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Implement Fixed-Point Square Root Using Lookup Table

This example shows how to implement fixed-point square root using a lookup table. Lookup tables generate efficient code for embedded devices.

Setup

To ensure that this example does not change your preferences or settings, this code stores the original state.

originalFormat = get(0,'format'); format long g
originalWarningState = warning('off','fixed:fi:underflow');
originalFiprefState = get(fipref); reset(fipref)

You will restore this state at the end of the example.

Square Root Implementation

The square root algorithm is summarized here.

  1. Declare the number of bits in a byte, B, as a constant. In this example, B = 8.

  2. Use the function fi_normalize_unsigned_8_bit_byte(), described in the example Normalize Data for Lookup Tables, to normalize the input u > 0 such that u = x*2^n, 0.5 <= x < 2, and n is even.

  3. Extract the upper B bits of x. Let x_B denote the upper B bits of x.

  4. Generate a lookup table, SQRTLUT, such that the integer i = x_B- 2^(B-2) + 1 is used as an index to SQRTLUT so that sqrt(x_B) can be evaluated by looking up the index sqrt(x_B) = SQRTLUT(i).

  5. Use the remainder, r = x - x_B, interpreted as a fraction, to linearly interpolate between SQRTLUT(i) and the next value in the table SQRTLUT(i+1). The remainder, r, is created by extracting the lower w - B bits of x, where w denotes the wordlength of x. It is interpreted as a fraction by using function reinterpretcast().

  6. Finally, compute the output using the lookup table and linear interpolation:

sqrt(u) = sqrt(x*2^n)

= sqrt(x)*2^(n/2)

= (SQRTLUT(i) + r*(SQRTLUT(i+1) - SQRTLUT(i)))*2^(n/2)

The function fi_sqrtlookup_8_bit_byte(), defined at the end of this example, implements this algorithm.

Example

Use fi_sqrtlookup_8_bit_byte() to compute the fixed-point square root using a lookup table. Compare the fixed-point lookup table result to the square root calculated using sqrt and double precision.

u = fi(linspace(0,128,1000),0,16,12);
y = fi_sqrtlookup_8_bit_byte(u);
y_expected = sqrt(double(u));

Plot the results.

clf
subplot(211)
plot(u,y,u,y_expected)
legend('Output','Expected output','Location','Best')

subplot(212)
plot(u,double(y)-y_expected,'r')
legend('Error')

Figure contains 2 axes. Axes 1 contains 2 objects of type line. These objects represent Output, Expected output. Axes 2 contains an object of type line. This object represents Error.

figure(gcf)

Cleanup

Restore the original state.

set(0,'format',originalFormat);
warning(originalWarningState);
fipref(originalFiprefState);

sqrt_lookup_table Function Definition

The function sqrt_lookup_table loads the lookup table of square-root values. You can create the table by running:

sqrt_table = sqrt((2^(B-2):2^(B))/2^(B-1));

function SQRTLUT = sqrt_lookup_table()
    B = 8;  % Number of bits in a byte
    % sqrt_table = sqrt((2^(B-2):2^(B))/2^(B-1))
    sqrt_table = [0.707106781186548   0.712609640686961   0.718070330817254   0.723489806424389 ...
                  0.728868986855663   0.734208757779421   0.739509972887452   0.744773455488312 ...
                  0.750000000000000   0.755190373349661   0.760345316287277   0.765465544619743 ...
                  0.770551750371122   0.775604602874429   0.780624749799800   0.785612818123533 ...
                  0.790569415042095   0.795495128834866   0.800390529679106   0.805256170420320 ...
                  0.810092587300983   0.814900300650331   0.819679815537750   0.824431622392057 ...
                  0.829156197588850   0.833854004007896   0.838525491562421   0.843171097702003 ...
                  0.847791247890659   0.852386356061616   0.856956825050130   0.861503047005639 ...
                  0.866025403784439   0.870524267324007   0.875000000000000   0.879452954966893 ...
                  0.883883476483184   0.888291900221993   0.892678553567856   0.897043755900458 ...
                  0.901387818865997   0.905711046636840   0.910013736160065   0.914296177395487 ...
                  0.918558653543692   0.922801441264588   0.927024810886958   0.931229026609459 ...
                  0.935414346693485   0.939581023648307   0.943729304408844   0.947859430506444 ...
                  0.951971638232989   0.956066158798647   0.960143218483576   0.964203038783845 ...
                  0.968245836551854   0.972271824131503   0.976281209488332   0.980274196334883 ...
                  0.984250984251476   0.988211768802619   0.992156741649222   0.996086090656827 ...
                  1.000000000000000   1.003898650263063   1.007782218537319   1.011650878514915 ...
                  1.015504800579495   1.019344151893756   1.023169096484056   1.026979795322186 ...
                  1.030776406404415   1.034559084827928   1.038327982864759   1.042083250033317 ...
                  1.045825033167594   1.049553476484167   1.053268721647045   1.056970907830485 ...
                  1.060660171779821   1.064336647870400   1.068000468164691   1.071651762467640 ...
                  1.075290658380328   1.078917281352004   1.082531754730548   1.086134199811423 ...
                  1.089724735885168   1.093303480283494   1.096870548424015   1.100426053853688 ...
                  1.103970108290981   1.107502821666834   1.111024302164449   1.114534656257938 ...
                  1.118033988749895   1.121522402807898   1.125000000000000   1.128466880329237 ...
                  1.131923142267177   1.135368882786559   1.138804197393037   1.142229180156067 ...
                  1.145643923738960   1.149048519428140   1.152443057161611   1.155827625556683 ...
                  1.159202311936963   1.162567202358642   1.165922381636102   1.169267933366857 ...
                  1.172603939955857   1.175930482639174   1.179247641507075   1.182555495526531 ...
                  1.185854122563142   1.189143599402528   1.192424001771182   1.195695404356812 ...
                  1.198957880828180   1.202211503854459   1.205456345124119   1.208692475363357 ...
                  1.211919964354082   1.215138880951474   1.218349293101120   1.221551267855754 ...
                  1.224744871391589   1.227930169024281   1.231107225224513   1.234276103633219 ...
                  1.237436867076458   1.240589577579950   1.243734296383275   1.246871083953750 ...
                  1.250000000000000   1.253121103485214   1.256234452640111   1.259340104975618 ...
                  1.262438117295260   1.265528545707287   1.268611445636527   1.271686871835988 ...
                  1.274754878398196   1.277815518766305   1.280868845744950   1.283914911510884 ...
                  1.286953767623375   1.289985465034393   1.293010054098575   1.296027584582983 ...
                  1.299038105676658   1.302041665999979   1.305038313613819   1.308028096028522 ...
                  1.311011060212689   1.313987252601790   1.316956719106592   1.319919505121430 ...
                  1.322875655532295   1.325825214724777   1.328768226591831   1.331704734541407 ...
                  1.334634781503914   1.337558409939543   1.340475661845451   1.343386578762792 ...
                  1.346291201783626   1.349189571557681   1.352081728298996   1.354967711792425 ...
                  1.357847561400027   1.360721316067327   1.363589014329464   1.366450694317215 ...
                  1.369306393762915   1.372156150006259   1.375000000000000   1.377837980315538 ...
                  1.380670127148408   1.383496476323666   1.386317063301177   1.389131923180804 ...
                  1.391941090707505   1.394744600276337   1.397542485937369   1.400334781400505 ...
                  1.403121520040228   1.405902734900249   1.408678458698081   1.411448723829527 ...
                  1.414213562373095];
    % Cast to fixed point with the most accurate rounding method
    WL = 4*B;  % Word length
    FL = 2*B;  % Fraction length
    SQRTLUT = fi(sqrt_table,1,WL,FL,'RoundingMethod','Nearest');
    % Set fimath for the most efficient math operations
    F = fimath('OverflowAction','Wrap',...
               'RoundingMethod','Floor',...
               'SumMode','KeepLSB',...
               'SumWordLength',WL,...
               'ProductMode','KeepLSB',...
               'ProductWordLength',WL);
    SQRTLUT = setfimath(SQRTLUT,F);
end

fi_sqrtlookup_8_bit_byte() Function Definition

function y = fi_sqrtlookup_8_bit_byte(u)
    % Load the lookup table
    SQRTLUT = sqrt_lookup_table();
    % Remove fimath from the input to insulate this function from math
    % settings declared outside this function.
    u = removefimath(u);
    % Declare the output
    y = coder.nullcopy(fi(zeros(size(u)),numerictype(SQRTLUT),fimath(SQRTLUT)));
    B = 8; % Number of bits in a byte
    w = u.WordLength;
    for k = 1:numel(u)
        assert(u(k)>=0,'Input must be non-negative.');
        if u(k)==0
            y(k)=0;
        else
            % Normalize the input such that u = x*2^n and 0.5 <= x < 2
            [x,n] = fi_normalize_unsigned_8_bit_byte(u(k));
            isodd = storedInteger(bitand(fi(1,1,8,0),fi(n)));
            x = bitsra(x,isodd);
            n = n + isodd;
            % Extract the high byte of x
            high_byte = storedInteger(bitsliceget(x,w,w-B+1));
            % Convert the high byte into an index for SQRTLUT
            i =  high_byte - 2^(B-2) + 1;
            % The upper byte was used for the index into SQRTLUT.
            % The remainder, r, interpreted as a fraction, is used to
            % linearly interpolate between points.
            T_unsigned_fraction = numerictype(0,w-B,w-B);
            r = reinterpretcast(bitsliceget(x,w-B,1),T_unsigned_fraction);
            y(k) = bitshift((SQRTLUT(i) + r*(SQRTLUT(i+1) - SQRTLUT(i))),...
                            bitsra(n,1));
        end
    end
    % Remove fimath from the output to insulate the caller from math settings
    % declared inside this function.
    y = removefimath(y);
end