diff --git a/lib/solver.dart b/lib/solver.dart index aa86916..04b0c2a 100644 --- a/lib/solver.dart +++ b/lib/solver.dart @@ -277,10 +277,9 @@ class SolverService { final deltaDouble = delta.toDouble(); if (deltaDouble > 0) { - // Keep sqrt symbolic instead of evaluating to decimal - final sqrtDeltaStr = _formatSqrtExpression(delta.toDouble()); - final x1Expr = _formatQuadraticRoot(-b, sqrtDeltaStr, 2 * a, true); - final x2Expr = _formatQuadraticRoot(-b, sqrtDeltaStr, 2 * a, false); + // Pass delta directly to maintain precision + final x1Expr = _formatQuadraticRoot(-b, delta, 2 * a, true); + final x2Expr = _formatQuadraticRoot(-b, delta, 2 * a, false); steps.add( CalculationStep( @@ -319,8 +318,9 @@ class SolverService { ), ); - // Keep sqrt symbolic for complex roots - final sqrtNegDeltaStr = _formatSqrtExpression(-delta.toDouble()); + // For complex roots, we need to handle -delta + final negDelta = -delta; + final sqrtNegDeltaStr = _formatSqrtFromRational(negDelta); final realPart = -b / (2 * a); final imagPartExpr = _formatImaginaryPart(sqrtNegDeltaStr, 2 * a); @@ -1079,49 +1079,113 @@ ${b1}y &= ${c1 - a1 * x.toDouble()} int gcd(int a, int b) => b == 0 ? a : gcd(b, a % b); - /// 格式化平方根表达式，保持符号形式 - String _formatSqrtExpression(double value) { - if (value == 0) return '0'; + /// 格式化 Rational 值的平方根表达式，保持符号形式 + String _formatSqrtFromRational(Rational value) { + if (value == Rational.zero) return '0'; // 处理负数（用于复数根） - if (value < 0) { - return '\\sqrt{${(-value).toInt()}}'; + if (value < Rational.zero) { + return '\\sqrt{${(-value).toBigInt()}}'; } - // 检查是否为完全平方数 - final sqrtValue = sqrt(value); - final rounded = sqrtValue.round(); - if ((sqrtValue - rounded).abs() < 1e-10) { - return rounded.toString(); - } + // 尝试将 Rational 转换为完全平方数的形式 + // 例如: 4/9 -> 2/3, 9/4 -> 3/2, 25/16 -> 5/4 等 - // 寻找最大的完全平方数因子 - int maxSquareFactor = 1; - int intValue = value.toInt(); - for (int i = 2; i * i <= intValue; i++) { - if (intValue % (i * i) == 0) { - maxSquareFactor = i * i; + // 首先简化分数 + final simplified = value; + + // 检查分子和分母是否都是完全平方数 + final numerator = simplified.numerator; + final denominator = simplified.denominator; + + // 寻找分子和分母的平方根因子 + BigInt sqrtNumerator = _findSquareRootFactor(numerator); + BigInt sqrtDenominator = _findSquareRootFactor(denominator); + + // 计算剩余的分子和分母 + final remainingNumerator = numerator ~/ (sqrtNumerator * sqrtNumerator); + final remainingDenominator = + denominator ~/ (sqrtDenominator * sqrtDenominator); + + // 构建结果 + String result = ''; + + // 处理系数部分 + if (sqrtNumerator > BigInt.one || sqrtDenominator > BigInt.one) { + if (sqrtNumerator > sqrtDenominator) { + final coeff = sqrtNumerator ~/ sqrtDenominator; + if (coeff > BigInt.one) { + result += '$coeff'; + } + } else if (sqrtDenominator > sqrtNumerator) { + // 这会导致分母，需要用分数表示 + final coeffNum = sqrtNumerator; + final coeffDen = sqrtDenominator; + if (coeffNum == BigInt.one) { + result += '\\frac{1}{$coeffDen}'; + } else { + result += '\\frac{$coeffNum}{$coeffDen}'; + } } } - final coefficient = sqrt(maxSquareFactor).round(); - final remaining = intValue ~/ maxSquareFactor; - - if (remaining == 1) { - return coefficient == 1 - ? '\\sqrt{$intValue}' - : '$coefficient\\sqrt{$remaining}'; - } else if (coefficient == 1) { - return '\\sqrt{$remaining}'; + // 处理根号部分 + if (remainingNumerator == BigInt.one && + remainingDenominator == BigInt.one) { + // 没有根号部分 + if (result.isEmpty) { + return '1'; + } + } else if (remainingNumerator == remainingDenominator) { + // 根号部分约分后为1 + if (result.isEmpty) { + return '1'; + } } else { - return '$coefficient\\sqrt{$remaining}'; + // 需要根号 + String sqrtContent = ''; + if (remainingDenominator == BigInt.one) { + sqrtContent = '$remainingNumerator'; + } else { + sqrtContent = '\\frac{$remainingNumerator}{$remainingDenominator}'; + } + + if (result.isEmpty) { + result = '\\sqrt{$sqrtContent}'; + } else { + result += '\\sqrt{$sqrtContent}'; + } } + + return result.isEmpty ? '1' : result; + } + + /// 寻找一个大整数的平方根因子 + BigInt _findSquareRootFactor(BigInt n) { + if (n <= BigInt.one) return BigInt.one; + + BigInt factor = BigInt.one; + BigInt i = BigInt.two; + + while (i * i <= n) { + BigInt count = BigInt.zero; + while (n % (i * i) == BigInt.zero) { + n = n ~/ (i * i); + count += BigInt.one; + } + if (count > BigInt.zero) { + factor = factor * i; + } + i += BigInt.one; + } + + return factor; } /// 格式化二次方程的根：(-b ± sqrt(delta)) / (2a) String _formatQuadraticRoot( double b, - String sqrtExpr, + Rational delta, double denominator, bool isPlus, ) { @@ -1133,22 +1197,23 @@ ${b1}y &= ${c1 - a1 * x.toDouble()} : '(${b.toInt()})'; final denomStr = denominator == 2 ? '2' : denominator.toString(); + // Format sqrt(delta) symbolically using the Rational value + final sqrtExpr = _formatSqrtFromRational(delta); + if (b == 0) { // 简化为 ±sqrt(delta)/denominator if (denominator == 2) { - return isPlus - ? '\\frac{\\sqrt{${sqrtExpr.replaceAll('\\sqrt{', '').replaceAll('}', '')}}}{2}' - : '-\\frac{\\sqrt{${sqrtExpr.replaceAll('\\sqrt{', '').replaceAll('}', '')}}}{2}'; + return isPlus ? '\\frac{$sqrtExpr}{2}' : '-\\frac{$sqrtExpr}{2}'; } else { return isPlus - ? '\\frac{\\sqrt{${sqrtExpr.replaceAll('\\sqrt{', '').replaceAll('}', '')}}}{$denomStr}' - : '-\\frac{\\sqrt{${sqrtExpr.replaceAll('\\sqrt{', '').replaceAll('}', '')}}}{$denomStr}'; + ? '\\frac{$sqrtExpr}{$denomStr}' + : '-\\frac{$sqrtExpr}{$denomStr}'; } } else { // 完整的表达式：(-b ± sqrt(delta))/denominator final numerator = b > 0 - ? '-$bStr $sign \\sqrt{${sqrtExpr.replaceAll('\\sqrt{', '').replaceAll('}', '')}}' - : '(${b.toInt()}) $sign \\sqrt{${sqrtExpr.replaceAll('\\sqrt{', '').replaceAll('}', '')}}'; + ? '-$bStr $sign $sqrtExpr' + : '(${b.toInt()}) $sign $sqrtExpr'; if (denominator == 2) { return '\\frac{$numerator}{2}'; @@ -1256,33 +1321,6 @@ ${b1}y &= ${c1 - a1 * x.toDouble()} return coeffs; } - /// 规范化系数字符串 - String _normalizeCoefficientString(String coeff) { - if (coeff.isEmpty || coeff == '+') return '1'; - if (coeff == '-') return '-1'; - - // 处理类似 "2sqrt(3)" 的情况 - coeff = coeff.replaceAll(' ', ''); // 移除空格 - - // 检查是否是纯数字 - final numValue = double.tryParse(coeff); - if (numValue != null) { - return coeff; - } - - // 检查是否包含 sqrt - if (coeff.contains('sqrt(') || coeff.contains('\\sqrt{')) { - // 如果前面没有数字系数，默认为 1 - if (coeff.startsWith('sqrt(') || coeff.startsWith('\\sqrt{')) { - return coeff.startsWith('-') ? coeff : '1' + coeff; - } - // 如果前面有数字，保持原样 - return coeff; - } - - return coeff; - } - /// 合并系数，保持符号形式 String _combineCoefficients(String? existing, String newCoeff) { if (existing == null || existing == '0') return newCoeff;