Algorithm accuracy vs calculator precision

[dm319]

OK quote from Kahan (stolen from one of Steve's posts of course:

Kahan through the medium of Steve Wrote:It’s true that the HP-45’s arithmetic was somewhat grotty in spots, but it wasn’t that bad. But what TI was doing was clever. You see, the 45 did its arithmetic to ten significant decimals, period. Everything was done to ten significant decimals, including the internal algorithms that computer logs and exponentials. TI was doing their arithmetic internally carrying 13 significant decimals, but they only showed you ten. So that meant that, though you type ten digits in, as soon as you did some arithmetic, you had 13 decimal digits. But you only saw ten significant decimals. Well, that could hide a lot of sins, couldn’t it? The TI thing was cheaper, but that’s because Hewlett-Packard can’t do anything that’s cheap there. Their whole culture is such that, whatever they do, it’s going to be expensive. So I discovered that if you did this log exponential thing seven times, then the last digit would change. You see, their arithmetic at the 13th digit was grottier, if anything could be grottier, than the 45.

And this is my plot of digit efficiency in TVM calculations: