Talk:Primorial numbers: Difference between revisions

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say "First ten primorials: ", join ", ", map { pn_primorial($_) } 0..9;</lang> and all is done ;-)
say "First ten primorials: ", join ", ", map { pn_primorial($_) } 0..9;</lang> and all is done ;-)
[[user#Horsth|Horsth]]
[[user#Horsth|Horsth]]
: I just added a version using the new vecprod command. It is quite a bit faster. It uses a product tree for efficient large vector products -- the old comment was misleading about using a factor tree. [[User:Danaj|Danaj]] ([[User talk:Danaj|talk]]) 16:50, 9 October 2018 (UTC)
::Once I figured out what a "product tree" is, simply multiplying in pairs (technically adjacent, not that my Phix entry does that) I was positively ''shocked'' to see a better than 50-fold performance improvement (6 mins => 6 secs). --[[User:Petelomax|Pete Lomax]] ([[User talk:Petelomax|talk]]) 04:28, 22 March 2020 (UTC)


== a faster-than-exponential growth rate in CPU time stated by dinosaur ==
== a faster-than-exponential growth rate in CPU time stated by dinosaur ==
===Story===

school - multiplication ist quadratic in runtime O(count of digits) ~ (count of digits)^2
school - multiplication ist quadratic in runtime O(count of digits) ~ (count of digits)^2
<pre>Prime(100000) = 1299709, Primorial(100000) ~ 0.1909604E+563921 121.547 seconds.
<pre>Prime(100000) = 1299709, Primorial(100000) ~ 0.1909604E+563921 121.547 seconds.
Line 20: Line 22:
The implementation of primorial I am using, running on this laptop, took 0.002909 seconds for primorial 100000 and 0.049093 seconds for primorial 1000000. With digit counts, that became 0.004267 seconds and 0.04369 seconds. I guess that's faster than exponential growth rate if by fast you mean the execution time... but ... actually, I'm not sure where I'm going with this. I am mildly surprised that digit count could be faster than the raw primorial but that might just be random variation --[[User:Rdm|Rdm]] ([[User talk:Rdm|talk]]) 20:41, 8 April 2016 (UTC) Edit: System Info claims I'm working with an i7: 2.8GHz clock, 256KB L2, 6MB L3 - no mention of L1. --[[User:Rdm|Rdm]] ([[User talk:Rdm|talk]]) 09:26, 9 April 2016 (UTC)
The implementation of primorial I am using, running on this laptop, took 0.002909 seconds for primorial 100000 and 0.049093 seconds for primorial 1000000. With digit counts, that became 0.004267 seconds and 0.04369 seconds. I guess that's faster than exponential growth rate if by fast you mean the execution time... but ... actually, I'm not sure where I'm going with this. I am mildly surprised that digit count could be faster than the raw primorial but that might just be random variation --[[User:Rdm|Rdm]] ([[User talk:Rdm|talk]]) 20:41, 8 April 2016 (UTC) Edit: System Info claims I'm working with an i7: 2.8GHz clock, 256KB L2, 6MB L3 - no mention of L1. --[[User:Rdm|Rdm]] ([[User talk:Rdm|talk]]) 09:26, 9 April 2016 (UTC)


:According to System Info, the computer has 64KB of L1 I-cache, 16KB L1 D-cache, and 2048KB of L2 cache, with further jargon about 2 way set associative, 64 byte line size, etc. and no mention of L3. However, I have no idea how much of each resource is devoted to the various activities such as running the GIMP crunch and windows itself. This is why experimental timings are tricky to perform as well as being a load on the patience. I am however surprised by the speeds quoted by some contributors, given that towards the end the number has millions of digits.
:According to System Info, the computer has 64KB of L1 I-cache, 16KB L1 D-cache, and 2048KB of L2 cache, with further jargon about 2 way set associative, 64 byte line size, etc. and no mention of L3. Later, I see that the GIMP calculation gives 8MB for L3 on this AMD k10. However, I have no idea how much of each resource is devoted to the various activities such as running the GIMP crunch and windows itself. This is why experimental timings are tricky to perform as well as being a load on the patience. I am however surprised by the speeds quoted by some contributors, given that towards the end the number has millions of digits.
:As for the standard multiplication procedure of n digits by m digits consuming time proportional to n.m, remember that my version cheats by using a one digit multiplicand (err, the second number in a*b) even though it is much larger than the base of arithmetic. Computer crunching of integers is unaffected by whether x is multiplied by 12 or 1234567, except for overflows. Thus, its running time should be proportional to n, the number of digits in the big number. In principle, and ignoring caches. If successive primorials were attained by multiplying by a constant, then that number of digits would increase linearly, but of course the multiplicand is increasing in size itself. If that increase was the same as with factorials then the number of digits would increase according to log(x!) which is proportional to x*log(x) via Stirling's approximation, which is to say a linear growth (the x) multiplied by a factor that grows ever more slowly (the log(x))
:As for the standard multiplication procedure of n digits by m digits consuming time proportional to n.m, remember that my version cheats by using a one digit multiplicand (err, the second number in a*b) even though it is much larger than the base of arithmetic. Computer crunching of integers is unaffected by whether x is multiplied by 12 or 1234567, except for overflows. Thus, its running time should be proportional to n, the number of digits in the big number. In principle, and ignoring caches. If successive primorials were attained by multiplying by a constant, then that number of digits would increase linearly, but of course the multiplicand is increasing in size itself. If that increase was the same as with factorials then the number of digits would increase according to log(x!) which is proportional to x*log(x) via Stirling's approximation, which is to say a linear growth (the x) multiplied by a factor that grows ever more slowly (the log(x))
:But Primorial(x) grows even faster than x! However, I didn't look and think long about the cpu timings before making the remark about "faster than exponential". I shall twiddle the prog. to produce more reports on timing... [[User:Dinosaur|Dinosaur]] ([[User talk:Dinosaur|talk]]) 08:51, 9 April 2016 (UTC)
:But Primorial(x) grows even faster than x! However, I didn't look and think long about the cpu timings before making the remark about "faster than exponential". I shall twiddle the prog. to produce more reports on timing... [[User:Dinosaur|Dinosaur]] ([[User talk:Dinosaur|talk]]) 08:51, 9 April 2016 (UTC)

:: On reading http://rosettacode.org/mw/index.php?title=Primorial_numbers&type=revision&diff=225184&oldid=225110 I think I'm beginning to understand your point of view. I guess the issue is "in proportion to what?" For example, if we consider resource consumption (quite a lot) in proportion to the number of arguments (always just one), resource growth is infinite. If we think of resource growth in proportion to the number of digits in the argument we get "resource consumption (time to complete) growth faster than exponential". If we think of resource growth in proportion to the numerical value of the argument time consumed growth grows faster than linear.
:: Anyways, I guess a general issue here is that we tend to think in "shorthand" and when relating those thought to other people it often takes some extra work to make sure they understand our thinking. --[[User:Rdm|Rdm]] ([[User talk:Rdm|talk]]) 01:58, 10 April 2016 (UTC)

:Our edits clashed! In the article I have rephrased the usage of "exponential" to note that it is the task that is growing exponentially (n = 10^i for i = 1, 2, 3, ...) so that even for a linear exercise, the work would grow exponentially. But here there is a further growth factor beyond the linear, because the number size is growing faster than linear, because n is growing... The supposition being that a multiply is linear in (number of digits) given that the multiplicand is one (embolismic) digit only. Except, we know that sooner or later, its digit count will grow. The unit of "sizeness" is not always obvious or convenient. To calculate Primorial(n) the task size is obviously determined by the value of n, from which is deduced log(n) for the number of digits both of n itself and Primorial(n) and one argues about the time for a multiply with those sizes of digit strings, and then perhaps the integration of those times so as to reach n from 1. All of which depends on n. But, since digit string length is so important, one might choose to talk about log(n) as being the unit of size. In the factoring of numbers, for example, I (and Prof. Knuth) speak of n and sqrt(n), but, with ever larger n, it seems now it is fashionable to use log(n) as the unit, and speak of "polynomial time in N" or similar, when N is actually log(n)

:Righto, another overnight run, this time with a report every thousand for n, d, and t where t is the cpu time to attain n after reaching n - 1000. So, from 99000 to attain 100000 the digit count went from 557809 to 563921 and the cpu time was 2·625. For 999000 to 1000000 the digit count went from 6715619 to 6722809 and the cpu time was 33·969. Plotting showed almost linear growth rates ''across this limited span'':
Correlation
0·99990 t = d*5·636E-6 -0·41458
0·99973 d = n*6·6983 -125797·8
0·99943 t = n*3·7744E-5 - 1·1205
:These were for data only up to n = 800,000 because there was a change in slope subsequently, probably because I stopped using the computer for other minor activity and thus didn't disrupt its various buffers. The Mersenne prime calculations continued. The points wobble above and below the fitted line across the span, and there is a bend visible at the start where the overall line is too low. That is, at the start the calculation is going a little faster and presumably, one could talk about not yet overflowing some on-chip memory buffer as another reason for non-linearity. Similarly, at the high end the points start to rise above the fitted line, and one could talk about a growth factor such as log(n) rather than a constant. This is obscured by the greater speed later on when I had walked away from the toiling computer. Nevertheless, one can see that a slight curve is being approximated by a straight line over a limited span, which means that using the start to attempt to identify the effect of a cpu buffer size will be tricky. Even so, a log(n) vs. log(t) plot shows a change of slope at the start - except that is for only the first few points. Alas, with "only" a few thousand digits to the big number, the resolution of the CPU timing function is insufficient to show finer details, and repeating the calculation a hundred times (say) will require auxiliary storage to enable the repetition and this will change the workings of the presumed buffering processes away from what was intended to be investigated. A frustrating business.
:Unfortunately, there is no provision for posting images, and anyway I've had a lot of trouble with Octave producing damaged plot files (.svg, .png, etc.) having to resort to screen image captures, so instead, here are the data:

:: You should be able to post images now. Look for an "Upload File" link at the bottom of the page. Also, you might want to look at where the time is going in your implementation - I am going to guess that it's in your code to generate the prime numbers. --[[User:Rdm|Rdm]] ([[User talk:Rdm|talk]]) 03:00, 10 April 2016 (UTC)

:::Pox! I have just created file Primorial.cpuTime.svg, but, "Error creating thumbnail: /bin/bash: rsvg-convert: command not found" results. Inkscape however draws the plot without difficulty, though at 657KB the file is a bit large for just a thousand points, a line, and some annotation. And now that it is uploaded, I can't see how to delete it. I shall attempt to replace it by a screen capture. As for Octave, it won't "Print" the plot as either a .png nor a .jpg due to some complaint of its own "'C:\Program' is not recognized as an internal or external command, operable program or batch file." so a tar pit there.
:::The calculation of the primes is not a problem, because all million are prepared in array PRIME before any Primorial attempts are made, and the time for this is somewhere in the blink of starting the run and the first Primorial results appearing even as the screen window manifests. [[User:Dinosaur|Dinosaur]] ([[User talk:Dinosaur|talk]]) 06:30, 10 April 2016 (UTC)

::::And I can't replace the .svg file by a .jpg because I can't change the name of the file. So, a new file (only 38KB in this format), and damnit, I forgot to give the name the prefix "Primorial.cpu" [[File:Time.jpg]]
And here is a plot of cpu time against the number of digits in the big number, which is successively being multiplied by a ''single'' digit value. [[File:Primorial.CPUtime.digits.jpg]]

:::::Only admins are allowed to delete files. I've removed "Primorial.cpuTime.svg". --[[User:AndiPersti|Andreas Perstinger]] ([[User talk:AndiPersti|talk]]) 18:36, 10 April 2016 (UTC)

===Grist===
<pre>
1000, 3393, 0.000
2000, 7483, 0.016
3000, 11829, 0.047
4000, 16340, 0.047
5000, 20975, 0.078
6000, 25706, 0.094
7000, 30519, 0.109
8000, 35400, 0.141
9000, 40342, 0.156
10000, 45337, 0.172
11000, 50381, 0.219
12000, 55468, 0.203
13000, 60595, 0.250
14000, 65759, 0.266
15000, 70957, 0.250
16000, 76187, 0.328
17000, 81447, 0.344
18000, 86735, 0.359
19000, 92050, 0.391
20000, 97389, 0.406
21000, 102753, 0.422
22000, 108139, 0.469
23000, 113547, 0.484
24000, 118975, 0.500
25000, 124424, 0.531
26000, 129891, 0.562
27000, 135377, 0.578
28000, 140881, 0.609
29000, 146401, 0.625
30000, 151937, 0.656
31000, 157490, 0.703
32000, 163057, 0.688
33000, 168640, 0.750
34000, 174237, 0.766
35000, 179849, 0.750
36000, 185473, 0.797
37000, 191111, 0.812
38000, 196762, 0.859
39000, 202425, 0.891
40000, 208100, 0.906
41000, 213787, 0.938
42000, 219486, 0.953
43000, 225196, 1.000
44000, 230916, 1.016
45000, 236648, 1.047
46000, 242390, 1.062
47000, 248143, 1.094
48000, 253905, 1.156
49000, 259678, 1.156
50000, 265460, 1.156
51000, 271251, 1.203
52000, 277052, 1.234
53000, 282862, 1.234
54000, 288681, 1.297
55000, 294508, 1.297
56000, 300345, 1.328
57000, 306189, 1.359
58000, 312042, 1.406
59000, 317903, 1.422
60000, 323772, 1.453
61000, 329649, 1.484
62000, 335534, 1.516
63000, 341426, 1.531
64000, 347326, 1.562
65000, 353233, 1.594
66000, 359148, 1.609
67000, 365069, 1.641
68000, 370998, 1.688
69000, 376933, 1.719
70000, 382876, 1.719
71000, 388825, 1.719
72000, 394780, 1.766
73000, 400743, 1.812
74000, 406711, 1.844
75000, 412686, 1.875
76000, 418668, 1.922
77000, 424656, 1.938
78000, 430649, 1.938
79000, 436649, 1.969
80000, 442655, 2.016
81000, 448667, 2.047
82000, 454684, 2.062
83000, 460708, 2.109
84000, 466737, 2.141
85000, 472772, 2.156
86000, 478812, 2.281
87000, 484857, 2.234
88000, 490908, 2.281
89000, 496965, 2.297
90000, 503026, 2.328
91000, 509093, 2.359
92000, 515165, 2.391
93000, 521242, 2.422
94000, 527324, 2.438
95000, 533411, 2.500
96000, 539504, 2.516
97000, 545601, 2.547
98000, 551703, 2.578
99000, 557809, 2.594
100000, 563921, 2.625
101000, 570037, 2.656
102000, 576158, 2.672
103000, 582283, 2.719
104000, 588413, 2.750
105000, 594548, 2.797
106000, 600687, 2.812
107000, 606831, 2.859
108000, 612978, 2.891
109000, 619131, 2.922
110000, 625287, 2.969
111000, 631448, 2.984
112000, 637613, 3.047
113000, 643782, 3.094
114000, 649956, 3.094
115000, 656133, 3.156
116000, 662315, 3.156
117000, 668500, 3.203
118000, 674690, 3.281
119000, 680884, 3.297
120000, 687081, 3.344
121000, 693283, 3.359
122000, 699488, 3.422
123000, 705698, 3.453
124000, 711911, 3.500
125000, 718128, 3.516
126000, 724349, 3.547
127000, 730573, 3.625
128000, 736801, 3.672
129000, 743033, 3.703
130000, 749268, 3.734
131000, 755507, 3.797
132000, 761750, 3.828
133000, 767996, 3.875
134000, 774246, 3.922
135000, 780499, 3.953
136000, 786756, 3.984
137000, 793016, 4.047
138000, 799280, 4.062
139000, 805547, 4.141
140000, 811817, 4.172
141000, 818091, 4.188
142000, 824368, 4.250
143000, 830648, 4.281
144000, 836932, 4.312
145000, 843219, 4.375
146000, 849509, 4.391
147000, 855802, 4.422
148000, 862099, 4.438
149000, 868398, 4.500
150000, 874701, 4.547
151000, 881007, 4.578
152000, 887316, 4.641
153000, 893628, 4.688
154000, 899943, 4.688
155000, 906261, 4.750
156000, 912583, 4.766
157000, 918907, 4.781
158000, 925234, 4.859
159000, 931564, 4.875
160000, 937898, 4.922
161000, 944234, 4.984
162000, 950573, 4.984
163000, 956915, 5.016
164000, 963259, 5.078
165000, 969607, 5.094
166000, 975958, 5.109
167000, 982311, 5.188
168000, 988667, 5.203
169000, 995026, 5.156
170000,1001388, 5.219
171000,1007752, 5.312
172000,1014119, 5.328
173000,1020489, 5.375
174000,1026862, 5.391
175000,1033238, 5.484
176000,1039616, 5.500
177000,1045997, 5.484
178000,1052380, 5.562
179000,1058766, 5.594
180000,1065155, 5.656
181000,1071546, 5.656
182000,1077940, 5.703
183000,1084336, 5.688
184000,1090735, 5.750
185000,1097137, 5.812
186000,1103541, 5.875
187000,1109948, 5.906
188000,1116357, 5.922
189000,1122768, 5.891
190000,1129182, 6.000
191000,1135599, 6.016
192000,1142018, 6.047
193000,1148439, 6.109
194000,1154863, 6.094
195000,1161290, 6.172
196000,1167718, 6.188
197000,1174149, 6.250
198000,1180583, 6.250
199000,1187018, 6.297
200000,1193457, 6.344
201000,1199897, 6.359
202000,1206340, 6.406
203000,1212785, 6.438
204000,1219233, 6.469
205000,1225682, 6.516
206000,1232134, 6.547
207000,1238588, 6.578
208000,1245045, 6.609
209000,1251504, 6.609
210000,1257965, 6.703
211000,1264428, 6.750
212000,1270893, 6.734
213000,1277361, 6.828
214000,1283831, 6.859
215000,1290303, 6.859
216000,1296777, 6.938
217000,1303254, 6.953
218000,1309732, 7.000
219000,1316213, 7.047
220000,1322696, 7.047
221000,1329181, 7.141
222000,1335668, 7.125
223000,1342158, 7.188
224000,1348649, 7.203
225000,1355143, 7.281
226000,1361638, 7.266
227000,1368136, 7.328
228000,1374635, 7.344
229000,1381137, 7.406
230000,1387641, 7.406
231000,1394147, 7.484
232000,1400654, 7.516
233000,1407164, 7.531
234000,1413676, 7.594
235000,1420189, 7.609
236000,1426705, 7.609
237000,1433223, 7.703
238000,1439742, 7.719
239000,1446264, 7.734
240000,1452788, 7.812
241000,1459313, 7.828
242000,1465841, 7.891
243000,1472370, 7.938
244000,1478902, 7.875
245000,1485435, 7.938
246000,1491970, 8.031
247000,1498507, 8.078
248000,1505046, 8.094
249000,1511587, 8.094
250000,1518130, 8.203
251000,1524675, 8.219
252000,1531221, 8.250
253000,1537770, 8.266
254000,1544320, 8.328
255000,1550873, 8.344
256000,1557427, 8.391
257000,1563983, 8.422
258000,1570540, 8.484
259000,1577100, 8.500
260000,1583661, 8.531
261000,1590224, 8.562
262000,1596789, 8.609
263000,1603355, 8.656
264000,1609924, 8.672
265000,1616494, 8.719
266000,1623066, 8.672
267000,1629640, 8.750
268000,1636215, 8.812
269000,1642792, 8.875
270000,1649371, 8.859
271000,1655952, 8.906
272000,1662534, 8.984
273000,1669118, 9.031
274000,1675704, 9.047
275000,1682291, 9.109
276000,1688881, 9.109
277000,1695472, 9.141
278000,1702064, 9.188
279000,1708658, 9.250
280000,1715254, 9.297
281000,1721852, 9.297
282000,1728451, 9.312
283000,1735052, 9.375
284000,1741655, 9.406
285000,1748259, 9.391
286000,1754865, 9.516
287000,1761472, 9.516
288000,1768082, 9.578
289000,1774692, 9.594
290000,1781305, 9.625
291000,1787919, 9.703
292000,1794534, 9.734
293000,1801151, 9.750
294000,1807770, 9.781
295000,1814391, 9.828
296000,1821013, 9.859
297000,1827636, 9.844
298000,1834261, 9.906
299000,1840888, 9.938
300000,1847516, 9.953
301000,1854146, 9.969
302000,1860777,10.047
303000,1867410,10.094
304000,1874045,10.109
305000,1880681,10.203
306000,1887318,10.047
307000,1893958,10.250
308000,1900598,10.281
309000,1907240,10.250
310000,1913884,10.328
311000,1920529,10.422
312000,1927176,10.438
313000,1933824,10.500
314000,1940474,10.547
315000,1947125,10.516
316000,1953777,10.359
317000,1960431,10.547
318000,1967087,10.578
319000,1973744,10.719
320000,1980402,10.688
321000,1987062,10.781
322000,1993724,10.719
323000,2000387,10.828
324000,2007051,10.875
325000,2013717,10.594
326000,2020384,10.859
327000,2027052,11.016
328000,2033722,11.062
329000,2040394,11.031
330000,2047067,11.156
331000,2053741,11.078
332000,2060417,11.125
333000,2067094,11.188
334000,2073772,11.297
335000,2080452,11.297
336000,2087133,11.344
337000,2093816,11.297
338000,2100500,11.359
339000,2107186,11.391
340000,2113873,11.453
341000,2120561,11.516
342000,2127250,11.562
343000,2133941,11.438
344000,2140633,11.625
345000,2147327,11.609
346000,2154022,11.656
347000,2160718,11.688
348000,2167416,11.641
349000,2174115,11.781
350000,2180815,11.859
351000,2187517,11.812
352000,2194220,11.938
353000,2200924,11.953
354000,2207630,11.984
355000,2214337,12.062
356000,2221045,12.078
357000,2227755,12.109
358000,2234466,11.953
359000,2241178,12.172
360000,2247891,12.141
361000,2254606,12.219
362000,2261322,12.344
363000,2268040,12.156
364000,2274758,12.281
365000,2281478,12.375
366000,2288199,12.406
367000,2294922,12.250
368000,2301646,12.484
369000,2308371,12.500
370000,2315097,12.641
371000,2321825,12.484
372000,2328553,12.453
373000,2335283,12.766
374000,2342015,12.734
375000,2348747,12.750
376000,2355481,12.844
377000,2362216,12.812
378000,2368952,12.750
379000,2375690,12.859
380000,2382429,12.875
381000,2389169,12.906
382000,2395910,13.031
383000,2402652,13.031
384000,2409396,13.094
385000,2416141,13.125
386000,2422887,13.250
387000,2429635,13.250
388000,2436383,13.219
389000,2443133,13.312
390000,2449884,13.312
391000,2456636,13.203
392000,2463389,13.469
393000,2470144,13.344
394000,2476900,13.453
395000,2483657,13.547
396000,2490415,13.594
397000,2497174,13.609
398000,2503935,13.641
399000,2510696,13.672
400000,2517459,13.781
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1000000,6722809,33.969
</pre>
:Happy plotting! [[User:Dinosaur|Dinosaur]] ([[User talk:Dinosaur|talk]]) 02:45, 10 April 2016 (UTC)

Latest revision as of 04:28, 22 March 2020

PARI/GP you speak about factor tree, but only primes are used, where is the advantage??? I love such solutions like in perl <lang perl>use ntheory qw(pn_primorial); say "First ten primorials: ", join ", ", map { pn_primorial($_) } 0..9;</lang> and all is done ;-) Horsth

I just added a version using the new vecprod command. It is quite a bit faster. It uses a product tree for efficient large vector products -- the old comment was misleading about using a factor tree. Danaj (talk) 16:50, 9 October 2018 (UTC)
Once I figured out what a "product tree" is, simply multiplying in pairs (technically adjacent, not that my Phix entry does that) I was positively shocked to see a better than 50-fold performance improvement (6 mins => 6 secs). --Pete Lomax (talk) 04:28, 22 March 2020 (UTC)

a faster-than-exponential growth rate in CPU time stated by dinosaur

Story

school - multiplication ist quadratic in runtime O(count of digits) ~ (count of digits)^2 

Prime(100000) = 1299709, Primorial(100000) ~ 0.1909604E+563921                      121.547 seconds.
Prime(1000000) = 15485863, Primorial(1000000) ~ 0.1147175E+6722809                17756.797 seconds.

10-times Numbers -> ~100-times runtime 121*100-> 12100

the digitcount of the numbers one multiplies grew be a factor of lg10(15.45Mio)/lg10(1.299Mio) =1,17

so you get 10-> (1.17*10) ^ 2 = 136,89*121,547= 16638 thats not far from 17756

Now take into account that the number gets so big, that it doesn't fit in level3 Cache ,which slow things down.Only 3% will be enough 121,547*(1.17*1.03*10)^2 s = 17651 s Horsth

The implementation of primorial I am using, running on this laptop, took 0.002909 seconds for primorial 100000 and 0.049093 seconds for primorial 1000000. With digit counts, that became 0.004267 seconds and 0.04369 seconds. I guess that's faster than exponential growth rate if by fast you mean the execution time... but ... actually, I'm not sure where I'm going with this. I am mildly surprised that digit count could be faster than the raw primorial but that might just be random variation --Rdm (talk) 20:41, 8 April 2016 (UTC) Edit: System Info claims I'm working with an i7: 2.8GHz clock, 256KB L2, 6MB L3 - no mention of L1. --Rdm (talk) 09:26, 9 April 2016 (UTC)

According to System Info, the computer has 64KB of L1 I-cache, 16KB L1 D-cache, and 2048KB of L2 cache, with further jargon about 2 way set associative, 64 byte line size, etc. and no mention of L3. Later, I see that the GIMP calculation gives 8MB for L3 on this AMD k10. However, I have no idea how much of each resource is devoted to the various activities such as running the GIMP crunch and windows itself. This is why experimental timings are tricky to perform as well as being a load on the patience. I am however surprised by the speeds quoted by some contributors, given that towards the end the number has millions of digits.
As for the standard multiplication procedure of n digits by m digits consuming time proportional to n.m, remember that my version cheats by using a one digit multiplicand (err, the second number in a*b) even though it is much larger than the base of arithmetic. Computer crunching of integers is unaffected by whether x is multiplied by 12 or 1234567, except for overflows. Thus, its running time should be proportional to n, the number of digits in the big number. In principle, and ignoring caches. If successive primorials were attained by multiplying by a constant, then that number of digits would increase linearly, but of course the multiplicand is increasing in size itself. If that increase was the same as with factorials then the number of digits would increase according to log(x!) which is proportional to x*log(x) via Stirling's approximation, which is to say a linear growth (the x) multiplied by a factor that grows ever more slowly (the log(x))
But Primorial(x) grows even faster than x! However, I didn't look and think long about the cpu timings before making the remark about "faster than exponential". I shall twiddle the prog. to produce more reports on timing... Dinosaur (talk) 08:51, 9 April 2016 (UTC)
On reading http://rosettacode.org/mw/index.php?title=Primorial_numbers&type=revision&diff=225184&oldid=225110 I think I'm beginning to understand your point of view. I guess the issue is "in proportion to what?" For example, if we consider resource consumption (quite a lot) in proportion to the number of arguments (always just one), resource growth is infinite. If we think of resource growth in proportion to the number of digits in the argument we get "resource consumption (time to complete) growth faster than exponential". If we think of resource growth in proportion to the numerical value of the argument time consumed growth grows faster than linear.
Anyways, I guess a general issue here is that we tend to think in "shorthand" and when relating those thought to other people it often takes some extra work to make sure they understand our thinking. --Rdm (talk) 01:58, 10 April 2016 (UTC)
Our edits clashed! In the article I have rephrased the usage of "exponential" to note that it is the task that is growing exponentially (n = 10^i for i = 1, 2, 3, ...) so that even for a linear exercise, the work would grow exponentially. But here there is a further growth factor beyond the linear, because the number size is growing faster than linear, because n is growing... The supposition being that a multiply is linear in (number of digits) given that the multiplicand is one (embolismic) digit only. Except, we know that sooner or later, its digit count will grow. The unit of "sizeness" is not always obvious or convenient. To calculate Primorial(n) the task size is obviously determined by the value of n, from which is deduced log(n) for the number of digits both of n itself and Primorial(n) and one argues about the time for a multiply with those sizes of digit strings, and then perhaps the integration of those times so as to reach n from 1. All of which depends on n. But, since digit string length is so important, one might choose to talk about log(n) as being the unit of size. In the factoring of numbers, for example, I (and Prof. Knuth) speak of n and sqrt(n), but, with ever larger n, it seems now it is fashionable to use log(n) as the unit, and speak of "polynomial time in N" or similar, when N is actually log(n)
Righto, another overnight run, this time with a report every thousand for n, d, and t where t is the cpu time to attain n after reaching n - 1000. So, from 99000 to attain 100000 the digit count went from 557809 to 563921 and the cpu time was 2·625. For 999000 to 1000000 the digit count went from 6715619 to 6722809 and the cpu time was 33·969. Plotting showed almost linear growth rates across this limited span:
Correlation
 0·99990       t = d*5·636E-6  -0·41458
 0·99973       d = n*6·6983    -125797·8
 0·99943       t = n*3·7744E-5 - 1·1205
These were for data only up to n = 800,000 because there was a change in slope subsequently, probably because I stopped using the computer for other minor activity and thus didn't disrupt its various buffers. The Mersenne prime calculations continued. The points wobble above and below the fitted line across the span, and there is a bend visible at the start where the overall line is too low. That is, at the start the calculation is going a little faster and presumably, one could talk about not yet overflowing some on-chip memory buffer as another reason for non-linearity. Similarly, at the high end the points start to rise above the fitted line, and one could talk about a growth factor such as log(n) rather than a constant. This is obscured by the greater speed later on when I had walked away from the toiling computer. Nevertheless, one can see that a slight curve is being approximated by a straight line over a limited span, which means that using the start to attempt to identify the effect of a cpu buffer size will be tricky. Even so, a log(n) vs. log(t) plot shows a change of slope at the start - except that is for only the first few points. Alas, with "only" a few thousand digits to the big number, the resolution of the CPU timing function is insufficient to show finer details, and repeating the calculation a hundred times (say) will require auxiliary storage to enable the repetition and this will change the workings of the presumed buffering processes away from what was intended to be investigated. A frustrating business.
Unfortunately, there is no provision for posting images, and anyway I've had a lot of trouble with Octave producing damaged plot files (.svg, .png, etc.) having to resort to screen image captures, so instead, here are the data:
You should be able to post images now. Look for an "Upload File" link at the bottom of the page. Also, you might want to look at where the time is going in your implementation - I am going to guess that it's in your code to generate the prime numbers. --Rdm (talk) 03:00, 10 April 2016 (UTC)
Pox! I have just created file Primorial.cpuTime.svg, but, "Error creating thumbnail: /bin/bash: rsvg-convert: command not found" results. Inkscape however draws the plot without difficulty, though at 657KB the file is a bit large for just a thousand points, a line, and some annotation. And now that it is uploaded, I can't see how to delete it. I shall attempt to replace it by a screen capture. As for Octave, it won't "Print" the plot as either a .png nor a .jpg due to some complaint of its own "'C:\Program' is not recognized as an internal or external command, operable program or batch file." so a tar pit there.
The calculation of the primes is not a problem, because all million are prepared in array PRIME before any Primorial attempts are made, and the time for this is somewhere in the blink of starting the run and the first Primorial results appearing even as the screen window manifests. Dinosaur (talk) 06:30, 10 April 2016 (UTC)
And I can't replace the .svg file by a .jpg because I can't change the name of the file. So, a new file (only 38KB in this format), and damnit, I forgot to give the name the prefix "Primorial.cpu"

And here is a plot of cpu time against the number of digits in the big number, which is successively being multiplied by a single digit value.

Only admins are allowed to delete files. I've removed "Primorial.cpuTime.svg". --Andreas Perstinger (talk) 18:36, 10 April 2016 (UTC)

Grist

   1000,   3393, 0.000
   2000,   7483, 0.016
   3000,  11829, 0.047
   4000,  16340, 0.047
   5000,  20975, 0.078
   6000,  25706, 0.094
   7000,  30519, 0.109
   8000,  35400, 0.141
   9000,  40342, 0.156
  10000,  45337, 0.172
  11000,  50381, 0.219
  12000,  55468, 0.203
  13000,  60595, 0.250
  14000,  65759, 0.266
  15000,  70957, 0.250
  16000,  76187, 0.328
  17000,  81447, 0.344
  18000,  86735, 0.359
  19000,  92050, 0.391
  20000,  97389, 0.406
  21000, 102753, 0.422
  22000, 108139, 0.469
  23000, 113547, 0.484
  24000, 118975, 0.500
  25000, 124424, 0.531
  26000, 129891, 0.562
  27000, 135377, 0.578
  28000, 140881, 0.609
  29000, 146401, 0.625
  30000, 151937, 0.656
  31000, 157490, 0.703
  32000, 163057, 0.688
  33000, 168640, 0.750
  34000, 174237, 0.766
  35000, 179849, 0.750
  36000, 185473, 0.797
  37000, 191111, 0.812
  38000, 196762, 0.859
  39000, 202425, 0.891
  40000, 208100, 0.906
  41000, 213787, 0.938
  42000, 219486, 0.953
  43000, 225196, 1.000
  44000, 230916, 1.016
  45000, 236648, 1.047
  46000, 242390, 1.062
  47000, 248143, 1.094
  48000, 253905, 1.156
  49000, 259678, 1.156
  50000, 265460, 1.156
  51000, 271251, 1.203
  52000, 277052, 1.234
  53000, 282862, 1.234
  54000, 288681, 1.297
  55000, 294508, 1.297
  56000, 300345, 1.328
  57000, 306189, 1.359
  58000, 312042, 1.406
  59000, 317903, 1.422
  60000, 323772, 1.453
  61000, 329649, 1.484
  62000, 335534, 1.516
  63000, 341426, 1.531
  64000, 347326, 1.562
  65000, 353233, 1.594
  66000, 359148, 1.609
  67000, 365069, 1.641
  68000, 370998, 1.688
  69000, 376933, 1.719
  70000, 382876, 1.719
  71000, 388825, 1.719
  72000, 394780, 1.766
  73000, 400743, 1.812
  74000, 406711, 1.844
  75000, 412686, 1.875
  76000, 418668, 1.922
  77000, 424656, 1.938
  78000, 430649, 1.938
  79000, 436649, 1.969
  80000, 442655, 2.016
  81000, 448667, 2.047
  82000, 454684, 2.062
  83000, 460708, 2.109
  84000, 466737, 2.141
  85000, 472772, 2.156
  86000, 478812, 2.281
  87000, 484857, 2.234
  88000, 490908, 2.281
  89000, 496965, 2.297
  90000, 503026, 2.328
  91000, 509093, 2.359
  92000, 515165, 2.391
  93000, 521242, 2.422
  94000, 527324, 2.438
  95000, 533411, 2.500
  96000, 539504, 2.516
  97000, 545601, 2.547
  98000, 551703, 2.578
  99000, 557809, 2.594
 100000, 563921, 2.625
 101000, 570037, 2.656
 102000, 576158, 2.672
 103000, 582283, 2.719
 104000, 588413, 2.750
 105000, 594548, 2.797
 106000, 600687, 2.812
 107000, 606831, 2.859
 108000, 612978, 2.891
 109000, 619131, 2.922
 110000, 625287, 2.969
 111000, 631448, 2.984
 112000, 637613, 3.047
 113000, 643782, 3.094
 114000, 649956, 3.094
 115000, 656133, 3.156
 116000, 662315, 3.156
 117000, 668500, 3.203
 118000, 674690, 3.281
 119000, 680884, 3.297
 120000, 687081, 3.344
 121000, 693283, 3.359
 122000, 699488, 3.422
 123000, 705698, 3.453
 124000, 711911, 3.500
 125000, 718128, 3.516
 126000, 724349, 3.547
 127000, 730573, 3.625
 128000, 736801, 3.672
 129000, 743033, 3.703
 130000, 749268, 3.734
 131000, 755507, 3.797
 132000, 761750, 3.828
 133000, 767996, 3.875
 134000, 774246, 3.922
 135000, 780499, 3.953
 136000, 786756, 3.984
 137000, 793016, 4.047
 138000, 799280, 4.062
 139000, 805547, 4.141
 140000, 811817, 4.172
 141000, 818091, 4.188
 142000, 824368, 4.250
 143000, 830648, 4.281
 144000, 836932, 4.312
 145000, 843219, 4.375
 146000, 849509, 4.391
 147000, 855802, 4.422
 148000, 862099, 4.438
 149000, 868398, 4.500
 150000, 874701, 4.547
 151000, 881007, 4.578
 152000, 887316, 4.641
 153000, 893628, 4.688
 154000, 899943, 4.688
 155000, 906261, 4.750
 156000, 912583, 4.766
 157000, 918907, 4.781
 158000, 925234, 4.859
 159000, 931564, 4.875
 160000, 937898, 4.922
 161000, 944234, 4.984
 162000, 950573, 4.984
 163000, 956915, 5.016
 164000, 963259, 5.078
 165000, 969607, 5.094
 166000, 975958, 5.109
 167000, 982311, 5.188
 168000, 988667, 5.203
 169000, 995026, 5.156
 170000,1001388, 5.219
 171000,1007752, 5.312
 172000,1014119, 5.328
 173000,1020489, 5.375
 174000,1026862, 5.391
 175000,1033238, 5.484
 176000,1039616, 5.500
 177000,1045997, 5.484
 178000,1052380, 5.562
 179000,1058766, 5.594
 180000,1065155, 5.656
 181000,1071546, 5.656
 182000,1077940, 5.703
 183000,1084336, 5.688
 184000,1090735, 5.750
 185000,1097137, 5.812
 186000,1103541, 5.875
 187000,1109948, 5.906
 188000,1116357, 5.922
 189000,1122768, 5.891
 190000,1129182, 6.000
 191000,1135599, 6.016
 192000,1142018, 6.047
 193000,1148439, 6.109
 194000,1154863, 6.094
 195000,1161290, 6.172
 196000,1167718, 6.188
 197000,1174149, 6.250
 198000,1180583, 6.250
 199000,1187018, 6.297
 200000,1193457, 6.344
 201000,1199897, 6.359
 202000,1206340, 6.406
 203000,1212785, 6.438
 204000,1219233, 6.469
 205000,1225682, 6.516
 206000,1232134, 6.547
 207000,1238588, 6.578
 208000,1245045, 6.609
 209000,1251504, 6.609
 210000,1257965, 6.703
 211000,1264428, 6.750
 212000,1270893, 6.734
 213000,1277361, 6.828
 214000,1283831, 6.859
 215000,1290303, 6.859
 216000,1296777, 6.938
 217000,1303254, 6.953
 218000,1309732, 7.000
 219000,1316213, 7.047
 220000,1322696, 7.047
 221000,1329181, 7.141
 222000,1335668, 7.125
 223000,1342158, 7.188
 224000,1348649, 7.203
 225000,1355143, 7.281
 226000,1361638, 7.266
 227000,1368136, 7.328
 228000,1374635, 7.344
 229000,1381137, 7.406
 230000,1387641, 7.406
 231000,1394147, 7.484
 232000,1400654, 7.516
 233000,1407164, 7.531
 234000,1413676, 7.594
 235000,1420189, 7.609
 236000,1426705, 7.609
 237000,1433223, 7.703
 238000,1439742, 7.719
 239000,1446264, 7.734
 240000,1452788, 7.812
 241000,1459313, 7.828
 242000,1465841, 7.891
 243000,1472370, 7.938
 244000,1478902, 7.875
 245000,1485435, 7.938
 246000,1491970, 8.031
 247000,1498507, 8.078
 248000,1505046, 8.094
 249000,1511587, 8.094
 250000,1518130, 8.203
 251000,1524675, 8.219
 252000,1531221, 8.250
 253000,1537770, 8.266
 254000,1544320, 8.328
 255000,1550873, 8.344
 256000,1557427, 8.391
 257000,1563983, 8.422
 258000,1570540, 8.484
 259000,1577100, 8.500
 260000,1583661, 8.531
 261000,1590224, 8.562
 262000,1596789, 8.609
 263000,1603355, 8.656
 264000,1609924, 8.672
 265000,1616494, 8.719
 266000,1623066, 8.672
 267000,1629640, 8.750
 268000,1636215, 8.812
 269000,1642792, 8.875
 270000,1649371, 8.859
 271000,1655952, 8.906
 272000,1662534, 8.984
 273000,1669118, 9.031
 274000,1675704, 9.047
 275000,1682291, 9.109
 276000,1688881, 9.109
 277000,1695472, 9.141
 278000,1702064, 9.188
 279000,1708658, 9.250
 280000,1715254, 9.297
 281000,1721852, 9.297
 282000,1728451, 9.312
 283000,1735052, 9.375
 284000,1741655, 9.406
 285000,1748259, 9.391
 286000,1754865, 9.516
 287000,1761472, 9.516
 288000,1768082, 9.578
 289000,1774692, 9.594
 290000,1781305, 9.625
 291000,1787919, 9.703
 292000,1794534, 9.734
 293000,1801151, 9.750
 294000,1807770, 9.781
 295000,1814391, 9.828
 296000,1821013, 9.859
 297000,1827636, 9.844
 298000,1834261, 9.906
 299000,1840888, 9.938
 300000,1847516, 9.953
 301000,1854146, 9.969
 302000,1860777,10.047
 303000,1867410,10.094
 304000,1874045,10.109
 305000,1880681,10.203
 306000,1887318,10.047
 307000,1893958,10.250
 308000,1900598,10.281
 309000,1907240,10.250
 310000,1913884,10.328
 311000,1920529,10.422
 312000,1927176,10.438
 313000,1933824,10.500
 314000,1940474,10.547
 315000,1947125,10.516
 316000,1953777,10.359
 317000,1960431,10.547
 318000,1967087,10.578
 319000,1973744,10.719
 320000,1980402,10.688
 321000,1987062,10.781
 322000,1993724,10.719
 323000,2000387,10.828
 324000,2007051,10.875
 325000,2013717,10.594
 326000,2020384,10.859
 327000,2027052,11.016
 328000,2033722,11.062
 329000,2040394,11.031
 330000,2047067,11.156
 331000,2053741,11.078
 332000,2060417,11.125
 333000,2067094,11.188
 334000,2073772,11.297
 335000,2080452,11.297
 336000,2087133,11.344
 337000,2093816,11.297
 338000,2100500,11.359
 339000,2107186,11.391
 340000,2113873,11.453
 341000,2120561,11.516
 342000,2127250,11.562
 343000,2133941,11.438
 344000,2140633,11.625
 345000,2147327,11.609
 346000,2154022,11.656
 347000,2160718,11.688
 348000,2167416,11.641
 349000,2174115,11.781
 350000,2180815,11.859
 351000,2187517,11.812
 352000,2194220,11.938
 353000,2200924,11.953
 354000,2207630,11.984
 355000,2214337,12.062
 356000,2221045,12.078
 357000,2227755,12.109
 358000,2234466,11.953
 359000,2241178,12.172
 360000,2247891,12.141
 361000,2254606,12.219
 362000,2261322,12.344
 363000,2268040,12.156
 364000,2274758,12.281
 365000,2281478,12.375
 366000,2288199,12.406
 367000,2294922,12.250
 368000,2301646,12.484
 369000,2308371,12.500
 370000,2315097,12.641
 371000,2321825,12.484
 372000,2328553,12.453
 373000,2335283,12.766
 374000,2342015,12.734
 375000,2348747,12.750
 376000,2355481,12.844
 377000,2362216,12.812
 378000,2368952,12.750
 379000,2375690,12.859
 380000,2382429,12.875
 381000,2389169,12.906
 382000,2395910,13.031
 383000,2402652,13.031
 384000,2409396,13.094
 385000,2416141,13.125
 386000,2422887,13.250
 387000,2429635,13.250
 388000,2436383,13.219
 389000,2443133,13.312
 390000,2449884,13.312
 391000,2456636,13.203
 392000,2463389,13.469
 393000,2470144,13.344
 394000,2476900,13.453
 395000,2483657,13.547
 396000,2490415,13.594
 397000,2497174,13.609
 398000,2503935,13.641
 399000,2510696,13.672
 400000,2517459,13.781
 401000,2524223,13.766
 402000,2530988,13.859
 403000,2537755,13.828
 404000,2544522,13.812
 405000,2551291,13.797
 406000,2558061,13.906
 407000,2564832,13.812
 408000,2571604,14.000
 409000,2578377,14.047
 410000,2585151,14.016
 411000,2591927,14.125
 412000,2598703,14.094
 413000,2605481,14.125
 414000,2612260,14.234
 415000,2619040,14.250
 416000,2625821,14.266
 417000,2632603,14.375
 418000,2639387,14.312
 419000,2646171,14.375
 420000,2652957,14.500
 421000,2659743,14.484
 422000,2666531,14.438
 423000,2673320,14.578
 424000,2680110,14.531
 425000,2686901,14.578
 426000,2693694,14.641
 427000,2700487,14.719
 428000,2707281,14.781
 429000,2714077,14.750
 430000,2720873,14.812
 431000,2727671,14.844
 432000,2734470,14.766
 433000,2741270,14.906
 434000,2748071,14.969
 435000,2754873,14.969
 436000,2761676,14.938
 437000,2768480,14.984
 438000,2775285,15.000
 439000,2782091,15.266
 440000,2788899,15.219
 441000,2795707,15.359
 442000,2802516,15.312
 443000,2809327,15.375
 444000,2816139,15.406
 445000,2822951,15.422
 446000,2829765,15.484
 447000,2836580,15.562
 448000,2843395,15.625
 449000,2850212,15.656
 450000,2857030,15.672
 451000,2863849,15.766
 452000,2870669,15.672
 453000,2877490,15.781
 454000,2884312,15.750
 455000,2891135,15.859
 456000,2897959,15.906
 457000,2904784,15.797
 458000,2911610,15.594
 459000,2918437,15.547
 460000,2925265,15.609
 461000,2932094,15.891
 462000,2938924,16.109
 463000,2945755,16.094
 464000,2952587,16.172
 465000,2959420,16.172
 466000,2966255,16.297
 467000,2973090,16.281
 468000,2979926,16.203
 469000,2986763,16.328
 470000,2993601,16.375
 471000,3000440,16.422
 472000,3007280,16.484
 473000,3014121,16.516
 474000,3020963,16.578
 475000,3027806,16.641
 476000,3034650,16.594
 477000,3041495,16.547
 478000,3048341,16.703
 479000,3055188,16.688
 480000,3062036,16.797
 481000,3068885,16.812
 482000,3075734,16.750
 483000,3082585,16.828
 484000,3089437,16.922
 485000,3096290,16.875
 486000,3103143,16.969
 487000,3109998,17.047
 488000,3116853,17.031
 489000,3123710,17.031
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 950000,6363904,35.844
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 952000,6378237,35.922
 953000,6385405,35.922
 954000,6392572,36.016
 955000,6399741,36.047
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 957000,6414079,36.078
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 964000,6464277,36.328
 965000,6471450,35.078
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 968000,6492972,32.750
 969000,6500147,32.812
 970000,6507322,32.828
 971000,6514498,32.859
 972000,6521675,32.875
 973000,6528852,32.922
 974000,6536029,32.938
 975000,6543207,33.000
 976000,6550386,33.000
 977000,6557564,33.078
 978000,6564744,33.109
 979000,6571924,33.172
 980000,6579104,33.172
 981000,6586285,33.203
 982000,6593466,33.234
 983000,6600648,33.312
 984000,6607830,33.281
 985000,6615012,33.375
 986000,6622196,33.422
 987000,6629379,33.453
 988000,6636563,33.453
 989000,6643748,33.516
 990000,6650933,33.562
 991000,6658118,33.578
 992000,6665304,33.641
 993000,6672490,33.688
 994000,6679677,33.703
 995000,6686865,33.750
 996000,6694053,33.766
 997000,6701241,33.859
 998000,6708430,33.859
 999000,6715619,33.875
1000000,6722809,33.969
Happy plotting! Dinosaur (talk) 02:45, 10 April 2016 (UTC)
Retrieved from "https://rosettacode.org/wiki/Talk:Primorial_numbers?oldid=266984"