Visual walkthrough — Performance profiling — CPU, memory, I - O profiling

We assume nothing. If you can read a ruler, you can follow line one.

Step 1 — Draw the whole program as one bar

WHAT we did: turned "how long the program takes" into a picture — a ruler of length $1$.

WHY: every symbol we introduce next will be a piece of this bar. If we can point at a piece, we can never sneak in an undefined quantity. Fractions of the bar are exactly the "80/20" language the parent note keeps using — now they have a picture.

PICTURE: the amber bar below is the entire run. Its full width means $T=1=100\%$.

Step 2 — Cut the bar where your optimization can reach

WHAT we did: split the single bar into "attackable" ($p$) and "leftover" ($1-p$).

WHY the subtraction $1-p$ and nothing fancier? Because the pieces are complementary — they tile the bar with no gap and no overlap. The moment you name one piece $p$, the other is forced to be $1-p$. No extra tool needed; it's just "the rest of the ruler."

PICTURE: the cyan piece is $p$ (your target), the white piece is $1-p$ (untouchable). Together they still span the full bar.

Step 3 — Shrink only the slice you can reach

WHAT we did: replaced the cyan piece $p$ with a shorter cyan piece $p/s$.

WHY divide, not subtract? Because "faster" is multiplicative, not additive. Running twice as fast doesn't remove a fixed chunk of time — it halves whatever was there. Halving is $÷2$; running $s\times$ faster is $÷s$. This is the one place a division (a ratio tool) enters, and it enters precisely because speed is defined as work-per-time — a ratio.

PICTURE: watch the cyan piece collapse from $p$ to $p/s$ while the white piece refuses to budge.

Step 4 — Measure the new, shorter bar

WHAT we did: added the two current pieces to get the length of the new bar.

WHY add? Because time laid end to end simply adds — that is what "total elapsed time" means on a ruler. No overlap between the pieces means no double counting.

PICTURE: the two bars stacked — original ($=1$) on top, shrunken ($T_{\text{new}}$) below — so you can literally see how much shorter you made the run.

Step 5 — Turn "shorter" into a speedup number

WHAT we did: divided the original bar ($1$) by the new bar ($T_{\text{new}}$).

WHY this is the payoff: every quantity in this boxed formula was drawn as a piece of a ruler before it appeared. Nothing was assumed. This is the central result the parent note quotes — and now you own its picture.

Step 6 — The edge cases (never leave the reader guessing)

Every honest derivation walks its own extremes. Here they are, each a corner of the same picture.

PICTURE: the ceiling curve. Height is the best possible speedup $1/(1-p)$ as the target fraction $p$ slides from $0$ to $1$. See how flat it stays for small $p$ and how it explodes only as $p\to 1$.

Step 7 — Plug in real numbers (the parent's Example 1)

Reveal-to-test:

The one-picture summary

Everything above, compressed into a single diagram: the original bar of length $1$, its split into white $(1-p)$ and cyan $p$, the cyan piece collapsing to $p/s$, and the resulting speedup $S=1/[(1-p)+p/s]$ — with the four edge cases pinned to their corners.