(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>
