自己模拟实现math.h中的函数

　　之前一直都很迷惑pow()函数时怎么实现的，对于整数次的幂我还能很容易做到，但是对于分数次幂就不是那么好做了。需要一些高等数学级数的知识。

　　我这里实现了求ln(x), pow(double x, double y), exp(x), sin(x), cos(x), sinh(x), cosh(x), tanh(x), arctanh(x)等一些常见的函数功能。具体请看代码

 /*============================================================================*\
  *
  *                     模拟实现 math.h 库中的函数
  *
  *                         樊列龙 2013/7/1
  *
 \*============================================================================*/

 #include <iostream>
 #include <iomanip>
 #include <string>
 #include <cstdlib>
 #include <cstdio>

 using namespace std;

 typedef int MODE;                                    // 计算sin(x)时采用弧度还是角度
 const int BY_RADIDAN = ;                           // 弧度
 const int BY_ANGLE = ;                                // 角度

 const double RATIO = 0.01745329251994;                 // 角度转弧度的转换率

 long Factorial( long x );                           // 阶乘
 double Pow( double x, double y =  );               // 求x^y
 double Square( double x );                          // 求平方
 double Cube( double x );                            // 求立方
 double Sin( double x, MODE mode = BY_RADIDAN );     // 计算sin(x)
 double Cos( double x , MODE mode = BY_RADIDAN );    // 计算cos(x)
 double Tan( double x , MODE mode = BY_RADIDAN );    // 计算tan(x)
 int Int(double x);                                  // 取整, 不进行四舍五入
 double Fabs( double x );                            // 求绝对值
 double Arctanh(double x);                           // 求achtanh()
 double Ln(double x);                                // 求自然对数
 double Exp( double x );                             // 求e的x次幂
 double Sinh( double x );                            // 求双曲正弦 sinh()
 double Cosh( double x );                            // 求双曲余弦 cosh()
 double Tanh(double x);                              // 求双曲正切 tanh()
 double PI();                                        // 求圆周率
 double gen(double x, double y);                     // 求根

 // 计算阶乘
 long Factorial( long x )
 {
     if(  == x ||  == x )
     {
         return ;
     }

     long ans = ;
     for( int i = ; i <= x; i++ )
     {
         ans *= i;
     }

     return ans;
 }

 // 计算求幂, 默认是平方
 // x^y=e^(ln(x)*y)
 double Pow(double x, double y)
 {
     return Exp( (Ln(x)*y) );
 }

 // 求根
 double gen(double x, double y)
 {
     return Pow(x,/y);
 }

 // 计算绝对值
 double Fabs( double x )
 {
     if( x < -1e-)
         return -x;
     return x;
 }

 int Int(double x)
 {
     return static_cast<int>(Fabs( x ));
 }

 // 利用泰勒级数算 sin(x) , 默认传入弧度
 // sinx ≈x-(x^3)/3!+(x^5)/5!-(x^7)/7!-(x^9)/9!...
 // 计算精度为7位
 double Sin( double x, MODE mode )
 {
     // 如果是角度则转换成弧度计算
     if( BY_ANGLE ==  mode )
     {
         x *= RATIO;
     }
     x = x - Int( (x/(  * PI() ) ) ) *  * PI();
     if(  == x )
         return ;

     double sum = , fs = x, fm = , t, t1 = ;  //求和、分子、分母、精度、t1表示符号位
     int n=;                                    //项数

     t = fs / fm;                                //精度赋值
     while( Fabs(t) >= 1e-)                     //循环条件
     {
         sum = sum + t;                          //求和
         n++;                                    //基数增加
         fs = fs * x * x;                        //新分子
         fm = fm * ( *n - ) * ( *n -  );     //新分母
         t1= -t1;                                //符号位　一正一负
         t = t1 * ( fs / fm );                   //精度
     }

     return sum;
 }

 // cos(x) = 1 - (x^2)/2! + (x^4)/4!......
 // 结果精度为7位
 double Cos( double x , MODE mode)
 {
     double term,sum;
     int n = ;

     sum = 1.0;
     term = 1.0;

     while(Fabs(term) > 1e-)
     {
         term = term * x * x / (n * (n-)) * (-1.0);
         sum = sum + term;
         n += ;
     }

     return sum;
 }

 // tanθ = sinθ / cosθ
 // 结果精度为7位
 double Tan( double x , MODE mode)
 {
     return ( Sin(x) / Cos(x) );
 }

 // 计算 x^2
 double Square( double x )
 {
     return ( x*x );
 }

 // 计算 x^3
 double Cube( double x )
 {
     return ( x*x*x );
 }

 // arctanh(x)= x + x^3/3 + x^5/5 + ...       (x≤1)
 // 求sinh(y) 会用到
 // 精度为7位
 double Arctanh(double x)
 {
     int n = ;
     double sum = , term = x,  numer = x, denom = ;
     while(Fabs(term) > 1e-)
     {
         sum += term;                            // 求和
         numer = numer * x * x;                  // 分子
         denom = n+;                            // 分母
         term = numer/denom;                     // 精度
         n += ;                                 // 计数
     }

     return sum;
 }

 // ln(x) = 2 arctanh((x-1)/(x+1))
 // 调用了Arctanh(double) 方法
 double Ln(double x)
 {
     return  * Arctanh((x-)/(x+));
 }

 // 求e^x 用于Pow( double, double )调用
 // e^x = 1+x+(x^2)/2!+(x^3)/3!+...
 // 精度为7位
 double Exp( double x )
 {
     double ret = , term = x, numer = x, denom = ;;
     int n = ;
     while(Fabs(term) > 1e-)
     {
         ret += term;                            // 求和
         numer *= x;                             // 分子
         denom = denom * (n+);                  // 分母
         n++;                                    // 累加
         term = numer/denom;                     // 精度
     }

     return ret;

 }

 // sinh(x)=(exp(x) - exp(-x)) / 2.0
 // 精度为7位
 double Sinh(double x)
 {
     return ( ( Exp(x) - Exp(-x) ) / 2.0 );
 }

 // cosh(x)=(exp(x) + exp(-x)) / 2.0;
 // 精度为7位
 double Cosh(double x)
 {
     return ( ( Exp(x) + Exp(-x) ) / 2.0 );
 }

 // tanh(x) = sinh(x) / cosh(x);
 // 精度为7位
 double Tanh(double x)
 {
     return ( Sinh(x) / Cosh(x) );
 }

 // 求PI: pi/4 = 1 - 1/3 + 1/5 - 1/7...
 double PI()
 {
     int a = , b = ;
     double m = 1.0, sum = 1.0;

     while( Fabs(m) > 1e- )
     {
         b = -b;
         m = double(b) / a;
         sum = sum + m;
         a = a + ;
     }
     return  * sum;
 }

 //==============================================================================
 void drow_line(int i = ,string s = "━", bool newline = false)
 {
     if(newline) cout << endl;
     while(i--)
         cout << s;
     cout << endl;
 }

 void cls()
 {
     system("cls");
 }

 void pause()
 {
     system("pause");
     //getchar();
 }

 // 菜单选项
 void menu_option(int& menu_op,
                  int max = ,
                  int min = ,
                  string s = " 请选择相应的菜单选项>>> ",
                  string error = " ERROR:请选择正确的菜单选项! >>> ")
 {
     cout << s;
     cin >> menu_op;
     while(cin.fail() || menu_op < min || menu_op > max)
     {
         cout << error;
         cin.clear();
         fflush(stdin);
         cin >> menu_op;
     }
 }

 // 选择逻辑处理
 bool choos()
 {
     string contin;
     while(true)
     {
         getline(cin, contin);
         if(contin == "Y" || contin == "y" || contin == "是")
         {
             return true;
         }
         else if(contin == "N" || contin == "n" || contin == "否")
         {
             return false;
         }
         else
         {
             cout << " 提示：请输入(Y/N)！>>> ";
         }
     }

 }

 bool check_input(int& x,
                  string str = " 请输入数据x：",
                  string err = " 输入数据格式有误！")
 {
     cout << str;
     cin >> x;
     if(cin.good())
         return true;
     cout << err << endl;
     return false;
 }

 bool check_input(double& x,
                  string str = " 请输入数据 x：",
                  string err = " 输入数据格式有误！")
 {
     cout << str;
     cin >> x;
     if(cin.good())
         return true;
         cout << err << endl;
     return false;
 }

 void print_result(double ans, string str )
 {
     cout << str << " = " << ans << endl;
     pause();
 }

 //==============================================================================

 /**
  *
  * 主菜单
  */
 void menu_main()
 {
     cls();
     cout << endl << endl << endl;

     cout << "\t";
     drow_line();

     cout << "\t\t\t\t*** 主菜单选项 ***" << endl;

     cout << "\t";
     drow_line();

     cout << left;
     cout << "\t        " << setw() << "1. 取整 Int( x )";
     cout << setw() << "2. 计算 sinh( x )" << endl;
     cout << "\t        " << setw() << "3. 计算 sin( x )";
     cout << setw() << "4. 计算 n^2 " << endl;
     cout << "\t        " << setw() << "5. 计算 n!";
     cout << setw() << "6. 计算 cosh( x ) " << endl;
     cout << "\t        " << setw() << "7. 计算 cos( x )";
     cout << setw() << "8. 计算 x^y" << endl;
     cout << "\t        " << setw() << "9. 计算 y√x ";
     cout << setw() << "10. 计算 tanh( x ) " << endl;
     cout << "\t        " << setw() << "11. 计算 Π";
     cout << setw() << "12. 计算 tan( x )" << endl;
     cout << "\t        " << setw() << "13. 计算 x^3";
     cout << setw() << "14. 计算 3√x" << endl;
     cout << "\t        " << setw() << "0. 退出系统"  << endl;

     cout << "\t";
     drow_line();

     cout << endl << endl;
     drow_line(,"><");
     cout << endl;

 }

 int main()
 {
     int ansInt;
     double ans;
     int xi;
     double x,y;

     while(true)
     {
         cls();
         menu_main();
         int option;
         menu_option(option);                            // 选择功能
         switch(option)
         {
         case  :
             cout << "是否退出？(Y/N)";
             if(choos())
                 exit();
             break;
         case :
             check_input(x);
             ansInt = Int(x);
             drow_line(,"**");
             cout << "Int(" << x << ") = " << ansInt << endl;
             pause();
             break;
         case :
             check_input(x);
             ans = Sinh(x);
             drow_line(,"**");
             cout << "sinh(" << x << ") = " << ans << endl;
             pause();
             break;
         case :
             check_input(x);
             ans = Sin(x);
             drow_line(,"**");
             cout << "sin(" << x << ") = " << ans << endl;
             pause();
             break;
         case :
             check_input(x);
             ans = Square(x);
             drow_line(,"**");
             cout << x << "^2 = " << ans << endl;
             pause();
             break;
         case :
             check_input(xi, " 请输入整数n: ");
             ansInt = Factorial(xi);
             drow_line(,"**");
             cout << xi << "! = " << ansInt << endl;
             pause();
             break;
         case :
             check_input(x);
             ans = Cosh(x);
             drow_line(,"**");
             cout << "cosh(" << x << ") = " << ans << endl;
             pause();
             break;
         case :
             check_input(x);
             ans = Cos(x);
             drow_line(,"**");
             cout << "cos(" << x << ") = " << ans << endl;
             pause();
             break;
         case :
             check_input(x, " 请输入数据 x y : ");
             check_input(y, "");
             ans = Pow(x,y);
             drow_line(,"**");
             cout << x << "^" << y << " = " << ans << endl;
             pause();
             break;
         case :
             check_input(x, " 请输入数据 x y : ");
             check_input(y, "");
             ans = gen(x,y);
             drow_line(,"**");
             cout << y << "√" << x << " = " << ans << endl;
             pause();
             break;
         case :
             check_input(x);
             ans = Tanh(x);
             drow_line(,"**");
             cout << "tanh(" << x << ") = " << ans << endl;
             pause();
             break;
         case :
             ans = PI();
             drow_line(,"**");
             cout << "π = " << ans << endl;
             pause();
             break;
         case :
             check_input(x);
             ans = Tan(x);
             drow_line(,"**");
             cout << "tan(" << x << ") = " << ans << endl;
             pause();
             break;
         case :
             check_input(x);
             ans = Cube(x);
             drow_line(,"**");
             cout << x << "^3 = " << ans << endl;
             pause();
             break;
         case :
             check_input(x);
             ans = gen(x,);
             drow_line(,"**");
             cout << "3√" << x << " = " << ans << endl;
             pause();
             break;
         default:
             break;
         }
     }
     return ;
 }

执行结果：