Does it make any sense to use inline keyword with templates?

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It is not irrelevant. And no, not every function template is inline by default. The standard is even explicit about it in Explicit specialization ([temp.expl.spec])

Have the following:

a.cc

#include "tpl.h"

b.cc

#include "tpl.h"

tpl.h (taken from Explicit Specialization):

#ifndef TPL_H
#define TPL_H
template<class T> void f(T) {}
template<class T> inline T g(T) {}

template<> inline void f<>(int) {} // OK: inline
template<> int g<>(int) {} // error: not inline
#endif

Compile this, et voila:

g++ a.cc b.cc
/tmp/ccfWLeDX.o: In function `int g<int>(int)':
inlinexx2.cc:(.text+0x0): multiple definition of `int g<int>(int)'
/tmp/ccUa4K20.o:inlinexx.cc:(.text+0x0): first defined here
collect2: ld returned 1 exit status

Not stating inline when doing explicit instantiation may also lead to issues.

So in summary: For non fully specialized function templates, i.e. ones that carry at least one unknown type, you can omit inline, and not receive errors, but still they are not inline. For full specializations, i.e. ones that use only known types, you cannot omit it.

Proposed rule of thumb: Write inline if you mean it and just be consistent. It makes you think less about whether to or not to just because you can. (This rule of thumb is conforming to Vandevoorde’s/Josuttis’s C++ Template: The Complete Guide).

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