41–50 of 64 posts
And yet it incorrectly simplifies f(x) = x/x with f(x) = 1
This is completely different from the phenomenon described in the article: arccosh discontinuity can’t be dealt the same way. In fact complex analysis prefers to deal with it my making functions path-dependent (multi-valued).
Earlier quoted context omitted.
Source available? The license is still proprietary, right?
Yes, it is all proprietary, but there are still ways to inspect most of the WL-implemented functions since the system does not go to extreme pains to keep them hidden from introspection. It is not unlike Maple in that sense.
More generally it's not at all clear what 'simplify' means. Is x*x simpler than x^2? Probably? Is sqrt(5)^3 simpler than 5^(3/2)? I don't know. It entirely depends on what you're going to be doing with the expression later.
In this case, a heuristic like "less parameters, less operators and less function calls" covers all the cases.
I’m algebraist at heart and training, but I remember beautiful many-layered surfaces of ordinary complex functions in books and on blackboards.
More generally it's not at all clear what 'simplify' means. Is x*x simpler than x^2? Probably? Is sqrt(5)^3 simpler than 5^(3/2)? I don't know. It entirely depends on what you're going to be doing with the expression later.
I think "simplify" is pretty clear here. For trigonometric functions you would expect a trig function and an inverse trig function to be simplified. We all know what we'd expect if we saw sin(arcsin(x)) (ie x). If we saw cos(arcsin(x)) I'll spoil it for you: it simplifies to sqrt(1-x^2). Hyperbolic functions aren't used as much but the same principle applies. Here the core identity is cosh^2(x) = sinh^2(x) = 1 so: si…
The first time I looked at the comment above, there was a reply, a reply to that reply, and a reply to the reply to the reply.
Later I came back and this time there were no replies. Since HN won't let you delete a comment that has a reply the only ways a comment chain should be able to go away are (1) the participants delete them in reverse order, or (2) a moderator intervenes.
I came back again and the comments are back!
I wonder if this is related to another comment problem I've seen many times in the past few weeks? I'll be using the "next" or "prev" links on top level comments to move through the comment and will come to a point where that breaks. Next reaches a comment that it will not go past. Coming from below prev will also not go past that point. Examining the links, next and prev are pointing to a nonexistent comment.
This is a general pattern in CAS. For a more basic case, it’s not obvious sqrt(square(x)) will simplify to x without any further assumptions on x.
That's not what it simplifies to using a real or complex number domains for x, it's abs(x). CAS need type inference assumptions and/or type qualifiers to be more powerful. Edit: Fixed stuff.
More generally it's not at all clear what 'simplify' means. Is x*x simpler than x^2? Probably? Is sqrt(5)^3 simpler than 5^(3/2)? I don't know. It entirely depends on what you're going to be doing with the expression later.
In this case, a heuristic like "less parameters, less operators and less function calls" covers all the cases.
Earlier quoted context omitted.
How is going from two functions with one variable to three functions with a variable and a constant a simplification?
If you can't recognize how much simpler the simplified version is, I'm not sure exactly what to tell you. But let's think about it in terms of assembly steps: 1. Multiply the input by itself 2. Add 1 3. Take the square root. There is often a fast square root function available. The above is a fairly simply sequence of SIMD instructions. You can even do it without SIMD if you want. Compare this to sinh being (e^x - e^…
Earlier quoted context omitted.
How is going from two functions with one variable to three functions with a variable and a constant a simplification?
If you can't recognize how much simpler the simplified version is, I'm not sure exactly what to tell you. But let's think about it in terms of assembly steps: 1. Multiply the input by itself 2. Add 1 3. Take the square root. There is often a fast square root function available. The above is a fairly simply sequence of SIMD instructions. You can even do it without SIMD if you want. Compare this to sinh being (e^x - e^…
"Simplify" is a very old term (>50y) in computer algebra. Its meaning has become kind of layered in that time.
This is a general pattern in CAS. For a more basic case, it’s not obvious sqrt(square(x)) will simplify to x without any further assumptions on x.
I think you would get sqrt(x^2) = x, if x belonged to the natural domain of sqrt , which is a Riemann surface, that may also be defined using the language of "sheaves". I don't know how to connect this to the article or Mathematica.