[原创] “相对合理”的误差有多大？

愚工688

连乘式的素对计算值的相对误差范围：
相对误差δ(m)的统计计算:
μ——平均值；
σχ——标准偏差；

M=[ 6 , 100 ]         r= 7    n= 48    μ=-.2418  σχ= .2292  δ(min)=-.625  δ(max)= .3429
M=[ 6 , 10000 ]       r= 97   n= 4998  μ=-.075   σχ= .0736  δ(min)=-.625  δ(max)= .3429
M=[ 10002 , 20000 ]   r= 139  n= 5000  μ=-.0315  σχ= .0361  δ(min)=-.1603 δ(max)= .1017
M=[ 20002 , 30000 ]   r= 173  n= 5000  μ=-.0100  σχ= .0288  δ(min)=-.1145 δ(max)= .1245  
M=[ 30002 , 40000 ]   r= 199  n= 5000  μ=-.0037  σχ= .0263  δ(min)=-.1034 δ(max)= .1101
M=[ 40002 , 50000 ]   r= 223  n= 5000  μ= .005   σχ= .0253  δ(min)=-.1021 δ(max)= .1131
M=[ 50002 , 60000 ]   r= 241  n= 5000  μ= .0082  σχ= .0219  δ(min)=-.0688 δ(max)= .1064
M=[ 60002 , 70000 ]   r= 263  n= 5000  μ= .0139  σχ= .0213  δ(min)=-.0681 δ(max)= .0993
M=[ 70002 , 80000 ]   r= 281  n= 5000  μ= .0145  σχ= .0202  δ(min)=-.051  δ(max)= .1006  
M=[ 80002 , 90000 ]   r= 293  n= 5000  μ= .0129  σχ= .0196  δ(min)=-.0597 δ(max)= .0976
M=[ 90002 , 100000 ]  r= 313  n= 5000  μ= .0218  σχ= .0174  δ(min)=-.038  δ(max)= .112
M=[ 100002 , 110000 ] r= 331  n= 5000  μ= .0233  σχ= .017   δ(min)=-.0381 δ(max)= .0906

愚工688

更大偶数范围的样本的相对误差统计计算：
100000000 -   100000098 : n= 50 μ= .1192  σx= .0013  δmin= .1156  δmax= .1224
1000000000 - 1000000098 : n= 50 μ= .1368  σx= .0004  δmin= .1356  δmax= .138
10000000000-10000000098 : n= 50 μ= .1494  σx= .0002  δmin= .1491  δmax= .1497
30000000002-30000000100 : n= 50 μ= .15494 σx= .0001  δmin= .15474 δmax= .15519
50000000002-50000000100 : n= 50 μ= .1571  σx= .0001  δmin= .1569  δmax= .1573
50000000000 - 50000000048 : n= 25 μ= .157047 σx = .000095  δmin = .15688  δmax = .15725
70000000000 - 70000000048 : n= 25 μ= .158689 σx = .000061  δmin = .158571 δmax = .158863
80000000000 - 80000000048 : n= 25 μ= .159080 σx = .000052  δmin = .158896 δmax = .159196
100000000000-100000000048 : n= 25 μ= .160175 σx = .000049  δmin = .16005  δmax = .16026
200000000000-200000000048 : n= 25 μ= .162808 σx = .000041  δmin = .16272  δmax = .16289
400000000000-400000000038 : n= 20 μ= .16544  σx = .000024  δmin = .165403 δmax = .165486

对于大偶数区域，连乘式计算值的相对误差趋于0.20附近，而标准偏差则趋小，样本的极大值与极小值趋近。

愚工688

计算实例：3500亿区域的偶数的素数对计算

G(350000000000) = 566240377；
Sp( 350000000000 *)≈  565994611.8 , Δ≈-0.000434, k(m)= 1.6
G(350000000002) = 353889363；
Sp( 350000000002 *)≈  353746632.4 , Δ≈-0.000403, k(m)= 1
G(350000000004) = 717784873；
Sp( 350000000004 *)≈  717457958.6 , Δ≈-0.000455, k(m)= 2.02817
G(350000000006) = 362950888；
Sp( 350000000006 *)≈  362817058.8 , Δ≈-0.000369, k(m)= 1.02564
G(350000000008) = 374985721；
Sp( 350000000008 *)≈  374813393.7 , Δ≈-0.000460, k(m)= 1.05955
G(350000000010) = 972138855
Sp( 350000000010 *)≈  971724198.1 , Δ≈-0.000427, k(m)= 2.74695
G(350000000012) = 395467221；
Sp( 350000000012 *)≈  395317744.4 , Δ≈-0.000378, k(m)= 1.11752
G(350000000014) = 424688917；
Sp( 350000000014 *)≈  424520117.7 , Δ≈-0.000397, k(m)= 1.20007
G(350000000016) = 707801727；
Sp( 350000000016 *)≈  707493264.7 , Δ≈-0.000436, k(m)= 2
G(350000000018) = 353914007；
Sp( 350000000018 *)≈  353748339 , Δ≈-0.000468, k(m)= 1
G(350000000020) = 539549822；
Sp( 350000000020 *)≈  539306176.5 , Δ≈-0.000452, k(m)= 1.52455
G(350000000022) = 734215313；
Sp( 350000000022 *)≈  733903026 , Δ≈-0.000425, k(m)= 2.07466
start time :09:52:32, end time:10:21:59use time :

计算式：
Sp( 350000000000 *) = 1/(1+ .16544 )*( 350000000000 /2 -2)*p(m) ≈ 565994611.8 , k(m)= 1.6
Sp( 350000000002 *) = 1/(1+ .16544 )*( 350000000002 /2 -2)*p(m) ≈ 353746632.4 , k(m)= 1
Sp( 350000000004 *) = 1/(1+ .16544 )*( 350000000004 /2 -2)*p(m) ≈ 717457958.6 , k(m)= 2.02817
Sp( 350000000006 *) = 1/(1+ .16544 )*( 350000000006 /2 -2)*p(m) ≈ 362817058.8 , k(m)= 1.02564
Sp( 350000000008 *) = 1/(1+ .16544 )*( 350000000008 /2 -2)*p(m) ≈ 374813393.7 , k(m)= 1.05955
Sp( 350000000010 *) = 1/(1+ .16544 )*( 350000000010 /2 -2)*p(m) ≈ 971724198.1 , k(m)= 2.74695
Sp( 350000000012 *) = 1/(1+ .16544 )*( 350000000012 /2 -2)*p(m) ≈ 395317744.4 , k(m)= 1.11752
Sp( 350000000014 *) = 1/(1+ .16544 )*( 350000000014 /2 -2)*p(m) ≈ 424520117.7 , k(m)= 1.20007
Sp( 350000000016 *) = 1/(1+ .16544 )*( 350000000016 /2 -2)*p(m) ≈ 707493264.7 , k(m)= 2
Sp( 350000000018 *) = 1/(1+ .16544 )*( 350000000018 /2 -2)*p(m) ≈ 353748339 , k(m)= 1
Sp( 350000000020 *) = 1/(1+ .16544 )*( 350000000020 /2 -2)*p(m) ≈ 539306176.5 , k(m)= 1.52455
Sp( 350000000022 *) = 1/(1+ .16544 )*( 350000000022 /2 -2)*p(m) ≈ 733903026 , k(m)= 2.07466

*p(m)——展开即素数连乘式。相对误差修正系数 1/(1+ .16544 )的μ取0.16544 。