%%
%% %CopyrightBegin%
%%
%% Copyright Ericsson AB 2015. All Rights Reserved.
%%
%% The contents of this file are subject to the Erlang Public License,
%% Version 1.1, (the "License"); you may not use this file except in
%% compliance with the License. You should have received a copy of the
%% Erlang Public License along with this software. If not, it can be
%% retrieved online at http://www.erlang.org/.
%%
%% Software distributed under the License is distributed on an "AS IS"
%% basis, WITHOUT WARRANTY OF ANY KIND, either express or implied. See
%% the License for the specific language governing rights and limitations
%% under the License.
%%
%% %CopyrightEnd%
%%
%% =====================================================================
%% Multiple PRNG module for Erlang/OTP
%% Copyright (c) 2015 Kenji Rikitake
%% =====================================================================
-module(rand).
-export([seed_s/1, seed_s/2, seed/1, seed/2,
export_seed/0, export_seed_s/1,
uniform/0, uniform/1, uniform_s/1, uniform_s/2,
normal/0, normal_s/1
]).
-compile({inline, [exs64_next/1, exsplus_next/1,
exs1024_next/1, exs1024_calc/2,
get_52/1, normal_kiwi/1]}).
-define(DEFAULT_ALG_HANDLER, exsplus).
-define(SEED_DICT, rand_seed).
%% =====================================================================
%% Types
%% =====================================================================
%% This depends on the algorithm handler function
-type alg_seed() :: exs64_state() | exsplus_state() | exs1024_state().
%% This is the algorithm handler function within this module
-type alg_handler() :: #{type => alg(),
max => integer(),
next => fun(),
uniform => fun(),
uniform_n => fun()}.
%% Internal state
-opaque state() :: {alg_handler(), alg_seed()}.
-type alg() :: exs64 | exsplus | exs1024.
-opaque export_state() :: {alg(), alg_seed()}.
-export_type([alg/0, state/0, export_state/0]).
%% =====================================================================
%% API
%% =====================================================================
%% Return algorithm and seed so that RNG state can be recreated with seed/1
-spec export_seed() -> undefined | export_state().
export_seed() ->
case seed_get() of
{#{type:=Alg}, Seed} -> {Alg, Seed};
_ -> undefined
end.
-spec export_seed_s(state()) -> export_state().
export_seed_s({#{type:=Alg}, Seed}) -> {Alg, Seed}.
%% seed(Alg) seeds RNG with runtime dependent values
%% and return the NEW state
%% seed({Alg,Seed}) setup RNG with a previously exported seed
%% and return the NEW state
-spec seed(AlgOrExpState::alg() | export_state()) -> state().
seed(Alg) ->
R = seed_s(Alg),
_ = seed_put(R),
R.
-spec seed_s(AlgOrExpState::alg() | export_state()) -> state().
seed_s(Alg) when is_atom(Alg) ->
seed_s(Alg, {erlang:phash2([{node(),self()}]),
erlang:system_time(),
erlang:unique_integer()});
seed_s({Alg0, Seed}) ->
{Alg,_SeedFun} = mk_alg(Alg0),
{Alg, Seed}.
%% seed/2: seeds RNG with the algorithm and given values
%% and returns the NEW state.
-spec seed(Alg :: alg(), {integer(), integer(), integer()}) -> state().
seed(Alg0, S0) ->
State = seed_s(Alg0, S0),
_ = seed_put(State),
State.
-spec seed_s(Alg :: alg(), {integer(), integer(), integer()}) -> state().
seed_s(Alg0, S0 = {_, _, _}) ->
{Alg, Seed} = mk_alg(Alg0),
AS = Seed(S0),
{Alg, AS}.
%%% uniform/0, uniform/1, uniform_s/1, uniform_s/2 are all
%%% uniformly distributed random numbers.
%% uniform/0: returns a random float X where 0.0 < X < 1.0,
%% updating the state in the process dictionary.
-spec uniform() -> X::float().
uniform() ->
{X, Seed} = uniform_s(seed_get()),
_ = seed_put(Seed),
X.
%% uniform/1: given an integer N >= 1,
%% uniform/1 returns a random integer X where 1 =< X =< N,
%% updating the state in the process dictionary.
-spec uniform(N :: pos_integer()) -> X::pos_integer().
uniform(N) ->
{X, Seed} = uniform_s(N, seed_get()),
_ = seed_put(Seed),
X.
%% uniform_s/1: given a state, uniform_s/1
%% returns a random float X where 0.0 < X < 1.0,
%% and a new state.
-spec uniform_s(state()) -> {X::float(), NewS :: state()}.
uniform_s(State = {#{uniform:=Uniform}, _}) ->
Uniform(State).
%% uniform_s/2: given an integer N >= 1 and a state, uniform_s/2
%% uniform_s/2 returns a random integer X where 1 =< X =< N,
%% and a new state.
-spec uniform_s(N::pos_integer(), state()) -> {X::pos_integer(), NewS::state()}.
uniform_s(N, State = {#{uniform_n:=Uniform, max:=Max}, _})
when 0 < N, N =< Max ->
Uniform(N, State);
uniform_s(N, State0 = {#{uniform:=Uniform}, _})
when is_integer(N), 0 < N ->
{F, State} = Uniform(State0),
{trunc(F * N) + 1, State}.
%% normal/0: returns a random float with standard normal distribution
%% updating the state in the process dictionary.
-spec normal() -> float().
normal() ->
{X, Seed} = normal_s(seed_get()),
_ = seed_put(Seed),
X.
%% normal_s/1: returns a random float with standard normal distribution
%% The Ziggurat Method for generating random variables - Marsaglia and Tsang
%% Paper and reference code: http://www.jstatsoft.org/v05/i08/
-spec normal_s(state()) -> {float(), NewS :: state()}.
normal_s(State0) ->
{Sign, R, State} = get_52(State0),
Idx = R band 16#FF,
Idx1 = Idx+1,
{Ki, Wi} = normal_kiwi(Idx1),
X = R * Wi,
case R < Ki of
%% Fast path 95% of the time
true when Sign =:= 0 -> {X, State};
true -> {-X, State};
%% Slow path
false when Sign =:= 0 -> normal_s(Idx, Sign, X, State);
false -> normal_s(Idx, Sign, -X, State)
end.
%% =====================================================================
%% Internal functions
-define(UINT21MASK, 16#00000000001fffff).
-define(UINT32MASK, 16#00000000ffffffff).
-define(UINT33MASK, 16#00000001ffffffff).
-define(UINT39MASK, 16#0000007fffffffff).
-define(UINT58MASK, 16#03ffffffffffffff).
-define(UINT64MASK, 16#ffffffffffffffff).
-type uint64() :: 0..16#ffffffffffffffff.
-type uint58() :: 0..16#03ffffffffffffff.
-spec seed_put(state()) -> undefined | state().
seed_put(Seed) ->
put(?SEED_DICT, Seed).
seed_get() ->
case get(?SEED_DICT) of
undefined -> seed(?DEFAULT_ALG_HANDLER);
Old -> Old % no type checking here
end.
%% Setup alg record
mk_alg(exs64) ->
{#{type=>exs64, max=>?UINT64MASK, next=>fun exs64_next/1,
uniform=>fun exs64_uniform/1, uniform_n=>fun exs64_uniform/2},
fun exs64_seed/1};
mk_alg(exsplus) ->
{#{type=>exsplus, max=>?UINT58MASK, next=>fun exsplus_next/1,
uniform=>fun exsplus_uniform/1, uniform_n=>fun exsplus_uniform/2},
fun exsplus_seed/1};
mk_alg(exs1024) ->
{#{type=>exs1024, max=>?UINT64MASK, next=>fun exs1024_next/1,
uniform=>fun exs1024_uniform/1, uniform_n=>fun exs1024_uniform/2},
fun exs1024_seed/1}.
%% =====================================================================
%% exs64 PRNG: Xorshift64*
%% Algorithm by Sebastiano Vigna
%% Reference URL: http://xorshift.di.unimi.it/
%% =====================================================================
-type exs64_state() :: uint64().
exs64_seed({A1, A2, A3}) ->
{V1, _} = exs64_next(((A1 band ?UINT32MASK) * 4294967197 + 1)),
{V2, _} = exs64_next(((A2 band ?UINT32MASK) * 4294967231 + 1)),
{V3, _} = exs64_next(((A3 band ?UINT32MASK) * 4294967279 + 1)),
((V1 * V2 * V3) rem (?UINT64MASK - 1)) + 1.
%% Advance xorshift64* state for one step and generate 64bit unsigned integer
-spec exs64_next(exs64_state()) -> {uint64(), exs64_state()}.
exs64_next(R) ->
R1 = R bxor (R bsr 12),
R2 = R1 bxor ((R1 band ?UINT39MASK) bsl 25),
R3 = R2 bxor (R2 bsr 27),
{(R3 * 2685821657736338717) band ?UINT64MASK, R3}.
exs64_uniform({Alg, R0}) ->
{V, R1} = exs64_next(R0),
{V / 18446744073709551616, {Alg, R1}}.
exs64_uniform(Max, {Alg, R}) ->
{V, R1} = exs64_next(R),
{(V rem Max) + 1, {Alg, R1}}.
%% =====================================================================
%% exsplus PRNG: Xorshift116+
%% Algorithm by Sebastiano Vigna
%% Reference URL: http://xorshift.di.unimi.it/
%% 58 bits fits into an immediate on 64bits erlang and is thus much faster.
%% Modification of the original Xorshift128+ algorithm to 116
%% by Sebastiano Vigna, a lot of thanks for his help and work.
%% =====================================================================
-type exsplus_state() :: nonempty_improper_list(uint58(), uint58()).
exsplus_seed({A1, A2, A3}) ->
{_, R1} = exsplus_next([(((A1 * 4294967197) + 1) band ?UINT58MASK)|
(((A2 * 4294967231) + 1) band ?UINT58MASK)]),
{_, R2} = exsplus_next([(((A3 * 4294967279) + 1) band ?UINT58MASK)|
tl(R1)]),
R2.
%% Advance xorshift116+ state for one step and generate 58bit unsigned integer
-spec exsplus_next(exsplus_state()) -> {uint58(), exsplus_state()}.
exsplus_next([S1|S0]) ->
%% Note: members s0 and s1 are swapped here
S11 = (S1 bxor (S1 bsl 24)) band ?UINT58MASK,
S12 = S11 bxor S0 bxor (S11 bsr 11) bxor (S0 bsr 41),
{(S0 + S12) band ?UINT58MASK, [S0|S12]}.
exsplus_uniform({Alg, R0}) ->
{I, R1} = exsplus_next(R0),
{I / (?UINT58MASK+1), {Alg, R1}}.
exsplus_uniform(Max, {Alg, R}) ->
{V, R1} = exsplus_next(R),
{(V rem Max) + 1, {Alg, R1}}.
%% =====================================================================
%% exs1024 PRNG: Xorshift1024*
%% Algorithm by Sebastiano Vigna
%% Reference URL: http://xorshift.di.unimi.it/
%% =====================================================================
-type exs1024_state() :: {list(uint64()), list(uint64())}.
exs1024_seed({A1, A2, A3}) ->
B1 = (((A1 band ?UINT21MASK) + 1) * 2097131) band ?UINT21MASK,
B2 = (((A2 band ?UINT21MASK) + 1) * 2097133) band ?UINT21MASK,
B3 = (((A3 band ?UINT21MASK) + 1) * 2097143) band ?UINT21MASK,
{exs1024_gen1024((B1 bsl 43) bor (B2 bsl 22) bor (B3 bsl 1) bor 1),
[]}.
%% Generate a list of 16 64-bit element list
%% of the xorshift64* random sequence
%% from a given 64-bit seed.
%% Note: dependent on exs64_next/1
-spec exs1024_gen1024(uint64()) -> list(uint64()).
exs1024_gen1024(R) ->
exs1024_gen1024(16, R, []).
exs1024_gen1024(0, _, L) ->
L;
exs1024_gen1024(N, R, L) ->
{X, R2} = exs64_next(R),
exs1024_gen1024(N - 1, R2, [X|L]).
%% Calculation of xorshift1024*.
%% exs1024_calc(S0, S1) -> {X, NS1}.
%% X: random number output
-spec exs1024_calc(uint64(), uint64()) -> {uint64(), uint64()}.
exs1024_calc(S0, S1) ->
S11 = S1 bxor ((S1 band ?UINT33MASK) bsl 31),
S12 = S11 bxor (S11 bsr 11),
S01 = S0 bxor (S0 bsr 30),
NS1 = S01 bxor S12,
{(NS1 * 1181783497276652981) band ?UINT64MASK, NS1}.
%% Advance xorshift1024* state for one step and generate 64bit unsigned integer
-spec exs1024_next(exs1024_state()) -> {uint64(), exs1024_state()}.
exs1024_next({[S0,S1|L3], RL}) ->
{X, NS1} = exs1024_calc(S0, S1),
{X, {[NS1|L3], [S0|RL]}};
exs1024_next({[H], RL}) ->
NL = [H|lists:reverse(RL)],
exs1024_next({NL, []}).
exs1024_uniform({Alg, R0}) ->
{V, R1} = exs1024_next(R0),
{V / 18446744073709551616, {Alg, R1}}.
exs1024_uniform(Max, {Alg, R}) ->
{V, R1} = exs1024_next(R),
{(V rem Max) + 1, {Alg, R1}}.
%% =====================================================================
%% Ziggurat cont
%% =====================================================================
-define(NOR_R, 3.6541528853610087963519472518).
-define(NOR_INV_R, 1/?NOR_R).
%% return a {sign, Random51bits, State}
get_52({Alg=#{next:=Next}, S0}) ->
{Int,S1} = Next(S0),
{((1 bsl 51) band Int), Int band ((1 bsl 51)-1), {Alg, S1}}.
%% Slow path
normal_s(0, Sign, X0, State0) ->
{U0, S1} = uniform_s(State0),
X = -?NOR_INV_R*math:log(U0),
{U1, S2} = uniform_s(S1),
Y = -math:log(U1),
case Y+Y > X*X of
false ->
normal_s(0, Sign, X0, S2);
true when Sign =:= 0 ->
{?NOR_R + X, S2};
true ->
{-?NOR_R - X, S2}
end;
normal_s(Idx, _Sign, X, State0) ->
Fi2 = normal_fi(Idx+1),
{U0, S1} = uniform_s(State0),
case ((normal_fi(Idx) - Fi2)*U0 + Fi2) < math:exp(-0.5*X*X) of
true -> {X, S1};
false -> normal_s(S1)
end.
%% Tables for generating normal_s
%% ki is zipped with wi (slightly faster)
normal_kiwi(Indx) ->
element(Indx,
{{2104047571236786,1.736725412160263e-15}, {0,9.558660351455634e-17},
{1693657211986787,1.2708704834810623e-16},{1919380038271141,1.4909740962495474e-16},
{2015384402196343,1.6658733631586268e-16},{2068365869448128,1.8136120810119029e-16},
{2101878624052573,1.9429720153135588e-16},{2124958784102998,2.0589500628482093e-16},
{2141808670795147,2.1646860576895422e-16},{2154644611568301,2.2622940392218116e-16},
{2164744887587275,2.353271891404589e-16},{2172897953696594,2.438723455742877e-16},
{2179616279372365,2.5194879829274225e-16},{2185247251868649,2.5962199772528103e-16},
{2190034623107822,2.6694407473648285e-16},{2194154434521197,2.7395729685142446e-16},
{2197736978774660,2.8069646002484804e-16},{2200880740891961,2.871905890411393e-16},
{2203661538010620,2.9346417484728883e-16},{2206138681109102,2.9953809336782113e-16},
{2208359231806599,3.054303000719244e-16},{2210361007258210,3.111563633892157e-16},
{2212174742388539,3.1672988018581815e-16},{2213825672704646,3.2216280350549905e-16},
{2215334711002614,3.274657040793975e-16},{2216719334487595,3.326479811684171e-16},
{2217994262139172,3.377180341735323e-16},{2219171977965032,3.4268340353119356e-16},
{2220263139538712,3.475508873172976e-16},{2221276900117330,3.523266384600203e-16},
{2222221164932930,3.5701624633953494e-16},{2223102796829069,3.616248057159834e-16},
{2223927782546658,3.661569752965354e-16},{2224701368170060,3.7061702777236077e-16},
{2225428170204312,3.75008892787478e-16},{2226112267248242,3.7933619401549554e-16},
{2226757276105256,3.836022812967728e-16},{2227366415328399,3.8781025861250247e-16},
{2227942558554684,3.919630085325768e-16},{2228488279492521,3.9606321366256378e-16},
{2229005890047222,4.001133755254669e-16},{2229497472775193,4.041158312414333e-16},
{2229964908627060,4.080727683096045e-16},{2230409900758597,4.119862377480744e-16},
{2230833995044585,4.1585816580828064e-16},{2231238597816133,4.1969036444740733e-16},
{2231624991250191,4.234845407152071e-16},{2231994346765928,4.272423051889976e-16},
{2232347736722750,4.309651795716294e-16},{2232686144665934,4.346546035512876e-16},
{2233010474325959,4.383119410085457e-16},{2233321557544881,4.4193848564470665e-16},
{2233620161276071,4.455354660957914e-16},{2233906993781271,4.491040505882875e-16},
{2234182710130335,4.52645351185714e-16},{2234447917093496,4.561604276690038e-16},
{2234703177503020,4.596502910884941e-16},{2234949014150181,4.631159070208165e-16},
{2235185913274316,4.665581985600875e-16},{2235414327692884,4.699780490694195e-16},
{2235634679614920,4.733763047158324e-16},{2235847363174595,4.767537768090853e-16},
{2236052746716837,4.8011124396270155e-16},{2236251174862869,4.834494540935008e-16},
{2236442970379967,4.867691262742209e-16},{2236628435876762,4.900709524522994e-16},
{2236807855342765,4.933555990465414e-16},{2236981495548562,4.966237084322178e-16},
{2237149607321147,4.998759003240909e-16},{2237312426707209,5.031127730659319e-16},
{2237470176035652,5.0633490483427195e-16},{2237623064889403,5.095428547633892e-16},
{2237771290995388,5.127371639978797e-16},{2237915041040597,5.159183566785736e-16},
{2238054491421305,5.190869408670343e-16},{2238189808931712,5.222434094134042e-16},
{2238321151397660,5.253882407719454e-16},{2238448668260432,5.285218997682382e-16},
{2238572501115169,5.316448383216618e-16},{2238692784207942,5.34757496126473e-16},
{2238809644895133,5.378603012945235e-16},{2238923204068402,5.409536709623993e-16},
{2239033576548190,5.440380118655467e-16},{2239140871448443,5.471137208817361e-16},
{2239245192514958,5.501811855460336e-16},{2239346638439541,5.532407845392784e-16},
{2239445303151952,5.56292888151909e-16},{2239541276091442,5.593378587248462e-16},
{2239634642459498,5.623760510690043e-16},{2239725483455293,5.65407812864896e-16},
{2239813876495186,5.684334850436814e-16},{2239899895417494,5.714534021509204e-16},
{2239983610673676,5.744678926941961e-16},{2240065089506935,5.774772794756965e-16},
{2240144396119183,5.804818799107686e-16},{2240221591827230,5.834820063333892e-16},
{2240296735208969,5.864779662894365e-16},{2240369882240293,5.894700628185872e-16},
{2240441086423386,5.924585947256134e-16},{2240510398907004,5.95443856841806e-16},
{2240577868599305,5.984261402772028e-16},{2240643542273726,6.014057326642664e-16},
{2240707464668391,6.043829183936125e-16},{2240769678579486,6.073579788423606e-16},
{2240830224948980,6.103311925956439e-16},{2240889142947082,6.133028356617911e-16},
{2240946470049769,6.162731816816596e-16},{2241002242111691,6.192425021325847e-16},
{2241056493434746,6.222110665273788e-16},{2241109256832602,6.251791426088e-16},
{2241160563691400,6.281469965398895e-16},{2241210444026879,6.311148930905604e-16},
{2241258926538122,6.34083095820806e-16},{2241306038658137,6.370518672608815e-16},
{2241351806601435,6.400214690888025e-16},{2241396255408788,6.429921623054896e-16},
{2241439408989313,6.459642074078832e-16},{2241481290160038,6.489378645603397e-16},
{2241521920683062,6.519133937646159e-16},{2241561321300462,6.548910550287415e-16},
{2241599511767028,6.578711085350741e-16},{2241636510880960,6.608538148078259e-16},
{2241672336512612,6.638394348803506e-16},{2241707005631362,6.668282304624746e-16},
{2241740534330713,6.698204641081558e-16},{2241772937851689,6.728163993837531e-16},
{2241804230604585,6.758163010371901e-16},{2241834426189161,6.78820435168298e-16},
{2241863537413311,6.818290694006254e-16},{2241891576310281,6.848424730550038e-16},
{2241918554154466,6.878609173251664e-16},{2241944481475843,6.908846754557169e-16},
{2241969368073071,6.939140229227569e-16},{2241993223025298,6.969492376174829e-16},
{2242016054702685,6.999906000330764e-16},{2242037870775710,7.030383934552151e-16},
{2242058678223225,7.060929041565482e-16},{2242078483339331,7.091544215954873e-16},
{2242097291739040,7.122232386196779e-16},{2242115108362774,7.152996516745303e-16},
{2242131937479672,7.183839610172063e-16},{2242147782689725,7.214764709364707e-16},
{2242162646924736,7.245774899788387e-16},{2242176532448092,7.276873311814693e-16},
{2242189440853337,7.308063123122743e-16},{2242201373061537,7.339347561177405e-16},
{2242212329317416,7.370729905789831e-16},{2242222309184237,7.4022134917658e-16},
{2242231311537397,7.433801711647648e-16},{2242239334556717,7.465498018555889e-16},
{2242246375717369,7.497305929136979e-16},{2242252431779415,7.529229026624058e-16},
{2242257498775893,7.561270964017922e-16},{2242261571999416,7.5934354673958895e-16},
{2242264645987196,7.625726339356756e-16},{2242266714504453,7.658147462610487e-16},
{2242267770526109,7.690702803721919e-16},{2242267806216711,7.723396417018299e-16},
{2242266812908462,7.756232448671174e-16},{2242264781077289,7.789215140963852e-16},
{2242261700316818,7.822348836756411e-16},{2242257559310145,7.855637984161084e-16},
{2242252345799276,7.889087141441755e-16},{2242246046552082,7.922700982152271e-16},
{2242238647326615,7.956484300529366e-16},{2242230132832625,7.99044201715713e-16},
{2242220486690076,8.024579184921259e-16},{2242209691384458,8.058900995272657e-16},
{2242197728218684,8.093412784821501e-16},{2242184577261310,8.128120042284501e-16},
{2242170217290819,8.163028415809877e-16},{2242154625735679,8.198143720706533e-16},
{2242137778609839,8.23347194760605e-16},{2242119650443327,8.26901927108847e-16},
{2242100214207556,8.304792058805374e-16},{2242079441234906,8.340796881136629e-16},
{2242057301132135,8.377040521420222e-16},{2242033761687079,8.413529986798028e-16},
{2242008788768107,8.450272519724097e-16},{2241982346215682,8.487275610186155e-16},
{2241954395725356,8.524547008695596e-16},{2241924896721443,8.562094740106233e-16},
{2241893806220517,8.599927118327665e-16},{2241861078683830,8.638052762005259e-16},
{2241826665857598,8.676480611245582e-16},{2241790516600041,8.715219945473698e-16},
{2241752576693881,8.754280402517175e-16},{2241712788642916,8.793671999021043e-16},
{2241671091451078,8.833405152308408e-16},{2241627420382235,8.873490703813135e-16},
{2241581706698773,8.913939944224086e-16},{2241533877376767,8.954764640495068e-16},
{2241483854795281,8.9959770648911e-16},{2241431556397035,9.037590026260118e-16},
{2241376894317345,9.079616903740068e-16},{2241319774977817,9.122071683134846e-16},
{2241260098640860,9.164968996219135e-16},{2241197758920538,9.208324163262308e-16},
{2241132642244704,9.252153239095693e-16},{2241064627262652,9.296473063086417e-16},
{2240993584191742,9.341301313425265e-16},{2240919374095536,9.38665656618666e-16},
{2240841848084890,9.432558359676707e-16},{2240760846432232,9.479027264651738e-16},
{2240676197587784,9.526084961066279e-16},{2240587717084782,9.57375432209745e-16},
{2240495206318753,9.622059506294838e-16},{2240398451183567,9.671026058823054e-16},
{2240297220544165,9.720681022901626e-16},{2240191264522612,9.771053062707209e-16},
{2240080312570155,9.822172599190541e-16},{2239964071293331,9.874071960480671e-16},
{2239842221996530,9.926785548807976e-16},{2239714417896699,9.980350026183645e-16},
{2239580280957725,1.003480452143618e-15},{2239439398282193,1.0090190861637457e-15},
{2239291317986196,1.0146553831467086e-15},{2239135544468203,1.0203941464683124e-15},
{2238971532964979,1.0262405372613567e-15},{2238798683265269,1.0322001115486456e-15},
{2238616332424351,1.03827886235154e-15},{2238423746288095,1.044483267600047e-15},
{2238220109591890,1.0508203448355195e-15},{2238004514345216,1.057297713900989e-15},
{2237775946143212,1.06392366906768e-15},{2237533267957822,1.0707072623632994e-15},
{2237275200846753,1.0776584002668106e-15},{2237000300869952,1.0847879564403425e-15},
{2236706931309099,1.0921079038149563e-15},{2236393229029147,1.0996314701785628e-15},
{2236057063479501,1.1073733224935752e-15},{2235695986373246,1.1153497865853155e-15},
{2235307169458859,1.1235791107110833e-15},{2234887326941578,1.1320817840164846e-15},
{2234432617919447,1.140880924258278e-15},{2233938522519765,1.1500027537839792e-15},
{2233399683022677,1.159477189144919e-15},{2232809697779198,1.169338578691096e-15},
{2232160850599817,1.17962663529558e-15},{2231443750584641,1.190387629928289e-15},
{2230646845562170,1.2016759392543819e-15},{2229755753817986,1.2135560818666897e-15},
{2228752329126533,1.2261054417450561e-15},{2227613325162504,1.2394179789163251e-15},
{2226308442121174,1.2536093926602567e-15},{2224797391720399,1.268824481425501e-15},
{2223025347823832,1.2852479319096109e-15},{2220915633329809,1.3031206634689985e-15},
{2218357446087030,1.3227655770195326e-15},{2215184158448668,1.3446300925011171e-15},
{2211132412537369,1.3693606835128518e-15},{2205758503851065,1.397943667277524e-15},
{2198248265654987,1.4319989869661328e-15},{2186916352102141,1.4744848603597596e-15},
{2167562552481814,1.5317872741611144e-15},{2125549880839716,1.6227698675312968e-15}}).
normal_fi(Indx) ->
element(Indx,
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6.2445001554702606e-01,6.1941436060583399e-01,6.1442072388891344e-01,
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3.9609881851583223e-01,3.9249506145931540e-01,3.8890886001878855e-01,
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5.5224032992509968e-03,4.0379725933630305e-03,2.6090727461021627e-03,
1.2602859304985975e-03}).
