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GF256: two hundred fifty-six bits

You will learn

How the rule splits 256 bits into 1 + 97 + 158.

GF256 has 256 bits: 1 sign, 97 exponent, 158 mantissa. The family spec has a test for these counts too. At this width rounding costs little, and the ratio E / M is 0.614, 0.004 from 1 / phi.

Try it

Find EM_RATIO and PHI_DIST in gf256.t27. Then in goldenfloat_family.t27 find the test that checks the GF256 bit counts, and check that both specs agree.

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GoldenFloat 21: GF256
GoldenFloat 21: GF256 ↗

GF256: 256 bits as the spec lays them out, read from gf256.t27. Lesson 21 of the GoldenFloat course.

specs/numeric/gf256.t27

// SPDX-License-Identifier: Apache-2.0
; gf256.t27 -- GoldenFloat256 Encode/Decode
; GF256: 256-bit floating point with 1 sign + 97 exponent + 158 mantissa
; Bit layout: [S(1) E(97) M(158)] = [255:255][254:158][157:0]
; phi^2 + 1/phi^2 = 3 | TRINITY
; Generated by the closed-form rule e = round((N-1)/phi^2), m = N-1-e.
; STATUS: Conj (spec-only; no RTL on this repo's path).

module triformat-gf256;

// ============================================================================
// Constants -- derived from the closed-form rule
// ============================================================================

pub const TOTAL_BITS : u16 = 256;
pub const SIGN_BITS  : u8 = 1;
pub const EXP_BITS   : u8 = 97;
pub const MANT_BITS  : u16 = 158;

pub const SIGN_SHIFT : u16 = 255;
pub const EXP_SHIFT  : u16 = 158;
pub const MANT_SHIFT : u16 = 0;

; BIAS = 2^(96) - 1   -- multi-word constant (exceeds u64); see codegen
; EXP_MAX = 2^97 - 1   -- multi-word constant (exceeds u64)
pub const BIAS_EXPR    : str = "2^(97-1) - 1";
pub const EXP_MAX_EXPR : str = "2^97 - 1";

// E/M ratio (target: 1/phi ~ 0.6180339887)
pub const EM_RATIO   : f64 = 0.613924050633;
pub const PHI_DIST   : f64 = 0.004109938117;

pub const PHI_BIAS_STATUS : str = "OPEN -- not derivable from closed form; empirical per format";

// PHI_BIAS for this rung is NOT defined. The published formula
// PHI_BIAS = EXP_MAX - BIAS reproduces GF64 only and is RETRACTED as a general law.
// Do NOT invent a value via Fibonacci/Lucas/square coincidence; those are
// descriptive, not prescriptive.

// ============================================================================
// Invariants -- the Fpath below, made executable (W601)
//
// This file declared its own falsification path in a comment and nothing
// checked it. W600's per-test measurement found 38 specs that compile while
// asserting nothing; this is one, and the rule it is derived from is stated
// precisely enough to be a test.
// ============================================================================

invariant gf256_field_widths_partition_the_word {
    @compileAssert(SIGN_BITS + EXP_BITS + MANT_BITS == TOTAL_BITS);
}

invariant gf256_closed_form_mantissa {
    // m = N - 1 - e, the second half of the generating rule
    @compileAssert(MANT_BITS == TOTAL_BITS - 1 - EXP_BITS);
}

invariant gf256_closed_form_exponent {
    // e = round((N-1)/phi^2)  <=>  (e - 1/2)*phi^2 <= N-1 <= (e + 1/2)*phi^2
    // Stated as bounds because the rule rounds; phi^2 = 2.618033988749895.
    @compileAssert((EXP_BITS as f64 - 0.5) * 2.618033988749895 <= TOTAL_BITS as f64 - 1.0);
    @compileAssert(TOTAL_BITS as f64 - 1.0 <= (EXP_BITS as f64 + 0.5) * 2.618033988749895);
}

invariant gf256_shifts_follow_the_layout {
    @compileAssert(SIGN_SHIFT == TOTAL_BITS - 1);
    @compileAssert(EXP_SHIFT == MANT_BITS);
    @compileAssert(MANT_SHIFT == 0);
}

; ============================================================================
; Claim-status: Conj
; Fpath: closed-form rule mis-applied (verify e = round((256-1)/phi^2) = 97, m = 158)
;        or RTL emission diverges from this constant set.
;        As of W601 the Fpath above is CHECKED by the invariants in this file.
; ============================================================================

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