Posit: Tapered Floating-Point (UNUM Type III)

Why

IEEE-754 floating-point distributes its precision uniformly across the entire exponent range. A 32-bit float has the same number of fraction bits whether the value is 10^-30 or 10^30. But most real-world computations cluster their values near 1.0 — neural network weights, probabilities, normalized sensor readings, physical quantities in natural units. IEEE-754 wastes precision on extreme exponents that are rarely used while providing no more precision where it matters most.

Posits, invented by John Gustafson as UNUM Type III, solve this with tapered precision: values near 1.0 get the most fraction bits (highest precision), while values at the extremes of the range trade fraction bits for exponent bits (widest range). The result is a format that provides more useful precision than IEEE-754 at the same bit width, with a cleaner encoding that eliminates the redundancy and complexity of NaN, denormals, and signed zeros.

What

posit<nbits, es, bt> is a tapered-precision floating-point:

Parameter	Type	Default	Description
nbits	unsigned	—	Total bits (2 to 256+)
es	unsigned	—	Maximum exponent bits (0 to 9+)
bt	typename	uint8_t	Storage block type

Encoding

A posit encodes four fields in a variable-length format:

[sign : 1] [regime : variable] [exponent : ≤ es] [fraction : remaining]

• Sign: 0 = positive, 1 = negative (2’s complement for the entire encoding)
• Regime: run-length encoded scale factor
  • Run of k ones followed by a zero: regime value = k (scale = useed^k)
  • Run of k zeros followed by a one: regime value = -k (scale = useed^(-k))
• Exponent: up to es bits of binary exponent
• Fraction: remaining bits encode the significand

The useed = 2^(2^es) is the regime scale factor:

• es=0: useed = 2 (each regime step doubles/halves)
• es=1: useed = 4
• es=2: useed = 16
• es=3: useed = 256

Tapered Precision

For values near 1.0, the regime is short (1-2 bits), leaving most bits for fraction (high precision). For extreme values, the regime consumes most bits, providing enormous dynamic range at the cost of precision:

posit<32, 2>:
  Near 1.0:  regime=2 bits, exponent=2 bits, fraction=27 bits  (~8 decimal digits)
  Near max:  regime=29 bits, exponent=2 bits, fraction=0 bits  (range only)

Key Properties

• Two special values only: all-zeros = 0, MSB-only = NaR (Not a Real)
• No NaN, no signed zero, no denormals: cleaner than IEEE-754
• Reciprocal symmetry: posit(x) and posit(1/x) have the same precision
• Exact integers: small integers near 1.0 are represented exactly
• More useful precision: typically 1-2 extra decimal digits vs same-size IEEE float
• Fused operations: atomic fma(), fam(), fmma() using blocktriple intermediate

Dynamic Range

Configuration	Dynamic Range
posit<8, 0>	2^(-6) to 2^6
posit<16, 1>	2^(-28) to 2^28
posit<32, 2>	2^(-120) to 2^120
posit<64, 3>	2^(-496) to 2^496

How It Works

The regime field is the key innovation. It uses run-length encoding to create a variable-width exponent:

1. Decode regime: count the run of identical bits after the sign bit
2. Compute regime scale: useed^regime_value
3. Decode fixed exponent: next es bits (if available)
4. Decode fraction: remaining bits form the significand with an implicit leading 1

The value is: (-1)^sign × useed^regime × 2^exponent × (1 + fraction)

This variable-length structure is what creates tapered precision: when the regime is short (value near 1), more bits remain for fraction; when the regime is long (extreme value), precision is sacrificed for range.

Arithmetic uses a blocktriple intermediate representation that carries sufficient precision for exact computation before rounding back to the target posit format.

How to Use It

Include

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

Basic Usage

posit<32, 2> a(3.14159265358979);
posit<32, 2> b(2.71828182845905);
posit<32, 2> c = a * b;

std::cout << "pi * e = " << c << std::endl;
std::cout << "encoding: " << to_binary(c) << std::endl;
std::cout << "components: " << components(c) << std::endl;

Comparing Precision with IEEE-754

// posit<32,2> has more useful precision than float32 near 1.0
float f = 1.0f;
posit<32, 2> p(1.0);

for (int i = 0; i < 10; ++i) {
    f += 1.0e-7f;
    p += posit<32, 2>(1.0e-7);
}
// posit preserves more digits near 1.0 due to tapered precision

Mixed-Precision Deep Learning

template<typename Real>
Real sigmoid(Real x) {
    return Real(1) / (Real(1) + exp(-x));
}

// 8-bit posit for inference: more precision near 0-1 range than fp8
using Inference = posit<8, 0>;
Inference weight(0.5);
Inference input(0.3);
Inference output = sigmoid(weight * input);

Plug-in Replacement

template<typename Real>
Real newton_sqrt(Real x, int iterations = 10) {
    Real guess = x / Real(2);
    for (int i = 0; i < iterations; ++i) {
        guess = (guess + x / guess) / Real(2);
    }
    return guess;
}

// Compare convergence: posit vs float
auto sqrt_f = newton_sqrt(2.0f);
auto sqrt_p = newton_sqrt(posit<32, 2>(2.0));
// posit converges to more accurate result at same bit width

Quire: Exact Dot Products

#include <universal/number/posit/posit.hpp>
// The quire provides exact accumulation with no intermediate rounding
using Posit = posit<32, 2>;

// Fused dot product: exact accumulation in quire, single rounding at the end
Posit a[] = { Posit(1.0), Posit(2.0), Posit(3.0) };
Posit b[] = { Posit(4.0), Posit(5.0), Posit(6.0) };

quire<32, 2> q;
for (int i = 0; i < 3; ++i) {
    q += quire_mul(a[i], b[i]);  // No rounding between multiplications
}
Posit dot_product;
convert(q.to_value(), dot_product);  // Single rounding at the end

Problems It Solves

Problem	How posit Solves It
IEEE float wastes precision on extreme values	Tapered precision: most bits near 1.0 where values cluster
NaN propagation corrupts entire computation	Only one non-real value (NaR), explicit and predictable
Signed zero and multiple NaN encodings waste bit patterns	Two special values only: 0 and NaR
8-bit float has too little precision for ML inference	posit<8,0> has more precision near 0-1 than fp8
Dot product rounding errors accumulate	Quire provides exact accumulation, single final rounding
Need better precision without increasing bit width	posit<32,2> typically provides 1-2 more decimal digits than float32
Reproducible linear algebra across platforms	Quire + posit = same result on every platform