Data Representation: Floating Point for Real Numbers Computer Organization and Assembly Language:...
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Transcript of Data Representation: Floating Point for Real Numbers Computer Organization and Assembly Language:...
Data Representation: Floating Point for Real Numbers
Computer Organization and Assembly Language: Module 11
Floating Point Representation The IEEE-754 Floating Point Standard is a widely used floating
point representation from among the many alternative formats The representation of floating point numbers contains:
a mantissa (variant of a scaled, sign magnitude integer) an exponent (8-bit, biased-127 integer)
In this way floating point representation resembles scientific notation Any number N can be represented as M*10^e, where
e = floor(log10N) M = N/(10^e)
1 < M < 10
A number N represented in floating point is determined by the mantissa m, an exponent e, and its sign, s
N = (-1)s * m * 2e
If the sign is negative, s = 1. If the sign is positive, s = 0. The mantissa is normalized, i.e., 1 m < 2 In the IEEE-754 single precision format, the mantissa is
represented with 23 bits (only the fractional part is stored m = (+/-) 1.f22f21…f1f0
Double precision floating point works the same way, but the bit fields are larger: 1-bit sign, 11-bit exponent, 52 bits for the fractional part of the mantissa
Floating Point Representation
Conversion to base-2
1.Break the decimal number into two parts: an integer and a fraction
2.Convert the integer into binary and place it to the left of the binary point
3.Convert the fraction into binary and place it to the right of the binary point
4.Write it in base-2 scientific notation and normalize
Convert 22.625 to floating point representation
1. Convert 22 to binary. 2210 = 101102
2. Convert .625 to binary2*.625 = 1 + .252*.25 = 0 + .52*.5 = 1 + 0
3. Thus 22.62510 = 10110.1012
4. In base –2 scientific notation: 10110.101*20
Normalized form: 1.0110101*24
Example
.62510=.1012
IEEE-754 SPFP Representation
Given the floating point representation N = (-1)s * m * 2e where m = 1.f22f21…f1f0
we can convert it to the IEEE-754 SPFP format using the relations:
F = (m-1)*223 (hence F is an integer) E = e + 127S = s
S E F
Single-Precision Floating Point
The IEEE-754 single precision format has 32 bits distributed as
0 E 255, thus the actual exponent e (interpreted as biased-127) is restricted so that -127 e 128 But e = -127 and e = 128 have special meaning
S E F
1 8 23
Special values and the hidden bit
In IEEE-754 , zero is represented by setting E = F = 0 regardless of the sign bit, thus there are two representations for zero: +0 and -0.
+ by S=0, E=255, F=0 - by S=1, E=255, F=0 NaN or Not-a-Number by E=255, F0
(may result from 0 divided by 0) The leading 1 in the fraction is not represented. It is
the hidden bit.
Converting to IEEE-754 SPFP
1.Convert into a normalized base-2 representation
2.Bias the exponent. The result will be E.
3.Put the values into the correct field. Note that only the fractional part of the mantissa is stored in F.
Example
Convert 22.625 to IEEE-754 SPFP format1. In scientific notation: 10110.101*20
Normalized form: 1.0110101*24
2. Bias the exponent: 4 + 127 = 131
13110 = 100000112
3. Place into the correct fields. S = 0 E = 10000011 F = 011 0101 0000 0000 0000 0000
100000110
S E F
01101010000000000000000
Example
Convert 17.15 to IEEE FPS format 17.1510 = 10001.0010 0110 0110 0110 011*20
1. Normalized form: 1. 0001 0010 0110 0110 0110 011 * 24
2. Bias the exponent: 4 + 127 = 131
13110 = 100000112
3. Place into the correct fields. S = 0 E = 10000011 F = 000 1001 0011 0011 0011 0011
00010010011001100110011100000110
S E F
Example
Convert -83.7 to IEEE FPS format (single precision)
2*.7 = 1 + .42*.4 = 0 + .82*.8 = 1 + .62*.6 = 1 + .22*.2 = 0 + .42*.4 = 0 + .82*.8 = 1 + .62*.6 = 1 + .22*.2 = 0 + .4. . .
-83.710=-1010011.101100110
1. In binary scientific notation:
-1010011.10110011001100110 * 20
Normalized: -1.01001110110011001100110 * 26
2. Bias the exponent: 6 + 127 = 133
13310 = 100001012
3. Place into the correct fields
S = 1
E = 10000101
F = 01001110110011001100110
01001110110011001100110100001011
S E F
Representing as hexadecimal
It is difficult for people to read binary one bit pattern looks much like another
Raw data, which is not being interpreted as representing a particular data type, is often displayed using hexadecimal instead of binary
The final step in many IEEE-754 SPFP problems will be to convert the result to hexadecimal 11000010101001110110011001100110 C2A76666
Given a single precision floating point number with bit fields S, E, and F (interpreted as unsigned integers), the value of the number is normally calculated as
N = (-1)S(1 + F/223)2E-127
This interpretation is not used when E = 255 (+, - , or NaN) E = 0, F = 0 (+0 or –0) What about E 0, F 0?
Graceful underflow
Given a single precision floating point number with bit fields S, E = 0, and F (interpreted as unsigned integers), the value of the number is calculated as
N = (-1)S(0 + F/223)2-126
This allows representation of numbers as small as 2-149, though each order of magnitude below 2-126 results in loss of one bit of precision.
Graceful underflow
Normal interpretation: N = 2(1 – 127) = 2-126
24 bits of precision (counting the hidden bit)
E = 0 interpretation: N = 2-126 (.12) = 2-126 (.5) = 2-127
Only 23 bits of precision
E =0 interpretation: N = 2-126 (.00012) = 2-126(.0625) = 2-130
Only 20 bits of precision
Graceful underflow
0 00000001 00000000000000000000000
0 00000000 10000000000000000000000
0 00000000 00010000000000000000000