Number theory
While the word "algorithm" goes back only to certain readers of al-Khwārizmī, careful descriptions of methods of solution are older than proofs: such methods (that is, algorithms) are as old as any recognisable mathematics—ancient Egyptian, Babylonian, Vedic, Chinese—whereas proofs appeared only with the Greeks of the classical period.
An interesting early case is that of what we now call the Euclidean algorithm. In its basic form (namely, as an algorithm for computing the greatest common divisor) it appears as Proposition 2 of Book VII in "Elements", together with a proof of correctness. However, in the form that is often used in number theory (namely, as an algorithm for finding integer solutions to an equation formula_78,
or, what is the same, for finding the quantities whose existence is assured by the Chinese remainder theorem) it first appears in the works of Āryabhaṭa (5th–6th century CE) as an algorithm called
"kuṭṭaka" ("pulveriser"), without a proof of correctness.
Google Hummingbird
The "Hummingbird" update was the first major update to Google's search algorithm since the 2010 "Caffeine" search architecture upgrade, but even that was limited primarily to improving the indexing of information rather than sorting through information. Amit Singhal, then-search chief at Google, told "Search Engine Land" that "Hummingbird" was the most dramatic change of the algorithm since 2001, when he first joined Google. Unlike previous search algorithms, which would focus on each individual word in the search query, "Hummingbird" considers the context of the different words together, with the goal that pages matching the meaning do better, rather than pages matching just a few words. The name is derived from the speed and accuracy of the hummingbird animal.