From 3795b659f61032cc3fbc048bd68679b152113875 Mon Sep 17 00:00:00 2001 From: LittleSheep Date: Sat, 13 Sep 2025 22:56:01 +0800 Subject: [PATCH] :lipstick: Improve accurate of sqrt calculation --- lib/solver.dart | 463 +++++++++++++++++++++++++++++++++++++++++++----- pubspec.lock | 8 + pubspec.yaml | 1 + 3 files changed, 425 insertions(+), 47 deletions(-) diff --git a/lib/solver.dart b/lib/solver.dart index 8eec81c..54532a3 100644 --- a/lib/solver.dart +++ b/lib/solver.dart @@ -1,6 +1,7 @@ import 'dart:math'; import 'package:flutter/foundation.dart'; // For kDebugMode import 'package:math_expressions/math_expressions.dart'; +import 'package:rational/rational.dart'; import 'models/calculation_step.dart'; /// 帮助解析一元一次方程 ax+b=cx+d 的辅助类 @@ -103,8 +104,8 @@ class SolverService { final parts = _parseLinearEquation(input); final a = parts.a, b = parts.b, c = parts.c, d = parts.d; - final newA = a - c; - final newD = d - b; + final newA = _rationalFromDouble(a) - _rationalFromDouble(c); + final newD = _rationalFromDouble(d) - _rationalFromDouble(b); steps.add( CalculationStep( @@ -121,14 +122,15 @@ class SolverService { stepNumber: 2, title: '合并同类项', explanation: '合并等式两边的项。', - formula: '\$\$${newA}x = $newD\$\$', + formula: + '\$\$${newA.toDouble().toStringAsFixed(4)}x = ${newD.toDouble().toStringAsFixed(4)}\$\$', ), ); - if (newA == 0) { + if (newA == Rational.zero) { return CalculationResult( steps: steps, - finalAnswer: newD == 0 ? '有无穷多解' : '无解', + finalAnswer: newD == Rational.zero ? '有无穷多解' : '无解', ); } @@ -152,9 +154,29 @@ class SolverService { final eqParts = input.split('='); if (eqParts.length != 2) throw Exception("方程格式错误，应包含一个 '='。"); + // Keep original equation for display + final originalEquation = _formatOriginalEquation(input); + + // Parse coefficients symbolically + final leftCoeffsSymbolic = _parsePolynomialSymbolic(eqParts[0]); + final rightCoeffsSymbolic = _parsePolynomialSymbolic(eqParts[1]); + + final aSymbolic = _subtractCoefficients( + leftCoeffsSymbolic[2] ?? '0', + rightCoeffsSymbolic[2] ?? '0', + ); + final bSymbolic = _subtractCoefficients( + leftCoeffsSymbolic[1] ?? '0', + rightCoeffsSymbolic[1] ?? '0', + ); + final cSymbolic = _subtractCoefficients( + leftCoeffsSymbolic[0] ?? '0', + rightCoeffsSymbolic[0] ?? '0', + ); + + // Also get numeric values for calculations final leftCoeffs = _parsePolynomial(eqParts[0]); final rightCoeffs = _parsePolynomial(eqParts[1]); - final a = (leftCoeffs[2] ?? 0) - (rightCoeffs[2] ?? 0); final b = (leftCoeffs[1] ?? 0) - (rightCoeffs[1] ?? 0); final c = (leftCoeffs[0] ?? 0) - (rightCoeffs[0] ?? 0); @@ -168,8 +190,7 @@ class SolverService { stepNumber: 1, title: '整理方程', explanation: r'将方程整理成标准形式 $ax^2+bx+c=0$。', - formula: - '\$\$${a}x^2 ${b >= 0 ? '+' : ''} ${b}x ${c >= 0 ? '+' : ''} $c = 0\$\$', + formula: originalEquation, ), ); @@ -213,36 +234,47 @@ class SolverService { ), ); - final delta = b * b - 4 * a * c; + // Calculate delta symbolically + final deltaSymbolic = _calculateDeltaSymbolic( + aSymbolic, + bSymbolic, + cSymbolic, + ); + final delta = + _rationalFromDouble(b).pow(2) - + Rational.fromInt(4) * _rationalFromDouble(a) * _rationalFromDouble(c); + steps.add( CalculationStep( stepNumber: 3, title: '计算判别式 (Delta)', - explanation: - '\$\$\\Delta = b^2 - 4ac = ($b)^2 - 4 \\cdot ($a) \\cdot ($c) = $delta\$\$', - formula: '\$\$\\Delta = $delta\$\$', + explanation: '\$\$\\Delta = b^2 - 4ac = $deltaSymbolic\$\$', + formula: + '\$\$\\Delta = $deltaSymbolic = ${delta.toDouble().toStringAsFixed(4)}\$\$', ), ); - if (delta > 0) { - final x1 = (-b + sqrt(delta)) / (2 * a); - final x2 = (-b - sqrt(delta)) / (2 * a); + 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); + steps.add( CalculationStep( stepNumber: 4, title: '应用求根公式', explanation: r'因为 $\Delta > 0$，方程有两个不相等的实数根。公式: $x = \frac{-b \pm \sqrt{\Delta}}{2a}$。', - formula: - '\$\$x_1 = ${x1.toStringAsFixed(4)}, \\quad x_2 = ${x2.toStringAsFixed(4)}\$\$', + formula: '\$\$x_1 = $x1Expr, \\quad x_2 = $x2Expr\$\$', ), ); return CalculationResult( steps: steps, - finalAnswer: - '\$\$x_1 = ${x1.toStringAsFixed(4)}, \\quad x_2 = ${x2.toStringAsFixed(4)}\$\$', + finalAnswer: '\$\$x_1 = $x1Expr, \\quad x_2 = $x2Expr\$\$', ); - } else if (delta == 0) { + } else if (deltaDouble == 0) { final x = -b / (2 * a); steps.add( CalculationStep( @@ -266,9 +298,10 @@ class SolverService { ), ); - final sqrtDelta = sqrt(-delta); + // Keep sqrt symbolic for complex roots + final sqrtNegDeltaStr = _formatSqrtExpression(-delta.toDouble()); final realPart = -b / (2 * a); - final imagPart = sqrtDelta / (2 * a); + final imagPartExpr = _formatImaginaryPart(sqrtNegDeltaStr, 2 * a); steps.add( CalculationStep( @@ -282,7 +315,7 @@ class SolverService { return CalculationResult( steps: steps, finalAnswer: - '\$\$x_1 = ${realPart.toStringAsFixed(4)} + ${imagPart.toStringAsFixed(4)}i, \\quad x_2 = ${realPart.toStringAsFixed(4)} - ${imagPart.toStringAsFixed(4)}i\$\$', + '\$\$x_1 = ${realPart.toStringAsFixed(4)} + $imagPartExpr, \\quad x_2 = ${realPart.toStringAsFixed(4)} - $imagPartExpr\$\$', ); } } @@ -316,16 +349,23 @@ ${a2}x ${b2 >= 0 ? '+' : ''} ${b2}y = $c2 & (2) ), ); - final det = a1 * b2 - a2 * b1; - if (det == 0) { + final det = + _rationalFromDouble(a1) * _rationalFromDouble(b2) - + _rationalFromDouble(a2) * _rationalFromDouble(b1); + if (det == Rational.zero) { + final infiniteCheck = + _rationalFromDouble(a1) * _rationalFromDouble(c2) - + _rationalFromDouble(a2) * _rationalFromDouble(c1); return CalculationResult( steps: steps, - finalAnswer: a1 * c2 - a2 * c1 == 0 ? '有无穷多解' : '无解', + finalAnswer: infiniteCheck == Rational.zero ? '有无穷多解' : '无解', ); } - final newA1 = a1 * b2, newC1 = c1 * b2; - final newA2 = a2 * b1, newC2 = c2 * b1; + final newA1 = _rationalFromDouble(a1) * _rationalFromDouble(b2); + final newC1 = _rationalFromDouble(c1) * _rationalFromDouble(b2); + final newA2 = _rationalFromDouble(a2) * _rationalFromDouble(b1); + final newC2 = _rationalFromDouble(c2) * _rationalFromDouble(b1); steps.add( CalculationStep( @@ -336,8 +376,8 @@ ${a2}x ${b2 >= 0 ? '+' : ''} ${b2}y = $c2 & (2) ''' \$\$ \\begin{cases} -${newA1}x ${b1 * b2 >= 0 ? '+' : ''} ${b1 * b2}y = $newC1 & (3) \\\\ -${newA2}x ${b1 * b2 >= 0 ? '+' : ''} ${b1 * b2}y = $newC2 & (4) +${newA1.toDouble().toStringAsFixed(2)}x ${b1 * b2 >= 0 ? '+' : ''} ${(b1 * b2).toStringAsFixed(2)}y = ${newC1.toDouble().toStringAsFixed(2)} & (3) \\\\ +${newA2.toDouble().toStringAsFixed(2)}x ${b1 * b2 >= 0 ? '+' : ''} ${(b1 * b2).toStringAsFixed(2)}y = ${newC2.toDouble().toStringAsFixed(2)} & (4) \\end{cases} \$\$ ''', @@ -353,7 +393,7 @@ ${newA2}x ${b1 * b2 >= 0 ? '+' : ''} ${b1 * b2}y = $newC2 & (4) title: '相减', explanation: '将方程(3)减去方程(4)，得到一个只含 x 的方程。', formula: - '\$\$($newA1 - $newA2)x = $newC1 - $newC2 \\Rightarrow ${xCoeff}x = $constCoeff\$\$', + '\$\$(${newA1.toDouble().toStringAsFixed(2)} - ${newA2.toDouble().toStringAsFixed(2)})x = ${newC1.toDouble().toStringAsFixed(2)} - ${newC2.toDouble().toStringAsFixed(2)} \\Rightarrow ${xCoeff.toDouble().toStringAsFixed(2)}x = ${constCoeff.toDouble().toStringAsFixed(2)}\$\$', ), ); @@ -369,21 +409,21 @@ ${newA2}x ${b1 * b2 >= 0 ? '+' : ''} ${b1 * b2}y = $newC2 & (4) if (b1.abs() < 1e-9) { final yCoeff = b2; - final yConst = c2 - a2 * x; + final yConst = c2 - a2 * x.toDouble(); final y = yConst / yCoeff; steps.add( CalculationStep( stepNumber: 4, title: '回代求解 y', - explanation: '将 x = $x 代入原方程(2)中。', + explanation: '将 x = ${x.toDouble().toStringAsFixed(4)} 代入原方程(2)中。', formula: ''' \$\$ \\begin{aligned} -$a2($x) + ${b2}y &= $c2 \\\\ -${a2 * x} + ${b2}y &= $c2 \\\\ -${b2}y &= $c2 - ${a2 * x} \\\\ -${b2}y &= ${c2 - a2 * x} +$a2(${x.toDouble().toStringAsFixed(4)}) + ${b2}y &= $c2 \\\\ +${a2 * x.toDouble()} + ${b2}y &= $c2 \\\\ +${b2}y &= $c2 - ${a2 * x.toDouble()} \\\\ +${b2}y &= ${c2 - a2 * x.toDouble()} \\end{aligned} \$\$ ''', @@ -394,30 +434,31 @@ ${b2}y &= ${c2 - a2 * x} stepNumber: 5, title: '解出 y', explanation: '求解得到 y 的值。', - formula: '\$\$y = $y\$\$', + formula: '\$\$y = ${y.toStringAsFixed(4)}\$\$', ), ); return CalculationResult( steps: steps, - finalAnswer: '\$\$x = $x, \\quad y = $y\$\$', + finalAnswer: + '\$\$x = ${x.toDouble().toStringAsFixed(4)}, \\quad y = ${y.toStringAsFixed(4)}\$\$', ); } else { final yCoeff = b1; - final yConst = c1 - a1 * x; + final yConst = c1 - a1 * x.toDouble(); final y = yConst / yCoeff; steps.add( CalculationStep( stepNumber: 4, title: '回代求解 y', - explanation: '将 x = $x 代入原方程(1)中。', + explanation: '将 x = ${x.toDouble().toStringAsFixed(4)} 代入原方程(1)中。', formula: ''' \$\$ \\begin{aligned} -$a1($x) + ${b1}y &= $c1 \\\\ -${a1 * x} + ${b1}y &= $c1 \\\\ -${b1}y &= $c1 - ${a1 * x} \\\\ -${b1}y &= ${c1 - a1 * x} +$a1(${x.toDouble().toStringAsFixed(4)}) + ${b1}y &= $c1 \\\\ +${a1 * x.toDouble()} + ${b1}y &= $c1 \\\\ +${b1}y &= $c1 - ${a1 * x.toDouble()} \\\\ +${b1}y &= ${c1 - a1 * x.toDouble()} \\end{aligned} \$\$ ''', @@ -428,12 +469,13 @@ ${b1}y &= ${c1 - a1 * x} stepNumber: 5, title: '解出 y', explanation: '求解得到 y 的值。', - formula: '\$\$y = $y\$\$', + formula: '\$\$y = ${y.toStringAsFixed(4)}\$\$', ), ); return CalculationResult( steps: steps, - finalAnswer: '\$\$x = $x, \\quad y = $y\$\$', + finalAnswer: + '\$\$x = ${x.toDouble().toStringAsFixed(4)}, \\quad y = ${y.toStringAsFixed(4)}\$\$', ); } } @@ -940,4 +982,331 @@ ${b1}y &= ${c1 - a1 * x} } int gcd(int a, int b) => b == 0 ? a : gcd(b, a % b); + + /// 格式化平方根表达式，保持符号形式 + String _formatSqrtExpression(double value) { + if (value == 0) return '0'; + + // 处理负数（用于复数根） + if (value < 0) { + return '\\sqrt{${(-value).toInt()}}'; + } + + // 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(); + } + + // 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; + } + } + + 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}'; + } else { + return '$coefficient\\sqrt{$remaining}'; + } + } + + /// 格式化二次方程的根：(-b ± sqrt(delta)) / (2a) + String _formatQuadraticRoot( + double b, + String sqrtExpr, + double denominator, + bool isPlus, + ) { + final sign = isPlus ? '+' : '-'; + final bStr = b == 0 + ? '' + : b > 0 + ? '${b.toInt()}' + : '(${b.toInt()})'; + final denomStr = denominator == 2 ? '2' : denominator.toString(); + + 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}'; + } else { + return isPlus + ? '\\frac{\\sqrt{${sqrtExpr.replaceAll('\\sqrt{', '').replaceAll('}', '')}}}{$denomStr}' + : '-\\frac{\\sqrt{${sqrtExpr.replaceAll('\\sqrt{', '').replaceAll('}', '')}}}{$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('}', '')}}'; + + if (denominator == 2) { + return '\\frac{$numerator}{2}'; + } else { + return '\\frac{$numerator}{$denomStr}'; + } + } + } + + /// 格式化复数根的虚部：sqrt(-delta)/(2a) + String _formatImaginaryPart(String sqrtExpr, double denominator) { + final denomStr = denominator == 2 ? '2' : denominator.toString(); + + if (denominator == 2) { + return '\\frac{\\sqrt{${sqrtExpr.replaceAll('\\sqrt{', '').replaceAll('}', '')}}}{2}i'; + } else { + return '\\frac{\\sqrt{${sqrtExpr.replaceAll('\\sqrt{', '').replaceAll('}', '')}}}{$denomStr}i'; + } + } + + /// 格式化原始方程，保持符号形式 + String _formatOriginalEquation(String input) { + // Simply return the original equation with proper LaTeX formatting + // This avoids complex parsing issues and preserves the original symbolic form + String result = input.replaceAll(' ', ''); + + // 确保方程格式正确 + if (!result.contains('=')) { + result = '$result=0'; + } + + // Replace sqrt with LaTeX format + result = result.replaceAll('sqrt(', '\\sqrt{'); + result = result.replaceAll(')', '}'); + + return '\$\$$result\$\$'; + } + + /// 解析多项式，保持符号形式 + Map _parsePolynomialSymbolic(String side) { + final coeffs = {}; + + // Use a simpler approach: split by terms and parse each term individually + var s = side.replaceAll(' ', ''); // Remove spaces + if (!s.startsWith('+') && !s.startsWith('-')) { + s = '+$s'; + } + + // Split by + and - but be more careful about parentheses and functions + final terms = []; + int start = 0; + int parenDepth = 0; + + for (int i = 0; i < s.length; i++) { + final char = s[i]; + + if (char == '(') + parenDepth++; + else if (char == ')') + parenDepth--; + + // Only split on + or - when not inside parentheses + if (parenDepth == 0 && (char == '+' || char == '-') && i > start) { + terms.add(s.substring(start, i)); + start = i; + } + } + terms.add(s.substring(start)); + + for (final term in terms) { + if (term.isEmpty) continue; + + // Parse each term + final termPattern = RegExp(r'^([+-]?)(.*?)x(?:\^(\d+))?$|^([+-]?)(.*?)$'); + final match = termPattern.firstMatch(term); + + if (match != null) { + if (match.group(5) != null) { + // Constant term + final sign = match.group(4) ?? '+'; + final value = match.group(5)!; + final coeffStr = sign == '+' && value.isNotEmpty + ? value + : '$sign$value'; + coeffs[0] = _combineCoefficients(coeffs[0], coeffStr); + } else { + // x term + final sign = match.group(1) ?? '+'; + final coeffPart = match.group(2) ?? ''; + final power = match.group(3) != null ? int.parse(match.group(3)!) : 1; + + String coeffStr; + if (coeffPart.isEmpty) { + coeffStr = sign == '+' ? '1' : '-1'; + } else { + coeffStr = sign == '+' ? coeffPart : '$sign$coeffPart'; + } + + coeffs[power] = _combineCoefficients(coeffs[power], coeffStr); + } + } + } + + return coeffs; + } + + /// 规范化系数字符串 + String _normalizeCoefficientString(String coeff) { + if (coeff.isEmpty || coeff == '+') return '1'; + if (coeff == '-') return '-1'; + + // 处理类似 "2sqrt(3)" 的情况 + coeff = coeff.replaceAll(' ', ''); // 移除空格 + + // 检查是否是纯数字 + final numValue = double.tryParse(coeff); + if (numValue != null) { + return coeff; + } + + // 检查是否包含 sqrt + if (coeff.contains('sqrt(') || coeff.contains('\\sqrt{')) { + // 如果前面没有数字系数，默认为 1 + if (coeff.startsWith('sqrt(') || coeff.startsWith('\\sqrt{')) { + return coeff.startsWith('-') ? coeff : '1' + coeff; + } + // 如果前面有数字，保持原样 + return coeff; + } + + return coeff; + } + + /// 合并系数，保持符号形式 + String _combineCoefficients(String? existing, String newCoeff) { + if (existing == null || existing == '0') return newCoeff; + if (newCoeff == '0') return existing; + + // 简化逻辑：如果都是数字，可以相加；否则保持原样 + final existingNum = double.tryParse(existing); + final newNum = double.tryParse(newCoeff); + + if (existingNum != null && newNum != null) { + final sum = existingNum + newNum; + return sum.toString(); + } + + // 如果包含符号表达式，直接连接 + return '$existing+$newCoeff'.replaceAll('+-', '-'); + } + + /// 减去系数 + String _subtractCoefficients(String a, String b) { + if (a == '0') return b.startsWith('-') ? b.substring(1) : '-$b'; + if (b == '0') return a; + + final aNum = double.tryParse(a); + final bNum = double.tryParse(b); + + if (aNum != null && bNum != null) { + final result = aNum - bNum; + return result.toString(); + } + + // 符号表达式相减 + return '$a-${b.startsWith('-') ? b.substring(1) : b}'; + } + + /// 计算判别式，保持符号形式 + String _calculateDeltaSymbolic(String a, String b, String c) { + // Delta = b^2 - 4ac + + // 计算 b^2 + String bSquared; + if (b == '0') { + bSquared = '0'; + } else if (b == '1') { + bSquared = '1'; + } else if (b == '-1') { + bSquared = '1'; + } else if (b.startsWith('-')) { + final absB = b.substring(1); + bSquared = '$absB^2'; + } else { + bSquared = '$b^2'; + } + + // 计算 4ac + String fourAC; + if (a == '0' || c == '0') { + fourAC = '0'; + } else { + // 处理符号 + String aCoeff = a; + String cCoeff = c; + + // 如果 a 或 c 是负数，需要处理符号 + bool aNegative = a.startsWith('-'); + bool cNegative = c.startsWith('-'); + + if (aNegative) aCoeff = a.substring(1); + if (cNegative) cCoeff = c.substring(1); + + String acProduct; + if (aCoeff == '1' && cCoeff == '1') { + acProduct = '1'; + } else if (aCoeff == '1') { + acProduct = cCoeff; + } else if (cCoeff == '1') { + acProduct = aCoeff; + } else { + acProduct = '$aCoeff \\cdot $cCoeff'; + } + + // 确定 4ac 的符号 + bool productNegative = aNegative != cNegative; + String fourACValue = '4 \\cdot $acProduct'; + + if (productNegative) { + fourAC = '-$fourACValue'; + } else { + fourAC = fourACValue; + } + } + + // 计算 Delta = b^2 - 4ac + if (bSquared == '0' && fourAC == '0') { + return '0'; + } else if (bSquared == '0') { + return fourAC.startsWith('-') ? fourAC.substring(1) : '-$fourAC'; + } else if (fourAC == '0') { + return bSquared; + } else { + String sign = fourAC.startsWith('-') ? '+' : '-'; + String absFourAC = fourAC.startsWith('-') ? fourAC.substring(1) : fourAC; + return '$bSquared $sign $absFourAC'; + } + } + + Rational _rationalFromDouble(double value, {int maxPrecision = 12}) { + // 限制小数精度，避免无限循环小数 + final str = value.toStringAsFixed(maxPrecision); + + if (!str.contains('.')) { + return Rational.parse(str); + } + + final parts = str.split('.'); + final integerPart = parts[0]; + final fractionalPart = parts[1]; + + final numerator = BigInt.parse(integerPart + fractionalPart); + final denominator = BigInt.from(10).pow(fractionalPart.length); + + return Rational(numerator, denominator); + } } diff --git a/pubspec.lock b/pubspec.lock index 65a1eda..e38716b 100644 --- a/pubspec.lock +++ b/pubspec.lock @@ -424,6 +424,14 @@ packages: url: "https://pub.dev" source: hosted version: "6.1.5+1" + rational: + dependency: "direct main" + description: + name: rational + sha256: cb808fb6f1a839e6fc5f7d8cb3b0a10e1db48b3be102de73938c627f0b636336 + url: "https://pub.dev" + source: hosted + version: "2.2.3" sky_engine: dependency: transitive description: flutter diff --git a/pubspec.yaml b/pubspec.yaml index ba6acd4..2d1b13c 100644 --- a/pubspec.yaml +++ b/pubspec.yaml @@ -39,6 +39,7 @@ dependencies: google_fonts: ^6.3.1 go_router: ^14.2.0 url_launcher: ^6.3.0 + rational: ^2.2.3 dev_dependencies: flutter_test: