Task 6 of 6

Payoff: Mean and RMS, Fused

The payoff. Two statistics over 4,096 values: the mean (sum ÷ n) and the RMS — root-mean-square, √(sum of squares ÷ n) — the standard "how big is this signal" measure in audio and physics.

RMS needs every value squared first. The rookie move is a separate squaring kernel — a whole extra pass over memory. The pro move is fusion: square each value in the same statement that reads it, inside the partial-sum kernel. Map and reduce, one pass over the data.

Stack the whole module: strided partials (task 2) shrink 4,096 values to 64, then a single shared halving ladder (task 4) finishes both totals.

Goal: compute and log the mean and the RMS of data — two partial-sum kernels (one fused with squaring) plus one shared dynamic halving ladder.

Requirements

partialSums: 64 strided partial sums of data, as in task 2
partialSquares: same shape, but square each value as it is read — no separate squaring pass
One dynamic halving-ladder kernel rides both 64-value arrays down to scalars
mean = total / 4096, rms = Math.sqrt(totalSq / 4096) — log both

Hint 1 — the fused body

Read once, use twice:

const v = data[i * this.constants.threads + this.thread.x];
sum += v * v;

Hint 2 — one ladder, two rides

The ladder kernel doesn't care what its 64 inputs mean. Wrap the driver loop in a function and call it once with each partials array.

Hint 3 — the whole shape

const total = await ladder(await partialSums(data));
const totalSq = await ladder(await partialSquares(data));

then divide, square-root, and log.

Same idea elsewhere

Fusing the map into the reduce is a marquee optimization on every platform: thrust::transform_reduce exists precisely for it, CUDA programmers hand-fuse to halve their memory traffic, and WebGPU/Metal kernels bake the transform into the accumulation loop. Memory bandwidth is the budget — fusion is the discount.

All tasks in Reductions

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