Topic 3d Representation of Real Numbers Introduction to Computer Systems Engineering (CPEG 323) 2019/5/1 cpeg323-05F\Topic3d-323

Recap What can be represented in n bits? But, what about? Unsigned 0 to 2n - 1 2’s Complement -2n-1 to 2n-1 - 1 BCD 0 to 10n/4 - 1 But, what about? very large numbers 9,349,398,989,787,762,244,859,087,678 very small number 0.0000000000000000000000045691 rationals 2/3 Irrationals 2.71828…., 3.1415926….., 2019/5/1 cpeg323-05F\Topic3d-323

Conceptual Overview: Finite-precision numbers Negative underflow Positive underflow Expressible negative numbers zero Expressible positive numbers Negative overflow Positive overflow ………………. Is there any “underflow” region in integer representation? Between two adjacent integer numbers, there is no other integer. 2019/5/1 cpeg323-05F\Topic3d-323

Representing Real Numbers How to represent fractional parts to the right of the ``decimal'' point? A number like 0.12 (i.e., (1/2)10) not represented well by integers 0 or 1! Two ways to represent real numbers better: Fixed point Floating point 2019/5/1 cpeg323-05F\Topic3d-323

Fixed-Point Data Format 4 2 1 1/2 1/4 …… 1/8 S 1 1 1 . 1 Sign Integer Fractional When representing a real value, you have two choices: the fixed-point format or the floating-point format. Any way, to avoid nonlinear effects introduced by overflow, underflow, saturation, and wrapping during computation, control of each variable’s range has to be done by appropriate scaling of operands. In the floating-point case, the exponent of a variable is part of the run-time representation and is computed automatically by the floating-point ALU. But in the fixed-point case, the exponent of each variable is implicit and determined by the programmer off-line using the predicted dynamic range of the variable. Therefore, the scaling of fixed-point operands has to be done explicitly by the programmer. This is generally a tedious and error-prone process. In fact, you may regard the fixed-point as a form of integer with an additional hypothetical binary point. In our research, the number of bits to the left of the binary-point excluding the sign bit is called Integer Word-Length or IWL. hypothetical binary point 2019/5/1 cpeg323-05F\Topic3d-323

Fixed Point Pros Cons Add two reals just by adding the integers Much less logic than floating point Faster Often used in digital signal processing Cons The range of numbers is narrow number=400 000 000 000 000 000 000 000 000. 000 It is much more economical to represent as 4*1026 2019/5/1 cpeg323-05F\Topic3d-323

Recall Scientific Notation decimal point exponent -23 6.02 x 10 Significand radix (base) Issues: Arithmetic (+, -, *, / ) Representation, Normal form Range and Precision(Determined by?) Rounding and errors Exceptions (e.g., divide by zero, overflow, underflow) Properties 2019/5/1 cpeg323-05F\Topic3d-323

Scientific Notation: Normalized 12.35 x 10^-9 ? 1.235 x 10^-8 ? scientific notation: has a single digit to the left of the decimal point Normalized scientific notation: such a single digit must be non-zero. 2019/5/1 cpeg323-05F\Topic3d-323

Floating Point Representation Numerical Form: (–1)s M 2E Sign bit s determines whether number is negative or positive Significand M normally a fractional value in range [1.0,2.0). Exponent E weights value by power of two Encoding MSB is sign bit: S=0/1 exp field encodes E frac field encodes M s exp frac 2019/5/1 cpeg323-05F\Topic3d-323

IEEE 754 standard Three formats: single/double/extended precision (32,64,80 bits). Single precision: s exp frac 1 bit 8 bits 23 bits Double precision: see page 192 in P&H book 2019/5/1 cpeg323-05F\Topic3d-323

IEEE 754 Standard Floating Point Representation the representation: (–1)s (1+ Fraction) 2E-bias where E is the exponent representation in the exp field Sign bit s determines whether number is negative or positive Fraction is normally a fractional value in range between 0 and 1 A leading 1 added to the fraction is “implicit” Exponent using a “biased” notation For single precision – the bias is 127 2019/5/1 cpeg323-05F\Topic3d-323

Advantage of using the biased notation for exponents Under the single-precision IEEE 754 standard: bias = 127 If the real exponent is +1, what is the biased exponent ? How about if the real exponent is -1 ? Answer: 1+127 = 128! (Note 128-127 = +1!) Answer: -1+127 = 126! (Note 126-127 = -1!) How about if the real exponent is -127 ? 2019/5/1 cpeg323-05F\Topic3d-323

Normalized Encoding Example Value: Float F = 15213.010; 1521310 = 1.1101101101101 X 213 Significand M = 1.1101101101101 frac = 11011011011010000000000 (23 bits! With leading 1 hiding!) Exponent E = 1310, Bias = 12710 Exp = 14010 = 10001100 Floating Point Representation: Binary:01000110011011011011010000000000 exp frac 2019/5/1 cpeg323-05F\Topic3d-323

Denormalized Values Condition Significand value M = 0.xxx…x Cases  exp = 000…0 Significand value M = 0.xxx…x xxx…x: bits of frac Cases frac = 000…0 Represents value 0 Note that have distinct values +0 and –0 frac  000…0 Numbers very close to 0.0 Lose precision as get smaller “Gradual underflow” 2019/5/1 cpeg323-05F\Topic3d-323

Special Values (infinity) Condition Not-a-Number (NaN) exp = 111…1 Operation that overflows Both positive and negative E.g., 1.0/0.0 = 1.0/0.0 = +, 1.0/0.0 =  Not-a-Number (NaN) Represents case when no numeric value can be determined E.g., sqrt(–1),  2019/5/1 cpeg323-05F\Topic3d-323

IEEE 754: Summary Normalized: ± 0<exp<max Any bit pattern ± Denormalized: ± Any nonzero bit pattern zero: ± Infinite: ± 11…1 NaN: ± 11…1 Any nonzero bit pattern 2019/5/1 cpeg323-05F\Topic3d-323

IEEE754: Summary (Cont.) + 0 +0  -Normalized -Denorm +Denorm NaN NaN 0 +0 Negative overflow Positive underflow Negative underflow Positive overflow 2019/5/1 cpeg323-05F\Topic3d-323

Special Properties of Encoding FP Zero Same as Integer Zero All bits = 0 Can (Almost) Use Unsigned Integer Comparison Must first compare sign bits Must consider -0 = 0 NaNs problematic Will be greater than any other values Otherwise OK Denorm vs. normalized Normalized vs. infinity 2019/5/1 cpeg323-05F\Topic3d-323

FP Addition Operands Exact Result: (–1)s M 2E (–1)s1 M1 2E1 Assume E1 > E2 Exact Result: (–1)s M 2E Exponent E: E1 Sign s, significand M: Result of signed align & add E1–E2 (–1)s1 M1 (–1)s2 M2 + (–1)s M 2019/5/1 cpeg323-05F\Topic3d-323

Decimal Number Conversion Convert Binary to Decimal (base 2 to base 10) x x x x x. d d d d d d …2 24 23 22 21 20 2-1 2-2 2-3 2-4 2-5 2-6 … … 1101.0112 = 1*23+1*22+0*21+1*20+0*2-1+1*2-2+1*2-3 = 13.37510 2019/5/1 cpeg323-05F\Topic3d-323

Decimal Number Conversion (Cont.) Convert Decimal Integer to binary Integer divide the decimal value by 2 and then write down the remainder from bottom to top (Divide 2 and Get the Remainders) 3710 = ?2 Quotient reminder 37÷2 = 18 … 1 18÷2 = 9 … 0 9÷2 = 4 … 1 4÷2 = 2 … 0 2÷2= 1 … 0 1÷2 = 0 … 1 1001012 Can it always be converted into an accurate binary number? 2019/5/1 YES! cpeg323-05F\Topic3d-323

Decimal Number Conversion (Cont.) Convert Decimal Fraction to Binary Fraction multiply the decimal value by 2 and then write down the integer number from top to bottom (Multiply 2 and Get the Integers) . 0.37510 = ?2 .375 * 2 = 0.75 .75 * 2 = 1.5 .5 * 2 = 1.0 Where to put the binary point? Prior to the First number! 0.0112 Can it always be converted into an accurate binary number? 2019/5/1 NO! cpeg323-05F\Topic3d-323

Decimal Number Conversion (Cont.) Convert Decimal Number to Binary Binary Put Together: Integer Part . Fraction Part 37.37510 = 100101.011 2019/5/1 cpeg323-05F\Topic3d-323

Summary Computer arithmetic is constrained by limited precision Bit patterns have no inherent meaning but standards do exist two’s complement IEEE 754 floating point OPcode determines “meaning” of the bit patterns Performance and accuracy are important so there are many complexities in real machines (i.e., algorithms and implementation). 2019/5/1 cpeg323-05F\Topic3d-323

IEEE 754 Floating point Review (1)Normalized scientific notation (-1)ssignificand*2exp (2) IEEE 754 Representation (-1)s(1+fraction)*2E-bias E = exp+bias (127) s E fraction 2019/5/1 cpeg323-05F\Topic3d-323