Checking the maths
I still use my HP-15C. Had to put new batteries into it just a few months ago. Should be good for another 5 years!!
It's interesting that the paper was written 10 years before Intel got bitten by the FDIV bug. By that time the 754 standard was already agreed on. So just having a standard doesn't fix the problem.
Interesting chunk of history though. I remember getting the p754 draft and writing code for a 6809 because 80 bit wasn't available for it. Long ago and far away.....
It's interesting that the paper was written 10 years before Intel got bitten by the FDIV bug. By that time the 754 standard was already agreed on. So just having a standard doesn't fix the problem.
Interesting chunk of history though. I remember getting the p754 draft and writing code for a 6809 because 80 bit wasn't available for it. Long ago and far away.....
I'm reminded of two episodes in my life when it the issue appeared in the flesh, once a couple of decades ago I was doing my then girlfriends homework and she got a B after her tutor told her the answers she gave was wrong, as I had done them all by hand I told her to tell him he was wrong!
He duly came back with the answer that his calculator coudn't be wrong.. so after checking the various models we had in the house, I noticed some of them didn't give the right answer! so got her to trot back and get everyone in the glass to test theirs, and low and beyond, about 1 in 10 got the wrong answer. After checking her (my..) work by hand, she got an A..
Another time when I was in college, we was using Turbo Pascal on the PC to do some maths stuff and I noticed its math routines was screwed, so fixed that and was the only one to notice, every one else was happy to miss out on one penny in every pound!
I also got to point someone who wrote some calculator software just the other day towards this article and they are going to issue another version of their software without bugs...
He duly came back with the answer that his calculator coudn't be wrong.. so after checking the various models we had in the house, I noticed some of them didn't give the right answer! so got her to trot back and get everyone in the glass to test theirs, and low and beyond, about 1 in 10 got the wrong answer. After checking her (my..) work by hand, she got an A..
Another time when I was in college, we was using Turbo Pascal on the PC to do some maths stuff and I noticed its math routines was screwed, so fixed that and was the only one to notice, every one else was happy to miss out on one penny in every pound!
I also got to point someone who wrote some calculator software just the other day towards this article and they are going to issue another version of their software without bugs...
The best book I ever found for numerical approximation formulas is "Computer Approximations" published by Krieger. For each formula they give the numerical accuracy and a complete analysis of the error. Plus slow formulas for testing the fast ones!
The details actually do matter. Sometimes it's hard to explain, but when you get a whole class that gets different answers on their calculators, it's easy to show!
Thanks for that book reference, I'm so glad the internet is around these days to ask people questions all over the world, back in the past when I asked my maths teacher how to work out square roots, and he just replied use the calculator.. When I pointed out I actually wanted to code the calculator function he seemed bemused at the very idea, and then went on to tell us briefly how each year they take out a little bit more out of the maths curriculum and how in his day you had to know a lot more than you do now.