math.htm

对于学习很有帮助

  (C) Copyright 1989, 1995-1996 Earl F. Glynn, Overland Park, KS.
  All Rights Reserved.

  This UNIT was derived from the CRCT FORTRAN 77 program given in
  "Byte-wise CRC Calculations" by Aram Perez in IEEE Micro, June 1983,
  pp. 40-50.  The constants here are for the CRC-32 generator polynomial,
  as defined in the Microsoft Systems Journal, March 1995, pp. 107-108

  This CRC algorithm emphasizes speed at the expense of the 512 element
  lookup table.}

INTERFACE

  PROCEDURE CalcCRC32 (p:  pointer; nbyte:  WORD; VAR CRCvalue:  LongInt);
  PROCEDURE CalcFileCRC32 (FromName:  STRING; VAR CRCvalue:  LongInt;
              VAR IOBuffer:  pointer;  BufferSize:  WORD; VAR TotalBytes:  LongInt;
              VAR error:  WORD);

IMPLEMENTATION

  CONST
    table:  ARRAY[0..255] OF LongInt =
   ($00000000, $77073096, $EE0E612C, $990951BA,
    $076DC419, $706AF48F, $E963A535, $9E6495A3,
    $0EDB8832, $79DCB8A4, $E0D5E91E, $97D2D988,
    $09B64C2B, $7EB17CBD, $E7B82D07, $90BF1D91,
    $1DB71064, $6AB020F2, $F3B97148, $84BE41DE,
    $1ADAD47D, $6DDDE4EB, $F4D4B551, $83D385C7,
    $136C9856, $646BA8C0, $FD62F97A, $8A65C9EC,
    $14015C4F, $63066CD9, $FA0F3D63, $8D080DF5,
    $3B6E20C8, $4C69105E, $D56041E4, $A2677172,
    $3C03E4D1, $4B04D447, $D20D85FD, $A50AB56B,
    $35B5A8FA, $42B2986C, $DBBBC9D6, $ACBCF940,
    $32D86CE3, $45DF5C75, $DCD60DCF, $ABD13D59,
    $26D930AC, $51DE003A, $C8D75180, $BFD06116,
    $21B4F4B5, $56B3C423, $CFBA9599, $B8BDA50F,
    $2802B89E, $5F058808, $C60CD9B2, $B10BE924,
    $2F6F7C87, $58684C11, $C1611DAB, $B6662D3D,

    $76DC4190, $01DB7106, $98D220BC, $EFD5102A,
    $71B18589, $06B6B51F, $9FBFE4A5, $E8B8D433,
    $7807C9A2, $0F00F934, $9609A88E, $E10E9818,
    $7F6A0DBB, $086D3D2D, $91646C97, $E6635C01,
    $6B6B51F4, $1C6C6162, $856530D8, $F262004E,
    $6C0695ED, $1B01A57B, $8208F4C1, $F50FC457,
    $65B0D9C6, $12B7E950, $8BBEB8EA, $FCB9887C,
    $62DD1DDF, $15DA2D49, $8CD37CF3, $FBD44C65,
    $4DB26158, $3AB551CE, $A3BC0074, $D4BB30E2,
    $4ADFA541, $3DD895D7, $A4D1C46D, $D3D6F4FB,
    $4369E96A, $346ED9FC, $AD678846, $DA60B8D0,
    $44042D73, $33031DE5, $AA0A4C5F, $DD0D7CC9,
    $5005713C, $270241AA, $BE0B1010, $C90C2086,
    $5768B525, $206F85B3, $B966D409, $CE61E49F,
    $5EDEF90E, $29D9C998, $B0D09822, $C7D7A8B4,
    $59B33D17, $2EB40D81, $B7BD5C3B, $C0BA6CAD,

    $EDB88320, $9ABFB3B6, $03B6E20C, $74B1D29A,
    $EAD54739, $9DD277AF, $04DB2615, $73DC1683,
    $E3630B12, $94643B84, $0D6D6A3E, $7A6A5AA8,
    $E40ECF0B, $9309FF9D, $0A00AE27, $7D079EB1,
    $F00F9344, $8708A3D2, $1E01F268, $6906C2FE,
    $F762575D, $806567CB, $196C3671, $6E6B06E7,
    $FED41B76, $89D32BE0, $10DA7A5A, $67DD4ACC,
    $F9B9DF6F, $8EBEEFF9, $17B7BE43, $60B08ED5,
    $D6D6A3E8, $A1D1937E, $38D8C2C4, $4FDFF252,
    $D1BB67F1, $A6BC5767, $3FB506DD, $48B2364B,
    $D80D2BDA, $AF0A1B4C, $36034AF6, $41047A60,
    $DF60EFC3, $A867DF55, $316E8EEF, $4669BE79,
    $CB61B38C, $BC66831A, $256FD2A0, $5268E236,
    $CC0C7795, $BB0B4703, $220216B9, $5505262F,
    $C5BA3BBE, $B2BD0B28, $2BB45A92, $5CB36A04,
    $C2D7FFA7, $B5D0CF31, $2CD99E8B, $5BDEAE1D,

    $9B64C2B0, $EC63F226, $756AA39C, $026D930A,
    $9C0906A9, $EB0E363F, $72076785, $05005713,
    $95BF4A82, $E2B87A14, $7BB12BAE, $0CB61B38,
    $92D28E9B, $E5D5BE0D, $7CDCEFB7, $0BDBDF21,
    $86D3D2D4, $F1D4E242, $68DDB3F8, $1FDA836E,
    $81BE16CD, $F6B9265B, $6FB077E1, $18B74777,
    $88085AE6, $FF0F6A70, $66063BCA, $11010B5C,
    $8F659EFF, $F862AE69, $616BFFD3, $166CCF45,
    $A00AE278, $D70DD2EE, $4E048354, $3903B3C2,
    $A7672661, $D06016F7, $4969474D, $3E6E77DB,
    $AED16A4A, $D9D65ADC, $40DF0B66, $37D83BF0,
    $A9BCAE53, $DEBB9EC5, $47B2CF7F, $30B5FFE9,
    $BDBDF21C, $CABAC28A, $53B39330, $24B4A3A6,
    $BAD03605, $CDD70693, $54DE5729, $23D967BF,
    $B3667A2E, $C4614AB8, $5D681B02, $2A6F2B94,
    $B40BBE37, $C30C8EA1, $5A05DF1B, $2D02EF8D);


  TYPE
    buffer = ARRAY[1..65521] OF BYTE;  {largest buffer that can be}
                                       {allocated on heap         }
  VAR
    i:  WORD;
    q:  ^buffer;

  PROCEDURE CalcCRC32 (p:  pointer; nbyte:  WORD; VAR CRCvalue:  LongInt);
   {The following is a little cryptic (but executes very quickly).
    The algorithm is as follows:
      1.  exclusive-or the input byte with the low-order portion of
          the CRC register to get an INDEX
      2.  shift the CRC register eight bits to the right
      3.  exclusive-or the CRC register with the contents of
          Table[INDEX]
      4.  repeat steps 1 through 3 for all bytes}
  BEGIN
    q := p;
    FOR   i := 1 TO nBYTE DO
      CRCvalue := (CRCvalue SHR 8)  XOR
                  Table[ q^[i] XOR (CRCvalue AND $000000FF) ]
  END {CalcCRC32};


  PROCEDURE CalcFileCRC32 (FromName:  STRING; VAR CRCvalue:  LongInt;
            VAR IOBuffer:  pointer;  BufferSize:  WORD; VAR TotalBytes:  LongInt;
            VAR error:  WORD);
    VAR
      BytesRead:  WORD;
      FromFile :  FILE;
      i        :  WORD;
  BEGIN
    FileMode := 0;  {Turbo default is 2 for R/W; 0 is for R/O}
    CRCValue := $FFFFFFFF;
    ASSIGN (FromFile,FromName);
    {$I-} RESET (FromFile,1); {$I+}
    error := IOResult;
    IF   error = 0
    THEN BEGIN
      TotalBytes := 0;

      REPEAT
        BlockRead (FromFile,IOBuffer^,BufferSize,BytesRead);
        CalcCRC32 (IOBuffer,BytesRead,CRCvalue);
        INC (TotalBytes, BytesRead)
      UNTIL BytesRead = 0;
      CLOSE (FromFile)
    END;
    CRCvalue := NOT CRCvalue
  END {CalcFileCRC32};

END {CRC}.
</PRE><HR>


<P><H1><A NAME="math3">Raising a number to a power</P></A></H1>
 <P><I>Paul Cunningham ([e-mail])</I></P>

<PRE> Question[ 
This may sound trivial, but how do you raise a value to a power?
   eg). 2^12 = 4095
          ]
</PRE>

The question is not that trivial. The trouble is that the power function 
is not that simple. Several distinct situations need to be considered for 
the function Power(X, N) i.e X^N. <p>

<OL>
<LI>  X don't care, N = 0
<LI>  X = 1, N don't care
<LI>   X = 0 and N &gt; 0
<LI>   X = 0 and N &lt; 0
<LI>   X &gt; 0
<LI>   X &lt; 0 and N is an odd integer
<LI>   X &lt; 0 and N is an even integer
<LI>   X &lt; 0 and N not an integer
</OL>

Consider the following robust (though not necessarily the most 
efficient!) power function. <P>

<HR><PRE>interface

type
   EPowerException = class(Exception)
end;


implementation

function Power(X, N : real) : extended;
var
  t : longint;
  r : real;
  isInteger : boolean;
begin

   if N = 0 then begin
      result := 1.0;
      exit;
   end;

   if X = 1.0 then begin
      result := 1.0;
      exit;
   end;

   if X = 0.0 then begin
      if N &gt; 0.0 then
         begin
           result := 0.0;
           exit;
         end
      else
        raise EPowerException.Create('Infinite Result');
   end;

   if (X &gt; 0) then
      try
         result := exp(N * ln(X));
         exit;
      except
         raise EPowerException.Create('Overflow/Underflow Result');
   end;


{ X is negative but we still may compute the result if n is an integer}
{ try and get integer portion of n into a longint, it will be quicker to 
}       { compute odd n}
  try
     t := trunc(n);
     if (n - t) = 0 then
        isInteger := true
     else
        isInteger := False;
  except
     {Bit superfluous as result will probably underflow/overflow anyway}
     r := int(n);
     if (n - r) = 0 then
        begin
           isInteger := true;
           if frac(r/2) = 0.5 then
              t := 1
           else
              t := 2;
        end
     else
        isInteger := False;
  end;

  if isInteger then
      begin
         {n is an integer}
         if odd(t) then
            {n is odd}
            try
               result := -exp(N * ln(-X));
               exit;
            except
               raise EPowerException.Create('Overflow/Underflow Result');
            end
         else
            {n is even}
            try
               result := exp(N * ln(-X));
               exit;
            except
               raise EPowerException.Create('Overflow/Underflow Result');
            end;
      end   
   else
      raise EPowerException.Create('Complex Result');

end;
</PRE><HR>

<H1><A NAME="math4">Number to String</A></H1>

<I>From: "Joe E. Healy " [e-mail] (Advanced Technology Group)</I>

Here's a base that will go up to a million.  You can expand it past there 
if you like.  Give credit where credit is due, and dont resell the code! <p>

<hr><pre>
unit uNum2Str;

// Possible enhancements
// Move strings out to resource files
// Put in a general num2str utility

interface

function Num2Dollars( dNum: double ) : String;

implementation

uses SysUtils;

function LessThan99( dNum: double ) : String; forward;

// floating point modulus
function FloatMod( i,j: double ): double;
begin
    result := i - (Int(i/j) * j);
end;

function Hundreds( dNum: double ) : String;
var
    workVar: double;
begin
    if ( dNum < 100 ) or ( dNum > 999 ) then
        raise Exception.Create( 'hundreds range exceeded' );

    result := '';

    workVar := Int( dNum / 100 );
    if workVar > 0 then
        result := LessThan99(workVar) + ' Hundred';
end;

function OneToNine( dNum: Double ) : String;
begin
    if ( dNum < 1 ) or (dNum > 9 ) then
        raise exception.create( 'onetonine: value out of range' );

    result := 'woops';

    if dNum = 1 then result := 'One'
    else if dNum = 2 then result := 'Two'
    else if dNum = 3 then result := 'Three'
    else if dNum = 4 then result := 'Four'
    else if dNum = 5.0 then result := 'Five'
    else if dNum = 6 then result := 'Six'
    else if dNum = 7 then result := 'Seven'
    else if dNum = 8 then result := 'Eight'
    else if dNum = 9 then result := 'Nine';

end;

function ZeroTo19( dNum: double ) : String;
begin
    if (dNum < 0) or (dNum > 19) then
        raise Exception.Create( 'Bad value in dNum' );

    result := '';

    if dNum = 0 then result := 'Zero'
    else if (dNum >= 1) and (dNum <= 9) then result := OneToNine( dNum )
    else if dNum = 10 then result := 'Ten'
    else if dNum = 11 then result := 'Eleven'
    else if dNum = 12 then result := 'Twelve'
    else if dNum = 13 then result := 'Thirteen'
    else if dNum = 14 then result := 'Fourteen'
    else if dNum = 15 then result := 'Fifteen'
    else if dNum = 16 then result := 'Sixteen'
    else if dNum = 17 then result := 'Seventeen'
    else if dNum = 18 then result := 'Eighteen'
    else if dNum = 19 then result := 'Nineteen'
    else result := 'woops!';
end;

function TwentyTo99( dNum: double ) : String;
var
    BigNum: String;
begin
    if ( dNum < 20 ) or ( dNum > 99 ) then
        raise exception.Create( 'TwentyTo99: dNum out of range!' );

    BigNum := 'woops';

    if dNum >= 90 then BigNum := 'Ninety'
    else if dNum >= 80 then BigNum := 'Eighty'
    else if dNum >= 70 then BigNum := 'Seventy'
    else if dNum >= 60 then BigNum := 'Sixty'
    else if dNum >= 50 then BigNum := 'Fifty'
    else if dNum >= 40 then BigNum := 'Forty'
    else if dNum >= 30 then BigNum := 'Thirty'
    else if dNum >= 20 then BigNum := 'Twenty';

    // lose the big num
    dNum := FloatMod( dNum, 10 );
    
    if dNum > 0.00 then
        result := BigNum + ' ' + OneToNine( dNum )
    else
        result := BigNum;
end;

function LessThan99( dNum: double ) : String;
begin
    if dNum <= 19 then
        result := ZeroTo19(dNum)
    else
        result := TwentyTo99(dNum);
end;

function Num2Dollars( dNum: double ) : String;
var
    centsString: String;
    cents: double;
    workVar: double;
begin
    result := '';

    if dNum < 0 then
        raise Exception.Create( 'Negative numbers not supported' );

    if dNum > 999999999.99 then
        raise Exception.Create( 'Num2Dollars only supports up to the millions at this point!' );



    cents := (dNum - Int( dNum )) * 100.0;
    if cents = 0.0 then
        centsString := 'and 00/100 Dollars'
    else if cents < 10 then
        centsString := Format( 'and 0%1.0f/100 Dollars', [cents] )
    else
        centsString := Format( 'and %2.0f/100 Dollars', [cents] );

    dNum := Int( dNum - (cents / 100.0) ); // lose the cents

    // deal with million's
    if (dNum >= 1000000 ) and ( dNum <= 999999999 ) then
    begin
        workVar := dNum / 1000000;
        workVar := Int( workVar );
        if (workVar <= 9) then
            result := ZeroTo19(workVar)
        else if ( workVar <= 99 ) then
            result := LessThan99( workVar )
        else if ( workVar <= 999 ) then
            result := Hundreds( workVar )
        else
            result := 'mill fubar';

        result := result + ' Million';

        dNum := dNum - ( workVar * 1000000 );
    end;

    // deal with 1000's
    if (dNum >= 1000 ) and ( dNum <= 999999.99 ) then
    begin
        // doing the two below statements in one line of code yields some really
        // freaky floating point errors
        workVar := dNum/1000;
        workVar := Int( workVar );
        if (workVar <= 9) then
            result := ZeroTo19(workVar)
        else if ( workVar <= 99 ) then
            result := LessThan99( workVar )
        else if ( workVar <= 999 ) then
            result := Hundreds( workVar )
        else
            result := 'thou fubar';

        result := result + ' Thousand';

        dNum := dNum - ( workVar * 1000 );
    end;

    // deal with 100's
    if (dNum >= 100.00 ) and (dNum <= 999.99) then
    begin
        result := result + ' ' + Hundreds( dNum );
        dNum := FloatMod( dNum, 100 );
    end;

    // format in anything less than 100
    if ( dNum > 0) or ((dNum = 0) and (Length( result ) = 0)) then
    begin
        result := result + ' ' + LessThan99( dNum );
    end;
    result := result + ' ' + centsString;
end;

end.
</pre><hr>


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