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feat(v4.0.0): Q.SIN and (Q.REDUCE), bit for bit with v3

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(Q.REDUCE) is v3's q48_reduce_angle: the angle as one cell in [-pi, pi].
x / 2pi does not fit a cell, but only the remainder is needed, and
|x| mod 2pi is two UM/MOD steps (high cell first, its remainder leading
the low cell); then the sign of x and one step of 2pi back into range.

Q.SIN is v3's q48_sin_approx on the reduced angle: the odd Taylor series
to n = 11, each term the last times x^2 over n(n-1), stopping below 10
ulp. After the reduction every value fits one cell, so the 6-cell
variable (QT) holds single cells. The loop is laid out for Q.COS to jump
into, so the pair share it without an extra call level.

Checked bit for bit against v3 (both functions ported into the test) on
33 edge angles -- around +-pi/2, +-pi, +-2pi, the 32-bit seam and both
64-bit extremes -- and 3000 pseudo-random angles of every size and sign,
at 32- and 64-bit cells, optimised and ASan+UBSan (`make test`,
`make sanitize`). Mutating either range boundary, the series length or
the sign alternation fails. Moving the 10-ulp stop to 11 fails nothing,
and a scan of all 205888 reduced angles in C shows why: no input's
result depends on it, for sine or cosine.

Headroom (data under arg / return): (Q.REDUCE) 5/5, Q.SIN 3/2.

Test results on the amd64 host only. This is a development check, not
acceptance (JUSTIFICATION.md section 16).

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
This commit is contained in:
rajames
2026-10-03 10:44:42 -04:00
co-authored by Claude Opus 5.5
parent e117333a87
commit 1ce705783c
2 changed files with 180 additions and 5 deletions
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docs/v4.0.0/DECOMPOSITION.md
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@@ -147,8 +147,8 @@ IF body1 ELSE body2 THEN → if L1 drop body1 jump L2
times, as on the F18. `FOR ... UNEXT` is the same but the body must fit in one instruction word.
**Register conventions.** `A` and `B` are caller-saved. A word that uses them says so. Words in this
document that clobber `A`: `@ ! +! -! 2@ 2! C@ C! UM* * UM/MOD Q.FROM-INT Q.TO-INT Q.* Q./ Q.EXP Q.SQRT Q.LOG SEND RECV`. Words that clobber `B`:
`Q./ Q.EXP Q.SQRT Q.LOG SEND RECV`.
document that clobber `A`: `@ ! +! -! 2@ 2! C@ C! UM* * UM/MOD Q.FROM-INT Q.TO-INT Q.* Q./ Q.EXP Q.SQRT Q.LOG Q.SIN SEND RECV`. Words that clobber `B`:
`Q./ Q.EXP Q.SQRT Q.LOG Q.SIN SEND RECV`.
**Return-stack words** (`>R R> R@ 2>R 2R> 2R@ I J UNLOOP` and the loop runtimes) are always IN.
@@ -664,7 +664,10 @@ word. Per D-10 it occupies two cells on a 64-bit node too. Per D-8, v4 Q values
| `D2*C` | CAP | `( lo hi cin -- lo' hi' cout )`: the double shifted left one bit, `cin` entering at the bottom and `cout` the bit leaving the top (both 0 or 1). `a! dup -if L1 drop 1 jump L2 L1: drop 0 L2: push over -if L3 drop 1 jump L4 L3: drop 0 L4: over + + push 2* a + pop pop` — `cin` waits in `A`, and `2hi + m` is `over + +`, so it needs one return entry. Executed on the golden model (2026-10-02). Clobbers `A`. |
| `(Q/)` | CAP | Variable, 5 cells: `Q./`'s quotient register, divisor and result sign. Like `BASE` and the hold buffer, it lives in node memory. |
| `Q.ABS` `Q.NEG` | CAP | `DABS`, `DNEGATE` — executed on the golden model (2026-10-02) against v3's `q48_abs` and `0 - q`: bit-for-bit at 32-bit cells, including v3's wrap of Q min to itself; at 64-bit cells the low cell is v3's and the high cell is the true sign, per D-10. |
| `Q.SIN` `Q.COS` | CAP | Algorithms ported from `q48_words.c`; the hosted C versions are the golden model. |
| `Q.COS` | CAP | Algorithm ported from `q48_words.c`; the hosted C version is the golden model. |
| `Q.SIN` | CAP | v3's `q48_sin_approx`: on `r = (Q.REDUCE) x` and `xu = |r|`, `xu - xu^3/3! + xu^5/5! - …` to `n = 11`, each term the last times `xu^2` over `n(n-1)`, stopping once a term is below 10 ulp; negated if `r < 0`. After the reduction every value fits one cell, so the variable `(QT)` holds single cells and the only calls are `(Q.REDUCE)`, `Q.*`, `Q./` and `UM*`. The Taylor loop is written so that `Q.COS` can jump into it. Executed on the golden model (2026-10-03): bit for bit v3's result on angles of every size and sign. Leaves its caller 3 data cells under its argument and 2 return entries. Clobbers `A` and `B`. |
| `(Q.REDUCE)` | CAP | `( x -- r )`, v3's `q48_reduce_angle`: the angle reduced to one cell, `-pi <= r <= pi` with `pi = 205887`, `2pi = 411774` — the remainder of `x / 2pi` with the sign of `x`, then one step of `2pi` back into range. `x / 2pi` does not fit a cell, but only the remainder is wanted: `|x| mod 2pi` is two `UM/MOD` steps, the high cell first and its remainder then leading the low cell. `dup (QT) b! !b -if L1 DNEGATE L1: 0 411774 UM/MOD drop 411774 UM/MOD drop (QT) b! @b -if P1 drop inv 1 + jump J1 P1: drop J1: dup -205888 + -if BIG drop dup 205887 + -if OK drop 411774 + ; OK: drop ; BIG: drop -411774 + ;` Executed on the golden model (2026-10-03), bit for bit v3's. Clobbers `A` and `B`. |
| `(QT)` | CAP | Variable, 6 cells: `Q.SIN` / `Q.COS`'s sign, `x^2`, term, sum, `n` and subtract flag. |
| `Q.LOG` | CAP | v3's `q48_log_approx`: `x = 2^k * m` with `1.0 <= m < 2.0`; `ln m` by up to 6 Newton rounds on `e^y = m` from `y = m - 1.0`, stopping once `|m - e^y| < 100` ulp; result `y + k * ln 2` with `ln 2 = 45426`. `e^y` is `Q.EXP`'s Taylor loop **written in line** (flattened, ruled 2026-10-02), so `Q.LOG` calls only `Q.*`, `Q./` and `UM*` and the call chain is one level shorter than through `Q.EXP`. After the reduction `m`, `y`, the Taylor term and its sum each fit one cell at either width, so the variable `(QL)` holds single cells. `x <= 0` returns 0 and sets `NODE-ERROR` (D-12). Executed on the golden model (2026-10-03): bit for bit v3's result for every `x > 0` tested, including a sweep of the reduced `m`. v3's clamp `y = max(y - corr, 0)` is kept for fidelity but cannot fire: over all 65536 values of `m` it is never reached, and the iteration never needs more than 4 rounds. Leaves its caller 3 data cells under its argument and 2 return entries. Clobbers `A` and `B`. |
| `(QL)` | CAP | Variable, 7 cells: `Q.LOG`'s `m`, `y`, `k`, rounds left, Taylor term (then `delta`), sum, and `n * 1.0`. |
| `Q.SQRT` | CAP | v3's `q48_sqrt_approx`: Newton from `x0 = q/2 + 0.25`, up to 8 rounds of `x' = (x + q/x) / 2`, returning `x` as soon as `|x' - x| < 10` ulp; `sqrt(0) = 0`, `sqrt(1.0) = 1.0`. `q < 0` returns 0 and sets `NODE-ERROR` (D-12). `q`, `x` and the rounds left live in the variable `(QR)`. The halving is a logical double shift, `push a! 0 pop +* MAXHI and push drop a pop`. Executed on the golden model (2026-10-02): bit for bit v3's result for `0 <= q < 2^48`; above that v3's `q48_div` saturates and v4 divides correctly, so they part. Leaves its caller 3 data cells under its argument and 2 return entries. Clobbers `A` and `B`. |