notation
Exponent
Mantissa
1.0two x 2-1
“binary point”
radix (base)
Floating point arithmetic
Binary point is not fixed (as it is for integers)
Declare such variable in C as float

6. Floating Point Representation
Normal format: +1.xxxxxxxxxxtwo*2yyyytwo
Multiple of Word Size (32 bits) 31 S
Exponent 30 23 22
Significand
1 bit
8 bits
23 bits
S represents Sign Exponent represents y’s Significand represents x’s
Represent numbers as small as 2.0 x to as large as 2.0 x 1038

7. Overflow and Underflow
Result is too large (> 2.0x1038 )
Exponent larger than represented in 8-bit Exponent field
Underflow
Result is too small
>0, < 2.0x10-38
Negative exponent larger than represented in 8-bit Exponent field
How to reduce chances of overflow or underflow?

8. Double Precision FP Use two words (64 bits)31 30 20 19 S
Exponent
Significand
1 bit
11 bits
20 bits
Significand (cont’d)
32 bits
C variable declared as double
Represent numbers almost as small as 2.0 x to almost as large as 2.0 x 10308
Primary advantage is greater accuracy (52 bits)

9. Floating Point Representation
Normalized scientific notation: +1.xxxxtwo*2yyyytwo 31 S
Exponent 30 23 22
Significand
1 bit
8 bits
23 bits
Single
Precision
Significand (cont’d) S
Exponent 20 19
Significand
1 bit
11 bits
20 bits
32 bits 31 30
Double
Precision
Exponent: biased notation
Significand: sign – magnitude notation
Bias
127 (SP)
1023 (DP)

10. IEEE 754 FP Standard Used in almost all computers (since 1980)
Porting of FP programs
Quality of FP computer arithmetic
Sign bit:
Significand:
Leading 1 implicit for normalized numbers
bits single, bits double
always true: 0 < Significand < 1
0 has no leading 1
Reserve exponent value 0 just for number 0
1 means negative
0 means positive
(-1)S * (1 + Significand) * 2Exp

11. IEEE 754 Exponent Use FP numbers even without FP hardware
Sort records with FP numbers using integer compares
Break FP number into 3 parts: compare signs, then compare exponents, then compare significands
Faster (single comparison, ideally)
Highest order bit is sign ( negative < positive)
Exponent next, so big exponent => bigger #
Significand last: exponents same => bigger #

12. Biased Notation for Exponents
Two’s complement does not work for exponent
Most negative exponent: two
Most positive exponent: two
Bias: number added to real exponent
127 for single precision
1023 for double precision
1.0 * 2-1
(-1)S * (1 + Significand) * 2(Exponent - Bias)

13. Significand Method 1 (Fractions): Method 2 (Place Values):
In decimal: => 34010/ => 3410/10010
In binary: => 1102/10002 (610/810) => 112/1002 (310/410)
Helps understand the meaning of the significand
Method 2 (Place 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
Good for quickly calculating significand value
Use this method for translating FP numbers

14. Binary to Decimal FP Sign: 0 => positive Exponent: Significand:
Sign: 0 => positive
Exponent:
two = 104ten
Bias adjustment: = -23
Significand:
1 + 1x2-1+ 0x x x x = =
Represents: *2-23 ~ 1.986*10-7

15. Decimal to Binary FP (1/2)
Simple Case: If denominator is a power of 2 (2, 4, 8, 16, etc.), then it’s easy.
Example: Binary FP 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

16. Decimal to Binary FP (2/2)
Denominator is not an exponent of 2
Number can not be represented precisely
Lots of significand bits for precision
Difficult part: get the significand
Rational numbers have a repeating pattern
Conversion
Write out binary number with repeating pattern.
Cut it off after correct number of bits (different for single vs. double precision).
Derive sign, exponent and significand fields.

17. Decimal to Binary Significand: 101 0101 0101 0101 0101 0101…
x 2 -
x 2
x 2
=> x 21
Significand:
Sign: negative => 1
Exponent: = 128ten = two 1

18. Types and Data 1.986 *10-7 878,003,010 “4UCB” ori $s5, $v0, 17218
0011
0100
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!
Power/danger of unrestricted addresses/pointers:
Use ASCII as FP, instructions as data, integers as instructions, ...
Security holes in programs

19. Expressible Negative Numbers Expressible Positive Numbers
IEEE 754 Special Values
Negative
Underflow
Positive
Underflow
Negative
Overflow
Expressible Negative Numbers
Expressible Positive Numbers
Positive
Overflow
-(1-2-24)*2128
-.5*2-127
.5*2-127
(1-2-24)*2128
Special Value
Exponent
Significand
+/- 0
Denormalized number
Nonzero
NaN
+/- infinity

20. Value: Not a Number Don’t use NaN
What is the result of: sqrt(-4.0)or 0/0?
If infinity is not an error, these shouldn’t be either.
Called Not a Number (NaN)
Exponent = 255, Significand nonzero
Applications
NaNs help with debugging
They contaminate: op(NaN, X) = NaN
Don’t use NaN
Ask math majors

21. Value: Denorms
Problem: There’s a gap among representable FP numbers around 0
Smallest pos num: a = 1.0… 2 * = 2-126
2nd smallest pos num: b = * =
a - 0 = 2-126
b - a = 2-150
Solution:
Denormalized numbers: no leading 1
Smallest pos num: a = 2-150
2nd smallest num: b = 2-149 b a + -
Gap! + -

22. Rounding Math on real numbers => rounding
Rounding also occurs when converting types
Double  single precision  integer
Round towards +infinity
ALWAYS round “up”: => 3; => -2
Round towards -infinity
ALWAYS round “down”: => 1; => -2
Truncate
Just drop the last bits (round towards 0)
Round to (nearest) even (default)
2.5 => 2; 3.5 => 4

23. FP Fallacy FP Add, subtract associative: FALSE!
x = – 1.5 x 1038, y = 1.5 x 1038, and z = 1.0
x + (y + z) = –1.5x (1.5x ) = –1.5x (1.5x1038) = 0.0
(x + y) + z = (–1.5x x1038) = (0.0) = 1.0
Floating Point add, subtract are not associative!
Why? FP result approximates real result
1.5 x 1038 is so much larger than 1.0 that 1.5 x in floating point representation is still 1.5 x 1038

24. Computational Errors with FP
Coalition vs. Iraq (91/92)
Military base in Dhahran (Saudi Arabia) was protected by 6 US Patriot Missile batteries (mobile, operates only a few hours). Tracks and intercepts enemy missiles. One hit US Army barracks 02/05/91, killing 28 americans
Lost of precision int -> float

25. FP Addition / Subtraction
Much more difficult than with integers
Can’t just add significands
Algorithm
De-normalize to match exponents
Add (subtract) significands to get resulting one
Keep the same exponent
Normalize (possibly changing exponent)
Note: If signs differ, just perform a subtract instead.

26. MIPS FP Architecture (1/2)
Separate floating point instructions:
Single Precision: add.s, sub.s, mul.s, div.s
Double Precision: add.d, sub.d, mul.d, div.d
These instructions are far more complicated than their integer counterparts
Problems:
It’s inefficient to have different instructions take vastly differing amounts of time.
Generally, a particular piece of data will not change from FP to int, or vice versa, within a program.
Some programs do not do floating point calculations
It takes lots of hardware relative to integers to do FP fast

27. MIPS FP Architecture (2/2)
1990 Solution: separate chip that handles only FP.
Coprocessor 1: FP chip
Contains bit registers: $f0, $f1, …
Most registers specified in .s and .d instructions ($f)
Separate load and store: lwc1 and swc1 (“load word coprocessor 1”, “store …”)
Double Precision: by convention, even/odd pair contain one DP FP number: $f0/$f1, … , $f30/$f31
1990 Computer contains multiple separate chips:
Processor: handles all the normal stuff
Coprocessor 1: handles FP and only FP;
Move data between main processor and coprocessors:
mfc0, mtc0, mfc1, mtc1, etc.

28. Floating Point Hardware (FP Add)

29. C => MIPS float f2c (float fahr) {
return ((5.0 / 9.0) * (fahr – 32.0)); }
F2c:
lwc1 $f16, const5($gp) # $f16 = 5.0
lwc1 $f18, const9($gp) # $f18 = 9.0
div.s $f16, $f16, $f18 # $f16 = 5.0/9.0
lwc1 $f20, const32($gp) # $f20 = 32.0
sub.s $f20, $f12, $f20 # $f20 = fahr – 32.0
mul.s $f0, $f16, $f20 # $f0 = (5/9)*(fahr-32)
jr $ra # return

30. Conclusion
Floating Point numbers approximate values that we want to use.
IEEE 754 Floating Point Standard is most widely accepted attempt to standardize FP arithmetic
New MIPS architectural elements
Registers ($f0-$f31)
Single Precision (32 bits, 2x10-38… 2x1038)
add.s, sub.s, mul.s, div.s
Double Precision (64 bits , 2x10-308…2x10308)
add.d, sub.d, mul.d, div.d
Type is not associated with data, bits have no meaning unless given in context (e.g., int vs. float)