使用Python以尽可能快的速度从Madhava–Leibniz系列生成子序列

Question

Madhava–Leibniz系列： 我只需要创建一个从元素k到元素n的子序列： 从这个系列中创建并返回越快越好。你知道吗

我从“莱布尼兹”一线性函数开始，用试错法一步一步地发展到“马达瓦”函数，速度快了一倍。你知道吗

更新：感谢Andrej Kesely的建议和两个破纪录的功能：“madhavaèu leibniz”（比上一个记录提高20%）和“madhavaèu leibnizèu starmap”（另外15%）。你知道吗

最快的时间现在属于一个新的函数“madhava\u leibniz”。这是一个2.8比第一个'莱布尼兹'版本快！你知道吗

现在我的脚本输出：

('3', '7', '3')
macOS version 10.14.5
Darwin-18.6.0-x86_64-i386-64bit
Python ('v3.7.3:ef4ec6ed12', 'Mar 25 2019 16:52:21') Clang 6.0 (clang-600.0.57)
Executing in 64bit

chunks * elements = m * n = 4 * 25,000,000 = 100,000,000

Time Pi 3.141592653589793 Error of calculation   function
21.997  3.141592643589326 1.0000467121074053e-08 leibniz
11.489  3.141592643589326 1.0000467121074053e-08 madhava
 9.269  3.141592643589326 1.0000467121074053e-08 madhava_leibniz
 7.834  3.141592643589326 1.0000467121074053e-08 madhava_leibniz_starmap

时间以秒为单位。你知道吗

我选择了4作为块的数量，所以稍后我可以针对多处理/多线程优化这些函数。n=250\u 000\u 000，但我建议在调试期间设置n=25\u 000\u 000，以缩短测试时间

Pi计算和精度只是为了测试所创建的子序列。时间才是最重要的。你知道吗

我仍然在学习Python，可能错过了一些让它更快的方法。你能推荐一个比“madhavaèu leibniz”更快的版本吗？

剧本：

import math
import time
import platform
import sys
from itertools import cycle, starmap
from operator import truediv

# Functions with '*leibniz*' name must be Pythonic stanza (one-liner)
# but the statement can span multiple lines if over "Maximum Line Length"
# [79 characters in PEP 8 -- Style Guide for Python Code]


def madhava_leibniz_starmap(k, n):
    return starmap(truediv, zip(cycle([1, -1] if k & 1 else [1, -1]),
                                range(2*k+1, 2*n+2, 2)))


def madhava_leibniz(k, n):
    return [s / d for s, d in zip(cycle([1, -1] if k & 1 else [1, -1]),
                                  range(2*k+1, 2*n+2, 2))]


def leibniz(k, n):
    return [[1.0, -1.0][i % 2] / (2 * i + 1) for i in range(k, n+1)]


# Functions without '*leibniz*' but with 'madhava' pattern in the name are
# optimized for speed any way you like. Can be multiple statements.


def madhava(k, n):
    series = [0.0] * (n - k + 1)
    first_divisor = 2 * k + 1
    last_divisor_plus_1 = 2 * n + 2
    i = 0
    if k & 1:
        for divisor in range(first_divisor, last_divisor_plus_1, 4):
            series[i] = -1 / divisor
            i += 2
        i = 1
        for divisor in range(first_divisor + 2, last_divisor_plus_1, 4):
            series[i] = 1 / divisor
            i += 2
    else:
        for divisor in range(first_divisor, last_divisor_plus_1, 4):
            series[i] = 1 / divisor
            i += 2
        i = 1
        for divisor in range(first_divisor + 2, last_divisor_plus_1, 4):
            series[i] = -1 / divisor
            i += 2
    return series


def report(function, m, n):  # Test a function: time and values in data series
    t = time.time()
    series = []
    for i in range(m):
        series += function(i * n, (i + 1) * n - 1)
    p = sum(series) * 4.0
    print('{:6.3f}{:19.15f} {} {}'.format(time.time() - t, p, math.pi-p,
                                          function.__name__))
    if len(series) != n * m:
        print('Error! actual length {}, requested {}'.format(len(series), n*m))
        exit(1)
    sign = [1.0, -1.0]
    for i, a in enumerate(series):
        e = sign[i % 2] / (2*i + 1)
        if a != e:
            print('Error! @ {} actual value {}, expected {}'.format(i, a, e))
            exit(1)


if __name__ == '__main__':  # Testing ... #####################################
    def main():
        print(platform.node())
        (mac_ver, _, _) = platform.mac_ver()
        print(platform.python_version_tuple())
        if mac_ver is not None and mac_ver != "":
            print("macOS version", mac_ver)
        print(platform.platform())
        print("Python", platform.python_build(), platform.python_compiler())
        print("Executing in", "64bit" if sys.maxsize > 2 ** 32 else "32bit")
        m = 4
        n = 25_000_000
        print('\nchunks * elements = m * n = {:,} * {:,} = {:,}\n'.format(m, n,
                                                                          m*n))
        print('Time Pi%18.15f Error of calculation   function' % math.pi)
        for f in [leibniz, madhava, madhava_leibniz, madhava_leibniz_starmap]:
            report(f, m, n)

    main()

Answer 1

我使用itertools.cycle稍微更改了madhava()函数：

def madhava(k, n):  # Optimized for speed any way you like.
    # you don't need to allocate the array in advance, so comment it out
    # series = [0.0] * (n - k + 1)
    first_divisor = 2 * k + 1
    last_divisor_plus_1 = 2 * n + 2

    if k & 1:
        series = [what / divisor for what, divisor in zip(cycle([-1, 1]), range(first_divisor, last_divisor_plus_1, 2))]
    else:
        series = [what / divisor for what, divisor in zip(cycle([1, -1]), range(first_divisor, last_divisor_plus_1, 2))]
    return series

我的机器上的原始版本（AMD 2400G，Ubuntu 18.04）：

('3', '6', '7')
Linux-5.0.20-050020-generic-x86_64-with-Ubuntu-18.04-bionic
Python ('default', 'Oct 22 2018 11:32:17') GCC 8.2.0
Executing in 64bit

Time Pi 3.141592653589793 Error of calculation   function
m * n = 4 * 25 = 100
 0.000  3.131592903558554 0.00999975003123943 madhava
 0.000  3.131592903558554 0.00999975003123943 leibniz
m * n = 4 * 250 = 1,000
 0.000  3.140592653839794 0.000999999749998981 madhava
 0.000  3.140592653839794 0.000999999749998981 leibniz
m * n = 4 * 2,500 = 10,000
 0.001  3.141492653590034 9.99999997586265e-05 madhava
 0.002  3.141492653590034 9.99999997586265e-05 leibniz
m * n = 4 * 25,000 = 100,000
 0.009  3.141582653589720 1.0000000073340232e-05 madhava
 0.016  3.141582653589720 1.0000000073340232e-05 leibniz
m * n = 4 * 250,000 = 1,000,000
 0.091  3.141591653589774 1.0000000187915248e-06 madhava
 0.150  3.141591653589774 1.0000000187915248e-06 leibniz
m * n = 4 * 2,500,000 = 10,000,000
 0.890  3.141592553589792 1.0000000161269895e-07 madhava
 1.546  3.141592553589792 1.0000000161269895e-07 leibniz
m * n = 4 * 25,000,000 = 100,000,000
 9.002  3.141592643589326 1.0000467121074053e-08 madhava
15.699  3.141592643589326 1.0000467121074053e-08 leibniz

使用itertools.cycle的版本：

('3', '6', '7')
Linux-5.0.20-050020-generic-x86_64-with-Ubuntu-18.04-bionic
Python ('default', 'Oct 22 2018 11:32:17') GCC 8.2.0
Executing in 64bit

Time Pi 3.141592653589793 Error of calculation   function
m * n = 4 * 25 = 100
 0.000  3.131592903558554 0.00999975003123943 madhava
 0.000  3.131592903558554 0.00999975003123943 leibniz
m * n = 4 * 250 = 1,000
 0.000  3.140592653839794 0.000999999749998981 madhava
 0.000  3.140592653839794 0.000999999749998981 leibniz
m * n = 4 * 2,500 = 10,000
 0.001  3.141492653590034 9.99999997586265e-05 madhava
 0.002  3.141492653590034 9.99999997586265e-05 leibniz
m * n = 4 * 25,000 = 100,000
 0.007  3.141582653589720 1.0000000073340232e-05 madhava
 0.016  3.141582653589720 1.0000000073340232e-05 leibniz
m * n = 4 * 250,000 = 1,000,000
 0.061  3.141591653589774 1.0000000187915248e-06 madhava
 0.152  3.141591653589774 1.0000000187915248e-06 leibniz
m * n = 4 * 2,500,000 = 10,000,000
 0.608  3.141592553589792 1.0000000161269895e-07 madhava
 1.530  3.141592553589792 1.0000000161269895e-07 leibniz
m * n = 4 * 25,000,000 = 100,000,000
 6.167  3.141592643589326 1.0000467121074053e-08 madhava
15.617  3.141592643589326 1.0000467121074053e-08 leibniz

另一个版本，使用itertools.starmap和operator.truediv：

def madhava_leibniz_starmap(k, n):
    return starmap(truediv, zip(cycle([1, -1] if k & 1 else [1, -1]), range(2*k+1, 2*n+2, 2)))

在我的电脑上，这会产生：

('3', '6', '7')
Linux-5.0.20-050020-generic-x86_64-with-Ubuntu-18.04-bionic
Python ('default', 'Oct 22 2018 11:32:17') GCC 8.2.0
Executing in 64bit

chunks * elements = m * n = 4 * 25,000,000 = 100,000,000

Time Pi 3.141592653589793 Error of calculation   function
15.569  3.141592643589326 1.0000467121074053e-08 leibniz
 8.968  3.141592643589326 1.0000467121074053e-08 madhava
 6.182  3.141592643589326 1.0000467121074053e-08 madhava_leibniz
 5.238  3.141592643589326 1.0000467121074053e-08 madhava_leibniz_starmap

比列表理解快一秒。你知道吗