These formulas have grown better at reading the minds of users to interpret a very short query. Are the users looking for a job, a purchase or a fact? The formulas can tell that people who type “apples” are likely to be thinking about fruit, while those who type “Apple” are mulling computers or iPods. They can even compensate for vaguely worded queries or outright mistakes.

“Search over the last few years has moved from ‘Give me what I typed’ to ‘Give me what I want,’ ” says Mr. Singhal, a 39-year-old native of India who joined Google in 2000 and is now a Google Fellow, the designation the company reserves for its elite engineers.

Google recently allowed a reporter from The New York Times to spend a day with Mr. Singhal and others in the search-quality team, observing some internal meetings and talking to several top engineers. There were many questions that Google wouldn’t answer. But the engineers still explained more than they ever have before in the news media about how their search system works.

As Google constantly fine-tunes its search engine, one challenge it faces is sheer scale. It is now the most popular Web site in the world, offering its services in 112 languages, indexing tens of billons of Web pages and handling hundreds of millions of queries a day.

Even more daunting, many of those pages are shams created by hucksters trying to lure Web surfers to their sites filled with ads, pornography or financial scams. At the same time, users have come to expect that Google can sift through all that data and find what they are seeking, with just a few words as clues.

“Expectations are higher now,” said Udi Manber, who oversees Google’s entire search-quality group. “When search first started, if you searched for something and you found it, it was a miracle. Now, if you don’t get exactly what you want in the first three results, something is wrong.”

Google’s approach to search reflects its unconventional management practices. It has hundreds of engineers, including leading experts in search lured from academia, loosely organized and working on projects that interest them. But when it comes to the search engine — which has many thousands of interlocking equations — it has to double-check the engineers’ independent work with objective, quantitative rigor to ensure that new formulas don’t do more harm than good.