Module tower

Nested square roots: values in a tower of adjoined radicals.

crate::Root2 holds one square root over plain numbers. Arc constructions need more: a hit on a circle is a + b*sqrt(D), and a distance or a second construction from that hit takes another square root of an expression that already contains sqrt(D). This module represents such values exactly.

Representation

A Tower is a list of radicands r_1, ..., r_k, each built only from the radicals before it. A Nested value at level k is a + b*sqrt(r_k) with a and b at level k - 1, stored flat as a coefficient vector of length 2^k (low half a, high half b).

Why no canonical form is needed

Radicals need not be independent: sqrt(8) and sqrt(2) may both be adjoined, and then sqrt(8) - 2*sqrt(2) has non-zero coefficients but value zero. That is fine. Both the product rule (a + b√r)(c + d√r) = (ac + bd·r) + (ad + bc)√r and the sign rule below are identities about real numbers, true whatever the coefficients are. So signs are exact, including exact zeros, without ever reducing to a basis, which is what keeps this module small.

Sign

sign(a + b*sqrt(r)), recursively by level: if a and b*sqrt(r) agree in sign (or one is zero) that is the answer; otherwise it is sign(a) * sign(a^2 - b^2*r), a value one level down. Level 0 asks the arithmetic T directly, so the same code runs as the interval filter and as the exact fallback (see crate::certify()).

Cost

Each level squares once, so the polynomial degree in the inputs, and with it exact-tier mantissa length, doubles per level; a product costs five products one level down. Depth is capped at MAX_DEPTH.

Structs

Nested

A value in a Tower: 2^level coefficients over T.

Tower

The radicals adjoined so far.

Constants

MAX_DEPTH

Most radicals one tower may hold. A product at depth k costs 5^k base products, and exact mantissas grow 2^k-fold in degree.