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Computer Science 210
Computer Organization
Floating Point Representation
Real Numbers
Format 1: <whole
part>.<fractional part>
Examples: 0.25, 3.1415 …
Format 2 (normalized form): <digit>.<fractional
part> × <exponent>
Example: 2.5 × 10-1
In mathematics, infinite range and infinite precision (“uncountably
infinite”)
math.pi
>>> import math
>>> math.pi
3.141592653589793
>>> print(math.pi)
3.14159265359
>>> print("%.50f" % math.pi)
3.14159265358979311599796346854418516159057617187500
Looks like about 48 places of precision (in base10)
IEEE Standard
Single precision: 32 bits
Double precision: 64 bits
2.5 × 10-1
Reserve some bits for the significand (the digits to the left of
×) and some for the exponent (the stuff to the right of ×)
Double precision uses 52 bits for the significand, 11 bits for
the exponent, and one sign bit
Approximate double precision range is 10-308 to 10308
IEEE Single Precision Format
32 bits
S
1
Exponent
8
Significand
23
Roughly (-1)S x F x 2E
F is related to the significand
E is related to the exponent
Rough range
Small fractions 2 x 10-38
Large fractions 2 x 1038
Fractions in Binary
In general, 2-N = 1/2N
0.12 = 1 × 2-1 = 1 × ½ = 0.510
0.012 = 1 × 2-2 = 1 × ¼ = 0.2510
0.112 = 1 × ½ + 1 × ¼ = 0.7510
Decimal to Binary Conversion
(Whole Numbers)
•
While N > 0 do
Set N to N/2 (whole part)
Record the remainder (1 or 0)
Set A to remainders in reverse order
Decimal to Binary - Example
• Example: Convert 32410 to binary
N
Rem
N Rem
324
162
0
5
0
81
0
2
1
40
1
1
0
20
0
0
1
10
0
• 32410 = 1010001002
Decimal to Binary - Fractions
•
While N > 0 (or enough bits) do
Set N to N*2 (whole part)
Record the whole number part (1 or 0)
Set N to fraction part
Set bits to sequence of whole number parts
(in order obtained)
Decimal fraction to binary - Example
• Example: Convert .6562510 to binary
N
Whole Part
.65625
1.31250
1
0.6250
0
1.250
1
0.50
0
1.0
1
.6562510 = .101012
Decimal fraction to binary - Example
• Example: Convert .4510 to binary
N
Whole Part
.45
0.9
0
1.8
1
1.6
1
1.2
1
0.4
0
0.8
0
1.6
1
.4510 = .011100110011…2
Round-Off Errors
>>> 0.1
0.1
>>> print("%.48f" % 0.1)
0.100000000000000005551115123125782702118158340454
>>> print("%.48f" % 0.25)
0.250000000000000000000000000000000000000000000000
>>> print("%.48f" % 0.3)
0.299999999999999988897769753748434595763683319092
Caused by conversion of decimal fractions to binary
Scientific Notation - Decimal
•
Number
0.000000001
5,326,043,000
Normalized Scientific
1.0 x 10-9
5.326043 x 109
Floating Point
• IEEE Single Precision Standard (32 bits)
S
1
Exponent
8
Significand
23
• Roughly (-1)S x F x 2E
– F is related to significand
– E is related to exponent
• Rough range
– Small fractions 2 x 10-38
– Large fractions 2 x 1038
Floating Point – Exponent Field
• This comes before significand for sorting purposes
• With 8 bit exponent range would be –128 to 127
• Note: -1 would be 11111111 and with simple
sorting would appear largest.
• For this reason, we take the exponent, add 127 and
represent this as unsigned. This is called bias 127.
• Then exponent field 11111111 (255) would
represent 255 - 127 = 128. Also 00000000 (0)
would represent 0 - 127 = -127.
• Range of exponents is -127 to 128
Floating Point – Significand
• Normalized form:
1.1011… x 2E
• Hidden bit trick:
Since the bit to left of binary point is
always 1, why store it?
• We don’t.
• Number = (-1)S x (1+Significand) x 2E-127
Floating Point Example:
Convert 312.875 to IEEE
Step 1. Convert to binary:
100111000.111
Step 2. Normalize:
1.00111000111 x 28
Step 3. Compute biased exponent in binary:
8 + 127 = 135  10000111
Step 4. Write the floating point representation:
0 10000111 00111000111000000000000
or 439C7000 in hexadecimal
Floating Point Example: Convert IEEE
11000000101000… to decimal
Step 1. Sign bit is 1; so number is negative
Step 2. Exponent field is 10000001 or 129; so actual
exponent is 2
Step 3. Significand is 010000…; so 1 + Significand is
1.0100…
Step 4. Number = (-1)S x (1+Significand) x 2E-127
= (-1)1 x (1.010) x 22
= -101
= -5