BKM algorithm
The BKM algorithm is a shift-and-add algorithm for computing elementary functions, first published in 1994 by Jean-Claude Bajard, Sylvanus Kla, and Jean-Michel Muller. BKM is based on computing complex logarithms (L-mode) and exponentials (E-mode) using a method similar to the algorithm Henry Briggs used to compute logarithms. By using a precomputed table of logarithms of negative powers of two, the BKM algorithm computes elementary functions using only integer add, shift, and compare operations.
BKM is similar to CORDIC, but uses a table of logarithms rather than a table of arctangents. On each iteration, a choice of coefficient is made from a set of nine complex numbers, 1, 0, −1, i, −i, 1+i, 1−i, −1+i, −1−i, rather than only −1 or +1 as used by CORDIC. BKM provides a simpler method of computing some elementary functions, and unlike CORDIC, BKM needs no result scaling factor. The convergence rate of BKM is approximately one bit per iteration, like CORDIC, but BKM requires more precomputed table elements for the same precision because the table stores logarithms of complex operands.
As with other algorithms in the shift-and-add class, BKM is particularly well-suited to hardware implementation. The relative performance of software BKM implementation in comparison to other methods such as polynomial or rational approximations will depend on the availability of fast multi-bit shifts (i.e. a barrel shifter) or hardware floating point arithmetic.
Overview
In order to solve the equation
- [math]\displaystyle{ \ln(x) = y }[/math]
the BKM algorithm takes advantage of a basic property of logarithms
- [math]\displaystyle{ \ln(ab) = \ln(a)+\ln(b) }[/math]
Using Pi notation, this identity generalizes to
- [math]\displaystyle{ \ln\left(\prod_{k=0}^n a_k\right) = \sum_{k=0}^n\ln(a_k) }[/math]
Because any number can be represented by a product, this allows us to choose any set of values [math]\displaystyle{ a_k }[/math] which multiply to give the value we started with. In computer systems, it's much faster to multiply and divide by multiples of 2, but because not every number is a multiple of 2, using [math]\displaystyle{ a_k = 1+2^m }[/math] is a better option than a more simple choice of [math]\displaystyle{ a_k = 2^m }[/math]. Since we want to start with large changes and get more accurate as [math]\displaystyle{ k }[/math] increases, we can more specifically use [math]\displaystyle{ a_k = 1+2^{-k} }[/math], allowing the product to approach any value between 1 and ~4.768, depending on which subset of [math]\displaystyle{ a_k }[/math] we use in the final product. At this point, the above equation looks like this:
- [math]\displaystyle{ \ln\left(\prod_{k\in K\subset \mathbb{Z}_0^+} 1+2^{-k}\right) = \sum_{k\in K\subset \mathbb{Z}_0^+}\ln(1+2^{-k}) }[/math]
This choice of [math]\displaystyle{ a_k }[/math] reduces the computational complexity of the product from repeated multiplication to simple addition and bit-shifting depending on the implementation. Finally, by storing the values [math]\displaystyle{ \ln(1+2^{-k}) }[/math] in a table, calculating the solution is also a simple matter of addition. Iteratively, this gives us two separate sequences. One sequence approaches the input value [math]\displaystyle{ x }[/math] while the other approaches the output value [math]\displaystyle{ \ln(x)=y }[/math]:
[math]\displaystyle{ x_k = \begin{cases} 1 & \text{if } k = 0 \\ x_{k-1}\cdot (1+2^{-k}) & \text{if } x_k \text{ would be} \leq x \\ x_{k-1} & \text{otherwise} \end{cases} }[/math]
Given this recursive definition and because [math]\displaystyle{ x_k }[/math] is strictly increasing, it can be shown by induction and convergence that
- [math]\displaystyle{ \lim_{k\to\infty}x_k = x }[/math]
for any [math]\displaystyle{ 1 \leq x \lesssim 4.768 }[/math]. For calculating the output, we first create the reference table
- [math]\displaystyle{ A_k = \ln(1+2^{-k}) }[/math]
Then the output is computed iteratively by the definition [math]\displaystyle{ y_k = \begin{cases} 0 & \text{if } k = 0 \\ y_{k-1} + A_k & \text{if } x_k \text{ would be} \leq x \\ y_{k-1} & \text{otherwise} \end{cases} }[/math] The conditions in this iteration are the same as the conditions for the input. Similar to the input, this sequence is also strictly increasing, so it can be shown that
- [math]\displaystyle{ \lim_{k\to\infty}y_k = y }[/math]
for any [math]\displaystyle{ 0 \leq y \lesssim 1.562 }[/math].
Because the algorithm above calculates both the input and output simultaneously, it's possible to modify it slightly so that [math]\displaystyle{ y }[/math] is the known value and [math]\displaystyle{ x }[/math] is the value we want to calculate, thereby calculating the exponential instead of the logarithm. Since x becomes an unknown in this case, the conditional changes from
- [math]\displaystyle{ \dots \text{if } x_k \text{ would be} \leq x }[/math]
to
- [math]\displaystyle{ \dots \text{if } y_k \text{ would be} \leq y }[/math]
Logarithm function
To calculate the logarithm function (L-mode), the algorithm in each iteration tests if [math]\displaystyle{ x_n \cdot (1+2^{-n}) \le x }[/math]. If so, it calculates [math]\displaystyle{ x_{n+1} }[/math] and [math]\displaystyle{ y_{n+1} }[/math]. After [math]\displaystyle{ N }[/math] iterations the value of the function is known with an error of [math]\displaystyle{ \Delta \ln(x) \le 2^{-N} }[/math].
Example program for natural logarithm in C++ (see A e
for table):
double log_e (const double Argument, const int Bits = 53) // 1 <= Argument <= 4.768462058 { double x = 1.0, y = 0.0, s = 1.0; for (int k = 0; k < Bits; k++) { double const z = x + x*s; if (z <= Argument) { x = z; y += A_e[k]; } s *= 0.5; } return y; }
Logarithms for bases other than e can be calculated with similar effort.
Example program for binary logarithm in C++ (see A 2
for table):
double log_2 (const double Argument, const int Bits = 53) // 1 <= Argument <= 4.768462058 { double x = 1.0, y = 0.0, s = 1.0; for (int k = 0; k < Bits; k++) { double const z = x + x*s; if (z <= Argument) { x = z; y += A_2[k]; } s *= 0.5; } return y; }
The allowed argument range is the same for both examples (1 ≤ Argument
≤ 4.768462058…). In the case of the base-2 logarithm the exponent can be split off in advance (to get the integer part) so that the algorithm can be applied to the remainder (between 1 and 2). Since the argument is smaller than 2.384231…, the iteration of k can start with 1. Working in either base, the multiplication by s can be replaced with direct modification of the floating point exponent, subtracting 1 from it during each iteration. This results in the algorithm using only addition and no multiplication.
Exponential function
To calculate the exponential function (E-mode), the algorithm in each iteration tests if [math]\displaystyle{ y_n + \ln(1+2^{-n}) \le y }[/math]. If so, it calculates [math]\displaystyle{ x_{n+1} }[/math] and [math]\displaystyle{ y_{n+1} }[/math]. After [math]\displaystyle{ N }[/math] iterations the value of the function is known with an error of [math]\displaystyle{ \Delta \exp(x) \le 2^{-N} }[/math].
Example program in C++ (see A e
for table):
double exp (const double Argument, const int Bits = 54) // 0 <= Argument <= 1.5620238332 { double x = 1.0, y = 0.0, s = 1.0; for (int k = 0; k < Bits; k++) { double const z = y + A_e[k]; if (z <= Argument) { y = z; x = x + x*s; } s *= 0.5; } return x; }
Tables for examples
static const double A_e [] = ...
|
---|
static const double A_e [] = // A_e[k] = ln (1 + 0.5^k) { 0.693147180559945297099404706000, 0.405465108108164392935428259000, 0.223143551314209769962616701000, 0.117783035656383447138088388000, 0.060624621816434840186291518000, 0.030771658666753686222134530000, 0.015504186535965253358272343000, 0.007782140442054949100825041000, 0.003898640415657322636221046000, 0.001951220131261749216850870000, 0.000976085973055458892686544000, 0.000488162079501351186957460000, 0.000244110827527362687853374000, 0.000122062862525677363338881000, 0.000061033293680638525913091000, 0.000030517112473186377476993000, 0.000015258672648362398138404000, 0.000007629365427567572417821000, 0.000003814689989685889381171000, 0.000001907346813825409407938000, 0.000000953673861659188260005000, 0.000000476837044516323000000000, 0.000000238418550679858000000000, 0.000000119209282445354000000000, 0.000000059604642999033900000000, 0.000000029802321943606100000000, 0.000000014901161082825400000000, 0.000000007450580569168250000000, 0.000000003725290291523020000000, 0.000000001862645147496230000000, 0.000000000931322574181798000000, 0.000000000465661287199319000000, 0.000000000232830643626765000000, 0.000000000116415321820159000000, 0.000000000058207660911773300000, 0.000000000029103830456310200000, 0.000000000014551915228261000000, 0.000000000007275957614156960000, 0.000000000003637978807085100000, 0.000000000001818989403544200000, 0.000000000000909494701772515000, 0.000000000000454747350886361000, 0.000000000000227373675443206000, 0.000000000000113686837721610000, 0.000000000000056843418860806400, 0.000000000000028421709430403600, 0.000000000000014210854715201900, 0.000000000000007105427357600980, 0.000000000000003552713678800490, 0.000000000000001776356839400250, 0.000000000000000888178419700125, 0.000000000000000444089209850063, 0.000000000000000222044604925031, 0.000000000000000111022302462516, 0.000000000000000055511151231258, 0.000000000000000027755575615629, 0.000000000000000013877787807815, 0.000000000000000006938893903907, 0.000000000000000003469446951954, 0.000000000000000001734723475977, 0.000000000000000000867361737988, 0.000000000000000000433680868994, 0.000000000000000000216840434497, 0.000000000000000000108420217249, 0.000000000000000000054210108624, 0.000000000000000000027105054312 }; |
static const double A_2 [] = ...
|
---|
static const double A_2 [] = // A_2[k] = log_2 (1 + 0.5^k) { 1.0000000000000000000000000000000000000000000000000000000000000000000000000000, 0.5849625007211561814537389439478165087598144076924810604557526545410982276485, 0.3219280948873623478703194294893901758648313930245806120547563958159347765589, 0.1699250014423123629074778878956330175196288153849621209115053090821964552970, 0.0874628412503394082540660108104043540112672823448206881266090643866965081686, 0.0443941193584534376531019906736094674630459333742491317685543002674288465967, 0.0223678130284545082671320837460849094932677948156179815932199216587899627785, 0.0112272554232541203378805844158839407281095943600297940811823651462712311786, 0.0056245491938781069198591026740666017211096815383520359072957784732489771013, 0.0028150156070540381547362547502839489729507927389771959487826944878598909400, 0.0014081943928083889066101665016890524233311715793462235597709051792834906001, 0.0007042690112466432585379340422201964456668872087249334581924550139514213168, 0.0003521774803010272377989609925281744988670304302127133979341729842842377649, 0.0001760994864425060348637509459678580940163670081839283659942864068257522373, 0.0000880524301221769086378699983597183301490534085738474534831071719854721939, 0.0000440268868273167176441087067175806394819146645511899503059774914593663365, 0.0000220136113603404964890728830697555571275493801909791504158295359319433723, 0.0000110068476674814423006223021573490183469930819844945565597452748333526464, 0.0000055034343306486037230640321058826431606183125807276574241540303833251704, 0.0000027517197895612831123023958331509538486493412831626219340570294203116559, 0.0000013758605508411382010566802834037147561973553922354232704569052932922954, 0.0000006879304394358496786728937442939160483304056131990916985043387874690617, 0.0000003439652607217645360118314743718005315334062644619363447395987584138324, 0.0000001719826406118446361936972479533123619972434705828085978955697643547921, 0.0000000859913228686632156462565208266682841603921494181830811515318381744650, 0.0000000429956620750168703982940244684787907148132725669106053076409624949917, 0.0000000214978311976797556164155504126645192380395989504741781512309853438587, 0.0000000107489156388827085092095702361647949603617203979413516082280717515504, 0.0000000053744578294520620044408178949217773318785601260677517784797554422804, 0.0000000026872289172287079490026152352638891824761667284401180026908031182361, 0.0000000013436144592400232123622589569799954658536700992739887706412976115422, 0.0000000006718072297764289157920422846078078155859484240808550018085324187007, 0.0000000003359036149273187853169587152657145221968468364663464125722491530858, 0.0000000001679518074734354745159899223037458278711244127245990591908996412262, 0.0000000000839759037391617577226571237484864917411614198675604731728132152582, 0.0000000000419879518701918839775296677020135040214077417929807824842667285938, 0.0000000000209939759352486932678195559552767641474249812845414125580747434389, 0.0000000000104969879676625344536740142096218372850561859495065136990936290929, 0.0000000000052484939838408141817781356260462777942148580518406975851213868092, 0.0000000000026242469919227938296243586262369156865545638305682553644113887909, 0.0000000000013121234959619935994960031017850191710121890821178731821983105443, 0.0000000000006560617479811459709189576337295395590603644549624717910616347038, 0.0000000000003280308739906102782522178545328259781415615142931952662153623493, 0.0000000000001640154369953144623242936888032768768777422997704541618141646683, 0.0000000000000820077184976595619616930350508356401599552034612281802599177300, 0.0000000000000410038592488303636807330652208397742314215159774270270147020117, 0.0000000000000205019296244153275153381695384157073687186580546938331088730952, 0.0000000000000102509648122077001764119940017243502120046885379813510430378661, 0.0000000000000051254824061038591928917243090559919209628584150482483994782302, 0.0000000000000025627412030519318726172939815845367496027046030028595094737777, 0.0000000000000012813706015259665053515049475574143952543145124550608158430592, 0.0000000000000006406853007629833949364669629701200556369782295210193569318434, 0.0000000000000003203426503814917330334121037829290364330169106716787999052925, 0.0000000000000001601713251907458754080007074659337446341494733882570243497196, 0.0000000000000000800856625953729399268240176265844257044861248416330071223615, 0.0000000000000000400428312976864705191179247866966320469710511619971334577509, 0.0000000000000000200214156488432353984854413866994246781519154793320684126179, 0.0000000000000000100107078244216177339743404416874899847406043033792202127070, 0.0000000000000000050053539122108088756700751579281894640362199287591340285355, 0.0000000000000000025026769561054044400057638132352058574658089256646014899499, 0.0000000000000000012513384780527022205455634651853807110362316427807660551208, 0.0000000000000000006256692390263511104084521222346348012116229213309001913762, 0.0000000000000000003128346195131755552381436585278035120438976487697544916191, 0.0000000000000000001564173097565877776275512286165232838833090480508502328437, 0.0000000000000000000782086548782938888158954641464170239072244145219054734086, 0.0000000000000000000391043274391469444084776945327473574450334092075712154016, 0.0000000000000000000195521637195734722043713378812583900953755962557525252782, 0.0000000000000000000097760818597867361022187915943503728909029699365320287407, 0.0000000000000000000048880409298933680511176764606054809062553340323879609794, 0.0000000000000000000024440204649466840255609083961603140683286362962192177597, 0.0000000000000000000012220102324733420127809717395445504379645613448652614939, 0.0000000000000000000006110051162366710063906152551383735699323415812152114058, 0.0000000000000000000003055025581183355031953399739107113727036860315024588989, 0.0000000000000000000001527512790591677515976780735407368332862218276873443537, 0.0000000000000000000000763756395295838757988410584167137033767056170417508383, 0.0000000000000000000000381878197647919378994210346199431733717514843471513618, 0.0000000000000000000000190939098823959689497106436628681671067254111334889005, 0.0000000000000000000000095469549411979844748553534196582286585751228071408728, 0.0000000000000000000000047734774705989922374276846068851506055906657137209047, 0.0000000000000000000000023867387352994961187138442777065843718711089344045782, 0.0000000000000000000000011933693676497480593569226324192944532044984865894525, 0.0000000000000000000000005966846838248740296784614396011477934194852481410926, 0.0000000000000000000000002983423419124370148392307506484490384140516252814304, 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0.0000000000000000000000000000364187429092330340379920387564158411083803465567, 0.0000000000000000000000000000182093714546165170189960193783228378441837282509, 0.0000000000000000000000000000091046857273082585094980096891901482445902524441, 0.0000000000000000000000000000045523428636541292547490048446022564529197237262, 0.0000000000000000000000000000022761714318270646273745024223029238091160103901 }; |
Notes
References
- "BKM: A new hardware algorithm for complex elementary functions". IEEE Transactions on Computers 43 (8): 955–963. August 1994. doi:10.1109/12.295857. ISSN 0018-9340. http://perso.ens-lyon.fr/jean-michel.muller/BKM94.pdf. Retrieved 2017-12-21.
- "On-Line Hardware Implementation for Complex Exponential and Logarithm". ESSCIRC '94: Twientieth European Solid-State Circuits Conference. Ulm, Germany. 20–22 September 1994. ISBN 2-86332-160-9. https://ieeexplore.ieee.org/document/5468485. Retrieved 2021-08-23. (5 pages) [1][2]
- Luk, Franklin T., ed (1999-11-02). "Evaluation of Complex Elementary Functions: A New Version of BKM". SPIE Proceedings, Advanced Signal Processing Algorithms, Architectures, and Implementations IX. Advanced Signal Processing Algorithms, Architectures, and Implementations IX (Society of Photo-Optical Instrumentation Engineers (SPIE)) 3807: 2–9. doi:10.1117/12.367631. Bibcode: 1999SPIE.3807....2B. https://webusers.imj-prg.fr/~jean-claude.bajard/MesPublis/Spie1999.pdf. Retrieved 2020-06-09. (8 pages) [3]
- "Radix-10 BKM Algorithm for Computing Transcendentals on Pocket Computers". Journal of VLSI Signal Processing (Kluwer Academic Publishers / Institut national de recherche en informatique et en automatique (INRIA)) 25 (2): 179–186. 2006-05-24. doi:10.1023/A:1008127208220. RR-3754. INRIA-00072908. Theme 2 ISSN 0249-6399. ISSN 0922-5773. https://hal.inria.fr/inria-00072908/document. Retrieved 2018-07-11. (1+15 pages) [4][5][6]
- "High radix BKM algorithm with Selection by Rounding". 2002-01-21. http://ftp.jussieu.fr/lip6/reports/2002/lip6.2002.009.pdf. [7] (1+11 pages)
- "High Radix BKM Algorithm". Numerical Algorithms. SCAN'2002 International Conference (Springer Science+Business Media, LLC) 37 (1–4 [4]): 113–125. 2004-12-01. doi:10.1023/B:NUMA.0000049459.69390.ff. ISSN 1017-1398. Bibcode: 2004NuAlg..37..113D. https://www.researchgate.net/publication/263307443_High_Radix_BKM_Algorithm_SCAN%272002_International_Conference_Guest_Editors_Rene_Alt_and_Jean-Luc_Lamotte. [8]
- Elementary Functions: Algorithms and Implementation (2 ed.). Boston, MA, USA: Birkhäuser. 2006. ISBN 978-0-8176-4372-0.
- Elementary Functions: Algorithms and Implementation (3 ed.). Boston, MA, USA: Birkhäuser. 2016-12-12. ISBN:1-4899-7981-6. ISBN 978-1-4899-7981-0.
Further reading
- (in de) Arithmetische Algorithmen der Mikrorechentechnik (1 ed.). Berlin, Germany: VEB Verlag Technik. 1989. pp. 280–282. ISBN:978-3-34100515-6. EAN 9783341005156. MPN 5539165. License 201.370/4/89. ISBN 3-34100515-3. https://books.google.com/books?id=DqYWAQAAMAAJ. Retrieved 2015-12-01.
- "Pseudo Division and Pseudo Multiplication Processes". IBM Journal of Research and Development (Riverton, New Jersey, USA: IBM Corporation) 6 (2): 210–226, 287. 1961-08-29. April 1962. doi:10.1147/rd.62.0210. http://dl.acm.org/citation.cfm?id=1661979.1661985. Retrieved 2015-12-01.
- "Automatic computation of exponentials, logarithms, ratios and square roots". IBM Journal of Research and Development (San Jose, California, USA; Riverton, New Jersey, USA: IBM San Jose Research Laboratory; IBM Corporation) 16 (4): 380–388. July 1972. doi:10.1147/rd.164.0380. http://dl.acm.org/citation.cfm?id=1661877. Retrieved 2015-12-01.
- "Accelerated Shift-and-Add algorithms". Reliable Computing (Boston, USA: Laboratoire d'Analyse Numérique et d'Optimisation (ANO) de l'Université des Sciences et Technologies de Lille; Kluwer Academic Publishers) 6 (2): 193–205. 2000-05-01. doi:10.1023/A:1009921407000. ISSN 1385-3139. OCLC 67306353. https://perso.math.univ-toulouse.fr/yak/files/2018/09/RC-2000.pdf. Retrieved 2021-08-23. (14 pages) [9]
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