class QQ { constructor(p, q = 1n, reduced = false) { if (q === 0n) throw new Error(`Denominator cannot be zero: ${p}/${q}`); this.p = q < 0n ? -BigInt(p) : BigInt(p); this.q = q < 0n ? -BigInt(q) : BigInt(q); if (!reduced) this.simplify(); this.value = Number(this.p) / Number(this.q); this.cf = null; this.cfrac = this.continuedFraction(); } static toQQ(value) { if (value instanceof QQ) return value; if (typeof value === 'bigint') return new QQ(value, 1n); if (typeof value === 'number' && Number.isInteger(value)) { return new QQ(BigInt(value), 1n); } throw new Error(`Invalid input type (${typeof value}). Expected QQ, bigint, or integer number.`); } simplify() { const gcd = (a, b) => { while (b !== 0n) [a, b] = [b, a % b]; return a; }; const divisor = gcd(this.p < 0n ? -this.p : this.p, this.q); this.p /= divisor; this.q /= divisor; } sign() { return this.p < 0n ? -1n : 1n } add(other) { other = QQ.toQQ(other); return new QQ(this.p * other.q + other.p * this.q, this.q * other.q); } subtract(other) { other = QQ.toQQ(other); return new QQ(this.p * other.q - other.p * this.q, this.q * other.q); } multiply(other) { other = QQ.toQQ(other); return new QQ(this.p * other.p, this.q * other.q); } divide(other) { return this.multiply(QQ.toQQ(other).inverse()); } inverse() { return new QQ(this.q, this.p, true); } power(k) { if (k === 0n) return new QQ(1n, 1n); // Any number to the power of 0 is 1 if (k > 0n) return new QQ(this.p ** k, this.q ** k, true); // For negative k, invert the fraction and use the positive exponent return new QQ(this.q ** -k, this.p ** -k, true); } lessThan(other) { other = QQ.toQQ(other); return this.p * other.q < other.p * this.q; } equalTo(other) { other = QQ.toQQ(other); return this.p * other.q === other.p * this.q; } greaterThan(other) { other = QQ.toQQ(other); return this.p * other.q > other.p * this.q; } continuedFraction() { let p = this.p < 0n ? this.p % this.q + this.q : this.p; let q = this.q; const cfrac = []; while (q !== 0n) { cfrac.push(p / q); [p, q] = [q, p % q]; } if (this.p < 0n) cfrac[0] = this.p / this.q - 1n; this.cf = `[${cfrac[0]};${cfrac.slice(1).join(',')}]`; this.cfrac = cfrac; return cfrac; } static from_cfrac(cfrac) { if (cfrac.length === 0) throw new Error('Continued fraction list cannot be empty'); let p = 1n, q = 0n; // Initialize p and q for backward calculation for (let i = cfrac.length - 1; i >= 0; i--) { [p, q] = [BigInt(cfrac[i]) * p + q, p]; // Correct order of swapping } return new QQ(p, q); // Return the rational number QQ(p, q) } approximant(max, direction = null) { if (this.q <= max) { return new QQ(this.p, this.q, true); } const a = this.continuedFraction(); const c = []; let pM2 = 0n, pM1 = 1n, qM2 = 1n, qM1 = 0n; for (const ai of a) { const steps = ai < 0n ? -ai : ai; for (let m = 1n; m <= steps; m++) { const p = m * pM1 + pM2, q = m * qM1 + qM2; if (q > max) break; c.push([p, q]); } const p0 = ai * pM1 + pM2, q0 = ai * qM1 + qM2; pM2 = pM1; qM2 = qM1; pM1 = p0; qM1 = q0; } if (!c.length) { return new QQ(0n, 1n); } let [bestN, bestD] = c[0], abs = (x) => x < 0n ? -x : x; let bestErr = abs(this.p * bestD - this.q * bestN); for (let i = 1; i < c.length; i++) { const [n, d] = c[i], diff = this.p * d - this.q * n; if (direction === 'down' && diff < 0n) continue; if (direction === 'up' && diff > 0n) continue; const err = abs(diff); if (err < bestErr || (err === bestErr && d < bestD)) { bestErr = err; bestN = n; bestD = d; } } return new QQ(bestN, bestD, true); } relax(max) { const bound = max < 0n ? -max : max; if (bound >= this.q) return this; const a = (this.p * bound) / this.q; if (max * this.p < 0n) return new QQ(a, bound); return new QQ(a + (this.p > 0n ? 1n : -1n), bound); } }

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