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diff --git a/lib/stdlib/test/rand_Xoroshiro928ss_dev.txt b/lib/stdlib/test/rand_Xoroshiro928ss_dev.txt new file mode 100644 index 0000000000..150f37fcfa --- /dev/null +++ b/lib/stdlib/test/rand_Xoroshiro928ss_dev.txt @@ -0,0 +1,343 @@ +%CopyrightBegin% + +Copyright Ericsson AB 2015-2017. All Rights Reserved. + +Licensed under the Apache License, Version 2.0 (the "License"); +you may not use this file except in compliance with the License. +You may obtain a copy of the License at + + http://www.apache.org/licenses/LICENSE-2.0 + +Unless required by applicable law or agreed to in writing, software +distributed under the License is distributed on an "AS IS" BASIS, +WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. +See the License for the specific language governing permissions and +limitations under the License. + +%CopyrightEnd% + + +Memorable facts from designing the Xoroshiro928** generator +=========================================================== +AKA: exro928ss in the rand module + +Author: Raimo Niskanen, for the Erlang/OTP team @ Ericsson. + +Reference URL: http://vigna.di.unimi.it/ftp/papers/ScrambledLinear.pdf +i.e the Xoroshiro1024 generator with ** scrambler: + + int p; + uint64_t s[16]; + + const int q = p; + const uint64_t s0 = s[p = (p + 1) & 15]; + + uint64_t s15 = s[q]; + + const uint64_t result_starstar = rotl(s0 * S, R) * T; + + s15 ^= s0; + s[q] = rotl(s0, A) ^ s15 ^ (s15 << B); + s[p] = rotl(s15, C); + +Where {S, R, T} = {5, 7, 9} as recommended in the paper. + +We want to scale down to 58 bit words (16 of them) +so we get a generator with period 2^928 - 1. + +{A, B, C} were deduced as follows +--------------------------------- + +First, to find which triplets that give a full period generator +one have to factor 2^928 - 1. + +https://www.alpertron.com.ar/ECM.HTM actually could do that +and gave the result: + + Value + 2^928 - 1 + + 2269 007733 883335 972287 082669 296112 915239 349672 942191 252221 331572 + 442536 403137 824056 312817 862695 551072 066953 619064 625508 194663 + 368599 769448 406663 254670 871573 830845 597595 897613 333042 429214 + 224697 474472 410882 236254 024057 110212 260250 671521 235807 709272 + 244389 361641 091086 035023 229622 419455 (280 digits) = 3 × 5 × 17 × 59 × + 233 × 257 × 929 × 1103 × 2089 × 5569 × 8353 × 59393 × 65537 × 3 033169 × + 39 594977 × 107 367629 × 536 903681 × 748 264961 × 2245 984577 × 239 + 686663 718401 × 15 929619 591127 520827 829953 × 82280 195167 144119 + 832390 568177 × 6033 312171 721035 031651 315652 130497 (34 digits) × 18 + 774318 450142 955120 650303 957350 521748 903233 (44 digits) × 15 694604 + 006012 505869 851221 169365 594050 637743 819041 (50 digits) + +Sebastiano Vigna from that calculated all full period triplets, at the +end of this document, and the ones with the highest degree were: + + 23-25-12 411 + 36-2-35 411 + 55-5-54 411 + 14-19-11 415 + 34-37-5 415 + 37-3-56 415 + 55-11-54 417 + 30-3-41 419 + 11-45-50 423 + 50-19-47 423 + 52-27-13 427 + 54-9-25 433 + 56-43-35 433 + 44-9-45 441 + +All these candidates were tested with TestU01-1.2.3: + + http://simul.iro.umontreal.ca/testu01/tu01.html + +A plugin was created with parameters for {A, B, C} and, since +TestU01 is a 32-bit test tool, with parameters to reverse +the generated bits or not, and to take the 32 highest or lowest +bits from the reversed or non-reversed 58 bit output. + +The generators were seeded with a SplitMix64 generator like the +one used for seeding this generator in the rand module, +taking the 58 lowest bits and wasting all zero values. + +3 runs were made with all candidates and all four bit selection variants. +For these runs the seeder was initialized with 12345678, 876543212345678 +and 1234567890. + +After all these runs the candidate with the highest degree: 44-9-45 +had not gotten any suspicious p-value at all. All the other +got p-values around 5.0e-4 worst 9.8e-6 suggesting only random +failures, so they would probably have worked about as well. + +Finally 44-9-45 was run through PractRand-0.93: + + http://pracrand.sourceforge.net/ + +Again, all 4 bit selection variants of 32 bits were run. +Random failures with p-values around e-3..e-4 for the "smaller" +tests, but for 8, 16 and 32 TB tests no anomalies were found +(with the internal seeder masked to 58 bits): + + Xoroshiro928High seed: 0x3178ec5d + Xoroshiro928Low seed: 0xa9a04fb9 + Xoroshiro928ReverseHigh seed: 0xfa0bdbab + Xoroshiro928ReverseLow seed: 0xada51705 + + +Then, S. Vigna calculated the 2^512 jump coefficient as well +as a 2^20 jump coefficient (for testing purposes) for 44-9-45. + +2^512: + { 0x44085302f77130ca, 0xba05381fdfd14902, 0x10a1de1d7d6813d2, + 0xb83fe51a1eb3be19, 0xa81b0090567fd9f0, 0x5ac26d5d20f9b49f, + 0x4ddd98ee4be41e01, 0x0657e19f00d4b358, 0xf02f778573cf0f0a, + 0xb45a3a8a3cef3cc0, 0x6e62a33cc2323831, 0xbcb3b7c4cc049c53, + 0x83f240c6007e76ce, 0xe19f5fc1a1504acd, 0x00000000b10773cb } + +2^20: + { 0xbdb966a3daf905e6, 0x644807a56270cf78, 0xda90f4a806c17e9e, + 0x4a426866bfad3c77, 0xaf699c306d8e7566, 0x8ebc73c700b8b091, + 0xc081a7bf148531fb, 0xdc4d3af15f8a4dfd, 0x90627c014098f4b6, + 0x06df2eb1feaf0fb6, 0x5bdeb1a5a90f2e6b, 0xa480c5878c3549bd, + 0xff45ef33c82f3d48, 0xa30bebc15fefcc78, 0x00000000cb3d181c } + +Standard jump function pseudocode: + + Jump constant j = 0xb10773cb...44085302f77130ca + Generator state: s + New generator state: t = 0 + foreach bit in j, low to high: + if the bit is one: + t ^= s + next s + s = t + + +The complete list of full period constants +------------------------------------------ + +29-48-54 27 x^928 + x^874 + x^870 + x^840 + x^814 + x^784 + x^759 + x^750 + x^724 + x^720 + x^634 + x^630 + x^600 + x^574 + x^544 + x^519 + x^510 + x^484 + x^480 + x^390 + x^360 + x^270 + x^240 + x^150 + x^120 + x^30 + 1 + +29-18-56 65 x^928 + x^870 + x^850 + x^840 + x^832 + x^821 + x^802 + x^791 + x^761 + x^750 + x^734 + x^731 + x^720 + x^712 + x^701 + x^686 + x^674 + x^671 + x^663 + x^641 + x^630 + x^611 + x^610 + x^603 + x^600 + x^596 + x^577 + x^566 + x^547 + x^524 + x^517 + x^510 + x^487 + x^480 + x^476 + x^464 + x^457 + x^446 + x^427 + x^423 + x^397 + x^390 + x^367 + x^363 + x^360 + x^356 + x^352 + x^341 + x^326 + x^322 + x^311 + x^281 + x^270 + x^251 + x^240 + x^236 + x^232 + x^221 + x^191 + x^161 + x^150 + x^131 + x^120 + x^30 + 1 + +29-32-36 81 x^928 + x^874 + x^870 + x^840 + x^820 + x^819 + x^794 + x^790 + x^770 + x^765 + x^764 + x^760 + x^754 + x^750 + x^740 + x^739 + x^735 + x^734 + x^730 + x^709 + 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