Talk:CORDIC - Knowledge

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FIXED(0.447614), FIXED(0.223811), FIXED(0.111906), FIXED(0.055953), FIXED(0.027977) ]; var X; var Y; var TargetAngle; var CurrAngle; var Step; X = FIXED(AG_CONST); /* AG_CONST * cos(0) */ Y = 0; /* AG_CONST * sin(0) */ TargetAngle = FIXED(28.027); CurrAngle = 0; for (Step = 0; Step < 12; Step++) { var NewX; if (TargetAngle : -->

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I came here to mention this - the article discusses a lot how the algorithm avoids the need for hardware multiply, but all the implementations use multiplication to more closely match the maths. I would prefer to see them use the appropriate shifts or ldexp and remove all uses of multiplication (with

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Not so strange, since almost anybody can easily verify up to about 14 digits. More than that requires much more specialized skill, or a good reference. However, also note that having more digits will not enable one to implement a CORDIC that accurate; one would also have to have trig functions that

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It seems a little strange that Volder should publish his first paper describing the CORDIC in the IRE Transactions on Electronic Computers, September 1959 and in the very same journal Daggett publishes his work on CORDIC. I reckon Daggett was peer reviewing for the journal and stole some ideas!!

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added reading "In the meantime it has been learned that Hewlett Packard and other calculator manufacturers employ the decimal CORDIC techniques in their scientific calculators." It is unclear, if this footnote was added in the 1983 reprint edition only, or if it is already present in the 1974 edition.

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I hate to belabor this but I think the definition of K = 1.647 is correct. The 1.647 is a result of the 2^-i shifts (I think). Then you must undo this gain by dividing by K at the end to remove this gain. I've done the algorithm on my computer and you do indeed up with a value K times greater than it

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Yeah, the value of K in the implentation is indeed correct. However, the equation for K given in previous edition was wrong, which led to my confusion and my changing of K in the implentation before. K is defined as cos(arctan(2^(-i))), you can simply check that K should be 1/sqrt(1+2^(-2i)), instead

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The article mixes in n and i for subscripts and superscripts and it is confusing. n should be used to refer to the total number of iterations while i should refer to the current iteration. This pops up in the beta(n+1) equation. I don't know if I am right so I am hesitant to change it. Maybe someone

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found also in DSP forum from Andraka "Sounds about right. THe worst case angular error after each iteration is the rotation angle of that iteration (which happens if the previous iteration landed exactly on your target angle), so the max error angle is atan(2^-i) after iteration i. Iterations start

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I don't fully agree with this. It would be true, if for each iteration the angle adjustment was half the previous adjustment. As anyone can see from the angle table in the article, this is not so. Though the number of bits required can be used as an indication of the number of iterations needed, it

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The value in the implentation section should really be what it was before someone changed it, 0.607. My comment wasn't correct which may have led to the confusion. The inverse of K is needed to get the true magnitude because all the shifts result in a gain of K = 1.647 which then must be removed to

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I agree. It gets even worse than that: the main purpose of CORDIC is to avoid multiplications and divisions in the main loop, while all implementations so far that I seen do several of these in each iteration, they do 2**-i in main loop instead of precomputing it, etc. The implementations provided

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The Motorola 68000 Floating Point Coprocessor, the MC68881, internally uses CORDIC. The MC68882 does as well and improves on its predecessor by generating a result step on each clock edge (doubling the speed). I think the 6809's Floating Point ROM M6839 also uses CORDIC as it was developed by the

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At least the 1983 reprint edition of Schmid's 1974 "Decimal Computation" contains the following sentence on page 162: "So far CORDIC has been known to be implemented only in binary form. But, as will be demonstrated here, the algorithm can be easily modified for a decimal system." with a footnote

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Maybe I've not expressed myself clearly. You indeed needs to multiply 0.6073(or divide 1.647) after the iterations. But the problem is we define K as "product of cos(arctan(2^(-i)))", so its value should be "product of 1/sqrt(1+2^(-2i))", instead of "product of sqrt(1+2^(-2i))", and it approaches

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I just added the last sentence quoted above. It partially duplicates the first sentence above (grammatically exploded), but 95% of readers will barely notice this, as their parse recursion is already busy dropping the factoids embedded in the first sentence into the buffer-overrun bit bucket. —

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should be equal the wordlength of the computer. Unlimited n corresponds to unlimited wordlength. But it is very hard to imagine a computer with the unlimited the length of word. Modern computers have 32, 64, 128 bits. Can you imagine the weight and the size of the computer with :

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Step) + Y; X = NewX; CurrAngle -= Angles; } } print('CORDIC: ' + FLOAT(TargetAngle) + ' ' + FLOAT(Y) + ' ' + FLOAT(X) + '\n' ); print('FP: ' + FLOAT(TargetAngle) + ' ' + Math.sin(DEG2RAD(FLOAT(TargetAngle))) + ' ' + Math.cos(DEG2RAD(FLOAT(TargetAngle))) + '\n' );

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Quite interesting, even Jean-Michel Muller's "Elementary Functions - Algorithms and Implementations" (2nd edition, 2006, page 156) can be read as if Hermann Schmid & Anthony Bogacki were the first to suggest decimal CORDIC in their 1973 paper, although he doesn't say so

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0.6073 when n approaches infinite. On the other hand, the system gain is A="product of sqrt(1+2^(-2i))"=1.647=1/K. I think that you just mixed up A and K. You can check the beignning part of section 3 of the following article, which explains this issue quite clearly.

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should be. For example, if I ignore K, a 90 deg. shift of (1 , 0) becomes (0, 1.647). So you indeed must either multiply by inv. K or divide by K to obtain the true result. I'll wait until you respond to this so we don't end up reverting each other's edits.

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Actually the number of iterations for CORDIC must be equal to the wordlength. For example, if the wordlength is 16 there is no sense to do 40 iterations , as said unknown author. In this case the number of iterations must be 16. Not more and not less!

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In the other issue of precision, it is true, but not the reason you gave. Higher precision arithmetics usually are implemented on hardware with multipliers and square root circuitry, and then the quadratically converging AGM becomes more efficient. --

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RDIC Gain (cumulative magnitude) of about 1.647. Therefore, to get the true value of magnitude we must multiply by the reciprocal of 1.647, which is 0.607. (Remember, the exact CORDIC Gain is a function of the how many iterations you do."

515:"Now, having rotated our complex number to have a phase of zero, we end up with "C = Ic + j0". The magnitude of this complex value is just "Ic" (since "Qc" is zero.) However, in the rotation process, C has been multiplied by a CO 2474:

Additional note: It is even more strange to ask for source for the additional digits, while the initial approximation is technically. unsourced itself. Not everybody calculates with single or double precision hardware only. --

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1000 if you call it with beta = 5000 which is not an extremely large value! (In addition, you get rounding errors from performing more than 1000 additions of pi.) A simple "mod" instruction should really be preferred. —

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A note regarding numerical approximation: Since this is an reference work, it is foolish to assume a few digits would suite everyone. In fact, the first digits are listed on J. M. Muller, page 134. --

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gives you the result on J.-M. Mueller, "Elementary functions", p. 134. The author probably used Maple, but it is certainly not difficult. Most commercial Fortran Compiler also supports REAL*16. --

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n - word length (number of iterations equals to n for sin, cos, OR equals to 2n for asin, acos, ln, exp) r - the number of additional bits of the words

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of sqrt(1+2^(-2i)) as given before. And by this definition, lim(K)=0.6073, again not 1.647 as given before. I've corrected it and removed the "inv" in the implentation to avoid further confusion.

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on page 13 you can see the table contained the formulae of the CORDIC errors. naturally you cant read russian but the formulaes language is international. in this table:

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01:00, 2 January 2007 Dicklyon (Talk | contribs) (→Implementation in C - remove section, as the algorithm is not useful either pedagogically or practically in this form)

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Well, I don't happen to have Mathematica nor a Fortran compiler handy, but I take your point. If you'd like to put the number back again, please do add the reference.

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Small update: Even the original 1974 edition of Schmid's book contains the above-mentioned footnote, so it must have been a "last-minute" correction in 1974. --

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In computer science, CORDIC is often used to implement floating-point arithmetic when the target platform lacks hardware multiply for cost or space reasons.

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any idea of the accuracy of this algorithm and connections with the number of iterations ? any comparison between this algorithm an some others ? --

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I presume by 'it', you mean who is credited with the original development of the algorithm? Andraka says the original work is credited to Volder.

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Somebody has written: "For most ordinary purposes, 40 iterations (n = 40) is sufficient to obtain the correct result to the 10th decimal place."

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The reference number was made by Muller only to show very doubtful or more precisely speaking absolute senseless the limit value of

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The More Formalized section of this article uses sin and cos in the rotation matrix to develop the ability to calculate sin and cos.

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When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.

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at 0. The last iteration needed to get an error of less than 0.1 degrees is therefore log2(tan(0.1)) rounded away from zero =: -->

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Volder conceived and wrote about CORDIC in 1956 already, but this was an internal report only. Daggett was a co-worker of Volder.

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The following is a C implementation of CORDIC using floats. Beta is the input angle in radians. We start with vector v = (1,0).

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This is an encyclopedia, not a implementation manual. Constants are interesting themselves just for their mathematical interest.

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of CORDIC. In 1966 in "Electronics" was published the paper of J.Parini about DIVIC computer, based on CORDIC technique.

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The python code above seems a little complex given that a port of the C implementation to JS can be written like this:

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related articles on Knowledge. If you would like to participate, please visit the project page, where you can join

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I can't figure out who developed it, and the articles creator left us with two choices, hence the "dispute"

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accurate to generate the needed angles. So the additional digits are really quite pointless, aren't they?

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If you have discovered URLs which were erroneously considered dead by the bot, you can report them with

2587: 2171:$ python cordic.py 30 40 K = 0.607252935009 Sin(30.0) = 0.500000000000 and Cos(30.0) = 0.866025403785 368: 39: 3275:

https://web.archive.org/web/20150309055201/http://www.jacques-laporte.org/Briggs%20and%20the%20HP35.htm

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https://web.archive.org/web/20160818122704/http://www.jacques-laporte.org/TheSecretOfTheAlgorithms.htm

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I disagree. Let them come in the order that they are referred to, like most wikipedia articles do.

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on Knowledge. If you would like to participate, please visit the project page, where you can join

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10, and counting from 0 you have 11 iterations to complete not counting your angle reduction." --

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by J. E. Meggitt, "Pseudo Division and Pseudo Multiplication Processes", IBM Journal, April 1962

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But seriously had Volder written an earlier or paper or were the two actually working together?

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This is not the main purpose of the page (although the main reason of existence of CORDIC is

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Andraka is right, Jack Volder was the first (1959), to say nothing about Henry Briggs (1624).

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https://web.archive.org/web/20160818121038/http://www.jacques-laporte.org/digit_by_digit.htm

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Step) + Y; X = NewX; CurrAngle += Angles; } else { NewX = X + (Y : -->

1191: 339: 191: 3349: 2370:?! On the other hand, if the number of iterations is, for example, 32, then the value of 926:

On the other side, when if the number of iterations is, for example, 32, then the value of

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https://web.archive.org/web/20150309055210/http://www.jacques-laporte.org/Trigonometry.htm

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computation of tric functions), but IMHO it's unreasonable to do "beta = beta mod pi" by

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https://web.archive.org/web/20160818120118/http://www.jacques-laporte.org/Logarithm_1.htm

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And Hewlett-Packard is known to have had decimal CORDIC implemented in 1966 internally.

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If you found an error with any archives or the URLs themselves, you can fix them with

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If it comes back or someone can find the new URL (if there is one) - please update it.

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Requested articles/Applied arts and sciences/Computer science, computing, and Internet

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C is more known than MATLAB. Please allow the example at least here. Thanks. :)

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should be exactly equal to the number of iterations. Not more and not less!

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Just for the record, here is the souce code in Python programming language

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Hermann Schmid and Anthony Bogacki described decimal CORDIC only in 1973.

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Ahh I see. You are right on all accounts. Thanks for the clarification.

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I have removed the following external link as it appears to be broken:

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An editor has reviewed this edit and fixed any errors that were found.

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has neither practical nor theoretical sense! So reference number is

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Of course, this can be included, but we need reliable references. --

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the more digits you use, the more precisely result you get. But for

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As such, they all belong to the class of shift-and-add algorithms.

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comments showing what the now somewhat obscured code is is doing).

2816:// If beta is negative, we need to do a counter-clockwise rotation: 1264: 1252: 579:

It seems interesting that the gain for the hyperbolic mode (m=-1)

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same engineers, before they worked on the 68000's co-processor.

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adding or subtracting pi! It requires a recursion of depth : -->

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is quoted in J.S. Walther's paper and in many other places as

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Find pictures for the biographies of computer scientists (see

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http://www.jacques-laporte.org/Briggs%20and%20the%20HP35.htm

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http://www.jacques-laporte.org/TheSecretOfTheAlgorithms.htm

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with a little more experience w/the algorithm can edit it.

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for additional information. I made the following changes:

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should be equal the wordlength of the computer. Unlimited

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beginning from the first publications to the recent ones.

660:{\displaystyle K(n)=\prod _{i=1}^{n-1}{\sqrt {1-2^{-2i}}}} 3183:

The distinction is not universally followed, but ideally

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you can find many other ideas incl. cordic accuracy in

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oh nvm I didn't read the bitshift implementation : -->

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It would be much more logical to put the References

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additions, subtractions, bitshift and lookup tables.

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What is the difference between NOTES and REFERENCES?

190:, a collaborative effort to improve the coverage of 101:, a collaborative effort to improve the coverage of 3150:I don`t understand what is the difference between 2405: 2342: 2199:00:55, 19 February 2007 (UTC)richmoore44@gmail.com 1008: 986: 961: 911: 875: 785: 722: 659: 302:Computer science articles needing expert attention 3309:http://www.jacques-laporte.org/digit_by_digit.htm 3100:I removed five dead links from "external links": 3289:http://www.jacques-laporte.org//Trigonometry.htm 2307: 840: 750: 687: 1260:and here we say about root mean square errors. 3319:http://www.jacques-laporte.org/Logarithm_1.htm 883:. It is well known, that number of iterations 442:WikiProject Computer science/Unreferenced BLPs 558:It's nice to have a discussion. Best wishes! 8: 3518:(e.g. in simple microcontrollers and FPGAs). 3083:http://www.dspguru.com/info/faqs/cordic2.htm 1094:I have clarified the WP article accordingly. 510:http://www.dspguru.com/info/faqs/cordic2.htm 796:Has anyone else ran into this discrepency? 508:See the article in the references section: 359:Computer science articles without infoboxes 297:Computer science articles needing attention 2524:About verifying, just type in Mathematica: 263:Here are some tasks awaiting attention: 237: 152: 47: 3515:when no hardware multiplier is available 3257:I have just modified 5 external links on 2392: 2387: 2310: 2298: 1001: 979: 948: 943: 898: 893: 843: 831: 820:There is no sense to say of the limit of 753: 741: 690: 678: 643: 631: 619: 608: 587: 556:http://www.andraka.com/files/crdcsrvy.pdf 3620:Mid-importance Computer science articles 733:If you run it it seems to converge to : 3523:as the only operations it requires are 2633:// Number of times to run the algorithm 154: 49: 19: 3560:More Formalized can't be the algorithm 2378:. If instead of that we use suggested 2376:be necessary to evaluate just for n=32 2366:. Did anybody see a computer with the 2184:CurrAngle) { NewX = X - (Y : --> 934:. If instead of that we use suggested 204:Knowledge:WikiProject Computer science 3625:WikiProject Computer science articles 207:Template:WikiProject Computer science 7: 184:This article is within the scope of 95:This article is within the scope of 2416:In the other words the coefficient 1356:# convert from degrees into radians 932:necessary to evaluate just for n=32 38:It is of interest to the following 3406:suck way more than you realize. :) 2399: 2389: 2317: 955: 945: 905: 895: 850: 760: 697: 378:Timeline of computing 2020–present 14: 3615:B-Class Computer science articles 3610:Low-priority mathematics articles 3261:. Please take a moment to review 2382:=0.6072529350088812561694 (for : 938:=0.6072529350088812561694 (for : 404:Computing articles needing images 115:Knowledge:WikiProject Mathematics 3559: 3327: 2406:{\displaystyle \_{n\to \infty }} 962:{\displaystyle \_{n\to \infty }} 912:{\displaystyle \_{n\to \infty }} 254: 177: 156: 118:Template:WikiProject Mathematics 82: 72: 51: 20: 2771:// To store the new value of x; 2681:// Vector v; x and y components 224:This article has been rated as 135:This article has been rated as 3400:13:42, 11 September 2018 (UTC) 3124:Usenet discussion about CORDIC 2396: 2331: 2325: 2314: 952: 902: 864: 858: 847: 774: 768: 757: 711: 705: 694: 598: 592: 1: 3555:13:25, 3 September 2020 (UTC) 3095:11:20, 12 November 2010 (UTC) 2595:This is copied from history: 2413:) then error would be large! 2204:00:55, 19 February 2007 (UTC) 1157:08:28, 22 November 2007 (UTC) 458:Tag all relevant articles in 198:and see a list of open tasks. 109:and see a list of open tasks. 3605:B-Class mathematics articles 3451:and closely related methods 3201:23:48, 12 October 2013 (UTC) 3177:13:41, 12 October 2013 (UTC) 3140:23:05, 20 January 2016 (UTC) 2282:01:36, 13 January 2007 (UTC) 2270:21:37, 12 January 2007 (UTC) 2219:07:46, 10 October 2006 (UTC) 1173:10:52, 2 February 2011 (UTC) 1121:17:58, 6 February 2016 (UTC) 1107:23:24, 20 January 2016 (UTC) 815:04:35, 1 February 2011 (UTC) 467:WikiProject Computer science 243:WikiProject Computer science 187:WikiProject Computer science 3244:01:06, 4 January 2016 (UTC) 3228:20:28, 8 October 2014 (UTC) 3207:(Co)Processors using Cordic 2566:09:05, 2 January 2007 (UTC) 2549:09:03, 2 January 2007 (UTC) 2514:08:59, 2 January 2007 (UTC) 2491:08:56, 2 January 2007 (UTC) 2480:08:51, 2 January 2007 (UTC) 2470:08:49, 2 January 2007 (UTC) 2252:in the chronological order: 2240:01:05, 4 January 2016 (UTC) 1243:17:23, 11 August 2006 (UTC) 1227:10:58, 11 August 2006 (UTC) 500:Value of K in implentations 398:List of computer scientists 3641: 3590:02:34, 17 March 2023 (UTC) 3574:02:28, 17 March 2023 (UTC) 3377:inefficient implementation 3254:Hello fellow Wikipedians, 2580:About the following edit: 2190:Step); Y = -(X : --> 1203:21:49, Jul 26, 2004 (UTC) 573:07:03, 24 March 2006 (UTC) 563:04:59, 24 March 2006 (UTC) 548:21:08, 23 March 2006 (UTC) 537:11:16, 23 March 2006 (UTC) 527:19:56, 21 March 2006 (UTC) 230:project's importance scale 3434:06:19, 20 July 2023 (UTC) 3372:17:50, 28 July 2017 (UTC) 3119:Another USENET discussion 2534:N, {n, 0, Infinity}], 22] 2456:21:25, 20 July 2010 (UTC) 2438:21:25, 18 July 2010 (UTC) 2368:unlimited length of word 2354:The number of iterations 2186:Step); Y = (X : --> 1251:you can have a look at: 1078:19:29, 29 July 2010 (UTC) 1050:01:59, 4 March 2006 (UTC) 1040:20:01, 3 March 2006 (UTC) 505:get the true magnitude. 460:Category:Computer science 236: 223: 210:Computer science articles 172: 134: 67: 46: 3419:17:18, 8 July 2022 (UTC) 2600: 2591:08:39, 18 May 2007 (UTC) 2338:0.6072529350088812561694 1281: 871:0.6072529350088812561694 462:and sub-categories with 141:project's priority scale 3250:External links modified 3187:are clarifications and 2444:history and application 98:WikiProject Mathematics 3547: 2407: 2344: 1010: 988: 963: 913: 877: 787: 724: 661: 630: 423:Computer science stubs 28:This article is rated 3467:pseudo-multiplication 3445: 3077:External Link Removed 2408: 2345: 2260:comment was added by 2168:Here is a sample run 1011: 989: 964: 914: 878: 788: 725: 662: 604: 3160:chronological order! 3109:The CORDIC Algorithm 2386: 2364:unlimited wordlength 2297: 1009:{\displaystyle \pi } 1000: 987:{\displaystyle \pi } 978: 942: 892: 830: 740: 677: 586: 241:Things you can help 121:mathematics articles 2576:Implementation in C 1271:vladimir@baykov.de 1061:Decimal CORDIC was 973:is not the Number 3512:are commonly used 3360:InternetArchiveBot 3104:CORDIC information 2403: 2340: 2321: 1268:Vladimir Baykov 1006: 984: 959: 909: 873: 854: 783: 764: 720: 701: 657: 90:Mathematics portal 34:content assessment 3218:comment added by 3167:comment added by 3114:USENET discussion 2306: 2273: 1284:#!/usr/bin/python 1159: 1143:comment added by 1129:Strange statement 839: 805:comment added by 749: 686: 655: 497: 496: 493: 492: 489: 488: 485: 484: 481: 480: 151: 150: 147: 146: 3632: 3497:factor combining 3370: 3361: 3334: 3331: 3330: 3230: 3179: 3072: 3069: 3066: 3063: 3060: 3057: 3054: 3051: 3048: 3045: 3042: 3039: 3036: 3033: 3030: 3027: 3024: 3021: 3018: 3015: 3012: 3009: 3006: 3003: 3000: 2997: 2994: 2991: 2988: 2985: 2982: 2979: 2976: 2973: 2970: 2967: 2964: 2961: 2958: 2955: 2952: 2949: 2946: 2943: 2940: 2937: 2934: 2931: 2928: 2925: 2922: 2919: 2916: 2913: 2910: 2907: 2904: 2901: 2898: 2895: 2892: 2889: 2886: 2883: 2880: 2877: 2874: 2871: 2868: 2865: 2862: 2859: 2856: 2853: 2850: 2847: 2844: 2841: 2838: 2835: 2832: 2829: 2826: 2823: 2820: 2817: 2814: 2811: 2808: 2805: 2802: 2799: 2796: 2793: 2790: 2787: 2784: 2781: 2778: 2775: 2772: 2769: 2766: 2763: 2760: 2757: 2754: 2751: 2748: 2745: 2742: 2739: 2736: 2733: 2730: 2727: 2724: 2721: 2718: 2715: 2712: 2709: 2706: 2703: 2700: 2697: 2694: 2691: 2688: 2685: 2682: 2679: 2676: 2673: 2670: 2667: 2664: 2661: 2658: 2655: 2652: 2649: 2646: 2643: 2640: 2637: 2634: 2631: 2628: 2625: 2622: 2619: 2616: 2613: 2610: 2607: 2604: 2535: 2412: 2410: 2409: 2404: 2402: 2362:corresponds to 2349: 2347: 2346: 2341: 2320: 2255: 2164: 2161: 2158: 2155: 2152: 2149: 2146: 2143: 2140: 2137: 2134: 2131: 2128: 2125: 2122: 2119: 2116: 2113: 2110: 2107: 2104: 2101: 2098: 2095: 2092: 2089: 2086: 2083: 2080: 2077: 2074: 2071: 2068: 2065: 2062: 2059: 2056: 2053: 2050: 2047: 2044: 2041: 2038: 2035: 2032: 2029: 2026: 2023: 2020: 2017: 2014: 2011: 2008: 2005: 2002: 1999: 1996: 1993: 1990: 1987: 1984: 1981: 1978: 1975: 1972: 1969: 1966: 1963: 1960: 1957: 1954: 1951: 1948: 1945: 1942: 1939: 1936: 1933: 1930: 1927: 1924: 1921: 1918: 1915: 1912: 1909: 1906: 1903: 1900: 1897: 1894: 1891: 1888: 1885: 1882: 1879: 1876: 1873: 1870: 1867: 1864: 1861: 1858: 1855: 1852: 1849: 1846: 1843: 1840: 1837: 1834: 1831: 1828: 1825: 1822: 1819: 1816: 1813: 1810: 1807: 1804: 1801: 1798: 1795: 1792: 1789: 1786: 1783: 1780: 1777: 1774: 1771: 1768: 1765: 1762: 1759: 1756: 1753: 1750: 1747: 1744: 1741: 1738: 1735: 1732: 1729: 1726: 1723: 1720: 1717: 1714: 1711: 1708: 1705: 1702: 1699: 1696: 1693: 1690: 1687: 1684: 1681: 1678: 1675: 1672: 1669: 1666: 1663: 1660: 1657: 1654: 1651: 1648: 1645: 1642: 1639: 1636: 1633: 1630: 1627: 1624: 1621: 1618: 1615: 1612: 1609: 1606: 1603: 1600: 1597: 1594: 1591: 1588: 1585: 1582: 1579: 1576: 1573: 1570: 1567: 1564: 1561: 1558: 1555: 1552: 1549: 1546: 1543: 1540: 1537: 1534: 1531: 1528: 1525: 1522: 1519: 1516: 1513: 1510: 1507: 1504: 1501: 1498: 1495: 1492: 1489: 1486: 1483: 1480: 1477: 1474: 1471: 1468: 1465: 1462: 1459: 1456: 1453: 1450: 1447: 1444: 1441: 1438: 1435: 1432: 1429: 1426: 1423: 1420: 1417: 1414: 1411: 1408: 1405: 1402: 1399: 1396: 1393: 1390: 1387: 1384: 1381: 1378: 1375: 1372: 1369: 1366: 1363: 1360: 1357: 1354: 1351: 1348: 1345: 1342: 1339: 1336: 1333: 1330: 1327: 1324: 1321: 1318: 1315: 1312: 1309: 1306: 1303: 1300: 1297: 1294: 1291: 1288: 1285: 1201: 1196: 1189: 1163:is not exact. -- 1138: 1015: 1013: 1012: 1007: 993: 991: 990: 985: 968: 966: 965: 960: 958: 918: 916: 915: 910: 908: 882: 880: 879: 874: 853: 817: 792: 790: 789: 784: 763: 729: 727: 726: 721: 700: 666: 664: 663: 658: 656: 654: 653: 632: 629: 618: 471: 465: 340:Computer science 269:Article requests 258: 251: 250: 238: 212: 211: 208: 205: 202: 201:Computer science 192:Computer science 181: 174: 173: 168: 164:Computer science 160: 153: 123: 122: 119: 116: 113: 92: 87: 86: 76: 69: 68: 63: 55: 48: 31: 25: 24: 16: 3640: 3639: 3635: 3634: 3633: 3631: 3630: 3629: 3595: 3594: 3562: 3483:pseudo-division 3444: 3379: 3364: 3359: 3332: 3328: 3267:this simple FaQ 3252: 3213: 3209: 3162: 3148: 3079: 3074: 3073: 3070: 3067: 3064: 3061: 3058: 3055: 3052: 3049: 3046: 3043: 3040: 3037: 3034: 3031: 3028: 3025: 3022: 3019: 3016: 3013: 3010: 3007: 3004: 3001: 2998: 2995: 2992: 2989: 2986: 2983: 2980: 2977: 2974: 2971: 2968: 2965: 2962: 2959: 2956: 2953: 2950: 2947: 2944: 2941: 2938: 2935: 2932: 2929: 2926: 2923: 2920: 2917: 2914: 2911: 2908: 2905: 2902: 2899: 2896: 2893: 2890: 2887: 2884: 2881: 2878: 2875: 2872: 2869: 2866: 2863: 2860: 2857: 2854: 2851: 2848: 2845: 2842: 2839: 2836: 2833: 2830: 2827: 2824: 2821: 2818: 2815: 2812: 2809: 2806: 2803: 2800: 2797: 2794: 2791: 2788: 2785: 2782: 2779: 2776: 2773: 2770: 2767: 2764: 2761: 2758: 2755: 2752: 2749: 2746: 2743: 2740: 2737: 2734: 2731: 2728: 2725: 2722: 2719: 2716: 2713: 2710: 2707: 2704: 2701: 2698: 2695: 2692: 2689: 2686: 2683: 2680: 2677: 2674: 2671: 2668: 2665: 2662: 2659: 2656: 2653: 2650: 2647: 2644: 2641: 2638: 2635: 2632: 2629: 2626: 2623: 2620: 2617: 2614: 2611: 2608: 2605: 2602: 2578: 2533: 2463: 2461:Numerical value 2448:Vladimir Baykov 2430:Vladimir Baykov 2384: 2383: 2295: 2294: 2256:—The preceding 2248: 2211: 2194: 2177: 2172: 2166: 2165: 2162: 2159: 2156: 2153: 2150: 2147: 2144: 2141: 2138: 2135: 2132: 2129: 2126: 2123: 2120: 2117: 2114: 2111: 2108: 2105: 2102: 2099: 2096: 2093: 2090: 2087: 2084: 2081: 2078: 2075: 2072: 2069: 2066: 2063: 2060: 2057: 2054: 2051: 2048: 2045: 2042: 2039: 2036: 2033: 2030: 2027: 2024: 2021: 2018: 2015: 2012: 2009: 2006: 2003: 2000: 1997: 1994: 1991: 1988: 1985: 1982: 1979: 1976: 1973: 1970: 1967: 1964: 1961: 1958: 1955: 1952: 1949: 1946: 1943: 1940: 1937: 1934: 1931: 1928: 1925: 1922: 1919: 1916: 1913: 1910: 1907: 1904: 1901: 1898: 1895: 1892: 1889: 1886: 1883: 1880: 1877: 1874: 1871: 1868: 1865: 1862: 1859: 1856: 1853: 1850: 1847: 1844: 1841: 1838: 1835: 1832: 1829: 1826: 1823: 1820: 1817: 1814: 1811: 1808: 1805: 1802: 1799: 1796: 1793: 1790: 1787: 1784: 1781: 1778: 1775: 1772: 1769: 1766: 1763: 1760: 1757: 1754: 1751: 1748: 1745: 1742: 1739: 1736: 1733: 1730: 1727: 1724: 1721: 1718: 1715: 1712: 1709: 1706: 1703: 1700: 1697: 1694: 1691: 1688: 1685: 1682: 1679: 1676: 1673: 1670: 1667: 1664: 1661: 1658: 1655: 1652: 1649: 1646: 1643: 1640: 1637: 1634: 1631: 1628: 1625: 1622: 1619: 1616: 1613: 1610: 1607: 1604: 1601: 1598: 1595: 1592: 1589: 1586: 1583: 1580: 1577: 1574: 1571: 1568: 1565: 1562: 1559: 1556: 1553: 1550: 1547: 1544: 1541: 1538: 1535: 1532: 1529: 1526: 1523: 1520: 1517: 1514: 1511: 1508: 1505: 1502: 1499: 1496: 1493: 1490: 1487: 1484: 1481: 1478: 1475: 1472: 1469: 1466: 1463: 1460: 1457: 1454: 1451: 1448: 1445: 1442: 1439: 1436: 1433: 1430: 1427: 1424: 1421: 1418: 1415: 1412: 1409: 1406: 1403: 1400: 1397: 1394: 1391: 1388: 1385: 1382: 1379: 1376: 1373: 1370: 1367: 1364: 1361: 1358: 1355: 1352: 1349: 1346: 1343: 1340: 1337: 1334: 1331: 1328: 1325: 1322: 1319: 1316: 1313: 1310: 1307: 1304: 1301: 1298: 1295: 1292: 1289: 1286: 1283: 1277: 1258: 1248:about accuracy 1220: 1199: 1192: 1187: 1180: 1145:Vladimir Baikov 1131: 1070:Vladimir Baykov 1063:first suggested 1059: 1032: 998: 997: 976: 975: 940: 939: 890: 889: 828: 827: 800: 738: 737: 675: 674: 639: 584: 583: 502: 477: 474: 469: 463: 451:Project-related 446: 427: 408: 382: 363: 344: 325: 306: 282: 209: 206: 203: 200: 199: 166: 120: 117: 114: 111: 110: 88: 81: 61: 32:on Knowledge's 29: 12: 11: 5: 3638: 3636: 3628: 3627: 3622: 3617: 3612: 3607: 3597: 3596: 3593: 3592: 3561: 3558: 3543: 3542: 3536: 3535: 3532: 3531: 3530: 3529: 3528: 3527: 3521: 3520: 3519: 3509: 3508: 3507: 3506: 3505: 3504: 3503: 3502: 3501: 3500: 3492: 3491: 3487: 3486: 3485: 3484: 3476: 3475: 3471: 3470: 3469: 3468: 3458: 3457: 3456: 3455: 3443: 3440: 3439: 3438: 3437: 3436: 3407: 3378: 3375: 3354: 3353: 3346: 3322: 3321: 3313:Added archive 3311: 3303:Added archive 3301: 3293:Added archive 3291: 3283:Added archive 3281: 3273:Added archive 3251: 3248: 3247: 3246: 3220:131.107.147.56 3208: 3205: 3204: 3203: 3191:are sources. — 3147: 3144: 3143: 3142: 3128: 3127: 3126: 3121: 3116: 3111: 3106: 3087:194.72.120.131 3078: 3075: 2601: 2577: 2574: 2573: 2572: 2571: 2570: 2569: 2568: 2554: 2553: 2552: 2551: 2539: 2538: 2537: 2536: 2528: 2527: 2526: 2525: 2519: 2518: 2517: 2516: 2503: 2502: 2501: 2500: 2494: 2493: 2462: 2459: 2420:for unlimited 2401: 2398: 2395: 2391: 2352: 2351: 2339: 2336: 2333: 2330: 2327: 2324: 2319: 2316: 2313: 2309: 2305: 2302: 2285: 2284: 2247: 2244: 2243: 2242: 2228: 2225: 2210: 2207: 2201:86.152.208.199 2197:86.152.208.199 2181: 2176: 2173: 2170: 1282: 1276: 1273: 1256: 1247: 1231: 1219: 1216: 1214: 1212: 1211: 1208: 1179: 1176: 1130: 1127: 1126: 1125: 1124: 1123: 1095: 1092: 1088: 1084: 1058: 1057:Decimal CORDIC 1055: 1054: 1053: 1052: 1031: 1030:Mixing i and n 1028: 1027: 1005: 995:!! For Number 983: 957: 954: 951: 947: 921: 920: 907: 904: 901: 897: 872: 869: 866: 863: 860: 857: 852: 849: 846: 842: 838: 835: 798: 794: 793: 782: 779: 776: 773: 770: 767: 762: 759: 756: 752: 748: 745: 731: 730: 719: 716: 713: 710: 707: 704: 699: 696: 693: 689: 685: 682: 668: 667: 652: 649: 646: 642: 638: 635: 628: 625: 622: 617: 614: 611: 607: 603: 600: 597: 594: 591: 577: 576: 575: 551: 550: 517: 516: 501: 498: 495: 494: 491: 490: 487: 486: 483: 482: 479: 478: 476: 475: 473: 472: 455: 447: 445: 444: 438: 428: 426: 425: 419: 409: 407: 406: 401: 393: 383: 381: 380: 374: 364: 362: 361: 355: 345: 343: 342: 336: 326: 324: 323: 317: 307: 305: 304: 299: 293: 283: 281: 280: 274: 262: 260: 259: 247: 246: 234: 233: 226:Mid-importance 222: 216: 215: 213: 196:the discussion 182: 170: 169: 167:Mid‑importance 161: 149: 148: 145: 144: 133: 127: 126: 124: 107:the discussion 94: 93: 77: 65: 64: 56: 44: 43: 37: 26: 13: 10: 9: 6: 4: 3: 2: 3637: 3626: 3623: 3621: 3618: 3616: 3613: 3611: 3608: 3606: 3603: 3602: 3600: 3591: 3587: 3583: 3582:174.63.44.188 3578: 3577: 3576: 3575: 3571: 3567: 3566:174.63.44.188 3557: 3556: 3553: 3546: 3541: 3540: 3539: 3534: 3533: 3525: 3524: 3522: 3517: 3516: 3514: 3513: 3511: 3510: 3499: 3496: 3495: 3494: 3493: 3489: 3488: 3482: 3481: 3480: 3479: 3478: 3477: 3473: 3472: 3466: 3465: 3464: 3463: 3462: 3461: 3460: 3459: 3453: 3452: 3450: 3449: 3448: 3442:Bury the lede 3441: 3435: 3431: 3427: 3422: 3421: 3420: 3416: 3412: 3408: 3404: 3403: 3402: 3401: 3398: 3394: 3388: 3384: 3376: 3374: 3373: 3368: 3363: 3362: 3351: 3347: 3344: 3340: 3339: 3338: 3337: 3325: 3320: 3316: 3312: 3310: 3306: 3302: 3300: 3296: 3292: 3290: 3286: 3282: 3280: 3276: 3272: 3271: 3270: 3268: 3264: 3260: 3255: 3249: 3245: 3241: 3237: 3233: 3232: 3231: 3229: 3225: 3221: 3217: 3206: 3202: 3198: 3194: 3190: 3186: 3182: 3181: 3180: 3178: 3174: 3170: 3169:134.99.136.28 3166: 3161: 3157: 3153: 3145: 3141: 3137: 3133: 3129: 3125: 3122: 3120: 3117: 3115: 3112: 3110: 3107: 3105: 3102: 3101: 3099: 3098: 3097: 3096: 3092: 3088: 3084: 3076: 2645:// in Radians 2599: 2596: 2593: 2592: 2589: 2588:130.83.161.82 2584: 2581: 2575: 2567: 2564: 2560: 2559: 2558: 2557: 2556: 2555: 2550: 2547: 2543: 2542: 2541: 2540: 2532: 2531: 2530: 2529: 2523: 2522: 2521: 2520: 2515: 2512: 2507: 2506: 2505: 2504: 2498: 2497: 2496: 2495: 2492: 2489: 2484: 2483: 2482: 2481: 2478: 2472: 2471: 2468: 2460: 2458: 2457: 2453: 2449: 2445: 2440: 2439: 2435: 2431: 2427: 2423: 2419: 2414: 2393: 2381: 2377: 2373: 2369: 2365: 2361: 2357: 2337: 2334: 2328: 2322: 2311: 2303: 2300: 2293: 2292: 2291: 2290: 2283: 2280: 2276: 2275: 2274: 2271: 2267: 2263: 2262:80.133.27.236 2259: 2253: 2245: 2241: 2237: 2233: 2229: 2226: 2223: 2222: 2221: 2220: 2217: 2208: 2206: 2205: 2202: 2198: 2180: 2174: 2169: 2148:main_function 1332:main_function 1280: 1274: 1272: 1269: 1266: 1265: 1261: 1255: 1253: 1249: 1245: 1244: 1241: 1234: 1233: 1229: 1228: 1225: 1217: 1215: 1209: 1206: 1205: 1204: 1202: 1197: 1195: 1190: 1184: 1177: 1175: 1174: 1170: 1166: 1160: 1158: 1154: 1150: 1146: 1142: 1134: 1128: 1122: 1118: 1114: 1110: 1109: 1108: 1104: 1100: 1096: 1093: 1089: 1085: 1082: 1081: 1080: 1079: 1075: 1071: 1066: 1064: 1056: 1051: 1048: 1044: 1043: 1042: 1041: 1038: 1029: 1026: 1024: 1020: 1016: 1003: 994: 981: 972: 949: 937: 933: 929: 924: 899: 886: 870: 867: 861: 855: 844: 836: 833: 826: 825: 824: 823: 818: 816: 812: 808: 807:71.228.178.91 804: 797: 780: 777: 771: 765: 754: 746: 743: 736: 735: 734: 717: 714: 708: 702: 691: 683: 680: 673: 672: 671: 650: 647: 644: 640: 636: 633: 626: 623: 620: 615: 612: 609: 605: 601: 595: 589: 582: 581: 580: 574: 571: 567: 566: 565: 564: 561: 557: 549: 546: 541: 540: 539: 538: 535: 529: 528: 525: 521: 514: 513: 512: 511: 506: 499: 468: 461: 457: 456: 454: 452: 448: 443: 440: 439: 437: 435: 434: 429: 424: 421: 420: 418: 416: 415: 410: 405: 402: 399: 395: 394: 392: 390: 389: 384: 379: 376: 375: 373: 371: 370: 365: 360: 357: 356: 354: 352: 351: 346: 341: 338: 337: 335: 333: 332: 327: 322: 319: 318: 316: 314: 313: 308: 303: 300: 298: 295: 294: 292: 290: 289: 284: 279: 276: 275: 273: 271: 270: 265: 264: 261: 257: 253: 252: 249: 248: 244: 240: 239: 235: 231: 227: 221: 218: 217: 214: 197: 193: 189: 188: 183: 180: 176: 175: 171: 165: 162: 159: 155: 142: 138: 132: 129: 128: 125: 108: 104: 100: 99: 91: 85: 80: 78: 75: 71: 70: 66: 60: 57: 54: 50: 45: 41: 35: 27: 23: 18: 17: 3563: 3548: 3544: 3537: 3498: 3446: 3386: 3382: 3380: 3358: 3355: 3335: 3326: 3323: 3256: 3253: 3236:Matthiaspaul 3214:— Preceding 3210: 3188: 3184: 3163:— Preceding 3159: 3155: 3151: 3149: 3132:Matthiaspaul 3080: 2597: 2594: 2585: 2582: 2579: 2473: 2464: 2443: 2441: 2425: 2421: 2417: 2415: 2379: 2375: 2371: 2367: 2363: 2359: 2355: 2353: 2288: 2286: 2251: 2249: 2232:Matthiaspaul 2212: 2195: 2178: 2167: 1278: 1270: 1267: 1262: 1259: 1250: 1246: 1235: 1230: 1221: 1213: 1193: 1181: 1165:89.146.28.90 1161: 1135: 1132: 1113:Matthiaspaul 1099:Matthiaspaul 1067: 1062: 1060: 1033: 1022: 1018: 970: 935: 931: 927: 925: 922: 884: 821: 819: 799: 795: 732: 669: 578: 552: 530: 522: 518: 507: 503: 450: 449: 433:Unreferenced 431: 430: 412: 411: 386: 385: 367: 366: 348: 347: 329: 328: 310: 309: 286: 285: 267: 266: 225: 185: 137:Low-priority 136: 96: 62:Low‑priority 40:WikiProjects 3411:WhyYouAskMe 3387:recursively 3026:arctanTable 2924:arctanTable 2726:arctanTable 2639:arctanTable 2426:unnecessary 2224:Both... ;-) 1830:ArcTanTable 1722:ArcTanTable 1407:ArcTanTable 1380:ArcTanTable 1139:—Preceding 1091:explicitly. 1021:the number 801:—Preceding 112:Mathematics 103:mathematics 59:Mathematics 3599:Categories 3367:Report bug 3189:References 3156:REFERENCES 2798:iterations 2708:iterations 2621:iterations 2246:References 2175:JavaScript 1971:'__main__' 1884:iterations 1602:iterations 1485:iterations 1401:iterations 1344:iterations 1290:__future__ 923:Nonsense! 3454:known as: 3426:-- Axman6 3383:efficient 3350:this tool 3343:this tool 1037:Addps4cat 570:Addps4cat 545:Addps4cat 524:Addps4cat 321:Computing 3447:CORDIC 3356:Cheers.— 3216:unsigned 3165:unsigned 2563:Dicklyon 2488:Dicklyon 2279:Dicklyon 2258:unsigned 2216:Graemec2 1965:__name__ 1917:and Cos( 1296:division 1240:bloublou 1224:bloublou 1218:accuracy 1178:Disputed 1153:contribs 1141:unsigned 803:unsigned 369:Maintain 312:Copyedit 3263:my edit 3193:Tamfang 2374:should 2209:Strange 1950:degrees 1938:degrees 1926:%14.12f 1914:%14.12f 1872:%14.12f 1365:degrees 1338:degrees 930:should 560:Cdpango 534:Cdpango 350:Infobox 288:Cleanup 228:on the 139:on the 30:B-class 3580:.< 3552:MaxEnt 3259:CORDIC 3041:vnew_x 2942:vnew_x 2840:vnew_x 2765:vnew_x 2657:0.6073 2442:About 1548:append 1413:append 1323:import 1305:import 1293:import 1275:Python 1047:Henrik 331:Expand 36:scale. 3185:Notes 3152:NOTES 2762:float 2666:float 2648:float 2636:float 2191:: --> 2189:: --> 2187:: --> 2185:: --> 2124:print 2076:float 2013:float 1920:%4.1f 1908:%4.1f 1905:"Sin( 1902:print 1869:"K = 1866:print 1596:range 1560:value 1497:value 1491:value 1479:range 1461:value 1395:range 1377:180.0 781:0.828 414:Stubs 388:Photo 245:with: 3586:talk 3570:talk 3430:talk 3415:talk 3397:Talk 3240:talk 3224:talk 3197:talk 3173:talk 3154:and 3136:talk 3091:talk 3020:beta 2936:else 2918:beta 2828:< 2825:beta 2795:< 2732:atan 2705:< 2663:// K 2546:Ylai 2511:Ylai 2477:Ylai 2467:Ylai 2452:talk 2434:talk 2266:talk 2236:talk 2160:iter 2139:iter 2136:17.0 2118:else 2112:argv 2094:iter 2088:argv 2055:argv 2040:elif 2031:iter 2025:argv 1992:argv 1923:) = 1911:) = 1824:beta 1818:beta 1725:else 1716:beta 1710:beta 1614:< 1611:beta 1503:sqrt 1419:atan 1359:beta 1320:sqrt 1308:atan 1302:math 1299:from 1287:from 1188:siro 1169:talk 1149:talk 1117:talk 1103:talk 1074:talk 811:talk 718:0.80 3474:and 3409:~~ 3393:MFH 3317:to 3307:to 3297:to 3287:to 3277:to 3062:v_y 3050:v_x 3035:v_x 3017:)); 2999:pow 2993:v_x 2984:v_y 2981:)); 2963:pow 2957:v_y 2948:v_x 2915:)); 2897:pow 2891:v_x 2882:v_y 2879:)); 2861:pow 2855:v_y 2846:v_x 2774:for 2756:)); 2738:pow 2684:for 2675:v_y 2669:v_x 2618:int 2603:int 2428:. 2308:lim 2154:deg 2130:deg 2106:sys 2100:int 2082:sys 2070:deg 2049:sys 2043:len 2019:sys 2007:deg 1986:sys 1980:len 1794:2.0 1755:2.0 1686:2.0 1647:2.0 1587:for 1584:0.0 1578:1.0 1554:1.0 1515:2.0 1509:1.0 1470:for 1467:1.0 1425:2.0 1386:for 1329:def 1326:sys 841:lim 751:lim 688:lim 220:Mid 131:Low 3601:: 3588:) 3572:) 3490:or 3432:) 3417:) 3242:) 3226:) 3199:) 3175:) 3138:) 3130:-- 3093:) 3065:*= 3053:*= 3023:-= 2987:+= 2921:+= 2885:-= 2819:if 2807:++ 2717:++ 2454:) 2436:) 2400:∞ 2397:→ 2390:_ 2335:≈ 2318:∞ 2315:→ 2268:) 2238:) 2230:-- 2061:== 1998:== 1977:if 1968:== 1962:if 1956:Vx 1944:Vy 1896:KN 1863:KN 1857:Vy 1851:KN 1845:Vx 1839:Vy 1833:Vx 1797:** 1788:Vx 1782:Vy 1758:** 1749:Vy 1743:Vx 1737:Vy 1731:Vx 1689:** 1680:Vx 1674:Vy 1650:** 1641:Vy 1635:Vx 1629:Vy 1623:Vx 1608:if 1605:): 1593:in 1572:Vy 1566:Vx 1542:KN 1518:** 1488:): 1476:in 1455:KN 1428:** 1404:): 1392:in 1371:pi 1353:): 1314:pi 1171:) 1155:) 1151:• 1119:) 1105:) 1097:-- 1076:) 1004:π 982:π 956:∞ 953:→ 946:_ 906:∞ 903:→ 896:_ 868:≈ 851:∞ 848:→ 813:) 778:≈ 761:∞ 758:→ 715:≈ 698:∞ 695:→ 645:− 637:− 624:− 606:∏ 470:}} 464:{{ 3584:( 3568:( 3428:( 3413:( 3395:: 3369:) 3365:( 3352:. 3345:. 3333:Y 3238:( 3222:( 3195:( 3171:( 3134:( 3089:( 3071:; 3068:K 3059:; 3056:K 3047:} 3044:; 3038:= 3032:} 3029:; 3014:i 3011:- 3008:, 3005:2 3002:( 2996:* 2990:( 2978:i 2975:- 2972:, 2969:2 2966:( 2960:* 2954:( 2951:- 2945:= 2939:{ 2930:} 2927:; 2912:i 2909:- 2906:, 2903:2 2900:( 2894:* 2888:( 2876:i 2873:- 2870:, 2867:2 2864:( 2858:* 2852:( 2849:+ 2843:= 2837:{ 2834:) 2831:0 2822:( 2813:{ 2810:) 2804:i 2801:; 2792:i 2789:; 2786:0 2783:= 2780:i 2777:( 2768:; 2759:} 2753:i 2750:- 2747:, 2744:2 2741:( 2735:( 2729:= 2723:{ 2720:) 2714:i 2711:; 2702:i 2699:; 2696:0 2693:= 2690:i 2687:( 2678:; 2672:, 2660:; 2654:= 2651:K 2642:; 2630:; 2627:0 2624:= 2615:; 2612:0 2609:= 2606:i 2450:( 2432:( 2422:n 2418:K 2394:n 2380:K 2372:K 2360:n 2356:n 2350:. 2332:) 2329:n 2326:( 2323:K 2312:n 2304:= 2301:K 2289:K 2272:. 2264:( 2234:( 2163:) 2157:, 2151:( 2145:8 2142:= 2133:= 2121:: 2115:) 2109:. 2103:( 2097:= 2091:) 2085:. 2079:( 2073:= 2067:: 2064:3 2058:) 2052:. 2046:( 2037:8 2034:= 2028:) 2022:. 2016:( 2010:= 2004:: 2001:2 1995:) 1989:. 1983:( 1974:: 1959:) 1953:, 1947:, 1941:, 1935:( 1932:% 1929:" 1899:) 1893:, 1890:1 1887:- 1881:( 1878:% 1875:" 1860:* 1854:, 1848:* 1842:= 1836:, 1827:- 1821:= 1815:) 1812:i 1809:* 1806:1 1803:- 1800:( 1791:* 1785:+ 1779:, 1776:) 1773:i 1770:* 1767:1 1764:- 1761:( 1752:* 1746:- 1740:= 1734:, 1728:: 1719:+ 1713:= 1707:) 1704:i 1701:* 1698:1 1695:- 1692:( 1683:* 1677:- 1671:, 1668:) 1665:i 1662:* 1659:1 1656:- 1653:( 1644:* 1638:+ 1632:= 1626:, 1620:: 1617:0 1599:( 1590:i 1581:, 1575:= 1569:, 1563:) 1557:/ 1551:( 1545:. 1539:) 1536:) 1533:i 1530:* 1527:2 1524:- 1521:( 1512:+ 1506:( 1500:* 1494:= 1482:( 1473:i 1464:= 1458:= 1452:) 1449:) 1446:) 1443:i 1440:* 1437:1 1434:- 1431:( 1422:( 1416:( 1410:. 1398:( 1389:i 1383:= 1374:/ 1368:* 1362:= 1350:8 1347:= 1341:, 1335:( 1317:, 1311:, 1200:o 1194:χ 1185:— 1167:( 1147:( 1115:( 1101:( 1072:( 1023:n 1019:K 971:K 950:n 936:K 928:K 919:? 900:n 885:n 865:) 862:n 859:( 856:K 845:n 837:= 834:K 822:K 809:( 775:) 772:n 769:( 766:K 755:n 747:= 744:K 712:) 709:n 706:( 703:K 692:n 684:= 681:K 651:i 648:2 641:2 634:1 627:1 621:n 616:1 613:= 610:i 602:= 599:) 596:n 593:( 590:K 453:: 436:: 417:: 400:) 391:: 372:: 353:: 334:: 315:: 291:: 272:: 232:. 143:. 42::

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