1. CSCI206 - Computer Organization & Programming
Floating Point Limits
zyBook: 10.9, 10.10

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

3. IEEE754 - Reserved Values
Not a Number =

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

5. 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
......

6. step 1 write in binary
Need 24 bits for single (52 for double). In this example, 2 bits before point, 22 bits after.
3.14 ==

7. 3.14 - step 2 normalize binary
Normalized form is 1.yyyyy
3.14 == ==
Note a total of 24 bits.

8. 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!!).

9. 3.14 - step 4 encode exponent 3.14 == Exponent = 1, B = 127, (8 bits)
E (biased exponent) = 128 =

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

11. 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 on Intel: c3f5 4840, on MIPS: 4048 f5c3

12. Review IEEE754 S Exponent Mantissa
Special values, else normalized numbers
Exponent
Mantissa (fraction)
Value
+/- zero
nonzero
denormalized number
all 1’s
+/- infinity
NaN (not a number)

13. Largest Single Precision Float
8 bit exponent (bias = 127), 23 bit fraction
All 1’s in the exponent is reserved for NaN and infinity
Maximum biased exponent is = 254
Maximum fraction is 23 1’s

14. Largest Single Precision Float
= 254
= 127

15. Largest Single Precision Float
Move the decimal point 23 digits to the right
subtract 23 from exponent

16. Largest Single Precision Float
Convert mantissa

17. Smallest Nonzero Single
What we want is:
But that has exponent & fraction = 0
That value is reserved for zero!
Therefore, the closest we can get is:
either or

18. Smallest Nonzero Single
Normalized
In this case, using a normalized number is not ideal, if we could use a denormalized number we could get a much smaller value:
This is equivalent to:
An extra 22 bits of precision!
Denormalized

19. Smallest Nonzero Single
The IEEE realized this and when the exponent is zero and the fraction is > 0, the value is treated as a denormalized number.
The smallest nonzero normalized:
The smallest nonzero denormalized:
exp =
exp =
m = 0000….1

20. Smallest Nonzero Normalized Single
biased exponent = 1
fraction = 0

21. Smallest Nonzero Denormalized Single
biased exponent = 0
fraction =
*Note, even though the exponent is encoded as -127, it is computed using the smallest “valid” exponent, which is -126.