CHAPTER 2:

Floating Point

6 to represent real numbers on computers.

6 reals in scientific notation which represents numbers as a base number and an exponent. The base for the scaling is normally 2, 10 or 16.

} Scientific notation

◦     ±___.___ x 10^__

◦     –2.34      × 10⁵⁶       (normalized)

◦     +0.002    × 10^–4     (not normalized)

◦     +987.02  × 10⁹     (not normalized)

} Scientific notation in binary

◦     ±1.______₂ × 2^___

  Example:

1.0101₂    x  2⁴

-1.10011₂ x  2¹²

1.0001₂     x  2^-3

A. Floating Point Standard

} Defined by IEEE Std 754

} Developed in response to divergence of representations

◦     Portability issues for scientific code

} Two representations

◦     Single precision (32-bit)

◦     Double precision (64-bit)

B. IEEE Floating-Point Format

Bit	s	Exponent	Fraction
single(32)	1	8	23
double(64)	1	11	52

} x = (-1)^s x (1 + Fraction ) x 2^{(Exponent-Bias)}

}S: sign bit (0 Þ positive, 1 Þ negative)

} Normalize significand:

1.0 ≤ |significand| < 2.0

} Exponent: excess representation: actual exponent + Bias

◦     Ensures exponent is unsigned

◦     Single: Bias = 127; Double: Bias = 1203

C. Floating-Point Example

} Represent is –45.75

1^st: change 45.75 to binary form

         (a)                               (b)

45=32+8+4+1              45/2 = 22 r 1

=2⁵+2³+2²+2⁰         22/2 = 11 r 0

=101101₂                 11/2 = 5 r   1 

                              5/2 = 2 r   1  

                              2/2 = 1 r   0   

                              1/2 = 0 r   1  ↑

 45 = 101101₂

            (a)                              (b)

0.75 = 0.5 + 025      0.75 x 2 = 1.5  1 ↓

= 2^-1 + 2^-2        0.5 x 2 = 1.0   1

= 0.11₂             

        0.75 = 0.11₂

45.75 = 101101₂ + 0.11₂

= 101101.11₂

2^nd: change the –45.75 to formula below

 x = (-1)^s x (1 + Fraction ) x 2^{(Exponent-Bias)}

–45.75 = –1 x (101101.11₂)

      = –1 x 1.0110111₂ x 2⁵

      = –1¹ x (1 + 0.0110111₂) x 2⁵

(the exponent is 5 because the decimal are move to the left hand side with 5 digit places.)

3^rd: list out the information

S = 1                          ( negative )

Fraction = 0110111……00₂

Exponent = 5+ Bias   ( Bias of single = 127)

                         = 5 + 127  

                         = 132

= 10000100₂

(the exponent must in the base 2)

s	Exponent	Fraction
1	10000100	0110111……00

※ Single: 1100001000110111……00

D. Floating-Point Example

} What number is represented by the single-precision float

               010000011011100……00

 1^st: list out the information

S = 0

Fraction = 01110…00₂

Exponent = 10000011₂

        = 2⁷ + 2¹ + 2⁰

        = 131

2^nd: replace the information into the equation

}x = (-1)^s x (1 + Fraction ) x 2^{(Exponent-Bias)}

} x = (–1)⁰× (1 + 0.0111₂) × 2^{(131 – 127)}

           = (1) × 1.4375 x 2⁴

※         =  23

the end ^^

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Prepared    By      :    TEE SONG WEI

Contact Number :       0173403160 

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