There is probably a bit of a misconception here. In many application the signal is runnning all the time or is VERY long: modem, audio stream, video etc. In this case you can't really define the "length" of the signal. The relevant metric is here "number of operations per input sample" not the "total number of operations". If you watch a streaming movie, it's obvious that a two hour movie takes twice the operations of a one hour movie. But that's okay since you have twice the time to process it. The key metric is if your processor can keep up with the stream , so that's why "ops per sample" or "ops per second" are used.
In this case, you don't do an FFT over the entire input signal but you apply "overlap add", i.e. you chop the input signals into blocks that are about the same size as the filter length and then you process one block at a time.
So the effectiveness of the algorithm is only a function of the filter length. And it increases linearly for direct convoltuion but only logarithmically for Overlap Add (which is FFT based). The "break even" point is typically at "many tens of samples".
We can run some numbers assuming that the filter length is $N$ and lets also assume its a power of 2.
An FFT of length $M$ has $log_2(M) \cdot M/2$ butterflies, each with one complex multiplies and two complex adds. That's a total of 10 real operations. So the price for an FFT is roughly $5 \cdot log_2(M) \cdot M/2$
Overlap add requires zero padding to 2N and that's our FFT size. Hence overlap add costs
- Forward FFT: $5 \cdot log_2(2N) \cdot 2N/2$
- Spectral Multiply: $3*N$ (N/2 complex multiplies)
- Inverse FFT: $5 \cdot log_2(2N) \cdot 2N/2$
- Overlap adding: $N$
- Total: $N\cdot (10 \cdot log_2(N) + 10 + 3 + 1)$
So overlap add cost per sample is roughly $ 10 \cdot log_2(N) + 14$ as compared to $2N$ for direct convolution. The break even point is a filter length of 32.
DISCLAIMER: I made a lot of simplying assumptions and in a practical implementation the break even point can vary wildly based on processor architecture and smartness of the algorithm. The main cost is in the FFT and for a single channel real system you would deploy a core complex FFT of N/2. I also skipped over bit reverse ordering and other overheads.