BKM algorithm

<h2 id="overview">Overview</h2>
<p>In order to solve the equation
</p>

ln
        ⁡
        (
        x
        )
        =
        y
      
    
    {\displaystyle \ln(x)=y}

<p>the BKM algorithm takes advantage of a basic <a href="/facts/List_of_logarithmic_identities/5T8DX7d8">property of logarithms</a>
</p>

ln
        ⁡
        (
        a
        b
        )
        =
        ln
        ⁡
        (
        a
        )
        +
        ln
        ⁡
        (
        b
        )
      
    
    {\displaystyle \ln(ab)=\ln(a)+\ln(b)}

<p>Using <a href="/facts/Pi_notation/VtcUjpNv">Pi notation</a>, this identity generalizes to
</p>

ln
        ⁡
        
          (
          
            
              ∏
              
                k
                =
                0
              
              
                n
              
            
            
              a
              
                k
              
            
          
          )
        
        =
        
          ∑
          
            k
            =
            0
          
          
            n
          
        
        ln
        ⁡
        (
        
          a
          
            k
          
        
        )
      
    
    {\displaystyle \ln \left(\prod _{k=0}^{n}a_{k}\right)=\sum _{k=0}^{n}\ln(a_{k})}

<p>Because any number can be represented by a product, this allows us to choose any set of values 
  
    
      
        
          a
          
            k
          
        
      
    
    {\displaystyle a_{k}}
  
 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 
  
    
      
        
          a
          
            k
          
        
        =
        1
        +
        
          2
          
            m
          
        
      
    
    {\displaystyle a_{k}=1+2^{m}}
  
 is a better option than a more simple choice of 
  
    
      
        
          a
          
            k
          
        
        =
        
          2
          
            m
          
        
      
    
    {\displaystyle a_{k}=2^{m}}
  
. Since we want to start with large changes and get more accurate as 
  
    
      
        k
      
    
    {\displaystyle k}
  
 increases, we can more specifically use 
  
    
      
        
          a
          
            k
          
        
        =
        1
        +
        
          2
          
            −
            k
          
        
      
    
    {\displaystyle a_{k}=1+2^{-k}}
  
, allowing the product to approach any value between 1 and ~4.768, depending on which subset of 
  
    
      
        
          a
          
            k
          
        
      
    
    {\displaystyle a_{k}}
  
 we use in the final product. At this point, the above equation looks like this:
</p>

ln
        ⁡
        
          (
          
            
              ∏
              
                k
                ∈
                K
                ⊂
                
                  
                    Z
                  
                  
                    0
                  
                  
                    +
                  
                
              
            
            1
            +
            
              2
              
                −
                k
              
            
          
          )
        
        =
        
          ∑
          
            k
            ∈
            K
            ⊂
            
              
                Z
              
              
                0
              
              
                +
              
            
          
        
        ln
        ⁡
        (
        1
        +
        
          2
          
            −
            k
          
        
        )
      
    
    {\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})}

<p>This choice of 
  
    
      
        
          a
          
            k
          
        
      
    
    {\displaystyle a_{k}}
  
 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 
  
    
      
        ln
        ⁡
        (
        1
        +
        
          2
          
            −
            k
          
        
        )
      
    
    {\displaystyle \ln(1+2^{-k})}
  
 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 
  
    
      
        x
      
    
    {\displaystyle x}
  
 while the other approaches the output value 
  
    
      
        ln
        ⁡
        (
        x
        )
        =
        y
      
    
    {\displaystyle \ln(x)=y}
  
:
</p><p>
  
    
      
        
          x
          
            k
          
        
        =
        
          
            {
            
              
                
                  1
                
                
                  
                    if 
                  
                  k
                  =
                  0
                
              
              
                
                  
                    x
                    
                      k
                      −
                      1
                    
                  
                  ⋅
                  (
                  1
                  +
                  
                    2
                    
                      −
                      k
                    
                  
                  )
                
                
                  
                    if 
                  
                  
                    x
                    
                      k
                    
                  
                  
                     would be
                  
                  ≤
                  x
                
              
              
                
                  
                    x
                    
                      k
                      −
                      1
                    
                  
                
                
                  
                    otherwise
                  
                
              
            
            
          
        
      
    
    {\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}}}

</p><p>Given this recursive definition and because 
  
    
      
        
          x
          
            k
          
        
      
    
    {\displaystyle x_{k}}
  
 is strictly increasing, it can be shown by <a href="/facts/Mathematical_induction/KxAPqNrH">induction</a> and <a href="/facts/Convergent_series/DTok170z">convergence</a> that
</p>

lim
          
            k
            →
            ∞
          
        
        
          x
          
            k
          
        
        =
        x
      
    
    {\displaystyle \lim _{k\to \infty }x_{k}=x}

<p>for any 
  
    
      
        1
        ≤
        x
        ≲
        4.768
      
    
    {\displaystyle 1\leq x\lesssim 4.768}
  
. For calculating the output, we first create the reference table
</p>

A
          
            k
          
        
        =
        ln
        ⁡
        (
        1
        +
        
          2
          
            −
            k
          
        
        )
      
    
    {\displaystyle A_{k}=\ln(1+2^{-k})}

<p>Then the output is computed iteratively by the definition

y
          
            k
          
        
        =
        
          
            {
            
              
                
                  0
                
                
                  
                    if 
                  
                  k
                  =
                  0
                
              
              
                
                  
                    y
                    
                      k
                      −
                      1
                    
                  
                  +
                  
                    A
                    
                      k
                    
                  
                
                
                  
                    if 
                  
                  
                    x
                    
                      k
                    
                  
                  
                     would be
                  
                  ≤
                  x
                
              
              
                
                  
                    y
                    
                      k
                      −
                      1
                    
                  
                
                
                  
                    otherwise
                  
                
              
            
            
          
        
      
    
    {\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}}}

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
</p>

lim
          
            k
            →
            ∞
          
        
        
          y
          
            k
          
        
        =
        y
      
    
    {\displaystyle \lim _{k\to \infty }y_{k}=y}

<p>for any 
  
    
      
        0
        ≤
        y
        ≲
        1.562
      
    
    {\displaystyle 0\leq y\lesssim 1.562}
  
.
</p><p>Because the algorithm above calculates both the input and output simultaneously, it's possible to modify it slightly so that 
  
    
      
        y
      
    
    {\displaystyle y}
  
 is the known value and 
  
    
      
        x
      
    
    {\displaystyle x}
  
 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
</p>

…
        
          if 
        
        
          x
          
            k
          
        
        
           would be
        
        ≤
        x
      
    
    {\displaystyle \dots {\text{if }}x_{k}{\text{ would be}}\leq x}

…
        
          if 
        
        
          y
          
            k
          
        
        
           would be
        
        ≤
        y
      
    
    {\displaystyle \dots {\text{if }}y_{k}{\text{ would be}}\leq y}

<h2 id="logarithm-function">Logarithm function</h2>
<p>To calculate the logarithm function (L-mode), the algorithm in each iteration tests if 
  
    
      
        
          x
          
            n
          
        
        ⋅
        (
        1
        +
        
          2
          
            −
            n
          
        
        )
        ≤
        x
      
    
    {\displaystyle x_{n}\cdot (1+2^{-n})\leq x}
  
. If so, it calculates 
  
    
      
        
          x
          
            n
            +
            1
          
        
      
    
    {\displaystyle x_{n+1}}
  
 and 
  
    
      
        
          y
          
            n
            +
            1
          
        
      
    
    {\displaystyle y_{n+1}}
  
. After 
  
    
      
        N
      
    
    {\displaystyle N}
  
 iterations the value of the function is known with an error of 
  
    
      
        Δ
        ln
        ⁡
        (
        x
        )
        ≤
        
          2
          
            −
            N
          
        
      
    
    {\displaystyle \Delta \ln(x)\leq 2^{-N}}
  
.
</p><p>Example program for natural logarithm in C++ (see A_e for table):
</p>
 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;
 }

<p>Logarithms for bases other than <a href="/facts/E_(number)/coEtSBk8">e</a> can be calculated with similar effort.
</p><p>Example program for binary logarithm in C++ (see A_2 for table):
</p>
 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;
 }

<p>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 <i>k</i> 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.
</p>
<h2 id="exponential-function">Exponential function</h2>
<p>To calculate the exponential function (E-mode), the algorithm in each iteration tests if 
  
    
      
        
          y
          
            n
          
        
        +
        ln
        ⁡
        (
        1
        +
        
          2
          
            −
            n
          
        
        )
        ≤
        y
      
    
    {\displaystyle y_{n}+\ln(1+2^{-n})\leq y}
  
. If so, it calculates 
  
    
      
        
          x
          
            n
            +
            1
          
        
      
    
    {\displaystyle x_{n+1}}
  
 and 
  
    
      
        
          y
          
            n
            +
            1
          
        
      
    
    {\displaystyle y_{n+1}}
  
. After 
  
    
      
        N
      
    
    {\displaystyle N}
  
 iterations the value of the function is known with an error of 
  
    
      
        Δ
        exp
        ⁡
        (
        x
        )
        ≤
        
          2
          
            −
            N
          
        
      
    
    {\displaystyle \Delta \exp(x)\leq 2^{-N}}
  
.
</p><p>Example program in C++ (see A_e for table):
</p>
 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;
 }

<h2 id="tables-for-examples">Tables for examples</h2>

<table><tbody><tr><th>static const double A_e [] = ...</th></tr><tr><td> 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 };</td></tr></tbody></table>

<table><tbody><tr><th>static const double A_2 [] = ...</th></tr><tr><td> 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,   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<h2 id="notes">Notes</h2>

<ul><li>Bajard, Jean-Claude; Kla, Sylvanus; Muller, Jean-Michel (August 1994). <a href="http://perso.ens-lyon.fr/jean-michel.muller/BKM94.pdf">"BKM: A new hardware algorithm for complex elementary functions"</a> (PDF). <i><a href="/facts/IEEE_Transactions_on_Computers/jjc4KLle">IEEE Transactions on Computers</a></i>. 43 (8): 955–963. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1109%2F12.295857">10.1109/12.295857</a>. <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0018-9340">0018-9340</a>. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:1073.68501">1073.68501</a>. <a href="https://web.archive.org/web/20181103210201/http://perso.ens-lyon.fr/jean-michel.muller/BKM94.pdf">Archived</a> (PDF) from the original on 2018-11-03. Retrieved 2017-12-21.</li>
<li>Skaf, Ali; Muller, Jean-Michel; Guyot, Alain (20–22 September 1994). <a href="https://ieeexplore.ieee.org/document/5468485"><i>On-Line Hardware Implementation for Complex Exponential and Logarithm</i></a>. ESSCIRC '94: Twientieth European Solid-State Circuits Conference. Ulm, Germany. <a href="/facts/CiteSeerX_(identifier)/SceDmd3c">CiteSeerX</a> <a href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.47.7521">10.1.1.47.7521</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 2-86332-160-9. Retrieved 2021-08-23. (5 pages) <a href="https://hal.archives-ouvertes.fr/hal-00014939">[1]</a><a href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.47.7521&rep=rep1&type=pdf">[2]</a></li>
<li>Bajard, Jean-Claude; Imbert, Laurent (1999-11-02). <a href="https://webusers.imj-prg.fr/~jean-claude.bajard/MesPublis/Spie1999.pdf">"Evaluation of complex elementary functions: A new version of BKM"</a> (PDF). In Luk, Franklin T. (ed.). <i>Advanced Signal Processing Algorithms, Architectures, and Implementations IX</i>. SPIE Proceedings. Vol. 3807. <a href="/facts/Society_of_Photo-Optical_Instrumentation_Engineers/2fVIRgKp">Society of Photo-Optical Instrumentation Engineers</a> (SPIE). pp. 2–9. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1999SPIE.3807....2B">1999SPIE.3807....2B</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1117%2F12.367631">10.1117/12.367631</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:121001818">121001818</a>. <a href="https://web.archive.org/web/20200609182503/https://webusers.imj-prg.fr/~jean-claude.bajard/MesPublis/Spie1999.pdf">Archived</a> (PDF) from the original on 2020-06-09. Retrieved 2020-06-09. (8 pages) <a href="https://www.spiedigitallibrary.org/conference-proceedings-of-spie/3807/1/Evaluation-of-complex-elementary-functions--a-new-version-of/10.1117/12.367631.short">[3]</a></li>
<li>Imbert, Laurent; Muller, Jean-Michel; Rico, Fabien (2006-05-24) [2000-06-01, September 1999]. <a href="https://hal.inria.fr/inria-00072908/document">"Radix-10 BKM Algorithm for Computing Transcendentals on Pocket Computers"</a>. <i>Journal of VLSI Signal Processing</i> (Research report). 25 (2). <a href="/facts/Kluwer_Academic_Publishers/nAesf6nT">Kluwer Academic Publishers</a> / <a href="/facts/Institut_national_de_recherche_en_informatique_et_en_automatique/ruIkcapf">Institut national de recherche en informatique et en automatique</a> (INRIA): 179–186. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1023%2FA%3A1008127208220">10.1023/A:1008127208220</a>. <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0922-5773">0922-5773</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:392036">392036</a>. RR-3754. INRIA-00072908. Theme 2 <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://www.worldcat.org/search?fq=x0:jrnl&q=n2:0249-6399">0249-6399</a>. <a href="https://web.archive.org/web/20180711091703/https://hal.inria.fr/inria-00072908/document">Archived</a> from the original on 2018-07-11. Retrieved 2018-07-11. (1+15 pages) <a href="https://www.researchgate.net/profile/Fabien_Rico/publication/226548529_A_Radix-10_BKM_Algorithm_for_Computing_Transcendentals_on_Pocket_Computers/links/00b49517b782924cd8000000/A-Radix-10-BKM-Algorithm-for-Computing-Transcendentals-on-Pocket-Computers.pdf">[4]</a><a href="https://dl.acm.org/doi/10.1023/A%3A1008127208220">[5]</a><a href="https://link.springer.com/article/10.1023/A:1008127208220">[6]</a></li>
<li>Didier, Laurent-Stéphane; Rico, Fabien (2002-01-21). <a href="http://ftp.jussieu.fr/lip6/reports/2002/lip6.2002.009.pdf">"High radix BKM algorithm with Selection by Rounding"</a> (PDF). <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:17750192">17750192</a>. lip6.2002.009. hal-02545612. <a href="https://web.archive.org/web/20210823130525/http://ftp.jussieu.fr/lip6/reports/2002/lip6.2002.009.pdf">Archived</a> (PDF) from the original on 2021-08-23. Retrieved 2021-08-23. <a href="https://hal.archives-ouvertes.fr/hal-02545612">[7]</a> (1+11 pages)</li>
<li>Didier, Laurent-Stéphane; Rico, Fabien (2004-12-01). <a href="https://www.researchgate.net/publication/263307443_High_Radix_BKM_Algorithm_SCAN%272002_International_Conference_Guest_Editors_Rene_Alt_and_Jean-Luc_Lamotte">"High Radix BKM Algorithm"</a>. <i>Numerical Algorithms</i>. SCAN'2002 International Conference. 37 (1–4 [4]). <a href="/facts/Springer_Science%2BBusiness_Media,_LLC/nAesf6nT">Springer Science+Business Media, LLC</a>: 113–125. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2004NuAlg..37..113D">2004NuAlg..37..113D</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1023%2FB%3ANUMA.0000049459.69390.ff">10.1023/B:NUMA.0000049459.69390.ff</a>. <a href="/facts/EISSN_(identifier)/DPAflDvU">eISSN</a> <a href="https://search.worldcat.org/issn/1572-9265">1572-9265</a>. <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/1017-1398">1017-1398</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:2761452">2761452</a>. <a href="https://web.archive.org/web/20180602213505/https://link.springer.com/article/10.1023%2FB%3ANUMA.0000049459.69390.ff">[8]</a></li>
<li>Muller, Jean-Michel (2006). <i>Elementary Functions: Algorithms and Implementation</i> (2 ed.). Boston, MA, USA: <a href="/facts/Birkh%C3%A4user/brvgnrnJ">Birkhäuser</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8176-4372-0. <a href="/facts/LCCN_(identifier)/5hb5Cw2V">LCCN</a> <a href="https://lccn.loc.gov/2005048094">2005048094</a>.</li>
<li>Muller, Jean-Michel (2016-12-12). <i>Elementary Functions: Algorithms and Implementation</i> (3 ed.). Boston, MA, USA: <a href="/facts/Birkh%C3%A4user/brvgnrnJ">Birkhäuser</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4899-7981-0. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 1-4899-7981-6.</li></ul>
<h2 id="further-reading">Further reading</h2>
<ul><li>Jorke, Günter; Lampe, Bernhard; Wengel, Norbert (1989). <a href="https://books.google.com/books?id=DqYWAQAAMAAJ"><i>Arithmetische Algorithmen der Mikrorechentechnik</i></a> (in German) (1 ed.). Berlin, Germany: VEB Verlag Technik. pp. 280–282. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-34100515-3. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-34100515-6. <a href="/facts/EAN_(identifier)/1BqwBFU2">EAN</a> <a href="https://eandata.com/lookup/9783341005156">9783341005156</a>. MPN 5539165. License 201.370/4/89. Retrieved 2015-12-01.</li>
<li>Meggitt, John E. (1961-08-29). <a href="http://dl.acm.org/citation.cfm?id=1661979.1661985">"Pseudo Division and Pseudo Multiplication Processes"</a>. <i><a href="/facts/IBM_Journal_of_Research_and_Development/LswQR28R">IBM Journal of Research and Development</a></i>. 6 (2). Riverton, New Jersey, USA: <a href="/facts/IBM_Corporation/QszA7nwd">IBM Corporation</a> (published April 1962): 210–226, 287. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1147%2Frd.62.0210">10.1147/rd.62.0210</a>. Retrieved 2015-12-01.</li>
<li>Chi Chen, Tien (July 1972). <a href="http://dl.acm.org/citation.cfm?id=1661877">"Automatic computation of exponentials, logarithms, ratios and square roots"</a>. <i><a href="/facts/IBM_Journal_of_Research_and_Development/LswQR28R">IBM Journal of Research and Development</a></i>. 16 (4). San Jose, California, USA; Riverton, New Jersey, USA: <a href="/facts/IBM_San_Jose_Research_Laboratory/4DaaL58U">IBM San Jose Research Laboratory</a>; <a href="/facts/IBM_Corporation/QszA7nwd">IBM Corporation</a>: 380–388. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1147%2Frd.164.0380">10.1147/rd.164.0380</a>. Retrieved 2015-12-01.</li>
<li><a href="/facts/Nathalie_Revol/bwQdhl06">Revol, Nathalie</a>; Yakoubsohn, Jean-Claude (2000-05-01). <a href="https://perso.math.univ-toulouse.fr/yak/files/2018/09/RC-2000.pdf">"Accelerated Shift-and-Add algorithms"</a> (PDF). <i>Reliable Computing</i>. 6 (2). Boston, USA: Laboratoire d'Analyse Numérique et d'Optimisation (ANO) de l'<a href="/facts/Universit%C3%A9_des_Sciences_et_Technologies_de_Lille/fg5QwPQR">Université des Sciences et Technologies de Lille</a>; <a href="/facts/Kluwer_Academic_Publishers/nAesf6nT">Kluwer Academic Publishers</a>: 193–205. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1023%2FA%3A1009921407000">10.1023/A:1009921407000</a>. <a href="/facts/EISSN_(identifier)/DPAflDvU">eISSN</a> <a href="https://search.worldcat.org/issn/1573-1340">1573-1340</a>. <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/1385-3139">1385-3139</a>. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/67306353">67306353</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:10716391">10716391</a>. <a href="https://web.archive.org/web/20210823132202/https://perso.math.univ-toulouse.fr/yak/files/2018/09/RC-2000.pdf">Archived</a> (PDF) from the original on 2021-08-23. Retrieved 2021-08-23. (14 pages) <a href="http://perso.ens-lyon.fr/nathalie.revol/publis/RY00.pdf">[9]</a></li></ul>

BKM algorithm open-in-new

BKM algorithm