1. Introduction to Numerical Analysis I
MATH/CMPSC 455
Introduction to Numerical Analysis I
Floating Point Representation of Real Numbers

2. Floating Point Representation of Real Numbers
This is about how computers represent and operate real numbers.
Helps us to understand rounding errors
We consider IEEE 754 Floating Point Standard
Representing binary numbers in computer:
format
machine representation

3. Floating Point Format Formats for decimal system Standard Notation
Scientific Notation
Normalized Scientific Notation

4. Floating Point Format
Format for floating point number (binary representation)
Normalized IEEE floating point standard:
sign (+ or -)
mantissa , which contains the significant bits. (N b’s)
exponent (p, M-bit binary number) …

5. Precision
sign
Exponent (M)
Mantissa
(N)
single 1 8 23
double 11 52
Long double 15 64
Definition (machine epsilon, ): It is the distance between 1 and the smallest floating point number greater than 1. Gives a bound on the relative error due to rounding.
For the IEEE double precision floating point standard:

6. Rounding
How do we fit a given binary number in a finite number of bits?
IEEE Rounding to Nearest Rule:
For double precision, if the 53rd bit to the right of the binary point is 0, then round down (truncate after the 52nd bit). If the 53rd bit is 1, then round up (add 1 to 52 bit), unless all known bits to the right of the 1 are 0’s, in which case 1 is added to bit 52 if and only if bit 52 is 1.

7. Rounding
Notation: Denote the IEEE double precision floating point number associated to x, using the Rounding to the Nearest Rule, by fl(x).
Definition (absolute error & relative error): Let be a computed version of the exact quantity .

8. Rounding
Example:
Example:
Relative rounding error:

9. Machine Representation…
Sign: 1 bit, 0 for positive, 1 for negative;
Mantissa: 52 bits, …
Exponent: 11 bits so 0 < e < = 2047 and
p = e
1~2046  ~ 1023
2 values reserved for infinity / NaN and 0
2047  infinity if the mantissa is allzeros, NaN otherwise;
0  small numbers including 0 11

10. Addition and Rounding of Floating Point Numbers
Step 1: line up the two numbers
Double Precision
Step 2: add them
Higher Precision
Step 3: store the result as a floating point number
Double Precision

11. Example :
Example :