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Number Representation Part 2 Little-Endian vs. Big-Endian Representations Floating Point Representations ECE 645: Lecture 5.

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Presentation on theme: "Number Representation Part 2 Little-Endian vs. Big-Endian Representations Floating Point Representations ECE 645: Lecture 5."— Presentation transcript:

1 Number Representation Part 2 Little-Endian vs. Big-Endian Representations Floating Point Representations ECE 645: Lecture 5

2 Required Reading Endianness, from Wikipedia, the free encyclopedia Behrooz Parhami, Computer Arithmetic: Algorithms and Hardware Design Chapter 17, Floating-Point Representations

3 Little-Endian vs. Big-Endian Representation of Integers

4 Little-Endian vs. Big-Endian Representation A0 B1 C2 D3 E4 F LSB MSB MSB = A0 B1 C2 D3 E4 F5 67 LSB = 89 Big-Endian Little-Endian LSB = 89 0 MAX 67 F5 E4 D3 C2 B1 MSB = A0 address

5 Little-Endian vs. Big-Endian Camps Big-Endian Little-Endian 0 MAX address MSB LSB... LSB MSB... Motorola 68xx, 680x0 Intel IBM Hewlett-Packard DEC VAX Internet TCP/IP Sun SuperSPARC Bi-Endian Motorola Power PC Silicon Graphics MIPS RS 232 AMD

6 Origin of the terms Little-Endian vs. Big-Endian Jonathan Swift, Gulliver’s Travels A law requiring all citizens of Lilliput to break their soft-eggs at the little ends only A civil war breaking between the Little Endians and the Big-Endians, resulting in the Big Endians taking refuge on a nearby island, the kingdom of Blefuscu Satire over holy wars between Protestant Church of England and the Catholic Church of France

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8 Little-Endian vs. Big-Endian Big-EndianLittle-Endian easier to determine a sign of the number easier to compare two numbers easier to divide two numbers easier to print easier addition and multiplication of multiprecision numbers Advantages and Disadvantages

9 Pointers (1) F5 E4 D3 C2 B1 A0 Big-Endian Little-Endian 0 MAX address int * iptr; (* iptr) = 8967;(* iptr) = 6789; iptr+1

10 Pointers (2) F5 E4 D3 C2 B1 A0 Big-Endian Little-Endian 0 MAX address long int * lptr; (* lptr) = 8967F5E4;(* lptr) = E4F56789; lptr + 1

11 Floating Point Representations

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16 The ANSI/IEEE standard floating- point number representation formats Originally IEEE Superseded by IEEE Standard.

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19 Table 17.1 Some features of the ANSI/IEEE standard floatingpoint number representation formats

20 00017FFEFF7E – –1+1 Decimal code Hex code Exponent value f = 0: Representation of  0 f  0: Representation of denormals, 0.f  2 –126 f = 0: Representation of  f  0: Representation of NaNs Exponent encoding in 8 bits for the single/short (32-bit) ANSI/IEEE format 1.f  2 e Exponent Encoding

21 Fig Denormals in the IEEE single-precision format.

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24 The IEEE standard includes five rounding modes: Round to nearest, ties away from 0 (rtna) Round to nearest, ties to even (rtne) [default rounding mode] Round toward zero (inward) Round toward +  (upward) Round toward –  (downward) Rounding Modes

25 Round to Nearest Number Fig Rounding of a signed- magnitude value to the nearest number. Rounding has a slight upward bias. Consider rounding (x k–1 x k–2... x 1 x 0. x –1 x –2 ) two to an integer (y k–1 y k–2... y 1 y 0. ) two The four possible cases, and their representation errors are: x –1 x –2 Round Error 00 down 0 01 down– up up 0.25 With equal prob., mean = For certain calculations, the probability of getting a midpoint value can be much higher than 2 –l rtna(x)

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27 Directed Rounding: Motivation We may need result errors to be in a known direction Example: in computing upper bounds, larger results are acceptable, but results that are smaller than correct values could invalidate the upper bound This leads to the definition of directed rounding modes upward-directed rounding (round toward +  ) and downward-directed rounding (round toward –  ) (required features of IEEE floating-point standard)

28 Directed Rounding: Visualization Fig Upward-directed rounding or rounding toward + . Fig Truncation or chopping of a 2’s-complement number (same as downward- directed rounding).

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30 Requirements for Arithmetic Results of the 4 basic arithmetic operations (+, , ,  ) as well as square- rooting must match those obtained if all intermediate computations were infinitely precise That is, a floating-point arithmetic operation should introduce no more imprecision than the error attributable to the final rounding of a result that has no exact representation (this is the best possible) Example: (1 + 2  1 )  (1 + 2  23 ) Rounded result1 + 2   22 Error = ½ ulp Exact result1 + 2    24

31 New IEEE Standard Basic Formats

32 New IEEE Standard Binary Interchange Formats


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