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Pi to 5,000 places from a 16,000 character 1401
by Ed Thelen January 28, 2005
this is my first 1401 program, it ain't pretty, but it works
the language is AutocoderA bit of background - this method of approximating Pi is from Machin, about 300 years old.
Pi = 16*arctan(1/5)-4*arctan(1/239)
His sequence converges relatively rapidly compared to older schemes, and was reasonable to use through the 1960s.Machin's method uses the previously discovered series for arctan:
arctan(x) = x - x3/3 + x5/5 - x7/7 + ...
which is conveniently calculated in three large fields.
Note that the three 5050 character fields are really crammed into the 16,000 character maximum memory for a 1401. Index registers X2 and X3 and the card punch buffer have working storage and instructions to make room - The printer buffer is free so the program can run again after the halt :-)) The AutoCoder source code is here
The AutoCoder assembly listing is here
The "object deck", program cards to be loaded into the 1401 is here .
And the result is here. :-))Comment by Van Snyder
JOB SubRoutine PI * SO - WE HAVE 5,000 VALID PI CHARACTERS :-)) * NOW LETS PRETTY IT UP A BIT, ORGANIZE SUBROUTINES * CALL A SUBROUTINE DEPENDS UPON * IF THE INDEX REGISTER OPTION IS INSTALLED, * ANY BRANCH LEAVES THE ADDRESS OF THE NEXT SEQUENTIAL * INSTRUCTION IF THE BRANCH NOT TAKEN IN THE * B ADDRESS REGISTER :-)) * THE ARCTAN ROUTINE IS WRITTEN AS A CRUDE SUBROUTINE. * CTL 6611 *6=16,000C;6=16,000T;1=OBJDECK;,1=MODADD * 1 2 3 4 5 6 7 8 *78901234567890123456789012345678901234567890123456789012345678901234567890 * label | op | OPERATION |xxxxxxx ORG 87 X1 DSA 0 index register 1 ORG 92 * X2 DSA 0 index register 2 * ORG 97 * X3 DSA 0 index register 3 * * start storage areas * small areas * ORG 100 * PUT SMALL STUFF IN PUNCH AREA ITCNT DCW 000000 * ITERATION COUNTER, STARTS AT ZERO DIVBAS DCW 000000 * DIVISOR FOR BASET IX2P1 DCW 000000 * ITCNT times 2 plus 1, DIVISOR FOR INTER C0 DCW 000000 C1 DCW 000001 C2 DCW 000002 C25 DCW 000025 * 5^2 C239P2 DCW 057121 * 239^2 CNTZRO DCW 000000 * COUNT LEADING ZEROS, DONE YET? CF DCW 1 * "1"=ADD, "0" OR OTHER = SUBTRACT LC0L9 DCW 000000000 * * COMPUTE 16*ARCTAN(1/5) START MCW @+@,ACCUM&5049 * ZERO ACCUMULATOR, FORCE POSITIVE MCW @0@,ACCUM&5048 B START1 * SKIP PRINT AREA ORG 335 START1 MCW ACCUM&5048,ACCUM&5047 MCW @0@,BASET&5048 * ZERO BASE MCW BASET&5048,BASET&5047 * BIG REGISTERS SET MCW @80@,BASET&8 * SET A HIGH ORDER TO 1*5*16 MCW @1@,CF * SET NEXT TO ADD MCW C25,DIVBAS * SET NEXT TO ADD B ATAN *** CALL ATAN SUBROUTINE * * NOW, ADD MINUS 4*ARCTAN(1/239) TO ABOVE 16*ARCTAN(1/5) * DO NOT ZERO ACCUMULATOR START2 MCW @0@,BASET&5048 * ZERO BASE MCW BASET&5048,BASET&5047 MCW @956@,BASET&8 * SET A HIGH ORDER TO 1*4*239 MCW @0@,CF * SET NEXT SUBTRACT MCW C239P2,DIVBAS * SET SET BASE DIVISOR B ATAN *** CALL ATAN SUBROUTINE * WE NOW HAVE PI IN MEMORY, PRINT IT OUT B PRINT * CALL PRINT ROUTINE HALT H START *** HALT HERE - * ATAN SBR ATANX&3 * SET RETURN MCW C0,CNTZRO * CLEAR LEADING ZERO COUNTER MCW C0,ITCNT * ZERO ITERATION CNTR MCW C1,IX2P1 * INIT INTER DIVISOR MCW @000@,X1 * CLEAR ASSOCIATED INDEX REGISTER ATANL MCW @+@,BASET&5049 * SET POSITIVE SIGN D DIVBAS,BASET&6 * TRIAL ARITH MCW BASET&5042,BASET&5049 * SHIFT QUOTIENT MCW LC0L9,BASET&6 * REMOVE UNSHIFTED MCW BASET&5049,INTER&5049 * MOVE QUOTIENT TO NEXT MCW @+@,INTER&5049 * SET POSITIVE SIGN D IX2P1,INTER&6 DIVIDE INTERMEDIATE MCW INTER&5042,INTER&5049 * SHIFT INTERMEDIATE MCW LC0L9,INTER&6 * REMOVE UNSHIFTED MCW @+@,INTER&5049 * SET UP BASE BCE ATANA,CF,1 * COMPARE FOR ADD ATANS MCW @1@,CF * SET NEXT ADD S INTER&5049,ACCUM&5049 * DO THE SUBTRACTION B ATANW * GO TO WRAP-UP ATANA A INTER&5049,ACCUM&5049 * ADD TO ACCUMULATOR MCW @0@,CF * SET NEXT SUBTRACT ATANW A C1,ITCNT * INC ITERATION COUNT A C2,IX2P1 * FORM NEXT DIVISOR ATANB BCE ATANI,BASET&X1,0 * CHAR = 0, DONE? B ATANL * GO DO ANOTHER LOOP PASS ATANI A C1,CNTZRO * INCREMENT # LEADING ZEROS SBR X1,1&X1 * INCREMENT X1 C CNTZRO,@5049@ * CHECK END BU ATANB * NO, CHECK MORE ZEROS ATANX B 0 *** RETURN TO CALL * * * ------------------------------------------------------------- * * NOW WE PRINT THAT BABY OUT :-)) * PROPOSED FORMAT * 2 * 0 1 2 3 4 5 6 7 8 * 12345678901234567890123456789012345678901234567890123456789012345678901234567890 * 3. * 10E-00000 * + NNNNNNNNNN NNNNNNNNNN NNNNNNNNNN NNNNNNNNNN NNNNNNNNNN * 10E-00050 * + NNNNNNNNNN NNNNNNNNNN NNNNNNNNNN NNNNNNNNNN NNNNNNNNNN * 10E-00100 PRINT SBR PRINTX&3 PRINTD MCW @008@,X1 * CLEAR PICKUP X MCW C0,ITCNT * CLEAR TOTAL CH MOVED CS 332 * start clearing down to 200, PRINT AREA CS MCW @. * 10E-00000@,269 SW 265 * SET WORD MARK IN EXPONENT FIELD MCW ACCUM&X1,256 MOVE 5TH FIELD SBR X1,10&X1 * STEP X1 W *write the print area to the printer * PR1LOP CS 252 * CLEAR PREVIOUS NUMERICS SW 203,214 * SET WORD MARKS FOR B FIELD SW 225,236 SW 247,265 A @00050@,269 * ADD 50 TO EXPONENT MCW ACCUM&X1,212 MOVE 1ST FIELD SBR X1,10&X1 * STEP X1 MCW ACCUM&X1,223 MOVE 2ND FIELD SBR X1,10&X1 * STEP X1 MCW ACCUM&X1,234 MOVE 3RD FIELD SBR X1,10&X1 * STEP X1 MCW ACCUM&X1,245 MOVE 4TH FIELD SBR X1,10&X1 * STEP X1 MCW ACCUM&X1,256 MOVE 5TH FIELD SBR X1,10&X1 * STEP X1 W *write the print area to the printer A @000050@,ITCNT * ADD 50 TO TOTAL DIGITS C @005000@,ITCNT * TEST FOR END BL PR1LOP * END IS HIGHER THAN COUNT PRINTX B 0 *** RETURN TO CALL * * * large areas Ron Mak says that blanks process as zeros * ORG 780 *FLDLEN EQU 5050 * LENGTH OF THE 3 BIG FIELDS BASET DA 1X5050,C * BASE, 1ST DIVISION HERE INTER DA 1X5050,C * INTERMEDIATE BUFFER, 2ND DIVISION HERE ACCUM DA 1X5050,C * ACCUMULATOR, ANSWER IS FORMED HERE * 1 2 3 4 5 6 7 8 *78901234567890123456789012345678901234567890123456789012345678901234567890 * label | op | OPERATION |xxxxxxx * *** Nov 15 * Machin's method - * PI = 16arctan(1/5) - 4arctan(1/239) * An arctan series is 1/n - 1/(3xn^3) + 1/(5xn^5) - ... * *** Nov 16 * In any case, *Lets do arctan(x), x = 1/5 * 0.19739555984988075837004976519479 *then 16ARCTAN(1/5) * 3.1583289575980921339207962431166 * then aectan(1/239) * 0.0041840760020747238645382149592855 * times 4 * 0.016736304008298895458152859837142 * Pi = * 3.1415926535897932384626433832795 * :-)) * * Set N to 1/x = 5 (this is the fancy footwork :-)) * * 1) a) form three large as practical equal sized areas in memory. * lets call them "Accumulator", ACCUM * "Intermediate", INTER * "Base of next term" BASET * b) form smaller working areas * iteration counter ITCNT used for observation * starts at zero, counts up after a pass * divisor of BASET, DIVBAS, usually 5^2 or 239^2 * 5^2 = 25, 239^2 = 57121 * add/subtract control flag, CF , toggled after a pass * 0 means add, other is subtract * divisor of INTER is IX2P1 , ITCNT times 2 plus 1 * * 2) figure where the decimal point will be * (same relative place in each area) * ( several characters or words to the "right" of * the top significance.) * * 3) a) zero the ACCUM * b) ( INTER does not need initialization) * c) set X (5.0 in our example) into BASET * d) set counter N to 1 * * 4) While BASET is non-zero * a) Divide BASET by X^2 ( 25 in our example ) giving BASET * b) Divide BASET by (2xN - 1) giving INTER * c) If N is odd, add Intermediate to Accumulator * else, subtract Intermediate from Accumulator * d) Add 1 to N (assure that machine can divide by 2N) * * 5) Accumulator is a fine approximation of arctan(x) ;-)) END START * LAST CARD IN DECK, TRANSFER ADDRESS
And the result (simulation running time about 7 seconds on a fast PC)000000003. * 10E-00000 1415926535 8979323846 2643383279 5028841971 6939937510. * 10E-00050 5820974944 5923078164 0628620899 8628034825 3421170679. * 10E-00100 8214808651 3282306647 0938446095 5058223172 5359408128. * 10E-00150 4811174502 8410270193 8521105559 6446229489 5493038196. * 10E-00200 4428810975 6659334461 2847564823 3786783165 2712019091. * 10E-00250 4564856692 3460348610 4543266482 1339360726 0249141273. * 10E-00300 7245870066 0631558817 4881520920 9628292540 9171536436. * 10E-00350 7892590360 0113305305 4882046652 1384146951 9415116094. * 10E-00400 3305727036 5759591953 0921861173 8193261179 3105118548. * 10E-00450 0744623799 6274956735 1885752724 8912279381 8301194912. * 10E-00500 9833673362 4406566430 8602139494 6395224737 1907021798. * 10E-00550 6094370277 0539217176 2931767523 8467481846 7669405132. * 10E-00600 0005681271 4526356082 7785771342 7577896091 7363717872. * 10E-00650 1468440901 2249534301 4654958537 1050792279 6892589235. * 10E-00700 4201995611 2129021960 8640344181 5981362977 4771309960. * 10E-00750 5187072113 4999999837 2978049951 0597317328 1609631859. * 10E-00800 5024459455 3469083026 4252230825 3344685035 2619311881. * 10E-00850 7101000313 7838752886 5875332083 8142061717 7669147303. * 10E-00900 5982534904 2875546873 1159562863 8823537875 9375195778. * 10E-00950 1857780532 1712268066 1300192787 6611195909 2164201989. * 10E-01000 3809525720 1065485863 2788659361 5338182796 8230301952. * 10E-01050 0353018529 6899577362 2599413891 2497217752 8347913151. * 10E-01100 5574857242 4541506959 5082953311 6861727855 8890750983. * 10E-01150 8175463746 4939319255 0604009277 0167113900 9848824012. * 10E-01200 8583616035 6370766010 4710181942 9555961989 4676783744. * 10E-01250 9448255379 7747268471 0404753464 6208046684 2590694912. * 10E-01300 9331367702 8989152104 7521620569 6602405803 8150193511. * 10E-01350 2533824300 3558764024 7496473263 9141992726 0426992279. * 10E-01400 6782354781 6360093417 2164121992 4586315030 2861829745. * 10E-01450 5570674983 8505494588 5869269956 9092721079 7509302955. * 10E-01500 3211653449 8720275596 0236480665 4991198818 3479775356. * 10E-01550 6369807426 5425278625 5181841757 4672890977 7727938000. * 10E-01600 8164706001 6145249192 1732172147 7235014144 1973568548. * 10E-01650 1613611573 5255213347 5741849468 4385233239 0739414333. * 10E-01700 4547762416 8625189835 6948556209 9219222184 2725502542. * 10E-01750 5688767179 0494601653 4668049886 2723279178 6085784383. * 10E-01800 8279679766 8145410095 3883786360 9506800642 2512520511. * 10E-01850 7392984896 0841284886 2694560424 1965285022 2106611863. * 10E-01900 0674427862 2039194945 0471237137 8696095636 4371917287. * 10E-01950 4677646575 7396241389 0865832645 9958133904 7802759009. * 10E-02000 9465764078 9512694683 9835259570 9825822620 5224894077. * 10E-02050 2671947826 8482601476 9909026401 3639443745 5305068203. * 10E-02100 4962524517 4939965143 1429809190 6592509372 2169646151. * 10E-02150 5709858387 4105978859 5977297549 8930161753 9284681382. * 10E-02200 6868386894 2774155991 8559252459 5395943104 9972524680. * 10E-02250 8459872736 4469584865 3836736222 6260991246 0805124388. * 10E-02300 4390451244 1365497627 8079771569 1435997700 1296160894. * 10E-02350 4169486855 5848406353 4220722258 2848864815 8456028506. * 10E-02400 0168427394 5226746767 8895252138 5225499546 6672782398. * 10E-02450 6456596116 3548862305 7745649803 5593634568 1743241125. * 10E-02500 1507606947 9451096596 0940252288 7971089314 5669136867. * 10E-02550 2287489405 6010150330 8617928680 9208747609 1782493858. * 10E-02600 9009714909 6759852613 6554978189 3129784821 6829989487. * 10E-02650 2265880485 7564014270 4775551323 7964145152 3746234364. * 10E-02700 5428584447 9526586782 1051141354 7357395231 1342716610. * 10E-02750 2135969536 2314429524 8493718711 0145765403 5902799344. * 10E-02800 0374200731 0578539062 1983874478 0847848968 3321445713. * 10E-02850 8687519435 0643021845 3191048481 0053706146 8067491927. * 10E-02900 8191197939 9520614196 6342875444 0643745123 7181921799. * 10E-02950 9839101591 9561814675 1426912397 4894090718 6494231961. * 10E-03000 5679452080 9514655022 5231603881 9301420937 6213785595. * 10E-03050 6638937787 0830390697 9207734672 2182562599 6615014215. * 10E-03100 0306803844 7734549202 6054146659 2520149744 2850732518. * 10E-03150 6660021324 3408819071 0486331734 6496514539 0579626856. * 10E-03200 1005508106 6587969981 6357473638 4052571459 1028970641. * 10E-03250 4011097120 6280439039 7595156771 5770042033 7869936007. * 10E-03300 2305587631 7635942187 3125147120 5329281918 2618612586. * 10E-03350 7321579198 4148488291 6447060957 5270695722 0917567116. * 10E-03400 7229109816 9091528017 3506712748 5832228718 3520935396. * 10E-03450 5725121083 5791513698 8209144421 0067510334 6711031412. * 10E-03500 6711136990 8658516398 3150197016 5151168517 1437657618. * 10E-03550 3515565088 4909989859 9823873455 2833163550 7647918535. * 10E-03600 8932261854 8963213293 3089857064 2046752590 7091548141. * 10E-03650 6549859461 6371802709 8199430992 4488957571 2828905923. * 10E-03700 2332609729 9712084433 5732654893 8239119325 9746366730. * 10E-03750 5836041428 1388303203 8249037589 8524374417 0291327656. * 10E-03800 1809377344 4030707469 2112019130 2033038019 7621101100. * 10E-03850 4492932151 6084244485 9637669838 9522868478 3123552658. * 10E-03900 2131449576 8572624334 4189303968 6426243410 7732269780. * 10E-03950 2807318915 4411010446 8232527162 0105265227 2111660396. * 10E-04000 6655730925 4711055785 3763466820 6531098965 2691862056. * 10E-04050 4769312570 5863566201 8558100729 3606598764 8611791045. * 10E-04100 3348850346 1136576867 5324944166 8039626579 7877185560. * 10E-04150 8455296541 2665408530 6143444318 5867697514 5661406800. * 10E-04200 7002378776 5913440171 2749470420 5622305389 9456131407. * 10E-04250 1127000407 8547332699 3908145466 4645880797 2708266830. * 10E-04300 6343285878 5698305235 8089330657 5740679545 7163775254. * 10E-04350 2021149557 6158140025 0126228594 1302164715 5097925923. * 10E-04400 0990796547 3761255176 5675135751 7829666454 7791745011. * 10E-04450 2996148903 0463994713 2962107340 4375189573 5961458901. * 10E-04500 9389713111 7904297828 5647503203 1986915140 2870808599. * 10E-04550 0480109412 1472213179 4764777262 2414254854 5403321571. * 10E-04600 8530614228 8137585043 0633217518 2979866223 7172159160. * 10E-04650 7716692547 4873898665 4949450114 6540628433 6639379003. * 10E-04700 9769265672 1463853067 3609657120 9180763832 7166416274. * 10E-04750 8888007869 2560290228 4721040317 2118608204 1900042296. * 10E-04800 6171196377 9213375751 1495950156 6049631862 9472654736. * 10E-04850 4252308177 0367515906 7350235072 8354056704 0386743513. * 10E-04900 6222247715 8915049530 9844489333 0963408780 7693259939. * 10E-04950 7805419341 4473774418 4263129860 8099888687 4132604721. * 10E-05000