Simplification of expression with sum of absolute values

I try to simplify following expression

1/2*(Abs[a1] + Abs[b1] + Abs[c1] + Abs[d1]),


where a1, b1, c1, d1 are quite long:

a1=1/16 E^(-4 I t bb[1]) (-1 + E^(4 I t bb[1]))^2;
b1=1/48 E^(-30 I t bb[
1]) (-E^(
26 I t bb[1]) (-1 + E^(4 I t bb[1]))^2 + (2 E^(
52 I t bb[1]) (1 + 8 E^(4 I t bb[1]) + 78 E^(8 I t bb[1]) +
8 E^(12 I t bb[1]) + E^(16 I t bb[1])))/(E^(78 I t bb[1]) +
12 E^(82 I t bb[1]) - 129 E^(86 I t bb[1]) +
232 E^(90 I t bb[1]) - 129 E^(94 I t bb[1]) +
12 E^(98 I t bb[1]) + E^(102 I t bb[1]) +
6 Sqrt[3]
Sqrt[-E^(
164 I t bb[1]) (1 + E^(4 I t bb[1]))^2 (5 +
54 E^(4 I t bb[1]) - 5 E^(8 I t bb[1]) +
1940 E^(12 I t bb[1]) - 5 E^(16 I t bb[1]) +
54 E^(20 I t bb[1]) + 5 E^(24 I t bb[1]))])^(1/3) +
2 (E^(78 I t bb[1]) + 12 E^(82 I t bb[1]) - 129 E^(86 I t bb[1]) +
232 E^(90 I t bb[1]) - 129 E^(94 I t bb[1]) +
12 E^(98 I t bb[1]) + E^(102 I t bb[1]) +
6 Sqrt[3]
Sqrt[-E^(
164 I t bb[1]) (1 + E^(4 I t bb[1]))^2 (5 +
54 E^(4 I t bb[1]) - 5 E^(8 I t bb[1]) +
1940 E^(12 I t bb[1]) - 5 E^(16 I t bb[1]) +
54 E^(20 I t bb[1]) + 5 E^(24 I t bb[1]))])^(1/3));
c1=1/48 E^(-30 I t bb[
1]) (-E^(
26 I t bb[1]) (-1 + E^(4 I t bb[1]))^2 - (I (-I + Sqrt[3]) E^(
52 I t bb[1]) (1 + 8 E^(4 I t bb[1]) + 78 E^(8 I t bb[1]) +
8 E^(12 I t bb[1]) + E^(16 I t bb[1])))/(E^(78 I t bb[1]) +
12 E^(82 I t bb[1]) - 129 E^(86 I t bb[1]) +
232 E^(90 I t bb[1]) - 129 E^(94 I t bb[1]) +
12 E^(98 I t bb[1]) + E^(102 I t bb[1]) +
6 Sqrt[3]
Sqrt[-E^(
164 I t bb[1]) (1 + E^(4 I t bb[1]))^2 (5 +
54 E^(4 I t bb[1]) - 5 E^(8 I t bb[1]) +
1940 E^(12 I t bb[1]) - 5 E^(16 I t bb[1]) +
54 E^(20 I t bb[1]) + 5 E^(24 I t bb[1]))])^(1/3) +
I (I + Sqrt[3]) (E^(78 I t bb[1]) + 12 E^(82 I t bb[1]) -
129 E^(86 I t bb[1]) + 232 E^(90 I t bb[1]) -
129 E^(94 I t bb[1]) + 12 E^(98 I t bb[1]) + E^(
102 I t bb[1]) +
6 Sqrt[3]
Sqrt[-E^(
164 I t bb[1]) (1 + E^(4 I t bb[1]))^2 (5 +
54 E^(4 I t bb[1]) - 5 E^(8 I t bb[1]) +
1940 E^(12 I t bb[1]) - 5 E^(16 I t bb[1]) +
54 E^(20 I t bb[1]) + 5 E^(24 I t bb[1]))])^(1/3));
d1=1/48 E^(-30 I t bb[
1]) (-E^(
26 I t bb[1]) (-1 + E^(4 I t bb[1]))^2 + (I (I + Sqrt[3]) E^(
52 I t bb[1]) (1 + 8 E^(4 I t bb[1]) + 78 E^(8 I t bb[1]) +
8 E^(12 I t bb[1]) + E^(16 I t bb[1])))/(E^(78 I t bb[1]) +
12 E^(82 I t bb[1]) - 129 E^(86 I t bb[1]) +
232 E^(90 I t bb[1]) - 129 E^(94 I t bb[1]) +
12 E^(98 I t bb[1]) + E^(102 I t bb[1]) +
6 Sqrt[3]
Sqrt[-E^(
164 I t bb[1]) (1 + E^(4 I t bb[1]))^2 (5 +
54 E^(4 I t bb[1]) - 5 E^(8 I t bb[1]) +
1940 E^(12 I t bb[1]) - 5 E^(16 I t bb[1]) +
54 E^(20 I t bb[1]) + 5 E^(24 I t bb[1]))])^(
1/3) - (1 + I Sqrt[3]) (E^(78 I t bb[1]) + 12 E^(82 I t bb[1]) -
129 E^(86 I t bb[1]) + 232 E^(90 I t bb[1]) -
129 E^(94 I t bb[1]) + 12 E^(98 I t bb[1]) + E^(
102 I t bb[1]) +
6 Sqrt[3]
Sqrt[-E^(
164 I t bb[1]) (1 + E^(4 I t bb[1]))^2 (5 +
54 E^(4 I t bb[1]) - 5 E^(8 I t bb[1]) +
1940 E^(12 I t bb[1]) - 5 E^(16 I t bb[1]) +
54 E^(20 I t bb[1]) + 5 E^(24 I t bb[1]))])^(1/3))


I have tried to do it with a following code

Simplify[1/
2*(RealAbs[a1] + RealAbs[b1] + RealAbs[c1] +
RealAbs[d1]), {Element[bb[1], PositiveReals],
Element[t, PositiveReals]}],


as I know that at least for positive t and bb[1] (and this I assume) the a1, b1, c1, d1 are real (even if in the expressions above there are some imaginary units, but these should cancel out).

However, in the output the RealAbs are still there and the form is still ugly. The FullSimplify takes forever.

How should I proceed?

To say something about the origin of the quantities: a1, b1, c1, d1 are the eigenvalues of the "harmless looking" 4x4 hermitian matrix

$$-\frac{1}{4}\begin{pmatrix} -1 & -2 e^{-2 I bb[1] t} + \cos[2 bb[1] t]& \cos[2 bb[1] t]& \cos[2 bb[1] t]^2\\ -2 e^{2 I bb[1] t} + \cos[2 bb[1] t]& -1& \cos[2 bb[1] t]^2& \cos[2 bb[1] t]\\ \cos[ 2 bb[1] t]& \cos[2 bb[1] t]^2& 1& \cos[2 bb[1] t]\\ \cos[2 bb[1] t]^2& \cos[ 2 bb[1] t]& \cos[2 bb[1] t]& 1 \end{pmatrix}$$

• Do you believe that a simpler form exists? I can't see any reason why it should. – mikado May 12 '20 at 17:02
• At least the absolute values can be explicitly calculated and one should be able to bring a1, b1, ... in a form from which one sees that they are real. – Agnieszka May 12 '20 at 19:53