Universal Number Systems Guide

This directory contains comprehensive documentation for each number system in the Universal library. Each document explains why the number system exists, what it does, and how to use it to solve specific problems.

Number Systems by Category

Integer and Fixed-Point

Type	Bits	Description	Best For
integer	N	Arbitrary-width signed integer	Cryptography, combinatorics, wide counters
fixpnt	N	Binary fixed-point with configurable radix	DSP, control systems, embedded (no FPU)
dfixpnt	N	Decimal fixed-point with configurable radix	Financial ledgers, COBOL migration, tax/POS
rational	2N	Exact numerator/denominator fraction	Symbolic math, exact geometry, financial

Configurable Floating-Point

Type	Bits	Description	Best For
cfloat	4-256	Fully parameterized IEEE-compatible float	Mixed-precision research, custom HW design
bfloat16	16	Google Brain Float (8-bit exponent, 7-bit fraction)	Neural network training, TPU workloads
areal	N	Faithful float with uncertainty bit	Verified computing, uncertainty tracking
dfloat	N	Decimal floating-point (base-10)	Financial systems, regulatory compliance
hfloat	N	Hexadecimal floating-point (IBM System/360)	Legacy mainframe validation, data archaeology

Micro-Precision and Block-Scaled (AI Quantization)

Type	Bits	Description	Best For
microfloat	4-8	OCP MX element types (e2m1, e4m3, e5m2)	AI model elements, quantization validation
e8m0	8	Exponent-only power-of-two scale	Block scale factor for MX format
mxfloat	Block	OCP Microscaling block format	AI inference, model compression (OCP)
nvblock	Block	NVIDIA NVFP4 block format	GPU inference, NVIDIA accelerators

Posit Family (UNUM Type III)

Type	Bits	Description	Best For
posit	N	Tapered-precision floating-point (current v2)	General numeric, more precision than IEEE
posit1	N	Original posit implementation (legacy v1)	Backward compatibility
posito	N	Experimental posit variant	Differential testing, research
takum	N	Bounded-range tapered float	General computing, predictable range

Kulisch Super-Accumulator

Type	Bits	Description	Best For
quire	Wide	Generalized super-accumulator for exact dot products	Reproducible linear algebra, BLAS

The quire is a number-system-agnostic accumulator that provides exact dot products for cfloat, posit, fixpnt, lns, dbns, integer, native float, and native double.

Interval and Uncertainty Arithmetic

Type	Bits	Description	Best For
valid	2N	Interval arithmetic with posit-encoded bounds	Verified computing with posit precision
interval	2N	Generic interval over any scalar type	Tolerance analysis, uncertainty propagation
sorn	N	Set of operand range numbers	Rigorous uncertainty, safety-critical bounds
unum2	N	Configurable exact-value lattice	Research, custom value distributions

Logarithmic Number Systems

Type	Bits	Description	Best For
lns	N	Single-base logarithmic (base 2)	DSP, multiply-heavy workloads, low-power HW
dbns	N	Double-base logarithmic (base 0.5 and 3)	Research, mixed-radix applications

Extended Precision (Multi-Component)

Type	Bits	Decimal Digits	Description	Best For
dd	128	~31	Double-double (2 doubles)	Extended precision, ill-conditioned systems
qd	256	~64	Quad-double (4 doubles)	Ultra-high precision, constant computation
dd_cascade	128	~31	DD via unified cascade framework	Consistent API across precision tiers
td_cascade	192	~48	Triple-double (3 doubles)	Intermediate precision tier
qd_cascade	256	~64	QD via unified cascade framework	Consistent API across precision tiers

Adaptive (Elastic) Precision

Type	Description	Best For
einteger	Adaptive-precision integer	Cryptography, arbitrary-width counters
edecimal	Adaptive-precision decimal	Exact decimal arithmetic
erational	Adaptive-precision rational	Symbolic math, exact fractions
efloat	Adaptive-precision multi-digit float	Variable-precision scientific computing
ereal	Adaptive-precision multi-component real	Highest-precision computation

Compressed Floating-Point

Type	Description	Best For
zfpblock	ZFP block-based float compression (1D/2D/3D)	Scientific data storage, simulation checkpoints

Complex Number Support

Type	Description	Best For
complex	Complex arithmetic for any Universal scalar	FFT, signal processing, quantum computing

Choosing a Number System

By Application Domain

Domain	Recommended Types
Deep Learning Inference	microfloat, mxfloat, nvblock, bfloat16, cfloat(fp8)
Deep Learning Training	bfloat16, cfloat(fp16/fp32), posit
DSP / Signal Processing	fixpnt, lns, complex
Financial / Accounting	dfixpnt, dfloat, rational, fixpnt
Embedded (no FPU)	fixpnt, integer
Scientific HPC	dd, qd, posit, cfloat
Verified / Validated Computing	interval, valid, areal, sorn
Reproducible Linear Algebra	any type + quire (cfloat, posit, fixpnt, lns, float, double)
Cryptography / Big Numbers	integer
Data Compression	zfpblock
Custom Hardware Design	cfloat, posit, takum, lns

By Precision Need

Precision	Type	Decimal Digits
2 digits	bfloat16	~2
3 digits	cfloat(fp8), microfloat	~2-3
7 digits	cfloat(fp32), posit<32,2>	~7-8
16 digits	cfloat(fp64), double	~16
31 digits	dd, dd_cascade	~31
48 digits	td_cascade	~48
64 digits	qd, qd_cascade	~64
Exact (decimal)	dfixpnt	Configurable (ndigits)
Exact	rational, integer, quire	Unlimited (within nbits)

Quick Start

Every number system is header-only. Include the type and start computing:

#include <universal/number/posit/posit.hpp>  // or any type
using namespace sw::universal;

// Plug-in replacement pattern
template<typename Real>
Real my_algorithm(Real a, Real b) {
    return (a + b) * (a - b);
}

// Use with any Universal type
auto r1 = my_algorithm(posit<32,2>(3.0), posit<32,2>(4.0));
auto r2 = my_algorithm(cfloat<16,5,uint16_t,true,false,false>(3.0),
                        cfloat<16,5,uint16_t,true,false,false>(4.0));
auto r3 = my_algorithm(dd(3.0), dd(4.0));

For detailed usage patterns, see the api/api.cpp test file in each number system’s regression test directory under static/.