[jeany55]

Step 4 of the way off eudplib: every number a program works with, in a map of our own. All 59 of the ledger's number rows are taken. Each part has its own section in docs/v4-plan.md.

Follows #42 (merged first; this PR was stacked on it).

Ledger	Before	After
3 Arrays of a fixed length	53 of 54	54 of 54
4 Numbers	0 of 59	59 of 59

No other row changed.

What it adds

1. Products, quotients and remainders of two values and by a constant, signed and u32. A product adds one side Horner's way in a doubling trigger. A division doubles the remainder, with one shared step holding the divisor.
2. & | ^ as values, and shifts by a value. Bits are masked writes: an action's mask field holds the other side, as eudplib does it, without the calls. Shift counts of 32 or more saturate, as TrigScript has it.
3. Fixed point: products and quotients by a constant and of two values, a constant over a value, and a fixed number printed.
4. Random numbers. random(n) and random() draw the game's own random bits through Set Switch to a random state. A draw that comes out at n or more is made again, so every number below n is exactly as likely. eudplib's f_dwrand % n leans slightly towards small numbers.
5. Math.sqrt, sin, cos, atan2, and u8 / u16.
   • The square root keeps the root at the place its bits end up, so it only gains constants, and the remainder doubles instead. A shared compare trigger steps its own next pointer into a pair of triggers laid out for each step.
   • Sine, cosine and arc tangent are statements the emitter already makes. The arc tangent's quotient takes 17 steps, not 47.
   • Stores into u8 and u16 stop at their top.

Measured

Each probe was built both ways and played in the lifted 1.16.1 game, weighed with Remastered's costs. Every line printed is the same both ways and ends "ok". Times are for 200 of each:

	Ours	eudplib
products of two values	0.677 ms	0.943 ms
quotients of two values	0.825 ms	1.175 ms
shifts by a value (three kinds)	0.834 ms	2.399 ms
& | ^ of two values	0.086 ms	0.217 ms
fixed x * y	0.524 ms	0.677 ms
fixed x / y	0.693 ms	1.093 ms
random(j % 10 + 1)	0.257 ms	1.836 ms
Math.sqrt(x), x fixed	0.389 ms	1.348 ms
Math.sqrt(n), n whole	0.653 ms	3.672 ms
Math.sin / Math.cos	1.085 ms	2.261 ms
Math.atan2	1.343 ms	2.300 ms

The probes are probes/own-numbers.ts, own-bits.ts, own-fixed.ts, own-random.ts and own-math.ts.

Fixed on the way, in both builds

• A whole number's square root below zero. It is read from 0 up, so the answer is the largest fixed number, but its value range was 0. Math.sqrt(w) == … was then worked out wrong when the map was built (compiler/ranges.ts; test in tests/fixed.test.ts).

Fixed on the way, in our own build

• Shift counts of 32 or more now saturate; the existing x << k took the count's remainder.
• a >> 0 and a shift whose count always works out to the same number were refused; both are now taken.
• xs[a >> 1], the last refused row of step 3, is now taken.

Checked

• Emitter tests: random programs with every operation above, as graphs and from the laid-out bytes. 3,000 of 3,000 programs with functions are taken. The test interpreters now apply the game's masked-write semantics.
• Random numbers: each value below the bound comes up within a fifth of its share, over 400 draws a value. The ledger's random scripts were run 200 times each way, against the simulator, with the same shares.
• Ledger: every taken number script against the simulator.
• Suite: 1,246 passing.
• Test time: a random program now holds four heavy operations at most (products, quotients, shifts by a value, square roots, sines), and these tests may take 300 s. With step 4's operations they had run past the 120 s limit on CI (283 s). Here they take 23 s and 47 s, against 12 s and 25 s before step 4.

Left for later (in the plan)

• Sine time: a sine is still mostly copies. A table of differences, or a read that gives two cells at once, would remove them.
• Map size: each square root, sine and arc tangent is written out where it is used, so own-math.ts is 9 MB. A routine shared by every use, as the digits routine is, is the fix.

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