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Beating Posits at Their Own Game: Takum Arithmetic

Laslo Hunhold

TL;DR

The paper introduces Takum Arithmetic, a base-$\sqrt{e}$ logarithmic tapered-precision number format designed for general-purpose computing. It formalizes a lossless Takum encoding, a NaR convention, and a logarithmic significand, proving key properties such as uniqueness, monotonicity, and inversion, while detailing rounding and hardware-friendly encoding. Takums are shown to deliver a constant dynamic range for all $n\ge 12$, offer favorable relative-precision bounds, and exhibit strong arithmetic-closure properties—often outperforming posits and standard IEEE formats in both dynamic range and accuracy, particularly for large and small magnitudes. The work argues for an emerging field of tapered-precision numerical analysis and demonstrates practical hardware advantages, enabling efficient, mixed-precision workflows and potentially superior general-purpose arithmetic when compared to existing standards.

Abstract

Recent evaluations have highlighted the tapered posit number format as a promising alternative to the uniform precision IEEE 754 floating-point numbers, which suffer from various deficiencies. Although the posit encoding scheme offers superior coding efficiency at values close to unity, its efficiency markedly diminishes with deviation from unity. This reduction in efficiency leads to suboptimal encodings and a consequent diminution in dynamic range, thereby rendering posits suboptimal for general-purpose computer arithmetic. This paper introduces and formally proves 'takum' as a novel general-purpose logarithmic tapered-precision number format, synthesising the advantages of posits in low-bit applications with high encoding efficiency for numbers distant from unity. Takums exhibit an asymptotically constant dynamic range in terms of bit string length, which is delineated in the paper to be suitable for a general-purpose number format. It is demonstrated that takums either match or surpass existing alternatives. Moreover, takums address several issues previously identified in posits while unveiling novel and beneficial arithmetic properties.

Beating Posits at Their Own Game: Takum Arithmetic

TL;DR

The paper introduces Takum Arithmetic, a base- logarithmic tapered-precision number format designed for general-purpose computing. It formalizes a lossless Takum encoding, a NaR convention, and a logarithmic significand, proving key properties such as uniqueness, monotonicity, and inversion, while detailing rounding and hardware-friendly encoding. Takums are shown to deliver a constant dynamic range for all , offer favorable relative-precision bounds, and exhibit strong arithmetic-closure properties—often outperforming posits and standard IEEE formats in both dynamic range and accuracy, particularly for large and small magnitudes. The work argues for an emerging field of tapered-precision numerical analysis and demonstrates practical hardware advantages, enabling efficient, mixed-precision workflows and potentially superior general-purpose arithmetic when compared to existing standards.

Abstract

Recent evaluations have highlighted the tapered posit number format as a promising alternative to the uniform precision IEEE 754 floating-point numbers, which suffer from various deficiencies. Although the posit encoding scheme offers superior coding efficiency at values close to unity, its efficiency markedly diminishes with deviation from unity. This reduction in efficiency leads to suboptimal encodings and a consequent diminution in dynamic range, thereby rendering posits suboptimal for general-purpose computer arithmetic. This paper introduces and formally proves 'takum' as a novel general-purpose logarithmic tapered-precision number format, synthesising the advantages of posits in low-bit applications with high encoding efficiency for numbers distant from unity. Takums exhibit an asymptotically constant dynamic range in terms of bit string length, which is delineated in the paper to be suitable for a general-purpose number format. It is demonstrated that takums either match or surpass existing alternatives. Moreover, takums address several issues previously identified in posits while unveiling novel and beneficial arithmetic properties.
Paper Structure (33 sections, 14 theorems, 94 equations, 22 figures, 6 tables, 5 algorithms)

This paper contains 33 sections, 14 theorems, 94 equations, 22 figures, 6 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Gustafson Criteria
  3. Dynamic Range Criteria
  4. IEEE 754 Floating-Point Numbers
  5. Posits
  6. Takum Arithmetic
  7. Definition
  8. Rounding
  9. Logarithmic Significand
  10. Choice of Base $\sqrt{\mathrm{e}}$
  11. Formal Analysis
  12. $\mathrm{NaR}$ Convention
  13. Linear Takums
  14. Evaluation
  15. Gustafson Criteria
  16. ...and 18 more sections

Key Result

proposition thmcounterproposition

Assume an IEEE 754 floating-point format with $n_e$ exponent bits and $n_f$ fraction bits. The ratio of redundant bit strings and bit strings representing numbers exceeding $\pm\left(\sqrt{\mathrm{e}}^{-255},\sqrt{\mathrm{e}}^{255}\right)$ is approximately

Figures (22)

  • Figure 1: IEEE 754 floating-point number bit string format.
  • Figure 2: A comparison of the linear and logarithmic significands.
  • Figure 3: Bit requirements for encoding a given value with different methods. The Elias codes are both defined in lindstrom-2018.
  • Figure 4: Dynamic range comparison between various number formats relative to the bit string length $n$.
  • Figure 5: Comparison of negative logarithmic relative approximation error upper bounds within an 8-bit budget for encoded numbers $x$. Higher values indicate lower relative error and, consequently, higher precision.
  • ...and 17 more figures

Theorems & Definitions (36)

  • proposition thmcounterproposition
  • proof
  • definition thmcounterdefinition: posit encoding
  • proposition thmcounterproposition
  • proof
  • definition thmcounterdefinition: takum encoding
  • proposition thmcounterproposition: Takum Uniqueness
  • proof
  • definition thmcounterdefinition: Unsigned Integer Mapping
  • definition thmcounterdefinition: Bit String Incrementation
  • ...and 26 more