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.
Declare the number of bits in a byte,
B, as a constant. In this example,B = 8.Use the function
fi_normalize_unsigned_8_bit_byte(), described in the example Normalize Data for Lookup Tables, to normalize the inputu > 0such thatu = x*2^n,0.5 <= x < 2, andnis even.Extract the upper
Bbits ofx. Letx_Bdenote the upperBbits ofx.Generate a lookup table,
SQRTLUT, such that the integeri = x_B- 2^(B-2) + 1is used as an index toSQRTLUTso thatsqrt(x_B)can be evaluated by looking up the indexsqrt(x_B) = SQRTLUT(i).Use the remainder,
r = x - x_B, interpreted as a fraction, to linearly interpolate betweenSQRTLUT(i)and the next value in the tableSQRTLUT(i+1). The remainder,r, is created by extracting the lowerw - Bbits ofx, wherewdenotes the wordlength ofx. It is interpreted as a fraction by using functionreinterpretcast().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(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