binary16 (fp16, half)

status: Verified

use_case: GPU activations, inference

gf_relation: competitor


binary128 (fp128, quad)

use_case: high-precision simulations


binary256 (octuple)

use_case: astronomy, cryptography


x87 FP80

cluster: ExtendedFloat

status: Historical

standard: Intel x87 (explicit integer bit)

use_case: legacy long double on x86

gf_relation: orthogonal

source: Intel SDM


double-double

standard: Bailey/Hida (software)

use_case: software extended precision

source: Bailey-Hida 2001


quad-double

use_case: astrophysics, quad-precision sims


bfloat16 (BF16)

cluster: MlLowPrecision

standard: Google Brain

use_case: training (range > precision)

source: Wang-Kanwar 2019


TensorFloat-32 (TF32)

standard: NVIDIA Ampere

use_case: A100/H100 mixed precision

source: NVIDIA Ampere whitepaper


FP8 E4M3

standard: OCP / NVIDIA / Arm / Intel

use_case: inference, gradient ranges

source: Micikevicius 2022 (arXiv:2209.05433)


FP8 E5M2

standard: OCP / NVIDIA

use_case: activations, wide range

source: Micikevicius 2022


FP6 E3M2

standard: OCP MX

use_case: aggressive quant inference


FP6 E2M3

use_case: mantissa-heavy quant

gf_relation: ally


FP4 E2M1

use_case: extreme quant inference


cluster: Microscaling

standard: OCP MX v1.0

use_case: LLM inference


use_case: aggressive inference

source: Rouhani 2023


use_case: extreme quant


NF4 (NormalFloat 4-bit)

cluster: QuantTuned

standard: Dettmers 2023 (QLoRA)

use_case: LLM weight quantization (quantile-based on N(0,1))


AFP (Adaptive Floating-Point)

standard: Tambe 2020

use_case: efficient training

source: Tambe 2020 (DAC)


cluster: PositUnumIII

standard: Posit Standard 2022 (es=2)

use_case: inference

source: Posit Standard 2022 (posithub.org)


use_case: mixed-precision training

source: Posit Standard 2022


use_case: f32 replacement


use_case: f64 replacement


standard: Hunhold 2024 (tapered-precision)

use_case: IEEE-754 backward-compatible tapered


standard: Hunhold 2024

use_case: single-rule ladder counterexample


use_case: tapered fp32-class

source: Hunhold 2024


use_case: tapered fp64-class


cluster: Lns

standard: Arnold 1990; LNS-Madam (2021)

use_case: DSP, signal processing


standard: LNS-Madam (2021)

use_case: log-domain training (mul -> add)

source: Alam 2021


use_case: log-domain DSP


use_case: scientific log-domain


GFTernary

cluster: GoldenFloat

standard: this work; {-phi, 0, +phi}

use_case: bulk layers (hybrid)

gf_relation: self


status: Experimental

standard: this work; F0 minimal

use_case: proof-of-concept


standard: this work; L1 Lucas

use_case: edge / sensors

source: BENCH-007 (specs/numeric/gf8.t27)


standard: this work; L0/F3

use_case: mid-range / audio

source: BENCH-007 (specs/numeric/gf12.t27)


standard: this work; PHI_BIAS=60; FPGA 35/35 at 323 MHz Artix-7

use_case: training and inference (production)

source: specs/numeric/gf16.t27; zenodo 10.5281/zenodo.19227877 (HW archive)


standard: this work; 17-squared empirical PHI_BIAS=289

use_case: high-precision edge

source: specs/numeric/gf20.t27 (spec only)


standard: this work; rule e=round(23/phi^2)=9; normative bias=2^(e-1)-1=255; empirical PHI_BIAS=1364 (=L15) OPEN

use_case: server inference

source: specs/numeric/gf24.t27 (spec only)


standard: this work; F0 resolved

use_case: fp32 drop-in

source: BENCH-012 (specs/numeric/gf32.t27)


standard: this work; EXP_MAX - BIAS

use_case: scientific / double

source: BENCH-007b (specs/numeric/gf64.t27)


GF6 (rule-derived)

status: Open

standard: this work; rule e=round(5/phi^2)=2; FP6 E2M3 bridge

use_case: OPEN R&D: bridge GF4-GF8; FP6 E2M3 hint

gf_relation: experimental


GF10 (rule-derived)

standard: this work; rule e=round(9/phi^2)=3; bridge GF8-GF12

use_case: OPEN R&D: tight-precision activations


GF14 (rule-derived)

standard: this work; rule e=round(13/phi^2)=5; bridge GF12-GF16; lowest phi-dist below GF48

use_case: OPEN R&D: drop-in for fp16 with tighter phi alignment


GF48 (rule-derived)

standard: this work; rule e=round(47/phi^2)=18

use_case: OPEN R&D: between GF32 and GF64; tightest phi-dist of the wide rungs


GF96 (rule-derived)

standard: this work; rule e=round(95/phi^2)=36

use_case: OPEN R&D: between GF64 and GF128 (phi-aligned extended)


GF128 (rule-derived)

standard: this work; rule e=round(127/phi^2)=49 (corrects v1.1 typo e=48)

use_case: OPEN R&D: phi-aligned binary128 alternative


GF256 (rule-derived)

standard: this work; rule e=round(255/phi^2)=97; normative bias=2^96-1

use_case: OPEN R&D: phi-aligned binary256 alternative


GF512 (rule-derived)

standard: this work; rule e=round(511/phi^2)=195

use_case: OPEN R&D: ultra-wide phi-aligned (extrapolation, no RTL)


GF1024 (rule-derived)

standard: this work; rule e=round(1023/phi^2)=391; lowest phi-distance in the ladder

use_case: OPEN R&D: limit-of-ladder phi alignment (extrapolation, no RTL)


GF8-BFP (block FP atop GF8)

standard: this work; per-tile shared exponent

use_case: OPEN R&D: LLM-quantization-friendly GF8


GF + LNS hybrid (dual-space)

standard: this work; mul in log-space, accumulate Lucas-closed

use_case: OPEN R&D: dual-space arithmetic


MXGF6 (microscaling GF6)

standard: this work; OCP MX block + GF6

use_case: OPEN R&D: phi-aligned MX-6 candidate


MXGF4 (microscaling GF4)

standard: this work; OCP MX block + GF4

use_case: OPEN R&D: phi-aligned MX-4 candidate


INT4 / UINT4

cluster: IntegerFixed

standard: two complement

use_case: aggressive quantization

source: ISO/IEC 9899


INT8 / UINT8

use_case: INT8 inference, per-channel scale


INT16 / UINT16

use_case: DSP, embedded ML


INT32 / UINT32

use_case: general CPU integer


INT64 / UINT64

use_case: databases, timestamps


INT128 / UINT128

use_case: crypto, big-int

source: Rust/Clang u128


Q-format (Qm.n)

storage: varies

standard: TI fixed-point

use_case: audio DSP, fixed-point ML

source: TI SPRA704


BCD (binary-coded decimal)

use_case: calculators, GAAP

source: ISO/IEC 8859


IBM HFP (single)

cluster: HistoricalVendor

standard: IBM System/360 (1964); base-16 exponent

use_case: legacy mainframe

source: IBM POO


IBM HFP (extended)

standard: IBM z/Architecture


Microsoft MBF (single)

standard: MS BASIC / MS-DOS (pre-IEEE)

use_case: MS BASIC legacy

source: MS-DOS docs


Microsoft MBF (double)

standard: MS BASIC


VAX F-float

standard: DEC VAX

use_case: DEC legacy

source: VAX Architecture Reference


VAX D-float

use_case: DEC legacy double


VAX G-float

standard: DEC VAX (IEEE-like)


VAX H-float

use_case: DEC quad


Cray float

standard: Cray-1 (1976); no NaN/Inf, unrounded mul

use_case: Cray legacy

source: Cray-1 Hardware Reference


minifloat (arbitrary E:M, <=16 bits)

cluster: Theoretical

standard: parametric framework

use_case: design space of GF4/GF8/GF12/GF16

source: Higham 1996


Unum I (tapered + ubound)

standard: Gustafson 2015 (predecessor to posit)

use_case: interval arithmetic

source: Gustafson 2015 (The End of Error)


Unum II (SORN projective)

standard: Gustafson 2016

use_case: lookup-table real arithmetic; not GF-comparable

source: Gustafson 2016


tapered floating point

standard: Morris 1971; posit ancestor

use_case: variable mantissa via regime bits

source: Morris 1971 (IEEE TC)


block floating point (BFP)

cluster: CompressionTrick

standard: Wilkinson 1965; modern revivals

use_case: per-tile shared exponent

source: Darvish-Rouhani 2020


shared-exponent formats

standard: generalised BFP

use_case: LLM quantization


INT8 with per-channel scale

standard: Jacob 2018 (TFLite)

use_case: standard quant inference

source: Jacob 2018 (CVPR)


stochastic rounding (technique)

standard: Gupta 2015

use_case: training small networks at low precision

source: Gupta 2015 (ICML)