Sistem Digital : Sistem Bilangan III ( Representasi Floating-Point dengan Single Precision )
Floating-Point Representation
Principles
With a fixed-point notation it is possible to represent a range of positive and negative integers centered on or near 0. By assuming a fixed binary or radix point, this format allows the representation of numbers with a fractional component as well
Limitations :
- Very large numbers cannot be represented nor can very small fractions
- The fractional part of the quotient in a division of two large numbers could be lost
Typical 32-Bit Floating-Point Format
Floating-Point
The final portion of the word. Any floating-point number can be expressed in many ways
Normal number
The most significant digit of the significand is nonzero
Untuk lebih sederhana operasi bilangan floating point perlu di normalkan. Bilangan normal adalah dimana MSB dari significand adalah nonzero. Untuk representasi binary, normal number adalah MSB dari significand adalah 1.
Berikut bentuk normal nonzero number
Dimana b adalah digit biner ( 0 atau 1)
MSB selalu 1, karena itu tidak perlu disimpan (secara implisit). Jadi secara implisit, 23 bit untuk menyimpan 24 bit significand. Sehingga nilainya berada di antara dan sama dengan 1 s/d < 2.
Contoh bentuk yang dinormalkan
Format 32-bit floating point
Sign disimpan pada bit pertama (dari 32 bit). Bit pertama dari significand adalah 1 dan tidak perlu disimpan pada field. Nilai 127 ditambahkan pada true exponent untuk disimpan di field exponent – disebut biased exponent representation
Base adalah 2
E = exponent
S = Significand
Contoh
Binary 1011010010001
Dibentuk normal 1.011010010001 * 212
Format 32 bit
Sign = 0 (positif)
Biased = 12 + 127 = 10001011
Significand/Mantissa (Fractional) = 011010010001
Bentuk floating pointnya dalam format 32
Floating Point 32 bit ke Decimal
1 10010001 10001110001000000000000
Sign = 1 ( negative )
Exponent = 10010001 => 145
Bilangan = (-1)1 (1.10001110001)(2145-127) = -1100011100010000000 = 407680 (decimal)
Latihan
1. Konversi bilangan decimal 3,248 x 104 ke single-precision floating-point binary?
2. Tentukan nilai binary dan decimal dari bilangan binary floating-point berikut :
0 10011000 10000100010100110000000
Expressible Numbers of Floating-point
Produce one of these conditions
Exponent overflow : A positive exponent exceeds the maximum possible exponent value. In some systems, this may be designated as +∞ or - ∞.
Exponent underflow : A negative exponent is less than the minimum possible exponent value (e.g., - 200 is less than -127)
Significand underflow : In the process of aligning significands, digits may flow off the right end of the significand. As we will discuss, some form of rounding is required.
Significand overflow : The addition of two significands of the same sign may result in a carry out of the most significant bit. This can be fixed by realignment.
IEEE 754 Formats
Arithmetic Operations
Addition and Subtraction
Ada 4 fase dari algoritma untuk penambahan dan pengurangan
Phase 1. Zero check : Because addition and subtraction are identical except for a sign change, the process begins by changing the sign of the subtrahend if it is a subtract operation. Next, if either operand is 0, the other is reported as the result.
Phase 2. Significand alignment : The next phase is to manipulate the numbers so that the two exponents are equal.
Phase 3. Addition : Next, the two significands are added together, taking into account their signs. Because the signs may differ, the result may be 0. There is also the possibility of significand overflow by 1 digit. If so, the significand of the result is shifted right and the exponent is incremented. An exponent overflow could occur as a result; this would be reported and the operation halted.
Phase 4. Normalization : The final phase normalizes the result. Normalization consists of shifting significand digits left until the most significant digit (bit, or 4 bits for base-16 exponent) is nonzero. Each shift causes a decrement of the exponent and thus could cause an exponent underflow. Finally, the result must be rounded off and then reported.
Floating-Point Addition and Subtraction Flowchart
Floating-Point Multiplication Flowchart
Floating-Point Division Flowchart
Sumber
Slide AOK : Representasi Bilangan Biner
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