I believe that I got the first external cassette drive for the PET on the west coast. Only used it a very few times. The big boost was when I got the floppy disk drive. I could do anything!

For a nice introduction to relativity, look at The Einstein Theory of Relativity: A Trip to the Fourth Dimension by Lillian R. Lieber. $15 on Amazon. Written in 1945 and still quite good.

I enjoyed this article. I always find it interesting how things came to be and how individual contributions made things happen.

If you believe them.

As Victor said, his parents were very upset when they came home and found him in front of a roaring fire, because they did not have a fireplace.

Looks interesting. Should start with a definition of the Hyperbolic Tangent. It is only about 2/3 of the way that the definition occurs in a discussion of computing exp(x).

Damn. Never knew that. TIL

I will wait for you to discover these Keyboard Shortcuts - Press the `fn + ^` (that globe key + control) and then try `c`, `f`, and all of the four arrow keys.

In the mid 70's, I was a graduate CS student at USC's Information Sciences Institute. I remember my feeling of awe when I used Arpanet (or was it Darpanet) to log into London and do stuff there. Wow!

Trump is clearly winning his war on America.

Not if the ratio between the largest and smallest floats is very large (2^(2^n)) where n is the number of bits in the exponent.

I think you either haven't thought about this or you did your math wrong.

You need (2^e)+m+1 bits. That is more bits than would fit in the cheap machine integer type you just have lying around, but it's not that many in real terms.

Let's do a tiny one to see though first, the "half-precision" or f16 type, 5 bits of exponent, 10 bits of fraction, 1 sign bit. We need 43 bits. This will actually fit in the 64-bit signed integer type on a modern CPU.

Now lets try f64, the big daddy, 11 exponent, 52 fraction, 1 sign bit so total 2048 + 52 + 1 = 2101 bits. As I said it doesn't fit in our machine integer types but it's much smaller than a kilobyte of RAM.

Edited: I can't count, though it doesn't make a huge difference.

You also need some extra bits at the top so that it doesn't overflow (e.g., on an adversarial input filled with copies of the max finite value, followed by just as many copies of its negation, so that it sums to 0). The exact number will depend on the maximum input length, but for arrays stored in addressible memory it will add up to no more than 64 or so.

Thanks, you're right there for accumulating excess, I don't think you can actually get to 64 extra bits but sure, lets say 64 extra bits if you want a general purpose in memory algorithm, it's not cheap but we shouldn't be surprised it can be done.