a * b = qs[a + b] - qs[|a - b|] where qs[n] is n squared over four
because (a+b)^2/4 minus (a-b)^2/4 is exactly a*b, and the halves the
flooring throws away cancel between the two terms. A multiply is two
lookups and a subtract.
AND THE TABLE IS BUILT BY ADDING, which is the part that makes it fit a
machine with no multiplier at all. A table of squares would need squaring
to fill; this one does not, because qs[n] = qs[n-1] + n/2, and n/2 goes 0,
1, 1, 2, 2, 3 - a number that steps up on every even n. So the whole thing
is a running total and a toggle, and nothing harder than an add appears
anywhere in building the thing that does the multiplying.
511 entries of two bytes, because a byte plus a byte reaches 510. That is
1,022 bytes of Data Memory, and it is the price: a kilobyte traded for an
operation the hardware has not got.
The operands go in memory rather than in registers. B cannot be stored and
a product does not fit in one byte anyway, so two in and two out would
spend more instructions shuffling than the multiply costs.
Checked against nought, the commutation both ways round, a square, and 255
times 255 - which is 0xFE01 and the largest product two bytes hold. The
square is the case the identity leans on hardest: the difference term is
nought and the whole answer comes out of one entry.
Wanted for Lunar Porter's orbit, where the trade between height and speed
has to be proportional to vx times vy and could not be. Useful well beyond
it: this is the routine every fixed point sum on this machine has been
doing without.
Co-Authored-By: Claude Opus 5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01E2JrLzFvuFX9fgi1LDRjrW
Twenty four places moved Q into A or B by pushing it and popping it back.
That is four bus cycles and two bytes to do what MVQA does in one of each,
and several of them are inside loops - Life, the calculator, int8. Nineteen
more loaded zero with INIA 0d0 where RSTA says the same thing in one byte.
Both are equivalent at the CPU rather than by assertion: RSTA and INIA both
leave Status alone, and PSHQ followed by POPA nets to A = Q with the Stack
Pointer where it started. The one difference is that the pair leaves a copy
of Q in memory just below the Stack Pointer and MVQA does not, which
nothing here reads.
Five recorded outputs moved and every one of them says the change worked:
- 16x16Life fits five more generations into the same cycle budget, the
first 457 lines identical, because the loop got cheaper.
- Life.sbx is 1409 bytes rather than 1411, in three tests that list it.
- Edit.sbx is 1995 rather than 1996.
That last one broke a check I added this morning, and the hole is worth
recording: the CosmOS README's claim about Edit's size did not have the
word "Edit" on the same line as the number, because the subject was in the
sentence before, so the check that measures quoted sizes skipped it
silently. The sentence now names what it is talking about, which makes it
both checkable and clearer, and the check fails on a wrong number there.
Comments on either half of a replaced pair are carried onto the
instruction that replaces them, so nothing anybody wrote was lost.
Thirteen build artifacts were tracked: ten .bin files, a disk image, and the
text file that had been put on it. Every one is produced by 'make' in
Programs/ from a source file that is also tracked.
NINE OF THE TEN BINARIES DISAGREED WITH THEIR OWN SOURCE, including
Programs/CosmOS/Bin/cosmos.bin - the operating system. Only hello.bin still
matched, and that by accident.
That is worse than shipping no binary at all. A repository holding both a
source file and a stale build of it has two answers to what a program does,
and the wrong one is the one that runs. Somebody reads the source, runs the
binary, and sees behaviour that is not in the code with nothing to tell them
why.
Nothing referenced any of them - not the makefiles, not the tests, not the
manuals. Checked before removing, and checked after: the suite passes, and a
clean clone with all of them gone builds both tools and assembles every
program from source alone.
.gitignore now covers *.bin, *.sbx and *.img, unanchored, because a build
output may appear anywhere a source file lives.
Programs/CosmOS/Bin/ held cosmos.bin, disk.img and a 17 byte hello.txt that
existed only to be put on that image. It was a snapshot of a demo from
before Tests/makedisks.sh built its own fixtures, and it goes with them.
Co-Authored-By: Claude Opus 5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01E2JrLzFvuFX9fgi1LDRjrW
Added DPUP and DPDN, which take a one byte operand specifying how far up or down to offset the Data Pointer.
Three Fibonacci generators using the print.asm library.
- 8 bit values printing in decimal representation.
- 16 bit values printing in hexadecimal representation.
- 32 bit values printing in hexadecimal representation.
AnachronautandClaude Opus 5
Anachronaut