Stories with Tag Addition

• Why Does Integer Addition Approximate Float Multiplication?

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  Posted: 2025-02-09 18:36:04

  Verbose tl;dr

  The blog post explores the surprising observation that repeated integer addition can approximate floating-point multiplication, specifically focusing on the case of multiplying by small floating-point numbers slightly greater than one. It explains this phenomenon by demonstrating how the accumulation of fractional parts during repeated addition mimics the effect of multiplication. When adding a floating-point number slightly larger than one to itself repeatedly, the fractional part grows with each addition, eventually getting large enough to increment the integer part. This stepping increase in the integer part, combined with the accumulating fractional component, closely resembles the scaling effect of multiplication by that same number. The post illustrates this relationship using both visual representations and mathematical explanations, linking the behavior to the inherent properties of floating-point numbers and their representation in binary.

  The blog post "Why Does Integer Addition Approximate Float Multiplication?" explores a seemingly counterintuitive relationship between integer addition and floating-point multiplication. It begins by presenting an observation: repeatedly adding a floating-point number to itself a certain number of times produces a result very close to multiplying that same floating-point number by the integer representing the number of additions. The author then delves into the underlying mechanics of floating-point representation and arithmetic to explain this phenomenon.

  The core of the explanation lies in the way floating-point numbers are stored in computer memory, specifically using the IEEE 754 standard. This standard represents floating-point numbers using three components: a sign bit, an exponent, and a significand (also known as the mantissa). The author meticulously details how floating-point addition is performed at the bit level, highlighting the process of aligning exponents, adding the significands, and then normalizing the result.

  The post then connects this floating-point addition process to the equivalent multiplication operation. Multiplying a floating-point number by an integer can be conceptually understood as repeated addition. When examining the bit-level operations involved in repeated floating-point addition, a pattern emerges that mimics the steps involved in floating-point multiplication. Specifically, repeatedly adding a floating-point number to itself shifts the exponent of that number in a way analogous to the exponent manipulation performed during multiplication by an integer.

  However, the author carefully points out that this approximation isn't perfect. The blog post demonstrates how rounding errors, inherent in floating-point arithmetic due to the limited precision of the significand, accumulate during repeated additions. This accumulation of rounding errors means the result of repeated addition can subtly diverge from the result obtained through direct multiplication, although the difference is often quite small. The author uses concrete examples to illustrate these subtle differences and emphasizes that while the two operations yield similar results, they are not strictly equivalent.

  Finally, the author concludes by reiterating that the apparent connection between integer addition and float multiplication stems from the underlying bit-level representation and manipulation defined by the IEEE 754 standard. The post emphasizes the importance of understanding these low-level details when working with floating-point numbers to avoid potential pitfalls related to precision and rounding. The close approximation between the two operations is a consequence of the way computers represent and process floating-point numbers, not a fundamental mathematical equivalence.

  • floating-point
  • integer
  • Arithmetic
  • multiplication
  • Addition
  • Approximation
  • mathematics
  • Computer Science
  • numerical analysis
  • binary representation
  • data types
  • programming

  Summary of Comments ( 29 )
  https://news.ycombinator.com/item?id=42992505

  Verbose tl;dr

  Hacker News commenters generally praised the article for clearly explaining a non-obvious relationship between integer addition and floating-point multiplication. Some highlighted the practical implications, particularly in older hardware or specialized situations where integer operations are significantly faster. One commenter pointed out the historical relevance to Quake III's fast inverse square root approximation, while another noted the connection to logarithms and how this technique could be extended to other operations. A few users discussed the limitations and boundary conditions, emphasizing the approximation's validity only within specific ranges and the importance of understanding those constraints. Some commenters provided further context by linking to related concepts like the "magic number" used in the Quake III algorithm and resources on floating-point representation.

  The Hacker News post "Why Does Integer Addition Approximate Float Multiplication?" with ID 42992505 has several comments discussing the article's core idea and expanding on its implications.

  Several commenters delve deeper into the mathematical underpinnings of the relationship between integer addition and floating-point multiplication. One explains it as a consequence of logarithms, where addition in the log domain corresponds to multiplication in the original domain. Integers, when spaced closely enough, can approximate a continuous logarithmic scale. Another comment points out that the described trick effectively implements multiplication by 2^x using only bit shifts and addition, which is faster than traditional floating-point multiplication in some contexts. They also discuss how this relates to generating MIDI note frequencies, where each semitone increase corresponds to multiplying the frequency by the 12th root of 2.

  Another thread discusses practical applications and limitations. One commenter mentions the use of this principle in embedded systems or older hardware where direct floating-point operations are expensive. However, they acknowledge the limitations in terms of accuracy, particularly for larger numbers or when high precision is required. Another user points out that this approach is related to the concept of "logarithmic number system" (LNS) which offers advantages in some specific computational domains.

  One commenter highlights that this concept is useful for understanding how some audio software algorithms work, where amplitude or frequency adjustments often rely on similar approximations for efficiency.

  Others discuss the pedagogical value of the article. One comment praises the author's ability to make a complex topic understandable and visually appealing.

  Finally, some comments offer corrections or minor clarifications to points made in the original article. For instance, one commenter suggests a more precise wording for a specific statement, while another points out a potential edge case where the approximation might break down.

• Pre-Trained Large Language Models Use Fourier Features for Addition (2024)

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  Posted: 2025-02-06 10:31:06

  Verbose tl;dr

  This paper investigates how pre-trained large language models (LLMs) perform integer addition. It finds that LLMs, despite lacking explicit training on arithmetic, learn to leverage positional encoding based on Fourier features to represent numbers internally. This allows them to achieve surprisingly good accuracy on addition tasks, particularly within the range of numbers present in their training data. The authors demonstrate this by analyzing attention patterns and comparing LLM performance with models using alternative positional encodings. They also show how manipulating or ablating these Fourier features directly impacts the models' ability to add, strongly suggesting that LLMs have implicitly learned a form of Fourier-based arithmetic.

  The preprint "Pre-Trained Large Language Models Use Fourier Features for Addition (2024)" by Michael Petrov, Hritik Bansal, and Micah Goldblum delves into the inner workings of pre-trained large language models (LLMs) and how they perform arithmetic operations, specifically focusing on addition. The authors hypothesize that LLMs leverage a mechanism similar to Fourier features, commonly used in signal processing and computer graphics, to represent and manipulate numerical information. This hypothesis stems from the observation that LLMs exhibit wave-like oscillatory behavior in their activation patterns when processing numbers.

  The research centers around analyzing the activations within LLMs, which are the internal representations of information as the model processes data. By probing these activations, the authors attempt to decode the internal mechanisms the model employs. They introduce a novel probing method specifically designed to detect the presence of Fourier features within the activations. This method involves fitting linear models to the activations and examining the frequency components present in these linear models. The presence of specific, predictable frequencies would suggest the utilization of a Fourier-like mechanism.

  Their experimental results across several popular LLMs, including Llama-2, GPT-NeoX, and Pythia, provide compelling evidence supporting their hypothesis. They demonstrate that the activations within these models, particularly in layers associated with numerical processing, indeed exhibit patterns consistent with the use of Fourier features. Furthermore, the observed frequencies within these activations correlate with the numerical values being processed, indicating a direct link between the Fourier-like representation and the actual arithmetic operations.

  The paper also explores the potential implications of these findings. The authors suggest that this Fourier-based representation might explain certain limitations observed in LLMs when dealing with large numbers or complex arithmetic tasks. The inherent periodicity of Fourier features might introduce ambiguities or inaccuracies when representing numbers outside a certain range or performing operations that require high precision. Understanding these limitations could pave the way for developing more robust and accurate LLMs for numerical reasoning.

  Finally, the study touches upon the broader significance of these discoveries within the context of understanding how LLMs represent and process information. The emergence of Fourier-like features, a concept borrowed from signal processing, suggests that LLMs might be developing internal representations that are surprisingly analogous to methods used in other fields. This unexpected connection could provide valuable insights into the underlying principles governing the learning and representation capabilities of these powerful models. The findings contribute to the ongoing effort to unravel the “black box” nature of LLMs and move towards a deeper understanding of their internal workings.

  • Large Language Models
  • LLMs
  • Pre-trained Models
  • Fourier Features
  • Fourier Transforms
  • Addition
  • Arithmetic
  • machine learning
  • deep learning
  • natural language processing
  • NLP
  • artificial intelligence
  • AI
  • arXiv
  • 2024

  Summary of Comments ( 15 )
  https://news.ycombinator.com/item?id=42960989

  Verbose tl;dr

  Hacker News users discussed the surprising finding that LLMs appear to use Fourier features internally to perform addition, as indicated by the linked paper. Several commenters expressed fascination with this emergent behavior, highlighting how LLMs discover and utilize mathematical concepts without explicit instruction. Some questioned the paper's methodology and the strength of its conclusions, suggesting alternative explanations or calling for further research to solidify the claims. A few users also discussed the broader implications of this discovery for understanding how LLMs function and how they might be improved. The potential link to the Fourier-based positional encoding used in Transformer models was also noted as a possible contributing factor.

  The Hacker News post titled "Pre-Trained Large Language Models Use Fourier Features for Addition (2024)" linking to the arXiv paper has generated a moderate amount of discussion with a few interesting threads.

  Several commenters focus on the implications of LLMs appearing to use Fourier transforms for addition. One commenter expresses surprise, stating they wouldn't have guessed this mechanism and questioning if it's a learned behavior or an emergent property of the architecture. This sparks further discussion about whether this behavior is specifically trained or a consequence of the training data's statistical properties. Some suggest it could be related to the positional encoding mechanisms already employed in transformer models, which use sinusoidal functions. Another commenter wonders if this Fourier-based approach to addition might offer advantages in terms of computational efficiency or generalization.

  Another thread delves into the limitations of the research. One commenter points out that the paper focuses specifically on addition and questions whether similar mechanisms are used for other arithmetic operations. They suggest investigating multiplication next. Another commenter questions the significance of the findings, arguing that demonstrating LLMs use Fourier transforms for addition doesn't necessarily reveal anything profound about their understanding of arithmetic. They argue it could simply be a pattern-matching technique that happens to be effective for addition.

  There's also a discussion about the interpretability of LLMs. One commenter expresses hope that research like this will eventually lead to a better understanding of how LLMs function internally. Another, however, is more skeptical, suggesting that even if we can identify specific mechanisms like the use of Fourier transforms, it might not provide a satisfying explanation of the overall emergent behavior of these complex models.

  Finally, a few comments offer tangential observations. One commenter notes the increasing prevalence of papers analyzing the internal workings of LLMs, highlighting the growing interest in this area of research. Another points out the connection to older research on neural networks and their ability to approximate functions, suggesting this work builds upon those foundations.