Visual walkthrough — Profiling — gprof, perf, Valgrind - Callgrind

Picture your program's run as a long ruler laid on the floor, painted in coloured stripes — each stripe is time spent inside one function, and the stripe we care about is red. We don't want to measure every stripe edge (too fussy, and touching it disturbs the timing). So instead we set a metronome that ticks every $\Delta t$ seconds. On each tick we glance down and note which colour we're standing on. Count the red ticks ($n_f$) and all the ticks ($N$). Since ticks are evenly spaced, each tick "owns" a $\Delta t$ slice of floor, so red time ≈ (red ticks) × $\Delta t$. Dividing by the whole thing lets $\Delta t$ vanish, leaving a clean fraction of ticks — that's the "% time" your profiler prints. It's only an estimate because a tick can miss a thin stripe entirely (that's why a function can show 0%); take more ticks and the wobble shrinks like $1/\sqrt{N}$; take infinitely many and you get Callgrind's exact answer.

Why does $\Delta t$ disappear from the final formula? ::: Both self time ($n_f\Delta t$) and total time ($N\Delta t$) carry one factor of $\Delta t$, so it cancels in the ratio $n_f/N$. A function shows 0 self samples — is its true time zero? ::: No. A fast function can fall entirely between two clicks and be missed; raise the rate or use exact-count Callgrind. To halve the statistical error of a percentage, how many more samples? ::: About 4× more, because error scales like $1/\sqrt{N}$. The $\Delta t\to 0$ limit of this picture corresponds to which tool? ::: Callgrind — exact, deterministic counts, at 20–100× slowdown.