Arithmetic
How are numbers represented in the computer?
2 Numbers*
Unsigned integers
10110110101101102 = 0*20 + 1*21 + 1*22 + 0*23 + ….
1332668 = 6*80 + 6*81 + 2*82 + 6*23 + ….
octal
b6b6 = 6*160 + 11*161 + 6*162 + 11*163
hexadecimal
Refer to least significant bit lsp
most significant bit msp
16 bit gives us 216 numbers, normally 0 to 216 – 1
65535
Signed integers
Can still only represent 216 distinct numbers
-215 to 215 – 1 (including 0)
Use twos complement
msp becomes a sign bit
1 goes to 0 and 0 goes to 1 then add 1
-n goes to n
or n goes to -n
Twos complement
3 Numbers*
Signed integers
0000000000000000
0000000000000001
0000000000000010
0000000000000011
Wraps round to
large negative and
decreases.
=
=
=
=
0
1
2
3
………………………………
0111111111111111
1000000000000000
1000000000000001
1000000000000010
=
=
=
=
32767
-32768
-32767
-32766
………………………
Two large positives
overflow to a
negative
1111111111111101 = -3
1111111111111110 = -2
1111111111111111 = -1
0000000000000000 = 0
Addition works check:
Subtraction also:
Overflow to zero
2 + (-1) = 1
Correctly handle
the overflow
-1 – (-2) = 1
-2 –(-1) = -1
And multiplication (an exercise for the student)
Positives have
leading 0s.
Negatives 1s
4 Sign extend
Moving to registers
A number whose word length is equal to the register
width can only be loaded in one way. Copy bits.
A shorter number : byte for a 16/32/64 bit machine
two byte : 32/64 machine bit.
Can
be loaded in two ways
Copy the byte(s) into the low order bytes.
THEN
Fill the higher order bytes with zero.
OR
Sign extend
Fill the higher order bytes with the value of the msp.
1000000000000000
11111111111111111000000000000000 = -32768
Sign extend preserves the value :
Add two positive numbers – if sign bit is 1, Overflow
Add two negative numbers – if sign bit is 0, Overflow
Positive – Negative: sign bit - if sign bit is 1, Overflow
Negative – Positive: sign bit - if sign bit is 0, Overflow
On overflow,
ignore OR
generate an
exception
5 Multiply
Multiply
Multiplicand
Addition sometimes end up with more
10 1 0 1 0
bits than either have individually.
10 0 1 1 0
Multiplication usually ends up with
00 0 0 0 0
more bits.
Multiplier
1 01 0 1 0 0
1 0 10 1 0 00
Multiplicand moves to the left on each multiply.
Need a 64 bit register
Product
0 0 0 00 0 000
0 0 0 0 00 0000
Multiplicand
Shift left
64 bits
1 0 1 0 1 00 0 0 0 0
1 1 0 0 0 11 1 1 0 0
Multiplier
Shift Right
64-bit ALU
32 bits
Product
Control
Test
64 bits
Naïve – move Multiplicand on a clock cycle, move
multiplier on a clock cycle. Add (if multiplier = 1) on
cycle. Can perform them all in parallel
Can also halve
register and ALU
length. Unused
portions.
6 Multiply
Signed Multiply
Convert both to positive. Multiply and then
apply sign using normal rules
Faster multiply
Use an adder for each bit.
Lowest order bit either 0 or 1 from lsb of both numbers
Highest order bit either 0 or 1 for msb of both
numbers.
Second bit needs input from two input bits, ditto
second to last.
Significant speed up – useful elsewhere.
Helps to parallelise instructions
Need extra
hardware
7 Division
Division
Quotient
1001
Easy algorithm
1000 1001010
Line up try to subtract.
-1000
If less than zero dividend is zero,
10
add back on, move along 1 place
101
If greater than zero, dividend is 1
1010
move along one place and repeat.
-1000
10
Dividend
Don’t worry too much – infrequent.
Hardware similar to simple adder
Speed up as for multiplication not possible. Need to
know sign of remainder from partial subtractions
Divisor
Remainder
8 Floating Point
Binary fractions
A decimal is 43.1415
4*101 + 3*100 + 1*10-1 + 4*10-2 + 1*10-3 + 5*10-4
A binary 10.0101
1*21 + 0*20 + 0*2-1 + 1*2-2 + 0*2-3 + 1*2-4
How to represent:
Standard form
197.16 becomes 0.19716*10 3
197.16, 3
Standard form
0.00197 becomes 0.19716*10 -2
197.16, -2
Division by 2 in
binary has the
same effect as by
10 in decimal
Binary
10.0101 = 0.100101 * 2-2 becomes 100101, 10
How do we store this?
One part of the word is the fraction
The other is the
exponent
Scientific notation
normalised
9 FP representation
Binary fractions
Need to split bits between fraction
&
s
exponent
exponent
accuracy
range
faction
31
30-23
22-0
position
1
8
23
bits
10 VAX FP
Binary fractions
Need to split bits between fraction
&
Fraction low
31-16
s
15
accuracy
exponent
exponent
14-7
range
Bit 15 is the sign
Fraction high
6-0
position
The exponent is stored in excess 128:
the value of the exponent is the number in the
exponent field-128. The exponent can run from
-127 to 127. (Only 254 values because the value 0
has a special meaning)
The fraction has the second most significant bit in 6
and the least significant bit (the 24 th) in 16
6Fraction
high0
31
Fraction low
15
The number is normalised to 0.1… x 2 n
This means the first bit is always 1, so it isn’t stored!
11 Precision
Larger numbers or greater precision
224 is about 4 Million – so even banks need more
precision and scientific calculations frequently use
larger numbers
Double Precision
Fraction low
medium
s
exp
high
Same range
31-0
G Float
Also 8 bytes, but some of the extra bits go to range and
not precision
H Float
16 Bytes …
1984 … 32 bit
operating system &
64bit arithmetic
about 33 decimal digits and 10-4932 to 104932
G and H are much slower, they are implemented in
microcode and not hardware
In the VAX11/780 and
VAX11/750. The
floating point hardware
was optional. And
costly
12 Standard
IEEE
Defined by IEEE Std 754-1985
Developed in response to divergence of
representations Portability issues for scientific
code
Now almost universally adopted
Two representations
Single precision (32-bit)
Double precision (64-bit)
Different representations were a real pain.
S
Even without big-endian, little-endian
confusion and of course varying word
lengths
Exponent
Fraction
x  ( 1)
S
 (1  Fraction)  2
(Exponent Bias)
S: sign bit (0  non-negative, 1  negative)
Normalize significand: 1.0 ≤ |significand| < 2.0
Always has a leading pre-binary-point 1 bit, so no need
to represent it explicitly (hidden bit)
Significand is Fraction with the “1.” restored
Exponent: excess representation: actual exponent + Bias
Ensures exponent is unsigned
Single: Bias = 127; Double: Bias = 1023
13 IEEE
Precision and Range
Exponents 000….000 & 111…111 reserved
Range power of 2 :-126 127
Equates to about 10 to the (-38  38)
Double precision
About 10 to the (-308  308)
Double precision arithmetic tends to be much
slower, but remember (a-b)*c can loose a lot of
precision is a is approximately equal b.
Precision
Single about 6 decimal digits
Double about 16 decimal digits
Double precision arithmetic tends to be much
slower, but remember (a-b)*c can loose a lot of
precision if a is approximately equal b.
Special numbers
Infinity – and NaN (not a number) : result of 0/0
Intel had a problem
in 1994 with
accuracy.
Essentially Intel
argued that the
error occuring in
ninth digit affected
only a few people.
IBM stopped
shipping PCs with
the offending chip.
Intel forced to
replace chips an
suffered a lot of bad
publicity
14 FP arithmetic
FP add and subtract
Multi-cycle (but can be pipelined)
Normalise numbers – make exponents equal.
Multiply numbers. Normalise
Step 1
Step 2
Step 3
Step 4
15 FP arithmetic
FP multiply etc
Multiply unit, as complicated as adder.
FP unit will also do integer  floating point
takes several cycles
In eg MIPS done by a separate co-processor.
VAX 11/700 series had a separate board, which
cost extra.
C code:
float f2c (float fahr) {
return ((5.0/9.0)*(fahr - 32.0));
}
fahr in $f12, result in $f0, literals in global
memory space
Compiled MIPS code:
lwc – load word to
coprocessor
f2c: lwc1
lwc2
div.s
lwc1
sub.s
mul.s
jr
$f16,
$f18,
$f16,
$f18,
$f18,
$f0,
$ra
const5($gp)
const9($gp)
$f16, $f18
const32($gp)
$f12, $f18
$f16, $f18
16 Operands
Types and Lengths
Data words also need to encode character sets
Character 8 bit
ASCII
Character 16 bit
Unicode
Also Ogham, Byzantine musical notation, ……
Integers .. Byte, half word, word, longword
Floating point, single, double.
Packed decimal (doesn’t suffer from rounding
errors)
17 Addressing
Alignment
Addressing is done via bytes, but computers work on words.
Starting at zero the first word is 0,1,2,3 bytes.
second word is 4,5,6,7
Can we use 1,2,3,4 or 2,3,4,5 or 3,4,5,6 as a word?
It depends ….
Some systems insist on stating at 0,4,8,12
This is referred to as alignment and if it is required is called
an alignment restriction
Word alignment is possible, as is half-word alignment. Some
machines allow unaligned access, but there may be a time
penalty.
sw $t0, 8($s1) stores from $t0
Addressable memory
The arguments of the instruction must give the memory
location and the register number.
The registers are numbered 0 to 31 and so need 5 bits
The memory is numbered 0 to 2n-1 bytes … it is this
restriction which determines the size of memory the computer
can address.
32 bit machine