昨日のExcelの検証用に作ったプログラムについて、書いていきます。
言語はC言語です。
まぁ、前々回、前回とチェビシェフの多項式がなんたるかは解っていらっしゃるでしょうから、いきなりですがソースです。
#include <stdio.h>
#include <stdlib.h>
int main()
{
long long int k[3][101];
int i, j;
k[0][0] = k[1][1] = 1;
k[0][1] = k[1][0] = 0;
for (j=2; j<101; j++) {
k[0][j] = 0;
k[1][j] = 0;
}
printf("T<sub>0</sub>(x)=+1<br>\nT<sub>1</sub>(x)=+x<br>\n");
i = 2;
while ( i < 101 ) {
k[i%3][0] = -k[(i+1)%3][0];
for (j=1; j<101; j++) {
if ( k[(i+2)%3][j-1] > 0 && 2*k[(i+2)%3][j-1] < 0 ) break;
if ( k[(i+2)%3][j-1] < 0 && 2*k[(i+2)%3][j-1] > 0 ) break;
if ( 2*k[(i+2)%3][j-1] > 0 && 2*k[(i+2)%3][j-1]-k[(i+1)%3][j] < 0 ) break;
if ( 2*k[(i+2)%3][j-1] < 0 && 2*k[(i+2)%3][j-1]-k[(i+1)%3][j] > 0 ) break;
k[i%3][j] = 2*k[(i+2)%3][j-1]-k[(i+1)%3][j];
}
if ( j < 101 ) break;
/*
for (j=0; j<99; j++) {
if ( k[i%3][j] > 0 && k[i%3][j+2] > 0 ) break;
if ( k[i%3][j] < 0 && k[i%3][j+2] < 0 ) break;
}
if ( j < 99 ) break;
*/
printf("T<sub>%d</sub>(x)=",i);
for (j=100; j>1; j--) {
if ( k[i%3][j] > 0 ) {
printf("+%lld",k[i%3][j]);
printf("x<sup>");
printf("%d",j);
printf("</sup>");
} else if ( k[i%3][j] < 0 ) {
printf("%lld",k[i%3][j]);
printf("x<sup>");
printf("%d",j);
printf("</sup>");
}
}
if ( k[i%3][j] > 1 ) {
printf("+%lld",k[i%3][j]);
printf("x");
} else if ( k[i%3][j] == 1 ) {
printf("+x");
} else if ( k[i%3][j] < 0 ) {
printf("%lld",k[i%3][j]);
printf("x");
}
j--;
if ( k[i%3][j] > 0 ) {
printf("+%lld",k[i%3][j]);
} else if ( k[i%3][j] < 0 ) {
printf("%lld",k[i%3][j]);
}
printf("<br>\n");
i++;
}
return EXIT_SUCCESS;
}
#include <stdlib.h>
int main()
{
long long int k[3][101];
int i, j;
k[0][0] = k[1][1] = 1;
k[0][1] = k[1][0] = 0;
for (j=2; j<101; j++) {
k[0][j] = 0;
k[1][j] = 0;
}
printf("T<sub>0</sub>(x)=+1<br>\nT<sub>1</sub>(x)=+x<br>\n");
i = 2;
while ( i < 101 ) {
k[i%3][0] = -k[(i+1)%3][0];
for (j=1; j<101; j++) {
if ( k[(i+2)%3][j-1] > 0 && 2*k[(i+2)%3][j-1] < 0 ) break;
if ( k[(i+2)%3][j-1] < 0 && 2*k[(i+2)%3][j-1] > 0 ) break;
if ( 2*k[(i+2)%3][j-1] > 0 && 2*k[(i+2)%3][j-1]-k[(i+1)%3][j] < 0 ) break;
if ( 2*k[(i+2)%3][j-1] < 0 && 2*k[(i+2)%3][j-1]-k[(i+1)%3][j] > 0 ) break;
k[i%3][j] = 2*k[(i+2)%3][j-1]-k[(i+1)%3][j];
}
if ( j < 101 ) break;
/*
for (j=0; j<99; j++) {
if ( k[i%3][j] > 0 && k[i%3][j+2] > 0 ) break;
if ( k[i%3][j] < 0 && k[i%3][j+2] < 0 ) break;
}
if ( j < 99 ) break;
*/
printf("T<sub>%d</sub>(x)=",i);
for (j=100; j>1; j--) {
if ( k[i%3][j] > 0 ) {
printf("+%lld",k[i%3][j]);
printf("x<sup>");
printf("%d",j);
printf("</sup>");
} else if ( k[i%3][j] < 0 ) {
printf("%lld",k[i%3][j]);
printf("x<sup>");
printf("%d",j);
printf("</sup>");
}
}
if ( k[i%3][j] > 1 ) {
printf("+%lld",k[i%3][j]);
printf("x");
} else if ( k[i%3][j] == 1 ) {
printf("+x");
} else if ( k[i%3][j] < 0 ) {
printf("%lld",k[i%3][j]);
printf("x");
}
j--;
if ( k[i%3][j] > 0 ) {
printf("+%lld",k[i%3][j]);
} else if ( k[i%3][j] < 0 ) {
printf("%lld",k[i%3][j]);
}
printf("<br>\n");
i++;
}
return EXIT_SUCCESS;
}
ざっと解説すると、
long long int型、つまり64ビット整数型ですね。
k[3][101]は係数の格納用です。
[101]はx0からx100の係数までを格納するため101個。
それに比べて[3]って少なすぎじゃない？と思われるかもしれません。
本来なら指数と同じようにT0(x)のT100(x)までの101個の配列を用意しなければならないでしょう。
ただね、そんなにあっても無駄なんですよ。
データ参照として2個前までで良いので、3個で十分なんです。
使わなくなったデータのところに新しいデータを上書きする方法です。
つまりリングバッファ的な作り込みです。
もっと言えば、[101]も[51]で十分なんですが、プログラムが汚くなりそうなので、やめておきました。
64ビット型ですから、1バイト=8ビットなので、4バイトです。
101×101も作ったら、40804バイト、約49KBメモリーを使うんです。
3×101ならば、1212バイト、約1.18KBメモリーで済みます。
まぁ、こういうテクニックもいずれは廃れていくのだろうか。
今回の肝はオーバーフローの検出だろう。
コメントアウトしてあるのが、最初に考えた検出方法で、
for (j=0; j<99; j++) {
if ( k[i%3][j] > 0 && k[i%3][j+2] > 0 ) break;
if ( k[i%3][j] < 0 && k[i%3][j+2] < 0 ) break;
}
if ( j < 99 ) break;
if ( k[i%3][j] > 0 && k[i%3][j+2] > 0 ) break;
if ( k[i%3][j] < 0 && k[i%3][j+2] < 0 ) break;
}
if ( j < 99 ) break;
これは、チェビシェフの多項式の性質で、降べきの順、昇べきの順、どちらであったとしても、符号が交互に入れ替わるというということに着目して、データ上、2つ後ろの項と符号が同じなら、breakしてforループから抜ける。
forループが最後まで行われなかったら、大外のwhileループからも抜ける。
現状、採用したのがこちらです。
for (j=1; j<101; j++) {
if ( k[(i+2)%3][j-1] > 0 && 2*k[(i+2)%3][j-1] < 0 ) break;
if ( k[(i+2)%3][j-1] < 0 && 2*k[(i+2)%3][j-1] > 0 ) break;
if ( 2*k[(i+2)%3][j-1] > 0 && 2*k[(i+2)%3][j-1]-k[(i+1)%3][j] < 0 ) break;
if ( 2*k[(i+2)%3][j-1] < 0 && 2*k[(i+2)%3][j-1]-k[(i+1)%3][j] > 0 ) break;
k[i%3][j] = 2*k[(i+2)%3][j-1]-k[(i+1)%3][j];
}
if ( j < 101 ) break;
if ( k[(i+2)%3][j-1] > 0 && 2*k[(i+2)%3][j-1] < 0 ) break;
if ( k[(i+2)%3][j-1] < 0 && 2*k[(i+2)%3][j-1] > 0 ) break;
if ( 2*k[(i+2)%3][j-1] > 0 && 2*k[(i+2)%3][j-1]-k[(i+1)%3][j] < 0 ) break;
if ( 2*k[(i+2)%3][j-1] < 0 && 2*k[(i+2)%3][j-1]-k[(i+1)%3][j] > 0 ) break;
k[i%3][j] = 2*k[(i+2)%3][j-1]-k[(i+1)%3][j];
}
if ( j < 101 ) break;
2倍するタイミングで符号が変わったら、breakしてforループから抜ける。
減算するタイミングで符号が変わったら、breakしてforループから抜ける。
forループが最後まで行われなかったら、大外のwhileループからも抜ける。
なぜ、こちらを採用したかと言えば、これが一般的なオーバーフロー検出だと私は考えているからです。
まぁ、どちらにしても同じところまでしか出力しなかったので、採用しなかったオーバーフロー検出も、今回に限って言えばそこまで悪くはなかったとも言えます。
また、Excelの回のときに使った方法、係数を全部足すと1になるという方法も良いかとは思います。
そもそも、このプログラムはExcelのデータの正しさを証明するために作ったいい加減なものですので、まさかブログに載せるとは考えてもいませんでした。
出力をtxtファイルにリダイレクトして、それをブログへ貼りました。
どうぞごらんあれ！
T0(x)=+1
T1(x)=+x
T2(x)=+2x2-1
T3(x)=+4x3-3x
T4(x)=+8x4-8x2+1
T5(x)=+16x5-20x3+5x
T6(x)=+32x6-48x4+18x2-1
T7(x)=+64x7-112x5+56x3-7x
T8(x)=+128x8-256x6+160x4-32x2+1
T9(x)=+256x9-576x7+432x5-120x3+9x
T10(x)=+512x10-1280x8+1120x6-400x4+50x2-1
T11(x)=+1024x11-2816x9+2816x7-1232x5+220x3-11x
T12(x)=+2048x12-6144x10+6912x8-3584x6+840x4-72x2+1
T13(x)=+4096x13-13312x11+16640x9-9984x7+2912x5-364x3+13x
T14(x)=+8192x14-28672x12+39424x10-26880x8+9408x6-1568x4+98x2-1
T15(x)=+16384x15-61440x13+92160x11-70400x9+28800x7-6048x5+560x3-15x
T16(x)=+32768x16-131072x14+212992x12-180224x10+84480x8-21504x6+2688x4-128x2+1
T17(x)=+65536x17-278528x15+487424x13-452608x11+239360x9-71808x7+11424x5-816x3+17x
T18(x)=+131072x18-589824x16+1105920x14-1118208x12+658944x10-228096x8+44352x6-4320x4+162x2-1
T19(x)=+262144x19-1245184x17+2490368x15-2723840x13+1770496x11-695552x9+160512x7-20064x5+1140x3-19x
T20(x)=+524288x20-2621440x18+5570560x16-6553600x14+4659200x12-2050048x10+549120x8-84480x6+6600x4-200x2+1
T21(x)=+1048576x21-5505024x19+12386304x17-15597568x15+12042240x13-5870592x11+1793792x9-329472x7+33264x5-1540x3+21x
T22(x)=+2097152x22-11534336x20+27394048x18-36765696x16+30638080x14-16400384x12+5637632x10-1208064x8+151008x6-9680x4+242x2-1
T23(x)=+4194304x23-24117248x21+60293120x19-85917696x17+76873728x15-44843008x13+17145856x11-4209920x9+631488x7-52624x5+2024x3-23x
T24(x)=+8388608x24-50331648x22+132120576x20-199229440x18+190513152x16-120324096x14+50692096x12-14057472x10+2471040x8-256256x6+13728x4-288x2+1
T25(x)=+16777216x25-104857600x23+288358400x21-458752000x19+466944000x17-317521920x15+146227200x13-45260800x11+9152000x9-1144000x7+80080x5-2600x3+25x
T26(x)=+33554432x26-218103808x24+627048448x22-1049624576x20+1133117440x18-825556992x16+412778496x14-141213696x12+32361472x10-4759040x8+416416x6-18928x4+338x2-1
T27(x)=+67108864x27-452984832x25+1358954496x23-2387607552x21+2724986880x19-2118057984x17+1143078912x15-428654592x13+109983744x11-18670080x9+1976832x7-117936x5+3276x3-27x
T28(x)=+134217728x28-939524096x26+2936012800x24-5402263552x22+6499598336x20-5369233408x18+3111714816x16-1270087680x14+361181184x12-69701632x10+8712704x8-652288x6+25480x4-392x2+1
T29(x)=+268435456x29-1946157056x27+6325010432x25-12163481600x23+15386804224x21-13463453696x19+8341487616x17-3683254272x15+1151016960x13-249387008x11+36095488x9-3281408x7+168896x5-4060x3+29x
T30(x)=+536870912x30-4026531840x28+13589544960x26-27262976000x24+36175872000x22-33426505728x20+22052208640x18-10478223360x16+3572121600x14-859955200x12+141892608x10-15275520x8+990080x6-33600x4+450x2-1
T31(x)=+1073741824x31-8321499136x29+29125246976x27-60850962432x25+84515225600x23-82239815680x21+57567870976x19-29297934336x17+10827497472x15-2870927360x13+533172224x11-66646528x9+5261568x7-236096x5+4960x3-31x
T32(x)=+2147483648x32-17179869184x30+62277025792x28-135291469824x26+196293427200x24-200655503360x22+148562247680x20-80648077312x18+32133218304x16-9313976320x14+1926299648x12-275185664x10+25798656x8-1462272x6+43520x4-512x2+1
T33(x)=+4294967296x33-35433480192x31+132875550720x29-299708186624x27+453437816832x25-485826232320x23+379364311040x21-218864025600x19+93564370944x17-29455450112x15+6723526656x13-1083543552x11+118243840x9-8186112x7+323136x5-5984x3+33x
T34(x)=+8589934592x34-73014444032x32+282930970624x30-661693399040x28+1042167103488x26-1167945891840x24+959384125440x22-586290298880x20+267776819200x18-91044118528x16+22761029632x14-4093386752x12+511673344x10-42170880x8+2108544x6-55488x4+578x2-1
T35(x)=+17179869184x35-150323855360x33+601295421440x31-1456262348800x29+2384042393600x27-2789329600512x25+2404594483200x23-1551944908800x21+754417664000x19-275652608000x17+74977509376x15-14910300160x13+2106890240x11-202585600x9+12403200x7-434112x5+7140x3-35x
T36(x)=+34359738368x36-309237645312x34+1275605286912x32-3195455668224x30+5429778186240x28-6620826304512x26+5977134858240x24-4063273943040x22+2095125626880x20-819082035200x18+240999137280x16-52581629952x14+8307167232x12-916844544x10+66977280x8-2976768x6+69768x4-648x2+1
T37(x)=+68719476736x37-635655159808x35+2701534429184x33-6992206757888x31+12315818721280x29-15625695002624x27+14743599316992x25-10531142369280x23+5742196162560x21-2392581734400x19+757650882560x17-180140769280x15+31524634624x13-3940579328x11+336540160x9-18356736x7+573648x5-8436x3+37x
T38(x)=+137438953472x38-1305670057984x36+5712306503680x34-15260018802688x32+27827093110784x30-36681168191488x28+36108024938496x26-27039419596800x24+15547666268160x22-6880289095680x20+2334383800320x18-601280675840x16+115630899200x14-16188325888x12+1589924864x10-103690752x8+4124064x6-86640x4+722x2-1
T39(x)=+274877906944x39-2680059592704x37+12060268167168x35-33221572034560x33+62646392979456x31-85678155104256x29+87841744879616x27-68822438510592x25+41626474905600x23-19502774353920x21+7061349335040x19-1960212234240x17+411402567680x15-63901286400x13+7120429056x11-543921664x9+26604864x7-746928x5+9880x3-39x
T40(x)=+549755813888x40-5497558138880x38+25426206392320x36-72155450572800x34+140552804761600x32-199183403319296x30+212364657950720x28-173752901959680x26+110292369408000x24-54553214976000x22+21002987765760x20-6254808268800x18+1424085811200x16-243433472000x14+30429184000x12-2677768192x10+156900480x8-5617920x6+106400x4-800x2+1
T41(x)=+1099511627776x41-11269994184704x39+53532472377344x37-156371169312768x35+314327181557760x33-461013199618048x31+510407471005696x29-435347548798976x27+289407177326592x25-150732904857600x23+61508749885440x21-19570965872640x19+4808383856640x17-898269511680x15+124759654400x13-12475965440x11+857722624x9-37840704x7+959728x5-11480x3+41x
T42(x)=+2199023255552x42-23089744183296x40+112562502893568x38-338168545017856x36+700809813688320x34-1062579203997696x32+1219998345330688x30-1083059755548672x28+752567256612864x26-411758179123200x24+177570714746880x22-60144919511040x20+15871575982080x18-3220624834560x16+492952780800x14-55381114880x12+4393213440x10-232581888x8+7537376x6-129360x4+882x2-1
T43(x)=+4398046511104x43-47278999994368x41+236394999971840x39-729869562413056x37+1557990796689408x35-2439485589553152x33+2901009890279424x31-2676526982103040x29+1940482062024704x27-1112923535572992x25+505874334351360x23-181798588907520x21+51314117836800x19-11249633525760x17+1884175073280x15-235521884160x13+21262392320x11-1322886400x9+52915456x7-1218448x5+13244x3-43x
T44(x)=+8796093022208x44-96757023244288x42+495879744126976x40-1572301627719680x38+3454150138396672x36-5579780992794624x34+6864598984556544x32-6573052309536768x30+4964023879598080x28-2978414327758848x26+1423506847825920x24-541167892561920x22+162773155184640x20-38370843033600x18+6988974981120x16-963996549120x14+97905899520x12-7038986240x10+338412800x8-9974272x6+155848x4-968x2+1
T45(x)=+17592186044416x45-197912092999680x43+1039038488248320x41-3380998255411200x39+7638169839206400x37-12717552782278656x35+16168683558666240x33-16047114509352960x31+12604574741299200x29-7897310717542400x27+3959937231224832x25-1588210119475200x23+507344899276800x21-128055803904000x19+25227583488000x17-3812168171520x15+431333683200x13-35340364800x11+1999712000x9-72864000x7+1530144x5-15180x3+45x
T46(x)=+35184372088832x46-404620279021568x44+2174833999740928x42-7257876254949376x40+16848641306132480x38-28889255702953984x36+37917148110127104x34-38958828003262464x32+31782201792135168x30-20758645314682880x28+10898288790208512x26-4599927086776320x24+1555857691115520x22-418884762992640x20+88826010009600x18-14613311324160x16+1826663915520x14-168586629120x12+11038410240x10-484140800x8+13034560x6-186208x4+1058x2-1
T47(x)=+70368744177664x47-826832744087552x45+4547580092481536x43-15554790998147072x41+37078280867676160x39-65416681245114368x37+88551849002532864x35-94086339565191168x33+79611518093623296x31-54121865370664960x29+29693888297959424x27-13159791404777472x25+4699925501706240x23-1345114425262080x21+305707823923200x19-54454206136320x17+7465496002560x15-768506941440x13+57417185280x11-2967993600x9+98933120x7-1902560x5+17296x3-47x
T48(x)=+140737488355328x48-1688849860263936x46+9499780463984640x44-33284415996035072x42+81414437990301696x40-147682003796361216x38+205992953708019712x36-226089827240509440x34+198181864190509056x32-140025932533465088x30+80146421910601728x28-37217871599763456x26+13999778090188800x24-4246086541639680x22+1030300410839040x20-197734422282240x18+29544303329280x16-3363677798400x14+283420999680x12-16974397440x10+682007040x8-16839680x6+220800x4-1152x2+1
T49(x)=+281474976710656x49-3448068464705536x47+19826393672056832x45-71116412084551680x43+178383666978750464x41-332442288460398592x39+477402588661153792x37-540731503483551744x35+490450067946209280x33-359663383160553472x31+214414709191868416x29-104129631497486336x27+41159347585155072x25-13192098584985600x23+3405715246940160x21-701176668487680x19+113542812794880x17-14192851599360x15+1335348940800x13-91365980160x11+4332007680x9-132612480x7+2344160x5-19600x3+49x
T50(x)=+562949953421312x50-7036874417766400x48+41341637204377600x46-151732604633088000x44+390051749953536000x42-746299014911098880x40+1102487181118668800x38-1287455960675123200x36+1206989963132928000x34-917508630511616000x32+568855350917201920x30-288405684905574400x28+119536566770073600x26-40383975260160000x24+11057517035520000x22-2432653747814400x20+424820047872000x18-57930006528000x16+6034375680000x14-466152960000x12+25638412800x10-947232000x8+21528000x6-260000x4+1250x2-1
T51(x)=+1125899906842624x51-14355223812243456x49+86131342873460736x47-323291602938232832x45+851219911991623680x43-1670981696800948224x41+2537416650697736192x39-3052314510011400192x37+2954711429749407744x35-2325467328969441280x33+1497374084994957312x31-791226079003017216x29+343202765037633536x27-121927298105475072x25+35307132656025600x23-8271022742568960x21+1550816764231680x19-229402825850880x17+26261602959360x15-2267654860800x13+142642805760x11-6226471680x9+175668480x7-2864160x5+22100x3-51x
T52(x)=+2251799813685248x52-29273397577908224x50+179299560164687872x48-687924843080843264x46+1854172428616335360x44-3732015143555432448x42+5821132316306571264x40-7207116201141469184x38+7196878820173938688x36-5857924621071810560x34+3912256800501530624x32-2151307508923236352x30+974811214980841472x28-363391162981023744x26+110998240572211200x24-27599562520657920x22+5534287276277760x20-883625699573760x18+110453212446720x16-10569685401600x14+751438571520x12-38091356160x10+1298568960x8-27256320x6+304200x4-1352x2+1
T1(x)=+x
T2(x)=+2x2-1
T3(x)=+4x3-3x
T4(x)=+8x4-8x2+1
T5(x)=+16x5-20x3+5x
T6(x)=+32x6-48x4+18x2-1
T7(x)=+64x7-112x5+56x3-7x
T8(x)=+128x8-256x6+160x4-32x2+1
T9(x)=+256x9-576x7+432x5-120x3+9x
T10(x)=+512x10-1280x8+1120x6-400x4+50x2-1
T11(x)=+1024x11-2816x9+2816x7-1232x5+220x3-11x
T12(x)=+2048x12-6144x10+6912x8-3584x6+840x4-72x2+1
T13(x)=+4096x13-13312x11+16640x9-9984x7+2912x5-364x3+13x
T14(x)=+8192x14-28672x12+39424x10-26880x8+9408x6-1568x4+98x2-1
T15(x)=+16384x15-61440x13+92160x11-70400x9+28800x7-6048x5+560x3-15x
T16(x)=+32768x16-131072x14+212992x12-180224x10+84480x8-21504x6+2688x4-128x2+1
T17(x)=+65536x17-278528x15+487424x13-452608x11+239360x9-71808x7+11424x5-816x3+17x
T18(x)=+131072x18-589824x16+1105920x14-1118208x12+658944x10-228096x8+44352x6-4320x4+162x2-1
T19(x)=+262144x19-1245184x17+2490368x15-2723840x13+1770496x11-695552x9+160512x7-20064x5+1140x3-19x
T20(x)=+524288x20-2621440x18+5570560x16-6553600x14+4659200x12-2050048x10+549120x8-84480x6+6600x4-200x2+1
T21(x)=+1048576x21-5505024x19+12386304x17-15597568x15+12042240x13-5870592x11+1793792x9-329472x7+33264x5-1540x3+21x
T22(x)=+2097152x22-11534336x20+27394048x18-36765696x16+30638080x14-16400384x12+5637632x10-1208064x8+151008x6-9680x4+242x2-1
T23(x)=+4194304x23-24117248x21+60293120x19-85917696x17+76873728x15-44843008x13+17145856x11-4209920x9+631488x7-52624x5+2024x3-23x
T24(x)=+8388608x24-50331648x22+132120576x20-199229440x18+190513152x16-120324096x14+50692096x12-14057472x10+2471040x8-256256x6+13728x4-288x2+1
T25(x)=+16777216x25-104857600x23+288358400x21-458752000x19+466944000x17-317521920x15+146227200x13-45260800x11+9152000x9-1144000x7+80080x5-2600x3+25x
T26(x)=+33554432x26-218103808x24+627048448x22-1049624576x20+1133117440x18-825556992x16+412778496x14-141213696x12+32361472x10-4759040x8+416416x6-18928x4+338x2-1
T27(x)=+67108864x27-452984832x25+1358954496x23-2387607552x21+2724986880x19-2118057984x17+1143078912x15-428654592x13+109983744x11-18670080x9+1976832x7-117936x5+3276x3-27x
T28(x)=+134217728x28-939524096x26+2936012800x24-5402263552x22+6499598336x20-5369233408x18+3111714816x16-1270087680x14+361181184x12-69701632x10+8712704x8-652288x6+25480x4-392x2+1
T29(x)=+268435456x29-1946157056x27+6325010432x25-12163481600x23+15386804224x21-13463453696x19+8341487616x17-3683254272x15+1151016960x13-249387008x11+36095488x9-3281408x7+168896x5-4060x3+29x
T30(x)=+536870912x30-4026531840x28+13589544960x26-27262976000x24+36175872000x22-33426505728x20+22052208640x18-10478223360x16+3572121600x14-859955200x12+141892608x10-15275520x8+990080x6-33600x4+450x2-1
T31(x)=+1073741824x31-8321499136x29+29125246976x27-60850962432x25+84515225600x23-82239815680x21+57567870976x19-29297934336x17+10827497472x15-2870927360x13+533172224x11-66646528x9+5261568x7-236096x5+4960x3-31x
T32(x)=+2147483648x32-17179869184x30+62277025792x28-135291469824x26+196293427200x24-200655503360x22+148562247680x20-80648077312x18+32133218304x16-9313976320x14+1926299648x12-275185664x10+25798656x8-1462272x6+43520x4-512x2+1
T33(x)=+4294967296x33-35433480192x31+132875550720x29-299708186624x27+453437816832x25-485826232320x23+379364311040x21-218864025600x19+93564370944x17-29455450112x15+6723526656x13-1083543552x11+118243840x9-8186112x7+323136x5-5984x3+33x
T34(x)=+8589934592x34-73014444032x32+282930970624x30-661693399040x28+1042167103488x26-1167945891840x24+959384125440x22-586290298880x20+267776819200x18-91044118528x16+22761029632x14-4093386752x12+511673344x10-42170880x8+2108544x6-55488x4+578x2-1
T35(x)=+17179869184x35-150323855360x33+601295421440x31-1456262348800x29+2384042393600x27-2789329600512x25+2404594483200x23-1551944908800x21+754417664000x19-275652608000x17+74977509376x15-14910300160x13+2106890240x11-202585600x9+12403200x7-434112x5+7140x3-35x
T36(x)=+34359738368x36-309237645312x34+1275605286912x32-3195455668224x30+5429778186240x28-6620826304512x26+5977134858240x24-4063273943040x22+2095125626880x20-819082035200x18+240999137280x16-52581629952x14+8307167232x12-916844544x10+66977280x8-2976768x6+69768x4-648x2+1
T37(x)=+68719476736x37-635655159808x35+2701534429184x33-6992206757888x31+12315818721280x29-15625695002624x27+14743599316992x25-10531142369280x23+5742196162560x21-2392581734400x19+757650882560x17-180140769280x15+31524634624x13-3940579328x11+336540160x9-18356736x7+573648x5-8436x3+37x
T38(x)=+137438953472x38-1305670057984x36+5712306503680x34-15260018802688x32+27827093110784x30-36681168191488x28+36108024938496x26-27039419596800x24+15547666268160x22-6880289095680x20+2334383800320x18-601280675840x16+115630899200x14-16188325888x12+1589924864x10-103690752x8+4124064x6-86640x4+722x2-1
T39(x)=+274877906944x39-2680059592704x37+12060268167168x35-33221572034560x33+62646392979456x31-85678155104256x29+87841744879616x27-68822438510592x25+41626474905600x23-19502774353920x21+7061349335040x19-1960212234240x17+411402567680x15-63901286400x13+7120429056x11-543921664x9+26604864x7-746928x5+9880x3-39x
T40(x)=+549755813888x40-5497558138880x38+25426206392320x36-72155450572800x34+140552804761600x32-199183403319296x30+212364657950720x28-173752901959680x26+110292369408000x24-54553214976000x22+21002987765760x20-6254808268800x18+1424085811200x16-243433472000x14+30429184000x12-2677768192x10+156900480x8-5617920x6+106400x4-800x2+1
T41(x)=+1099511627776x41-11269994184704x39+53532472377344x37-156371169312768x35+314327181557760x33-461013199618048x31+510407471005696x29-435347548798976x27+289407177326592x25-150732904857600x23+61508749885440x21-19570965872640x19+4808383856640x17-898269511680x15+124759654400x13-12475965440x11+857722624x9-37840704x7+959728x5-11480x3+41x
T42(x)=+2199023255552x42-23089744183296x40+112562502893568x38-338168545017856x36+700809813688320x34-1062579203997696x32+1219998345330688x30-1083059755548672x28+752567256612864x26-411758179123200x24+177570714746880x22-60144919511040x20+15871575982080x18-3220624834560x16+492952780800x14-55381114880x12+4393213440x10-232581888x8+7537376x6-129360x4+882x2-1
T43(x)=+4398046511104x43-47278999994368x41+236394999971840x39-729869562413056x37+1557990796689408x35-2439485589553152x33+2901009890279424x31-2676526982103040x29+1940482062024704x27-1112923535572992x25+505874334351360x23-181798588907520x21+51314117836800x19-11249633525760x17+1884175073280x15-235521884160x13+21262392320x11-1322886400x9+52915456x7-1218448x5+13244x3-43x
T44(x)=+8796093022208x44-96757023244288x42+495879744126976x40-1572301627719680x38+3454150138396672x36-5579780992794624x34+6864598984556544x32-6573052309536768x30+4964023879598080x28-2978414327758848x26+1423506847825920x24-541167892561920x22+162773155184640x20-38370843033600x18+6988974981120x16-963996549120x14+97905899520x12-7038986240x10+338412800x8-9974272x6+155848x4-968x2+1
T45(x)=+17592186044416x45-197912092999680x43+1039038488248320x41-3380998255411200x39+7638169839206400x37-12717552782278656x35+16168683558666240x33-16047114509352960x31+12604574741299200x29-7897310717542400x27+3959937231224832x25-1588210119475200x23+507344899276800x21-128055803904000x19+25227583488000x17-3812168171520x15+431333683200x13-35340364800x11+1999712000x9-72864000x7+1530144x5-15180x3+45x
T46(x)=+35184372088832x46-404620279021568x44+2174833999740928x42-7257876254949376x40+16848641306132480x38-28889255702953984x36+37917148110127104x34-38958828003262464x32+31782201792135168x30-20758645314682880x28+10898288790208512x26-4599927086776320x24+1555857691115520x22-418884762992640x20+88826010009600x18-14613311324160x16+1826663915520x14-168586629120x12+11038410240x10-484140800x8+13034560x6-186208x4+1058x2-1
T47(x)=+70368744177664x47-826832744087552x45+4547580092481536x43-15554790998147072x41+37078280867676160x39-65416681245114368x37+88551849002532864x35-94086339565191168x33+79611518093623296x31-54121865370664960x29+29693888297959424x27-13159791404777472x25+4699925501706240x23-1345114425262080x21+305707823923200x19-54454206136320x17+7465496002560x15-768506941440x13+57417185280x11-2967993600x9+98933120x7-1902560x5+17296x3-47x
T48(x)=+140737488355328x48-1688849860263936x46+9499780463984640x44-33284415996035072x42+81414437990301696x40-147682003796361216x38+205992953708019712x36-226089827240509440x34+198181864190509056x32-140025932533465088x30+80146421910601728x28-37217871599763456x26+13999778090188800x24-4246086541639680x22+1030300410839040x20-197734422282240x18+29544303329280x16-3363677798400x14+283420999680x12-16974397440x10+682007040x8-16839680x6+220800x4-1152x2+1
T49(x)=+281474976710656x49-3448068464705536x47+19826393672056832x45-71116412084551680x43+178383666978750464x41-332442288460398592x39+477402588661153792x37-540731503483551744x35+490450067946209280x33-359663383160553472x31+214414709191868416x29-104129631497486336x27+41159347585155072x25-13192098584985600x23+3405715246940160x21-701176668487680x19+113542812794880x17-14192851599360x15+1335348940800x13-91365980160x11+4332007680x9-132612480x7+2344160x5-19600x3+49x
T50(x)=+562949953421312x50-7036874417766400x48+41341637204377600x46-151732604633088000x44+390051749953536000x42-746299014911098880x40+1102487181118668800x38-1287455960675123200x36+1206989963132928000x34-917508630511616000x32+568855350917201920x30-288405684905574400x28+119536566770073600x26-40383975260160000x24+11057517035520000x22-2432653747814400x20+424820047872000x18-57930006528000x16+6034375680000x14-466152960000x12+25638412800x10-947232000x8+21528000x6-260000x4+1250x2-1
T51(x)=+1125899906842624x51-14355223812243456x49+86131342873460736x47-323291602938232832x45+851219911991623680x43-1670981696800948224x41+2537416650697736192x39-3052314510011400192x37+2954711429749407744x35-2325467328969441280x33+1497374084994957312x31-791226079003017216x29+343202765037633536x27-121927298105475072x25+35307132656025600x23-8271022742568960x21+1550816764231680x19-229402825850880x17+26261602959360x15-2267654860800x13+142642805760x11-6226471680x9+175668480x7-2864160x5+22100x3-51x
T52(x)=+2251799813685248x52-29273397577908224x50+179299560164687872x48-687924843080843264x46+1854172428616335360x44-3732015143555432448x42+5821132316306571264x40-7207116201141469184x38+7196878820173938688x36-5857924621071810560x34+3912256800501530624x32-2151307508923236352x30+974811214980841472x28-363391162981023744x26+110998240572211200x24-27599562520657920x22+5534287276277760x20-883625699573760x18+110453212446720x16-10569685401600x14+751438571520x12-38091356160x10+1298568960x8-27256320x6+304200x4-1352x2+1
64ビット整数で52次が限界のようですね。
128ビットCPUとか、
128ビットOSとか、
128ビットプログラミング言語とか、
そういう時代がそろそろ来てもよさそうな気がしますが、
そこまで求めないのだろうか。
本気でやる人は、CPUが128ビットになる前に、プログラミングで多倍長演算させてでも、桁数を増やしてくるでしょうね。
今回はやりませんよ。
ではでは