一般的なこと

割り算無しで円周率の計算

2009年7月2日

割り算は少し面倒なので、これを行わずに円周率を計算させる方法を考えてみた。

for ($max=2;$max<=2000000000;$max=$max*sqrt(10)) {
    $t=time();
    echo intval($max).': '.calc_pi(intval($max))."  (".(time()-$t).")\n";
}

function calc_pi($max) {
    $total=0;
    $max2=$max*$max;
    $prev_y=$max-1;
    for ($x=0;$x<$max;$x++) {
        $max2_x2=$max2-$x*$x;
        for ($y=$prev_y;0<=$y;$y--) {
            if ($y*$y < $max2_x2) break;
        }
        $total+=$y+1;
        $prev_y=$y;
    }
    return $total;
}


実行結果は、次のとおり。
2: 4  (0)
6: 33  (0)
20: 331  (0)
63: 3175  (0)
200: 31602  (0)
632: 314319  (0)
2000: 3143579  (0)
6324: 31416676  (0)
20000: 314179205  (0)
63245: 3141600733  (0)
200000: 31416126094  (1)
632455: 314159368708  (0)
2000000: 3141594652558  (3)
6324555: 31415929675774  (8)
20000000: 3.1415928535658E+14  (24)
63245553: 3.1415926966214E+15  (76)
200000000: 3.1415926756486E+16  (242)
632455532: 3.1415926602871E+17  (757)

ただ、２進数の数値を１０進数で表示させるときに、割り算は必要になるのだけれど…。ただしこの場合、１０で割るだけだから何か方法を考えてみよう。掛け算して、右シフトとか。

ちなみに、Wikipediaに載っている収束の早い方法で円周率を求めるスクリプトは、次の通り。
$prev='';
for ($max=1;$max<=1073741824;$max++) {
    $t=time();
    $pi=calc_pi($max);
    echo "$max: ".$pi."  (".(time()-$t).")\n";
    if ($pi===$prev) exit;
    $prev=$pi;
}

function calc_pi($max) {
    $fig=50;
    $pi='0';
    for($n=0;$n<$max;$n++){
        $y=bcdiv('3',bcadd('1',bcmul('2',$n,$fig),$fig),$fig);//$y=3/(2*$n+1);
        for($x=1;$x<=$n;$x++) $y=bcmul($y,bcdiv(bcadd($x,$n,$fig),bcmul('16',$x,$fig),$fig),$fig);//$y*=($x+$n)/(16*$x);
        $pi=bcadd($pi,$y,$fig);//$pi+=$y;
    }
    return $pi;
}

実行結果：
1: 3.00000000000000000000000000000000000000000000000000  (0)
2: 3.12500000000000000000000000000000000000000000000000  (0)
3: 3.13906250000000000000000000000000000000000000000000  (0)
4: 3.14115513392857142857142857142857142857142857142857  (0)
5: 3.14151117234002976190476190476190476190476190476190  (0)
6: 3.14157671577486640963203463203463203463203463203462  (0)
7: 3.14158942531912159292530386280386280386280386280385  (0)
8: 3.14159198235838245718311636280386280386280386280384  (0)
9: 3.14159251115786195576584413164577456856868621574501  (0)
10: 3.14159262287061749311263384304585680541079147890290  (0)
11: 3.14159264687556079607822377507885066701793433604575  (0)
12: 3.14159265210588688135876980275995259042146694474140  (0)
13: 3.14159265325873792265602348969466213937166225724139  (0)
14: 3.14159265351533760243194355890199599553044860706777  (0)
15: 3.14159265357293032612549212123332185179268852363902  (0)
16: 3.14159265358595061661642608384639847682401749402515  (0)
17: 3.14159265358891285600131896691248906315335157085779  (0)
18: 3.14159265358959056160849298995008835905936938801550  (0)
19: 3.14159265358974637793146374412070756654526875476065  (0)
20: 3.14159265358978236182926181471105468109455961729002  (0)
21: 3.14159265358979070504702871491957876055027172123928  (0)
22: 3.14159265358979264648516079676489008081121469559402  (0)
23: 3.14159265358979309973252269314519066555395251877355  (0)
24: 3.14159265358979320586410315569955105918300479582472  (0)
25: 3.14159265358979323078381629789582891946431607133556  (0)
26: 3.14159265358979323664972132283830569814426002942394  (0)
27: 3.14159265358979323803372159223919304405740718426681  (0)
28: 3.14159265358979323836096542698221430286801580025617  (0)
29: 3.14159265358979323843849617573672865968992142865277  (0)
30: 3.14159265358979323845689896669986550549021156502481  (0)
31: 3.14159265358979323846127465763857311206335432217720  (0)
32: 3.14159265358979323846231676887659993003945930165518  (0)
33: 3.14159265358979323846256533495254514882234155307033  (0)
34: 3.14159265358979323846262470806826589448932920075401  (0)
35: 3.14159265358979323846263890915077177829682640612762  (0)
36: 3.14159265358979323846264231012430349423281705978877  (0)
37: 3.14159265358979323846264312558794378057382395120362  (0)
38: 3.14159265358979323846264332133595004570496024509324  (0)
39: 3.14159265358979323846264336837467762778641281058918  (0)
40: 3.14159265358979323846264337968969774439469471016827  (0)
41: 3.14159265358979323846264338241412060503328604254763  (0)
42: 3.14159265358979323846264338307070811563605557427481  (0)
43: 3.14159265358979323846264338322908456383432165635004  (0)
44: 3.14159265358979323846264338326731865359081966634046  (0)
45: 3.14159265358979323846264338327655619908899808301619  (0)
46: 3.14159265358979323846264338327878973382315776673073  (0)
47: 3.14159265358979323846264338327933017039204440595583  (0)
48: 3.14159265358979323846264338327946102805957801577268  (1)
49: 3.14159265358979323846264338327949273420351838380846  (0)
50: 3.14159265358979323846264338327950042135844352612057  (0)
51: 3.14159265358979323846264338327950228625461732116515  (0)
52: 3.14159265358979323846264338327950273894369663020878  (0)
53: 3.14159265358979323846264338327950284889283679480685  (0)
54: 3.14159265358979323846264338327950287561187378816546  (0)
55: 3.14159265358979323846264338327950288210835387941934  (0)
56: 3.14159265358979323846264338327950288368871178204397  (0)
57: 3.14159265358979323846264338327950288407334337413938  (0)
58: 3.14159265358979323846264338327950288416700013995681  (0)
59: 3.14159265358979323846264338327950288418981569271129  (0)
60: 3.14159265358979323846264338327950288419537619160133  (0)
61: 3.14159265358979323846264338327950288419673194616024  (0)
62: 3.14159265358979323846264338327950288419706264059075  (0)
63: 3.14159265358979323846264338327950288419714333536557  (0)
64: 3.14159265358979323846264338327950288419716303377602  (0)
65: 3.14159265358979323846264338327950288419716784415123  (0)
66: 3.14159265358979323846264338327950288419716901927533  (0)
67: 3.14159265358979323846264338327950288419716930644645  (0)
68: 3.14159265358979323846264338327950288419716937664780  (1)
69: 3.14159265358979323846264338327950288419716939381476  (0)
70: 3.14159265358979323846264338327950288419716939801409  (0)
71: 3.14159265358979323846264338327950288419716939904164  (0)
72: 3.14159265358979323846264338327950288419716939929315  (0)
73: 3.14159265358979323846264338327950288419716939935472  (0)
74: 3.14159265358979323846264338327950288419716939936980  (0)
75: 3.14159265358979323846264338327950288419716939937349  (0)
76: 3.14159265358979323846264338327950288419716939937439  (0)
77: 3.14159265358979323846264338327950288419716939937461  (0)
78: 3.14159265358979323846264338327950288419716939937466  (0)
79: 3.14159265358979323846264338327950288419716939937467  (0)
80: 3.14159265358979323846264338327950288419716939937467  (0)

１秒以内に４０桁以上。速い。これは、先日作った、pi calculatorに入れてみたいアルゴリズムだ。

コメント

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