610 lines
15 KiB
Dart
610 lines
15 KiB
Dart
// === 在 abstract class Expr 中添加声明 ===
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import 'dart:math' show sqrt, cos, sin, tan;
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abstract class Expr {
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Expr simplify();
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/// 新增：对表达式进行“求值/数值化”——尽可能把可算的部分算出来
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Expr evaluate();
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/// Substitute variable with value
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Expr substitute(String varName, Expr value);
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@override
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String toString();
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MulExpr operator *(Expr other) => MulExpr(this, other);
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AddExpr operator +(Expr other) => AddExpr(this, other);
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SubExpr operator -(Expr other) => SubExpr(this, other);
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DivExpr operator /(Expr other) => DivExpr(this, other);
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}
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// === IntExpr ===
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class IntExpr extends Expr {
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final int value;
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IntExpr(this.value);
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@override
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Expr simplify() => this;
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@override
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Expr evaluate() => this;
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@override
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Expr substitute(String varName, Expr value) => this;
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@override
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String toString() => value.toString();
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}
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// === DoubleExpr ===
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class DoubleExpr extends Expr {
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final double value;
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DoubleExpr(this.value);
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@override
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Expr simplify() => this;
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@override
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Expr evaluate() => this;
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@override
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Expr substitute(String varName, Expr value) => this;
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@override
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String toString() => value.toString();
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}
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// === VarExpr ===
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class VarExpr extends Expr {
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final String name;
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VarExpr(this.name);
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@override
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Expr simplify() => this;
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@override
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Expr evaluate() => this;
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@override
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Expr substitute(String varName, Expr value) => name == varName ? value : this;
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@override
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String toString() => name;
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}
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// === FractionExpr.evaluate ===
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class FractionExpr extends Expr {
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final int numerator;
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final int denominator;
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FractionExpr(this.numerator, this.denominator) {
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if (denominator == 0) throw Exception("分母不能为0");
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}
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@override
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Expr simplify() {
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int g = _gcd(numerator.abs(), denominator.abs());
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int n = numerator ~/ g;
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int d = denominator ~/ g;
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// 分母负数转移到分子
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if (d < 0) {
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n = -n;
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d = -d;
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}
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if (d == 1) return IntExpr(n); // 化简成整数
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return FractionExpr(n, d);
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}
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@override
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Expr evaluate() => simplify();
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@override
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Expr substitute(String varName, Expr value) => this;
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@override
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String toString() => "$numerator/$denominator";
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}
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// === AddExpr.evaluate: 把可算的合并（整数、分数、以及同类 sqrt 项） ===
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class AddExpr extends Expr {
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final Expr left, right;
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AddExpr(this.left, this.right);
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@override
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Expr simplify() {
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var l = left.simplify();
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var r = right.simplify();
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if (l is FractionExpr && r is FractionExpr) {
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return FractionExpr(
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l.numerator * r.denominator + r.numerator * l.denominator,
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l.denominator * r.denominator,
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).simplify();
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}
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if (l is IntExpr && r is FractionExpr) {
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return FractionExpr(
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l.value * r.denominator + r.numerator,
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r.denominator,
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).simplify();
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}
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if (l is FractionExpr && r is IntExpr) {
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return FractionExpr(
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l.numerator + r.value * l.denominator,
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l.denominator,
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).simplify();
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}
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return AddExpr(l, r);
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}
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@override
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Expr evaluate() {
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var l = left.evaluate();
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var r = right.evaluate();
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// 纯整数相加 -> 整数
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if (l is IntExpr && r is IntExpr) {
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return IntExpr(l.value + r.value);
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}
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// 分数相加 / 分数与整数相加
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if (l is FractionExpr && r is FractionExpr) {
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return FractionExpr(
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l.numerator * r.denominator + r.numerator * l.denominator,
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l.denominator * r.denominator,
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).simplify();
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}
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if (l is IntExpr && r is FractionExpr) {
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return FractionExpr(
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l.value * r.denominator + r.numerator,
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r.denominator,
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).simplify();
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}
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if (l is FractionExpr && r is IntExpr) {
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return FractionExpr(
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l.numerator + r.value * l.denominator,
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l.denominator,
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).simplify();
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}
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// 合并同类的 sqrt 项: a*sqrt(X) + b*sqrt(X) = (a+b)*sqrt(X)
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var a = _asSqrtTerm(l);
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var b = _asSqrtTerm(r);
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if (a != null && b != null && a.inner.toString() == b.inner.toString()) {
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return MulExpr(IntExpr(a.coef + b.coef), SqrtExpr(a.inner)).simplify();
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}
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return AddExpr(l, r);
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}
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@override
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Expr substitute(String varName, Expr value) => AddExpr(
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left.substitute(varName, value),
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right.substitute(varName, value),
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);
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@override
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String toString() => "($left + $right)";
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}
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// === SubExpr.evaluate 类似 AddExpr，但做减法 ===
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class SubExpr extends Expr {
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final Expr left, right;
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SubExpr(this.left, this.right);
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@override
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Expr simplify() {
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var l = left.simplify();
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var r = right.simplify();
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if (l is IntExpr && r is IntExpr) {
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return IntExpr(l.value - r.value);
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}
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if (l is FractionExpr && r is FractionExpr) {
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return FractionExpr(
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l.numerator * r.denominator - r.numerator * l.denominator,
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l.denominator * r.denominator,
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).simplify();
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}
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return SubExpr(l, r);
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}
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@override
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Expr evaluate() {
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var l = left.evaluate();
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var r = right.evaluate();
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if (l is IntExpr && r is IntExpr) {
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return IntExpr(l.value - r.value);
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}
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if (l is FractionExpr && r is FractionExpr) {
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return FractionExpr(
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l.numerator * r.denominator - r.numerator * l.denominator,
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l.denominator * r.denominator,
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).simplify();
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}
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if (l is IntExpr && r is FractionExpr) {
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return FractionExpr(
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l.value * r.denominator - r.numerator,
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r.denominator,
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).simplify();
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}
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if (l is FractionExpr && r is IntExpr) {
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return FractionExpr(
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l.numerator - r.value * l.denominator,
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l.denominator,
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).simplify();
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}
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// 处理同类 sqrt 项: a*sqrt(X) - b*sqrt(X) = (a-b)*sqrt(X)
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var a = _asSqrtTerm(l);
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var b = _asSqrtTerm(r);
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if (a != null && b != null && a.inner.toString() == b.inner.toString()) {
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return MulExpr(IntExpr(a.coef - b.coef), SqrtExpr(a.inner)).simplify();
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}
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return SubExpr(l, r);
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}
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@override
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Expr substitute(String varName, Expr value) => SubExpr(
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left.substitute(varName, value),
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right.substitute(varName, value),
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);
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@override
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String toString() => "($left - $right)";
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}
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// === MulExpr.evaluate ===
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class MulExpr extends Expr {
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final Expr left, right;
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MulExpr(this.left, this.right);
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@override
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Expr simplify() {
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var l = left.simplify();
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var r = right.simplify();
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if (l is IntExpr && l.value == 1) return r;
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if (r is IntExpr && r.value == 1) return l;
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if (l is IntExpr && l.value == -1) return SubExpr(IntExpr(0), r).simplify();
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if (r is IntExpr && r.value == -1) return SubExpr(IntExpr(0), l).simplify();
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if (l is IntExpr && r is IntExpr) {
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return IntExpr(l.value * r.value);
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}
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if (l is FractionExpr && r is IntExpr) {
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return FractionExpr(l.numerator * r.value, l.denominator).simplify();
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}
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if (l is IntExpr && r is FractionExpr) {
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return FractionExpr(l.value * r.numerator, r.denominator).simplify();
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}
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if (l is FractionExpr && r is FractionExpr) {
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return FractionExpr(
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l.numerator * r.numerator,
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l.denominator * r.denominator,
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).simplify();
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}
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if (l is SqrtExpr && r is FractionExpr) {
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return FractionExpr(1, r.denominator).simplify() *
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MulExpr(l, IntExpr(r.numerator)).simplify();
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}
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return MulExpr(l, r);
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}
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@override
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Expr evaluate() {
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var l = left.evaluate();
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var r = right.evaluate();
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if (l is IntExpr && l.value == 1) return r;
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if (r is IntExpr && r.value == 1) return l;
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if (l is IntExpr && l.value == -1) return SubExpr(IntExpr(0), r).simplify();
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if (r is IntExpr && r.value == -1) return SubExpr(IntExpr(0), l).simplify();
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if (l is IntExpr && r is IntExpr) {
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return IntExpr(l.value * r.value);
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}
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if (l is FractionExpr && r is IntExpr) {
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return FractionExpr(l.numerator * r.value, l.denominator).simplify();
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}
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if (l is IntExpr && r is FractionExpr) {
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return FractionExpr(l.value * r.numerator, r.denominator).simplify();
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}
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if (l is FractionExpr && r is FractionExpr) {
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return FractionExpr(
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l.numerator * r.numerator,
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l.denominator * r.denominator,
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).simplify();
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}
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// sqrt * sqrt: sqrt(a)*sqrt(a) = a
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if (l is SqrtExpr &&
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r is SqrtExpr &&
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l.inner.toString() == r.inner.toString()) {
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return l.inner.simplify();
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}
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// int * sqrt -> 保留形式，之后 simplify() 再处理约分
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if ((l is IntExpr && r is SqrtExpr) || (l is SqrtExpr && r is IntExpr)) {
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return MulExpr(l, r).simplify();
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}
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return MulExpr(l, r);
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}
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@override
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Expr substitute(String varName, Expr value) => MulExpr(
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left.substitute(varName, value),
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right.substitute(varName, value),
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);
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@override
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String toString() => "($left * $right)";
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}
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// === DivExpr.evaluate ===
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class DivExpr extends Expr {
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final Expr left, right;
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DivExpr(this.left, this.right);
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@override
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Expr simplify() {
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var l = left.simplify();
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var r = right.simplify();
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if (l is IntExpr && r is IntExpr) {
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return FractionExpr(l.value, r.value).simplify();
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}
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if (l is FractionExpr && r is IntExpr) {
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return FractionExpr(l.numerator, l.denominator * r.value).simplify();
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}
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if (l is IntExpr && r is FractionExpr) {
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return FractionExpr(l.value * r.denominator, r.numerator).simplify();
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}
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if (l is FractionExpr && r is FractionExpr) {
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return FractionExpr(
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l.numerator * r.denominator,
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l.denominator * r.numerator,
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).simplify();
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}
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if (l is MulExpr &&
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l.left is IntExpr &&
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l.right is SqrtExpr &&
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r is IntExpr) {
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int coeff = (l.left as IntExpr).value;
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int denom = r.value;
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int g = _gcd(coeff.abs(), denom.abs());
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return MulExpr(
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IntExpr(coeff ~/ g),
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DivExpr(l.right, IntExpr(denom ~/ g)).simplify(),
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).simplify();
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}
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return DivExpr(l, r);
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}
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@override
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Expr evaluate() {
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var l = left.evaluate();
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var r = right.evaluate();
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if (l is IntExpr && r is IntExpr) {
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return FractionExpr(l.value, r.value).simplify();
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}
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if (l is FractionExpr && r is IntExpr) {
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return FractionExpr(l.numerator, l.denominator * r.value).simplify();
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}
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if (l is IntExpr && r is FractionExpr) {
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return FractionExpr(l.value * r.denominator, r.numerator).simplify();
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}
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if (l is FractionExpr && r is FractionExpr) {
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return FractionExpr(
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l.numerator * r.denominator,
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l.denominator * r.numerator,
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).simplify();
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}
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// handle (k * sqrt(X)) / d 约分
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if (l is MulExpr &&
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l.left is IntExpr &&
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l.right is SqrtExpr &&
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r is IntExpr) {
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int coeff = (l.left as IntExpr).value;
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int denom = r.value;
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int g = _gcd(coeff.abs(), denom.abs());
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return MulExpr(
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IntExpr(coeff ~/ g),
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DivExpr(l.right, IntExpr(denom ~/ g)).evaluate(),
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).evaluate();
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}
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return DivExpr(l, r);
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}
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@override
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Expr substitute(String varName, Expr value) => DivExpr(
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left.substitute(varName, value),
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right.substitute(varName, value),
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);
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@override
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String toString() => "($left / $right)";
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}
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// === SqrtExpr.evaluate ===
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class SqrtExpr extends Expr {
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final Expr inner;
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SqrtExpr(this.inner);
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@override
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Expr simplify() {
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var i = inner.simplify();
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if (i is IntExpr) {
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int n = i.value;
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int root = sqrt(n).floor();
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if (root * root == n) {
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return IntExpr(root); // 完全平方数
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}
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// 尝试拆分 sqrt，比如 sqrt(8) = 2*sqrt(2)
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for (int k = root; k > 1; k--) {
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if (n % (k * k) == 0) {
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return MulExpr(
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IntExpr(k),
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SqrtExpr(IntExpr(n ~/ (k * k))),
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).simplify();
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}
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}
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}
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return SqrtExpr(i);
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}
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@override
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Expr evaluate() {
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var i = inner.evaluate();
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if (i is IntExpr) {
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int n = i.value;
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int root = sqrt(n).floor();
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if (root * root == n) return IntExpr(root);
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// 拆平方因子并返回 k * sqrt(remain)
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for (int k = root; k > 1; k--) {
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if (n % (k * k) == 0) {
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return MulExpr(
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IntExpr(k),
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SqrtExpr(IntExpr(n ~/ (k * k))),
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).evaluate();
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}
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}
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}
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return SqrtExpr(i);
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}
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@override
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Expr substitute(String varName, Expr value) =>
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SqrtExpr(inner.substitute(varName, value));
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@override
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String toString() => "sqrt($inner)";
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}
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// === CosExpr ===
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class CosExpr extends Expr {
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final Expr inner;
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CosExpr(this.inner);
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@override
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Expr simplify() => CosExpr(inner.simplify());
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@override
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Expr evaluate() {
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var i = inner.evaluate();
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if (i is IntExpr) {
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return DoubleExpr(cos(i.value.toDouble()));
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}
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if (i is FractionExpr) {
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return DoubleExpr(cos(i.numerator / i.denominator));
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}
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if (i is DoubleExpr) {
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return DoubleExpr(cos(i.value));
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}
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return CosExpr(i);
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}
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@override
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Expr substitute(String varName, Expr value) =>
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CosExpr(inner.substitute(varName, value));
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@override
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String toString() => "cos($inner)";
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}
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// === SinExpr ===
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class SinExpr extends Expr {
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final Expr inner;
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SinExpr(this.inner);
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@override
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Expr simplify() => SinExpr(inner.simplify());
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@override
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Expr evaluate() {
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var i = inner.evaluate();
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if (i is IntExpr) {
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return DoubleExpr(sin(i.value.toDouble()));
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}
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if (i is FractionExpr) {
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return DoubleExpr(sin(i.numerator / i.denominator));
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}
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if (i is DoubleExpr) {
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return DoubleExpr(sin(i.value));
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}
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return SinExpr(i);
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}
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@override
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Expr substitute(String varName, Expr value) =>
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SinExpr(inner.substitute(varName, value));
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@override
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String toString() => "sin($inner)";
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}
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// === TanExpr ===
|
||
class TanExpr extends Expr {
|
||
final Expr inner;
|
||
TanExpr(this.inner);
|
||
|
||
@override
|
||
Expr simplify() => TanExpr(inner.simplify());
|
||
|
||
@override
|
||
Expr evaluate() {
|
||
var i = inner.evaluate();
|
||
if (i is IntExpr) {
|
||
return DoubleExpr(tan(i.value.toDouble()));
|
||
}
|
||
if (i is FractionExpr) {
|
||
return DoubleExpr(tan(i.numerator / i.denominator));
|
||
}
|
||
if (i is DoubleExpr) {
|
||
return DoubleExpr(tan(i.value));
|
||
}
|
||
return TanExpr(i);
|
||
}
|
||
|
||
@override
|
||
Expr substitute(String varName, Expr value) =>
|
||
TanExpr(inner.substitute(varName, value));
|
||
|
||
@override
|
||
String toString() => "tan($inner)";
|
||
}
|
||
|
||
// === 辅助：识别 a * sqrt(X) 形式 ===
|
||
class _SqrtTerm {
|
||
final int coef;
|
||
final Expr inner;
|
||
_SqrtTerm(this.coef, this.inner);
|
||
}
|
||
|
||
_SqrtTerm? _asSqrtTerm(Expr e) {
|
||
if (e is SqrtExpr) return _SqrtTerm(1, e.inner);
|
||
if (e is MulExpr) {
|
||
// 可能为 Int * Sqrt or Sqrt * Int
|
||
if (e.left is IntExpr && e.right is SqrtExpr) {
|
||
return _SqrtTerm((e.left as IntExpr).value, (e.right as SqrtExpr).inner);
|
||
}
|
||
if (e.right is IntExpr && e.left is SqrtExpr) {
|
||
return _SqrtTerm((e.right as IntExpr).value, (e.left as SqrtExpr).inner);
|
||
}
|
||
}
|
||
return null;
|
||
}
|
||
|
||
/// 辗转相除法求 gcd
|
||
int _gcd(int a, int b) => b == 0 ? a : _gcd(b, a % b);
|