# Single computation is fast, but 6 computations are extremely slow

I have an expression that involves many Sign functions. I want to expand this in the piecewise format.

My function is:

exp = 2 Sign[t] (1/2 (-1 + e) - 1/2 (1 + e) Sign[t]) + (1/2 (-1 + e1) +
1/2 (1 + e1) Sign[d - t1]) (-1 + Sign[t1]) - (1/2 (-e + e1) +
1/2 (e + e1) Sign[d + t - t1]) (-1 +
Sign[-t + t1]) - (1/2 (-1 + e2) + 1/2 (1 + e2) Sign[d - t2]) (-1 +
Sign[t2]) + (1/2 (-e + e2) + 1/2 (e + e2) Sign[d + t - t2]) (-1 +
Sign[-t + t2]) +
2 (1/2 (-e1 + e2) + 1/2 (e1 + e2) Sign[t1 - t2]) Sign[-t1 + t2]

Here, all variables can be regarded as real.

Using the answer provided in Converting HeavisideTheta[]s and Sign[]s functions to a single Piecewise[], I also borrow a user-defined function:

ToPiecewise[f_] :=
PiecewiseExpand[
f /. {Sign[x_] :> Piecewise[{{1, x >= 0}, {-1, x < 0}}]}]

What I want to eavluate is

ToPiecewise[exp]

but this is extremely slow! (Furthermore, aborting (Alt + .) does not work and I had to quit the kernel.) Surprisingly, if I consider a single term, for example the first one and evaluate

ToPiecewise[2 Sign[t] (1/2 (-1 + e) - 1/2 (1 + e) Sign[t])]

then the answer comes immediately. This happens for all other single terms. Why this strange behavior happen?

Consider:

exp = 2 Sign[t] (1/2 (-1 + e) - 1/2 (1 + e) Sign[t]) + (1/2 (-1 + e1) +
1/2 (1 + e1) Sign[d - t1]) (-1 + Sign[t1]) - (1/2 (-e + e1) +
1/2 (e + e1) Sign[d + t - t1]) (-1 +
Sign[-t + t1]) - (1/2 (-1 + e2) + 1/2 (1 + e2) Sign[d - t2]) (-1 +
Sign[t2]) + (1/2 (-e + e2) + 1/2 (e + e2) Sign[d + t - t2]) (-1 +
Sign[-t + t2]) +
2 (1/2 (-e1 + e2) + 1/2 (e1 + e2) Sign[t1 - t2]) Sign[-t1 + t2]

ToPiecewise[exp[[1]]]
ToPiecewise[exp[[2]]]
ToPiecewise[exp[[3]]] ...

All the terms have different conditions. This makes the overall condition extremely complicated.

A way around this is to give every term its own conditions like:

ToPiecewise /@ exp

• Thanks for your answer. However, it is sad since what I want is to simplify the whole expression into "single" piecewise command. I think this is not too hard, because the total case is $2^8=256$ which is not too large, where 8 comes from the different expressions inside Sign. Oct 9, 2021 at 15:54
• Counting and multiplying the different conditions gives 7840 cases. Oct 9, 2021 at 18:35
• I think your 7840 is obtained by $7840=2\times 4\times 4\times 5\times 7\times 7$. However, there is redundancy in some of the 6 terms. For example, for the 4th term, the condition t1-t2 >=0 and True is redundant. Oct 10, 2021 at 2:15
• Oh, I also forgot to mention that I am not interested when Sign evaluates to Sign[0]. In other words, I am only interested when t != 0, t1 - t2 != 0, t2 != 0, t1 != 0, d - t1 != 0, d - t2 != 0, -t + t1 != 0, d + t - t1 != 0, d + t - t2 != 0, -t + t2 != 0. Then, the total case is 256. Oct 10, 2021 at 2:36