Exact comparisons with reciprocals

compareReciprocal compare compares t⁻¹ with rational boundaries, given an exact comparator for a nonzero real t. The comparison with zero gives the sign of t. A boundary of the other sign, or zero, is then decided by that sign alone. Otherwise the boundary and t have the same sign, and inversion reverses the order on each half-line: t⁻¹ < b exactly when b⁻¹ < t. The result is the reversed comparison of t with b⁻¹, so no reciprocal enclosure is formed.

Secant, cosecant and cotangent are reciprocals of cosine, sine and tangent, and their hyperbolic counterparts are reciprocals of the hyperbolic cosine, sine and tangent. Their poles are special values that a wrapper handles before rounding, so this comparator is used only for t ≠ 0.

def FloatLib.Numerics.compareReciprocal (compare : ℚ → Ordering) (boundary : ℚ) : Ordering

Compare t⁻¹ with a rational boundary, given an exact comparator compare for a real t.

The contract requires t ≠ 0, so the result returned when compare 0 = .eq is never used.