Value-Aware Numerical Representation

Updated 16 January 2026

Value-aware numerical representation is a set of techniques that adjust encoding based on value magnitude, context, and task demands.
Methods include adaptive floating-point formats, context-sensitive bit assignment, and embedding-based strategies to optimize precision and resource allocation.
These approaches enhance computation efficiency and accuracy in hardware and neural networks while aligning with human-like numerical perception.

Value-aware numerical representation encompasses a broad class of techniques and formalisms in which the encoding, storage, or embedding of numbers is explicitly sensitive to the range, distribution, or semantic relevance of numerical values within specific computational, machine learning, or mathematical reasoning contexts. Rather than treating numbers as opaque tokens or relying exclusively on uniform-width bit strings, value-aware systems encode additional information derived from value magnitude, local context, or downstream task requirements. Such representations have emerged in hardware arithmetic, machine learning quantization, program analysis, table/text understanding, and LLMs, motivated by the need for increased efficiency, dynamic range, precision allocation, and semantic differentiation.

1. Principles and Taxonomy of Value-Aware Numerical Representations

Value-aware representations are characterized by their direct dependence on value magnitude, local numerical context, or application-driven precision needs, in contrast to static actor-agnostic schemes (e.g., canonical binary). Key forms include:

Dynamically partitioned floating-point formats: Encodings where the exponent/mantissa width or allocation varies adaptively with the represented value (e.g., Floating-Floating-Point, Morris variants).
Context-sensitive bit assignment: Number systems such as Adaptive Base Representation, where the positional "weight" or base is computed as a function of value or prior context.
Per-layer or per-block scaling: In neural architectures, quantization formats that select exponent bias or scale factors to fit observed value distributions in each layer or block at runtime or calibration (e.g., AdaptivFloat, ABFP).
Embedding-based methods: For unstructured inputs, neural models that inject value-sensitive features into input embeddings (e.g., digit splitting, explicit magnitude encodings, value-conditioned prefix tokens in Transformers).
Smooth and continuous integer encodings: Methods that embed discrete states or counts into invertible, smoothly varying real-valued signals, useful for differentiable optimization (e.g., integral-balance encodings).

Such methods are unified by the objective of non-uniform resolution: assigning more representational detail (e.g., denser grid points, wider mantissas) where numerical differences are semantically or statistically important, or by encoding value structure directly into feature or bit layouts.

2. Dynamic and Tapered Floating-Point Formats

A central class of value-aware representations is dynamic, tapered, or flexible-width floating-point schemes. Notable examples include:

Floating-Floating-Point (F2P): F2P numbers allow the exponent field width to "float" based on a hyper-exponent, trading off local precision for range in a value-aware manner. Bit allocations can be tuned so that, for instance, more mantissa bits are concentrated where values are small (SR/LR flavors for small/large real numbers). This enables precise representation in subranges where accuracy matters and a compressed encoding where it does not (Cohen et al., 2024).
Morris Tapered Floating-Point and Variants: MorrisHEB, MorrisBiasHEB, and MorrisUnaryHEB dynamically adjust the relative field widths for exponent and fraction based on either explicit field values or unary regime runs. These formats achieve significant increases in dynamic range (up to 149× posit16 for MorrisUnaryHEB) and higher accuracy for certain operations, with a "golden zone" of dense representable values centered around the origin (Ciocirlan et al., 2023). This adaptivity grants high local precision for magnitudes of greatest numerical or algorithmic interest.
AdaptivFloat and Adaptive Block Floating-Point (ABFP): AdaptivFloat assigns per-layer exponent biases at calibration or quantization, tightly fitting the dynamic range of weights or activations and performing optimal clipping. ABFP uses per-block exponents, dynamically recomputed to minimize quantization error over blocks of weights or activations (Tambe et al., 2019, Basumallik et al., 2022). Both approaches maximize information efficiency at low bit-width by matching dynamic range to data, outperforming uniform, IEEE, and posit encodings on DNN inference accuracy at ≤8 bits.

3. Value-Aware Embedding and Neural Representations

In LLMs and neural architectures operating on symbolic or tabular data, there is a recognized failure of traditional (token-based) embeddings to encode value structures. Value-aware embedding strategies include:

Prefix embeddings in Transformers: Models can prepend a computed, magnitude-conditioned embedding (e.g., a <num> token) to numeric expressions; the embedding is designed to inject numerical value information directly and continuously into the model's input space. This approach (NumValue-MLP, NumValue-RNN) improves arithmetic and comparison accuracy over standard token-based systems by aligning the representations with the underlying real numbers (Dutulescu et al., 14 Jan 2026).
Digit-level and magnitude decompositions: Systems like ForTaP for table modeling and Perfograph for program analysis employ digit splitting, magnitude binning, or similar feature engineering to create embeddings reflecting the structure of the number (e.g., length, significant digit, precision) (Cheng et al., 2021, TehraniJamsaz et al., 2023).
Latent numerical manifolds in pretrained LLMs: Analysis of LLM internal states reveals emergent sublinear (logarithmic-style) number lines, where representational distances between values compress with magnitude (AlquBoj et al., 22 Feb 2025). Although not explicitly engineered, this effect resembles value-aware scaling and has implications for embedding design.

4. Integer and Discrete Value-Aware Encodings

Value-aware strategies are not restricted to real-valued data. For discrete structures:

Smooth Integer Encoding via Integral Balance: This method encodes an integer $N$ as a real-valued function whose definite integral converges to zero as $N$ increases, with each $N$ linked to a unique "balance point." Recovery is achieved by searching for near-zero-crossings in the integral map (Semenov, 28 Apr 2025). This enables continuous, differentiable treatment of discrete counts in optimization and machine learning pipelines, providing robustness and embedding flexibility absent in one-hot encodings.
Adaptive Base Representation (ABR) for integers: ABR reassigns the weight of each bit in an $n$ -bit binary vector according to a corrective function and adaptive base sequence, yielding a unique and value-sensitive encoding with the same bit length and integer range as standard binary. ABR accommodates error detection/correction mechanisms and supports novel applications in data compression and steganography due to its uneven code distribution (Kumar, 16 Oct 2025).

5. Value-Aware Representations in Program, Table, and Knowledge Graph Models

Recent systems for program analysis, table understanding, and knowledge graph reasoning have adopted value-aware embeddings to address the limitations of symbolic or structure-agnostic encodings:

Perfograph: Introduces digit-level embeddings for numeric constant nodes and explicit graph stratification of aggregate data structures, resulting in improved performance on device mapping, parallelism discovery, and NUMA configuration prediction (up to 10% error rate reduction versus prior representations) (TehraniJamsaz et al., 2023).
ForTaP: In semi-structured tables, ForTaP's numeric feature decomposition and formula-based supervision yield state-of-the-art results in formula prediction and semantic table QA. The system's value-awareness arises both from its embedding layer and explicit numerical-reasoning objectives, which group and discriminate numbers by scale, precision, and context (Cheng et al., 2021).

6. Compression, Hardware, and Practical Implications

Value-aware numerical representations are often adopted for storage and hardware efficiency:

32-bit Encodings for 64-bit Values: Compact, table-driven schemes store the upper bits of a 64-bit double and reconstruct the rest by table lookup, enabling exact representation for numbers falling within predefined decimal digit patterns. The decoder's lookup table is value-aware in that its indexing uses the retained bits most salient for the specified subset, optimizing both cache behavior and representation fidelity (Neal, 2015).
Hardware Implementation and Trade-offs: Dynamic field width schemes (e.g., F2P, AdaptivFloat, Morris variants) entail minimal extra cost over standard floating-point units—primarily additional logic for field length decoding and value-dependent field extraction. These formats are highly efficient in environments demanding wide dynamic range and localized high precision (e.g., neural quantization, approximate counting, low-power DSP, analog inference) (Cohen et al., 2024, Ciocirlan et al., 2023, Tambe et al., 2019).

The implementation and selection of value-aware schemes must balance error characteristics, quantization behavior, energy consumption, and compatibility with downstream processing, motivating variable-width or value-sensitive field allocations and context-sensitive encoding functions.

7. Theoretical and Cognitive Parallels

Recent computational and neuroscientific evidence suggests a strong parallel between value-aware encoding strategies and human numerical cognition:

LLMs, when probed with standard dimensionality reduction, reveal that their internal representations of magnitude are sublinear, compressing with increasing value in a manner reminiscent of the Weber–Fechner law (AlquBoj et al., 22 Feb 2025).
The use of explicit log-scale or adaptive encodings in numerical processors and machine learning mirrors the logarithmic mapping observed in human numerosity perception, further motivating the adoption of value-awareness for improved alignment with natural processing biases.

In summary, value-aware numerical representation is an active and interdisciplinary area encompassing dynamic floating-point schemes, block- or layer-adaptive quantization, embedding architectures, and smooth integer mappings. These approaches are motivated by empirical distributions, semantic task requirements, or cognitive analysis, uniformly optimizing resource allocation and representational accuracy in both hardware and high-level neural processing pipelines.