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<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="UTF-8" />
<meta name="viewport" content="width=device-width, initial-scale=1.0" />
<title>9178 论证器</title>
<link rel="icon" href="favicon.ico" type="image/x-icon">
<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.16.9/dist/katex.min.css" crossorigin="anonymous">
<script defer src="https://cdn.jsdelivr.net/npm/katex@0.16.9/dist/katex.min.js" crossorigin="anonymous"></script>
<style>
:root {
--bg: #0f172a;
--panel: #111827;
--card: #1f2937;
--accent: #f97316;
--accent-2: #22d3ee;
--text: #e5e7eb;
--muted: #94a3b8;
--border: #1e293b;
}
* {
box-sizing: border-box;
}
body {
margin: 0;
min-height: 100vh;
font-family: "Sora", "Manrope", "Segoe UI", sans-serif;
background: radial-gradient(120% 120% at 20% 20%, rgba(34, 211, 238, 0.18), transparent),
radial-gradient(140% 140% at 80% 0%, rgba(249, 115, 22, 0.12), transparent),
var(--bg);
color: var(--text);
display: flex;
align-items: center;
justify-content: center;
padding: 32px 16px;
}
.shell {
width: min(1080px, 100%);
background: var(--panel);
border: 1px solid var(--border);
border-radius: 18px;
box-shadow: 0 30px 80px rgba(0, 0, 0, 0.35);
padding: 24px;
display: grid;
grid-template-columns: 1.1fr 0.9fr;
gap: 18px;
}
@media (max-width: 900px) {
.shell {
grid-template-columns: 1fr;
}
}
h1 {
margin: 0 0 8px;
font-size: 26px;
letter-spacing: 0.3px;
}
.muted {
color: var(--muted);
font-size: 14px;
}
.card {
background: var(--card);
border: 1px solid var(--border);
border-radius: 14px;
padding: 16px;
}
label {
display: block;
margin-bottom: 6px;
font-weight: 600;
}
input[type="number"],
select {
width: 100%;
padding: 10px 12px;
border-radius: 10px;
border: 1px solid var(--border);
background: #0b1220;
color: var(--text);
font-size: 15px;
}
button {
border: none;
background: linear-gradient(120deg, #40ffd4, #ff80dd);
color: #0f172a;
font-weight: 700;
padding: 12px 16px;
border-radius: 12px;
cursor: pointer;
font-size: 15px;
transition: transform 0.1s ease, box-shadow 0.1s ease;
width: 100%;
}
button:active {
transform: translateY(1px);
}
.row {
display: flex;
gap: 12px;
align-items: center;
}
.row>* {
flex: 1;
}
.output {
font-family: "JetBrains Mono", "SFMono-Regular", Consolas, monospace;
background: #0b1220;
padding: 12px;
border-radius: 10px;
border: 1px solid var(--border);
min-height: 80px;
line-height: 1.4;
overflow-wrap: anywhere;
}
.pill {
display: inline-flex;
align-items: center;
gap: 6px;
padding: 6px 10px;
border-radius: 999px;
background: rgba(34, 211, 238, 0.12);
color: var(--text);
font-size: 13px;
}
.grid {
display: grid;
gap: 10px;
}
.status {
font-size: 14px;
color: var(--muted);
}
.latex-box {
margin-top: 12px;
padding: 12px;
border-radius: 10px;
background: #0b1220;
border: 1px solid var(--border);
min-height: 60px;
}
.latex-card {
display: flex;
flex-direction: column;
height: 100%;
}
.footer-link {
align-self: flex-end;
margin-top: auto;
font-size: 13px;
}
.warn {
color: #fbbf24;
}
.error {
color: #f87171;
}
</style>
</head>
<body>
<div class="shell">
<div class="grid" style="gap:14px;">
<div>
<h1>9178 论证器</h1>
<div class="muted">用 9,1,7,8 + - * / sqrt exp ln 与整数幂 [-1,2] 近似表示目标数</div>
</div>
<div class="card grid">
<div>
<label for="target">目标值</label>
<input id="target" type="number" step="0.001" value="3.14" />
</div>
<div class="row">
<div>
<label for="blocks">最多块数</label>
<input id="blocks" type="number" min="1" max="8" step="1" value="7" />
</div>
<div>
<label for="tol">容差</label>
<input id="tol" type="number" step="0.0001" value="0.0001" />
</div>
</div>
<button id="solve">开始求解</button>
<div class="status" id="status">等待输入…</div>
</div>
<div class="card grid">
<div class="row">
<div class="pill" id="delta">Δ = —</div>
<div class="pill" id="value">≈ —</div>
</div>
<div class="output" id="expr">结果会显示在这里</div>
</div>
</div>
<div class="card latex-card">
<div style="margin-top:12px; font-weight:700;">LaTeX</div>
<div class="latex-box" id="latex-box">等待表达式…</div>
<div class="muted footer-link">https://github.com/LemonQu-GIT/Express9178</div>
</div>
</div>
<script>
// Core numeric helpers
const LIMIT = 10n ** 12n;
const EPS = 1e-3;
const MAX_PER_LEVEL = 50000;
const baseOps = ['+', '-', '*', '/'];
const powExponents = [-1, 2];
const unaryFns = ['sqrt', 'exp', 'ln'];
const prec = (op) => {
if (op === '+' || op === '-') return 1;
if (op === '*' || op === '/') return 2;
if (op === '^') return 3;
return 4;
};
const gcd = (a, b) => {
a = a < 0n ? -a : a;
b = b < 0n ? -b : b;
while (b !== 0n) {
const t = a % b;
a = b; b = t;
}
return a === 0n ? 1n : a;
};
const makeFrac = (n, d) => {
if (d === 0n) return null;
if (n === 0n) return { n: 0n, d: 1n };
const g = gcd(n, d);
n /= g; d /= g;
if (d < 0n) { n = -n; d = -d; }
if (n > LIMIT || n < -LIMIT || d > LIMIT || d < -LIMIT) return null;
return { n, d };
};
const fracToNumber = (f) => Number(f.n) / Number(f.d);
const applyFracBinary = (op, a, b) => {
let n = 0n, d = 0n;
switch (op) {
case '+': n = a.n * b.d + b.n * a.d; d = a.d * b.d; break;
case '-': n = a.n * b.d - b.n * a.d; d = a.d * b.d; break;
case '*': n = a.n * b.n; d = a.d * b.d; break;
case '/': if (b.n === 0n) return null; n = a.n * b.d; d = a.d * b.n; break;
default: return null;
}
return makeFrac(n, d);
};
const powFrac = (a, k) => {
if (k === 0) return { n: 1n, d: 1n };
let numPow = 1n, denPow = 1n;
const absK = Math.abs(k);
for (let i = 0; i < absK; i++) {
numPow *= a.n; denPow *= a.d;
if (numPow > LIMIT || numPow < -LIMIT || denPow > LIMIT || denPow < -LIMIT) return null;
}
const res = k > 0 ? { n: numPow, d: denPow } : { n: denPow, d: numPow };
return makeFrac(res.n, res.d);
};
const keyFromFrac = (f) => `F:${f.n}/${f.d}`;
const keyFromFloat = (v) => {
if (!Number.isFinite(v)) return null;
const q = Math.round(v / EPS) * EPS;
return `N:${q}`;
};
const combineExprBinary = (left, op, right) => {
const p = prec(op);
const needL = !left.atomic && left.prec < p;
const needR = !right.atomic && (right.prec < p || (right.prec === p && (op === '-' || op === '/' || op === '^')));
const le = needL ? `(${left.expr})` : left.expr;
const re = needR ? `(${right.expr})` : right.expr;
return { expr: `${le}${op}${re}`, prec: p, atomic: false };
};
const combineExprUnary = (fn, child) => ({ expr: `${fn}(${child.expr})`, prec: 4, atomic: false });
const applyUnary = (fn, node) => {
const v = node.val;
if (!Number.isFinite(v)) return null;
if (fn === 'sqrt' && v < 0) return null;
if (fn === 'ln' && v <= 0) return null;
let res;
switch (fn) {
case 'sqrt': res = Math.sqrt(v); break;
case 'exp': res = Math.exp(v); break;
case 'ln': res = Math.log(v); break;
default: return null;
}
if (!Number.isFinite(res)) return null;
const exprPart = combineExprUnary(fn, node);
return { expr: exprPart.expr, prec: exprPart.prec, atomic: false, val: res, frac: null, fromUnary: true };
};
const applyPow = (node, k) => {
if (!Number.isFinite(node.val)) return null;
const v = Math.pow(node.val, k);
if (!Number.isFinite(v)) return null;
const baseNeedsPar = !node.atomic && node.prec < 3;
const baseStr = baseNeedsPar ? `(${node.expr})` : node.expr;
const expr = `${baseStr}^${k}`;
let frac = null;
if (node.frac) frac = powFrac(node.frac, k);
return { expr, prec: 3, atomic: false, val: v, frac, fromUnary: true };
};
const normalizeNode = (raw) => {
const frac = raw.frac ?? null;
const val = frac ? fracToNumber(frac) : raw.val;
return { expr: raw.expr, prec: raw.prec, atomic: raw.atomic, val, frac, fromUnary: Boolean(raw.fromUnary) };
};
const keyForNode = (node) => node.frac ? keyFromFrac(node.frac) : keyFromFloat(node.val);
// Load precomputed single-block data
async function loadBase() {
const resp = await fetch('cache.json');
if (!resp.ok) throw new Error('无法加载 cache.json');
const wrapped = await resp.json();
const data = Array.isArray(wrapped) ? wrapped : wrapped.data;
return data.map((x) => ({
expr: String(x.expr),
prec: Number(x.prec ?? 2),
atomic: Boolean(x.atomic),
frac: { n: BigInt(x.n), d: BigInt(x.d) },
val: Number(BigInt(x.n)) / Number(BigInt(x.d)),
fromUnary: false,
}));
}
function expandUnaries(node, seen, bucket) {
if (node.fromUnary) return;
for (const fn of unaryFns) {
const u = applyUnary(fn, node);
if (!u) continue;
const key = keyForNode(u);
if (!key || seen.has(key)) continue;
seen.add(key);
bucket.push(u);
}
for (const k of powExponents) {
const p = applyPow(node, k);
if (!p) continue;
const key = keyForNode(p);
if (!key || seen.has(key)) continue;
seen.add(key);
bucket.push(p);
}
}
function computeFloatBinary(op, a, b) {
switch (op) {
case '+': return a + b;
case '-': return a - b;
case '*': return a * b;
case '/': return b === 0 ? NaN : a / b;
default: return NaN;
}
}
function solve(target, base, maxBlocks = 7, tol = 1e-4) {
const levels = Array.from({ length: maxBlocks + 1 }, () => new Map());
const seen = new Set();
const targetNum = Number(target);
const frontier = [];
for (const r of base) {
const k = keyFromFrac(r.frac);
levels[1].set(k, { ...r, atomic: false, fromUnary: false });
seen.add(k);
frontier.push({ ...r, atomic: false, fromUnary: false });
if (Math.abs(r.val - targetNum) < tol) return r.expr;
}
let idx = 0;
while (idx < frontier.length) {
const node = frontier[idx++];
const bucket = [];
expandUnaries(node, seen, bucket);
for (const u of bucket) {
const ku = keyForNode(u);
if (!ku) continue;
if (!levels[1].has(ku)) levels[1].set(ku, u);
frontier.push(u);
if (Math.abs(u.val - targetNum) < tol) return u.expr;
}
}
for (let b = 2; b <= maxBlocks; b++) {
for (const [, prev] of levels[b - 1]) {
for (const blk of base) {
for (const op of baseOps) {
const fracVal = prev.frac && blk.frac ? applyFracBinary(op, prev.frac, blk.frac) : null;
const valNum = fracVal ? fracToNumber(fracVal) : computeFloatBinary(op, prev.val, blk.val);
if (!Number.isFinite(valNum)) continue;
const exprPart = combineExprBinary(prev, op, blk);
const node = normalizeNode({ expr: exprPart.expr, prec: exprPart.prec, atomic: false, val: valNum, frac: fracVal, fromUnary: false });
const k = keyForNode(node);
if (!k || seen.has(k)) continue;
seen.add(k);
levels[b].set(k, node);
if (Math.abs(node.val - targetNum) < tol) return node.expr;
const bucket = [];
expandUnaries(node, seen, bucket);
for (const u of bucket) {
const ku = keyForNode(u);
if (!ku) continue;
levels[b].set(ku, u);
if (Math.abs(u.val - targetNum) < tol) return u.expr;
}
}
}
}
if (levels[b].size > MAX_PER_LEVEL) {
const arr = Array.from(levels[b].values());
arr.sort((a, bnode) => Math.abs(a.val - targetNum) - Math.abs(bnode.val - targetNum));
const trimmed = arr.slice(0, MAX_PER_LEVEL);
const m = new Map();
for (const n of trimmed) {
const kk = keyForNode(n);
if (kk) m.set(kk, n);
}
levels[b] = m;
}
}
return null;
}
// UI wiring
const targetEl = document.getElementById('target');
const blocksEl = document.getElementById('blocks');
const tolEl = document.getElementById('tol');
const statusEl = document.getElementById('status');
const exprEl = document.getElementById('expr');
const deltaEl = document.getElementById('delta');
const valueEl = document.getElementById('value');
const btn = document.getElementById('solve');
const latexBox = document.getElementById('latex-box');
let baseData = null;
async function init() {
statusEl.textContent = '加载单块枚举中…';
try {
baseData = await loadBase();
statusEl.textContent = `已加载 ${baseData.length} 条单块结果。`;
} catch (e) {
statusEl.textContent = '加载失败：' + e.message;
statusEl.classList.add('error');
}
}
function formatDelta(actual, target) {
const d = actual - target;
return `Δ=${d.toExponential(3)}`;
}
// --- Minimal parser to convert expression string -> LaTeX ---
function tokenizeExpr(src) {
const tokens = [];
const re = /\s+|\d+(?:\.\d+)?|[a-zA-Z]+|[+\-*/^()]/y;
let m;
while ((m = re.exec(src)) !== null) {
const [lex] = m;
if (/^\s+$/.test(lex)) continue;
if (/^\d/.test(lex)) tokens.push({ type: 'num', value: lex });
else if (/^[a-zA-Z]+$/.test(lex)) tokens.push({ type: 'id', value: lex });
else tokens.push({ type: 'op', value: lex });
}
tokens.push({ type: 'eof', value: '' });
return tokens;
}
function parseToAst(src) {
const tokens = tokenizeExpr(src);
let i = 0;
const peek = () => tokens[i];
const eat = (v) => {
if (peek().value === v) { i++; return true; }
return false;
};
function parsePrimary() {
const t = peek();
if (t.type === 'num') { i++; return { kind: 'num', value: t.value, prec: 4 }; }
if (t.type === 'id') {
i++;
if (eat('(')) {
const arg = parseExpr();
if (!eat(')')) throw new Error('缺少 )');
return { kind: 'call', name: t.value, arg, prec: 4 };
}
return { kind: 'id', name: t.value, prec: 4 };
}
if (eat('(')) {
const e = parseExpr();
if (!eat(')')) throw new Error('缺少 )');
return { kind: 'group', child: e, prec: e.prec };
}
throw new Error('无法解析 token ' + t.value);
}
function parseUnary() {
if (eat('+')) return { kind: 'unary', op: '+', child: parseUnary(), prec: 3 };
if (eat('-')) return { kind: 'unary', op: '-', child: parseUnary(), prec: 3 };
return parsePrimary();
}
function parsePow() {
let node = parseUnary();
while (eat('^')) {
const right = parseUnary();
node = { kind: 'bin', op: '^', left: node, right, prec: 3 };
}
return node;
}
function parseTerm() {
let node = parsePow();
while (peek().value === '*' || peek().value === '/') {
const op = peek().value; i++;
const right = parsePow();
node = { kind: 'bin', op, left: node, right, prec: 2 };
}
return node;
}
function parseExpr() {
let node = parseTerm();
while (peek().value === '+' || peek().value === '-') {
const op = peek().value; i++;
const right = parseTerm();
node = { kind: 'bin', op, left: node, right, prec: 1 };
}
return node;
}
const ast = parseExpr();
if (peek().type !== 'eof') throw new Error('多余输入');
return ast;
}
function wrap(child, parentPrec) {
if (!child || child.prec >= parentPrec) return child.tex;
return `\\left(${child.tex}\\right)`;
}
function astToLatex(node) {
if (node.kind === 'num') return { tex: node.value, prec: node.prec };
if (node.kind === 'id') return { tex: node.name, prec: node.prec };
if (node.kind === 'group') return { tex: `\\left(${astToLatex(node.child).tex}\\right)`, prec: node.child.prec };
if (node.kind === 'unary') {
const inner = astToLatex(node.child);
return { tex: `${node.op}${wrap(inner, 3)}`, prec: node.prec };
}
if (node.kind === 'call') {
const arg = astToLatex(node.arg);
if (node.name === 'sqrt') return { tex: `\\sqrt{${arg.tex}}`, prec: 4 };
if (node.name === 'exp') return { tex: `e^{${arg.tex}}`, prec: 4 };
if (node.name === 'ln') return { tex: `\\ln\\left(${arg.tex}\\right)`, prec: 4 };
return { tex: `${node.name}\\left(${arg.tex}\\right)`, prec: 4 };
}
if (node.kind === 'bin') {
const l = astToLatex(node.left);
const r = astToLatex(node.right);
if (node.op === '+') return { tex: `${wrap(l, 1)}+${wrap(r, 1)}`, prec: 1 };
if (node.op === '-') return { tex: `${wrap(l, 1)}-${wrap(r, 1)}`, prec: 1 };
if (node.op === '*') return { tex: `${wrap(l, 2)}\\cdot${wrap(r, 2)}`, prec: 2 };
if (node.op === '/') return { tex: `\\frac{${l.tex}}{${r.tex}}`, prec: 2 };
if (node.op === '^') return { tex: `${wrap(l, 3)}^{${r.tex}}`, prec: 3 };
}
throw new Error('未知节点');
}
function renderLatex(target, expr) {
if (!latexBox) return;
if (!window.katex) {
latexBox.textContent = `${target} = ${expr}（KaTeX 未加载，使用纯文本）`;
return;
}
try {
const ast = parseToAst(expr);
const tex = astToLatex(ast).tex;
const full = `${target} = ${tex}`;
window.katex.render(full, latexBox, { throwOnError: false });
} catch (e) {
latexBox.textContent = 'LaTeX 渲染失败: ' + e.message;
}
}
btn.addEventListener('click', () => {
if (!baseData) { statusEl.textContent = '未加载到基础数据。'; return; }
const target = Number(targetEl.value);
const maxBlocks = Number(blocksEl.value);
const tol = Number(tolEl.value);
if (!Number.isFinite(target) || !Number.isFinite(maxBlocks) || !Number.isFinite(tol)) {
statusEl.textContent = '输入不合法。';
return;
}
statusEl.textContent = '搜索中…';
exprEl.textContent = '计算中…';
deltaEl.textContent = 'Δ = …';
valueEl.textContent = '≈ …';
latexBox.textContent = '渲染中…';
// Slight delay to allow UI refresh
setTimeout(() => {
const expr = solve(target, baseData, maxBlocks, tol);
if (expr) {
exprEl.textContent = expr;
// Evaluate expression numerically (unsafe but controlled input)
try {
const jsExpr = expr.replace(/\^/g, '**');
const val = Function(`with (Math) { const ln = Math.log; return (${jsExpr}); }`)();
deltaEl.textContent = formatDelta(val, target);
valueEl.textContent = `≈ ${val}`;
statusEl.textContent = '完成';
renderLatex(target, expr);
} catch (e) {
deltaEl.textContent = 'Δ = —';
valueEl.textContent = '≈ —';
statusEl.textContent = '完成（表达式回算失败）';
latexBox.textContent = 'LaTeX 渲染失败';
}
} else {
statusEl.textContent = '未在当前限制内找到表达式。';
exprEl.textContent = '未找到';
deltaEl.textContent = 'Δ = —';
valueEl.textContent = '≈ —';
latexBox.textContent = '未找到表达式';
}
}, 10);
});
init();
</script>
</body>
</html>