Custom Floating-Point Formats

• Custom floating-point formats are non-standard numerical representations that allow tailored adjustments to precision, dynamic range, and hardware efficiency.
• They are implemented using methods such as bitslice arithmetic, tapered-precision encoding, and variable-length exponent fields to optimize energy use and performance.
• These formats provide crucial trade-offs between accuracy and hardware complexity, benefiting deep learning, scientific computing, and embedded systems.

Custom floating-point formats are non-standard numerical representations engineered to enable fine-grained control over precision, dynamic range, hardware efficiency, and algorithmic behavior. Unlike the rigidly defined IEEE 754 formats (such as binary32 or binary64), custom floating-point formats allow designers and software engineers to reshape the number of sign, exponent, and mantissa bits or adopt variable-length and entropy-based encodings. This flexibility is employed to increase performance, energy efficiency, computational density, or numerical fidelity tailored to the characteristics of a given hardware, algorithm, or workload.

1. Principles and Motivations

The core principle underlying custom floating-point formats is the ability to precisely match numeric representation to the requirements of a specific application or platform. The trade-off between accuracy, hardware complexity, power consumption, memory footprint, and dynamic range can be adjusted by customizing the number format beyond the conventional binary16/32/64 types. Motivations include:

• Energy efficiency and throughput: Lower bit-widths reduce data movement, storage requirements, and arithmetic latency, which is critical in energy-constrained and high-throughput systems (Tagliavini et al., 2017, Mach et al., 2020).
• Transprecision computing: Assigning different precisions to different parts of an algorithm for optimal performance and energy use (Tagliavini et al., 2017, Mach et al., 2020).
• Numerical robustness for specific domains: Ensuring the dynamic range and relative error are matched to deep learning, scientific computing, or multimedia processing (Bertaccini et al., 2022, Defour et al., 2020).
• Supporting reconfigurable/programmable hardware: Enabling FPGAs, ASICs, and custom accelerators to exploit the full design space of number representations (Xu et al., 2016, Campos et al., 9 Sep 2024).

2. Key Methodological Approaches

Custom floating-point formats are realized through several characteristic design and implementation strategies:

Methodology	Description	Typical Applications
Bitslice Vector Arithmetic	Splitting bitfields across wide registers for SIMD-parallel computation on arbitrary widths	Image processing, vectorized operations
Parameterized Format Templates	Software libraries allow specification of exponent/mantissa bit-widths per variable	Precision-tuned scientific codes
Entropy/Variable-length Coding	Exponents/signs are entropy-coded (e.g., Huffman/Limited Huffman), maximizing significand bits	Embedding compression, large-scale models
Tapered-Precision/Posit/Takum	Variable allocation between exponent and fraction based on value magnitude for wide dynamic range	Deep learning, scientific/hybrid workloads
Reconfigurable Hardware Paths	FPGA/ASIC datapaths adapt bit-widths dynamically at run time/mode select (for power/latency)	Mixed workloads in embedded systems
Shared Exponent/Block Scaling	Shared scaling factors for blocks of data (microscaling) to amortize dynamic range overhead	LLMs, quantized networks

• Bitslice methods (Xu et al., 2016, Garland et al., 2020) convert floating-point computation into bitwise logic, mapping arithmetic to O(n) bit-level operations efficiently vectorized in software or hardware.
• Parameterizable libraries such as FlexFloat (Tagliavini et al., 2017) or VPREC-libm (Defour et al., 2020) provide software interfaces for arbitrary exponent/mantissa widths.
• Tapered or block scaling encodings (Posit, Takum, HiFloat8, MXInt) (Hunhold, 28 Dec 2024, Hunhold, 18 Mar 2025, Luo et al., 25 Sep 2024, Cheng et al., 2023) devote bits variably to exponent or significand, maximizing precision near unity and maintaining range for outliers.
• Entropy coding (EFloat) (Bordawekar et al., 2021) leverages exponent value clustering for compressing the exponent field, reallocating “saved” bits to increase significand precision without sacrificing dynamic range.
• Layer- or domain-specific tuning (Tambe et al., 2019, Huang et al., 2021) involves analyzing weight and activation distributions per network layer to select optimal field sizes or scale-shared representations.

3. Comparison to Standard IEEE 754 Formats

Custom formats contrast sharply with IEEE 754, which strictly partitions a fixed field for exponent and a fixed field for mantissa for a single value:

• Precision versus Range: Standard types cannot optimize for non-uniform application demands; custom formats provide flexibility, e.g., binary8 (1s, 5e, 2m) or binary16alt (1s, 8e, 7m) (Tagliavini et al., 2017, Bertaccini et al., 2022).
• Hardware Complexity: Modern hardware often includes special-case handling for subnormals, NaN, or signed zeros, whereas custom formats may eschew some features for circuit simplicity (e.g., HiFloat8, Takum) (Luo et al., 25 Sep 2024, Hunhold, 18 Mar 2025).
• Blockwise and variable-length encodings: Techniques such as MXInt (Cheng et al., 2023) and EFloat (Bordawekar et al., 2021) operate at block granularity or use entropy coding, which is not possible in the rigid field mapping of IEEE 754.

Format	Range/Precision Control	Hardware Complexity	Applicability
IEEE 754	Fixed	High	General-purpose; lacks adaptability
Custom FP	Tunable	Potentially lower	Application/domain-specific tuning

4. Design Trade-offs and Experimental Insights

Custom floating-point formats present a complex landscape of empirical trade-offs:

• Performance versus Precision: Experimental results consistently show that for low-precision (≤8-bit, ≤16-bit) operands in deep learning or approximate computing, custom formats—tailored to application error tolerance—yield significant speed and energy improvements, sometimes with negligible accuracy loss (Tambe et al., 2019, Tagliavini et al., 2017, Huang et al., 2021).
• Area and Power: Hardware units operating on lower bit-widths (even via reconfigurability or bitslice techniques) reduce area and power consumption, especially when parallel SIMD is harnessed (Xu et al., 2016, Mach et al., 2020, Arish et al., 2019).
• Accuracy in Scientific/HPC: Posit and takum formats can improve decimal-digit accuracy by 0.6–1.4 digits over IEEE floats, though at the expense of increased software emulation overhead unless hardware support is available (Chien et al., 2019, Hunhold, 28 Dec 2024).
• Robustness and Stability: In FFT and PDE benchmarks, posit and takum formats demonstrate superior stability at low precision compared to IEEE and OFP8, with takum exhibiting better consistency and dynamic range at 8–16 bits (Hunhold et al., 29 Apr 2025).

5. Applications and Practical Use Cases

Custom floating-point formats find adoption across numerous domains:

• Deep Neural Network Inference/Training: Ultra-low bitwidth representations (FP8, binary8, FFP8) and block scaling approaches (MXInt, HiFloat8) maintain model accuracy while lowering bandwidth and storage (Tambe et al., 2019, Luo et al., 25 Sep 2024, Cheng et al., 2023).
• HPC Kernels and Scientific Computing: Posit and takum enable higher arithmetic precision per bit—improving solution stability in iterative solvers, eigensolvers, and spectral methods, provided efficient hardware is available (Chien et al., 2019, Hunhold et al., 29 Apr 2025).
• Embedded/Low-Power Platforms: Transprecision FPUs and reconfigurable datapaths permit dynamic adaptation of resource and accuracy trade-offs, directly reducing system energy (Tagliavini et al., 2017, Mach et al., 2020, Arish et al., 2019).
• Data Compression in Large Models: Entropy-coded EFloat enables high-precision vector embeddings at small bit budgets (e.g., 12–16 bits), outperforming BF16/FP16 in RMS error and ranking metrics (Bordawekar et al., 2021).
• Custom DSP Pipelines and Imaging: FPGA-based custom filters and image/video processing pipelines benefit from selectable mantissa/exponent fields, delivering real-time performance with hardware resource constraints (Campos et al., 9 Sep 2024).

6. Technical Formulations and Notational Conventions

Many custom formats retain standard floating-point semantics but generalize field sizing and representation. Common technical forms include:

• Generic floating-point formula:

$x = (-1)^s \times 2^{E - b} \times (1 + m/2^p)$

where $s$ is the sign bit, $E$ is the exponent (with offset $b$), $m$ is the integer mantissa with $p$ bits, as in classical FP but with $s$, $E$, $b$, $m$ field sizes variant per format.

• Bitslice adder logic (Xu et al., 2016, Garland et al., 2020):

$$\text{Sum:}\quad s_i = a_i \oplus b_i \oplus c_i \qquad \text{Carry:}\quad c_{i+1} = (a_i \wedge b_i) \vee (c_i \wedge (a_i \oplus b_i))$$

• Microscaling mean bitwidth per value (Cheng et al., 2023):

$p = \frac{e}{|B|} + m + 1$

where $e$ is shared exponent width, $|B|$ is block size, $m$ is local mantissa bits.

• Tapered-precision encoding (Posit/Takum) (Hunhold, 28 Dec 2024, Hunhold, 18 Mar 2025): A variable-length regime/exponent/fraction encoding where the allocation adapts with magnitude, maximizing density near $1$.

7. Limitations, Challenges, and Future Directions

• Scalability and Complexity: Software emulation suffers prohibitive overhead (4×–19×) compared to hardware IEEE 754 (Chien et al., 2019). This often restricts practical use to environments with hardware acceleration or special-purpose FPUs (Tagliavini et al., 2017, Mach et al., 2020).
• Tool and Compiler Support: While domain-specific languages and code generators can automate design (Campos et al., 9 Sep 2024, Garland et al., 2020), broad support for variable formats in mainstream compilers and libraries lags IEEE 754.
• Hardware Standardization: Efforts to unify SIMD instruction set extensions (e.g., by streamlining SIMD ISA with takum) could consolidate many specialized FP8/FP16 variants into a single, tapered-precision family, enhancing hardware and software ecosystem stability (Hunhold, 18 Mar 2025).
• Precision Tuning Automation: Profiling and optimizing code for per-call-site or per-layer precision (Defour et al., 2020, Huang et al., 2021) are active research areas enabling fine-grained energy/accuracy trade-offs.

Summary Table: Key Custom Floating-Point Approaches

Approach	Key Features	Typical Use Cases
Bitslice SIMD	Arbitrary precision, software/hardware	CNNs, low-precision multimedia
Tapered/Posit/Takum	Dynamic field widths, maximized density	Scientific computing, deep learning
Block Scaling	Shared exponent across tensor blocks	LLMs, quantization
Entropy-coded	Variable-length exponent, more fraction	Embedding/model compression
Hardware Reconfig	Run-time format change, low-power modes	Embedded, DSP, heterogeneous processors

Custom floating-point formats continue to expand the boundaries of precision-efficient computation, enabling energy savings and application specialization that traditional rigid standards cannot provide. Their practical impact critically depends on synergistic hardware/software co-design, precise application profiling, and domain-aware numerical analysis.