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Tekum: Balanced Ternary Tapered Precision Real Arithmetic

Laslo Hunhold

TL;DR

Tekum proposes a balanced ternary tapered-precision real arithmetic format addressing an underexplored area of real-number computation in ternary logic. It builds on three design filters to map ternary trits to real values, defines the tekum encoding with regime, exponent, and fraction trits, and evaluates its numerical properties against posits, takums, and conventional floats. The results show tekums deliver favorable accuracy, broad dynamic range (around 10^±87) with a truncation-based rounding, and a monotone, unique encoding that handles NaR and ∞ naturally. The work argues that balanced ternary real arithmetic can be viable and potentially transformative for energy-efficient, memory-bandwidth-limited computing, motivating further hardware and algorithmic exploration.

Abstract

In light of recent hardware advances, it is striking that real arithmetic in balanced ternary logic has received almost no attention in the literature. This is particularly surprising given ternary logic's promising properties, which could open new avenues for energy-efficient computing and offer novel strategies for overcoming the memory wall. This paper revisits the concept of tapered precision arithmetic, as used in posit and takum formats, and introduces a new scheme for balanced ternary logic: tekum arithmetic. Several fundamental design challenges are addressed along the way. The proposed format is evaluated and shown to exhibit highly promising characteristics. In many respects, it outperforms both posits and takums. As ternary hardware matures, this work represents a crucial step toward unlocking the full potential of real-number computation in ternary systems, laying the groundwork for a new class of number formats designed from the ground up for a new category of next-generation hardware.

Tekum: Balanced Ternary Tapered Precision Real Arithmetic

TL;DR

Tekum proposes a balanced ternary tapered-precision real arithmetic format addressing an underexplored area of real-number computation in ternary logic. It builds on three design filters to map ternary trits to real values, defines the tekum encoding with regime, exponent, and fraction trits, and evaluates its numerical properties against posits, takums, and conventional floats. The results show tekums deliver favorable accuracy, broad dynamic range (around 10^±87) with a truncation-based rounding, and a monotone, unique encoding that handles NaR and ∞ naturally. The work argues that balanced ternary real arithmetic can be viable and potentially transformative for energy-efficient, memory-bandwidth-limited computing, motivating further hardware and algorithmic exploration.

Abstract

In light of recent hardware advances, it is striking that real arithmetic in balanced ternary logic has received almost no attention in the literature. This is particularly surprising given ternary logic's promising properties, which could open new avenues for energy-efficient computing and offer novel strategies for overcoming the memory wall. This paper revisits the concept of tapered precision arithmetic, as used in posit and takum formats, and introduces a new scheme for balanced ternary logic: tekum arithmetic. Several fundamental design challenges are addressed along the way. The proposed format is evaluated and shown to exhibit highly promising characteristics. In many respects, it outperforms both posits and takums. As ternary hardware matures, this work represents a crucial step toward unlocking the full potential of real-number computation in ternary systems, laying the groundwork for a new class of number formats designed from the ground up for a new category of next-generation hardware.

Paper Structure

This paper contains 15 sections, 5 theorems, 20 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Balanced Ternary
  3. The Three Filters
  4. Filter 1: Asymmetry
  5. Filter 2: Misfit Tool
  6. Filter 3: Excess
  7. Tekum Definition
  8. Evaluation
  9. Comparing Binary with Ternary
  10. Format
  11. Accuracy
  12. Dynamic Range
  13. Formal Analysis
  14. Hardware Implementation
  15. Conclusion

Key Result

proposition 1

Let $n \in \mathbb{N}_0$. It holds $4 \mid (3^{2n} - 5)$ and $4 \nmid (3^{n} - 4)$.

Figures (5)

  • Figure 1: A visualisation of the real wheel algebra with the customary positioning of the bottom element $\mathrm{NaR}$ in the center as a "wheel hub" that distinguishes it from the projectively extended real numbers.
  • Figure 2: Mapping of ternary strings of lengths $n \in \{1,2,3\}$ to the real wheel algebra.
  • Figure 3: Mapping of ternary strings $\bm{t}$ of length $4$ to the real wheel algebra. The values of $\mathop{\mathrm{anc}}\nolimits_4(\bm{t})$ are given, with the first three trits, designated as the regime trits $\textcolor{regime}{\bm{r}}$, highlighted accordingly. The corresponding regime values $r$ are also indicated, partially with suffixed signs for better visual consistency.
  • Figure 4: The number of non-fraction bits, which can be considered as overhead, relative to the represented value $x$ in a selection of floating-point formats. The y-axis is inverted, thus meaning that higher values mean less overhead.
  • Figure 5: Dynamic range relative to the bit string length $n$ for tekum, (linear) takum, posit and a selection of floating-point formats.

Theorems & Definitions (18)

  • definition 1: balanced ternary strings
  • definition 2: concatenation
  • definition 3: integer mapping
  • definition 4: negation
  • definition 5: addition and subtraction
  • definition 6: modulus
  • proposition 1
  • proof
  • definition 7: anchor function
  • definition 8: tekum encoding
  • ...and 8 more