There is quite a bit of excitement on asm.js and how it will be able to run some very heavy applications. However, it is compiled from C++ code. Is it still possible to get the benefit of current improvements without knowing C++ or other low level languages?
Here is the thought that I had: Is it at all possible that we can write the code in Js, have it recompiled for asm.js for optimisation?
However, it is compiled from C++ code.
It is not. It's a language. Any program can emit text files which contain asm.js code. Emscripten compiles LLVM IR into asm.js, and there are compilers from C and C++ to LLVM IR, but this is only one possible way of getting asm.js code. Admittedly, it's currently the most mature, practical and popular way, but I would not be surprised at all if other asm.js compilers for other languages pop up some time in the future.
Is it still possible to get the benefit of current improvements without knowing C++ or other low level languages?
Well, in theory any language that can efficiently be compiled to machine code ahead-of-time can be implemented efficiently using asm.js, and that includes some rather high-level ones (e.g. Haskell). But currently, nobody has a working implementation, and I don't expect this to ever become very popular. Right now, if you want asm.js performance, you'd probably write C or C++ code and compile it to asm.js, yes.
Note that the above excludes (among many others) Javascript.The fact that asm.js is a subset of Javascript is convenient in that asm.js code will run on unmodified browsers, but it's not of much use for anyone writing Javascript. asm.js is basically just a thin layer above machine code, with some amends for security and JS interoperability. Compiling JS to asm.js is as hard as compiling it to machine code: Easy if you don't give a damn about performance (just always used boxed dynamically-typed values like an interpreter, and emit calls to runtime library functions), very hard when you do.
In fact, after decades of research into the subject, there's still no example of a highly dynamic language like Javascript, Ruby or Python being compiled into machine code ahead of time and running much faster than a clever interpreter. Just-in-time compilation, on the other hand, is very much practical -- but the major JS engines already do that, in a less roundabout way than compiling to asm.js, then parsing it again and compiling it to machine code.