1. CSCI206 - Computer Organization & Programming
Introduction to Floating Point Numbers
zyBook: 10.5

2. Real Numbers in Binary
Recall the equation describing a positional number system:

3. Real Numbers in Binary
Recall the equation describing a positional number system:
This can be extended to real numbers:
decimal point!

4. Examples
base 10
0.1 in binary is = 0.5 (base 10)

5. Real Numbers in Binary Fractions of a power of 2 are easy
Other values are represented using the sum of fractions with a power of 2 denominator

6. Algorithm to Convert Decimal to Binary
For decimal to binary integers we divided by 2 and record the remainder.
For decimal to binary real numbers we multiply by 2 and record the integer part.
Example: convert to binary.

7. binary digits beginning to the right of the decimal
to binary
binary digits beginning to the right of the decimal

8. binary digits beginning to the right of the decimal
to binary
binary digits beginning to the right of the decimal

9. binary digits beginning to the right of the decimal
to binary
binary digits beginning to the right of the decimal

10. binary digits beginning to the right of the decimal
to binary
binary digits beginning to the right of the decimal

11. binary digits beginning to the right of the decimal
to binary
binary digits beginning to the right of the decimal
only work with fractional part!

12. fraction part is zero, stop!
to binary =
It appears magic, but the reason behind the algorithm is to find k, such that v * 2^k = 1.0, as v is expressed as b0*2^(-1)+b1*2^(-2) …
fraction part is zero, stop!

13. Convert binary to decimal
Convert to decimal
== == 1/16 + 1/32 == 3/32 ==

14. Number Representation in Computing
For a given range of integers, there is a corresponding range of (exact) binary representations
Example: the range [0-15] corresponds to the 4-bit binary numbers 0000 through 1111.

15. Number Representation in Computing
Within a range of real numbers, there is no way to encode all possible values.
Example: the range [ ] has an infinite number of points, so we would need an infinite number of bits to represent all of the possible values!
As a result, in computing, real numbers are approximate.
Activity 24, question 1 - 2

16. An Observation
Many common base 10 real numbers generate an infinite number of binary digits

17. Fixed vs. Floating Point Approximations

18. Floating Point Representation
Decimal notation
Scientific notation 2
2×100
300
3×102
321.7
3.217×102
−53,000
−5.3×104
6,720,000,000
6.72×109
0.2
2×10−1
Floating Point Representation
Where
S is the sign bit
M is a fixed point number (precision of numbers)
E is a signed integer (range of numbers) S
Exponent
Mantissa
or Fraction
32 or 64 bit word

19. IEEE 754 Standard (1985) S Exponent Mantissa One bit for Sign
Single precision float (32 bits)
8 bit Exponent
23 bit Mantissa
Double precision float (64 bits)
11 bit Exponent
52 bit Mantissa

20. IEEE 754 Standard (1985) S Exponent Mantissa
Mantissa is normalized, meaning it is a fixed point number in the form 1.xxxxxx
to save one bit, the 1. is implicit (not represented)
Exponent is represented in biased form
B = 127 for single
B = 1023 for double

21. IEEE 754 Standard (1985) (normalized)
Exponent
Mantissa
S, E, and M are encoded in the binary word

22. IEEE754 - Reserved Values
Not a Number =

23. IEEE754 - Example
Show 3.14 as a single precision float

24. 3.14 - step 1 write in binary 3.14 == 3 + 0.14 0.14*2 = 0.28
0.28*2 = 0.56
0.56*2 = 1.12
0.12*2 = 0.24
......

25. 3.14 - step 1 write in binary need 24 bits for single (52 for double)
3.14 ==

26. 3.14 - step 2 normalize binary
Normalized form is 1.yyyyy
3.14 == ==
Note a total of 24 bits.

27. 3.14 - step 3 write mantissa & sign
3.14 ==
M =
S = 0 (positive)
Note that the mantissa keeps only 23 bits, the leading bit is always 1, so it is omitted in representation (only!!).

28. 3.14 - step 4 encode exponent 3.14 == Exponent = 1, B = 127, (8 bits)
E (biased exponent) = 128 =

29. step 5 write result
S = 0 (positive)
E =
M =
to hex = 0x4048f5c3

30. Endianness
On a little-endian system (Intel, etc), the IEEE754 value is byte & word swapped
0x f5 c3 (big endian)
c3f5
0x c3f (little endian)
Swap bytes and words!
float f = 3.14; unsigned char* p = (unsigned char*)&f; printf("%02x%02x %02x%02x\n", *p, *(p+1), *(p+2), *(p+3)); // result: c3f5 4840