将 PI 定义为小数点后 36 位
define PI to 36 decimal place
在 C++ 中可以做到这一点
ifndef M_PI
define M_PI 3.141592653589793238462643383279502884
endif
然而，在 C# 中，我尝试声明一个 double var 并为其分配具有 36 位小数的 PI 值。
这不起作用，因为 double 存储最多 16 位小数 (MSDN docx)。我尝试创建一个隐式类型的局部变量，但编译器只是将其声明为双精度类型，我又回到了原点。也许我可以在 C# 中包含 cmath？
如何在 C# 中将 PI 定义为 36 位小数
据我所知，C# 在 type decimal 中的精度最高，但它最多只能达到 28-29 位有效小数位。我不认为你可以在 C# 中覆盖多达 36 个十进制数字而不使用自定义库。
如果你真的需要那么高的精度，你可以使用任意精度的 C 库，比如 GNU MPC or GNU MPFR. As they are C libraries so you'd need to use interop 来使用它们...但这会限制 使用你的高精度pi到使用你选择的包...
Here is an article about calculating pi to 1 million digit in C#，你可以缩小到36位
这里有一个示例代码，我建议阅读全文及其背后的逻辑。
public static HighPrecision GetPi(int digits)
{
HighPrecision.Precision = digits;
HighPrecision first = 4 * Atan(5);
HighPrecision second = Atan(239);
return 4 * (first - second);
}
如果您只想存储 pi（而不是计算它），您可以从 here 复制值并将其存储在 HighPrecision 变量中。
在 C++ 中可以做而在 C# 中不能做的是为文字创建别名。用 c++ #define M_PI 3.141592653589793238462643383279502884 编写并没有将 PI 定义到 36 个位置。它只是文字的别名。
#define M_PI 3.141592653589793238462643383279502884
double pi = M_PI;
写的完全一样
double pi = 3.141592653589793238462643383279502884;
这在 C++ 和 C# 中的工作方式几乎相同。在 C++ 中，该值将具有与 C# 中相同的精度，因为文字将被解释为 double，我认为两者在本质上是相同的。
之所以将 M_PI 声明为最多 36 位数字，是因为这是 quad double 的最大精度。遗憾的是，在 C# 中没有如此精确的本机类型。 在 C++ 中，你不会从 M_PI 中得到任何特别的东西，除非你将它用作 quad double。
具有别名文字并没有多大帮助，它只允许写：
float pif = M_PI;
int pii = M_PI;
double pid = M_PI;
quad double piq = M_PI;
很遗憾，您需要使用自定义类型。
我发现这个问题很有趣，所以这是我的 2 美分。
我采用 Machin 公式 implementation, and applied it to the IntXLib implementation of big integers (which have optimized multiply and divide operators with Discrete Hartley transform).
最后，我将结果与 Pi value posted here 进行了比较。
这里，2500 位小数是在 i7 920 @2.66 Ghz 上用 138 毫秒计算出来的。
static void Main(string[] args)
{
String referencePi = "3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679821480865132823066470938446095505822317253594081284811174502841027019385211055596446229489549303819644288109756659334461284756482337867831652712019091456485669234603486104543266482133936072602491412737245870066063155881748815209209628292540917153643678925903600113305305488204665213841469519415116094330572703657595919530921861173819326117931051185480744623799627495673518857527248912279381830119491298336733624406566430860213949463952247371907021798609437027705392171762931767523846748184676694051320005681271452635608277857713427577896091736371787214684409012249534301465495853710507922796892589235420199561121290219608640344181598136297747713099605187072113499999983729780499510597317328160963185950244594553469083026425223082533446850352619311881710100031378387528865875332083814206171776691473035982534904287554687311595628638823537875937519577818577805321712268066130019278766111959092164201989380952572010654858632788659361533818279682303019520353018529689957736225994138912497217752834791315155748572424541506959508295331168617278558890750983817546374649393192550604009277016711390098488240128583616035637076601047101819429555961989467678374494482553797747268471040475346462080466842590694912933136770289891521047521620569660240580381501935112533824300355876402474964732639141992726042699227967823547816360093417216412199245863150302861829745557067498385054945885869269956909272107975093029553211653449872027559602364806654991198818347977535663698074265425278625518184175746728909777727938000816470600161452491921732172147723501414419735685481613611573525521334757418494684385233239073941433345477624168625189835694855620992192221842725502542568876717904946016534668049886272327917860857843838279679766814541009538837863609506800642251252051173929848960841284886269456042419652850222106611863067442786220391949450471237137869609563643719172874677646575739624138908658326459958133904780275900994657640789512694683983525957098258226205224894077267194782684826014769909026401363944374553050682034962524517493996514314298091906592509372216964615157098583874105978859597729754989301617539284681382686838689427741559918559252459539594310499725246808459872736446958486538367362226260991246080512438843904512441365497627807977156914359977001296160894416948685558484063534220722258284886481584560285060168427394522674676788952521385225499546667278239864565961163548862305774564980355936345681743241125";
Stopwatch watch = new Stopwatch();
watch.Restart();
DecimalX calculatedPi = PiHelper.Calculate(2500);
watch.Stop();
Console.WriteLine("Pi with 2500 decimals in " + watch.ElapsedMilliseconds + " ms");
String hmmmmm = calculatedPi.ToString();
if (hmmmmm == referencePi)
Console.WriteLine("Pi approximation found");
}
public class DecimalX
{
/// Integer represatation of the decimal
private readonly IntX _integerPart;
/// Power of 10 (10^X)
private readonly uint _scale;
public DecimalX(IntX integerPart, uint scale)
{
_integerPart = integerPart;
_scale = scale;
}
public override string ToString()
{
IntX afterPoint = null;
IntX beforePoint = IntX.DivideModulo(_integerPart, IntX.Pow(10, _scale, MultiplyMode.AutoFht), out afterPoint, DivideMode.AutoNewton);
return beforePoint.ToString() + "." + afterPoint.ToString();
}
}
public class PiHelper
{
public static IntX InverseTan(int denominator, int numberOfDigitsRequired)
{
int demonimatorSquared = denominator * denominator;
int degreeNeeded = GetDegreeOfPrecisionNeeded(demonimatorSquared, numberOfDigitsRequired);
IntX tenToNumberPowerOfDigitsRequired = IntX.Pow(10, (uint)numberOfDigitsRequired, MultiplyMode.AutoFht);
IntX s = IntX.Divide(tenToNumberPowerOfDigitsRequired, new IntX(2 * degreeNeeded + 1), DivideMode.AutoNewton); // s = (10^N)/c
int c = 2 * degreeNeeded + 1;
for (int i = 0; i < degreeNeeded; i++)
{
c = c - 2;
var temp1 = IntX.Divide(tenToNumberPowerOfDigitsRequired, new IntX(c), DivideMode.AutoNewton);
var temp2 = IntX.Divide(s, new IntX(demonimatorSquared), DivideMode.AutoNewton);
s = temp1 - temp2;
}
return IntX.Divide(s, new IntX(denominator), DivideMode.AutoNewton);
}
private static int GetDegreeOfPrecisionNeeded(int demonimatorSquared, int numberOfDigitsRequired)
{
int degreeNeeded = 0;
while ((Math.Log(2 * degreeNeeded + 3) + (degreeNeeded + 1) * Math.Log10(demonimatorSquared)) <= numberOfDigitsRequired * Math.Log(10))
degreeNeeded++;
return degreeNeeded;
}
public static DecimalX Calculate(int numberOfDigitsRequired)
{
int max = numberOfDigitsRequired + 8; // To be safe, compute 8 extra digits, to be dropped at end. The 8 is arbitrary
var a = IntX.Multiply(InverseTan(5, max), new IntX(16), MultiplyMode.AutoFht); //16 x arctan(1/5)
var b = IntX.Multiply(InverseTan(239, max), new IntX(4), MultiplyMode.AutoFht); //4 x arctan(1/239)
return new DecimalX(IntX.Divide(a - b, IntX.Pow(10, (uint)8), DivideMode.AutoNewton), (uint)numberOfDigitsRequired);
}
}
现在，您只需使用 "normal" 类型（如 float / double / int / 等）在 DecimalX 类型上实现所有运算符，但我认为这很容易！
在 C++ 中可以做到这一点
ifndef M_PI
define M_PI 3.141592653589793238462643383279502884
endif
然而，在 C# 中，我尝试声明一个 double var 并为其分配具有 36 位小数的 PI 值。
这不起作用，因为 double 存储最多 16 位小数 (MSDN docx)。我尝试创建一个隐式类型的局部变量，但编译器只是将其声明为双精度类型，我又回到了原点。也许我可以在 C# 中包含 cmath？
如何在 C# 中将 PI 定义为 36 位小数
据我所知，C# 在 type decimal 中的精度最高，但它最多只能达到 28-29 位有效小数位。我不认为你可以在 C# 中覆盖多达 36 个十进制数字而不使用自定义库。
如果你真的需要那么高的精度，你可以使用任意精度的 C 库，比如 GNU MPC or GNU MPFR. As they are C libraries so you'd need to use interop 来使用它们...但这会限制 使用你的高精度pi到使用你选择的包...
Here is an article about calculating pi to 1 million digit in C#，你可以缩小到36位
这里有一个示例代码，我建议阅读全文及其背后的逻辑。
public static HighPrecision GetPi(int digits)
{
HighPrecision.Precision = digits;
HighPrecision first = 4 * Atan(5);
HighPrecision second = Atan(239);
return 4 * (first - second);
}
如果您只想存储 pi（而不是计算它），您可以从 here 复制值并将其存储在 HighPrecision 变量中。
在 C++ 中可以做而在 C# 中不能做的是为文字创建别名。用 c++ #define M_PI 3.141592653589793238462643383279502884 编写并没有将 PI 定义到 36 个位置。它只是文字的别名。
#define M_PI 3.141592653589793238462643383279502884
double pi = M_PI;
写的完全一样
double pi = 3.141592653589793238462643383279502884;
这在 C++ 和 C# 中的工作方式几乎相同。在 C++ 中，该值将具有与 C# 中相同的精度，因为文字将被解释为 double，我认为两者在本质上是相同的。
之所以将 M_PI 声明为最多 36 位数字，是因为这是 quad double 的最大精度。遗憾的是，在 C# 中没有如此精确的本机类型。 在 C++ 中，你不会从 M_PI 中得到任何特别的东西，除非你将它用作 quad double。
具有别名文字并没有多大帮助，它只允许写：
float pif = M_PI;
int pii = M_PI;
double pid = M_PI;
quad double piq = M_PI;
很遗憾，您需要使用自定义类型。
我发现这个问题很有趣，所以这是我的 2 美分。
我采用 Machin 公式 implementation, and applied it to the IntXLib implementation of big integers (which have optimized multiply and divide operators with Discrete Hartley transform).
最后，我将结果与 Pi value posted here 进行了比较。
这里，2500 位小数是在 i7 920 @2.66 Ghz 上用 138 毫秒计算出来的。
static void Main(string[] args)
{
String referencePi = "3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679821480865132823066470938446095505822317253594081284811174502841027019385211055596446229489549303819644288109756659334461284756482337867831652712019091456485669234603486104543266482133936072602491412737245870066063155881748815209209628292540917153643678925903600113305305488204665213841469519415116094330572703657595919530921861173819326117931051185480744623799627495673518857527248912279381830119491298336733624406566430860213949463952247371907021798609437027705392171762931767523846748184676694051320005681271452635608277857713427577896091736371787214684409012249534301465495853710507922796892589235420199561121290219608640344181598136297747713099605187072113499999983729780499510597317328160963185950244594553469083026425223082533446850352619311881710100031378387528865875332083814206171776691473035982534904287554687311595628638823537875937519577818577805321712268066130019278766111959092164201989380952572010654858632788659361533818279682303019520353018529689957736225994138912497217752834791315155748572424541506959508295331168617278558890750983817546374649393192550604009277016711390098488240128583616035637076601047101819429555961989467678374494482553797747268471040475346462080466842590694912933136770289891521047521620569660240580381501935112533824300355876402474964732639141992726042699227967823547816360093417216412199245863150302861829745557067498385054945885869269956909272107975093029553211653449872027559602364806654991198818347977535663698074265425278625518184175746728909777727938000816470600161452491921732172147723501414419735685481613611573525521334757418494684385233239073941433345477624168625189835694855620992192221842725502542568876717904946016534668049886272327917860857843838279679766814541009538837863609506800642251252051173929848960841284886269456042419652850222106611863067442786220391949450471237137869609563643719172874677646575739624138908658326459958133904780275900994657640789512694683983525957098258226205224894077267194782684826014769909026401363944374553050682034962524517493996514314298091906592509372216964615157098583874105978859597729754989301617539284681382686838689427741559918559252459539594310499725246808459872736446958486538367362226260991246080512438843904512441365497627807977156914359977001296160894416948685558484063534220722258284886481584560285060168427394522674676788952521385225499546667278239864565961163548862305774564980355936345681743241125";
Stopwatch watch = new Stopwatch();
watch.Restart();
DecimalX calculatedPi = PiHelper.Calculate(2500);
watch.Stop();
Console.WriteLine("Pi with 2500 decimals in " + watch.ElapsedMilliseconds + " ms");
String hmmmmm = calculatedPi.ToString();
if (hmmmmm == referencePi)
Console.WriteLine("Pi approximation found");
}
public class DecimalX
{
/// Integer represatation of the decimal
private readonly IntX _integerPart;
/// Power of 10 (10^X)
private readonly uint _scale;
public DecimalX(IntX integerPart, uint scale)
{
_integerPart = integerPart;
_scale = scale;
}
public override string ToString()
{
IntX afterPoint = null;
IntX beforePoint = IntX.DivideModulo(_integerPart, IntX.Pow(10, _scale, MultiplyMode.AutoFht), out afterPoint, DivideMode.AutoNewton);
return beforePoint.ToString() + "." + afterPoint.ToString();
}
}
public class PiHelper
{
public static IntX InverseTan(int denominator, int numberOfDigitsRequired)
{
int demonimatorSquared = denominator * denominator;
int degreeNeeded = GetDegreeOfPrecisionNeeded(demonimatorSquared, numberOfDigitsRequired);
IntX tenToNumberPowerOfDigitsRequired = IntX.Pow(10, (uint)numberOfDigitsRequired, MultiplyMode.AutoFht);
IntX s = IntX.Divide(tenToNumberPowerOfDigitsRequired, new IntX(2 * degreeNeeded + 1), DivideMode.AutoNewton); // s = (10^N)/c
int c = 2 * degreeNeeded + 1;
for (int i = 0; i < degreeNeeded; i++)
{
c = c - 2;
var temp1 = IntX.Divide(tenToNumberPowerOfDigitsRequired, new IntX(c), DivideMode.AutoNewton);
var temp2 = IntX.Divide(s, new IntX(demonimatorSquared), DivideMode.AutoNewton);
s = temp1 - temp2;
}
return IntX.Divide(s, new IntX(denominator), DivideMode.AutoNewton);
}
private static int GetDegreeOfPrecisionNeeded(int demonimatorSquared, int numberOfDigitsRequired)
{
int degreeNeeded = 0;
while ((Math.Log(2 * degreeNeeded + 3) + (degreeNeeded + 1) * Math.Log10(demonimatorSquared)) <= numberOfDigitsRequired * Math.Log(10))
degreeNeeded++;
return degreeNeeded;
}
public static DecimalX Calculate(int numberOfDigitsRequired)
{
int max = numberOfDigitsRequired + 8; // To be safe, compute 8 extra digits, to be dropped at end. The 8 is arbitrary
var a = IntX.Multiply(InverseTan(5, max), new IntX(16), MultiplyMode.AutoFht); //16 x arctan(1/5)
var b = IntX.Multiply(InverseTan(239, max), new IntX(4), MultiplyMode.AutoFht); //4 x arctan(1/239)
return new DecimalX(IntX.Divide(a - b, IntX.Pow(10, (uint)8), DivideMode.AutoNewton), (uint)numberOfDigitsRequired);
}
}
现在，您只需使用 "normal" 类型（如 float / double / int / 等）在 DecimalX 类型上实现所有运算符，但我认为这很容易！