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什么是获取π值最快的方法？

performancealgorithmlanguage-agnosticunixpi
353
我正在寻找最快的方法来获取π的值，这是一个个人挑战。更具体地说，我正在使用不涉及使用#define常量，如M_PI或硬编码数字的方式。
下面的程序测试了我所知道的各种方式。理论上，内联汇编版本是最快的选择，尽管显然不可移植。我将其包含在内作为与其他版本进行比较的基线。在我的测试中，使用内置函数，4 * atan(1)版本在GCC 4.2上最快，因为它自动折叠atan(1)成一个常数。指定-fno-builtin后，atan2(0，-1)版本最快。
以下是主要测试程序（pitimes.c）：
#include <math.h>
#include <stdio.h>
#include <time.h>

#define ITERS 10000000
#define TESTWITH(x) {                                                       \
    diff = 0.0;                                                             \
    time1 = clock();                                                        \
    for (i = 0; i < ITERS; ++i)                                             \
        diff += (x) - M_PI;                                                 \
    time2 = clock();                                                        \
    printf("%s\t=> %e, time => %f\n", #x, diff, diffclock(time2, time1));   \
}

static inline double
diffclock(clock_t time1, clock_t time0)
{
    return (double) (time1 - time0) / CLOCKS_PER_SEC;
}

int
main()
{
    int i;
    clock_t time1, time2;
    double diff;

    /* Warmup. The atan2 case catches GCC's atan folding (which would
     * optimise the ``4 * atan(1) - M_PI'' to a no-op), if -fno-builtin
     * is not used. */
    TESTWITH(4 * atan(1))
    TESTWITH(4 * atan2(1, 1))

#if defined(__GNUC__) && (defined(__i386__) || defined(__amd64__))
    extern double fldpi();
    TESTWITH(fldpi())
#endif

    /* Actual tests start here. */
    TESTWITH(atan2(0, -1))
    TESTWITH(acos(-1))
    TESTWITH(2 * asin(1))
    TESTWITH(4 * atan2(1, 1))
    TESTWITH(4 * atan(1))

    return 0;
}

而内联汇编的东西（fldpi.c）只适用于x86和x64系统：

double
fldpi()
{
    double pi;
    asm("fldpi" : "=t" (pi));
    return pi;
}

还有一个构建脚本，用于构建我正在测试的所有配置（build.sh）：

#!/bin/sh
gcc -O3 -Wall -c           -m32 -o fldpi-32.o fldpi.c
gcc -O3 -Wall -c           -m64 -o fldpi-64.o fldpi.c

gcc -O3 -Wall -ffast-math  -m32 -o pitimes1-32 pitimes.c fldpi-32.o
gcc -O3 -Wall              -m32 -o pitimes2-32 pitimes.c fldpi-32.o -lm
gcc -O3 -Wall -fno-builtin -m32 -o pitimes3-32 pitimes.c fldpi-32.o -lm
gcc -O3 -Wall -ffast-math  -m64 -o pitimes1-64 pitimes.c fldpi-64.o -lm
gcc -O3 -Wall              -m64 -o pitimes2-64 pitimes.c fldpi-64.o -lm
gcc -O3 -Wall -fno-builtin -m64 -o pitimes3-64 pitimes.c fldpi-64.o -lm

除了在各种编译器标志之间进行测试（我也比较了32位和64位，因为优化不同），我还尝试过改变测试顺序。但是，每次都是atan2(0, -1)版本最优。

2
你为什么认为使用atan(1)与使用M_PI不同？如果你只使用算术运算，我可以理解你为什么这样做，但是对于atan，我看不出有什么意义。 - erikkallen
11
为什么你不想使用常量？比如由库定义或自己定义的常量？计算圆周率是浪费CPU周期，因为该问题已经被反复解决，并得出了比日常计算所需更多的有效数字。 - Tilo
1
除了预先计算常数π之外，只有一种解决方案更快：预先计算公式中出现的所有值，例如当需要周长时，您可以预先计算2*PI而不是在运行时每次将PI乘以2。 - ern0
5
在我所说的英语方言中，“optimisation”的拼写是带有“s”而不是“z”的（顺便提一下，“z”读作“zed”，而不是“zee” ;-))。如果您查看审查历史记录，您会发现这并不是我第一次撤销这种编辑。 - C. K. Young
3
@Pacerier 请见http://en.wiktionary.org/wiki/boggle 和 http://en.wiktionary.org/wiki/mindboggling。 - C. K. Young
显示剩余12条评论
24个回答
1

从圆的面积计算π :-)

<input id="range" type="range" min="10" max="960" value="10" step="50" oninput="calcPi()">
<br>
<div id="cont"></div>

<script>
function generateCircle(width) {
    var c = width/2;
    var delta = 1.0;
    var str = "";
    var xCount = 0;
    for (var x=0; x <= width; x++) {
        for (var y = 0; y <= width; y++) {
            var d = Math.sqrt((x-c)*(x-c) + (y-c)*(y-c));
            if (d > (width-1)/2) {
                str += '.';
            }
            else {
                xCount++;
                str += 'o';
            }
            str += "&nbsp;" 
        }
        str += "\n";
    }
    var pi = (xCount * 4) / (width * width);
    return [str, pi];
}

function calcPi() {
    var e = document.getElementById("cont");
    var width = document.getElementById("range").value;
    e.innerHTML = "<h4>Generating circle...</h4>";
    setTimeout(function() {
        var circ = generateCircle(width);
        e.innerHTML  = "<pre>" + "π = " + circ[1].toFixed(2) + "\n" + circ[0] +"</pre>";
    }, 200);
}
calcPi();
</script>

1

如果您不介意执行平方根和一些倒数操作，那么Chudnovsky算法非常快速。它只需要2次迭代就可以收敛到双精度。

/*
    Chudnovsky algorithm for computing PI
*/

#include <iostream>
#include <cmath>
using namespace std;

double calc_PI(int K=2) {

    static const int A = 545140134;
    static const int B = 13591409;
    static const int D = 640320;

    const double ID3 = 1./ (double(D)*double(D)*double(D));

    double sum = 0.;
    double b   = sqrt(ID3);
    long long int p = 1;
    long long int a = B;

    sum += double(p) * double(a)* b;

    // 2 iterations enough for double convergence
    for (int k=1; k<K; ++k) {
        // A*k + B
        a += A;
        // update denominator
        b *= ID3;
        // p = (-1)^k 6k! / 3k! k!^3
        p *= (6*k)*(6*k-1)*(6*k-2)*(6*k-3)*(6*k-4)*(6*k-5);
        p /= (3*k)*(3*k-1)*(3*k-2) * k*k*k;
        p = -p;

        sum += double(p) * double(a)* b;
    }

    return 1./(12*sum);
}

int main() {

    cout.precision(16);
    cout.setf(ios::fixed);

    for (int k=1; k<=5; ++k) cout << "k = " << k << "   PI = " << calc_PI(k) << endl;

    return 0;
}

结果：

k = 1   PI = 3.1415926535897341
k = 2   PI = 3.1415926535897931
k = 3   PI = 3.1415926535897931
k = 4   PI = 3.1415926535897931
k = 5   PI = 3.1415926535897931
0

更好的方法

要获取标准常量如pi或标准概念的输出，我们应该首先使用你正在使用的语言中可用的内置方法。这将以最快和最佳的方式返回值。我正在使用Python运行最快的方法来获取pi的值。

  • 数学库中的pi变量。 数学库将变量pi存储为常量。

math_pi.py

import math
print math.pi

使用Linux的时间工具运行脚本 /usr/bin/time -v python math_pi.py

输出：

Command being timed: "python math_pi.py"
User time (seconds): 0.01
System time (seconds): 0.01
Percent of CPU this job got: 91%
Elapsed (wall clock) time (h:mm:ss or m:ss): 0:00.03
  • 使用数学中的反余弦方法

acos_pi.py

import math
print math.acos(-1)

使用Linux的时间工具运行脚本 /usr/bin/time -v python acos_pi.py

输出：

Command being timed: "python acos_pi.py"
User time (seconds): 0.02
System time (seconds): 0.01
Percent of CPU this job got: 94%
Elapsed (wall clock) time (h:mm:ss or m:ss): 0:00.03

bbp_pi.py

from decimal import Decimal, getcontext
getcontext().prec=100
print sum(1/Decimal(16)**k * 
          (Decimal(4)/(8*k+1) - 
           Decimal(2)/(8*k+4) - 
           Decimal(1)/(8*k+5) -
           Decimal(1)/(8*k+6)) for k in range(100))

使用Linux的时间工具运行脚本 /usr/bin/time -v python bbp_pi.py

输出：

Command being timed: "python c.py"
User time (seconds): 0.05
System time (seconds): 0.01
Percent of CPU this job got: 98%
Elapsed (wall clock) time (h:mm:ss or m:ss): 0:00.06

因此，最好的方法是使用语言提供的内置方法，因为它们是最快和最好的获取输出的方式。在Python中使用math.pi。

0
比 GMPY2 和 MPmath 内置函数更快：45 分钟内完成十亿次运算。

我尝试了几种方法：Manchin、AGM和Chudnovsky Bros。使用二进制分裂的Chudnovsky是最快的：
我的Github链接：https://github.com/Overboard-code/Pi-Pourri

我的二进制分裂Chudnovsky速度大约是内置的gmpy2.const_pi()的两倍。MPmath.mp.pi()对于十亿位数需要50分钟，所以它几乎和Chudnovsky一样快。

我也非常感谢性能优化的建议。我不确定我的代码是否完美。它是100%准确的（所有公式都达到了1亿），但也许可以更快？

我尝试将gmpy2.const_pi()计算到1亿位数，需要300秒，而在同一台机器上，Chudnovsky只需要150秒。pi.txt和pi2.txt是相同的。

在我的旧i7 16GB笔记本电脑上，我在不到一个小时的时间内就得到了10亿位数。

这里是我尝试过的12种方法中最快的一段代码：

class PiChudnovsky:
    """Version of Chudnovsky Bros using Binary Splitting 
        So far this is the winner for fastest time to a million digits on my older intel i7
    """
    A = mpz(13591409)
    B = mpz(545140134)
    C = mpz(640320)
    D = mpz(426880)
    E = mpz(10005)
    C3_24  = pow(C, mpz(3)) // mpz(24)
    #DIGITS_PER_TERM = math.log(53360 ** 3) / math.log(10)  #=> 14.181647462725476
    DIGITS_PER_TERM = 14.181647462725476
    MMILL = mpz(1000000)

    def __init__(self,ndigits):
        """ Initialization
        :param int ndigits: digits of PI computation
        """
        self.ndigits = ndigits
        self.n      = mpz(self.ndigits // self.DIGITS_PER_TERM + 1)
        self.prec   = mpz((self.ndigits + 1) * LOG2_10)
        self.one_sq = pow(mpz(10),mpz(2 * ndigits))
        self.sqrt_c = isqrt(self.E * self.one_sq)
        self.iters  = mpz(0)
        self.start_time = 0

    def compute(self):
        """ Computation """
        try:
            self.start_time = time.time()
            logging.debug("Starting {} formula to {:,} decimal places"
                .format(name,ndigits) )
            __, q, t = self.__bs(mpz(0), self.n)  # p is just for recursion
            pi = (q * self.D * self.sqrt_c) // t
            logging.debug('{} calulation Done! {:,} iterations and {:.2f} seconds.'
                .format( name, int(self.iters),time.time() - self.start_time))
            get_context().precision= int((self.ndigits+10) * LOG2_10)
            pi_s = pi.digits() # digits() gmpy2 creates a string 
            pi_o = pi_s[:1] + "." + pi_s[1:]
            return pi_o,int(self.iters),time.time() - self.start_time
        except Exception as e:
            print (e.message, e.args)
            raise

    def __bs(self, a, b):
        """ PQT computation by BSA(= Binary Splitting Algorithm)
        :param int a: positive integer
        :param int b: positive integer
        :return list [int p_ab, int q_ab, int t_ab]
        """
        try:
            self.iters += mpz(1)
            if self.iters % self.MMILL  == mpz(0):
                logging.debug('Chudnovsky ... {:,} iterations and {:.2f} seconds.'
                    .format( int(self.iters),time.time() - self.start_time))
            if a + mpz(1) == b:
                if a == mpz(0):
                    p_ab = q_ab = mpz(1)
                else:
                    p_ab = mpz((mpz(6) * a - mpz(5)) * (mpz(2) * a - mpz(1)) * (mpz(6) * a - mpz(1)))
                    q_ab = pow(a,mpz(3)) * self.C3_24
                t_ab = p_ab * (self.A + self.B * a)
                if a & 1:
                    t_ab *= mpz(-1)
            else:
                m = (a + b) // mpz(2)
                p_am, q_am, t_am = self.__bs(a, m)
                p_mb, q_mb, t_mb = self.__bs(m, b)
                p_ab = p_am * p_mb
                q_ab = q_am * q_mb
                t_ab = q_mb * t_am + p_am * t_mb
            return [p_ab, q_ab, t_ab]
        except Exception as e:
            print (e.message, e.args)
            raise

以下是在45分钟内输出的10亿位数字：

python pi-pourri.py -v -d 1,000,000,000 -a 10 

[INFO] 2022-10-03 09:22:51,860 <module>: MainProcess Computing π to 1,000,000,000 digits.
[DEBUG] 2022-10-03 09:25:00,543 compute: MainProcess Starting   Chudnovsky brothers  1988 
    π = (Q(0, N) / 12T(0, N) + 12AQ(0, N))**(C**(3/2))
 formula to 1,000,000,000 decimal places
[DEBUG] 2022-10-03 09:25:04,995 __bs: MainProcess Chudnovsky ... 1,000,000 iterations and 4.45 seconds.
[DEBUG] 2022-10-03 09:25:10,836 __bs: MainProcess Chudnovsky ... 2,000,000 iterations and 10.29 seconds.
[DEBUG] 2022-10-03 09:25:18,227 __bs: MainProcess Chudnovsky ... 3,000,000 iterations and 17.68 seconds.
[DEBUG] 2022-10-03 09:25:24,512 __bs: MainProcess Chudnovsky ... 4,000,000 iterations and 23.97 seconds.
[DEBUG] 2022-10-03 09:25:35,670 __bs: MainProcess Chudnovsky ... 5,000,000 iterations and 35.13 seconds.
[DEBUG] 2022-10-03 09:25:41,376 __bs: MainProcess Chudnovsky ... 6,000,000 iterations and 40.83 seconds.
[DEBUG] 2022-10-03 09:25:49,238 __bs: MainProcess Chudnovsky ... 7,000,000 iterations and 48.69 seconds.
[DEBUG] 2022-10-03 09:25:55,646 __bs: MainProcess Chudnovsky ... 8,000,000 iterations and 55.10 seconds.
[DEBUG] 2022-10-03 09:26:15,043 __bs: MainProcess Chudnovsky ... 9,000,000 iterations and 74.50 seconds.
[DEBUG] 2022-10-03 09:26:21,437 __bs: MainProcess Chudnovsky ... 10,000,000 iterations and 80.89 seconds.
[DEBUG] 2022-10-03 09:26:26,587 __bs: MainProcess Chudnovsky ... 11,000,000 iterations and 86.04 seconds.
[DEBUG] 2022-10-03 09:26:34,777 __bs: MainProcess Chudnovsky ... 12,000,000 iterations and 94.23 seconds.
[DEBUG] 2022-10-03 09:26:41,231 __bs: MainProcess Chudnovsky ... 13,000,000 iterations and 100.69 seconds.
[DEBUG] 2022-10-03 09:26:52,972 __bs: MainProcess Chudnovsky ... 14,000,000 iterations and 112.43 seconds.
[DEBUG] 2022-10-03 09:26:59,517 __bs: MainProcess Chudnovsky ... 15,000,000 iterations and 118.97 seconds.
[DEBUG] 2022-10-03 09:27:07,932 __bs: MainProcess Chudnovsky ... 16,000,000 iterations and 127.39 seconds.
[DEBUG] 2022-10-03 09:27:14,036 __bs: MainProcess Chudnovsky ... 17,000,000 iterations and 133.49 seconds.
[DEBUG] 2022-10-03 09:27:51,629 __bs: MainProcess Chudnovsky ... 18,000,000 iterations and 171.09 seconds.
[DEBUG] 2022-10-03 09:27:58,176 __bs: MainProcess Chudnovsky ... 19,000,000 iterations and 177.63 seconds.
[DEBUG] 2022-10-03 09:28:06,704 __bs: MainProcess Chudnovsky ... 20,000,000 iterations and 186.16 seconds.
[DEBUG] 2022-10-03 09:28:13,376 __bs: MainProcess Chudnovsky ... 21,000,000 iterations and 192.83 seconds.
[DEBUG] 2022-10-03 09:28:18,737 __bs: MainProcess Chudnovsky ... 22,000,000 iterations and 198.19 seconds.
[DEBUG] 2022-10-03 09:28:31,095 __bs: MainProcess Chudnovsky ... 23,000,000 iterations and 210.55 seconds.
[DEBUG] 2022-10-03 09:28:37,789 __bs: MainProcess Chudnovsky ... 24,000,000 iterations and 217.25 seconds.
[DEBUG] 2022-10-03 09:28:46,171 __bs: MainProcess Chudnovsky ... 25,000,000 iterations and 225.63 seconds.
[DEBUG] 2022-10-03 09:28:52,933 __bs: MainProcess Chudnovsky ... 26,000,000 iterations and 232.39 seconds.
[DEBUG] 2022-10-03 09:29:13,524 __bs: MainProcess Chudnovsky ... 27,000,000 iterations and 252.98 seconds.
[DEBUG] 2022-10-03 09:29:19,676 __bs: MainProcess Chudnovsky ... 28,000,000 iterations and 259.13 seconds.
[DEBUG] 2022-10-03 09:29:28,196 __bs: MainProcess Chudnovsky ... 29,000,000 iterations and 267.65 seconds.
[DEBUG] 2022-10-03 09:29:34,720 __bs: MainProcess Chudnovsky ... 30,000,000 iterations and 274.18 seconds.
[DEBUG] 2022-10-03 09:29:47,075 __bs: MainProcess Chudnovsky ... 31,000,000 iterations and 286.53 seconds.
[DEBUG] 2022-10-03 09:29:53,746 __bs: MainProcess Chudnovsky ... 32,000,000 iterations and 293.20 seconds.
[DEBUG] 2022-10-03 09:29:59,099 __bs: MainProcess Chudnovsky ... 33,000,000 iterations and 298.56 seconds.
[DEBUG] 2022-10-03 09:30:07,511 __bs: MainProcess Chudnovsky ... 34,000,000 iterations and 306.97 seconds.
[DEBUG] 2022-10-03 09:30:14,279 __bs: MainProcess Chudnovsky ... 35,000,000 iterations and 313.74 seconds.
[DEBUG] 2022-10-03 09:31:31,710 __bs: MainProcess Chudnovsky ... 36,000,000 iterations and 391.17 seconds.
[DEBUG] 2022-10-03 09:31:38,454 __bs: MainProcess Chudnovsky ... 37,000,000 iterations and 397.91 seconds.
[DEBUG] 2022-10-03 09:31:46,437 __bs: MainProcess Chudnovsky ... 38,000,000 iterations and 405.89 seconds.
[DEBUG] 2022-10-03 09:31:53,285 __bs: MainProcess Chudnovsky ... 39,000,000 iterations and 412.74 seconds.
[DEBUG] 2022-10-03 09:32:05,602 __bs: MainProcess Chudnovsky ... 40,000,000 iterations and 425.06 seconds.
[DEBUG] 2022-10-03 09:32:12,220 __bs: MainProcess Chudnovsky ... 41,000,000 iterations and 431.68 seconds.
[DEBUG] 2022-10-03 09:32:20,708 __bs: MainProcess Chudnovsky ... 42,000,000 iterations and 440.17 seconds.
[DEBUG] 2022-10-03 09:32:27,552 __bs: MainProcess Chudnovsky ... 43,000,000 iterations and 447.01 seconds.
[DEBUG] 2022-10-03 09:32:32,986 __bs: MainProcess Chudnovsky ... 44,000,000 iterations and 452.44 seconds.
[DEBUG] 2022-10-03 09:32:53,904 __bs: MainProcess Chudnovsky ... 45,000,000 iterations and 473.36 seconds.
[DEBUG] 2022-10-03 09:33:00,832 __bs: MainProcess Chudnovsky ... 46,000,000 iterations and 480.29 seconds.
[DEBUG] 2022-10-03 09:33:09,198 __bs: MainProcess Chudnovsky ... 47,000,000 iterations and 488.66 seconds.
[DEBUG] 2022-10-03 09:33:16,000 __bs: MainProcess Chudnovsky ... 48,000,000 iterations and 495.46 seconds.
[DEBUG] 2022-10-03 09:33:27,921 __bs: MainProcess Chudnovsky ... 49,000,000 iterations and 507.38 seconds.
[DEBUG] 2022-10-03 09:33:34,778 __bs: MainProcess Chudnovsky ... 50,000,000 iterations and 514.24 seconds.
[DEBUG] 2022-10-03 09:33:43,298 __bs: MainProcess Chudnovsky ... 51,000,000 iterations and 522.76 seconds.
[DEBUG] 2022-10-03 09:33:49,959 __bs: MainProcess Chudnovsky ... 52,000,000 iterations and 529.42 seconds.
[DEBUG] 2022-10-03 09:34:29,294 __bs: MainProcess Chudnovsky ... 53,000,000 iterations and 568.75 seconds.
[DEBUG] 2022-10-03 09:34:36,176 __bs: MainProcess Chudnovsky ... 54,000,000 iterations and 575.63 seconds.
[DEBUG] 2022-10-03 09:34:41,576 __bs: MainProcess Chudnovsky ... 55,000,000 iterations and 581.03 seconds.
[DEBUG] 2022-10-03 09:34:50,161 __bs: MainProcess Chudnovsky ... 56,000,000 iterations and 589.62 seconds.
[DEBUG] 2022-10-03 09:34:56,811 __bs: MainProcess Chudnovsky ... 57,000,000 iterations and 596.27 seconds.
[DEBUG] 2022-10-03 09:35:09,382 __bs: MainProcess Chudnovsky ... 58,000,000 iterations and 608.84 seconds.
[DEBUG] 2022-10-03 09:35:16,206 __bs: MainProcess Chudnovsky ... 59,000,000 iterations and 615.66 seconds.
[DEBUG] 2022-10-03 09:35:24,295 __bs: MainProcess Chudnovsky ... 60,000,000 iterations and 623.75 seconds.
[DEBUG] 2022-10-03 09:35:31,095 __bs: MainProcess Chudnovsky ... 61,000,000 iterations and 630.55 seconds.
[DEBUG] 2022-10-03 09:35:52,139 __bs: MainProcess Chudnovsky ... 62,000,000 iterations and 651.60 seconds.
[DEBUG] 2022-10-03 09:35:58,781 __bs: MainProcess Chudnovsky ... 63,000,000 iterations and 658.24 seconds.
[DEBUG] 2022-10-03 09:36:07,399 __bs: MainProcess Chudnovsky ... 64,000,000 iterations and 666.86 seconds.
[DEBUG] 2022-10-03 09:36:12,847 __bs: MainProcess Chudnovsky ... 65,000,000 iterations and 672.30 seconds.
[DEBUG] 2022-10-03 09:36:19,763 __bs: MainProcess Chudnovsky ... 66,000,000 iterations and 679.22 seconds.
[DEBUG] 2022-10-03 09:36:32,351 __bs: MainProcess Chudnovsky ... 67,000,000 iterations and 691.81 seconds.
[DEBUG] 2022-10-03 09:36:39,078 __bs: MainProcess Chudnovsky ... 68,000,000 iterations and 698.53 seconds.
[DEBUG] 2022-10-03 09:36:47,830 __bs: MainProcess Chudnovsky ... 69,000,000 iterations and 707.29 seconds.
[DEBUG] 2022-10-03 09:36:54,701 __bs: MainProcess Chudnovsky ... 70,000,000 iterations and 714.16 seconds.
[DEBUG] 2022-10-03 09:39:39,357 __bs: MainProcess Chudnovsky ... 71,000,000 iterations and 878.81 seconds.
[DEBUG] 2022-10-03 09:39:46,199 __bs: MainProcess Chudnovsky ... 72,000,000 iterations and 885.66 seconds.
[DEBUG] 2022-10-03 09:39:54,956 __bs: MainProcess Chudnovsky ... 73,000,000 iterations and 894.41 seconds.
[DEBUG] 2022-10-03 09:40:01,639 __bs: MainProcess Chudnovsky ... 74,000,000 iterations and 901.10 seconds.
[DEBUG] 2022-10-03 09:40:14,219 __bs: MainProcess Chudnovsky ... 75,000,000 iterations and 913.68 seconds.
[DEBUG] 2022-10-03 09:40:19,680 __bs: MainProcess Chudnovsky ... 76,000,000 iterations and 919.14 seconds.
[DEBUG] 2022-10-03 09:40:26,625 __bs: MainProcess Chudnovsky ... 77,000,000 iterations and 926.08 seconds.
[DEBUG] 2022-10-03 09:40:35,212 __bs: MainProcess Chudnovsky ... 78,000,000 iterations and 934.67 seconds.
[DEBUG] 2022-10-03 09:40:41,914 __bs: MainProcess Chudnovsky ... 79,000,000 iterations and 941.37 seconds.
[DEBUG] 2022-10-03 09:41:03,218 __bs: MainProcess Chudnovsky ... 80,000,000 iterations and 962.68 seconds.
[DEBUG] 2022-10-03 09:41:10,213 __bs: MainProcess Chudnovsky ... 81,000,000 iterations and 969.67 seconds.
[DEBUG] 2022-10-03 09:41:18,344 __bs: MainProcess Chudnovsky ... 82,000,000 iterations and 977.80 seconds.
[DEBUG] 2022-10-03 09:41:25,261 __bs: MainProcess Chudnovsky ... 83,000,000 iterations and 984.72 seconds.
[DEBUG] 2022-10-03 09:41:37,663 __bs: MainProcess Chudnovsky ... 84,000,000 iterations and 997.12 seconds.
[DEBUG] 2022-10-03 09:41:44,680 __bs: MainProcess Chudnovsky ... 85,000,000 iterations and 1004.14 seconds.
[DEBUG] 2022-10-03 09:41:53,411 __bs: MainProcess Chudnovsky ... 86,000,000 iterations and 1012.87 seconds.
[DEBUG] 2022-10-03 09:41:58,926 __bs: MainProcess Chudnovsky ... 87,000,000 iterations and 1018.38 seconds.
[DEBUG] 2022-10-03 09:42:05,858 __bs: MainProcess Chudnovsky ... 88,000,000 iterations and 1025.32 seconds.
[DEBUG] 2022-10-03 09:42:46,163 __bs: MainProcess Chudnovsky ... 89,000,000 iterations and 1065.62 seconds.
[DEBUG] 2022-10-03 09:42:53,054 __bs: MainProcess Chudnovsky ... 90,000,000 iterations and 1072.51 seconds.
[DEBUG] 2022-10-03 09:43:02,030 __bs: MainProcess Chudnovsky ... 91,000,000 iterations and 1081.49 seconds.
[DEBUG] 2022-10-03 09:43:09,192 __bs: MainProcess Chudnovsky ... 92,000,000 iterations and 1088.65 seconds.
[DEBUG] 2022-10-03 09:43:21,533 __bs: MainProcess Chudnovsky ... 93,000,000 iterations and 1100.99 seconds.
[DEBUG] 2022-10-03 09:43:28,643 __bs: MainProcess Chudnovsky ... 94,000,000 iterations and 1108.10 seconds.
[DEBUG] 2022-10-03 09:43:37,372 __bs: MainProcess Chudnovsky ... 95,000,000 iterations and 1116.83 seconds.
[DEBUG] 2022-10-03 09:43:44,558 __bs: MainProcess Chudnovsky ... 96,000,000 iterations and 1124.02 seconds.
[DEBUG] 2022-10-03 09:44:06,555 __bs: MainProcess Chudnovsky ... 97,000,000 iterations and 1146.01 seconds.
[DEBUG] 2022-10-03 09:44:12,220 __bs: MainProcess Chudnovsky ... 98,000,000 iterations and 1151.68 seconds.
[DEBUG] 2022-10-03 09:44:19,278 __bs: MainProcess Chudnovsky ... 99,000,000 iterations and 1158.74 seconds.
[DEBUG] 2022-10-03 09:44:28,323 __bs: MainProcess Chudnovsky ... 100,000,000 iterations and 1167.78 seconds.
[DEBUG] 2022-10-03 09:44:35,211 __bs: MainProcess Chudnovsky ... 101,000,000 iterations and 1174.67 seconds.
[DEBUG] 2022-10-03 09:44:48,331 __bs: MainProcess Chudnovsky ... 102,000,000 iterations and 1187.79 seconds.
[DEBUG] 2022-10-03 09:44:54,835 __bs: MainProcess Chudnovsky ... 103,000,000 iterations and 1194.29 seconds.
[DEBUG] 2022-10-03 09:45:03,869 __bs: MainProcess Chudnovsky ... 104,000,000 iterations and 1203.33 seconds.
[DEBUG] 2022-10-03 09:45:10,967 __bs: MainProcess Chudnovsky ... 105,000,000 iterations and 1210.42 seconds.
[DEBUG] 2022-10-03 09:46:32,760 __bs: MainProcess Chudnovsky ... 106,000,000 iterations and 1292.22 seconds.
[DEBUG] 2022-10-03 09:46:39,872 __bs: MainProcess Chudnovsky ... 107,000,000 iterations and 1299.33 seconds.
[DEBUG] 2022-10-03 09:46:48,948 __bs: MainProcess Chudnovsky ... 108,000,000 iterations and 1308.41 seconds.
[DEBUG] 2022-10-03 09:46:54,611 __bs: MainProcess Chudnovsky ... 109,000,000 iterations and 1314.07 seconds.
[DEBUG] 2022-10-03 09:47:01,727 __bs: MainProcess Chudnovsky ... 110,000,000 iterations and 1321.18 seconds.
[DEBUG] 2022-10-03 09:47:14,525 __bs: MainProcess Chudnovsky ... 111,000,000 iterations and 1333.98 seconds.
[DEBUG] 2022-10-03 09:47:21,682 __bs: MainProcess Chudnovsky ... 112,000,000 iterations and 1341.14 seconds.
[DEBUG] 2022-10-03 09:47:30,610 __bs: MainProcess Chudnovsky ... 113,000,000 iterations and 1350.07 seconds.
[DEBUG] 2022-10-03 09:47:37,176 __bs: MainProcess Chudnovsky ... 114,000,000 iterations and 1356.63 seconds.
[DEBUG] 2022-10-03 09:47:59,642 __bs: MainProcess Chudnovsky ... 115,000,000 iterations and 1379.10 seconds.
[DEBUG] 2022-10-03 09:48:06,702 __bs: MainProcess Chudnovsky ... 116,000,000 iterations and 1386.16 seconds.
[DEBUG] 2022-10-03 09:48:15,483 __bs: MainProcess Chudnovsky ... 117,000,000 iterations and 1394.94 seconds.
[DEBUG] 2022-10-03 09:48:22,537 __bs: MainProcess Chudnovsky ... 118,000,000 iterations and 1401.99 seconds.
[DEBUG] 2022-10-03 09:48:35,714 __bs: MainProcess Chudnovsky ... 119,000,000 iterations and 1415.17 seconds.
[DEBUG] 2022-10-03 09:48:41,321 __bs: MainProcess Chudnovsky ... 120,000,000 iterations and 1420.78 seconds.
[DEBUG] 2022-10-03 09:48:48,408 __bs: MainProcess Chudnovsky ... 121,000,000 iterations and 1427.87 seconds.
[DEBUG] 2022-10-03 09:48:57,138 __bs: MainProcess Chudnovsky ... 122,000,000 iterations and 1436.60 seconds.
[DEBUG] 2022-10-03 09:49:04,328 __bs: MainProcess Chudnovsky ... 123,000,000 iterations and 1443.79 seconds.
[DEBUG] 2022-10-03 09:49:46,274 __bs: MainProcess Chudnovsky ... 124,000,000 iterations and 1485.73 seconds.
[DEBUG] 2022-10-03 09:49:52,833 __bs: MainProcess Chudnovsky ... 125,000,000 iterations and 1492.29 seconds.
[DEBUG] 2022-10-03 09:50:01,786 __bs: MainProcess Chudnovsky ... 126,000,000 iterations and 1501.24 seconds.
[DEBUG] 2022-10-03 09:50:08,975 __bs: MainProcess Chudnovsky ... 127,000,000 iterations and 1508.43 seconds.
[DEBUG] 2022-10-03 09:50:21,850 __bs: MainProcess Chudnovsky ... 128,000,000 iterations and 1521.31 seconds.
[DEBUG] 2022-10-03 09:50:28,962 __bs: MainProcess Chudnovsky ... 129,000,000 iterations and 1528.42 seconds.
[DEBUG] 2022-10-03 09:50:34,594 __bs: MainProcess Chudnovsky ... 130,000,000 iterations and 1534.05 seconds.
[DEBUG] 2022-10-03 09:50:43,647 __bs: MainProcess Chudnovsky ... 131,000,000 iterations and 1543.10 seconds.
[DEBUG] 2022-10-03 09:50:50,724 __bs: MainProcess Chudnovsky ... 132,000,000 iterations and 1550.18 seconds.
[DEBUG] 2022-10-03 09:51:12,742 __bs: MainProcess Chudnovsky ... 133,000,000 iterations and 1572.20 seconds.
[DEBUG] 2022-10-03 09:51:19,799 __bs: MainProcess Chudnovsky ... 134,000,000 iterations and 1579.26 seconds.
[DEBUG] 2022-10-03 09:51:28,824 __bs: MainProcess Chudnovsky ... 135,000,000 iterations and 1588.28 seconds.
[DEBUG] 2022-10-03 09:51:35,324 __bs: MainProcess Chudnovsky ... 136,000,000 iterations and 1594.78 seconds.
[DEBUG] 2022-10-03 09:51:48,419 __bs: MainProcess Chudnovsky ... 137,000,000 iterations and 1607.88 seconds.
[DEBUG] 2022-10-03 09:51:55,634 __bs: MainProcess Chudnovsky ... 138,000,000 iterations and 1615.09 seconds.
[DEBUG] 2022-10-03 09:52:04,435 __bs: MainProcess Chudnovsky ... 139,000,000 iterations and 1623.89 seconds.
[DEBUG] 2022-10-03 09:52:11,583 __bs: MainProcess Chudnovsky ... 140,000,000 iterations and 1631.04 seconds.
[DEBUG] 2022-10-03 09:52:17,222 __bs: MainProcess Chudnovsky ... 141,000,000 iterations and 1636.68 seconds.
[DEBUG] 2022-10-03 10:02:43,939 compute: MainProcess    Chudnovsky brothers  1988 
    π = (Q(0, N) / 12T(0, N) + 12AQ(0, N))**(C**(3/2))
 calulation Done! 141,027,339 iterations and 2263.39 seconds.
[INFO] 2022-10-03 10:09:07,119 <module>: MainProcess Last 5 digits of π were 45519 as expected at offset 999,999,995
[INFO] 2022-10-03 10:09:07,119 <module>: MainProcess Calculated π to 1,000,000,000 digits using a formula of:
 10     Chudnovsky brothers  1988 
    π = (Q(0, N) / 12T(0, N) + 12AQ(0, N))**(C**(3/2))
 
[INFO] 2022-10-03 10:09:07,120 <module>: MainProcess Calculation took 141,027,339 iterations and 0:44:06.398345.

math_pi.pi(b = 1000000) 比计算到一百万位更快，大约快了40倍。但是它无法计算到十亿位。一百万是最多的数字。

GMPY内置函数如下：

python pi-pourri.py -v -d 1,000,000,000 -a 11
[INFO] 2022-10-03 14:33:34,729 <module>: MainProcess Computing π to 1,000,000,000 digits.
[DEBUG] 2022-10-03 14:33:34,729 compute: MainProcess Starting   const_pi() function from the gmpy2 library formula to 1,000,000,000 decimal places
[DEBUG] 2022-10-03 15:46:46,575 compute: MainProcess    const_pi() function from the gmpy2 library calulation Done! 1 iterations and 4391.85 seconds.
[INFO] 2022-10-03 15:46:46,575 <module>: MainProcess Last 5 digits of π were 45519 as expected at offset 999,999,995
[INFO] 2022-10-03 15:46:46,575 <module>: MainProcess Calculated π to 1,000,000,000 digits using a formula of:
 11     const_pi() function from the gmpy2 library 
[INFO] 2022-10-03 15:46:46,575 <module>: MainProcess Calculation took 1 iterations and 1:13:11.845652.

MPmath内置函数的速度几乎与之相同。慢大约12％（6分钟）：

python pi-pourri.py -v -a 12 -d 1,000,000,000  
[INFO] 2022-10-04 09:10:37,085 <module>: MainProcess Computing π to 1,000,000,000 digits.
[DEBUG] 2022-10-04 09:10:37,085 compute: MainProcess Starting   mp.pi() function from the mpmath library formula to 1,000,000,000 decimal places
[DEBUG] 2022-10-04 10:01:25,321 compute: MainProcess    mp.pi() function from the mpmath library calulation Done! 1 iterations and 3048.22 seconds.
[INFO] 2022-10-04 10:01:25,338 <module>: MainProcess Last 5 digits of π were 45519 as expected at offset 999,999,995
[INFO] 2022-10-04 10:01:25,340 <module>: MainProcess Calculated π to 1,000,000,000 digits using a formula of:
 12     mp.pi() function from the mpmath library 
[INFO] 2022-10-04 10:01:25,343 <module>: MainProcess Calculation took 1 iterations and 0:50:48.250337.
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