stillwater-sc / universal / 23176044597

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fix(dd): improve exp/log accuracy for double-double and quad-double (#581)

* fix(dd): improve exp/log accuracy for double-double and quad-double

Fix multiple issues limiting dd/qd math function precision to ~14
digits instead of the expected 32 (dd) and 64 (qd) digits:

1. qd exp(): used qd_log2 (= log10(2) = 0.301) instead of qd_ln2
   (= ln(2) = 0.693) for argument reduction, producing completely
   wrong results. Fix: use qd_ln2.

2. qd exp(): used qd_eps = 2^-104 (dd precision) as convergence
   threshold instead of 2^-209 (qd precision). Fix: use correct
   qd epsilon = 1.22e-63.

3. dd/qd exp(): Taylor loop caps (i < 5 for dd, i < 9 for qd)
   could terminate before full precision convergence. Fix: raise
   to i < 14 (table has 15 entries).

4. dd log(): single Newton step from double initial approximation
   only doubles digits once (16 -> ~30). Fix: add second Newton
   step for full dd precision (~32 digits).

5. dd/qd exp2/exp10/expm1: delegated to native double, losing all
   extra precision. Fix: implement via exp(x * ln2), exp(x * ln10),
   and exp(x) - 1.0 in full dd/qd arithmetic.

Resolves #438

Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>

* fix(qd): add qd-precision regression tests for exp and log functions

Add proper regression tests that verify at full qd precision (~1e-60)
using mathematical identities instead of double-precision oracles:

exp tests:
- exp(0) == 1, exp(1) == qd_e (hardcoded constant)
- exp(log(2^i)) == 2^i (round-trip identity)
- exp2(i) == 2^i (exact via ldexp)
- exp10(i) == 10^i (exact via repeated multiplication)
- expm1(x) + 1 == exp(x) (cross-check identity)

log tests:
- log(exp(i)) == i (round-trip identity)
- log2(2^i) == i (exact powers of 2)
- log10(10^i) == i (exact powers of 10)
- log1p(exp(x)-1) == x (small argument identity)
- log1p(a) == log(1+a) (cross-check for larger arguments)

All tests pass at 1e-60 threshold, confirming full 64-digit precision.

Co-Authored-By: Claude Opus 4.6 ... (continued)

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