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sqrt_test's Introduction

6502 Integer Square Root - which is best?

The purpose of this page is to compare the performance and memory cost of several different implementations of a 16 bit integer square root on the 6502 CPU, to find out which is best. This function is sometimes known as isqrt, and conventionally it rounds down the result, so the result fits in 8 bits.

See the Wikipedia page for integer square root for details of algorithms.

We execute each routine exhaustively over all 65536 possible inputs, record the cycle count for each and graph the results.

Implementations tested

All implementations have been sourced from the internet and reformatted for the acme assembler. See here for the actual files.

file origin notes
sqrt1.a https://codebase64.org/doku.php?id=base:fast_sqrt
sqrt2.a http://www.6502.org/source/integers/root.htm
sqrt3.a http://www.txbobsc.com/aal/1986/aal8611.html#a1 a table based solution.
sqrt4.a http://www.txbobsc.com/aal/1985/aal8506.html#a2 adds successive odd numbers
sqrt5.a http://www.txbobsc.com/aal/1986/aal8609.html#a8
sqrt6.a https://www.bbcelite.com/master/main/subroutine/ll5.html from the BBC Micro game Elite.
sqrt7.a http://6502org.wikidot.com/software-math-sqrt tweaked by me and 0xC0DE
sqrt8.a https://mdfs.net/Info/Comp/6502/ProgTips/SqRoot adds successive odd numbers
sqrt9.a https://github.com/TobyLobster/sqrt_test/blob/main/sqrt/sqrt9.a a table based solution, my version of sqrt3.a tweaked for performance.
sqrt10.a https://github.com/TobyLobster/sqrt_test/blob/main/sqrt/sqrt10.a my version of sqrt1.a tweaked for performance.
sqrt11.a https://github.com/TobyLobster/sqrt_test/blob/main/sqrt/sqrt11.a a table based solution, using binary search. from here fixed and tweaked for performance.
sqrt12.a https://gitlab.riscosopen.org/RiscOS/Sources/Apps/Diversions/Meteors/-/blob/master/Srce6502/MetSrc2#L961 from the BBC Micro game Acornsoft Meteors
sqrt13.a https://stardot.org.uk/forums/viewtopic.php?p=367937#p367937 by hexwab
sqrt14.a https://stardot.org.uk/forums/viewtopic.php?p=367937#p367937 by hexwab
sqrt15.a https://stardot.org.uk/forums/viewtopic.php?p=367937#p367937 by hexwab
sqrt16.a https://github.com/TobyLobster/sqrt_test/blob/main/sqrt/sqrt16.a adds successive odd numbers
sqrt17.a https://github.com/TobyLobster/sqrt_test/blob/main/sqrt/sqrt17.a
sqrt18.a https://github.com/TobyLobster/sqrt_test/blob/main/sqrt/sqrt18.a

Python Script

After assembling each file using acme, we use py65mon to load and execute the binary 6502, check the results are accurate and record the cycle count. The results are then output to a CSV file for graphing using a separate python program.

Results

All algorithms provide the correct results. We graph the cycle count of each algorithm over all possible inputs.

SQRT Performance Comparison

We see immediately that three of the algorithms are much slower compared to the rest. sqrt4 and sqrt8 and sqrt16 each calculate squares simply by adding successive odd numbers. This is extremely slow for anything but small numbers. So we can get a more useful picture by omitting these three:

SQRT Performance Comparison

The straight line in the middle is sqrt7, which (remarkably) takes constant time.

file memory (bytes) worst case cycles average cycle count
sqrt1.a 59 354 317.7
sqrt2.a 73 923 846.5
sqrt3.a 796 138 43.8
sqrt4.a 36 7451 4989.1
sqrt5.a 67 766 731.0
sqrt6.a 55 574 522.9
sqrt7.a 38 465 465.0
sqrt8.a 37 9483 6342.4
sqrt9.a 847 129 39.8
sqrt10.a 168 262 227.4
sqrt11.a 595 333 268.8
sqrt12.a 79 1315 1198.5
sqrt13.a 140 491 264.4
sqrt14.a 205 217 194.1
sqrt15.a 512 87 33.7
sqrt16.a 33 8205 5488.6
sqrt17.a 377 484 135.4
sqrt18.a 298 510 142.8

All cycle counts include the final RTS, but not any initial JSR. Add 6 cycles for an initial 'JSR sqrt' instruction.

It is still crowded at the bottom of this graph. Here are the fastest, table based solutions:

SQRT Performance Comparison, table based solutions

Conclusion

It's a speed vs memory trade off.

  • If speed is all important and you can afford 512 bytes of memory then use the fastest routine sqrt15.a.
  • If every byte counts, choose sqrt7.a (38 bytes).
  • If every byte REALLY REALLY counts, choose sqrt16.a (33 bytes) or sqrt4.a (36 bytes, but 32 if input number is already in registers), but be aware that these are about eleven times slower than sqrt7.a (38 bytes), and twenty four times slower than sqrt10.a.

Memory vs Speed Comparison

The orange dots are good candidates to use. The grey dots are the also-rans, don't choose these because there are faster and smaller versions in orange.

Note however: sqrt9 and sqrt3 have two tables of squares (512 bytes). This memory cost can be shared with a fast multiply routine like https://everything2.com/user/eurorusty/writeups/Fast+6502+multiplication which uses the same tables.

sqrt_test's People

Contributors

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