#include <math.h>
#include "reg.h"

#define pi 3.1415926535897932384626433832795


static score_t r2_1000[] = {
1.0, 2.0, 2.5, 2.88888888889, 3.21875, 3.5104, 3.77469135802,
4.01813870071, 4.24501800537, 4.45831574489, 4.66021568,
4.85237205074, 5.0360736751, 5.21234791295, 5.38202942438,
5.54580728515, 5.70425824707, 5.8578708208, 6.00706309899,
6.15219620096, 6.293584586, 6.43150408335, 6.56619822786,
6.69788331895, 6.82675250392, 6.95297910625, 7.07671936269,
7.19811469288, 7.31729359534, 7.43437324249, 7.54946083103,
7.66265473218, 7.77404547667, 7.88371660286, 7.99174539033,
8.09820349737, 8.2031575174, 8.30666946646, 8.408797212,
8.50959485133, 8.60911304685, 8.70739932387, 8.80449833604,
8.90045210255, 8.9953002208, 9.08908005739, 9.1818269202,
9.27357421386, 9.36435358028, 9.45419502625, 9.54312703934,
9.63117669346, 9.71836974515, 9.80473072166, 9.89028300153,
9.97504888851, 10.0590496796, 10.1423057273, 10.2248364976,
10.3066606229, 10.3877959509, 10.4682595908, 10.5480679547,
10.627236797, 10.7057812508, 10.7837158615, 10.8610546184,
10.937810984, 11.0139979213, 11.0896279195, 11.1647130175,
11.239264827, 11.3132945527, 11.3868130127, 11.4598306561,
11.5323575813, 11.6044035514, 11.6759780103, 11.7470900966,
11.8177486575, 11.8879622617, 11.9577392113, 12.0270875534,
12.0960150909, 12.1645293929, 12.2326378041, 12.3003474542,
12.3676652666, 12.4345979668, 12.5011520898, 12.5673339879,
12.6331498379, 12.6986056474, 12.7637072616, 12.8284603691,
12.8928705079, 12.9569430709, 13.0206833111, 13.0840963465,
13.1471871653, 13.2099606302, 13.2724214826, 13.3345743473,
13.3964237358, 13.4579740508, 13.5192295895, 13.5801945472,
13.6408730205, 13.7012690107, 13.7613864269, 13.8212290886,
13.8808007292, 13.9401049979, 13.9991454631, 14.0579256142,
14.1164488646, 14.1747185536, 14.2327379487, 14.2905102481,
14.3480385822, 14.405326016, 14.4623755507, 14.5191901259,
14.5757726211, 14.6321258572, 14.6882525988, 14.7441555552,
14.799837382, 14.8553006829, 14.9105480108, 14.9655818692,
15.0204047137, 15.0750189533, 15.1294269511, 15.1836310264,
15.2376334551, 15.291436471, 15.3450422672, 15.3984529967,
15.4516707737, 15.5046976745, 15.5575357382, 15.6101869683,
15.6626533327, 15.7149367651, 15.7670391659, 15.8189624026,
15.870708311, 15.9222786955, 15.9736753304, 16.0248999601,
16.0759543002, 16.1268400377, 16.1775588321, 16.2281123159,
16.2785020951, 16.3287297498, 16.3787968348, 16.4287048805,
16.4784553929, 16.5280498543, 16.5774897242, 16.6267764392,
16.6759114139, 16.7248960413, 16.7737316932, 16.8224197206,
16.8709614543, 16.9193582051, 16.9676112644, 17.0157219046,
17.0636913794, 17.1115209241, 17.1592117562, 17.2067650757,
17.2541820653, 17.3014638909, 17.3486117017, 17.3956266311,
17.4425097961, 17.4892622985, 17.5358852248, 17.5823796464,
17.6287466201, 17.6749871882, 17.721102379, 17.7670932069,
17.8129606726, 17.8587057635, 17.9043294539, 17.9498327054,
17.9952164666, 18.0404816741, 18.0856292522, 18.130660113,
18.1755751573, 18.2203752741, 18.2650613411, 18.309634225,
18.3540947814, 18.3984438554, 18.4426822814, 18.4868108834,
18.5308304754, 18.5747418613, 18.6185458352, 18.6622431813,
18.7058346748, 18.749321081, 18.7927031564, 18.8359816483,
18.8791572952, 18.9222308268, 18.9652029643, 19.0080744204,
19.0508458994, 19.0935180976, 19.136091703, 19.1785673961,
19.2209458492, 19.2632277271, 19.3054136872, 19.3475043792,
19.3895004457, 19.431402522, 19.4732112364, 19.5149272102,
19.5565510578, 19.598083387, 19.6395247987, 19.6808758874,
19.7221372412, 19.7633094418, 19.8043930645, 19.8453886786,
19.8862968473, 19.9271181279, 19.9678530715, 20.0085022239,
20.0490661246, 20.0895453079, 20.1299403023, 20.1702516311,
20.2104798118, 20.250625357, 20.2906887737, 20.3306705641,
20.3705712249, 20.4103912481, 20.4501311206, 20.4897913245,
20.5293723369, 20.5688746305, 20.608298673, 20.6476449276,
20.6869138529, 20.7261059032, 20.7652215281, 20.804261173,
20.843225279, 20.8821142829, 20.9209286172, 20.9596687105,
20.9983349871, 21.0369278674, 21.0754477678, 21.1138951007,
21.1522702748, 21.1905736947, 21.2288057616, 21.2669668728,
21.3050574217, 21.3430777984, 21.3810283893, 21.4189095773,
21.4567217417, 21.4944652585, 21.5321405003, 21.5697478361,
21.6072876319, 21.6447602504, 21.6821660507, 21.7195053891,
21.7567786187, 21.7939860892, 21.8311281476, 21.8682051376,
21.9052174, 21.9421652726, 21.9790490904, 22.0158691853,
22.0526258864, 22.0893195201, 22.1259504099, 22.1625188766,
22.1990252382, 22.2354698102, 22.271852905, 22.3081748329,
22.3444359014, 22.3806364152, 22.4167766768, 22.4528569861,
22.4888776404, 22.5248389346, 22.5607411612, 22.5965846105,
22.63236957, 22.6680963252, 22.7037651592, 22.7393763528,
22.7749301845, 22.8104269307, 22.8458668654, 22.8812502606,
22.9165773859, 22.9518485091, 22.9870638955, 23.0222238086,
23.0573285098, 23.0923782582, 23.1273733112, 23.1623139241,
23.1972003501, 23.2320328405, 23.2668116448, 23.3015370104,
23.3362091828, 23.3708284059, 23.4053949214, 23.4399089693,
23.4743707879, 23.5087806136, 23.5431386809, 23.5774452228,
23.6117004703, 23.6459046528, 23.6800579981, 23.7141607321,
23.7482130793, 23.7822152621, 23.8161675019, 23.8500700178,
23.8839230279, 23.9177267482, 23.9514813936, 23.985187177,
24.0188443102, 24.0524530031, 24.0860134643, 24.1195259009,
24.1529905184, 24.1864075209, 24.2197771112, 24.2530994905,
24.2863748586, 24.3196034138, 24.3527853533, 24.3859208726,
24.4190101661, 24.4520534266, 24.4850508458, 24.5180026138,
24.5509089198, 24.5837699513, 24.6165858946, 24.6493569349,
24.682083256, 24.7147650405, 24.7474024696, 24.7799957236,
24.8125449812, 24.8450504203, 24.8775122172, 24.9099305473,
24.9423055847, 24.9746375025, 25.0069264724, 25.0391726652,
25.0713762504, 25.1035373964, 25.1356562707, 25.1677330394,
25.1997678678, 25.2317609198, 25.2637123586, 25.295622346,
25.3274910431, 25.3593186096, 25.3911052044, 25.4228509853,
25.4545561092, 25.4862207318, 25.517845008, 25.5494290916,
25.5809731355, 25.6124772915, 25.6439417106, 25.6753665429,
25.7067519372, 25.7380980417, 25.7694050037, 25.8006729694,
25.8319020841, 25.8630924924, 25.8942443377, 25.9253577628,
25.9564329095, 25.9874699187, 26.0184689304, 26.0494300839,
26.0803535176, 26.111239369, 26.1420877748, 26.1728988708,
26.2036727921, 26.234409673, 26.2651096468, 26.2957728462,
26.326399403, 26.3569894483, 26.3875431124, 26.4180605247,
26.448541814, 26.4789871082, 26.5093965346, 26.5397702197,
26.5701082891, 26.6004108679, 26.6306780804, 26.66091005,
26.6911068996, 26.7212687513, 26.7513957266, 26.781487946,
26.8115455297, 26.841568597, 26.8715572664, 26.9015116561,
26.9314318831, 26.9613180642, 26.9911703153, 27.0209887518,
27.0507734882, 27.0805246386, 27.1102423164, 27.1399266342,
27.1695777042, 27.1991956378, 27.2287805458, 27.2583325386,
27.2878517257, 27.3173382162, 27.3467921183, 27.3762135401,
27.4056025886, 27.4349593705, 27.4642839918, 27.4935765581,
27.5228371742, 27.5520659444, 27.5812629725, 27.6104283616,
27.6395622145, 27.6686646331, 27.697735719, 27.7267755733,
27.7557842963, 27.7847619879, 27.8137087475, 27.842624674,
27.8715098657, 27.9003644204, 27.9291884353, 27.9579820072,
27.9867452325, 28.0154782067, 28.0441810252, 28.0728537828,
28.1014965736, 28.1301094915, 28.1586926296, 28.1872460809,
28.2157699376, 28.2442642915, 28.2727292341, 28.3011648561,
28.329571248, 28.3579484998, 28.3862967009, 28.4146159405,
28.442906307, 28.4711678886, 28.499400773, 28.5276050474,
28.5557807987, 28.5839281132, 28.6120470769, 28.6401377751,
28.6682002931, 28.6962347154, 28.7242411262, 28.7522196094,
28.7801702483, 28.808093126, 28.8359883249, 28.8638559271,
28.8916960146, 28.9195086685, 28.9472939699, 28.9750519992,
29.0027828366, 29.0304865619, 29.0581632545, 29.0858129932,
29.1134358568, 29.1410319234, 29.1686012709, 29.1961439767,
29.2236601179, 29.2511497712, 29.2786130131, 29.3060499193,
29.3334605657, 29.3608450274, 29.3882033793, 29.415535696,
29.4428420516, 29.4701225201, 29.4973771748, 29.524606089,
29.5518093353, 29.5789869864, 29.6061391142, 29.6332657906,
29.6603670869, 29.6874430744, 29.7144938237, 29.7415194054,
29.7685198894, 29.7954953457, 29.8224458438, 29.8493714526,
29.8762722411, 29.9031482778, 29.9299996309, 29.9568263682,
29.9836285574, 30.0104062657, 30.03715956, 30.0638885071,
30.0905931733, 30.1172736246, 30.1439299267, 30.1705621452,
30.1971703452, 30.2237545916, 30.2503149489, 30.2768514815,
30.3033642532, 30.3298533279, 30.3563187689, 30.3827606395,
30.4091790024, 30.4355739203, 30.4619454555, 30.48829367,
30.5146186255, 30.5409203836, 30.5671990054, 30.593454552,
30.619687084, 30.6458966617, 30.6720833454, 30.698247195,
30.7243882701, 30.75050663, 30.7766023339, 30.8026754406,
30.8287260087, 30.8547540966, 30.8807597623, 30.9067430638,
30.9327040585, 30.958642804, 30.9845593572, 31.010453775,
31.0363261141, 31.0621764309, 31.0880047814, 31.1138112216,
31.1395958072, 31.1653585935, 31.1910996358, 31.216818989,
31.242516708, 31.268192847, 31.2938474606, 31.3194806026,
31.345092327, 31.3706826872, 31.3962517367, 31.4217995287,
31.4473261161, 31.4728315515, 31.4983158875, 31.5237791764,
31.5492214701, 31.5746428207, 31.6000432796, 31.6254228984,
31.6507817282, 31.67611982, 31.7014372247, 31.7267339927,
31.7520101746, 31.7772658204, 31.8025009802, 31.8277157038,
31.8529100406, 31.87808404, 31.9032377513, 31.9283712234,
31.9534845051, 31.9785776449, 32.0036506913, 32.0287036925,
32.0537366963, 32.0787497507, 32.1037429033, 32.1287162015,
32.1536696925, 32.1786034234, 32.2035174411, 32.2284117922,
32.2532865233, 32.2781416806, 32.3029773103, 32.3277934584,
32.3525901706, 32.3773674925, 32.4021254696, 32.426864147,
32.4515835699, 32.4762837831, 32.5009648313, 32.5256267592,
32.5502696109, 32.5748934309, 32.599498263, 32.6240841512,
32.6486511391, 32.6731992702, 32.6977285879, 32.7222391355,
32.7467309558, 32.7712040919, 32.7956585863, 32.8200944816,
32.8445118202, 32.8689106443, 32.8932909959, 32.9176529169,
32.9419964492, 32.9663216342, 32.9906285133, 33.0149171279,
33.0391875191, 33.0634397278, 33.0876737947, 33.1118897607,
33.1360876661, 33.1602675513, 33.1844294565, 33.2085734218,
33.2326994871, 33.256807692, 33.2808980763, 33.3049706794,
33.3290255406, 33.353062699, 33.3770821938, 33.4010840637,
33.4250683475, 33.4490350839, 33.4729843113, 33.4969160679,
33.520830392, 33.5447273217, 33.5686068947, 33.5924691489,
33.616314122, 33.6401418513, 33.6639523743, 33.6877457282,
33.71152195, 33.7352810768, 33.7590231452, 33.7827481922,
33.8064562541, 33.8301473674, 33.8538215684, 33.8774788933,
33.9011193781, 33.9247430588, 33.948349971, 33.9719401506,
33.9955136329, 34.0190704534, 34.0426106474, 34.0661342501,
34.0896412964, 34.1131318213, 34.1366058595, 34.1600634458,
34.1835046147, 34.2069294006, 34.2303378377, 34.2537299604,
34.2771058026, 34.3004653983, 34.3238087813, 34.3471359853,
34.370447044, 34.3937419907, 34.4170208589, 34.4402836819,
34.4635304926, 34.4867613243, 34.5099762097, 34.5331751816,
34.5563582728, 34.5795255158, 34.6026769431, 34.625812587,
34.6489324797, 34.6720366535, 34.6951251402, 34.7181979718,
34.7412551801, 34.7642967968, 34.7873228536, 34.8103333817,
34.8333284128, 34.8563079779, 34.8792721083, 34.9022208351,
34.9251541892, 34.9480722013, 34.9709749024, 34.993862323,
35.0167344937, 35.0395914449, 35.0624332071, 35.0852598103,
35.1080712847, 35.1308676605, 35.1536489675, 35.1764152355,
35.1991664944, 35.2219027737, 35.2446241031, 35.2673305119,
35.2900220296, 35.3126986853, 35.3353605082, 35.3580075275,
35.3806397721, 35.4032572708, 35.4258600525, 35.4484481458,
35.4710215794, 35.4935803816, 35.5161245811, 35.538654206,
35.5611692847, 35.5836698452, 35.6061559156, 35.6286275239,
35.6510846979, 35.6735274653, 35.695955854, 35.7183698915,
35.7407696053, 35.7631550228, 35.7855261715, 35.8078830784,
35.8302257708, 35.8525542758, 35.8748686204, 35.8971688314,
35.9194549357, 35.94172696, 35.963984931, 35.9862288753,
36.0084588193, 36.0306747894, 36.0528768121, 36.0750649133,
36.0972391195, 36.1193994565, 36.1415459505, 36.1636786272,
36.1857975127, 36.2079026325, 36.2299940124, 36.252071678,
36.2741356547, 36.296185968, 36.3182226433, 36.3402457058,
36.3622551806, 36.3842510931, 36.4062334681, 36.4282023307,
36.4501577057, 36.472099618, 36.4940280922, 36.5159431531,
36.5378448252, 36.559733133, 36.581608101, 36.6034697535,
36.6253181149, 36.6471532092, 36.6689750607, 36.6907836935,
36.7125791315, 36.7343613986, 36.7561305187, 36.7778865156,
36.7996294129, 36.8213592344, 36.8430760035, 36.8647797438,
36.8864704787, 36.9081482315, 36.9298130256, 36.951464884,
36.9731038301, 36.9947298868, 37.0163430771, 37.0379434241,
37.0595309504, 37.0811056791, 37.1026676327, 37.124216834,
37.1457533055, 37.1672770699, 37.1887881495, 37.2102865668,
37.231772344, 37.2532455036, 37.2747060675, 37.2961540581,
37.3175894974, 37.3390124074, 37.36042281, 37.3818207271,
37.4032061805, 37.4245791921, 37.4459397834, 37.4672879761,
37.4886237918, 37.5099472519, 37.531258378, 37.5525571913,
37.5738437133, 37.5951179651, 37.616379968, 37.6376297431,
37.6588673114, 37.680092694, 37.7013059119, 37.7225069858,
37.7436959367, 37.7648727854, 37.7860375524, 37.8071902586,
37.8283309244, 37.8494595705, 37.8705762172, 37.8916808851,
37.9127735945, 37.9338543657, 37.9549232189, 37.9759801743,
37.9970252521, 38.0180584723, 38.039079855, 38.0600894201,
38.0810871876, 38.1020731772, 38.1230474088, 38.1440099022,
38.1649606769, 38.1858997527, 38.2068271491, 38.2277428857,
38.2486469818, 38.269539457, 38.2904203306, 38.3112896218,
38.3321473499, 38.3529935342, 38.3738281938, 38.3946513477,
38.4154630151, 38.4362632149, 38.457051966, 38.4778292874,
38.4985951977, 38.519349716, 38.5400928608, 38.5608246508,
38.5815451047, 38.602254241, 38.6229520783, 38.643638635,
38.6643139297, 38.6849779805, 38.7056308059, 38.7262724241,
38.7469028534, 38.767522112, 38.7881302179, 38.8087271892,
38.829313044, 38.8498878003, 38.870451476, 38.8910040889,
38.9115456569, 38.9320761978, 38.9525957293, 38.9731042692,
38.993601835, 39.0140884444, 39.0345641149, 39.055028864,
39.0754827091, 39.0959256678, 39.1163577573, 39.136778995,
39.1571893981, 39.1775889839, 39.1979777695, 39.2183557721,
39.2387230088, 39.2590794966, 39.2794252526, 39.2997602936,
39.3200846366, 39.3403982984, 39.3607012959, 39.3809936458,
39.401275365, 39.4215464701, 39.4418069776, 39.4620569043,
39.4822962667, 39.5025250812, 39.5227433645, 39.5429511329,
39.5631484027, 39.5833351904, 39.6035115122, 39.6236773845,
39.6438328234, 39.6639778451, 39.6841124657, 39.7042367014,
39.7243505682, 39.744454082, 39.7645472589, 39.7846301149,
39.8047026656, 39.8247649272, 39.8448169153, 39.8648586456,
39.884890134, 39.9049113962, 39.9249224477, 39.9449233041,
39.9649139811, 39.9848944942, 40.0048648588, 40.0248250904,
40.0447752044, 40.0647152162, 40.0846451411, 40.1045649944,
40.1244747914, 40.1443745472, 40.164264277, 40.1841439961,
40.2040137194, 40.2238734621, 40.2437232392, 40.2635630656,
40.2833929563, 40.3032129263
};

static score_t reg2(int N){
  if (N<=1000)
    return r2_1000[N];
  else
    return exp(0.5*log(N*pi/2)
	       + sqrt(8/(9*N*pi))
	       + (1.0/12 - 4/(9*pi))/N);
}

score_t reg(int N, int K){

  if (K == 1)
    return 1.0 ;
  else if (K == 2)
    return reg2(N);
  else {
    int k = 0;
    score_t rk_2 = reg(N,1);
    score_t rk_1 = reg(N,2);
    score_t rk   = 0;
    for (k=3; k<=K;++k){
      rk = rk_1 + rk_2 /(k-2)*N;
      rk_2 = rk_1;
      rk_1 = rk;
    }
    return rk;
  }
}

#ifdef MAIN
#include <stdio.h>
#include <stdlib.h>

  int main(int argc, char* argv[]){
    int N, K;

    if (argc != 3) {
      fprintf(stderr, "Usage: reg N K\n");
      exit(2);
    }

    N = atoi(argv[1]);
    K = atoi(argv[2]);

    if (K<=0){
      fprintf(stderr, "K has to be a positive integer.\n");
      exit(2);
    }
     
    if (N<0){
      fprintf(stderr, "N has to be a non-negative integer.\n");
      exit(2);
    }


    printf("%f\n", reg(N,K));

    return 0;
  }
	
#endif