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documents:other:other-051

2進数の固定小数点表記

2017/05/02
ツイッターTLで2進数と10進数の錯誤をさせる、ありがちなひっかけのツイートを見たのでなんとなく作る。

固定小数点表現で有効桁8bitの一覧

10進数の1を1/(10^1)すれば0.1、1/(10^2)すれば0.01、1/(10^3)すれば0.001、…となる。

おなじように、2進数でも1を1/(2^1)すれば0.1、1/(2^2)すれば0.01、1/(2^3)すれば0.001、…となる。
そして、2進数の結果を10進数に換算すると、1/(2^1)=0.5、1/(2^2)=0.25、1/(2^3)=0.125、…となる。

8倍 4倍 2倍 1倍 1/2倍 1/4倍 1/8倍
2進数 1000 100 10 1 0.1 0.01 0.001
10進数 8 4 2 1 0.5 0.25 0.125

2進数の有効桁が小数点以下8ビットの場合、2進数と10進数の対応は以下となる。

2進数 10進数
1 0.1 0.5
2 0.01 0.25
3 0.001 0.125
4 0.0001 0.0625
5 0.00001 0.03125
6 0.000001 0.015625
7 0.0000001 0.0078125
8 0.00000001 0.00390625

この表から、例えば10進数で0.75を表す2進数を求めたければ、

 10進数の計算: 0.5 + 0.25 = 0.75
  2進数の計算: 0.1 + 0.01 = 0.11

で、答えは0.11000000になる。

以下は、10bit、12bit、16bitの比較が出てくるのでその値の確認用換算表。

2進数 10進数
10bitの場合
9 0.000000001 0.001953125
10 0.0000000001 0.0009765625
12bitの場合
11 0.00000000001 0.00048828125
12 0.000000000001 0.000244140625
16bitの場合
13 0.0000000000001 0.0001220703125
14 0.00000000000001 0.00006103515625
15 0.000000000000001 0.000030517578125
16 0.0000000000000001 0.0000152587890625

0~0.99までの換算

以下のプログラムを使い、10進数-2進数換算票を0.01刻みで出力する。

list1.pl
my @tbl = (0.5, 0.25, 0.125, 0.0625, 0.03125, 0.015625, 0.0078125, 0.00390625);
my $step = 0.01;
my $fmt  = "%.2f : 0.%s : %.8f : %.8f\n";
my $acm;
my $cal;
my $bin;
 
  for(my $i = 0; $i < 1; $i += $step)
  {
    $cal = $i;
    $acm = 0;
    $bin = "";
    for(my $m=0; $m<8; $m++)
    {
      $cal = $cal * 2;
      if ($cal >= 1)
      {
        $acm += $tbl[$m];
        $cal -= 1;
        $bin .= "1";
      }
      else
      {
        $bin .= "0";
      }
    }
    printf($fmt, $i, $bin, $acm, $i - $acm);
  }

以下は出力結果を整形したもの。2進数は精度8bitで表現。見ての通り、ピタリ一致するような結果はまずでない。

10進数 2進数 2進→10進計算結果 誤差
0.00 0.00000000 0.00000000 0.00000000
0.01 0.00000010 0.00781250 0.00218750
0.02 0.00000101 0.01953125 0.00046875
0.03 0.00000111 0.02734375 0.00265625
0.04 0.00001010 0.03906250 0.00093750
0.05 0.00001100 0.04687500 0.00312500
0.06 0.00001111 0.05859375 0.00140625
0.07 0.00010001 0.06640625 0.00359375
0.08 0.00010100 0.07812500 0.00187500
0.09 0.00010111 0.08984375 0.00015625
0.10 0.00011001 0.09765625 0.00234375
0.11 0.00011100 0.10937500 0.00062500
0.12 0.00011110 0.11718750 0.00281250
0.13 0.00100001 0.12890625 0.00109375
0.14 0.00100011 0.13671875 0.00328125
0.15 0.00100110 0.14843750 0.00156250
0.16 0.00101000 0.15625000 0.00375000
0.17 0.00101011 0.16796875 0.00203125
0.18 0.00101110 0.17968750 0.00031250
0.19 0.00110000 0.18750000 0.00250000
0.20 0.00110011 0.19921875 0.00078125
0.21 0.00110101 0.20703125 0.00296875
0.22 0.00111000 0.21875000 0.00125000
0.23 0.00111010 0.22656250 0.00343750
0.24 0.00111101 0.23828125 0.00171875
0.25 0.01000000 0.25000000 0.00000000
0.26 0.01000010 0.25781250 0.00218750
0.27 0.01000101 0.26953125 0.00046875
0.28 0.01000111 0.27734375 0.00265625
0.29 0.01001010 0.28906250 0.00093750
0.30 0.01001100 0.29687500 0.00312500
0.31 0.01001111 0.30859375 0.00140625
0.32 0.01010001 0.31640625 0.00359375
0.33 0.01010100 0.32812500 0.00187500
0.34 0.01010111 0.33984375 0.00015625
0.35 0.01011001 0.34765625 0.00234375
0.36 0.01011100 0.35937500 0.00062500
0.37 0.01011110 0.36718750 0.00281250
0.38 0.01100001 0.37890625 0.00109375
0.39 0.01100011 0.38671875 0.00328125
0.40 0.01100110 0.39843750 0.00156250
0.41 0.01101000 0.40625000 0.00375000
0.42 0.01101011 0.41796875 0.00203125
0.43 0.01101110 0.42968750 0.00031250
0.44 0.01110000 0.43750000 0.00250000
0.45 0.01110011 0.44921875 0.00078125
0.46 0.01110101 0.45703125 0.00296875
0.47 0.01111000 0.46875000 0.00125000
0.48 0.01111010 0.47656250 0.00343750
0.49 0.01111101 0.48828125 0.00171875
0.50 0.10000000 0.50000000 0.00000000
0.51 0.10000010 0.50781250 0.00218750
0.52 0.10000101 0.51953125 0.00046875
0.53 0.10000111 0.52734375 0.00265625
0.54 0.10001010 0.53906250 0.00093750
0.55 0.10001100 0.54687500 0.00312500
0.56 0.10001111 0.55859375 0.00140625
0.57 0.10010001 0.56640625 0.00359375
0.58 0.10010100 0.57812500 0.00187500
0.59 0.10010111 0.58984375 0.00015625
0.60 0.10011001 0.59765625 0.00234375
0.61 0.10011100 0.60937500 0.00062500
0.62 0.10011110 0.61718750 0.00281250
0.63 0.10100001 0.62890625 0.00109375
0.64 0.10100011 0.63671875 0.00328125
0.65 0.10100110 0.64843750 0.00156250
0.66 0.10101000 0.65625000 0.00375000
0.67 0.10101011 0.66796875 0.00203125
0.68 0.10101110 0.67968750 0.00031250
0.69 0.10110000 0.68750000 0.00250000
0.70 0.10110011 0.69921875 0.00078125
0.71 0.10110101 0.70703125 0.00296875
0.72 0.10111000 0.71875000 0.00125000
0.73 0.10111010 0.72656250 0.00343750
0.74 0.10111101 0.73828125 0.00171875
0.75 0.11000000 0.75000000 0.00000000
0.76 0.11000010 0.75781250 0.00218750
0.77 0.11000101 0.76953125 0.00046875
0.78 0.11000111 0.77734375 0.00265625
0.79 0.11001010 0.78906250 0.00093750
0.80 0.11001100 0.79687500 0.00312500
0.81 0.11001111 0.80859375 0.00140625
0.82 0.11010001 0.81640625 0.00359375
0.83 0.11010100 0.82812500 0.00187500
0.84 0.11010111 0.83984375 0.00015625
0.85 0.11011001 0.84765625 0.00234375
0.86 0.11011100 0.85937500 0.00062500
0.87 0.11011110 0.86718750 0.00281250
0.88 0.11100001 0.87890625 0.00109375
0.89 0.11100011 0.88671875 0.00328125
0.90 0.11100110 0.89843750 0.00156250
0.91 0.11101000 0.90625000 0.00375000
0.92 0.11101011 0.91796875 0.00203125
0.93 0.11101110 0.92968750 0.00031250
0.94 0.11110000 0.93750000 0.00250000
0.95 0.11110011 0.94921875 0.00078125
0.96 0.11110101 0.95703125 0.00296875
0.97 0.11111000 0.96875000 0.00125000
0.98 0.11111010 0.97656250 0.00343750
0.99 0.11111101 0.98828125 0.00171875

精度の話(0~0.999で計算してみる)

先のプログラムを0.001刻みにして8,10,12,16bitの各精度で出力してみる。

list2.pl
my @tbl = (  0.5,
             0.25,
             0.125,
             0.0625,
             0.03125,
             0.015625,
             0.0078125,
             0.00390625,
             0.001953125,
             0.0009765625,
             0.00048828125,
             0.000244140625,
             0.0001220703125,
             0.00006103515625,
             0.000030517578125,
             0.0000152587890625 );
 
my $step = 0.001;
my $fmt  = "%.3f : 0.%s : 0.%s : 0.%s : 0.%s : %.8f : %.10f : %.12f : %.16f\n";
my @acm;
my @cal;
my @bin;
my $scaleidx;
 
  for(my $i = 0; $i < 1; $i += $step)
  {
    @cal = ( $i, $i, $i, $i);
    @acm = (  0 , 0,  0, 0 );
    @bin = ( "", "", "", "");
    $scaleidx = 4;
 
    for(my $m = 0; $m < @tbl; $m++)
    {
      if ($m >  7) { $scaleidx = 3; }
      if ($m >  9) { $scaleidx = 2; }
      if ($m > 11) { $scaleidx = 1; }
 
      for(my $idx = 0; $idx < $scaleidx; $idx++)
      {
        $cal[$idx] = $cal[$idx] * 2;
        if ($cal[$idx] >= 1)
        {
          $acm[$idx] += $tbl[$m];
          $cal[$idx] -= 1;
          $bin[$idx] .= "1";
        }
        else
        {
          $bin[$idx] .= "0";
        }
      }
    }
    printf($fmt, $i,   $bin[3], $bin[2], $bin[1], $bin[0], $acm[3], $acm[2], $acm[1], $acm[0]);
  }

結果は以下の通り。例えば、精度8bitでは 10進数 0 ~ 0.003 の2進数表現が 0.0 になってしまう。10進数の小数点以下3桁を表現するには、8bitでは足りないという事がわかる。
※少なくとも10bitが必要。

10進数 2進数 2進→10進計算結果
精度8bit 精度10bit 精度12bit 精度16bit 精度8bit 精度10bit 精度12bit 精度16bit
0.000 0.00000000 0.0000000000 0.000000000000 0.0000000000000000 0.00000000 0.0000000000 0.000000000000 0.0000000000000000
0.001 0.00000000 0.0000000001 0.000000000100 0.0000000001000001 0.00000000 0.0009765625 0.000976562500 0.0009918212890625
0.002 0.00000000 0.0000000010 0.000000001000 0.0000000010000011 0.00000000 0.0019531250 0.001953125000 0.0019989013671875
0.003 0.00000000 0.0000000011 0.000000001100 0.0000000011000100 0.00000000 0.0029296875 0.002929687500 0.0029907226562500
0.004 0.00000001 0.0000000100 0.000000010000 0.0000000100000110 0.00390625 0.0039062500 0.003906250000 0.0039978027343750
0.005 0.00000001 0.0000000101 0.000000010100 0.0000000101000111 0.00390625 0.0048828125 0.004882812500 0.0049896240234375
0.006 0.00000001 0.0000000110 0.000000011000 0.0000000110001001 0.00390625 0.0058593750 0.005859375000 0.0059967041015625
0.007 0.00000001 0.0000000111 0.000000011100 0.0000000111001010 0.00390625 0.0068359375 0.006835937500 0.0069885253906250
0.008 0.00000010 0.0000001000 0.000000100000 0.0000001000001100 0.00781250 0.0078125000 0.007812500000 0.0079956054687500
0.009 0.00000010 0.0000001001 0.000000100100 0.0000001001001101 0.00781250 0.0087890625 0.008789062500 0.0089874267578125
0.010 0.00000010 0.0000001010 0.000000101000 0.0000001010001111 0.00781250 0.0097656250 0.009765625000 0.0099945068359375
0.011 0.00000010 0.0000001011 0.000000101101 0.0000001011010000 0.00781250 0.0107421875 0.010986328125 0.0109863281250000
0.012 0.00000011 0.0000001100 0.000000110001 0.0000001100010010 0.01171875 0.0117187500 0.011962890625 0.0119934082031250
0.013 0.00000011 0.0000001101 0.000000110101 0.0000001101010011 0.01171875 0.0126953125 0.012939453125 0.0129852294921875
0.014 0.00000011 0.0000001110 0.000000111001 0.0000001110010101 0.01171875 0.0136718750 0.013916015625 0.0139923095703125
0.015 0.00000011 0.0000001111 0.000000111101 0.0000001111010111 0.01171875 0.0146484375 0.014892578125 0.0149993896484375
0.016 0.00000100 0.0000010000 0.000001000001 0.0000010000011000 0.01562500 0.0156250000 0.015869140625 0.0159912109375000
0.017 0.00000100 0.0000010001 0.000001000101 0.0000010001011010 0.01562500 0.0166015625 0.016845703125 0.0169982910156250
0.018 0.00000100 0.0000010010 0.000001001001 0.0000010010011011 0.01562500 0.0175781250 0.017822265625 0.0179901123046875
0.019 0.00000100 0.0000010011 0.000001001101 0.0000010011011101 0.01562500 0.0185546875 0.018798828125 0.0189971923828125
0.020 0.00000101 0.0000010100 0.000001010001 0.0000010100011110 0.01953125 0.0195312500 0.019775390625 0.0199890136718750
0.021 0.00000101 0.0000010101 0.000001010110 0.0000010101100000 0.01953125 0.0205078125 0.020996093750 0.0209960937500000
0.022 0.00000101 0.0000010110 0.000001011010 0.0000010110100001 0.01953125 0.0214843750 0.021972656250 0.0219879150390625
0.023 0.00000101 0.0000010111 0.000001011110 0.0000010111100011 0.01953125 0.0224609375 0.022949218750 0.0229949951171875
0.024 0.00000110 0.0000011000 0.000001100010 0.0000011000100100 0.02343750 0.0234375000 0.023925781250 0.0239868164062500
0.025 0.00000110 0.0000011001 0.000001100110 0.0000011001100110 0.02343750 0.0244140625 0.024902343750 0.0249938964843750
0.026 0.00000110 0.0000011010 0.000001101010 0.0000011010100111 0.02343750 0.0253906250 0.025878906250 0.0259857177734375
0.027 0.00000110 0.0000011011 0.000001101110 0.0000011011101001 0.02343750 0.0263671875 0.026855468750 0.0269927978515625
0.028 0.00000111 0.0000011100 0.000001110010 0.0000011100101011 0.02734375 0.0273437500 0.027832031250 0.0279998779296875
0.029 0.00000111 0.0000011101 0.000001110110 0.0000011101101100 0.02734375 0.0283203125 0.028808593750 0.0289916992187500
0.030 0.00000111 0.0000011110 0.000001111010 0.0000011110101110 0.02734375 0.0292968750 0.029785156250 0.0299987792968750
0.031 0.00000111 0.0000011111 0.000001111110 0.0000011111101111 0.02734375 0.0302734375 0.030761718750 0.0309906005859375
0.032 0.00001000 0.0000100000 0.000010000011 0.0000100000110001 0.03125000 0.0312500000 0.031982421875 0.0319976806640625
0.033 0.00001000 0.0000100001 0.000010000111 0.0000100001110010 0.03125000 0.0322265625 0.032958984375 0.0329895019531250
0.034 0.00001000 0.0000100010 0.000010001011 0.0000100010110100 0.03125000 0.0332031250 0.033935546875 0.0339965820312500
0.035 0.00001000 0.0000100011 0.000010001111 0.0000100011110101 0.03125000 0.0341796875 0.034912109375 0.0349884033203125
0.036 0.00001001 0.0000100100 0.000010010011 0.0000100100110111 0.03515625 0.0351562500 0.035888671875 0.0359954833984375
0.037 0.00001001 0.0000100101 0.000010010111 0.0000100101111000 0.03515625 0.0361328125 0.036865234375 0.0369873046875000
0.038 0.00001001 0.0000100110 0.000010011011 0.0000100110111010 0.03515625 0.0371093750 0.037841796875 0.0379943847656250
0.039 0.00001001 0.0000100111 0.000010011111 0.0000100111111011 0.03515625 0.0380859375 0.038818359375 0.0389862060546875
0.040 0.00001010 0.0000101000 0.000010100011 0.0000101000111101 0.03906250 0.0390625000 0.039794921875 0.0399932861328125
0.041 0.00001010 0.0000101001 0.000010100111 0.0000101001111110 0.03906250 0.0400390625 0.040771484375 0.0409851074218750
0.042 0.00001010 0.0000101011 0.000010101100 0.0000101011000000 0.03906250 0.0419921875 0.041992187500 0.0419921875000000
0.043 0.00001011 0.0000101100 0.000010110000 0.0000101100000010 0.04296875 0.0429687500 0.042968750000 0.0429992675781250
0.044 0.00001011 0.0000101101 0.000010110100 0.0000101101000011 0.04296875 0.0439453125 0.043945312500 0.0439910888671875
0.045 0.00001011 0.0000101110 0.000010111000 0.0000101110000101 0.04296875 0.0449218750 0.044921875000 0.0449981689453125
0.046 0.00001011 0.0000101111 0.000010111100 0.0000101111000110 0.04296875 0.0458984375 0.045898437500 0.0459899902343750
0.047 0.00001100 0.0000110000 0.000011000000 0.0000110000001000 0.04687500 0.0468750000 0.046875000000 0.0469970703125000
0.048 0.00001100 0.0000110001 0.000011000100 0.0000110001001001 0.04687500 0.0478515625 0.047851562500 0.0479888916015625
0.049 0.00001100 0.0000110010 0.000011001000 0.0000110010001011 0.04687500 0.0488281250 0.048828125000 0.0489959716796875
0.050 0.00001100 0.0000110011 0.000011001100 0.0000110011001100 0.04687500 0.0498046875 0.049804687500 0.0499877929687500
0.051 0.00001101 0.0000110100 0.000011010000 0.0000110100001110 0.05078125 0.0507812500 0.050781250000 0.0509948730468750
0.052 0.00001101 0.0000110101 0.000011010100 0.0000110101001111 0.05078125 0.0517578125 0.051757812500 0.0519866943359375
0.053 0.00001101 0.0000110110 0.000011011001 0.0000110110010001 0.05078125 0.0527343750 0.052978515625 0.0529937744140625
0.054 0.00001101 0.0000110111 0.000011011101 0.0000110111010010 0.05078125 0.0537109375 0.053955078125 0.0539855957031250
0.055 0.00001110 0.0000111000 0.000011100001 0.0000111000010100 0.05468750 0.0546875000 0.054931640625 0.0549926757812500
0.056 0.00001110 0.0000111001 0.000011100101 0.0000111001010110 0.05468750 0.0556640625 0.055908203125 0.0559997558593750
0.057 0.00001110 0.0000111010 0.000011101001 0.0000111010010111 0.05468750 0.0566406250 0.056884765625 0.0569915771484375
0.058 0.00001110 0.0000111011 0.000011101101 0.0000111011011001 0.05468750 0.0576171875 0.057861328125 0.0579986572265625
0.059 0.00001111 0.0000111100 0.000011110001 0.0000111100011010 0.05859375 0.0585937500 0.058837890625 0.0589904785156250
0.060 0.00001111 0.0000111101 0.000011110101 0.0000111101011100 0.05859375 0.0595703125 0.059814453125 0.0599975585937500
0.061 0.00001111 0.0000111110 0.000011111001 0.0000111110011101 0.05859375 0.0605468750 0.060791015625 0.0609893798828125
0.062 0.00001111 0.0000111111 0.000011111101 0.0000111111011111 0.05859375 0.0615234375 0.061767578125 0.0619964599609375
0.063 0.00010000 0.0001000000 0.000100000010 0.0001000000100000 0.06250000 0.0625000000 0.062988281250 0.0629882812500000
0.064 0.00010000 0.0001000001 0.000100000110 0.0001000001100010 0.06250000 0.0634765625 0.063964843750 0.0639953613281250
0.065 0.00010000 0.0001000010 0.000100001010 0.0001000010100011 0.06250000 0.0644531250 0.064941406250 0.0649871826171875
0.066 0.00010000 0.0001000011 0.000100001110 0.0001000011100101 0.06250000 0.0654296875 0.065917968750 0.0659942626953125
0.067 0.00010001 0.0001000100 0.000100010010 0.0001000100100110 0.06640625 0.0664062500 0.066894531250 0.0669860839843750
0.068 0.00010001 0.0001000101 0.000100010110 0.0001000101101000 0.06640625 0.0673828125 0.067871093750 0.0679931640625000
0.069 0.00010001 0.0001000110 0.000100011010 0.0001000110101001 0.06640625 0.0683593750 0.068847656250 0.0689849853515625
0.070 0.00010001 0.0001000111 0.000100011110 0.0001000111101011 0.06640625 0.0693359375 0.069824218750 0.0699920654296875
0.071 0.00010010 0.0001001000 0.000100100010 0.0001001000101101 0.07031250 0.0703125000 0.070800781250 0.0709991455078125
0.072 0.00010010 0.0001001001 0.000100100110 0.0001001001101110 0.07031250 0.0712890625 0.071777343750 0.0719909667968750
0.073 0.00010010 0.0001001010 0.000100101011 0.0001001010110000 0.07031250 0.0722656250 0.072998046875 0.0729980468750000
0.074 0.00010010 0.0001001011 0.000100101111 0.0001001011110001 0.07031250 0.0732421875 0.073974609375 0.0739898681640625
0.075 0.00010011 0.0001001100 0.000100110011 0.0001001100110011 0.07421875 0.0742187500 0.074951171875 0.0749969482421875
0.076 0.00010011 0.0001001101 0.000100110111 0.0001001101110100 0.07421875 0.0751953125 0.075927734375 0.0759887695312500
0.077 0.00010011 0.0001001110 0.000100111011 0.0001001110110110 0.07421875 0.0761718750 0.076904296875 0.0769958496093750
0.078 0.00010011 0.0001001111 0.000100111111 0.0001001111110111 0.07421875 0.0771484375 0.077880859375 0.0779876708984375
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0.948 0.11110010 0.1111001010 0.111100101011 0.1111001010110000 0.94531250 0.9472656250 0.947998046875 0.9479980468750000
0.949 0.11110010 0.1111001011 0.111100101111 0.1111001011110001 0.94531250 0.9482421875 0.948974609375 0.9489898681640625
0.950 0.11110011 0.1111001100 0.111100110011 0.1111001100110011 0.94921875 0.9492187500 0.949951171875 0.9499969482421875
0.951 0.11110011 0.1111001101 0.111100110111 0.1111001101110100 0.94921875 0.9501953125 0.950927734375 0.9509887695312500
0.952 0.11110011 0.1111001110 0.111100111011 0.1111001110110110 0.94921875 0.9511718750 0.951904296875 0.9519958496093750
0.953 0.11110011 0.1111001111 0.111100111111 0.1111001111110111 0.94921875 0.9521484375 0.952880859375 0.9529876708984375
0.954 0.11110100 0.1111010000 0.111101000011 0.1111010000111001 0.95312500 0.9531250000 0.953857421875 0.9539947509765625
0.955 0.11110100 0.1111010001 0.111101000111 0.1111010001111010 0.95312500 0.9541015625 0.954833984375 0.9549865722656250
0.956 0.11110100 0.1111010010 0.111101001011 0.1111010010111100 0.95312500 0.9550781250 0.955810546875 0.9559936523437500
0.957 0.11110100 0.1111010011 0.111101001111 0.1111010011111101 0.95312500 0.9560546875 0.956787109375 0.9569854736328125
0.958 0.11110101 0.1111010100 0.111101010011 0.1111010100111111 0.95703125 0.9570312500 0.957763671875 0.9579925537109375
0.959 0.11110101 0.1111010110 0.111101011000 0.1111010110000001 0.95703125 0.9589843750 0.958984375000 0.9589996337890625
0.960 0.11110101 0.1111010111 0.111101011100 0.1111010111000010 0.95703125 0.9599609375 0.959960937500 0.9599914550781250
0.961 0.11110110 0.1111011000 0.111101100000 0.1111011000000100 0.96093750 0.9609375000 0.960937500000 0.9609985351562500
0.962 0.11110110 0.1111011001 0.111101100100 0.1111011001000101 0.96093750 0.9619140625 0.961914062500 0.9619903564453125
0.963 0.11110110 0.1111011010 0.111101101000 0.1111011010000111 0.96093750 0.9628906250 0.962890625000 0.9629974365234375
0.964 0.11110110 0.1111011011 0.111101101100 0.1111011011001000 0.96093750 0.9638671875 0.963867187500 0.9639892578125000
0.965 0.11110111 0.1111011100 0.111101110000 0.1111011100001010 0.96484375 0.9648437500 0.964843750000 0.9649963378906250
0.966 0.11110111 0.1111011101 0.111101110100 0.1111011101001011 0.96484375 0.9658203125 0.965820312500 0.9659881591796875
0.967 0.11110111 0.1111011110 0.111101111000 0.1111011110001101 0.96484375 0.9667968750 0.966796875000 0.9669952392578125
0.968 0.11110111 0.1111011111 0.111101111100 0.1111011111001110 0.96484375 0.9677734375 0.967773437500 0.9679870605468750
0.969 0.11111000 0.1111100000 0.111110000001 0.1111100000010000 0.96875000 0.9687500000 0.968994140625 0.9689941406250000
0.970 0.11111000 0.1111100001 0.111110000101 0.1111100001010001 0.96875000 0.9697265625 0.969970703125 0.9699859619140625
0.971 0.11111000 0.1111100010 0.111110001001 0.1111100010010011 0.96875000 0.9707031250 0.970947265625 0.9709930419921875
0.972 0.11111000 0.1111100011 0.111110001101 0.1111100011010100 0.96875000 0.9716796875 0.971923828125 0.9719848632812500
0.973 0.11111001 0.1111100100 0.111110010001 0.1111100100010110 0.97265625 0.9726562500 0.972900390625 0.9729919433593750
0.974 0.11111001 0.1111100101 0.111110010101 0.1111100101011000 0.97265625 0.9736328125 0.973876953125 0.9739990234375000
0.975 0.11111001 0.1111100110 0.111110011001 0.1111100110011001 0.97265625 0.9746093750 0.974853515625 0.9749908447265625
0.976 0.11111001 0.1111100111 0.111110011101 0.1111100111011011 0.97265625 0.9755859375 0.975830078125 0.9759979248046875
0.977 0.11111010 0.1111101000 0.111110100001 0.1111101000011100 0.97656250 0.9765625000 0.976806640625 0.9769897460937500
0.978 0.11111010 0.1111101001 0.111110100101 0.1111101001011110 0.97656250 0.9775390625 0.977783203125 0.9779968261718750
0.979 0.11111010 0.1111101010 0.111110101001 0.1111101010011111 0.97656250 0.9785156250 0.978759765625 0.9789886474609375
0.980 0.11111010 0.1111101011 0.111110101110 0.1111101011100001 0.97656250 0.9794921875 0.979980468750 0.9799957275390625
0.981 0.11111011 0.1111101100 0.111110110010 0.1111101100100010 0.98046875 0.9804687500 0.980957031250 0.9809875488281250
0.982 0.11111011 0.1111101101 0.111110110110 0.1111101101100100 0.98046875 0.9814453125 0.981933593750 0.9819946289062500
0.983 0.11111011 0.1111101110 0.111110111010 0.1111101110100101 0.98046875 0.9824218750 0.982910156250 0.9829864501953125
0.984 0.11111011 0.1111101111 0.111110111110 0.1111101111100111 0.98046875 0.9833984375 0.983886718750 0.9839935302734375
0.985 0.11111100 0.1111110000 0.111111000010 0.1111110000101000 0.98437500 0.9843750000 0.984863281250 0.9849853515625000
0.986 0.11111100 0.1111110001 0.111111000110 0.1111110001101010 0.98437500 0.9853515625 0.985839843750 0.9859924316406250
0.987 0.11111100 0.1111110010 0.111111001010 0.1111110010101100 0.98437500 0.9863281250 0.986816406250 0.9869995117187500
0.988 0.11111100 0.1111110011 0.111111001110 0.1111110011101101 0.98437500 0.9873046875 0.987792968750 0.9879913330078125
0.989 0.11111101 0.1111110100 0.111111010010 0.1111110100101111 0.98828125 0.9882812500 0.988769531250 0.9889984130859375
0.990 0.11111101 0.1111110101 0.111111010111 0.1111110101110000 0.98828125 0.9892578125 0.989990234375 0.9899902343750000
0.991 0.11111101 0.1111110110 0.111111011011 0.1111110110110010 0.98828125 0.9902343750 0.990966796875 0.9909973144531250
0.992 0.11111101 0.1111110111 0.111111011111 0.1111110111110011 0.98828125 0.9912109375 0.991943359375 0.9919891357421875
0.993 0.11111110 0.1111111000 0.111111100011 0.1111111000110101 0.99218750 0.9921875000 0.992919921875 0.9929962158203125
0.994 0.11111110 0.1111111001 0.111111100111 0.1111111001110110 0.99218750 0.9931640625 0.993896484375 0.9939880371093750
0.995 0.11111110 0.1111111010 0.111111101011 0.1111111010111000 0.99218750 0.9941406250 0.994873046875 0.9949951171875000
0.996 0.11111110 0.1111111011 0.111111101111 0.1111111011111001 0.99218750 0.9951171875 0.995849609375 0.9959869384765625
0.997 0.11111111 0.1111111100 0.111111110011 0.1111111100111011 0.99609375 0.9960937500 0.996826171875 0.9969940185546875
0.998 0.11111111 0.1111111101 0.111111110111 0.1111111101111100 0.99609375 0.9970703125 0.997802734375 0.9979858398437500
0.999 0.11111111 0.1111111110 0.111111111011 0.1111111110111110 0.99609375 0.9980468750 0.998779296875 0.9989929199218750
documents/other/other-051.txt · 最終更新: 2017/05/05 00:11 by k896951

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