Bear with me, this is a factual question so I suppose it goes here rather than Cafe Society.

In the original 1960s series it was canon that warp factors cubed equaled the superlight velocity. So warp 1 = lightspeed, warp 2 =x8, warp 3= x27, etc. Beginning with The Next Generation however, a new warp speed formula was introduced where warp 10 equals infinity. My personal fanwank is this: that the cubed formula gave accurate results up to about warp 8, but above that the deviations started becoming significant (episodes from TOS cite a breakdown of then-current warp theory above warp 8), enough so that the new formula where 10=infinity was introduced.

Now for my factual question: what mathematical function gives a curve that is very very close to y=x^3 for values of x up to about 8, but then has x=10 as an asymptote? In other words, if you graphed it logarithmically, it would be a straight line up to about x=8, then a sharp bend, and then a nearly vertical line.

Any continuous function can be approximated by polynomials, but the limit of those approximations need not be expressible in terms of elementary functions without specifying different rules for different parts of the domain. The function is just piece-wise defined. http://memory-beta.wikia.com/wiki/Warp_factor says that there actually is no formula, but the values above warp 9 were determined by physically drawing a curve.

I had an old article saved on my computer about this, but I couldn’t find it. Luckily, I found it on Google. It comes from the old TREK BBSes and USENET newsgroups. Unfortunately the answer is quite convoluted.

The simplified answer is that the drawn formula starts with v = W[sup]10/3[/sup], but the exponent gradually increases up to W = 9. After that, the exponent increases heavily. Hence, the true formula must have an additional factor that increases as W increases.

The most accurate answer (as of 2004) is best written in 5 different equations.

Each function increases the accuracy. You can just use a(W) to get a pretty good estimate, for example. Both a(W) and b(W) get you closer, and doing a(W), b(W), and c(W) gives you the exact answer.

Of course, this is unwieldy as heck, and maybe Dopers can do better. Here’s the page defining it all, which includes a chart showing the results from the curve in the book.

And here’s a set of Star Trek FAQs that led me back to that page. It includes the physics of Warp along with stuff about a Stardate system.

Finally, here’s an online converter made by Martin A. Shields, the guy who came up with the formula (found on archive.org).

Although a sometime fan, I had never thought much about this. At first I assumed that Warp was just a multiple of E, but then, I think , in the course of a conversation, we decided that it was logarithmic - W1=E, W2=Ex10 and so on.

I can’t see why they would need such a complicated scale?

You just need a function that is very close to 3 for x < 10 but with a vertical asymptote at x=10. Such a function is 3 + 1/(10-x). The effect of the asymptote can be lessened away from x=10 by using 3 + 1/k(10-x) for a large positive number k. I suggest using the warp function w(x)=x^{3+1/100(x-10)} with domain 0 <= x < 10.

Note that the proposed function doesn’t have to yield a velocity of exactly c*(x^3) for values of x between 1 and 8- just close enough that for 8 or less no one bothered about the difference, which starts becoming notable above 8.0, marked at 9.0 and extreme beyond that.
ETA: Lubricious Integument got it.