Residue Number Systems. Alternative number representations. TSTE 8 Digital Arithmetic Seminar 2. Residue Number Systems.

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1 TSTE8 Digital Arithmetic Seminar Oscar Gustafsson The idea is to use the residues of the numbers and perform operations on the residues Also called modular arithmetic since the residues are computed using the modulo function The residues will have shorter wordlengths so the computations are potentially faster Based on a set of relative prime modulis M M... MK } Relative prime so, gcdmp Mq) = p = q N The range of uniquely representable numbers is R = K k= M k Can represent any continuous range of R numbers Alternative number representations Logarithmic Number Systems Decimal Number Systems First, define and div M = M R = mod M = M div M) as the operations leading to the integer and remainder part of an integer division, respectively Denote X mod M as X M A basic identity is B M = M B M M where can be + or This means that we can perform the modular reduction at will Modular division can be performed using the multiplicative inverse defined as X = such that X M = A modular inverse of eists iff M = and gcd M) =

2 To determine the residue one can perform a normal integer division with remainder However, this is often to comple and faster methods eists for specific types of moduli For a moduli which is a power of the radi, e.g., N it is enough to take the N least significant bits digits) This also holds for two s radi) complement representations Other commonly used modulis are on the form N ± Here we can use M = mod M ) = mod M)+ div M)) mod M ) and M+ = mod M+) = mod M) div M)) mod M+) Eample: operations modulo 3 Addition + B 3 B Subtraction B 3 B Eample: reduce 755) = ) using the modulis Multiplication B 3 B Multiplicative inverse 3 -

3 Realization of modular additions and subtractions For M = N it is possible to simply use the N least significant bits of the addition/subtraction For an arbitrary M with N < M < N we first note that if + B M then + B M = + B M This condition is equivalent to + B + N M) N and + B M = + B M = + B + N M) N Therefore compute in parallel C = + B N and C = + B + N M) The correct output is C N if C div N =, i.e., + B + N M) N, i.e., the output carry bit is one, otherwise the correct output is C Eample: add 5 and 3 using the modulis Especially for M = N the correct result is either + B or + B + N This can be realized by performing + B and then take the output carry and add at the LSB position A multiplication is an addition of shifted terms, so this can be realized either by performing modular additions of the partial product results or by performing a normal multiplication followed by a modular reduction It should be noted that quantization is not possible in a straightforward way) using a residue number system, so the moduli set must be selected such that the moduli set range can cover all possible results and that moduli set should be used throughout all computations Conversion from RNS to binary can be performed using the Chinese Remainder Theorem Recall the range R = K k= M k For an RNS number... K } in a moduli set M M... MK } the corresponding binary representation can be computed as R = K k= Rk k Rk Mk R where Rk = R/Mk

4 Eample: Convert 3 4 5} in the moduli set 7 8 9} to radi- Challenging functions in RNS includes Reduce the magnitude/scale Range increase: can be handled by adding more modulis Comparison Sign-detection Eample: Compute in the moduli set } and convert back to radi- Logarithmic Number Systems In a base-l logarithmic number system LNS) a number is represented by the corresponding eponent as = ) S L E E = log L S = sign) Without loss of generality we will assume L = and not eplicitly state the base The LNS can be seen as a floating-point number without the significand The eponent typically consists of both an integer and a fractional part Large dynamic range and approimately constant relative representation error We need one more bit to represent a zero value

5 Logarithmic Number Systems Multiplication and division become additions/subtractions: C = B EC = E + EB C = /B EC = E EB Squaring and square roots become shifts: B = EB = E B = EB = E/ Eponents becomes multiplication: C = B EC = BE Logarithmic Number Systems An LNS adder/subtracter require both functions Φ + ) and Φ ) A B E > E A B lin lin log log E A S E B A S B E B E A E A E B Φ ( ) + Φ ( ) op S C E C log lin C Can perform both + and at the same time as the operations will always use different tables Logarithmic Number Systems Addition and subtraction become more complicated assuming > B ): C = ± B EC = log ± B = log ± B = E + log ± EB E = E + ΦEB E) In general we get EC = mae EB} + Φ E EB ) where Φ = Φ + ) = log + S Φ = Φ ) = log SΦ = and SΦ = S SB add/sub) It has been suggested to use LNS as a way to approimate multiplication, division etc. A simple way to determine the binary base-) logarithm of a number is to count the number of leading zeros forming the integer part of the logarithm and then determine the fractional part log + m) < for m < = e + m) log N) = e + log + m) ) A reverse conversion computing the eponential) is performed by determining m where m is the fractional part of the logarithm and then shift according to the integer part A simple approimation often referred to as Mitchell s logarithm) is log + m) m Similarly m + m, i.e., concatenating a one before the fractional bits

6 The approimation log + m) m can be seen as a linear approimation of the function log ().8.6 f() The same holds for the approimation m + m.8.6 f() Eample: consider the multiplication.. We get log.) = 3) +. =. and log.) = ) +. =. Adding the logarithms results in. +. =., leading to the final result. The eact answer is. Eample: Consider computing the square root of. log.) =., which when diving by two gives log.) =. and a final result of. The correct rounded) result is eactly the same in this case Consider the square root computation in the range < log ) = + = + In the range < 4 equivalent to ) log ) = log based.6 f()

7 ) = log based y.5.5 = log )log y ) )y ) = y + y y For division with y < + ) =.5 + 3)) = < so split in two regions log ) = Consider the square computation with < f() /y log based....3 y.5... y y ) = + y y + y + )) = + y y. Alternatively, as m + m deviates more for negative numbers close to minus one, y ) = + y +y + + y 3)) = + y 4 + y + y < so split in two regions y = log )+log y ) )+y ) = +y For multiplication with y < y log based /y log based

8 Logarithmic Number Systems If the application has a high dynamic range and/or many multiplications/division/eponentiations the LNS can be attractive The cost is conversion to and from LNS as well as the costly additions/subtractions Decimal Number Systems In some applications read money) the problem of representing decimal numbers using radi- representations is really unwanted You do not want to deposit 4.5 SEK to your bank account already containing SEK and end up with SEK Some observations related to this: Decimal numbers can be represented in BCD, requiring four bits per decimal digit, or in groups of three digits using bits ma 4) IEEE 754 defines three different decimal formats: decimal3, decimal64, and decimal8 containing 7, 6, and 34 decimal digits for the significand Two different ways to represent the significand The representation can be seen as in general de-normalized implicit prefi = ) However, for efficient use of the bits the eponent and signficand share some of the bits, leading to an implicit prefi of the significand in some cases