stillwater-sc / universal / 26224499819
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feat(elreal): Phase E.2 -- sqrt + hypot with depth-1 EFT refinement on sqrt (#894)

* feat(elreal): Phase E.2 -- sqrt + hypot with depth-1 EFT refinement on sqrt

Phase E.2 of epic #873 (resolves #888). The second math sub-issue:
square root via Newton-style EFT residual at depth 1, plus a depth-0
hypot via std::hypot for overflow safety.

sqrt(const elreal&):
  - Depth 0: c0 = std::sqrt(a.at(0)) (IEEE-754 correctly rounded)
  - Depth 1: Newton residual c1 ~= (a - c0^2) / (2*c0). a - c0^2 is
    computed with two_prod (giving c0^2 = prod_hi + prod_err exactly),
    then a single double subtraction picks up the operand's depth-1
    correction a.at(1). The division by 2*c0 is plain double arithmetic
    on the elreal-internal numerator, NOT a lazy elreal divide -- so we
    are not blocked by Phase F's reciprocal-stream work.
  - Depth 2+: 0.0 (documented limitation; full Heron iteration on the
    lazy stream lands in Phase F).
  - Negative argument: throws elreal_negative_sqrt_arg when
    ELREAL_THROW_ARITHMETIC_EXCEPTION is set; otherwise returns NaN
    via std::sqrt(negative double).

hypot(const elreal&, const elreal&):
  - Depth 0: c0 = std::hypot(a.at(0), b.at(0)) -- handles overflow,
    signed zero, infinities, and NaN per IEEE-754 conventions. The
    overflow-protected algebraic form (max(|a|,|b|) * sqrt(1 + (min/max)^2))
    is built into the standard library implementation.
  - Depth 1+: deferred. Either an overflow-protected algebraic form
    walked at depth 1, or sqrt(a*a + b*b) when neither operand is near
    overflow -- a small follow-up.

Tests under elastic/elreal/math/:
  sqrt.cpp        -- perfect squares, non-square cross-validation vs
                     std::sqrt, round-trip sqrt(a)^2 ~= a, sqrt(2)
                     matches std::numbers::sqrt2_v, depth-1 propagation
                     on rational input, negative -> NaN, special values
  sqrt_throw.cpp  -- elreal_negative_sqrt_arg in the throw mode (separate
           ... (continued)

46939 of 55729 relevant lines covered (84.23%)

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Coverage Regressions

Lines Coverage ∆ File
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 -0.55% include/sw/universal/number/posito/posito_impl.hpp
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