>Fifty years ago scientists were concerned with a different application of compression: reducing the size of mathematical tables. Guilty as charged. >Books of

>Fifty years ago scientists were concerned with a different application of compression: reducing the size of mathematical tables. Guilty as charged. >Books of tabulated functions are obsolete now, but the principles used in producing these tables are still very much relevant. Even more relevant back then in reverse, after computers came within reach. But you have to get there first. Going back more than 50 years, to arrive at the tables to begin with, an algorithm was calculated across a series of values, and the output tabulated according to increments that may or may not be the same as that of the original data collection. This could be especially true with physical properties or natural phenomena. OTOH with pure math functions which are well-defined to begin with, somebody has to pick the range and increments. Either way this needs to be done so that the table serves its intended purpose. Frequently one of the important features was the accuracy beyond what could be obtained from an engineering slide rule. In cases where a slide rule delivered adequate significant figures for the job, people were not motivated to generate a half-kilo book of numbers. But when more decimal places were needed it was tables, 4 to 6 decimal places or significant figures made it worth it lots of times. Although sometimes a "table or two" had been calculated using a slide rule in a very definitive way, very close to the highly valid raw data. The most trusted way it could be at the time. As the leading author of the table, numerically characterizing the property for publication, the best guess estimate of significant figures would be made beyond that which the slide rule was reliably capable. Simply because more decimal places were needed when that's the purpose of the table to begin with. Publish them and they become the best reference there is, regardless of their certainty. Further if tables find their way into an institutional workflow, they can be made official, and therefore required. To the full number of significant figures supported by the recognized weakest link in the calculation chain, often the table(s) itself. Which brings any deviations in precision to the forefront every time for scrutiny. Even though uncertainty can be well within the accuracy of the original slide rule, because it was too sloppy physically for the original task. Too bad, the table's been official forever, you can't use a computer to replace the table unless it outputs the exact same values as the table, to the full number of decimal places. The accuracy of a table can even be recognized as questionable across-the-board, that's OK if everybody agrees to use it anyway, so there. One of the things that helped to make progress was doing everything in integers that would have otherwise been done in floating point. Something like it was done with a slide rule where you had to keep good track of the decimal point your own self. It is all too common for floating point numbers to give the illusion that there are unlimited decimal places in everything. Getting back to the article, when building a table from an appropriate function using an adequate calculating device, ideally the repeatable number of significant figures to be published with complete certainty, combined with the size of the increment between entries, will allow interpolation between published values to be almost as trustworthy as the published values themselves. Even if you are not "publishing" the values, the number of decimal places accurately retained during critical calculations will be the same limitation, and the increment comes into play whether or not it is the same granularity going into a "database" or withdrawing from it.