Show HN: Compute polynomials twice as fast
summary
Polynomial evaluation can be performed more efficiently than the standard Horner's method by using rational preprocessing of coefficients. This approach reduces the number of required multiplications from n to approximately n/2. Such an optimization is crucial for high-performance computing tasks like function approximation and cryptography. The method works across various mathematical fields, making it widely applicable.
key points
- Rational preprocessing allows for a reduction in the total number of multiplications needed to calculate a polynomial's value. This significantly lowers the computational cost for high-degree polynomials.
- The technique is particularly useful for approximating transcendental functions such as sine and cosine. It also enhances the efficiency of operations in hashing and coding theory.
- A specialized tool and accompanying research paper provide the schedules necessary to implement these optimized multiplication sequences for any given polynomial.
who should read it
Software engineers and mathematicians interested in numerical analysis or cryptographic implementation. Those seeking to optimize performance in low-level computational libraries would find this useful.
discussion positive
consensus
Commenters generally agree that the project is a technically impressive demonstration of polynomial evaluation strategies. While some debate the practical utility of these methods for universal hashing compared to modern heuristic hashes or AES-NI instructions, there is a consensus that the technique is highly valuable for accelerating approximated functions like Taylor series. The sentiment is largely positive, though tempered by technical skepticism regarding real-world performance gains in specific hashing contexts.
top insight
The most significant insight is the distinction between using polynomials for MACs, where the message acts as coefficients and is evaluated once, versus k-independent hashing, where the polynomial is fixed and evaluated many times. This clarifies why preprocessing coefficients provides a massive speedup for the latter but is irrelevant for the most common network protocol hashes. The discussion emphasizes that provable collision guarantees are the primary trade-off against the raw speed of heuristic hashes.
Generated by an LLM from the article and thread. Check the sources before relying on it.