In the previous post I said I was trying remember where I’d seen the tangent sum applied. I mentioned a couple near misses, and it turns out that what I was trying to remember was another near miss. What I’d seen before was not the tangent sum but the hyperbolic tangent sum. Several people suggested that this was I probably had in mind.

The tangent sum of two numbers is defined as the tangent of the sum of their arc tangents:

tan(arctan *a* + arctan *b*).

The hyperbolic arctangent sum sticks an *h* on the end of the tans:

tanh(arctanh *a* + arctanh *b*).

We used ⊕ to denote tangent sum in the previous post; here we’ll use it to denote hyperbolic tangent sum:

*a* ⊕ *b* = tanh(arctanh *a* + arctanh *b*).

The hyperbolic tangent sum is defined on (−1, 1) because that is the range of hyperbolic tangent.

The hyperbolic tangent sum can be simplified to

*a* ⊕ *b* = (*a* + *b*) / (1 + *ab*).

Note the sign change in the denominator compared to the tangent sum. This means the hyperbolic tangent sum is better behaved than the tangent sum. If *a* and *b* are positive, then *a* ⊕ *b* is positive, and if *a* and *b* are in the domain of arctanh, then *a* ⊕ *b* is finite. The analogous statements weren’t true for the tangent sum.

I wrote about hyperbolic tangent sum three years ago, using the simplified form above and not mentioning hyperbolic tangents. At the time I called it relativistic addition because of the connection with special relativity.

Hyperbolic tangent sum is associative, and the expressions for *a* ⊕ *b* ⊕ *c* ⊕ … have the same form as for the tangent sum, except all the negative signs go away: the sums in the numerator and denominator do not alternate. See this post for more details and a proof.