Algorithms & Computation

Core Theoretical Principles

This subject explores the mathematical algorithms powering microprocessors, hardware calculators, and decentralized consensus networks.

1. Why Calculators Don’t Use Taylor Series

Many people assume calculators compute $\sin(x)$ or $\cos(x)$ using Taylor/Maclaurin series expansions:

$$ \sin(x) = \sum_{k=0}^\infty \frac{(-1)^k}{(2k+1)!} x^{2k+1} = x - \frac{x^3}{6} + \frac{x^5}{120} - \dots $$

However, evaluating higher-degree polynomials requires numerous multiplication operations and suffers from severe floating-point round-off errors at large angles. Hardware ALUs implement the CORDIC (Coordinate Rotation Digital Computer) algorithm instead.

2. The CORDIC Algorithm

CORDIC computes trigonometric functions iteratively using only bit-shifts and additions:

$$ \begin{pmatrix} x_{i+1} \\ y_{i+1} \end{pmatrix} = \begin{pmatrix} 1 & -d_i 2^{-i} \\ d_i 2^{-i} & 1 \end{pmatrix} \begin{pmatrix} x_i \\ y_i \end{pmatrix} $$

Where the rotation direction $d_i \in \{-1, +1\}$ is chosen to drive the residual angle $z_i \to 0$. Because multiplication by $2^{-i}$ is simply an arithmetic binary right-shift, CORDIC executes in simple hardware without requiring expensive floating-point multipliers.

3. Cryptographic Hash Difficulty (Bitcoin Proof of Work)

Bitcoin mining relies on finding a nonce $n$ such that the double SHA-256 hash of the block header $B$ falls strictly below a 256-bit target threshold $T$:

$$ \operatorname{SHA-256}(\operatorname{SHA-256}(B \parallel n)) < T $$