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William Sandqvist william@kth.se
It could be pedagogical to study a small 4 bit nibble computer in order to understand how real numbers are stored and handled!
Normaly 32 bit registers are used, but the principles from the nibble computer will remain the same.
William Sandqvist william@kth.se
The first microprocessor
Intel 4004 1971 was a 4 bit (nibble) processor. The customer Busicom wanted a dedicated calculator chip, but the disobedient engineers created a general purpose processor.
William Sandqvist william@kth.se
4.3 Imagine a ”Nibble” computer
• Computer registers are ”rings”.
A four bit register can hold 24 = 16 numbers.
Either 8 positive (+0…+7) and 8 negative (-1…-8) ”signed numbers”, or 16 (0…F) ”unsigned numbers”.
If a result is bigger than 16, the register overflows. (In some processors indicated with a Carry-flag).
William Sandqvist william@kth.se
Real numbers in Nibble computer
)2().1()1(][ 100123
123 bbb bbbbb
• Imagine a ”Nibble” floating point format!
• Imagine a ”Nibble” fixed point format, Q3!
125.025.05.0
2222][
0123
30
21
12
030123
bbbb
bbbbbbbb
William Sandqvist william@kth.se
Q3 [b3b2b1b0]
Fixed point format
The Q-format is the same as the signed numbers but downscaled:
-1 < Q3 < 1-b0
The step between numbers is the weight of b0 0.125
Fixed point circle!Fixed point number line!
William Sandqvist william@kth.se
Floating point (Nibble)Floating point format
Sign b3
Significand 1.b0
1.02 = 1.010
1.12 = 1.510
Exponent b2b1-1 (exess1)
200-1 = 0.5201-1 = 1210-1 = 2211-1 = 4
+ 1.0 0.5 = +0.5+ 1.0 1 = +1+ 1.0 2 = +2+ 1.0 4 = +4
+ 1.5 0.5 = +0.75+ 1.5 1 = +1.5+ 1.5 2 = +3+ 1.5 4 = +6
- 1.0 0.5 = -0.5- 1.0 1 = -1- 1.0 2 = -2- 1.0 4 = -4
- 1.5 0.5 = -0.75- 1.5 1 = -1.5- 1.5 2 = -3- 1.5 4 = -6
Floating point number line!
William Sandqvist william@kth.se
Quantisation error
Maximum Quantisation Error MQE:
Fixed point: (2-3)/2 = 2-4 = 0.125/2 = 0.0625
Float: (6-4)/2 = 1
William Sandqvist william@kth.se
The number Zero
In fixed point formats the number zero is represented as [0000]
In floating point format, there is no representation for zero!
We could use, 0000 as +0 and 1000 as –0, as in the IEEE standard format!
The IEEE 32 and 64 bit float standard
Floating point circle!
William Sandqvist william@kth.se
Converting Real numbers to Binary
Hawe you forgotten how?
William Sandqvist william@kth.se
4.4 Six bit float)2().1()1(][ 1
012012345345 bbb bbbbbbbbb
Significand 1.b2b1b0
1.0002 = 1.00010
1.0012 = 1.12510
1.0102 = 1.2510
1.0112 = 1.37510 1.1002 = 1.510 1.1012 = 1.62510 1.1102 = 1.7510
1.1112 = 1.87510
Exponent b4b3-1 (exess1)00-1 = -1 200-1 = 0.501-1 = 0 201-1 = 110-1 = 1 210-1 = 211-1 = 2 211-1 = 4
• 0.2510 = 0.012 = 1.02 2-2 exponent -2 not representable
• 0.812510 = 0.11012 = 1.10122-1 representable
• -1,37510 = -1.0112 = -1.0112 20 representable
• 4.2510 = 100.012 = 1.00012 22 significand 1.0001 not representable
• 7.510 = 111.12 = 1.1112 22 representable
William Sandqvist william@kth.se
Float AdditionAlgorithm:1. Check for zeroes2. Align significands3. Add/Sub significands4. Normalize result
a = 0011112 0 01 111 + 01 1.111b = 0100102 0 10 010 + 10 1.010
Align significands. 10 > 01 shift smaller number a to right to get same exponent:Significand 0.1111 exponent 10
Add significands: Normalize result:
10.0011 exp 10 1.00011 exp 11Rounding: 1.00011 ~ 1.001
Result: 0 11 001
a = 1.875 b = 2.5 a+b = 4.5 (4.375)
William Sandqvist william@kth.se
Float MultiplicationSimpler than addition!
Algorithm: 1. Check for zeroes2. Add exponents and subtract Bias3. Multiply significands4. Normalize
Exponents: 01 10 Bias = 1exp = 01+10 -1 = 10
Multiply significands
Normalize result:
10.010110 exp 10 1.0010110 exp 11
Rounding: 1.0010110 ~ 1.001
Result: 0 11 001
a = 1.875 b = 2.5 a*b = 4.5 (4.6875)
William Sandqvist william@kth.se
IEEE – 32 bit float
Dec IEEE-754
By using the exess–127 format for the exponent, we can compare floats as if they were sign and magnitude integers!
William Sandqvist william@kth.se
William Sandqvist william@kth.se
5-4 Cache conflict misses
for (i = 0; i < 50; i++) for(j = 0; j < 4; j++) x[i][j] = a[i][j] * c[i];
Data cache 1 k Word (Word = 32 bit). Cache line 4 Words.
Conflict misses depending on memory placement of x and a?
William Sandqvist william@kth.se
Memory – Cache adressmapping
Memorys are often Byte-organized, but we could draw it as if it was organized in Memory-lines with the same size as the Cache-line.
Direct addressmapping:
Memory-line: i Cache-line: j = i % K
If the CPU needs a data-item in a line, the whole line is transfered fom memory to cache.
William Sandqvist william@kth.se
a and x always conflict miss …
Conflict miss if the distans between the startaddress in memory of a and of x is a multiple of 1 k Words, 4096 Bytes.
for(j = 0; j < 4; j++) x[i][j] = a[i][j] * c[i];
William Sandqvist william@kth.se
a and x conflict miss 1:4 …for(j = 0; j < 4; j++) x[i][j] = a[i][j] * c[i];
Conflict miss 1:4 if the distans between the startaddress in memory of a and of x is a multiple of 1 k Words 3 Words, 4096 Bytes 12 Bytes.
William Sandqvist william@kth.se
a and x no conflict miss …for(j = 0; j < 4; j++) x[i][j] = a[i][j] * c[i];
No conflict miss if the distans between the startaddresses in memory of a and of x is not a multiple of 1 k Words, and not a multiple of 1 k Words 3, 2, 1 Words!
William Sandqvist william@kth.se
William Sandqvist william@kth.se
8-1 Arbitration3 Masters and 4 Slaves are connected to a classical bus.
Read access pattern:
M1 M2 M3
Rd S1 Rd S2 Rd S4
Rd S1 Rd S3 Rd S3
Rd S2
Sequential execution, 7 cycles needed.
1. M1 Rd S12. M2 Rd S23. M3 Rd S44. M2 Rd S35. M3 Rd S36. M2 Rd S27. M1 Rd S1
Many other alternatives are possible!
William Sandqvist william@kth.se
Slave side arbitrationWeighted round robin slave-side arbitration.
S1 S2 S3 S4
M1 1 - - -
M2 - 1 2 -
M3 - - 1 1
Best case execution time (3 cycles)
1. M1 Rd S1 ; M2 Rd S2 ; M3 Rd S4
2. M1 Rd S1 ; M2 Rd S3 ;
3. ; M2 Rd S2 ; M3 Rd S3
Worst case execution time (4 cycles)
1. M1 Rd S1 ; M2 Rd S2 ; M3 Rd S4
2. M1 Rd S1 ; ; M3 Rd S3
3. ; M2 Rd S3 ;
4. ; M2 Rd S2 ;
Connections and weights:
M1 M2 M3
Rd S1 Rd S2 Rd S4
Rd S1 Rd S3 Rd S3
Rd S2
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