Operations

The public seam is apply:

apply(op, Fr, ρ, xs...; entropy = nothing, index = 0) -> Fr

op is an operation singleton (Add, FMA, Sqrt, …), Fr the result format, ρ the projection. Operands are FloatByte values and may have different formats — result format and operand formats are all explicit. Special values follow the draft's pattern tables in their normative order; finite arguments are evaluated exactly (integer window arithmetic) and projected once.

julia> using FloatBytes

julia> ρ = DEFAULT_PROJECTION;

julia> apply(FMA, Binary8p4se, ρ, Binary8p4se(1.5), Binary8p4se(2.25), Binary8p4se(2.25))
Binary8p4se(0x53 ↦ 5.5)

julia> apply(Add, Binary8p3se, ρ, Binary8p4se(1.5), Binary5p2ue(0.5))  # mixed formats
Binary8p3se(0x44 ↦ 2.0)

Operation families

• Arithmetic (draft §4.10): Add, Subtract, Multiply, Divide, FMA, FAA, Sqrt, Recip, RSqrt, Hypot, Abs, Negate, CopySign, Convert.
• Extrema (§4.11): Minimum/Maximum and the Number, Magnitude, MagnitudeNumber, and Finite variants, plus Clamp. Ten min/max variants total, each with its own NaN and infinity rows.
• Comparisons and predicates (§4.12–4.13): compare_less and friends, total_order, isnan/isinf/isfinite/ issubnormal/iszero/isone/signbit, and the classifier floatclass. These return Booleans or an enum — no projection, no entropy.
• Neighbours (§4.16): nextgreaterthan, nextlessthan — format-preserving code-point walks with explicit terminal rows.
• Transcendentals (§4.10.9–.16): Exp … ArcTan2Pi — see below.

Fused operations project once

FMA and FAA assemble the complete expression exactly and round a single time. FAA is exactly associative — bit-identical for every operand order:

julia> x = Binary8p4se(224.0); y = Binary8p4se(-224.0); z = Binary8p4se(0.001953125);

julia> apply(FAA, Binary8p4se, ρ, x, y, z) == apply(FAA, Binary8p4se, ρ, z, x, y)
true

julia> Float64(apply(FAA, Binary8p4se, ρ, x, y, z))   # exact cancellation survives
0.001953125

Transcendentals: correct rounding, or an error

Each of the 25 unary transcendentals plus ArcTan2/ArcTan2Pi runs a rigorous enclosure loop: directed MPFR endpoints at doubling precision until the projection is decided. There is no undeclared approximation — if the loop cannot decide at its precision ceiling it throws InconclusiveProjectionError rather than writing a guess. Exact points (Exp(0) = 1, Log(1) = 0, SinPi at half-integers, TanPi(±1/4) = ±1, Tanh(±Inf) = ±1, …) are exact, and asymptotes keep their sidedness (Tanh of a large finite value is 1 − ε, never 1, so directed rounding toward zero stays below one):

julia> Float64(apply(TanPi, Binary8p4se, ρ, Binary8p4se(0.25)))
1.0

julia> apply(Tanh, Binary8p4se, Projection(TowardZero(), SatNone()), maxfinite(Binary8p4se))
Binary8p4se(0x3f ↦ 0.9375)