Tapered-Precision Representations (TPRs)

• TPRs are flexible number systems that dynamically adjust exponent and significand bits, optimizing the trade-off between precision and dynamic range.
• They underpin formats like posit, Morris TPRs, Takum, and Tekum, delivering superior golden zone precision and reduced bit pattern waste.
• Their regime-based encoding improves performance in low-precision hardware arithmetic, scientific computing, and ML quantization by tailoring representation to data.

Tapered-Precision Representations (TPRs) are a class of number systems designed to provide a flexible trade-off between dynamic range and precision by allocating the number of exponent and significand bits dynamically on a value-by-value basis. Unlike fixed-width IEEE 754 floating-point or classic fixed-point representations, TPRs utilize a regime or “tapering” field, allowing values closer to unity to enjoy higher precision at the cost of exponent range and, for large-magnitude values, to allocate more bits to exponent at the cost of precision. This approach enables encoding real numbers with greater flexibility, which is particularly advantageous in domains such as low-precision hardware arithmetic, scientific computing, and machine learning quantization. The TPR paradigm underpins formats such as posit, takum, tekum, the Morris family, and various ternary/balanced-radix and logarithmic number systems (Ciocirlan et al., 2023, Hunhold, 2024, Hunhold, 25 Nov 2025, Schoenbaum, 2021, Langroudi et al., 2021).

1. Foundational Principles and Historical Context

The TPR concept can be traced to Morris’s tapered floating-point (1971) which first encoded the exponent size within each word, dynamically determining the split between exponent and mantissa fields (Ciocirlan et al., 2023). Subsequent innovations include the posit format, which employs unary regime encoding; redundant signed-radix encodings (Schoenbaum, 2021); and logarithmic systems with hybrid characteristics such as takum (Hunhold, 2024) and tekum (Hunhold, 25 Nov 2025). The motivation for these designs derives from the limitations of fixed-width representations such as wasted bit-patterns, abrupt underflow/overflow, and suboptimal usage of representable values, particularly for low-precision arithmetic and near-unity values.

2. General Structure and Encoding Schemes

A canonical TPR partitions its bit or trit string into:

• A sign component (often a single bit or trit, but may be implicit in the digit set as in redundant signed-radix).
• A variable-length regime or size field, which encodes, commonly via unary or run-length or balanced-ternary count, the total bit/trit budget for the exponent.
• An exponent field of width determined by the regime/size field, which sets the value’s order of magnitude.
• A fraction (significand/mantissa) field whose width is the complement of regime plus exponent within the total register width.

This dynamic partitioning mechanism is implemented via various encoding approaches:

Format	Regime/Size Coding	Exponent Coding	Fraction Coding
Morris TPR	Binary/biased/unary field	Variable-length	Remainder bits
Posit/TFX	Unary regime	Fixed or implicit	Variable
Takum	Fixed 3-bit regime, direction	Characteristic bits	Variable
Tekum	Ternary regime (3 trits)	Variable length	Variable
Redundant radix-2	NAF regime	NAF, sparse	NAF, sparse

The value decoding generally follows the form:

$$x = (-1)^s \cdot \text{useed}^r \cdot 2^e \cdot (1+f)$$

for binary-based formats, or employs logarithmic or ternary analogues in more general settings (Hunhold, 25 Nov 2025, Hunhold, 2024, Ciocirlan et al., 2023).

3. Key Format Realizations and Comparative Metrics

Several contemporary TPR instances offer distinct attributes tailored to specific use cases:

• Morris-family (MorrisHEB, MorrisBiasHEB, MorrisUnaryHEB): Introduce hidden exponent bits and different regime codings, yielding better dynamic range and precision in the “golden zone” ([10^{-3},10^3]) especially for machine learning (Ciocirlan et al., 2023). For example, MorrisUnaryHEB with 16 bits provides 13.6% more distinct representable values in the golden zone than Posit<sub\>16,2</sub>.
• Takum: Logarithmic, direction-modulated regime with characteristic and mantissa, achieving an asymptotic constant dynamic range (e^{510}, independent of n for n ≥ 12), minimal bit pattern waste (1 codeword), and efficient arithmetic via log-domain transformations (Hunhold, 2024).
• Tekum: Balanced ternary variant with 3-trit regime, bias-controlled exponent, and truncation-only rounding, achieving a practical dynamic range of ≈10^{±87} for 40-trit representations, eliminating subnormals and carry-to-round circuits, and targeting emerging CNTFET ternary hardware (Hunhold, 25 Nov 2025).
• Redundant Signed Radix-2/NAF TPR: Employs carry-free, non-adjacent encoding, with the exponent and significand realized as sparse NAF sequences, supporting all standard data types and flexible “point signatures” for hardware-ISA safety (Schoenbaum, 2021).
• TFX (TENT framework): Unary regime, fixed-point fraction, and per-layer scalable dynamic range (TENT), optimized for TinyML quantization with 5–8 bits, yielding up to 31pp accuracy gains over fixed-point, with only ~17–30% energy-delay overhead (Langroudi et al., 2021).

A summary table of representative TPR features is below:

Scheme	Dynamic Range	“Golden Zone” Precision	Hardware/Loss Properties
IEEE754-16	≈[10^{-38},10^{38}]	typical	Many wasted patterns, subnormals
Posit-16	≈[10^{-17},10^{17}]	lower than MorrisUnary	regime-overhead at tails
Takum-16	≈[10^{-111},10^{110}]	constant by n≥12	log-domain, only 1 wasted pattern
Tekum-40	≈[10^{-87},10^{87}]	wide central plateau	balanced ternary, truncation rounding
MorrisUnaryHEB-16	≈10^{1233}	+13.6% vs posit-16	high exact-add, hardware unary regime

4. Mathematical Formalisms and Decoding Mappings

TPRs’ core versatility is evident in their mathematically precise encoding/decoding schemes. For instance, takum numbers of n bits are decoded via:

• Bitfield partition: sign (S), direction (D), regime (R, 3 bits), characteristic (C, r bits), mantissa (M, p bits).
• Regime decoding: r = 4R₂+2R₁+R₀ (D=1); r = 7–(4R₂+2R₁+R₀) (D=0).
• Characteristic: c = 2^r–1 + ∑C_i2^i (D=1); or c = –2^{r+1}+1 + ∑C_i2^i (D=0).
• Mantissa: m = 2^{–p}∑M_i2^i ∈ [0,1).
• Log value and real value: ℓ = (–1)^S(c+m); t = 0 if all fields zero; = (–1)^S·e^ℓ otherwise (Hunhold, 2024).

Tekum exploits balanced-ternary trits t_i∈{1,0,–1}, using anchoring and regime-based exponent scaling, with normalization ensuring every value is always anchored at 1.0 and rounding by simple truncation (Hunhold, 25 Nov 2025). Redundant signed-radix-2 TPRs use NAF coding so the exponent field’s length adjusts automatically, while precision at a given magnitude is determined by the digits left for the significand (Schoenbaum, 2021).

5. Precision-Dynamic Range Trade-offs and Application Implications

TPRs adapt their fraction/exponent trade-off to actual data or application requirements. For TinyML and edge-computing, per-layer quantization with TFX (in TENT) and similar formats results in reduced quantization error and enhanced classification accuracy, especially for low-bit scenarios (e.g., 8-bit TFX offers 27.5–33.8pp accuracy gains on ResNet-18 over fixed-point (Langroudi et al., 2021)). In benchmarks involving constants and scientific kernels, MorrisUnaryHEB and takum frequently achieve superior accuracy and dynamic range versus posit and IEEE 754 for equivalent word lengths (Hunhold, 2024, Ciocirlan et al., 2023).

In hardware, regime-based coding often leads to more efficient path lengths and removes much of the control complexity inherent to subnormal handling, sticky/guard bits, and dual-format encodings of legacy formats (such as IEEE-754). Balanced-ternary systems (tekum) and non-adjacent coding (redundant radix-2) are specifically motivated by emerging hardware with native multi-level logic, potentially offering further reductions in area and energy, contingent on technology maturation (Hunhold, 25 Nov 2025, Schoenbaum, 2021).

6. Limitations, Unresolved Issues, and Current Directions

Despite their strengths, TPRs present several trade-offs:

• Exact regime/fraction trade-offs, overflow/underflow endpoints, and mid-range “golden zone” width all depend on precise field layout and design choices; optimal parameters are context-dependent.
• For some TPRs (e.g., Morris TPRs), encoding uniqueness is only achieved in certain variants (MorrisHEB, MorrisBiasHEB), with under/overflow mapping to Not-Representable (NR) or reserved codewords (Ciocirlan et al., 2023).
• In highly quantized neural network hardware, the energy overhead, although modest (~17–30%), is not zero; regime and variable length fields introduce pipeline and decode variability, leading to modest area/control complexity (TENT (Langroudi et al., 2021), tekum (Hunhold, 25 Nov 2025)).
• While TPRs eliminate subnormals and many anomalous features of fixed-point and IEEE754, by design they may “waste” dynamic range at extremes (e.g., in tekum, dynamic range is trimmed empirically to avoid overspending representational states on rare or unnecessary values (Hunhold, 25 Nov 2025)).
• TPRs may also be less friendly to bit-level manipulation for cryptographic or legacy code interoperability due to nonuniform regimes and truncation/anchoring rounding rules, though this is format-specific.

A plausible implication is that the continuing proliferation of new TPRs—spanning unary, binary, ternary, and logarithmic hybrids—mirrors both emerging hardware and diverse application needs, rather than any universal superiority of a single format.

7. Software, Benchmarks, and Emerging Research

A unified Scala-based library supports rapid exploration of new TPR designs by encoding, decoding, and benchmarking arbitrary field layouts, including all major TPR families (Ciocirlan et al., 2023). The library evaluates decimal accuracy, exactness, and dynamic range both on unary/binary mathematical operations and on real-world constants—facilitating direct comparison between Posit, IEEE754, and new MorrisHEB/Bias/UnaryHEB formats.

Recent research evaluates TPRs against a spectrum of metrics: SI and cosmic constants representability, accuracy on classic numerical kernels (e.g., Wallis product, Rump’s polynomial), and machine learning performance (Hunhold, 2024, Ciocirlan et al., 2023, Langroudi et al., 2021). Experimental outcomes show that constant or plateaued dynamic range as in takum (Hunhold, 2024), and enhanced “golden zone” coverage as in MorrisUnaryHEB (Ciocirlan et al., 2023), are particularly beneficial in both scientific and ML contexts. Ongoing work explores application-specific, hardware-realizable TPRs for future non-binary computing substrates (Hunhold, 25 Nov 2025, Schoenbaum, 2021).