Happy Pi Day



#2

For the last few weekends I have been exploring faster and faster ways to compute Pi to 1000 digits on the 71B. The 71B is an incredible machine. I am always amazed at what this little hand-held computer can do.

If you are more interested in newer calculators, then check out my 50g Pi programs:

NOTE: This is a subset of a larger article I will finish someday. I will have more results, hopefully before the next Pi day (22/7).


Draft


Subject: 1000 Digits of Pi on Pre-1990 Hand-Held Calculators and Computers

The genesis of this article originated with the following statement from Gene Wright, "Looks like the race to 1000 digits on a machine prior to 1990 now belongs to TI? Can we beat the 5.81 hours on the TI95 with the HP71B?" (http://www.hpmuseum.org/cgi-sys/cgiwrap/hpmuseum/archv014.cgi?read=53988)

As of early 2009 this challenge has remained unanswered--until now...

I've spent the last few weekends exploring various ways to meet this challenge. The most amazing result is Ron Knapp's RPN code running on the 71B (entry 5). The quest to best 5.81 hours could have ended in 1985 with a HP-41 Translator Pac equipped 71B!


   Algorithm      Platform  Language  Memory  Digits  Time(s)                      
------------- -------- -------- ------ ------ -------
1. Atan Stormer1 HP-71B Forth 3360 B 1002 3172 (0.88 hr) (53.87 min)
2. Atan Stormer HP-71B Forth 3360 B 1002 3434 (0.95 hr) (57.23 min)
3. Atan Machin1 HP-71B Forth 3360 B 1003 3561 (0.99 hr) (59.35 min)
4. Atan Machin HP-71B Forth 3360 B 1003 3825 (1.06 hr) (63.75 min)
5. Atan Machin2 HP-71B RPN 220 R? 1001 6034 (1.68 hr)
6. Spigot HP-71B Forth 17.1 K 1002 6720 (1.87 hr)
7. Atan Machin3 TI-95 AOS 1001 ~20916 (5.81 hr)
8. Atan Machin2 HP-41CX RPN 1540 B 1001 ~30600 (8.50 hr)
11000 decimal digits represented as base 220. 2Ron Knapp implementation. 3Bob Fruit implementation.

Table 1 Summary of Results < 9 Hours


Entries 1 and 3 are the same as 2 and 4 respectively, just timed differently. All four were computed base 220 to get the most out of the 20-bit Saturn architecture, however entries 1 and 3 stopped the clock after 168 base 220 digits were computed, whereas entries 2 and 4 stopped the clock after the number was converted to base 10. Since it takes ~3.5 minutes just to display the results and without any clear rules on the subject I opted to report both. I expect that most will agree that 1000 decimal digits of Pi must be represented as 1000 decimal digits making entry 2 the fasted pre-1990 hand-held leader.

Entries 5 and 8 are based on Ron Knapp's original program published in 1981 (V8N6P69 of the PPC Calculator Journal).

Output and Code



  1. HP-71B Arctan Forth
  2. HP-71B Spigot Forth
  3. HP-71B Arctan RPN
  4. TI-95 Arctan AOS
  5. HP-41CX Arctan RPN

1. HP-71B Arctan Forth

Output of entries 2 and 4 (1 and 3 are the same but have different times reported. (see above)):

3.
141592653589793238462643383279502884197169399375105820974944592307816406286208
998628034825342117067982148086513282306647093844609550582231725359408128481117
450284102701938521105559644622948954930381964428810975665933446128475648233786
783165271201909145648566923460348610454326648213393607260249141273724587006606
315588174881520920962829254091715364367892590360011330530548820466521384146951
941511609433057270365759591953092186117381932611793105118548074462379962749567
351885752724891227938183011949129833673362440656643086021394946395224737190702
179860943702770539217176293176752384674818467669405132000568127145263560827785
771342757789609173637178721468440901224953430146549585371050792279689258923542
019956112129021960864034418159813629774771309960518707211349999998372978049951
059731732816096318595024459455346908302642522308253344685035261931188171010003
137838752886587533208381420617177669147303598253490428755468731159562863882353
787593751957781857780532171226806613001927876611195909216420198938
TIME: 3824.81 SEC
3.
141592653589793238462643383279502884197169399375105820974944592307816406286208
998628034825342117067982148086513282306647093844609550582231725359408128481117
450284102701938521105559644622948954930381964428810975665933446128475648233786
783165271201909145648566923460348610454326648213393607260249141273724587006606
315588174881520920962829254091715364367892590360011330530548820466521384146951
941511609433057270365759591953092186117381932611793105118548074462379962749567
351885752724891227938183011949129833673362440656643086021394946395224737190702
179860943702770539217176293176752384674818467669405132000568127145263560827785
771342757789609173637178721468440901224953430146549585371050792279689258923542
019956112129021960864034418159813629774771309960518707211349999998372978049951
059731732816096318595024459455346908302642522308253344685035261931188171010003
137838752886587533208381420617177669147303598253490428755468731159562863882353
787593751957781857780532171226806613001927876611195909216420198937
TIME: 3433.92 SEC

Code:

: PIATAN ;
79 6 / CONSTANT MAXL
1002 CONSTANT N
N 6 / 1 + CONSTANT M
5 CONSTANT S
M S * CONSTANT E
CREATE A E NALLOT
CREATE B E NALLOT
CREATE XX E NALLOT
CREATE YY E NALLOT
VARIABLE D
VARIABLE F
VARIABLE FF
VARIABLE G
VARIABLE H
: 1D. <# # #> TYPE ;
: 06D. <# # # # # # # #> TYPE ;
: MPSET OVER ! DUP E + SWAP S + DO 0 I ! S +LOOP ;
: MPCOPY 2DUP - SWAP DUP E + SWAP DO I @ OVER I + ! S +LOOP 2DROP ;
: UD+ ROT + ROT ROT + ;
: MPDIV
D ! DUP DUP M 0 DO DUP @ 0= IF S + ELSE LEAVE THEN LOOP DUP E - ROT = IF 2DROP 0 ELSE
0 SWAP ROT E + SWAP DO I @ SWAP D @ UM/MOD I ! S +LOOP DROP 1 THEN
;
: MPMULT
D ! DUP DUP E + S - M 0 DO DUP @ 0= IF S - ELSE LEAVE THEN LOOP > IF DROP ELSE
0 SWAP DUP E + S - DO I @ D @ UM* ROT 0 D+ SWAP I ! S NEGATE +LOOP DROP THEN
;
: MPSUB
2DUP - D !
0 SWAP DUP E + S - DO I D @ + DUP H ! @ 0 I @ 0 D- ROT 0 UD+ H @ !
S NEGATE +LOOP
2DROP
;
: MPADD
2DUP - D !
0 SWAP DUP E + S - DO I D @ + DUP H ! @ 0 I @ 0 D+ ROT 0 D+ SWAP H @ !
S NEGATE +LOOP
2DROP
;
: MPATAN
F ! G ! 1 FF ! XX 1 MPSET XX F @ MPDIV DROP G @ XX MPCOPY
F @ 1024 < IF F @ DUP * F ! 0 FF ! THEN
1 BEGIN
2 + XX F @ MPDIV DROP
FF @ 1 = IF XX F @ MPDIV DROP THEN
YY XX MPCOPY DUP YY SWAP MPDIV 1 =
WHILE
DUP 2 AND 0 > IF G @ YY MPSUB ELSE G @ YY MPADD THEN
REPEAT
DROP
;
: MPPRINT
XX SWAP MPCOPY XX @ 0 1D. ." ." CR
N 0 DO
0 XX ! XX 1000000 MPMULT XX @ 0 06D.
I 6 / 1 + MAXL MOD 0= IF CR THEN
6 +LOOP
;
: MACHIN
A 5 MPATAN A 16 MPMULT
B 239 MPATAN B 4 MPMULT A B MPSUB
;
: STORMER
A 57 MPATAN A 176 MPMULT
B 239 MPATAN B 28 MPMULT A B MPADD
B 682 MPATAN B 48 MPMULT A B MPSUB
B 12943 MPATAN B 96 MPMULT A B MPADD
;
: DOIT
CLOCK MACHIN A MPPRINT CR CLOCK
CR ." TIME: " X<>Y F- STD F. ." SEC" CR CR
  CLOCK STORMER A MPPRINT CR CLOCK
CR ." TIME: " X<>Y F- STD F. ." SEC" CR
;

2. HP-71B Spigot Forth

314159265358979323846264338327950288419716939937510582097494459230781640628620
899862803482534211706798214808651328230664709384460955058223172535940812848111
745028410270193852110555964462294895493038196442881097566593344612847564823378
678316527120190914564856692346034861045432664821339360726024914127372458700660
631558817488152092096282925409171536436789259036001133053054882046652138414695
194151160943305727036575959195309218611738193261179310511854807446237996274956
735188575272489122793818301194912983367336244065664308602139494639522473719070
217986094370277053921717629317675238467481846766940513200056812714526356082778
577134275778960917363717872146844090122495343014654958537105079227968925892354
201995611212902196086403441815981362977477130996051870721134999999837297804995
105973173281609631859502445945534690830264252230825334468503526193118817101000
313783875288658753320838142061717766914730359825349042875546873115956286388235
378759375195778185778053217122680661300192787661119590921642019893
TIME: 6720.22 SEC

Code:

: SPIGOT ;
: 06D. <# # # # # # # #> TYPE ;
79 6 / CONSTANT MAXL
1002 CONSTANT N
N 6 / 21 * CONSTANT M
5 CONSTANT S
CREATE A M S * NALLOT
1000000 CONSTANT F
VARIABLE E
VARIABLE K
VARIABLE P
: SPIGOTM
1 K !
0,
  1 M 1 - DO
I SWAP OVER UM* SWAP 2SWAP UM* D+
200000000000, D+
I 2* 1 - M/MOD
ROT A I S * + !
-1 +LOOP
F M/MOD 06D.
DUP E ! 0
  0 M 22 - DO
1 I DO
I SWAP OVER UM* SWAP 2SWAP UM* D+
A I S * + DUP P ! @ F UM* D+
I 2* 1 - M/MOD
ROT P @ !
-1 +LOOP
F M/MOD E @ 0 D+ 06D.
DUP E ! 0
1 K +! K @ MAXL MOD 0= IF CR THEN
-21 +LOOP
  2DROP 
;
: DOIT
CLOCK SPIGOTM CLOCK
CR CR ." TIME: " X<>Y F- STD F. ." SEC" CR
;

3. HP-71B Arctan RPN

This was originally written by Ron Knapp for the 41. Minor changes were made to address output formatting and timing.

Output:

3.141592653589793238462643383279502884197169399375105820974944592307816406286208
99862803482534211706798214808651328230664709384460955058223172535940812848111745
02841027019385211055596446229489549303819644288109756659334461284756482337867831
65271201909145648566923460348610454326648213393607260249141273724587006606315588
17488152092096282925409171536436789259036001133053054882046652138414695194151160
94330572703657595919530921861173819326117931051185480744623799627495673518857527
24891227938183011949129833673362440656643086021394946395224737190702179860943702
77053921717629317675238467481846766940513200056812714526356082778577134275778960
91736371787214684409012249534301465495853710507922796892589235420199561121290219
60864034418159813629774771309960518707211349999998372978049951059731732816096318
59502445945534690830264252230825334468503526193118817101000313783875288658753320
83814206171776691473035982534904287554687311595628638823537875937519577818577805
321712268066130019278766111959092164201989
TIME: 6033.60 SEC

Code (use RUN TRANS41 to input):

LBL "CPI2"      X<0?            R^              ST* T           RCL 05          
CLRG + ST+ T ST* Z +
219 RCL 01 X^2 * DSE Y
STO 00 RCL 04 R^ RCL 08 LBL 14
TIME ST/ Z + * STO IND 00
HR * + FRC RDN
STO IND 00 ENTER^ FC? 00 X<>Y DSE 03
FIX 03 LBL 99 GTO 04 LASTX DSE 00
4 RCL 08 RCL 13 INT GTO 13
STO 09 ST/ Z * R^ 219
1E5 MOD 3 + STO 00
ST/ Y R^ DSE 02 RCL 05 TIME
STO 04 INT GTO 05 - HR
X^2 LASTX LBL 04 + RCL IND 00
STO 05 FRC RCL 07 X>0? -
X<>Y RDN * ISG IND 00 STO IND 00
427 + RCL 06 X>0? LBL E
+ X<>Y LBL 05 GTO 11 SF 21
STO 02 INT X<>Y RCL 05 CLA
239 RCL 04 RDN + FIX 00
X^2 ST* T / LBL 11 14.114
STO 07 ST* Z STO 01 ISG 00 STO 00
LASTX * CLX GTO 10 SF 29
1E2 STO IND 00 R^ GTO J RCL IND 00
* RDN ENTER^ LBL 09 ACX
STO 13 ENTER^ GTO 06 RCL 03 STD
RDN LBL 99 LBL 07 STO 00 CF 29
X^2 RCL 08 RCL 01 RCL 10 ISG 00
STO 08 ST/ Z ST/ Z X^2 LBL 15
94E-5 MOD MOD 3 XEQ 16
STO 11 X<>Y X<>Y Y^X ISG 00
14.0139 INT INT LASTX FS? 00
STO 12 ST+ IND 00 X<>Y * GTO 17
25 RDN RCL 04 STO 08 XEQ 16
STO 10 + ST* Z CLX CLA
LBL 00 STO 01 * LBL 12 ISG 00
RCL 11 RCL 03 ENTER^ RCL IND 00 GTO 15
ST+ 12 RCL 04 LBL 06 X<>Y ACA
RCL 12 * RCL 01 RCL 04 GTO 17
RND ENTER^ ST/ Z ST/ Z LBL 16
STO 00 ISG 00 MOD * RCL IND 00
RCL 07 GTO 02 RDN ENTER^ RCL 04
RCL 02 DSE 02 INT LBL 99 /
INT GTO 00 + RCL 08 INT
ENTER^ 4096E-7 RCL IND 00 ST/ Z LASTX
ST* Z STO 08 - MOD FRC
2 1439.00006 X>0? R^ RCL 04
- STO 02 GTO 08 INT XEQ 18
ST- Z 837E-6 DSE 00 LASTX XEQ 18
* STO 11 LBL 99 FRC RTN
RCL 10 115.115 DSE IND 00 RDN LBL 18
* STO 12 ISG 00 + *
STO 06 80 RCL 05 X<>Y RCL Y
CLX STO 13 + INT X=0?
STO 01 5E6 LBL 08 RCL 04 GTO 19
X<>Y STO 07 STO IND 00 ST* T LOG
RCL 13 .75 R^ ST* Z INT
* STO 06 RCL 04 * LBL 19
ENTER^ LBL J * STO IND 00 RCL 09
GTO 01 RCL 11 ENTER^ RDN X<>Y
LBL 02 ST+ 12 ISG 00 ENTER^ X=Y?
RCL 06 RCL 12 GTO 07 LBL 99 GTO 20
ST/ Z RND RCL 03 RCL 08 -
MOD STO 00 STO 00 ST/ Z 0
X<>Y STO 03 FS?C 00 MOD LBL 21
INT SF 00 GTO 03 X<>Y ARCL X
X<>Y LBL 03 CLX INT DSE Y
RCL 04 RCL 02 ENTER^ ST+ IND 00 GTO 21
ST* Z INT DSE 02 RDN LBL 20
* ENTER^ FS? 00 + ARCL T
ENTER^ ENTER^ GTO 09 ISG 00 ACA
LBL 01 LBL 99 LBL 10 GTO 12 CLA
RCL 06 2 X<> IND 00 114.013 RTN
ST/ Z - RCL 04 STO 00 LBL 17
MOD ST* Z / 215 ADV
STO 03 RCL 10 FRC STO 03 ADV
RDN ST* Z LASTX CLX 219
INT X<>Y INT LBL 13 STO 00
+ * RCL 08 RCL IND 03 RCL IND 00
RCL 05 2 * + 3600
ST- Y ST- L FRC RCL IND 00 *
X<>Y CLX LASTX - FIX 02
RCL IND 00 LASTX INT 0 "TIME: "
+ ST* T ST+ IND 00 X<>Y ARCL X
X>0? ST- Y RDN X<0? "} SEC"
ISG 01 RDN X<>Y X>0? PRA
LBL 99 * RCL 05 GTO 14 END

4. TI-95 Arctan AOS

Code:

5. HP-41CX Arctan RPN

This was originally written by Ron Knapp. Minor changes were made to address output formatting, timing, and duplicate labels (labels were changed to increase readability).

NOTE: The timing was from EMU41, the orginal time was ~8.5 hours. The final article will have a 41CX retiming.

Output:

3.
14159 26535 89793 23846
26433 83279 50288 41971
69399 37510 58209 74944
59230 78164 06286 20899
86280 34825 34211 70679
82148 08651 32823 06647
09384 46095 50582 23172
53594 08128 48111 74502
84102 70193 85211 05559
64462 29489 54930 38196
44288 10975 66593 34461
28475 64823 37867 83165
27120 19091 45648 56692
34603 48610 45432 66482
13393 60726 02491 41273
72458 70066 06315 58817
48815 20920 96282 92540
91715 36436 78925 90360
01133 05305 48820 46652
13841 46951 94151 16094
33057 27036 57595 91953
09218 61173 81932 61179
31051 18548 07446 23799
62749 56735 18857 52724
89122 79381 83011 94912
98336 73362 44065 66430
86021 39494 63952 24737
19070 21798 60943 70277
05392 17176 29317 67523
84674 81846 76694 05132
00056 81271 45263 56082
77857 71342 75778 96091
73637 17872 14684 40901
22495 34301 46549 58537
10507 92279 68925 89235
42019 95611 21290 21960
86403 44181 59813 62977
47713 09960 51870 72113
49999 99837 29780 49951
05973 17328 16096 31859
50244 59455 34690 83026
42522 30825 33446 85035
26193 11881 71010 00313
78387 52886 58753 32083
81420 61717 76691 47303
59825 34904 28755 46873
11595 62863 88235 37875
93751 95778 18577 80532
17122 68066 13001 92787
66111 95909 21642 01989
TIME: 32.51 SEC

Code:

 01 LBL "CPI2"      111 STO IND 00      221 X>0?            331 +               
02 CLRG 112 RDN 222 GTO 09 332 ISG 00
03 219 113 ENTER 223 DSE 00 333 GTO 13
04 STO 00 114 LBL 99 224 LBL 99 334 114.013
05 TIME 115 RCL 08 225 DSE IND 00 335 STO 00
06 HR 116 ST/ Z 226 ISG 00 336 215
07 STO IND 00 117 MOD 227 RCL 05 337 STO 03
08 FIX 03 118 X<>Y 228 + 338 CLX
09 4 119 INT 229 LBL 09 339 LBL 14
10 STO 09 120 ST+ IND 00 230 STO IND 00 340 RCL IND 03
11 E5 121 RDN 231 R^ 341 +
12 ST/ Y 122 + 232 RCL 04 342 RCL IND 00
13 STO 04 123 STO 01 233 * 343 -
14 X^2 124 RCL 03 234 ENTER 344 0
15 STO 05 125 RCL 04 235 ISG 00 345 X<>Y
16 X<>Y 126 * 236 GTO 08 346 X<0?
17 427 127 ENTER 237 RCL 03 347 X>0?
18 + 128 ISG 00 238 STO 00 348 GTO 15
19 STO 02 129 GTO 03 239 FS?C 00 349 RCL 05
20 239 130 DSE 02 240 GTO 04 350 +
21 X^2 131 GTO 01 241 CLX 351 DSE Y
22 STO 07 132 4096 E-7 242 ENTER 352 LBL 15
23 LASTX 133 STO 08 243 DSE 02 353 STO IND 00
24 E2 134 1439.00006 244 FS? 00 354 RDN
25 * 135 STO 02 245 GTO 10 355 DSE 03
26 STO 13 136 837 E-6 246 LBL 11 356 DSE 00
27 RDN 137 STO 11 247 X<> IND 00 357 GTO 14
28 X^2 138 115.115 248 RCL 04 358 219
29 STO 08 139 STO 12 249 / 359 STO 00
30 94 E-5 140 80 250 FRC 360 TIME
31 STO 11 141 STO 13 251 LASTX 361 HR
32 14.0139 142 5 E6 252 INT 362 RCL IND 00
33 STO 12 143 STO 07 253 RCL 08 363 -
34 25 144 .75 254 * 364 STO IND 00
35 STO 10 145 STO 06 255 FRC 365 RTN
36 LBL 01 146 LBL J 256 LASTX 366 LBL E
37 RCL 11 147 RCL 11 257 INT 367 SF 21
38 ST+ 12 148 ST+ 12 258 ST+ IND 00 368 CLA
39 RCL 12 149 RCL 12 259 RDN 369 FIX 00
40 RND 150 RND 260 X<>Y 370 14.114
41 STO 00 151 STO 00 261 RCL 05 371 STO 00
42 RCL 07 152 STO 03 262 ST* T 372 SF 29
43 RCL 02 153 SF 00 263 ST* Z 373 RCL IND 00
44 INT 154 LBL 04 264 * 374 ARCL X
45 ENTER 155 RCL 02 265 RCL 08 375 ACA
46 ST* Z 156 INT 266 * 376 CLA
47 2 157 ENTER 267 FRC 377 PRBUF
48 - 158 ENTER 268 X<>Y 378 CF 29
49 ST- Z 159 LBL 99 269 LASTX 379 ISG 00
50 * 160 2 270 INT 380 LBL 16
51 RCL 10 161 - 271 R^ 381 XEQ 17
52 * 162 ST* Z 272 + 382 ISG 00
53 STO 06 163 RCL 10 273 RCL 05 383 FS? 00
54 CLX 164 ST* Z 274 - 384 GTO 18
55 STO 01 165 X<>Y 275 + 385 " "
56 X<>Y 166 * 276 X>0? 386 XEQ 17
57 RCL 13 167 2 277 ISG IND 00 387 PRBUF
58 * 168 ST- L 278 X>0? 388 CLA
59 ENTER 169 CLX 279 GTO 12 389 ISG 00
60 GTO 02 170 LASTX 280 RCL 05 390 GTO 16
61 LBL 03 171 ST* T 281 + 391 AVIEW
62 RCL 06 172 ST- Y 282 LBL 12 392 GTO 18
63 ST/ Z 173 RDN 283 ISG 00 393 LBL 17
64 MOD 174 * 284 GTO 11 394 RCL IND 00
65 X<>Y 175 R^ 285 GTO J 395 RCL 04
66 INT 176 ST+ T 286 LBL 10 396 /
67 X<>Y 177 X^2 287 RCL 03 397 INT
68 RCL 04 178 R^ 288 STO 00 398 LASTX
69 ST* Z 179 + 289 RCL 10 399 FRC
70 * 180 + 290 X^2 400 RCL 04
71 ENTER 181 FC? 00 291 3 401 XEQ 19
72 LBL 02 182 GTO 05 292 Y^X 402 " "
73 RCL 06 183 RCL 13 293 LASTX 403 XEQ 19
74 ST/ Z 184 * 294 * 404 RTN
75 MOD 185 3 295 STO 08 405 LBL 19
76 STO 03 186 DSE 02 296 CLX 406 *
77 RDN 187 GTO 06 297 LBL 13 407 RCL Y
78 INT 188 LBL 05 298 RCL IND 00 408 X=0?
79 + 189 RCL 07 299 X<>Y 409 GTO 20
80 RCL 05 190 * 300 RCL 04 410 LOG
81 ST- Y 191 RCL 06 301 ST/ Z 411 INT
82 X<>Y 192 LBL 06 302 * 412 LBL 20
83 RCL IND 00 193 X<>Y 303 ENTER 413 RCL 09
84 + 194 RDN 304 LBL 99 414 X<>Y
85 X>0? 195 / 305 RCL 08 415 X=Y?
86 ISG 01 196 STO 01 306 ST/ Z 416 GTO 21
87 LBL 99 197 CLX 307 MOD 417 -
88 X<0? 198 R^ 308 R^ 418 0
89 + 199 ENTER 309 INT 419 LBL 22
90 RCL 01 200 GTO 07 310 LASTX 420 ARCL X
91 RCL 04 201 LBL 08 311 FRC 421 DSE Y
92 ST/ Z 202 RCL 01 312 RDN 422 GTO 22
93 * 203 ST/ Z 313 + 423 LBL 21
94 ENTER 204 MOD 314 X<>Y 424 ARCL T
95 LBL 99 205 X<>Y 315 INT 425 ACA
96 RCL 08 206 INT 316 RCL 04 426 CLA
97 ST/ Z 207 X<>Y 317 ST* T 427 RTN
98 MOD 208 RCL 04 318 ST* Z 428 LBL 18
99 R^ 209 ST* Z 319 * 429 219
100 INT 210 * 320 STO IND 00 430 STO 00
101 LASTX 211 ENTER 321 RDN 431 RCL IND 00
102 FRC 212 LBL 07 322 ENTER 432 3600
103 RDN 213 RCL 01 323 LBL 99 433 *
104 + 214 ST/ Z 324 RCL 08 434 FIX 02
105 X<>Y 215 MOD 325 ST/ Z 435 "TIME: "
106 INT 216 RDN 326 MOD 436 ARCL X
107 RCL 04 217 INT 327 X<>Y 437 >" SEC"
108 ST* T 218 + 328 INT 438 PRA
109 ST* Z 219 RCL IND 00 329 ST+ IND 00 439 END
110 * 220 - 330 RDN

Edited: Minor bug in PIATAN Forth that may have left a value on the stack after run. Results and time unaffected, just clean up.

Edited: Removed TI-59 article.

Edited: 19 Mar 2009, 9:26 a.m. after one or more responses were posted


#3

Egan,

Wow, you've been a busy guy! I need some time to digest all this, but thank you for all your work on Vieta Accelerated. I didn't know about Kreminski's formula, very interesting.

-Katie

#4

Egan:

The article that you included with the TI-95 program was originally published as part of the documentation of a TI-59 program on V7N4/5P27-29 of TI PPC Notes back in mid-1982. You can access that at Viktor Toth's site. The procedures described for running the TI-59 program do not apply to the TI-95 program. There is also an error in the article where it states "... There is a linear relationship between the number of decimal places and the running time of the program. ..." Actually, the relationship should be, and usually is very close to a squared relationship.

The material in TI PPC Notes includes the TI-59 program. Unfortunately, the program listing is incomplete in that the first forty steps of program II-B are missing. If anyone would like to run the TI-59 program I can provide a listing of the missing steps.

The TI-59 program is limited to 460 digits by memory limitations. There was a program published in the French magazine Science et Vie which was substatially slower but which would find 1287 digits of pi. That is particularly impressive since the TI-59's user memory is only 1300 digits.

Palmer


#5

Palmer,

I was a bit confused myself about the pairing of that article and the source. They were sent to me as a single 3 page PDF. I've removed the article to avoid any ambiguity.

Do you have any links, articles, etc... about the TI-95 Pi program and its 5.81 hour result? Any corroborating or anecdotal information would be great.

Thanks.


#6

Could you please repost the link to the removed TI-59 article? My former printout of your post is incomplete and I'd like to read the document.

Marcus


#7

Here is the original PDF that was sent to me. It is higher resolution than the images posted here (MoHPC size limitation).

http://sense.net/~egan/ti-95pi.pdf


#8

Hi Egan,

thanks for the download. I might have caught the PI disease.

Marcus


#9

I do not find the number Pi that interesting. A handful of digits can solve many practical problems. E.g. with less than 100 digits you can calculate the circumference of the universe accurate to the radius of a hydrogen atom.

What I find interesting is all the different math that can be applied to computing Pi.

And, what I find most interesting is the computational methods used to compute Pi, especially on machines with limited resources. As I explore this history I am truly amazed at the ingenuity and determination of the individual programmers. Especially the TI programmers. AFAIK, the 41 and 71 outclass the 59 and 95 in terms of I/O and memory forcing the TI community to think smarter. Fascinating!

#10

Egan:

You wrote:

Quote:
Do you have any links, articles, etc... about the TI-95 Pi program and its 5.81 hour result? Any corroborating or anecdotal information would be great.

The following notes accompanied my transmission of the TI-95 program to Gene Wright:
Quote:
1. A direct translation would yield bad results if the number of digits asked for was greater than 770. I "fixed" it by changing from using ten digit registers with only the three guard digits available for carries from one register to the next to using nine digit registers with four digits available for the carries. The completed program is a hybrid which uses ten digit registers for up to 770 digits and nine digits for more than 770 digits. This yields about a ten per cent increase in speed for requests for less than 770 digits. Flag 3 controls whether 9 or 10 digit registers are used.

2. It would have been possible to simply print out the nine digit registers, but that made it difficult to check the answers against my pi table which was printed in groups of ten digits so I added a routine at the end which will either print or display the nine digit registers in a ten digit format. I could have used the same routine in another way to simply rearrange the 9 digit information back into the registers as 10 digit information but I didn't.

3. The TI-59 proram used two HIR registers to save two data registers for the calculation. My conversion replaced HIR 07 with data register 470 and HIR 08 with data register 471.

4. The TI-95 does not permit comparisons with the t register without using assembly language techniques. My conversion used data register 472 as a comparison register. Where the t register was used only as a place to park a value for later recovery I used the TI-95 t register as at lines 0408 and 0415.

5. I also used data register 474 to store the number of places desired and data register 473 to store the number of data registers to be used for the number of places desired.

6. In Bob's discussion of his program he says that "There is a linear relationship between the number of decimal places and the running time of the programs." Actually, the relationship is essentially a square relationship with an upward tail at the low end driven by the initialization routines and by the initializations for each iteration. You can see that in the plots.


That is all the supporting documentation that there is other than my handwritten programming notes and some run time results where the run time results are included in the graph that accompanied the transmission to Gene.

Some years ago I assembled a fairly thorough summary of the work with pi on the TI-59 including references to TI PPC Notes for details. It runs several pages. If you send an e-mail address I will send it to you.

Palmer


#11

Thank you very much. I find all of this history very interesting. You've got mail.


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