Intuition The one core ideaApproximate computing says: a perfect answer costs more energy, time, and chip-area than a "good enough" answer — so for tasks where nobody can tell the difference (photos, sound, AI guesses), we deliberately compute a little bit wrong to win a lot of savings. Everything on the parent page is just a careful answer to two questions: where can we afford to be wrong, and how much ?
This page assumes you have seen nothing . Before you can read the parent note, you need a small pile of tools. We build each one from a picture, say why the topic needs it, and only then use its symbol.
Definition Bit and bit-widthn bit is a single yes/no switch — it holds a 0 1 bit-width , written n how many of these switches we glue together to store one number .
Picture a row of light switches. With n 2 n n 2 n
Intuition Why the topic cares aboutn Fewer switches means a physically smaller, cheaper, lower-energy circuit. The whole idea of precision scaling (FP32 → FP16 → INT8) is "use a smaller n n
More on how these bit-patterns become real number formats: Precision and Number Formats (FP32, FP16, INT8) .
Not all switches matter equally. In the number 101 4 + 0 + 1 = 5 left 1 4 right 1 1
The most significant bit (MSB) is the leftmost switch — it controls the biggest chunk of the value. The least significant bit (LSB) is the rightmost — it controls the smallest , finest detail.
Intuition Why the topic needs this distinctionBeing wrong in the LSB changes a number by a tiny amount; being wrong in the MSB can double or halve it. Approximate computing puts all its deliberate errors in the LSBs and never touches the MSBs. This is exactly why the parent warns: "never approximate the exponent" — the exponent is like a super-MSB that controls magnitude. See Ripple-Carry vs Array Multipliers for where these bits live physically.
The parent's aggregation proof uses four Greek-flavoured symbols. Let us earn each.
ε ε "a small error" — the gap between the answer we computed and the exact answer. If exact = 5 4.9 ε = − 0.1
μ E [ ⋅ ] μ average (typical value) of a bunch of numbers. E [ X ] expected value of X E [ ε ] = 0 "on average, the error is zero" — sometimes too high, sometimes too low, cancelling out.
Definition Standard deviationσ σ how spread out the errors are — the typical distance from the average. Small σ σ
Intuition Why the topic needsE σ "errors average out" . That sentence only makes sense once you can say average = 0 (E [ ε ] = 0 spread = σ . These two knobs are exactly what decide whether a million tiny errors cancel or pile up.
E [ ε ] = 0 Why it feels right: zero average sounds like zero error. The fix: each individual ε i not zero — only the average over many is zero. Individual errors are alive and well; they just point in random directions.
Definition The sigma-sum symbol∑ i = 1 N ε i = ε 1 + ε 2 + ⋯ + ε N . "start the counter i 1 ε i N It is just a compact "add all of these" instruction.
N N N σ in random directions , they don't line up — they partly cancel, like N N N total error grows like N σ signal (the honest sum) grows like N This is the mathematical permission slip for approximate hardware.
Linked idea: this same "errors shrink when aggregated" is why Neural Network Quantization survives INT8.
∝ A ∝ B "A B — double B A shape of the relationship.
V 2 V voltage (electrical "push") fed to the chip. V 2 V × V V 0.8 V 2 0.64 36% 20%
Full derivation of this equation: Dynamic Power P = alpha C V^2 f . The area/energy version (E ∝ n 2 V 2 Ripple-Carry vs Array Multipliers .
Approximation is only allowed if we can measure how bad it got. That measuring stick is the quality metric .
Definition Three metrics you will meet
Relative error : ∣ exact ∣ ∣ approx − exact ∣ 2% Accuracy : for a classifier, the fraction of inputs it still labels correctly.PSNR (peak signal-to-noise ratio, in decibels/dB): a bigger number = a cleaner image. Above ≈ 40
number , not a feeling
"Looks fine to me" is not engineering. A quality metric turns "good enough" into a hard threshold ("PSNR ≥ 40 prove the error stays bounded. That word bounded is the entire promise of the field.
Definition What "the arrow" saysx N → ∞ 0 "as N x 0 It describes the destination of a trend , not a single value.
Intuition Why the topic needs the limitThe relative-error formula μ N σ exactly zero, but it heads there as we aggregate more. The limit arrow lets us state the crisp punchline: the more you aggregate, the more sloppiness you can afford.
Number formats FP32 FP16 INT8
Where to place error safely
Error epsilon mean mu spread sigma
Expectation E and variance
Proportional to and V squared
Power law P = alpha C V^2 f
Quality metrics PSNR accuracy
Related destinations once these foundations click: DRAM Refresh and Memory Reliability , Error-Correcting Codes , Dark Silicon and Energy-Efficient Architectures , and the parent 6.5.16 Approximate computing techniques (Hinglish) .
Cover the right side. If you can answer each, you are ready for the parent note.
What does the symbol n How many switch-wires build a number — the master dial for size, energy, and precision of the circuit.
Which bits are safe to corrupt and which are forbidden? LSBs (small value) are safe to approximate; MSBs and exponents (big value / magnitude) are forbidden.
What does E [ ε ] = 0 The error averages to zero across many operations — not that any single error is zero.
What does σ The typical spread (scatter) of the errors around their average.
Why does the total of N N N Random-direction errors partly cancel, like a random walk, so the crowd drifts only about
N steps.
What does A ∝ B A B
Why is V 2 Power depends on voltage squared, so a small voltage cut gives an outsized power saving.
What is a quality metric for, in one word? To make "good enough" a measurable, bounded, provable threshold.
What does x N → ∞ 0 The destination of a trend as N x 0
