Why 0.1 + 0.2 ≠ 0.3

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<!doctype html>
<html lang="en">
<head>
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Why 0.1 + 0.2 ≠ 0.3 — sloprun.dev</title>
<style>
/* sloprun design tokens — inline this block into every demo (self-contained rule).
   Identity: instrument-panel. Machine facts in mono; human words in sans.
   Green is EARNED: only for "it ran / it worked" states, never decoration. */
:root {
  --bg: #F6F6F3;          /* neutral paper, not cream */
  --surface: #FFFFFF;
  --ink: #1A1D21;
  --muted: #5A6068;
  --line: #E3E2DC;
  --accent: #C05600;      /* phosphor amber, darkened for light bg */
  --accent-soft: #FFF3E6;
  --run: #1A7F37;         /* earned green */
  --run-soft: #E7F4EA;
  --danger: #C93C37;
  --radius: 10px;
  --font-sans: system-ui, -apple-system, "Segoe UI", sans-serif;
  --font-mono: ui-monospace, "SF Mono", "Cascadia Code", Menlo, Consolas, monospace;
}
@media (prefers-color-scheme: dark) {
  :root:not([data-theme="light"]) {
    --bg: #14161A; --surface: #1C1F26; --ink: #E8E6E1; --muted: #9AA0A8;
    --line: #2A2E36; --accent: #FFB454; --accent-soft: #2A2118;
    --run: #3FB950; --run-soft: #16281B; --danger: #F47067;
  }
}
:root[data-theme="dark"] {
  --bg: #14161A; --surface: #1C1F26; --ink: #E8E6E1; --muted: #9AA0A8;
  --line: #2A2E36; --accent: #FFB454; --accent-soft: #2A2118;
  --run: #3FB950; --run-soft: #16281B; --danger: #F47067;
}
body { background: var(--bg); color: var(--ink); font-family: var(--font-sans); }

/* ---- derived surfaces (theme-paired, same rules as the tokens above) ---- */
:root { --off-face: #FBFBF9; --glow: rgba(192,86,0,.15); --wash: #F1F0EC; --dim: .45; --on-ink: #FFFFFF; --danger-soft: #FBEBEA; --man-on: #1A1D21; --man-ink: #FFFFFF; }
@media (prefers-color-scheme: dark) {
  :root:not([data-theme="light"]) { --off-face: #181B21; --glow: rgba(255,180,84,.14); --wash: #15181D; --dim: .40; --on-ink: #14161A; --danger-soft: #2C1A19; --man-on: #B4BAC2; --man-ink: #14161A; }
}
:root[data-theme="dark"] { --off-face: #181B21; --glow: rgba(255,180,84,.14); --wash: #15181D; --dim: .40; --on-ink: #14161A; --danger-soft: #2C1A19; --man-on: #B4BAC2; --man-ink: #14161A; }

/* ---- shell ---- */
* { box-sizing: border-box; }
html, body { margin: 0; padding: 0; }
body { font-size: 16px; line-height: 1.55; overflow-x: hidden; -webkit-text-size-adjust: 100%; }
.wrap { max-width: 1040px; margin: 0 auto; padding: 26px 16px 34px; }
h1 { font-size: 1.5rem; line-height: 1.2; margin: 0 0 7px; letter-spacing: -.015em; }
h2 { font-size: .95rem; margin: 0; }
.lede { margin: 0; color: var(--muted); font-size: .95rem; max-width: 70ch; }
.lede b { color: var(--ink); font-weight: 600; }
.mono, code { font-family: var(--font-mono); }
.sr-only {
  position: absolute; width: 1px; height: 1px; padding: 0; margin: -1px;
  overflow: hidden; clip: rect(0 0 0 0); white-space: nowrap; border: 0;
}
footer { margin-top: 22px; font-family: var(--font-mono); font-size: .7rem; color: var(--muted); }

.k-lab {
  font-family: var(--font-mono); font-size: .64rem; letter-spacing: .11em;
  text-transform: uppercase; color: var(--muted);
}
.panel {
  background: var(--surface); border: 1px solid var(--line);
  border-radius: var(--radius); padding: 14px; margin-top: 16px;
}
.panelhead { display: flex; flex-wrap: wrap; align-items: center; gap: 8px 12px; margin-bottom: 12px; }
.panelhead .k-lab.right { margin-left: auto; }

.btn {
  font-family: var(--font-mono); font-size: .73rem; letter-spacing: .02em;
  background: var(--bg); color: var(--ink); border: 1px solid var(--line);
  border-radius: 999px; padding: 6px 11px; min-height: 32px; cursor: pointer;
}
.btn:hover { border-color: var(--accent); color: var(--accent); }
.btn:focus-visible { outline: 2px solid var(--accent); outline-offset: 2px; }
.btn[aria-pressed="true"] { background: var(--accent-soft); border-color: var(--accent); color: var(--accent); }
.btn.ghost { background: transparent; color: var(--muted); }
.btn.ghost:hover { color: var(--accent); }
.btn:disabled { opacity: var(--dim); cursor: not-allowed; }
.btn:disabled:hover { border-color: var(--line); color: var(--ink); }

/* ---- input row ---- */
.inrow { display: flex; flex-wrap: wrap; align-items: flex-end; gap: 10px; }
.infield { flex: 1 1 210px; min-width: 0; }
.infield label { display: block; margin-bottom: 4px; }
.infield input {
  width: 100%; font-family: var(--font-mono); font-size: 1rem; color: var(--ink);
  background: var(--bg); border: 1px solid var(--line); border-radius: 8px;
  padding: 10px 11px; min-height: 44px;
}
.infield input:focus { outline: none; border-color: var(--accent); box-shadow: 0 0 0 3px var(--glow); }
.infield input:focus-visible { outline: 2px solid var(--accent); outline-offset: 2px; }
.infield.bad input { border-color: var(--danger); }
.steppers { display: flex; gap: 6px; }
.hint { margin: 8px 0 0; font-family: var(--font-mono); font-size: .68rem; color: var(--muted); }
.hint.bad { color: var(--danger); }
.presets { display: flex; flex-wrap: wrap; gap: 6px; margin-top: 11px; }

/* ---- chips + readouts ---- */
.chips { display: flex; flex-wrap: wrap; gap: 6px; margin-top: 12px; }
.chip {
  font-family: var(--font-mono); font-size: .7rem; border-radius: 999px;
  padding: 3px 10px; border: 1px solid var(--line); color: var(--muted); background: var(--bg);
}
.chip.amber { background: var(--accent-soft); border-color: var(--accent); color: var(--accent); }
.chip.good { background: var(--run-soft); border-color: var(--run); color: var(--run); }
.chip.bad { background: var(--danger-soft); border-color: var(--danger); color: var(--danger); }
.chip[hidden] { display: none; }

.rows { margin-top: 13px; border-top: 1px dashed var(--line); padding-top: 12px; display: grid; gap: 11px; }
.rw { display: grid; grid-template-columns: 132px minmax(0,1fr); gap: 4px 12px; align-items: baseline; }
.rw[hidden] { display: none; }
.rw > .v { font-family: var(--font-mono); font-size: .86rem; overflow-wrap: anywhere; min-width: 0; }
.rw .sub { display: block; font-family: var(--font-sans); font-size: .78rem; color: var(--muted); margin-top: 2px; }
.exact { display: block; max-height: 8.6em; overflow-y: auto; overflow-x: hidden; }
.exact:focus-visible { outline: 2px solid var(--accent); outline-offset: 3px; border-radius: 4px; }
.exact .tail { color: var(--accent); }
.exact .head { color: var(--ink); }
@media (max-width: 620px) { .rw { grid-template-columns: 1fr; } }

/* ---- the 64 switches ---- */
.bank { display: flex; flex-wrap: wrap; gap: 14px 20px; }
.band { min-width: 0; }
.band.sign { flex: 0 0 auto; }
.band.exp { flex: 0 1 340px; }
.band.grow { flex: 1 1 100%; }
.bandhead { display: flex; align-items: baseline; gap: 8px; margin-bottom: 5px; flex-wrap: wrap; }
.bandhead .nm { font-family: var(--font-mono); font-size: .68rem; letter-spacing: .06em; text-transform: uppercase; }
.bandhead .rng { font-family: var(--font-mono); font-size: .6rem; color: var(--muted); }
.band.sign .nm { color: var(--danger); }
.band.exp .nm { color: var(--accent); }
.band.man .nm { color: var(--ink); }
.grid { display: grid; gap: 3px; }
.band.sign .grid { grid-template-columns: 38px; }
.band.exp .grid { grid-template-columns: repeat(11, minmax(0,1fr)); width: 100%; }
.band.man .grid { grid-template-columns: repeat(26, minmax(0,1fr)); }
.bit {
  height: 28px; padding: 0; cursor: pointer; min-width: 0;
  border: 1px solid var(--line); background: var(--off-face); color: var(--muted);
  border-radius: 5px; font-family: var(--font-mono); font-size: .72rem; font-weight: 600;
  display: grid; place-items: center; line-height: 1;
}
.bit:hover { border-color: var(--accent); }
.bit:focus-visible { outline: 2px solid var(--accent); outline-offset: 2px; z-index: 2; }
.band.sign .bit[aria-checked="true"] { background: var(--danger); border-color: var(--danger); color: var(--on-ink); }
.band.exp  .bit[aria-checked="true"] { background: var(--accent); border-color: var(--accent); color: var(--on-ink); }
.band.man  .bit[aria-checked="true"] { background: var(--man-on); border-color: var(--man-on); color: var(--man-ink); }
.inspect {
  margin: 12px 0 0; padding: 9px 11px; background: var(--bg); border: 1px solid var(--line);
  border-radius: 8px; font-family: var(--font-mono); font-size: .72rem; color: var(--muted);
  min-height: 38px; overflow-wrap: anywhere;
}
.inspect b { color: var(--ink); font-weight: 600; }
.inspect .am { color: var(--accent); }
.formula {
  margin: 12px 0 0; padding: 11px 12px; background: var(--wash); border: 1px solid var(--line);
  border-radius: 8px; font-family: var(--font-mono); font-size: .78rem; color: var(--muted);
  overflow-wrap: anywhere; line-height: 1.7;
}
.formula b { color: var(--ink); font-weight: 600; }
.formula .am { color: var(--accent); }
.formula .dg { color: var(--danger); }
.bankfoot { display: flex; flex-wrap: wrap; align-items: center; gap: 8px; margin-top: 12px; }
.pattern {
  font-family: var(--font-mono); font-size: .72rem; color: var(--ink);
  background: var(--bg); border: 1px solid var(--line); border-radius: 999px; padding: 5px 11px;
}
.pattern span { color: var(--muted); }

/* ---- neighbours rail ---- */
.railwrap { display: grid; grid-template-columns: minmax(0,1fr); gap: 12px 26px; align-items: center; }
@media (min-width: 780px) { .railwrap { grid-template-columns: minmax(0,360px) minmax(0,1fr); } }
.rail { position: relative; height: 208px; margin: 4px 0 2px; }
.rail .stem {
  position: absolute; left: 9px; top: 12px; bottom: 12px; width: 2px;
  background: var(--line); border-radius: 2px;
}
.rail .zone {
  position: absolute; left: 3px; width: 14px; border-radius: 7px;
  background: var(--accent-soft); border: 1px dashed var(--accent);
}
.rail .node { position: absolute; left: 0; transform: translateY(-50%); display: flex; align-items: center; gap: 9px; max-width: 100%; }
.rail .dot { width: 22px; height: 10px; border-radius: 3px; background: var(--muted); opacity: .55; flex: none; }
.rail .node.self .dot { background: var(--accent); height: 14px; opacity: 1; }
.rail .lab { font-family: var(--font-mono); font-size: .7rem; color: var(--muted); overflow-wrap: anywhere; min-width: 0; }
.rail .node.self .lab { color: var(--ink); font-size: .78rem; }
.rail .lab em { font-style: normal; display: block; font-family: var(--font-sans); font-size: .66rem; color: var(--muted); }
.rail .mark { position: absolute; left: 2px; right: 0; transform: translateY(-50%); display: flex; align-items: center; gap: 8px; }
.rail .mark.pushright { justify-content: space-between; }
.rail .mark .ln { width: 26px; height: 0; border-top: 2px dashed var(--danger); flex: none; }
.rail .mark .tx { font-family: var(--font-mono); font-size: .66rem; color: var(--danger); }
.nnote { margin: 4px 0 0; font-size: .86rem; color: var(--muted); max-width: 66ch; }
.nnote b { color: var(--ink); font-weight: 600; }
.facts {
  list-style: none; margin: 12px 0 0; padding: 12px 0 0; border-top: 1px dashed var(--line);
  display: grid; grid-template-columns: repeat(auto-fit, minmax(150px, 1fr)); gap: 10px 14px;
}
.facts li { min-width: 0; }
.facts .v { display: block; font-family: var(--font-mono); font-size: .84rem; color: var(--ink); overflow-wrap: anywhere; }

/* ---- story ---- */
.dots { display: flex; gap: 5px; margin-left: auto; }
.dot-b {
  width: 26px; height: 26px; padding: 0; border-radius: 999px; cursor: pointer;
  border: 1px solid var(--line); background: var(--bg); color: var(--muted);
  font-family: var(--font-mono); font-size: .66rem;
}
.dot-b[aria-current="true"] { background: var(--accent-soft); border-color: var(--accent); color: var(--accent); }
.dot-b:hover { border-color: var(--accent); }
.dot-b:focus-visible { outline: 2px solid var(--accent); outline-offset: 2px; }
.step h3 { margin: 0 0 6px; font-size: 1.02rem; letter-spacing: -.01em; }
.step .say { margin: 0 0 11px; color: var(--muted); font-size: .92rem; max-width: 68ch; }
.step .say b { color: var(--ink); font-weight: 600; }
.slist { display: grid; gap: 9px; margin: 0; padding: 0; }
.sitem {
  border-left: 2px solid var(--line); padding: 2px 0 2px 11px;
}
.sitem.hi { border-left-color: var(--accent); }
.sitem.good { border-left-color: var(--run); }
.sitem.bad { border-left-color: var(--danger); }
.sitem .t.code { text-transform: none; letter-spacing: 0; font-size: .72rem; }
.sitem .t { display: block; font-family: var(--font-mono); font-size: .62rem; letter-spacing: .1em; text-transform: uppercase; color: var(--muted); }
.sitem .m { display: block; font-family: var(--font-mono); font-size: .8rem; color: var(--ink); overflow-wrap: anywhere; }
.sitem.hi .m { color: var(--accent); }
.sitem.good .m { color: var(--run); }
.sitem.bad .m { color: var(--danger); }
.stepfoot { display: flex; flex-wrap: wrap; align-items: center; gap: 8px; margin-top: 13px; }
.stepfoot .sp { margin-left: auto; font-family: var(--font-mono); font-size: .66rem; color: var(--muted); }

.lesson {
  margin: 16px 0 0; padding: 13px 15px; background: var(--surface);
  border: 1px solid var(--line); border-left: 3px solid var(--accent); border-radius: var(--radius);
  font-size: .96rem; line-height: 1.55;
}
.lesson b { font-weight: 600; }

/* ---- motion (opt-in only) ---- */
@media (prefers-reduced-motion: no-preference) {
  .btn, .chip, .bit, .dot-b, .infield input, .rail .node, .rail .zone, .rail .mark {
    transition: background-color .14s ease, border-color .14s ease, color .14s ease,
                box-shadow .14s ease, top .22s cubic-bezier(.3,.8,.35,1), height .14s ease;
  }
  .bit.flash { animation: ring .5s ease-out; }
  @keyframes ring {
    0% { box-shadow: 0 0 0 0 var(--glow); } 45% { box-shadow: 0 0 0 8px var(--glow); }
    100% { box-shadow: 0 0 0 0 var(--glow); }
  }
  .step.swap { animation: fade .22s ease-out both; }
  @keyframes fade { from { opacity: 0; transform: translateY(4px); } to { opacity: 1; transform: none; } }
}

/* ---- narrow ---- */
@media (max-width: 700px) {
  .band.man .grid { grid-template-columns: repeat(13, minmax(0,1fr)); }
}
@media (max-width: 560px) {
  .wrap { padding: 20px 12px 30px; }
  .panel { padding: 12px; }
  .bit { height: 26px; font-size: .66rem; }
  .band.sign .grid { grid-template-columns: 34px; }
  .rail { height: 232px; }
  .rail .lab { font-size: .64rem; }
  .rail .node.self .lab { font-size: .7rem; }
}
</style>
</head>
<body>
<main class="wrap">
  <header>
    <h1>Why 0.1 + 0.2 ≠ 0.3</h1>
    <p class="lede">Your computer stores every number in <b>64 switches</b>. Most decimals do not fit, so it keeps the nearest one it has. Type a number below — or flip the switches yourself — and watch the tidy decimal you asked for turn into the long one that is actually stored.</p>
  </header>

  <section class="panel" aria-labelledby="h-num">
    <div class="panelhead">
      <h2 id="h-num">The number</h2>
      <span class="k-lab right" id="klass">normal</span>
    </div>

    <div class="inrow">
      <div class="infield" id="fWrap">
        <label class="k-lab" for="numIn">type any number</label>
        <input id="numIn" type="text" autocomplete="off" spellcheck="false" autocapitalize="off"
               inputmode="decimal" aria-describedby="inHint">
      </div>
      <div class="steppers" role="group" aria-label="Move to a neighbouring number">
        <button class="btn" type="button" id="stepDn" title="the next number down that this format can hold">&#9660; one step down</button>
        <button class="btn" type="button" id="stepUp" title="the next number up that this format can hold">&#9650; one step up</button>
      </div>
    </div>
    <p class="hint" id="inHint">plain (0.1) · scientific (5e-324) · or a raw pattern (0x3FB999999999999A)</p>

    <div class="presets" id="presets" role="group" aria-label="Example numbers"></div>

    <div class="chips" id="chips"></div>

    <div class="rows">
      <div class="rw"><span class="k-lab">prints as</span><span class="v" id="rPrint"></span></div>
      <div class="rw">
        <span class="k-lab">actually stored</span>
        <span class="v mono" id="rExact"></span>
      </div>
      <div class="rw" id="rwDiff"><span class="k-lab">the gap</span><span class="v" id="rDiff"></span></div>
    </div>
    <p class="sr-only" id="live" aria-live="polite"></p>
  </section>

  <section class="panel" aria-labelledby="h-bits">
    <div class="panelhead">
      <h2 id="h-bits">The 64 switches</h2>
      <span class="k-lab right">click any switch to flip it</span>
    </div>

    <div class="bank" id="bank">
      <div class="band sign">
        <div class="bandhead"><span class="nm">sign</span><span class="rng">bit 63</span></div>
        <div class="grid" id="gSign" role="group" aria-label="Sign bit"></div>
      </div>
      <div class="band exp">
        <div class="bandhead"><span class="nm">exponent</span><span class="rng">bits 62&ndash;52 &middot; how big</span></div>
        <div class="grid" id="gExp" role="group" aria-label="Eleven exponent bits, from bit 62 down to bit 52"></div>
      </div>
      <div class="band man grow">
        <div class="bandhead"><span class="nm">fraction</span><span class="rng">bits 51&ndash;0 &middot; the digits</span></div>
        <div class="grid" id="gMan" role="group" aria-label="Fifty-two fraction bits, from bit 51 down to bit 0"></div>
      </div>
    </div>

    <p class="inspect" id="inspect"></p>
    <p class="formula" id="formula"></p>

    <div class="bankfoot">
      <span class="pattern" id="pattern"></span>
      <button class="btn ghost" type="button" id="copyBtn">copy pattern</button>
      <span class="chip" id="copyNote" hidden></span>
    </div>
  </section>

  <section class="panel" aria-labelledby="h-nb">
    <div class="panelhead">
      <h2 id="h-nb">Its nearest neighbours</h2>
      <span class="k-lab right">nothing exists in between</span>
    </div>

    <div class="railwrap">
      <div class="rail" id="rail"></div>
      <p class="nnote" id="railNote"></p>
    </div>

    <ul class="facts" id="facts"></ul>
  </section>

  <section class="panel" aria-labelledby="h-story">
    <div class="panelhead">
      <h2 id="h-story">The 0.1 + 0.2 story</h2>
      <div class="dots" id="dots" role="group" aria-label="Jump to a step"></div>
    </div>

    <div class="step" id="step"></div>

    <div class="stepfoot">
      <button class="btn" type="button" id="prevBtn">&larr; back</button>
      <button class="btn" type="button" id="nextBtn">next &rarr;</button>
      <button class="btn ghost" type="button" id="loadBtn">load this number into the switches</button>
      <span class="sp" id="stepCount"></span>
    </div>
    <p class="sr-only" id="stepLive" aria-live="polite"></p>
  </section>

  <p class="lesson"><b>The one-line lesson:</b> binary cannot write 0.1 exactly, the same way decimal cannot write one third exactly &mdash; so the machine stores the closest number it owns and quietly carries the difference forward.</p>

  <footer>demo &middot; sloprun.dev</footer>
</main>

<script>
(function () {
  "use strict";

  var $ = function (id) { return document.getElementById(id); };
  var dv = new DataView(new ArrayBuffer(8));
  var M64 = (1n << 64n) - 1n, SIGN = 1n << 63n, MANT = (1n << 52n) - 1n, HIDDEN = 1n << 52n;
  var INFBITS = 0x7FF0000000000000n;

  /* ---------- raw bits <-> numbers ---------- */
  function bitsOf(x) { dv.setFloat64(0, x); return dv.getBigUint64(0); }
  function numOf(u) { dv.setBigUint64(0, u & M64); return dv.getFloat64(0); }
  function expField(u) { return Number((u >> 52n) & 0x7FFn); }
  function mantOf(u) { return u & MANT; }
  function kindOf(u) {
    var eb = expField(u), m = mantOf(u);
    if (eb === 2047) return m === 0n ? "inf" : "nan";
    if (eb === 0) return m === 0n ? "zero" : "sub";
    return "norm";
  }
  function isFiniteBits(u) { var k = kindOf(u); return k !== "inf" && k !== "nan"; }

  /* exact value of a finite double as M * 2^e, M signed */
  function sigOf(u) {
    var neg = (u & SIGN) !== 0n, eb = expField(u), m = mantOf(u), M;
    if (eb === 0) { M = m; } else { M = m + HIDDEN; }
    return { M: neg ? -M : M, e: eb === 0 ? -1074 : eb - 1075 };
  }

  /* ---------- exact dyadic arithmetic (no precision lost) ---------- */
  function dyAdd(a, b) {
    var e = Math.min(a.e, b.e);
    return { M: (a.M << BigInt(a.e - e)) + (b.M << BigInt(b.e - e)), e: e };
  }
  function dySub(a, b) { return dyAdd(a, { M: -b.M, e: b.e }); }
  function dyHalf(a) { return { M: a.M, e: a.e - 1 }; }

  var p5 = {}, p10 = {};
  function pow5(k) { if (!p5[k]) p5[k] = 5n ** BigInt(k); return p5[k]; }
  function pow10(k) { if (!p10[k]) p10[k] = 10n ** BigInt(k); return p10[k]; }

  /* num / 10^k  ->  full decimal string, nothing rounded */
  function decFromScaled(num, k) {
    var neg = num < 0n; if (neg) num = -num;
    var t = num.toString();
    while (t.length <= k) t = "0" + t;
    var ip = t.slice(0, t.length - k);
    var fp = k ? t.slice(t.length - k).replace(/0+$/, "") : "";
    return (neg ? "-" : "") + ip + (fp ? "." + fp : "");
  }
  function exactDec(dy) {
    if (dy.e >= 0) return decFromScaled(dy.M << BigInt(dy.e), 0);
    var k = -dy.e;
    return decFromScaled(dy.M * pow5(k), k);
  }
  function sigDigits(s) {
    var t = s.replace(/^-/, "").replace(".", "").replace(/^0+/, "");
    return t.length || 1;
  }
  function clip(s, n) { return s.length > n ? s.slice(0, n) + "…" : s; }
  function decExpOf(s) {
    if (s.charAt(0) === "-") s = s.slice(1);
    var p = s.split("."), ip = p[0], all = ip + (p[1] || ""), i;
    for (i = 0; i < all.length; i++) if (all.charAt(i) !== "0") return ip.length - 1 - i;
    return null;
  }
  /* scientific notation straight from the exact digits — never underflows to 0 */
  function approx(s, d) {
    var dp = d === undefined ? 4 : d;
    var neg = s.charAt(0) === "-"; if (neg) s = s.slice(1);
    var p = s.split("."), ip = p[0], all = ip + (p[1] || ""), first = -1, i;
    for (i = 0; i < all.length; i++) if (all.charAt(i) !== "0") { first = i; break; }
    if (first < 0) return "0";
    var exp = ip.length - 1 - first;
    var digits = all.slice(first).replace(/0+$/, "") || "0";
    var keep = dp + 1;
    if (digits.length > keep) {
      var head = digits.slice(0, keep).split(""), j = keep - 1;
      if (digits.charCodeAt(keep) - 48 >= 5) {
        while (j >= 0) {
          if (head[j] === "9") { head[j] = "0"; j--; }
          else { head[j] = String(Number(head[j]) + 1); break; }
        }
        if (j < 0) { head.unshift("1"); head.pop(); exp++; }
      }
      digits = head.join("").replace(/0+$/, "") || "0";
    }
    var mant = digits.charAt(0) + (digits.length > 1 ? "." + digits.slice(1) : "");
    return (neg ? "-" : "") + mant + "e" + (exp >= 0 ? "+" : "-") + Math.abs(exp);
  }
  /* readable at human scales, scientific once the zeros take over */
  function fmtAt(s, d) {
    var e = decExpOf(s);
    if (e === null) return "0";
    return (e < -7 || e > 21) ? approx(s, d) : clip(s, d + 20);
  }
  /* two band edges must never print the same — widen until they differ */
  function friendlyPair(a, b) {
    for (var d = 6; d <= 26; d += 2) {
      var fa = fmtAt(a, d), fb = fmtAt(b, d);
      if (fa !== fb) return [fa, fb];
    }
    return [fmtAt(a, 26), fmtAt(b, 26)];
  }
  /* a long value inline: keep the real digits while they mean something,
     switch to scientific once a clip would show nothing but leading zeros */
  function fmtValue(s, n) {
    var e = decExpOf(s);
    if (e !== null && (e < -7 || e > 21)) return approx(s, 17);
    return clip(s, n);
  }
  /* plain digits while they stay readable, scientific once they do not */
  function niceNum(s) {
    var e = decExpOf(s);
    if (e === null) return "0";
    return (e >= -4 && e <= 9) ? clip(s, 24) : approx(s, 4);
  }
  function commas(n) { return String(n).replace(/\B(?=(\d{3})+(?!\d))/g, ","); }

  /* ---------- neighbours ---------- */
  function stepBits(u, dir) {
    if (kindOf(u) === "nan") return u;
    var mag = u & ~SIGN, neg = (u & SIGN) !== 0n;
    if (mag === 0n) return dir > 0 ? 1n : (SIGN | 1n);
    if (!neg) {
      if (dir > 0) { return mag === INFBITS ? u : u + 1n; }
      return u - 1n;
    }
    if (dir > 0) return u - 1n;
    return mag === INFBITS ? u : u + 1n;
  }

  /* ---------- decimal literals, kept exact ---------- */
  function parseLit(str) {
    var m = /^([+-]?)(\d*)(?:\.(\d*))?(?:[eE]([+-]?\d+))?$/.exec(str.trim());
    if (!m) return null;
    var ip = m[2] || "", fp = m[3] || "";
    if (!ip && !fp) return null;
    var ex = m[4] ? parseInt(m[4], 10) : 0;
    if (!isFinite(ex) || Math.abs(ex) > 20000 || ip.length + fp.length > 20000) return null;
    var n = BigInt((ip + fp).replace(/^0+(?=\d)/, "") || "0");
    var k = fp.length - ex;
    if (k < 0) { n = n * pow10(-k); k = 0; }
    if (m[1] === "-") n = -n;
    return { n: n, k: k, raw: str.trim() };
  }
  /* literal minus dyadic, exact, as num / 10^k */
  function litMinusDy(L, D) {
    var K = Math.max(0, -D.e);
    var lhs = L.n << BigInt(K);
    var rhs = (D.M << BigInt(D.e + K)) * pow10(L.k);
    return { num: (lhs - rhs) * pow5(K), k: L.k + K };
  }

  /* ---------- state ---------- */
  var state = { u: bitsOf(0.1), lit: parseLit("0.1"), focus: 52, step: 0 };
  var reduce = window.matchMedia && window.matchMedia("(prefers-reduced-motion: reduce)").matches;

  var PRESETS = [
    { label: "0.1", lit: "0.1" },
    { label: "0.2", lit: "0.2" },
    { label: "0.3", lit: "0.3" },
    { label: "0.1 + 0.2", val: 0.1 + 0.2 },
    { label: "1/3", val: 1 / 3 },
    { label: "0.5", lit: "0.5" },
    { label: "1", lit: "1" },
    { label: "2⁵³ + 1", lit: "9007199254740993" },
    { label: "1e21", lit: "1e21" },
    { label: "tiniest", val: 5e-324 },
    { label: "biggest", val: 1.7976931348623157e308 },
    { label: "− 0", val: -0 },
    { label: "Infinity", val: Infinity },
    { label: "NaN", val: NaN }
  ];

  /* ---------- build the switch bank ---------- */
  var byPos = [];
  (function buildBits() {
    function make(host, pos, band) {
      var b = document.createElement("button");
      b.type = "button";
      b.className = "bit";
      b.setAttribute("role", "switch");
      b.setAttribute("aria-checked", "false");
      b.setAttribute("tabindex", "-1");
      b.dataset.pos = String(pos);
      b.dataset.band = band;
      b.textContent = "0";
      host.appendChild(b);
      byPos[pos] = b;
    }
    make($("gSign"), 63, "sign");
    var i;
    for (i = 62; i >= 52; i--) make($("gExp"), i, "exp");
    for (i = 51; i >= 0; i--) make($("gMan"), i, "man");
  })();

  function bandName(pos) { return pos === 63 ? "sign" : pos >= 52 ? "exponent" : "fraction"; }

  function bitLabel(pos) {
    if (pos === 63) return "bit 63, sign";
    if (pos >= 52) return "bit " + pos + ", exponent, worth " + Math.pow(2, pos - 52) + " in the exponent field";
    return "bit " + pos + ", fraction, worth 2 to the power " + (pos - 52);
  }

  function setFocusPos(pos) {
    state.focus = pos;
    for (var i = 0; i < 64; i++) byPos[i].setAttribute("tabindex", i === pos ? "0" : "-1");
  }
  setFocusPos(52);

  function toggle(pos) {
    var nu = state.u ^ (1n << BigInt(pos));
    var x = numOf(nu);
    apply(nu, isFiniteBits(nu) ? parseLit(Object.is(x, -0) ? "-0" : String(x)) : null, false);
    if (!reduce) {
      var el = byPos[pos];
      el.classList.remove("flash");
      void el.offsetWidth;
      el.classList.add("flash");
    }
    say("Bit " + pos + " is now " + ((nu >> BigInt(pos)) & 1n) + ". The number is " + describe(x) + ".");
  }

  $("bank").addEventListener("click", function (e) {
    var t = e.target.closest ? e.target.closest(".bit") : null;
    if (!t) return;
    var pos = Number(t.dataset.pos);
    setFocusPos(pos);
    toggle(pos);
  });
  $("bank").addEventListener("keydown", function (e) {
    var t = e.target;
    if (!t || !t.classList || !t.classList.contains("bit")) return;
    var pos = Number(t.dataset.pos), d = 63 - pos, nd = null;
    if (e.key === "ArrowRight" || e.key === "ArrowDown") nd = d + 1;
    else if (e.key === "ArrowLeft" || e.key === "ArrowUp") nd = d - 1;
    else if (e.key === "Home") nd = 0;
    else if (e.key === "End") nd = 63;
    if (nd === null) return;
    e.preventDefault();
    nd = Math.max(0, Math.min(63, nd));
    setFocusPos(63 - nd);
    byPos[63 - nd].focus();
  });
  $("bank").addEventListener("focusin", function (e) {
    var t = e.target;
    if (!t.classList || !t.classList.contains("bit")) return;
    setFocusPos(Number(t.dataset.pos));
    inspect(Number(t.dataset.pos));
  });
  $("bank").addEventListener("focusout", function () { inspect(null); });
  $("bank").addEventListener("mouseover", function (e) {
    var t = e.target.closest ? e.target.closest(".bit") : null;
    if (t) inspect(Number(t.dataset.pos));
  });
  $("bank").addEventListener("mouseleave", function () {
    var a = document.activeElement;
    if (a && a.classList && a.classList.contains("bit")) inspect(Number(a.dataset.pos));
    else inspect(null);
  });

  function describe(x) {
    if (Number.isNaN(x)) return "not a number";
    if (x === Infinity) return "infinity";
    if (x === -Infinity) return "minus infinity";
    if (Object.is(x, -0)) return "negative zero";
    return String(x);
  }

  function inspect(pos) {
    var host = $("inspect");
    if (pos === null || pos === undefined) {
      host.innerHTML = "Point at a switch to see what it is worth. " +
        "<b>Sign</b> picks the direction, the <b>exponent</b> picks how big, the <b>fraction</b> holds the digits.";
      return;
    }
    var on = ((state.u >> BigInt(pos)) & 1n) === 1n;
    var flipped = numOf(state.u ^ (1n << BigInt(pos)));
    var what;
    if (pos === 63) what = "sign · on means negative";
    else if (pos >= 52) what = "exponent · adds " + commas(Math.pow(2, pos - 52)) + " to the exponent field";
    else what = "fraction · worth 2^" + (pos - 52) + " of the leading 1";
    host.innerHTML = "<b>bit " + pos + "</b> · " + what + " · currently <b>" + (on ? "1" : "0") +
      "</b> · flip it and the number becomes <span class=\"am\">" + esc(describe(flipped)) + "</span>";
  }

  function esc(s) {
    return String(s).replace(/&/g, "&amp;").replace(/</g, "&lt;").replace(/>/g, "&gt;");
  }

  /* ---------- announcements ---------- */
  var sayTimer = null;
  function say(msg) {
    if (sayTimer) window.clearTimeout(sayTimer);
    sayTimer = window.setTimeout(function () { $("live").textContent = msg; }, 320);
  }

  /* ---------- the main render ---------- */
  function apply(u, lit, fromInput) {
    state.u = u & M64;
    state.lit = lit || null;
    render(fromInput);
  }

  function render(fromInput) {
    var u = state.u, x = numOf(u), kind = kindOf(u), finite = isFiniteBits(u);
    var negZero = Object.is(x, -0);

    if (!fromInput) $("numIn").value = state.lit ? state.lit.raw : (finite ? (negZero ? "-0" : String(x)) : describeInput(x));
    $("fWrap").classList.remove("bad");

    /* classification */
    var klass = { norm: "normal number", sub: "subnormal · below the normal range",
                  zero: negZero ? "negative zero" : "zero", inf: x > 0 ? "infinity" : "minus infinity",
                  nan: "not a number" }[kind];
    $("klass").textContent = klass;

    /* prints as / stored as */
    $("rPrint").innerHTML = "<span>" + esc(negZero ? "0" : String(x)) + "</span>" +
      (negZero ? "<span class=\"sub\">JavaScript prints negative zero as plain 0, but the sign switch is on.</span>"
               : "<span class=\"sub\">the shortest text that reads back as exactly this number</span>");

    var dyad = finite ? sigOf(u) : null;
    var exactStr = finite ? (negZero ? "-0" : exactDec(dyad)) : (kind === "inf" ? (x > 0 ? "Infinity" : "-Infinity") : "NaN");
    var printStr = negZero ? "-0" : String(x);
    /* String(x) may be scientific (1e+21); expand it so the two strings line up digit for digit */
    var printFull = printStr;
    if (finite && !negZero) {
      var PL = parseLit(printStr);
      if (PL) printFull = decFromScaled(PL.n, PL.k);
    }
    var i = 0, lim = Math.min(printFull.length, exactStr.length);
    while (i < lim && printFull[i] === exactStr[i]) i++;
    var hasTail = finite && i < exactStr.length;
    $("rExact").innerHTML = finite
      ? "<span class=\"exact\"><span class=\"head\">" + esc(exactStr.slice(0, i)) + "</span><span class=\"tail\">" + esc(exactStr.slice(i)) + "</span></span>" +
        "<span class=\"sub\">" + commas(sigDigits(exactStr)) + (sigDigits(exactStr) === 1 ? " significant digit" : " significant digits") +
        (hasTail ? " — the amber ones are the digits you never asked for" : ", and every one of them is exact") + "</span>"
      : "<span class=\"head\">" + esc(exactStr) + "</span><span class=\"sub\">" +
        (kind === "inf" ? "the exponent switches are all on and the fraction is empty" :
                          "the exponent switches are all on and the fraction is not empty") + "</span>";

    /* chips */
    var chips = [];
    chips.push({ t: kind === "norm" ? "normal" : kind === "sub" ? "subnormal" : kind === "zero" ? "zero" : kind === "inf" ? "infinity" : "not a number",
                 c: kind === "norm" || kind === "zero" ? "" : "amber" });
    var exact = null, dstr = null, ddir = 0;
    if (finite && state.lit) {
      var d = litMinusDy(state.lit, dyad);
      exact = d.num === 0n;
      dstr = decFromScaled(d.num < 0n ? -d.num : d.num, d.k);
      ddir = d.num === 0n ? 0 : (d.num > 0n ? -1 : 1);
      chips.push(exact ? { t: "stored exactly", c: "good" } : { t: "rounded to the nearest one it has", c: "amber" });
    }
    if (kind === "sub") {
      var subBits = mantOf(u).toString(2).length;
      chips.push({ t: "only " + subBits + (subBits === 1 ? " bit" : " bits") + " of digits left, not 53", c: "amber" });
    } else if (finite) {
      chips.push({ t: "53 bits of digits ≈ 15.95 decimal digits", c: "" });
    }
    $("chips").innerHTML = chips.map(function (c) {
      return "<span class=\"chip " + c.c + "\">" + esc(c.t) + "</span>";
    }).join("");

    /* the gap between what you typed and what is kept */
    if (finite && state.lit) {
      $("rwDiff").hidden = false;
      if (exact) {
        $("rDiff").innerHTML = "<span style=\"color:var(--run)\">none — this decimal has an exact binary form</span>" +
          "<span class=\"sub\">every digit you typed survived the trip</span>";
      } else {
        $("rDiff").innerHTML = "<span class=\"tail\" style=\"color:var(--accent)\">" + esc(niceNum(dstr)) + "</span>" +
          "<span class=\"sub\">what is stored sits that far " + (ddir > 0 ? "above" : "below") +
          " the decimal you typed</span>";
      }
    } else {
      $("rwDiff").hidden = true;
    }

    /* a long stored value scrolls — so it has to be reachable from the keyboard,
       and it has to say so, or the clipped last line just looks like a bug */
    var exEl = $("rExact").querySelector(".exact");
    if (exEl) {
      if (exEl.scrollHeight > exEl.clientHeight + 2) {
        exEl.setAttribute("tabindex", "0");
        exEl.setAttribute("role", "region");
        exEl.setAttribute("aria-label", "the full stored value, " + sigDigits(exactStr) + " digits, scrollable");
        var subEl = $("rExact").querySelector(".sub");
        if (subEl) subEl.textContent += " — scroll the box for the rest";
      } else {
        exEl.removeAttribute("tabindex"); exEl.removeAttribute("role"); exEl.removeAttribute("aria-label");
      }
    }

    /* the two steppers must never be dead: at the ends of the format there is
       genuinely nowhere to go, so say that on the button instead of doing nothing */
    var upDead = stepBits(u, 1) === u, dnDead = stepBits(u, -1) === u;
    var noWhy = kind === "nan" ? "not a number has no neighbours to step to"
                               : "there is nothing past infinity";
    var wasFocused = document.activeElement;
    $("stepUp").disabled = upDead;
    $("stepDn").disabled = dnDead;
    $("stepUp").title = upDead ? noWhy : "the next number up that this format can hold";
    $("stepDn").title = dnDead ? noWhy : "the next number down that this format can hold";
    /* keyboard users must not be dropped when the button under them switches off */
    if (wasFocused === $("stepUp") && upDead && !dnDead) $("stepDn").focus();
    else if (wasFocused === $("stepDn") && dnDead && !upDead) $("stepUp").focus();

    renderBits(u);
    renderFormula(u, x, kind, dyad, exactStr);
    renderNeighbours(u, x, kind, finite, dyad);
    $("pattern").innerHTML = "<span>raw pattern</span> 0x" + u.toString(16).toUpperCase().padStart(16, "0");
    $("copyNote").hidden = true;
  }

  function describeInput(x) { return Number.isNaN(x) ? "NaN" : (x > 0 ? "Infinity" : "-Infinity"); }

  function renderBits(u) {
    for (var pos = 0; pos < 64; pos++) {
      var on = ((u >> BigInt(pos)) & 1n) === 1n;
      var b = byPos[pos];
      if (b.textContent !== (on ? "1" : "0")) b.textContent = on ? "1" : "0";
      b.setAttribute("aria-checked", on ? "true" : "false");
      b.setAttribute("aria-label", bitLabel(pos));
    }
  }

  function renderFormula(u, x, kind, dyad, exactStr) {
    var eb = expField(u), m = mantOf(u), neg = (u & SIGN) !== 0n;
    var out;
    if (kind === "nan") {
      out = "exponent = <span class=\"am\">all 11 switches on</span> and the fraction is not empty<br>" +
            "<b>→ not a number</b> — what you get from 0/0 or the square root of a negative";
    } else if (kind === "inf") {
      out = "exponent = <span class=\"am\">all 11 switches on</span>, fraction = empty<br>" +
            "<b>→ " + (neg ? "-" : "") + "infinity</b> — what a number becomes when it grows past the biggest one that fits";
    } else if (kind === "zero") {
      out = "exponent = <span class=\"am\">0</span>, fraction = <b>0</b><br><b>→ " + (neg ? "-0" : "0") +
            "</b> — the sign switch still works, which is why negative zero exists";
    } else if (kind === "sub") {
      var f = exactDec({ M: m, e: -52 });
      out = "(-1)<sup>" + (neg ? 1 : 0) + "</sup> × 2<sup class=\"am\">-1022</sup> × <b>0." +
            esc(f.split(".")[1] || "0") + "</b><br>= <b>" + esc(fmtValue(exactStr, 90)) + "</b>" +
            "<br><span class=\"am\">subnormal</span> — too small for the usual leading 1, so it trades digits for reach";
    } else {
      var lead = exactDec({ M: m + HIDDEN, e: -52 });
      out = "(-1)<sup>" + (neg ? 1 : 0) + "</sup> × 2<sup class=\"am\">(" + eb + " - 1023)</sup> × <b>" + esc(lead) + "</b>" +
            "<br>= (-1)<sup>" + (neg ? 1 : 0) + "</sup> × 2<sup class=\"am\">" + (eb - 1023) + "</sup> × <b>" + esc(lead) + "</b>" +
            "<br>= <b>" + esc(fmtValue(exactStr, 120)) + "</b>";
    }
    $("formula").innerHTML = out;
  }

  /* ---------- neighbours ---------- */
  function powLabel(dy) {
    var M = dy.M < 0n ? -dy.M : dy.M;
    if (M === 0n) return null;
    if ((M & (M - 1n)) !== 0n) return null;
    return "2^" + (dy.e + (M.toString(2).length - 1));
  }

  function renderNeighbours(u, x, kind, finite, dyad) {
    var rail = $("rail"), note = $("railNote"), facts = $("facts");
    var eb = expField(u), mh = mantOf(u).toString(16).toUpperCase().padStart(13, "0");
    var rows = [];

    if (!finite) {
      rail.innerHTML = "";
      rail.style.display = "none";
      note.innerHTML = kind === "nan"
        ? "<b>Not a number</b> has no neighbours — and it is not even equal to itself. Try <code>NaN === NaN</code> and you get <code>false</code>."
        : (x > 0
            ? "<b>Infinity</b> is the end of the line. One step down from here is the biggest number that fits; one step up does not exist."
            : "<b>Minus infinity</b> is the end of the line. One step up from here is the most negative number that fits; one step down does not exist.");
      rows.push(["exponent field", eb + " · all on"]);
      rows.push(["fraction field", "0x" + mh]);
      facts.innerHTML = rows.map(factRow).join("");
      return;
    }

    var dnB = stepBits(u, -1), upB = stepBits(u, 1);
    var dnFin = isFiniteBits(dnB), upFin = isFiniteBits(upB);
    rows.push(["exponent field", kind === "norm" ? eb + " → 2^" + (eb - 1023) : "0 → fixed at 2^-1022"]);
    rows.push(["fraction field", "0x" + mh]);

    if (!dnFin || !upFin) {
      rail.innerHTML = "";
      rail.style.display = "none";
      var neg = (u & SIGN) !== 0n;
      note.innerHTML = neg
        ? "This is the <b>most negative number the format can hold</b>. One step further down and it becomes minus infinity, so there is no neighbour to draw on that side."
        : "This is the <b>largest number the format can hold</b>. One step further up and it becomes infinity, so there is no neighbour to draw on that side.";
      facts.innerHTML = rows.map(factRow).join("");
      return;
    }

    rail.style.display = "";
    var dDn = sigOf(dnB), dUp = sigOf(upB);
    var gapUp = dySub(dUp, dyad), gapDn = dySub(dyad, dDn);
    var gUpS = exactDec(gapUp), gDnS = exactDec(gapDn);
    var pu = powLabel(gapUp), pd = powLabel(gapDn);
    rows.push(["step up from here", niceNum(gUpS) + (pu ? " · " + pu : "")]);
    rows.push(["step down from here", niceNum(gDnS) + (pd ? " · " + pd : "")]);
    facts.innerHTML = rows.map(factRow).join("");

    /* exact positions on the rail */
    var K = Math.max(0, -Math.min(dDn.e, dyad.e, dUp.e));
    var kL = state.lit ? state.lit.k : 0;
    var scale = function (d) { return (d.M << BigInt(d.e + K)) * pow10(kL); };
    var lo = scale(dDn), mid = scale(dyad), hi = scale(dUp);
    var span = hi - lo;
    var tOf = function (v) {
      if (span === 0n) return 0.5;
      return Number((v - lo) * 1000000n / span) / 1e6;
    };
    var tMid = tOf(mid);
    var zLo = tMid / 2, zHi = (tMid + 1) / 2;
    var top = function (t) { return (10 + (1 - t) * 80).toFixed(2) + "%"; };

    var midLow = dyHalf(dyAdd(dDn, dyad)), midHigh = dyHalf(dyAdd(dyad, dUp));
    var html = "<div class=\"stem\"></div>";
    html += "<div class=\"zone\" style=\"top:" + top(zHi) + ";height:" + ((zHi - zLo) * 80).toFixed(2) + "%\"></div>";
    html += node(top(1), clip(String(numOf(upB)), 26), "one step up", false);
    html += node(top(tMid), clip(Object.is(x, -0) ? "-0" : String(x), 26), "this number", true);
    html += node(top(0), clip(String(numOf(dnB)), 26), "one step down", false);

    var exactHit = false, tL = null;
    if (state.lit) {
      var dd = litMinusDy(state.lit, dyad);
      exactHit = dd.num === 0n;
      tL = tOf(state.lit.n << BigInt(K));
    }
    /* only draw the "you typed" marker where it will not sit on top of a node label */
    var markRoom = tL !== null && Math.abs(tL - tMid) > 0.055 && tL > 0.055 && tL < 0.945;
    if (state.lit && !exactHit && markRoom) {
      html += "<div class=\"mark\" title=\"your " + esc(clip(state.lit.raw, 18)) + " lands here\" style=\"top:" + top(tL) + "\"><div class=\"ln\"></div>" +
              "<div class=\"tx\">your " + esc(clip(state.lit.raw, 18)) + " lands here</div></div>";
    }
    rail.innerHTML = html;

    /* the tick line is always safe; its caption slides right if a node label is
       sitting there, and only disappears if even that does not clear it */
    var mkWrap = rail.querySelector(".mark"), mk = rail.querySelector(".mark .tx");
    if (mk) {
      var labs = rail.querySelectorAll(".node .lab");
      var clashes = function () {
        var mb = mk.getBoundingClientRect(), j;
        for (j = 0; j < labs.length; j++) {
          var lb = labs[j].getBoundingClientRect();
          if (mb.left < lb.right && lb.left < mb.right && mb.top < lb.bottom && lb.top < mb.bottom) return true;
        }
        return false;
      };
      if (clashes()) {
        mkWrap.classList.add("pushright");
        if (clashes()) mk.style.display = "none";
      }
    }

    var edges = friendlyPair(exactDec(midLow), exactDec(midHigh));
    var band = "Every decimal from <b>" + esc(edges[0]) + "</b> up to <b>" +
      esc(edges[1]) + "</b> is filed under this one number. ";
    if (state.lit && exactHit) {
      band += "Yours lands right on it — no rounding needed.";
    } else if (state.lit && tL !== null && tL >= 0 && tL <= 1) {
      band += "Your decimal sits inside that band, so it gets rounded to the tick beside it.";
    } else {
      band += "Nothing exists in the gaps: these three numbers are next-door neighbours.";
    }
    note.innerHTML = band;
  }

  function factRow(r) {
    return "<li><span class=\"k-lab\">" + esc(r[0]) + "</span><span class=\"v\">" + esc(r[1]) + "</span></li>";
  }
  function node(topPct, val, cap, self) {
    return "<div class=\"node" + (self ? " self" : "") + "\" style=\"top:" + topPct + "\">" +
      "<div class=\"dot\"></div><div class=\"lab\">" + esc(val) + "<em>" + esc(cap) + "</em></div></div>";
  }

  /* ---------- the 0.1 + 0.2 story, computed live ---------- */
  var STEPS = (function () {
    var A = 0.1, B = 0.2, C = 0.3, S = A + B;
    var dA = sigOf(bitsOf(A)), dB = sigOf(bitsOf(B)), dC = sigOf(bitsOf(C)), dS = sigOf(bitsOf(S));
    var exactSum = dyAdd(dA, dB);
    var trimmed = exactSum.M;
    while (trimmed % 2n === 0n && trimmed !== 0n) trimmed /= 2n;
    var width = trimmed.toString(2).length;
    var adjacent = stepBits(bitsOf(C), 1) === bitsOf(S);
    var tol = Math.abs(A + B - C) < 1e-9;
    var scaled = (A * 10 + B * 10) / 10 === C;
    var cents = (10 + 20) === 30;

    return [
      { title: "Two innocent decimals go in", load: A,
        say: "Neither 0.1 nor 0.2 has an exact form in binary — like one third in decimal, the digits never stop. Both get filed under the nearest number the format owns, and both land a hair <b>above</b> what you asked for.",
        items: [
          ["you write", "0.1", ""], ["the machine keeps", exactDec(dA), "hi"],
          ["you write", "0.2", ""], ["the machine keeps", exactDec(dB), "hi"]
        ] },
      { title: "Add the two stored values, exactly", load: null,
        say: "This next line is the true sum of the two numbers above — no rounding yet. The trouble is its digits need <b>" + width + " bits</b>, and a double only has room for 53.",
        items: [
          ["exact sum of what was stored", exactDec(exactSum), "hi"],
          ["bits needed for those digits", width + " — one too many", "bad"]
        ] },
      { title: "Squeeze it back into 53 bits", load: S,
        say: "The sum gets rounded to the nearest number that fits, which nudges it <b>up</b> one more time. Now print it and the extra digits finally surface.",
        items: [
          ["nearest number that fits", exactDec(dS), "hi"],
          ["so it prints as", String(S), "hi"]
        ] },
      { title: "Meanwhile, plain 0.3", load: C,
        say: "Typed on its own, 0.3 rounds <b>down</b> instead. Two different roads, two different numbers — and they land side by side with nothing in between.",
        items: [
          ["0.3 is stored as", exactDec(dC), "hi"],
          ["distance between them", adjacent ? "one single step — the smallest gap that exists here" : "one step", ""]
        ] },
      { title: "So the equality check fails", load: S,
        say: "Nothing is broken and no bug was introduced. The two values really are different numbers, by the smallest amount this format can express.",
        items: [
          ["0.1 + 0.2 === 0.3", String(A + B === C), "bad", 1],
          ["0.1 + 0.2", String(S), "", 1],
          ["0.1 + 0.2 === " + String(S), String(A + B === S), "good", 1]
        ] },
      { title: "What to do instead", load: null,
        say: "Compare with a tolerance, scale into whole units before you divide, or keep money in whole cents and only format it at the very end.",
        items: [
          ["Math.abs(0.1 + 0.2 - 0.3) < 1e-9", String(tol), tol ? "good" : "bad", 1],
          ["(0.1*10 + 0.2*10) / 10 === 0.3", String(scaled), scaled ? "good" : "bad", 1],
          ["count in whole cents: 10 + 20 === 30", String(cents), cents ? "good" : "bad", 1]
        ] }
    ];
  })();

  function renderStep() {
    var s = STEPS[state.step], host = $("step");
    var html = "<h3>" + esc(s.title) + "</h3><p class=\"say\">" + s.say + "</p><div class=\"slist\">";
    html += s.items.map(function (it) {
      return "<div class=\"sitem " + it[2] + "\"><span class=\"t" + (it[3] ? " code" : "") + "\">" + esc(it[0]) + "</span>" +
        "<span class=\"m\">" + esc(it[1]) + "</span></div>";
    }).join("");
    html += "</div>";
    host.innerHTML = html;
    if (!reduce) { host.classList.remove("swap"); void host.offsetWidth; host.classList.add("swap"); }

    $("prevBtn").disabled = state.step === 0;
    $("nextBtn").disabled = state.step === STEPS.length - 1;
    $("stepCount").textContent = "step " + (state.step + 1) + " of " + STEPS.length;
    var lb = $("loadBtn");
    if (s.load === null) { lb.hidden = true; }
    else { lb.hidden = false; lb.textContent = "load " + String(s.load) + " into the switches"; }
    var ds = $("dots").children;
    for (var i = 0; i < ds.length; i++) ds[i].setAttribute("aria-current", i === state.step ? "true" : "false");
    $("stepLive").textContent = "Step " + (state.step + 1) + " of " + STEPS.length + ": " + s.title;
  }

  function goStep(n) {
    state.step = Math.max(0, Math.min(STEPS.length - 1, n));
    renderStep();
  }

  /* ---------- reading what the visitor types ---------- */
  var HINT = "plain (0.1) · scientific (5e-324) · or a raw pattern (0x3FB999999999999A)";

  function fromInputText(raw) {
    raw = String(raw).trim();
    if (!raw) return null;
    if (/^0x[0-9a-fA-F]{1,16}$/.test(raw)) return { u: BigInt(raw) & M64, lit: null };
    if (/^[+-]?infinity$/i.test(raw)) return { u: bitsOf(raw.charAt(0) === "-" ? -Infinity : Infinity), lit: null };
    if (/^nan$/i.test(raw)) return { u: bitsOf(NaN), lit: null };
    var L = parseLit(raw);
    if (!L) return null;
    var x = Number(raw);
    if (!isFinite(x)) return { u: bitsOf(x), lit: null };
    return { u: bitsOf(x), lit: L };
  }

  $("numIn").addEventListener("input", function () {
    if (!this.value.trim()) {
      $("fWrap").classList.remove("bad");
      $("inHint").textContent = HINT;
      $("inHint").classList.remove("bad");
      return;
    }
    var r = fromInputText(this.value);
    if (!r) {
      $("fWrap").classList.add("bad");
      $("inHint").textContent = "cannot read that one — try 0.1, 5e-324, or 0x3FB999999999999A";
      $("inHint").classList.add("bad");
      return;
    }
    $("inHint").textContent = HINT;
    $("inHint").classList.remove("bad");
    apply(r.u, r.lit, true);
    say("Now showing " + describe(numOf(r.u)) + ".");
  });

  function setNumber(spec) {
    var r = spec.lit !== undefined ? fromInputText(spec.lit) : null;
    if (!r) {
      var v = spec.val;
      r = { u: bitsOf(v), lit: isFinite(v) ? parseLit(Object.is(v, -0) ? "-0" : String(v)) : null };
    }
    $("inHint").textContent = HINT;
    $("inHint").classList.remove("bad");
    apply(r.u, r.lit, false);
    say(describe(numOf(r.u)) + " loaded.");
  }

  (function buildPresets() {
    var host = $("presets");
    PRESETS.forEach(function (p) {
      var b = document.createElement("button");
      b.type = "button";
      b.className = "btn";
      b.textContent = p.label;
      b.addEventListener("click", function () { setNumber(p); });
      host.appendChild(b);
    });
  })();

  $("stepUp").addEventListener("click", function () { nudge(1); });
  $("stepDn").addEventListener("click", function () { nudge(-1); });
  function nudge(dir) {
    var nu = stepBits(state.u, dir);
    if (nu === state.u) { say("There is nothing beyond this one."); return; }
    var x = numOf(nu);
    apply(nu, isFiniteBits(nu) ? parseLit(Object.is(x, -0) ? "-0" : String(x)) : null, false);
    say("One step " + (dir > 0 ? "up" : "down") + ": " + describe(x) + ".");
  }

  /* ---------- story controls ---------- */
  (function buildDots() {
    var host = $("dots");
    STEPS.forEach(function (s, i) {
      var b = document.createElement("button");
      b.type = "button";
      b.className = "dot-b";
      b.textContent = String(i + 1);
      b.setAttribute("aria-label", "step " + (i + 1) + ": " + s.title);
      b.addEventListener("click", function () { goStep(i); });
      host.appendChild(b);
    });
  })();
  $("prevBtn").addEventListener("click", function () { goStep(state.step - 1); });
  $("nextBtn").addEventListener("click", function () { goStep(state.step + 1); });
  $("loadBtn").addEventListener("click", function () {
    var v = STEPS[state.step].load;
    if (v === null) return;
    setNumber({ val: v });
    var p = $("numIn");
    if (p.scrollIntoView) p.scrollIntoView({ block: "center", behavior: reduce ? "auto" : "smooth" });
  });

  /* ---------- copy the raw pattern ---------- */
  $("copyBtn").addEventListener("click", function () {
    var txt = "0x" + state.u.toString(16).toUpperCase().padStart(16, "0");
    var note = $("copyNote");
    function ok() { note.hidden = false; note.className = "chip good"; note.textContent = "copied " + txt; }
    function no() { note.hidden = false; note.className = "chip"; note.textContent = txt + " — select it to copy"; }
    try {
      if (navigator.clipboard && navigator.clipboard.writeText) {
        navigator.clipboard.writeText(txt).then(ok, no);
        return;
      }
    } catch (err) { /* fall through */ }
    var ta = document.createElement("textarea");
    ta.value = txt;
    ta.setAttribute("readonly", "");
    ta.style.position = "fixed";
    ta.style.opacity = "0";
    document.body.appendChild(ta);
    ta.select();
    var done = false;
    try { done = document.execCommand("copy"); } catch (err2) { done = false; }
    document.body.removeChild(ta);
    if (done) ok(); else no();
  });

  /* ---------- platform theme ---------- */
  window.addEventListener("message", function (e) {
    var d = e && e.data;
    if (d && d.type === "sloprun:theme" && (d.theme === "light" || d.theme === "dark")) {
      document.documentElement.setAttribute("data-theme", d.theme);
    }
  });

  /* ---------- boot ---------- */
  $("numIn").value = "0.1";
  inspect(null);
  render(false);
  renderStep();
})();
</script>
</body>
</html>