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

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

Little-Endian vs. Big-Endian Representation of Integers

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

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

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

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

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

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

Floating Point Representations

The ANSI/IEEE standard floating- point number representation formats Originally IEEE Superseded by IEEE Standard.

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

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

Fig Denormals in the IEEE single-precision format.

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

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)

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)

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).

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

New IEEE Standard Basic Formats

New IEEE Standard Binary Interchange Formats