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How to convert a very long binary number to decimal?

I have a binary number represented as 11.1111111 (similar to a decimal point). There are 2 bits before the dot and 1024 bits after the dot. It was an exercise in calculating e with a high level of precision, but now I am fixated on how to convert it to decimal. Just in case, you guys want to know the number, here it is:

10.1011011111100001010100010110001010001010111011010010101001101010101111110111000101011000100000001001110011110100111100111100011101100010111001110001011000001111001110001011010011011010010101101010011110000100110110010000010001010001100100001100111111101111001100100100111001110111001110001001001001101100111110111110010111110100101111111000110110001101100011000011000111010111011000111101101000000110110010000000101010111011000100011000010111101011010011110111110001111011010101110101011111110101100101011000010010010000110011111101010001111101011111000001100110111011010000100001010110001101100101010101010011110111101101000110101111001110110101010101110001001101011110011111110101011111001001001101011001100001001111000011000111000011100000111001101000101101110111111000101010011010001001110110101111001111101111111010000111001000011101111100010101100010100001001101101010110111100111001101010011000010101100110010100100111101001000001110100111100101111010101111000000101010110001100000101011001100100100111110110011101011

How can I convert this to 2.718 .... (there should be about 309 digits after the decimal point) I can’t just multiply each bit by 2 ^ x, because after a while the number 2 ^ x will be = 0 even when using double float accuracy. I am using Visual Basic, so I'm not sure if large variables exist.

[Edit by Spektre]

Just run your line with my code (based on the link in my comment), and the result:

e(bigdecimal)=2.71828182845904523536028747135266249775724709369995957496696762772407663035354759457138217852516642742746639193200305992181741359662904357290033429526059563073813232862794349076323382988075319525101901157383418793070215408914993488416750924476146066808226480016847741185374234544243710753907774499206955170189257927265177296267786175561825444670874889747782175809270565601486538810885558129926100522647929865142359038501319247028975364903531383896590857864585070203793060262761378008328322397393650711101939331201
e      (text)=2.71828182845904523536028747135266249775724709369995957496696762772407663035354759457138217852516642742746639193200305992181741359662904357290033429526059563073813232862794349076323382988075319525101901157383418793070215408914993488416750924476146066808226480016847741185374234544243710753907774499206955170189
e (reference)=2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427427466391932003059921817413596629043572900334295260595630738132328627943490763233829880753195251019011573834187930702154089149934884167509244761460668082264800168477411853742345442437107539077744992069551702761838606261331384583000752044933826560297606737113200709328709127443747047230696977209310141692836819025515108657463772111252389784425056953696770785449969967946864454905987931636889230098793127736178215424999229576351482208269895193668033182528869398496465105820939239829488793320362509443117301238197068416140397019837679320683282376464804295311802328782509819455815301756717361332069811250996181881593041690351598888519345807273866738589422879228499892086805825749279610484198444363463244968487560233624827041978623209002160990235304369941849146314093431738143640546253152096183690888707016768396424378140592714563549061303107208510383750510115747704171898610687396965521267154688957035035402123407849819334321068170121005627880235193033224745015853904730419957777093503660416997329725088687696640355570716226844716256079882651787134195124665201030592123667719432527867539855894489697096409754591856956380236370162112047742722836489613422516445078182442352948636372141740238893441247963574370263755294448337998016125492278509257782562092622648326277933386566481627725164019105900491644998289315056604725802778631864155195653244258698294695930801915298721172556347546396447910145904090586298496791287406870504895858671747985466775757320568128845920541334053922000113786300945560688166740016984205580403363795376452030402432256613527836951177883863874439662532249850654995886234281899707733276171783928034946501434558897071942586398772754710962953741521115136835062752602326484728703920764310059584116612054529703023647254929666938115137322753645098889031360205724817658511806303644281231496550704751025446501172721155519486685080036853228183152196003735625279449515828418829478761085263981395599006737648292244375287184624578036192981971399147564488262603903381441823262515097482798777996437308997038886778227138360577297882412561190717663946507063304527954661855096666185664709711344474016070462621568071748187784437143698821855967095910259686200235371858874856965220005031173439207321139080329363447972735595527734907178379342163701205005451326383544000186323991490705479778056697853358048966906295119432473099587655236812859041383241160722602998330535370876138939639177957454016137223618789365260538155841587186925538606164779834025435128

The first is converted from text to my data type arbnum, and then converted back to text, the average is a pure conversion of text to text (for example, in a link with conversion to hex code before), and the latter is a linke

Here is the hexadecimal string of your binary string:

e (hex)      =2.B7E151628AED2A6ABF7158809CF4F3C762E7160F38B4DA56A784D9045190CFEF324E7738926CFBE5F4BF8D8D8C31D763DA06C80ABB1185EB4F7C7B5757F5958490CFD47D7C19BB42158D9554F7B46BCED55C4D79FD5F24D6613C31C3839A2DDF8A9A276BCFBFA1C877C56284DAB79CD4C2B3293D20E9E5EAF02AC60ACC93ECEBh

I truncate down, make decimal sizes of the faces, so at the end, 1,2 or 3 bits can be left without processing ...

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3 answers

  • convert your binary string to hex

    the whole part is easily the "10." -> "2."fractional part is also simple:

    10.1011011111100001010100010110 bin
10.1011 0111 1110 0001 0101 0001 0110 bin
 2.B    7    E    1    5    1    6    hex

    , 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F ++:

    int i,j,l; char *t16="0123456789ABCDEF";
AnsiString s0="10.your binary number",s1="2.";
for (i=4,l=s0.Length();i<=l;)
    {
    j=0;
    if ((i<=l)&&(s0[i]=='1')) j+=8; i++;
    if ((i<=l)&&(s0[i]=='1')) j+=4; i++;
    if ((i<=l)&&(s0[i]=='1')) j+=2; i++;
    if ((i<=l)&&(s0[i]=='1')) j+=1; i++;
    s1+=char(t16[j]);
    } // here s1 holds the hex string

    preallocationg s1

  • hex to dec

    AnsiString str_hex2dec(const AnsiString &hex)
    {
    char c;
    AnsiString dec="",s;
    int i,j,l,ll,cy,val;
    int  i0,i1,i2,i3,sig;
    sig=+1; l=hex.Length();
    if (l) { c=hex[l]; if (c=='h') l--; if (c=='H') l--; }
    i0=0; i1=l; i2=0; i3=l;
    for (i=1;i<=l;i++)      // scan for parts of number
        {
        char c=hex[i];
        if (c=='-') sig=-sig;
        if ((c=='.')||(c==',')) i1=i-1;
        if ((c>='0')&&(c<='9')) { if (!i0) i0=i; if ((!i2)&&(i>i1)) i2=i; }
        if ((c>='A')&&(c<='F')) { if (!i0) i0=i; if ((!i2)&&(i>i1)) i2=i; }
        if ((c>='a')&&(c<='f')) { if (!i0) i0=i; if ((!i2)&&(i>i1)) i2=i; }
        }

    l=0; s=""; if (i0) for (i=i0;i<=i1;i++)
        {
        c=hex[i];
             if ((c>='0')&&(c<='9')) c-='0';
        else if ((c>='A')&&(c<='F')) c-='A'-10;
        else if ((c>='a')&&(c<='f')) c-='A'-10;
        for (cy=c,j=1;j<=l;j++)
            {
            val=(s[j]<<4)+cy;
            s[j]=val%10;
            cy  =val/10;
            }
        while (cy>0)
            {
            l++;
            s+=char(cy%10);
            cy/=10;
            }
        }
    if (s!="")
        {
        for (j=1;j<=l;j++) { c=s[j]; if (c<10) c+='0'; else c+='A'-10; s[j]=c; }
        for (i=l,j=1;j<i;j++,i--) { c=s[i]; s[i]=s[j]; s[j]=c; }
        dec+=s;
        }
    if (dec=="") dec="0";
    if (sig<0) dec="-"+dec;

    if (i2)
        {
        dec+='.';
        s=hex.SubString(i2,i3-i2+1);
        l=s.Length();
        for (i=1;i<=l;i++)
            {
            c=s[i];
                 if ((c>='0')&&(c<='9')) c-='0';
            else if ((c>='A')&&(c<='F')) c-='A'-10;
            else if ((c>='a')&&(c<='f')) c-='A'-10;
            s[i]=c;
            }
        ll=((l*1234)>>10);  // num of decimals to compute
        for (cy=0,i=1;i<=ll;i++)
            {
            for (cy=0,j=l;j>=1;j--)
                {
                val=s[j];
                val*=10;
                val+=cy;
                s[j]=val&15;
                cy=val>>4;
                }
            dec+=char(cy+'0');
            for (;;)
                {
                if (!l) break;;
                if (s[l]) break;
                l--;
                }
            if (!l) break;;
            }
        }

    return dec;
    }

    ++/VCL dec / ++

( ):

e (number)=2.7182818284590452353602874713526624977572470936999595749669676277240766303535475945713821785251664274274663919320030599218174135966290435729003342952605956307381323286279434907632338298807531952510190115738341879307021540891499348841675092447614606680822648001684774118537423454424371075390777449920695517018925792726517729626778617556182544467087488974778217580927056560148653881088555812992610052264792986514235903850131924702897536490353138389659085786458507020379306026276137800832832239739365071110193933120100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
e (text)  =2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427427466391932003059921817413596629043572900334295260595630738132328627943490763233829880753195251019011573834187930702154089149934884167509244761460668082264800168477411853742345442437107539077744992069551701892
e (const) =2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427427466391932003059921817413596629043572900334295260595630738132328627943490763233829880753195251019011573834187930702154089149934884167509244761460668082264800168477411853742345442437107539077744992069551702761838606261331384583000752044933826560297606737113200709328709127443747047230696977209310141692836819025515108657463772111252389784425056953696770785449969967946864454905987931636889230098793127736178215424999229576351482208269895193668033182528869398496465105820939239829488793320362509443117301238197068416140397019837679320683282376464804295311802328782509819455815301756717361332069811250996181881593041690351598888519345807273866738589422879228499892086805825749279610484198444363463244968487560233624827041978623209002160990235304369941849146314093431738143640546253152096183690888707016768396424378140592714563549061303107208510383750510115747704171898610687396965521267154688957035035402123407849819334321068170121005627880235193033224745015853904730419957777093503660416997329725088687696640355570716226844716256079882651787134195124665201030592123667719432527867539855894489697096409754591856956380236370162112047742722836489613422516445078182442352948636372141740238893441247963574370263755294448337998016125492278509257782562092622648326277933386566481627725164019105900491644998289315056604725802778631864155195653244258698294695930801915298721172556347546396447910145904090586298496791287406870504895858671747985466775757320568128845920541334053922000113786300945560688166740016984205580403363795376452030402432256613527836951177883863874439662532249850654995886234281899707733276171783928034946501434558897071942586398772754710962953741521115136835062752602326484728703920764310059584116612054529703023647254929666938115137322753645098889031360205724817658511806303644281231496550704751025446501172721155519486685080036853228183152196003735625279449515828418829478761085263981395599006737648292244375287184624578036192981971399147564488262603903381441823262515097482798777996437308997038886778227138360577297882412561190717663946507063304527954661855096666185664709711344474016070462621568071748187784437143698821855967095910259686200235371858874856965220005031173439207321139080329363447972735595527734907178379342163701205005451326383544000186323991490705479778056697853358048966906295119432473099587655236812859041383241160722602998330535370876138939639177957454016137223618789365260538155841587186925538606164779834025435128
e (hex)   =2.B7E151628AED2A6ABF7158809CF4F3C762E7160F38B4DA56A784D9045190CFEF324E7738926CFBE5F4BF8D8D8C31D763DA06C80ABB1185EB4F7C7B5757F5958490CFD47D7C19BB42158D9554F7B46BCED55C4D79FD5F24D6613C31C3839A2DDF8A9A276BCFBFA1C877C56284DAB79CD4C2B3293D20E9E5EAF02AC60ACC93ECEBh

AnsiString - VCL string , +,+=...

[Edit1] hex2dec

, hex, dec, .

  • hex O(n) , :

    • i0
    • i1 ( )
    • i2 ( )
    • i3
    • sig +1 -, -1

    , .

  • , :

    • 51Ah = 500h + 10h + Ah = 5*16^2 + 1*16^1 + 10*16^0 = 1306

    :

    • 51Ah = (((5)*16 +1)*16 +10) = 1306

    , BASE (16) BASE (10).

    , . . , 16 / . , .

    • if (i0) for (i=i0;i<=i1;i++),

    • c <0,F>hex -> <0,15>dec

    for (cy=c,j=1;j<=l;j++) s 16. s=""; l=0;, s , ASCII, {0,1,2,...9} {"0","1",...} l - s. VCL 1, for 1 not 0. , 16?

    123*16= 100*16 +  20*16 +  3*16
      =1600    + 320    + 48

    , <0,9> ( ) val. 16 ( - 4 , <<4 *16), cy , ( , ...). 1 , . s s[j]=val%10 val/10, 10^(j-1), , s [j], 10 j, , s [j] ... , .

    while (cy>0) s, , .

    s ASCII ( ) dec signum, . .

  • if (i2)

    BASE BASE (10) BASE (16). 10=Ah -...

        ------------
    0.B7Eh -> B7Eh
    ------------
    B7Eh * Ah = 7 2ECh
    2ECh * Ah = 1 D38h
    D38h * Ah = 8 430h
    430h * Ah = 2 9E0h
    -------------
    0.B7Eh -> 0.7182 dec
    ------------

    , , dec .. s ( ) ASCII <0,15> l . ll, , l ll=l*1234/1024. , , , ll tels, ...

    ll. for (cy=0,i=1;i<=ll;i++) . s 10, . s[j]=val%16 cy=val/16. AND Bit-shift, . , .

    for , s , ( l--), , . ...

, ...

+3

System.Numerics BigInteger. . https://msdn.microsoft.com/en-us/library/system.numerics.biginteger%28v=vs.110%29.aspx

EDIT: , . , 2 ^ 1, 1 . , 1.2 * 10 ^ 308. 2.12......

, 2 ^ -1 -1 2 ^ -1024. , 0.x, BigInteger.

+1

Each binary digit after the decimal point represents the decimal weight of 2 ^ -n, starting with n = 1.

This can be appreciated using any bignum library, using the Horner method like: (this is just pseudocode)

power_of_five = 1;
digits = 0;
while digits_left
  digits = digits * 10;
  power_of_five = power_of_five * 5;
  if (next_digit_is_set)
     digits = digits + power_of_five;
end

This will result in a 1024-digit bonus, of which only the first 309 are significant.

+1
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Source: https://habr.com/ru/post/1616461/

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