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[TI-57] Calculate Square Root
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[TI-57] Calculate Square Root
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[TI-57] Calculate Square Root
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10-24-2024, 07:47 AM
(This post was last modified: 10-25-2024 09:31 AM by Thomas Klemm.)
Post: #1
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[TI-57] Calculate Square Root
Based on a recent thread I took a closer look at the assembly code of the TI-57.
To analyze the code, I extended the RCL-57 emulator's command line tool to track the three registers A, B, and C. This is the relevant assembly code: Code: 0x0640: 0x13f4 CALL 0x03f4Here is a typical output: Code: (…)However, we simply list the line 0x0659 to get the relevant data when calculating \(\sqrt{41}\): A=1000100000000000 B=0000100000000000 C=0004100000000000 A=1000200000000000 B=0000100000000000 C=0003900000000000 A=1000300000000000 B=0000100000000000 C=0003500000000000 A=1000400000000000 B=0000100000000000 C=0002900000000000 A=1000500000000000 B=0000100000000000 C=0002100000000000 A=1000600000000000 B=0000100000000000 C=0001100000000000 A=1000700000000000 B=0000100000000000 C=0009900000000000 A=1000610000000000 B=0000010000000000 C=0004400000000000 A=1000620000000000 B=0000010000000000 C=0003180000000000 A=1000630000000000 B=0000010000000000 C=0001940000000000 A=1000640000000000 B=0000010000000000 C=0000680000000000 A=1000650000000000 B=0000010000000000 C=0009400000000000 A=1000641000000000 B=0000001000000000 C=0009760000000000 A=1000640100000000 B=0000000100000000 C=0003360000000000 A=1000640200000000 B=0000000100000000 C=0002079800000000 A=1000640300000000 B=0000000100000000 C=0000799400000000 A=1000640400000000 B=0000000100000000 C=0009518800000000 A=1000640310000000 B=0000000010000000 C=0000950700000000 A=1000640320000000 B=0000000010000000 C=0009670080000000 A=1000640311000000 B=0000000001000000 C=0002463590000000 A=1000640312000000 B=0000000001000000 C=0001182968000000 A=1000640313000000 B=0000000001000000 C=0009902344000000 A=1000640312100000 B=0000000000100000 C=0004786248000000 A=1000640312200000 B=0000000000100000 C=0003505623800000 A=1000640312300000 B=0000000000100000 C=0002224999400000 A=1000640312400000 B=0000000000100000 C=0000944374800000 A=1000640312500000 B=0000000000100000 C=0009663750000000 A=1000640312410000 B=0000000000010000 C=0002400311600000 A=1000640312420000 B=0000000000010000 C=0001119686780000 A=1000640312430000 B=0000000000010000 C=0009839061940000 A=1000640312421000 B=0000000000001000 C=0004153431180000 A=1000640312422000 B=0000000000001000 C=0002872806338000 A=1000640312423000 B=0000000000001000 C=0001592181494000 A=1000640312424000 B=0000000000001000 C=0000311556648000 A=1000640312423100 B=0000000000000100 C=0008878378287000 A=1000640312423200 B=0000000000000100 C=0007597753440800 A=1000640312423300 B=0000000000000100 C=0006317128594400 A=1000640312423400 B=0000000000000100 C=0005036503747800 A=1000640312423500 B=0000000000000100 C=0003755878901000 A=1000640312423600 B=0000000000000100 C=0002475254054000 A=1000640312423700 B=0000000000000100 C=0001194629206800 A=1000640312423800 B=0000000000000100 C=0009914004359400 A=1000640312423710 B=0000000000000010 C=0004902855407300 A=1000640312423720 B=0000000000000010 C=0003622230559880 A=1000640312423730 B=0000000000000010 C=0002341605712440 A=1000640312423740 B=0000000000000010 C=0001060980864980 A=1000640312423750 B=0000000000000010 C=0009780356017500 A=1000640312423741 B=0000000000000001 C=0003566371988660 A=1000640312423742 B=0000000000000001 C=0002285747141178 A=1000640312423743 B=0000000000000001 C=0001005122293694 A=1000640312423744 B=0000000000000001 C=0009724497446208 This is a largely literal translation of the code for the HP-42S: Code: 00 { 39-Byte Prgm } ; C=xExample 41 R/S 6.40312423743 For those who want to use this, I have added the relevant files: Archive: ti-57.zip Length Date Time Name --------- ---------- ----- ---- 364 10-24-2024 09:44 Makefile 5085 10-24-2024 09:44 cli/app_ti57.c 11280 10-24-2024 09:44 engine/ti57.c 83798 10-24-2024 09:44 engine/asm.c 122 10-24-2024 09:44 engine/asm.h --------- ------- 100649 5 files |
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10-24-2024, 06:24 PM
Post: #2
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RE: [TI-57] Calculate Square Root
(10-24-2024 07:47 AM)Thomas Klemm Wrote: Based on a recent thread I took a closer look at the assembly code of the TI-57. Hi Thomas, Thank you very much for sharing all of this with us! I love it! Keep yourself healthy! Laurent |
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10-25-2024, 09:09 AM
Post: #3
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RE: [TI-57] Calculate Square Root
Explanation
When the odd numbers are added, we get the squares. But we can split the odd number \(2k+1\) into \(k\) and \(k+1\). Therefore, each \(k\) but the last one is added twice: 0² 0 +0+1 1 +1+2 4 +2+3 9 +3+4 16 +4+5 25 +5+6 36 +6+7 49 We stop when \(41\) is overshot and shift the number by one place: 60² 3600 +60+61 3721 +61+62 3844 +62+63 3969 +63+64 4096 +64+65 4225 640² 409600 +640+641 410881 6400² 40960000 +6400+6401 40972801 +6401+6402 40985604 +6402+6403 40998409 +6403+6404 41011216 64030² 4099840900 +64030+64031 4099968961 +64031+64032 4100097024 640310² 409996896100 +640310+640311 409998176721 +640311+640312 409999457344 +640312+640313 410000737969 6403120² 40999945734400 +6403120+6403121 40999958540641 +6403121+6403122 40999971346884 +6403122+6403123 40999984153129 +6403123+6403124 40999996959376 +6403124+6403125 41000009765625 64031240² 4099999695937600 +64031240+64031241 4099999824000081 +64031241+64031242 4099999952062564 +64031242+64031243 4100000080125049 640312420² 409999995206256400 +640312420+640312421 409999996486881241 +640312421+640312422 409999997767506084 +640312422+640312423 409999999048130929 +640312423+640312424 410000000328755776 6403124230² 40999999904813092900 +6403124230+6403124231 40999999917619341361 +6403124231+6403124232 40999999930425589824 +6403124232+6403124233 40999999943231838289 +6403124233+6403124234 40999999956038086756 +6403124234+6403124235 40999999968844335225 +6403124235+6403124236 40999999981650583696 +6403124236+6403124237 40999999994456832169 +6403124237+6403124238 41000000007263080644 64031242370² 4099999999445683216900 +64031242370+64031242371 4099999999573745701641 +64031242371+64031242372 4099999999701808186384 +64031242372+64031242373 4099999999829870671129 +64031242373+64031242374 4099999999957933155876 +64031242374+64031242375 4100000000085995640625 640312423740² 409999999995793315587600 +640312423740+640312423741 409999999997073940435081 +640312423741+640312423742 409999999998354565282564 +640312423742+640312423743 409999999999635190130049 +640312423743+640312423744 410000000000915814977536 6403124237430² 40999999999963519013004900 But we can also split the summands into a constant and variable part. The sum of the variable parts is just \(b^2\), where \(b\) denotes the last digit: \( \begin{align} (60+4)^2 &= 60^2 \\ &+ 2 \cdot 60 \cdot 4 \\ &+ 4^2 \\ \end{align} \) \( \begin{align} (6400+3)^2 &= 6400^2 \\ &+ 2 \cdot 6400 \cdot 3 \\ &+ 3^2 \\ \end{align} \) Or then in general: \( \begin{align} (a+b)^2 &= a^2 \\ &+ 2ab \\ &+ b^2 \\ \end{align} \) Formally: \( \begin{align} \sum_{k=0}^{b-1} (a+k) + (a+k+1) &= \sum_{k=0}^{b-1} 2a + 2k+1 \\ \\ &= \sum_{k=0}^{b-1} (2a + 1) + \sum_{k=0}^{b-1} 2k \\ \\ &= 2(a+1)b + 2 \frac{b(b-1)}{2} \\ \\ &= 2ab + b² \\ \end{align} \) I wonder whether they were forced to implement this slightly more complicated algorithm due to David S. Cochran's patent US3576983A: Digital calculator system for computing square roots |
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10-25-2024, 09:35 AM
Post: #4
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RE: [TI-57] Calculate Square Root
(10-24-2024 06:24 PM)Nihotte(lma) Wrote: Thank you very much for sharing all of this with us! I would like to pass on the thanks to HrastProgrammer, pauln and whoever else contributed to the various TI-57 emulators. |
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10-25-2024, 07:41 PM
Post: #5
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RE: [TI-57] Calculate Square Root
Amazing work. Thank you for sharing!
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