Maximizing Memory-Level Parallelism via Integrated Stochastic Logic-in-Memory Architectures · Research brief

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research area: other
technical change: Stochastic arithmetic in MTJ logic-in-memory
published: 2026-04-25

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Central equations

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This appendix details the formulation of transcendental functions using truncated Maclaurin-series expansions, providing a hardware-friendly foundation for their realization in SC. ① $𝐬𝐢𝐧 (𝐱)$ . The sine function is a prominent transcendental function widely used in various applications. Implementing the sine function in hardware presents significant challenges. The 7th order truncated Maclaurin series expansion of $𝐬𝐢𝐧 (𝐱)$ is given by:

\sin (x) = \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{(2 ​ n + 1)!} ​ x^{2 ​ n + 1} \approx x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \frac{x^{7}}{7!}

\approx x (1 - \frac{x^{2}}{6} ​ (1 - \frac{x^{2}}{20} ​ (1 - \frac{x^{2}}{42}))) .

equation (3) · §Appendix · proposed method

This design also mirrors the structure used for implementing the $𝐬𝐢𝐧 (𝐱)$ function, with the key distinction being the use of different coefficient values. Such a modular representation enables a universal approach to designing transcendental functions within the same architectural framework, requiring only coefficient adjustments tailored to each specific function. ⑤ $𝐬𝐢𝐠𝐦𝐨𝐢𝐝 (𝐱)$ . The sigmoid function is a widely used transcendental activation function, particularly prominent in learning systems and neural networks. The 5th order truncated Maclaurin series expansion of the $𝐬𝐢𝐠𝐦𝐨𝐢𝐝 (𝐱)$ function is given by:

s ​ i ​ g ​ m ​ o ​ i ​ d ​ (x) \approx \frac{1}{2} + \frac{x}{4} - \frac{x^{3}}{48} + \frac{x^{5}}{480}

\approx 1 - \frac{1}{2} ​ (1 - \frac{x}{2} ​ (1 - \frac{x^{2}}{12} ​ (1 - \frac{x^{2}}{10}))) .

equation (10) · §Appendix · proposed method

The design of this function closely follows that of the $𝐬𝐢𝐧 (𝐱)$ function, with the primary distinction being the omission of the final stage in the polynomial expansion. ⑦ $𝐞^{- 𝐱}$ . The function $𝐞^{- 𝐱}$ , commonly referred to as the decay function, is a fundamental component in probabilistic modeling and statistical analysis. The 5th order truncated Maclaurin series expansion of the $𝐞 - x$ function is given by:

e^{- x} = \sum_{n = 0}^{\infty} \frac{{(- x)}^{n}}{n!} \approx 1 - x + \frac{x^{2}}{2!} - \frac{x^{3}}{3!} + \frac{x^{4}}{4!} - \frac{x^{5}}{5!}

\approx 1 - x (1 - \frac{x}{2} ​ (1 - \frac{x}{3} ​ (1 - \frac{x}{4} ​ (1 - \frac{x}{5})))) .

equation (12) · §Appendix · proposed method

equations from arXiv HTML · fetched 2026-09-12

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Replaces separate memory and processing in a von Neumann architecture with stochastic arithmetic inside MTJ memory arrays. It requires new hardware not available commercially, accepts precision limits from probabilistic bit streams, and lacks a conventional ML compilation pipeline. It cannot run on a GPU or through an API and requires fabricating magnetic devices.
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