Extended Precision Ln(X)

About the program

This program can calculate natural logarithms up to 1188 digits. It also prints the results, using the same method as used with EPP1. The program was written by myself and up to now it has not been published.

Method Used

To obtain the high precision this program delivers, the following method is used: :lq.

The program starts with the following series, which can be found in "Mathematical Tables and Formulas" by Robert D. Carmichael and Edwin R. Smith: .sk .ce Ln(y+1) = +

Part 1 in this series is: .sk .ce Ln(y) :fn id=pn.In the rest of this chapter I will use P(n) to refer to a Part 2, that has to be calculated for a specific\n. [n\=\y\+\1]:efn

Part 2:fnref refid=pn. is: .sk .ce 2 &ML [1/(2y+1) + 1/(3&ML.(2y+1)&S3.) + 1/(5&ML.(2y+1)&S5.) + ...] :fn id=ar.The only real difference between the two series is the fact that the terms in the series for Arctan(X) have alternating signs.:efn

If we take a closer look at Part 2 of the above series, we see that it is very similar to the series for Arctan(X) used to calculate &PI..:fnref refid=ar. Because of this close relationship, it was very easy to adapt the program used in the "EPP1" chapter for calculating &PI.. :elq

The most important modifications with respect to the program in that chapter are:

The TI-59/PC-100 Version

User Instructions

The instructions for this version of the program are:

  1. Enter the Calculate Part of the program.
  2. Enter n, a number for which you want to calculate P(n), and press A. It is important to realize that n = y\+\1, that is, if you want to calculate Ln(2), you have to enter n\=\2!
  3. Enter the number of digits you want and press R/S. After a short time the calculator will stop with a flashing\1 in the display. :note.The number you enter will be rounded up to the next multiple of\12 and if you enter a number\>\1188, it will return to the display flashing, to indicate an error.
  4. Now press 7 EE to start the program in Fast Mode.
    When the calculation of P(n) is completed the display will show a zero.
  5. Record banks 1, 2, 3 and 4 on magnetic cards. Mark these four banks A1 through A4.

    Optional, and to be repeated for every factor of the number:

  6. Reset the partitioning to 479.59. [6 OP 17]
  7. Enter the Sum & Print Part.
  8. If you have calculated two or more P(n)'s and want to sum them, you can enter the multiplication factor for the second P(n) and press X:T.
  9. Now enter the multiplication factor for the first P(n) and press B. .sk If you only calculated one P(n), or if you want to reprint an earlier calculated logarithm, you can skip step 8.
  10. When the display shows a -3, enter bank A4 and press R/S.
  11. When the display shows a -2, enter bank B4 and press R/S. :note.The -2 will not appear when you have omitted step 8.

    After a little while the printer will start printing. The digits obtained are not the first ones, but are digits 829-1188. After the printing has finished the display will once again show a\-3. You can now record bank\3 to save the result for later. Don't record it with\-3, but change the display temporarily into\3. After you have recorded bank 3, you can repeat steps 10\and 11 three more times with A3[/B3], digits 469-828, A2[/B2], digits 109-468 and finally A1[/B1], digits 1-108.

:note.The user instructions above assume that you enter 1188 as the number of digits wanted! If you want to calculate a smaller number of digits, you should refer to the "Some last notes about the program" section at the end of this chapter, for a detailed example.

Listing of the TI-59/PC-100 Version [Calculate Part]

000  92 RTN         060  67 X=T         120  64 PD*         180  08  8
001  01  1          061  00  00         121  99  99         181  69 OP
002  00  0          062  71  71         122  65  x          182  68  68
003  69 OP          063  82 HIR         123  82 HIR         183  22 INV
004  17  17         064  17  17         124  15  15         184  77 XòT
005  47 CMS         065  34 ûX          125  74 SM*         185  01  01
006  82 HIR         066  82 HIR         126  99  99         186  74  74
007  18  18         067  07  07         127  82 HIR         187  60 DEG
008  42 STO         068  22 INV         128  17  17         188  53  (
009  99  99         069  86 STF         129  54  )          189  32 X:T
010  02  2          070  04  04         130  97 DSZ         190  55  ö
011  53  (          071  02  2          131  99  99         191  01  1
012  52 EE          072  82 HIR         132  00  00         192  02  2
013  06  6          073  56  56         133  78  78         193  75  -
014  75  -          074  82 HIR         134  87 IFF         194  93  .
015  53  (          075  18  18         135  04  04         195  09  9
016  22 INV         076  48 EXC         136  00  00         196  02  2
017  52 EE          077  99  99         137  06  06         197  94 +/-
018  55  ö          078  53  (          138  01  1          198  82 HIR
019  82 HIR         079  53  (          139  52 EE          199  08  08
020  16  16         080  52 EE          140  06  6          200  54  )
021  54  )          081  06  6          141  22 INV         201  59 INT
022  59 INT         082  85  +          142  49 PRD         202  53  (
023  65  x          083  73 RC*         143  00  00         203  82 HIR
024  74 SM*         084  99  99         144  69 OP          204  38  38
025  99  99         085  75  -          145  20  20         205  65  x
026  82 HIR         086  53  (          146  82 HIR         206  01  1
027  16  16         087  24 CE          147  18  18         207  02  2
028  54  )          088  55  ö          148  59 INT         208  55  ö
029  53  (          089  82 HIR         149  22 INV         209  53  (
030  52 EE          090  17  17         150  28 LOG         210  01  1
031  06  6          091  54  )          151  52 EE          211  75  -
032  75  -          092  59 INT         152  22 INV         212  32 X:T
033  53  (          093  82 HIR         153  57 ENG         213  82 HIR
034  22 INV         094  05  05         154  49 PRD         214  17  17
035  52 EE          095  65  x          155  00  00         215  65  x
036  55  ö          096  82 HIR         156  00  0          216  02  2
037  82 HIR         097  17  17         157  81 RST         217  54  )
038  16  16         098  85  +          158  00  0          218  33 Xý
039  54  )          099  73 RC*         159  00  0          219  82 HIR
040  59 INT         100  99  99         160  00  0          220  07  07
041  65  x          101  22 INV         161  00  0          221  28 LOG
042  52 EE          102  59 INT         162  00  0          222  54  )
043  06  6          103  54  )          163  00  0          223  53  (
044  94 +/-         104  55  ö          164  00  0          224  59 INT
045  74 SM*         105  01  1          165  00  0          225  65  x
046  99  99         106  52 EE          166  00  0          226  02  2
047  82 HIR         107  06  6          167  00  0          227  85  +
048  16  16         108  94 +/-         168  00  0          228  01  1
049  54  )          109  72 ST*         169  00  0          229  54  )
050  22 INV         110  99  99         170  76 LBL         230  82 HIR
051  52 EE          111  75  -          171  11  A          231  06  06
052  97 DSZ         112  53  (          172  82 HIR         232  09  9
053  99  99         113  22 INV         173  07  07         233  69 OP
054  00  00         114  52 EE          174  92 RTN         234  17  17
055  11  11         115  55  ö          175  50 IxI         235  04  4
056  82 HIR         116  82 HIR         176  32 X:T         236  05  5
057  16  16         117  17  17         177  01  1          237  30 TAN
058  66 PAU         118  54  )          178  01  1          238  33 Xý
059  22 INV         119  59 INT         179  08  8          239  86 STF
<\pre>

Listing of the TI-59/PC-100 Version [Sum & Print Part]

000  29 CP          060  87 IFF         120  54  )          180  15  15
001  92 RTN         061  05  05         121  82 HIR         181  42 STO
002  76 LBL         062  01  01         122  56  56         182  05  05
003  12  B          063  30  30         123  86 STF         183  54  )
004  37 P/R         064  02  2          124  40 IND         184  59 INT
005  67 X=T         065  42 STO         125  01  01         185  65  x
006  00  00         066  07  07         126  87 IFF         186  48 EXC
007  09  09         067  53  (          127  00  00         187  05  05
008  22 INV         068  73 RC*         128  00  00         188  54  )
009  86 STF         069  08  08         129  01  01         189  74 SM*
010  04  04         070  55  ö          130  73 RC*         190  03  03
011  09  9          071  28 LOG         131  03  03         191  69 OP
012  69 OP          072  59 INT         132  53  (          192  23  23
013  17  17         073  82 HIR         133  22 INV         193  97 DSZ
014  47 CMS         074  06  06         134  59 INT         194  02  02
015  03  3          075  22 INV         135  65  x          195  01  01
016  00  0          076  28 LOG         136  82 HIR         196  30  30
017  10 E'          077  52 EE          137  18  18         197  22 INV
018  97 DSZ         078  54  )          138  85  +          198  57 ENG
019  01  01         079  72 ST*         139  73 RC*         199  69 OP
020  00  00         080  08  08         140  04  04         200  33  33
021  15  15         081  01  1          141  22 INV         201  53  (
022  82 HIR         082  52 EE          142  59 INT         202  73 RC*
023  16  16         083  06  6          143  65  x          203  03  03
024  10 E'          084  82 HIR         144  82 HIR         204  75  -
025  43 RCL         085  05  05         145  17  17         205  59 INT
026  05  05         086  64 PD*         146  85  +          206  99 PRT
027  99 PRT         087  08  08         147  43 RCL         207  54  )
028  00  0          088  22 INV         148  05  05         208  52 EE
029  81 RST         089  74 SM*         149  55  ö          209  06  6
030  76 LBL         090  08  08         150  82 HIR         210  22 INV
031  10 E'          091  03  3          151  15  15         211  57 ENG
032  42 STO         092  00  0          152  75  -          212  99 PRT
033  02  02         093  44 SUM         153  59 INT         213  97 DSZ
034  42 STO         094  08  08         154  42 STO         214  06  06
035  06  06         095  53  (          155  05  05         215  01  01
036  03  3          096  87 IFF         156  54  )          216  99  99
037  00  0          097  04  04         157  63 EX*         217  69 OP
038  42 STO         098  01  01         158  03  03         218  05  05
039  03  03         099  04  04         159  53  (          219  87 IFF
040  42 STO         100  97 DSZ         160  59 INT         220  05  05
041  08  08         101  07  07         161  65  x          221  00  00
042  06  6          102  00  00         162  82 HIR         222  01  01
043  00  0          103  68  68         163  18  18         223  82 HIR
044  42 STO         104  65  x          164  85  +          224  15  15
045  04  04         105  32 X:T         165  73 RC*         225  44 SUM
046  87 IFF         106  53  (          166  04  04         226  30  30
047  00  00         107  82 HIR         167  59 INT         227  22 INV
048  01  01         108  16  16         168  65  x          228  49 PRD
049  30  30         109  55  ö          169  82 HIR         229  30  30
050  03  3          110  22 INV         170  17  17         230  32 X:T
051  94 +/-         111  28 LOG         171  85  +          231  49 PRD
052  91 R/S         112  52 EE          172  43 RCL         232  30  30
053  87 IFF         113  32 X:T         173  05  05         233  86 STF
054  04  04         114  54  )          174  75  -          234  05  05
055  00  00         115  59 INT         175  53  (          235  92 RTN
056  60  60         116  42 STO         176  69 OP
057  02  2          117  01  01         177  24  24
058  94 +/-         118  75  -          178  55  ö
059  91 R/S         119  01  1          179  82 HIR
<\pre>

The TI-59 Version

User Instructions

The user instructions for this version of the program are exactly the same, except for the fact that the calculator will stop with every group of six digits in the display and that you have to press R/S to see the next six digits.

Listing of the TI-59 Version

To obtain the TI-59 version you only have to change steps 027, 206 and 212 of the Sum & Print part of the program into R/S.

Some last notes about the program