What kind of embedded system are you using? Are you running embedded Linux, for example? What kind of development environment are you using?

Some coding practices:
* Use fixed-point arithmetic instead of floating-point where precision is not critical.
* Replace computationally expensive operations with lookup tables when feasible, especially for trigonometric or logarithmic functions.

For specific tools, you need to use tools that are specific to your development environment. For example, if you use Linux, you can use gprof and Valgrind.

You can also use hardware tools like oscilloscopes and logic analyzers to measure execution times

For low-power modes, put the MCU in a low-power mode when idle and wake it up only when necessary.

Optimizations require a holistic approach; piecemeal optimization often results in poor performance overall.

Algorithmic optimization often requires multiple iterations on the algorithm and detailed code analysis; we once found a specific implementation of an application specific algorithm could be simplified by 98% by making use of intermediate results that could be used for multiple calculations further downstream. We probably spent 3 weeks on the looking for alternate implementations on the algorithm alone.

CPUs and MCUs are sometimes overpowered for the task at hand; careful algorithm analysis and optimization may reveal a simpler architecture for the processor. Sometimes, you may find that a software or firmware implementation of an algorithm can eliminate dedicated hardware, which may lower both cost and power. RISC vs. conventional processor trade offs look at things like that. Recursive algorithms often trade processor throughput for increased execution time with lower power hardware.

TTFN (ta ta for now)
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Optimizing performance?

"best approaches to improve efficiency without significantly increasing power consumption"

Improve efficiency is a little vague.
Are you looking to minimize power consumption?
Trying to get the greatest thruput of data?
Trying to minimize product cost?

Need to be a lot more specific on what you are trying to optimize.

More information is needed on what you want to do will get more meaningful results on how to do it. Typically a bare metal system can be optimized for power consumption better than something running a RTOS and the associated overhead. But still need to understand what your goals are first.

Jerry Twomey, Author of:
"Applied Embedded Electronics Design Essentials for Robust Systems"
https://effectiveelectrons.com/book/