How do temperature, top-p and other sampling parameters affect output?

// One fixed set of next-token logits, pushed through the sampling pipeline a
// serving engine runs on every step: penalties -> temperature -> softmax ->
// top-k / top-p / min-p truncation -> renormalise -> draw. The logits stand
// in for one forward pass of a model after the prompt "The weather today is".
const vocab = ["sunny", "cloudy", "rainy", "warm", "nice", "cold", "purple", "banana"];
const logits = [3.0, 2.5, 2.0, 1.5, 1.2, 0.8, -1.0, -2.5];
const JUNK = new Set(["purple", "banana"]);

function softmax(ls, T) {
  const m = Math.max(...ls.map(([, x]) => x));
  const e = ls.map(([w, x]) => [w, Math.exp((x - m) / T)]);
  const s = e.reduce((a, [, x]) => a + x, 0);
  return e.map(([w, x]) => [w, x / s]).sort((a, b) => b[1] - a[1]);
}

function distribution(o) {
  let l = vocab.map((w, i) => {                          // OpenAI-style penalties
    const c = (o.counts && o.counts[w]) || 0;
    return [w, logits[i] - c * (o.freq || 0) - (c > 0 ? o.pres || 0 : 0)];
  });
  if (!o.T) {                                            // temperature 0 = greedy
    const best = l.reduce((a, b) => (b[1] > a[1] ? b : a));
    return [[best[0], 1]];
  }
  // Truncation looks at the distribution at temperature T, unless the engine
  // is configured to truncate first and apply temperature last.
  let d = softmax(l, o.temperatureLast ? 1 : o.T);
  if (o.k) d = d.slice(0, o.k);                          // top-k: keep the k best
  if (o.p) {                                             // top-p: smallest set with mass >= p
    let acc = 0, n = 0;
    while (n < d.length && acc < o.p) acc += d[n++][1];
    d = d.slice(0, n);
  }
  if (o.minP) d = d.filter(([, p]) => p >= o.minP * d[0][1]); // min-p: relative to the top token
  const keep = new Set(d.map(([w]) => w));
  return softmax(l.filter(([w]) => keep.has(w)), o.T);   // renormalise survivors at T
}
const show = (d) => d.map(([w, p]) => w + "=" + p.toFixed(3)).join(" ");

for (const T of [0.2, 0.7, 1.0, 1.5, 3.0]) {
  const d = distribution({ T });
  const junk = d.filter(([w]) => JUNK.has(w)).reduce((a, [, p]) => a + p, 0);
  console.log("T=" + T.toFixed(1) + "  top3: " + show(d.slice(0, 3)) + " | P(purple or banana)=" + junk.toFixed(4));
}
console.log("T=0 (greedy) -> " + distribution({ T: 0 })[0][0]);

const cfgs = [
  ["T=1.0 top_k=3", { T: 1.0, k: 3 }],
  ["T=1.0 top_p=0.9", { T: 1.0, p: 0.9 }],
  ["T=3.0 top_p=0.9", { T: 3.0, p: 0.9 }],
  ["T=1.0 min_p=0.2", { T: 1.0, minP: 0.2 }],
  ["T=3.0 min_p=0.2", { T: 3.0, minP: 0.2 }],
  ["min_p=0.2, T=3 last", { T: 3.0, minP: 0.2, temperatureLast: true }],
];
console.log("candidates kept:");
for (const [name, o] of cfgs) {
  const d = distribution(o);
  console.log("  " + name.padEnd(20) + d.length + " tokens: " + d.map(([w]) => w).join(" "));
}

let seed = 42;
const rand = () => { seed = (seed * 48271) % 2147483647; return seed / 2147483647; };
function draw(d) {
  const r = rand();
  let acc = 0;
  for (const [w, p] of d) { acc += p; if (r < acc) return w; }
  return d[d.length - 1][0];
}
console.log("10000 seeded draws:");
for (const [name, o] of [["T=1.0", { T: 1.0 }], ["T=3.0", { T: 3.0 }], ...cfgs.slice(2, 3), ...cfgs.slice(4)]) {
  seed = 42;
  const d = distribution(o), tally = {};
  for (let i = 0; i < 10000; i++) { const w = draw(d); tally[w] = (tally[w] || 0) + 1; }
  const junk = (tally.purple || 0) + (tally.banana || 0);
  console.log("  " + name.padEnd(20) + "sunny=" + (tally.sunny || 0) + " distinct=" + Object.keys(tally).length + " junk=" + junk);
}

const counts = { sunny: 3, warm: 1 };      // already generated: "sunny sunny sunny warm"
for (const [name, o] of [["no penalty", {}], ["presence=0.5", { pres: 0.5 }], ["frequency=0.5", { freq: 0.5 }], ["both=0.5", { freq: 0.5, pres: 0.5 }]]) {
  const d = distribution({ T: 1.0, counts, ...o });
  console.log("penalty " + name.padEnd(14) + "top3: " + show(d.slice(0, 3)));
}