...one of the most highly
regarded and expertly designed C++ library projects in the
world.
— Herb Sutter and Andrei
Alexandrescu, C++
Coding Standards
Using a simple backend
class that wraps any fundamental
(builtin) arithmetic type we can measure the overhead involved in
wrapping a type inside the number
frontend, and the effect that turning on expression templates has. The following
table compares the performance between double
and a double
wrapped inside
class number
:
Table 1.12. Bessel Functions (16 digit precision)
Type 
Time 

arithmetic_backend<double> 
2.09301 (0.00133409s) 
arithmetic_backend<double>  no expression templates 
1 (0.000637403s) 
double 
1.07956 (0.000688113s) 
As you can see whether or not there is an overhead, and how large it is depends on the actual situation, but the overhead is in any cases small. Expression templates generally add a greater overhead the more complex the expression becomes due to the logic of figuring out how to best unpack and evaluate the expression, but of course this is also the situation where you save more temporaries. For a "trivial" backend like this, saving temporaries has no benefit, but for larger types it becomes a bigger win.
The following table compares arithmetic using either long
long
or number<arithmetic_backend<long long> >
for the voronoidiagram
builder test:
Type 
Relative time 


1.0(0.0128646s) 

1.005 (0.0129255s) 
This test involves mainly creating a lot of temporaries and performing a small amount of arithmetic on them, with very little difference in performance between the native and "wrapped" types.
Table 1.13. Platform Details
Version 


Compiler 
GNU C++ version 10.3.0 
GMP 
6.2.0 
MPFR 
262146 
Boost 
107800 
Run date 
Sep 30 2021 