Language Models Use Trigonometry to Do Addition
A model like Gemma-2-9B uses single-digit tokenization and likely requires additional algorithms to collate digits together into a single numerical representation. Conclusion We find that three mid-sized LLMs represent numbers as helices and manipulate them using the interpretable Clock algorithm to compute addition.
L MODELS USE TRIGONOMETRY TO D A DITION
lement complex mathematical algorithms . Understanding LLM algorithms is important for safe AI and can also provide valuable insight in o model errors, as shown in Appendix F. We hope that this work inspires additional investigations into LLM mathematical capabilities, especially as addition
A model like Gemma-2-9B uses single-digit tokenization and likely requires additional algorithms to collate digits together into a single numerical representation. Conclusion We find that three mid-sized LLMs represent numbers as helices and manipulate them using the interpretable Clock algorithm to compute addition.
IB A&I
Analysis & Interpretations Topic 3 - Trigonometry and Geometry. Original notes, exercises, videos on SL and HL content.
This paper proposes a novel hybrid arithmetic-trigonometric optimization algorithm (ATOA) using different trigonometric functions for complex and continuously evolving real-time problems. The proposed algorithm adopts different trigonometric functions, namely sin, cos, and tan, with the conventional sine cosine algorithm (SCA) and arithmetic optimization algorithm (AOA) to improve the ...
Efficient Algorithms for Trig Computations
The design of efficient algorithms for trigonometric computations is a rich area of study, balancing complexity, memory constraints, and the need for precision. As we have seen: CORDIC Algorithms are attractive for embedded systems due to their deterministic fixed-point arithmetic and minimal memory requirements, albeit with iterative ...
Hence, to obtain the effective parameters, a novel arithmetic-trigonometric optimization algorithm is developed by using the essential arithmetic operators and the basic trigonometric functions. Different combinations of the proposed technique are created with multiple combined variations from sin, cos, and tan functions.
Trigonometric Inference Providing Learning in Deep Neural Networks
We also tested a model with each layer having 10,000 neurons to check the difference in performance between the proposed algorithm and the classical inference. For small neural net models with large layer sizes, the two algorithms show almost no difference, and interestingly, trigonometric inference outperforms the baseline inference method.
A new research paper titled "Language Models Use Trigonometry to Do Addition" by Subhash Kantamneni and Max Tegmark from MIT provides fascinating insights into this question.
More Context About Algorithm Interpretation With Additional Trigonometry
⏰ Clockwork Algorithm: LLMs Use Trigonometry to do Simple math. This. This note connects the source idea with the visuals in a simple, reader-friendly way.
Unit Circle Chart Sin Cos Tan Sec Csc Cot Trigonometric Functions:. It gives the article a little more context before the image collection begins.
Trigonometric Formulas Class 11 - GeeksforGeeks. The extra context helps the page feel more useful without forcing the same phrase repeatedly.
