BPosits.jl

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BPosits.jl is a Julia package implementing Bounded Posits (BPosit) -- a posit variant with a capped regime and a guaranteed precision floor.

• Bounded Regime: |k| ≤ k_max = 1 / 7 / 13 / 19 for 8/16/32/64 bits, eliminating the precision collapse of standard posits.
• Exponent Size: es(n) = min(4, n/4) — 2 bits for BPosit8, 4 bits for all wider formats; the full exponent field is present at every regime.
• Precision Floor: p_min = n − 2 − es − k_max = 3 / 3 / 13 / 39 fraction bits guaranteed everywhere, giving a uniform log-relative-error bound λ ≤ ln(1 + 2^−(p_min+1)).
• Exact Quire: a 2048-bit fixed-point accumulator provides exact dot products for all widths; dot(a, b; quire=true).
• Lossless casting: BPosit8 → Float16, BPosit16 → Float32, BPosit32 → Float64 are exact; BPosit16 spans Float32's full dynamic range at half the width.

This package is modeled after Posits.jl and uses the libbposit C library (via libbposit_jll) for all numerics.

BPosits implement the BPosit format family (8/16/32/64-bit) with per-width parameters es = min(4, n/4) and k_max = 1/7/13/19, where the regime cap and fraction floor are dual constraints:

p_min = n − 2 − es − k_max = 3 / 3 / 13 / 39

The full es-bit exponent field is present at every regime (no truncation), which makes every bit pattern canonical and yields the uniform log-relative-error bound λ ≤ ln(1 + 2^−(p_min+1)) across the entire dynamic range. Values beyond the regime cap saturate to maxpos/minpos, matching Posit Standard (2022) rounding semantics (round-to-nearest, ties to even bit pattern; never to 0 or NaR).

The C backend (libbposit, ISO C99) provides:

• CLZ/shift codec: regime, exponent, and fraction fields are each decoded/encoded in O(1), no per-bit loops.
• Single-rounded direct arithmetic — add/sub/mul align the 61-bit significands in a 128-bit integer with sticky-bit (and borrow-aware) tracking, so the final encode performs the only rounding. ~19 ns/op, roughly 8× faster than the reference posit library at -O3, and verified bit-identical to the exact quire reference over exhaustive 8-bit pairs and 10⁷ random pairs per width.
• A single 2048-bit quire shared by all widths (BPosit64 products retain ≥383 carry bits) with fused dot, fma, sum, and mul! (matrix multiply) kernels that accumulate exactly on the C stack and round once — zero allocations from Julia.
• Single-rounded conversions: from Float64 via the IEEE bit pattern, and between BPosit widths by decode/re-encode (widening is exact; narrowing avoids the double rounding a Float64 round trip would introduce for BPosit64 sources).
• Elementary functions (sin, exp, log, ...) via double (8/16/32-bit) or x87 long double (64-bit, whose 58-bit significand exceeds double), plus exact sinpi/cospi. Domain errors return NaR; no exceptions.

The Julia layer defines the four widths as primitive subtypes of AbstractFloat with the complete numeric interface: comparisons (two's-complement bit-pattern order), rounding and integer conversion, exponent/significand/frexp/precision, hashing, exact BigFloat conversion, wider-wins promotion between widths, and rand.

The test suite (Pkg.test()) runs ~200k assertions in under 10 seconds, including exhaustive correctness checks over all 65,535 non-NaR BPosit16 bit patterns:

• Round-trip: every BPosit16 value satisfies Float64(x) == Float64(significand(x)) * 2.0^exponent(x)
• Widening: every BPosit16 value widens to BPosit64 exactly and narrows to BPosit8 identically to the Float64 route
• Arithmetic and quire: spot-checked at all widths; fused operations verified against the exact quire reference over 1,000 random trials per width

] add BPosits

using BPosits

x = BPosit32(1.0)
y = BPosit32(2.0)
z = x + y
printbits(z)                    # color-coded sign | regime | exponent | fraction
dot([x, y], [y, x], quire=true) # exact accumulation

MIT License. See LICENSE for details.

Quinlan, J. (2026). Bounded Posits (BPosit): a posit variant with a capped regime and a guaranteed precision floor. (Version v0.1.0) [Computer software]. Zenodo. https://doi.org/10.5281/zenodo.21532708

@software{quinlan2026bposit,
  author       = {Quinlan, James},
  title        = {Bounded Posits (BPosit): a posit variant with a
                   capped regime and a guaranteed precision floor.},
  month        = jul,
  year         = 2026,
  publisher    = {Zenodo},
  version      = {v0.1.0},
  doi          = {10.5281/zenodo.21532708},
  url          = {https://doi.org/10.5281/zenodo.21532708},
}