1. Floating Points & IEEE 754

2. Column Pattern What goes to the right of 1’s column? 23 8 22 4 21 2 20

3. 2-3 = 1 2 3 = 1 8 = 0.125 Column Pattern Negative powers of two: 23 8
2-3 = = = 0.125 23 8 22 4 21 2 20 1
2-1
0.5
2-2
0.25
2-3
0.125
2-4
0.0625

4. 10.112 = 2 + 0.5 + 0.25 = 2.7510 Number Binary decimal: 23 8 22 4 21 220 1
2-1
0.5
2-2
0.25
2-3
0.125
2-4
0.0625

5. Fixed Decimal Problems
Fixed decimal points waste space:
400,000,000,000,000, vs x 1017
vs x 10-14
In computers, space is precious
Computers use a floating decimal point (Like scientific notation)

6. Floating Point 1 Bits used to represent 3 parts: Sign Exponent
Fraction (or Mantissa)
Sign
Exponent
Mantissa 1

7. Sign
0 = positive, 1 = negative
Sign
Exponent
Mantissa 1

8. Exponent 1 Binary integer in excess notation
Gives power of 2 to multiply by
100 = 0
So 20
Sign
Exponent
Mantissa 1
Binary
Value
000 -4
001 -3
010 -2
011 -1
100
101 1
110 2
111 3

9. Mantissa 1 Fractional Value Always a decimal 1000 = 0.5 Sign Exponent1
2-1
0.5
2-2
0.25
2-3
0.125
2-4
0.0625 1

10. IEEE 754 Standards for 32bit and 64 bit floats
32 bit : float or single
64 bit : double

11. The Range The floating point number range: +  -Normalized -Denorm
NaN
NaN 0 +0

12. Sign
Sign bit
1 = negative
0 = positive

13. Exponent Exponent X bits = 2x different values Stored as biased value
8 bits = 256 values = 0-255
Stored as biased value
Bias value subtracted to get exponent
IEEE 32-bit float Biased 127: Exponent = value - 127
Binary
Value
Exponent 1
-126
128
137 10
255

14. Mantissa Mantissa Normalized Assumed 1.XXXXX Value in range [1-2)
Binary
Represents
Value … … 1 … …
1.5 … …
1.25 … …
1.375

15. Special Patterns
Exponent of all 1's/0's is special:

16. Mantissa If exponent 0 Mantissa is Denormalized Assumed 0.XXXXX
Value in range [0-1)
Binary
Represents
Value … … … …
0.5 … …
0.25 … …
0.375

17. Special Patterns
Exponent of all 1's/0's is special:

18. Issues
Can't count on absolute precision:

19. Issues Small values closer than large values
Accuracy expressed in digits not decimal places 32 bit : 7-8 decimal digits 64 bit : digits

20. Issues Can't count on absolute precision:
Proper epsilon depends on magnitude of x

21. Issues No associativity or commutitive property in floating point math
a*(b*c) and a*b*c can give different results

22. Issues
Errors compound with repeated calculations

23. Floating Point ARM
VFP unit optional in lower end chips

24. Floating Point ARM Special registers for coprocessor
32 32-bi registers s0-s31
Pairs can be used as 64-bit registers

25. Floating Point Instructions
Can declare float/double data values
VLDR to load VFP registers from address in ARM register

26. Moving and Converting
VMOV can move bits from regular to VFP registers:
Special instructions to:
Convert float   word
Convert float  double

27. x86 x86 processors had optional floating point coprocessor (x87)
All floating point functions stack based:

28. SSE SSE : Intel extension to Pentium chips
Added addressable registers for floating points

29. Integer vs FP Performance
Intel Haswell architecture:

30. Integer vs FP Performance
Intel Haswell architecture:

31. Awesomeness Fast Inverse Square Root