Fast and precise pow (Rebuilding _libm_sse2_pow_precise)

[guga]

Hello everyone, has anyone ever used fixed mathematical series tables to accurately calculate the pow of a number? I managed to recreate the _libm_sse2_pow_precise API from ucrtbased.dll but I'm having trouble understanding how the values in the tables were obtained. Has anyone seen these values before? I was able to more or less understand how the values in the first table were generated (I still need to understand a certain sequence used in this table). But I didn't understand how the values in the other tables were obtained. My idea is to be able to recreate these tables automatically, with a specific function that creates them. That way I can understand exactly how the function was created, and which mathematical equations were used to further improve the precision (which is around 15/16 digits).

The 1st table is:

Code Select Expand

; Array of 130 Real8 (In fact, it is 128, but it contains 2 extra Real8 seems to be used for alignment)
; 1st value = 16384/16384 = 16384/(256*64) = ((128*128)+0)/((128)*128) = ((128*128)+0/2)/((128)*128)
; 2nd value = 16257/16384 = 16257/(256*64) = ((127*128)+1)/((128)*128) = ((127*128)+2/2)/((128)*128) = ((127*128)+(128*(1/64))/2)/((128)*128)
; 3rd value = 16132/16384 = 16132/(256*64) = ((126*128)+4)/((128)*128) = ((126*128)+8/2)/((128)*128) = ((126*128)+(128*(4/64))/2)/((128)*128)
; 4th value = 16009/16384 = 16009/(256*64) = ((125*128)+9)/((128)*128) = ((125*128)+18/2)/((128)*128) = ((125*128)+(128*(9/64))/2)/((128)*128)
; 5th value = 15887.5/16384 = 15887.5/(256*64) = ((124*128)+15.5)/((128)*128) = ((124*128)+31/2)/((128)*128) = ((124*128)+(128*(15.5/64))/2)/((128)*128)
; 6th value = 15768/16384 = 15768/(256*64) = ((123*128)+24)/((128)*128) = ((123*128)+48/2)/((128)*128) = ((123*128)+(128*(24/64))/2)/((128)*128)
; 128th value = ((0*128)+16384/2)/((128)*128) ?
[<16 acrt_pow_rcp_t1:   R$ 1.0,                 0.99224853515625,   0.984619140625,     0.97711181640625,
                        R$ 0.969696044921875,   0.96240234375,      0.955230712890625,  0.948150634765625,
                        R$ 0.941162109375,      0.934295654296875,  0.92755126953125,   0.920867919921875,
                        R$ 0.914276123046875,   0.907806396484375,  0.901397705078125,  0.895111083984375,
                        R$ 0.888885498046875,   0.88275146484375,   0.876708984375,     0.870758056640625,
                        R$ 0.8648681640625,     0.85906982421875,   0.85333251953125,   0.847686767578125,
                        R$ 0.84210205078125,    0.83660888671875,   0.8311767578125,    0.8258056640625,
                        R$ 0.820526123046875,   0.815277099609375,  0.81011962890625,   0.805023193359375,
                        R$ 0.79998779296875,    0.7950439453125,    0.790130615234375,  0.7852783203125,
                        R$ 0.780487060546875,   0.7757568359375,    0.771087646484375,  0.7664794921875,
                        R$ 0.76190185546875,    0.75738525390625,   0.7529296875,       0.74853515625,
                        R$ 0.744171142578125,   0.739898681640625,  0.735626220703125,  0.731414794921875,
                        R$ 0.727264404296875,   0.723175048828125,  0.7191162109375,    0.715087890625,
                        R$ 0.71112060546875,    0.707183837890625,  0.70330810546875,   0.699462890625,
                        R$ 0.695648193359375,   0.69189453125,      0.68817138671875,   0.684478759765625,
                        R$ 0.68084716796875,    0.67724609375,      0.673675537109375,  0.670166015625,
                        R$ 0.666656494140625,   0.6632080078125,    0.6597900390625,    0.656402587890625,
                        R$ 0.653076171875,      0.649749755859375,  0.646453857421875,  0.643218994140625,
                        R$ 0.6400146484375,     0.636810302734375,  0.6336669921875,    0.63055419921875,
                        R$ 0.62744140625,       0.6243896484375,    0.621368408203125,  0.61834716796875,
                        R$ 0.615386962890625,   0.6124267578125,    0.609527587890625,  0.60662841796875,
                        R$ 0.603759765625,      0.6009521484375,    0.59814453125,      0.5953369140625,
                        R$ 0.59259033203125,    0.589874267578125,  0.587158203125,     0.58447265625,
                        R$ 0.581817626953125,   0.579193115234375,  0.576568603515625,  0.574005126953125,
                        R$ 0.571441650390625,   0.568878173828125,  0.566375732421875,  0.563873291015625,
                        R$ 0.5614013671875,     0.5589599609375,    0.5565185546875,    0.554107666015625,
                        R$ 0.551727294921875,   0.549346923828125,  0.5469970703125,    0.544677734375,
                        R$ 0.5423583984375,     0.540069580078125,  0.537811279296875,  0.535552978515625,
                        R$ 0.5333251953125,     0.5311279296875,    0.5289306640625,    0.526763916015625,
                        R$ 0.52459716796875,    0.5224609375,       0.52032470703125,   0.518218994140625,
                        R$ 0.516143798828125,   0.514068603515625,  0.511993408203125,  0.50994873046875,
                        R$ 0.5079345703125,     0.50592041015625,   0.503936767578125,  0.501953125,
                        R$ 0.5,                 0.0]

2nd table:

Code Select Expand

; Array of 258 Real8  (In fact, it is 256, but it contains 2 extra Real8 seems to be used for alignment)
; last = -0.005624549193726125, -1.51981979419044e-13 = log2(256/257). But, since we divided in 2, we have :
; a+b = log2(256/257)
; b = log2(256/257+1/257)
; a = log2(256/257) - log2(256/257+1/257)
[acrt_pow_log2_t1:      R$ 0,                   0,                      0.011226567492713,      1.016823494307948e-13,
                        R$ 0.0223623095939729,  1.509641361505043e-13,  0.0334044275305132,     2.101197734774185e-13,
                        R$ 0.0443954952202148,  1.016044049030341e-13,  0.05528793895632589,    1.883374902839381e-13,
                        R$ 0.06607887153541014, 2.138176331783674e-13,  0.07681181343741628,    5.709230722804341e-14,
                        R$ 0.08748485519754468, 5.868843569563728e-14,  0.09804893735872611,    3.619681935484368e-14,
                        R$ 0.1085010673270972,  1.543184264571083e-13,  0.1189338494700678,     1.860362057011582e-13,
                        R$ 0.1292981514975509,  7.554748217812214e-14,  0.1395434412183931,     1.516994721722207e-13,
                        R$ 0.1497643178722683,  1.051178722669094e-14,  0.159861361818912,      2.114869849339652e-13,
                        R$ 0.169930504897593,   4.40052858963378e-14,   0.1799207848823698,     2.240129032913967e-13,
                        R$ 0.1898300623352043,  1.375795440150339e-13,  0.1996561785576887,     7.145026853877997e-14,
                        R$ 0.2094478621945655,  5.367375028888726e-14,  0.2191526981448533,     1.924498639378365e-13,
                        R$ 0.2288200663576845,  5.936005594789268e-14,  0.2383968281446869,     1.433009328177132e-13,
                        R$ 0.2479330168987417,  1.684200910040635e-13,  0.2573747720703068,     8.862301254885962e-14,
                        R$ 0.2667727821483368,  1.013232499019477e-13,  0.2761257811359883,     1.12230539696592e-13,
                        R$ 0.2853788294112292,  1.016334320206645e-13,  0.294637603289857,      5.597048496184292e-15,
                        R$ 0.303793130980921,   1.853744221157845e-13,  0.3128977458679856,     1.60639003923233e-13,
                        R$ 0.3219501088344714,  1.549098793317993e-13,  0.3308934886636052,     9.419397015157054e-14,
                        R$ 0.3398369322621875,  1.80893087169476e-13,   0.3487240266531444,     2.169974414678604e-13,
                        R$ 0.3575533804798852,  6.149443240964241e-14,  0.3663235901076405,     3.826580964617253e-14,
                        R$ 0.3750332399847593,  1.955977358664581e-14,  0.3836809030231052,     2.947640342943602e-14,
                        R$ 0.3923229262338737,  2.112545050340321e-13,  0.4009007622883018,     2.098488527644043e-13,
                        R$ 0.409412950084743,   2.227235158820077e-13,  0.4178580183411214,     1.011301267316563e-13,
                        R$ 0.4262936480765802,  1.897728976533714e-13,  0.4346003667162677,     4.004267917640788e-14,
                        R$ 0.4429551907162477,  1.400196786904386e-13,  0.4512382853470172,     5.62719093585379e-14,
                        R$ 0.4594481290662316,  2.213963319159338e-14,  0.4675831925114835,     1.412876082947186e-13,
                        R$ 0.4757031623373678,  7.587747414268379e-15,  0.4838075221205145,     1.18664971091356e-13,
                        R$ 0.4918338344011772,  1.878064226732364e-13,  0.4998427913988053,     5.862732383910959e-15,
                        R$ 0.5077712507477372,  4.150897571512671e-14,  0.5156805763556349,     9.786938359853908e-14,
                        R$ 0.5235702112477156,  1.569667701192254e-13,  0.5313759570817638,     2.177088304752577e-13,
                        R$ 0.5391601869698661,  2.823731509563395e-14,  0.5469223213460737,     3.256485334128137e-14,
                        R$ 0.5545971068684139,  8.319730657207256e-14,  0.5622479276762533,     1.439645407616248e-13,
                        R$ 0.5698741825767684,  4.094263358904957e-14,  0.5774095661072352,     2.038623382754154e-13,
                        R$ -0.4150154853314234, -1.564570207123908e-13, -0.4075333316711749,    -1.492022497514329e-13,
                        R$ -0.4000789026219991, -1.351495278093843e-14, -0.3926528318520468,    -1.069202642230373e-13,
                        R$ -0.3853231761756888, -1.828572895052894e-13, -0.3779560917237177,    -1.547692563069645e-13,
                        R$ -0.3706193021482704, -7.800659110273396e-14, -0.3633819147821669,    -1.147403162887244e-13,
                        R$ -0.3561768305162332, -1.219956031355297e-13, -0.3489355820486253,    -1.476319661150002e-13,
                        R$ -0.3417967723917172, -1.113041314078495e-13, -0.3346922876937697,    -8.923071668805204e-14,
                        R$ -0.3275526440811518, -8.864338416218237e-14, -0.3205185246326892,    -1.844162847219179e-15,
                        R$ -0.3135207985078523, -1.902032350780128e-14, -0.3064889647284872,    -1.456359191949078e-13,
                        R$ -0.2995657852932254, -1.307094403586611e-14, -0.292609222764213,     -8.378060738742084e-14,
                        R$ -0.2857634254039567, -4.701821600676295e-14, -0.2788849888020195,    -1.511802016849652e-13,
                        R$ -0.2720465243899071, -3.076548904101923e-14, -0.2653220242348198,    -4.915978648845209e-14,
                        R$ -0.2585660338897924, -1.405312834892581e-13, -0.2517782570357667,    -1.009393730467223e-13,
                        R$ -0.2451069943811035, -1.033265339614818e-13, -0.2384793801425076,    -7.970918592730934e-14,
                        R$ -0.2318211786573556, -4.8659121779957e-14,   -0.2252074369434922,    -9.752275842785794e-15,
                        R$ -0.2186389106134357, -4.177604593923986e-14, -0.2121163585759405,    -1.87283555703194e-13,
                        R$ -0.2055641835243023, -1.534170470249947e-13, -0.1991355280706557,    -1.144747712045036e-13,
                        R$ -0.192678098233273,  -2.02774688785164e-13,  -0.1861916352681874,    -3.093545419374396e-14,
                        R$ -0.1798313565068383, -1.22416012559691e-13,  -0.1734429135508435,    -7.084811750820007e-14,
                        R$ -0.1671044823797274, -2.062854042969882e-13, -0.1608168496734379,    -1.092122852622241e-13,
                        R$ -0.1545016938646313, -1.337455005808682e-13, -0.1482382318115469,    -9.705478385564927e-14,
                        R$ -0.1420272600159933, -5.785496642520203e-14, -0.1357894336024401,    -1.543442716838467e-13,
                        R$ -0.1296050111523073, -1.980247712130743e-13, -0.123474798243933,     -2.075149641280211e-13,
                        R$ -0.1173184260098878, -1.160610222980619e-13, -0.1112171945496812,    -7.534408284934441e-14,
                        R$ -0.1051719176959978, -1.033034157098786e-13, -0.09910120289146107,   -8.226723868429274e-14,
                        R$ -0.09308739044399772, -2.197956159692524e-13, -0.08713130154046667,  -6.276674125004108e-14,
                        R$ -0.08115052127186573, -3.026990936508615e-15, -0.07522842768435112,  -1.286550323901214e-13,
                        R$ -0.06928192436521385, -2.095461474301054e-13, -0.06339508128849047,  -1.882220931891893e-14,
                        R$ -0.05748411879881132, -8.583738744817724e-14, -0.05163380031081033,  -4.815871164781571e-14,
                        R$ -0.04584496485881573, -7.18250842909487e-14,  -0.04003280800907305,  -4.664515782175133e-14,
                        R$ -0.03419714109190863, -1.428593977599921e-13, -0.0284241129484144,   -1.358834028958522e-13,
                        R$ -0.02271457304459545, -1.634509902670196e-13, -0.01698234753348515,  -1.627787964810823e-13,
                        R$ -0.01131462529701821, -4.846959270555319e-14, -0.005624549193726125, -1.51981979419044e-13,
                        R$ 0,                    0]


3rd table:

Code Select Expand


; array of 130 Real8 = 130*8 = 1040 bytes  (In fact, it is 128, but it contains 2 extra Real8 seems to be used for alignment)
[acrt_pow_rcp_t2:       R$ 1.0,                 0.999908447265625,  0.999786376953125,  0.999664306640625,
                        R$ 0.999542236328125,   0.999420166015625,  0.999298095703125,  0.999176025390625,
                        R$ 0.999053955078125,   0.998931884765625,  0.998809814453125,  0.998687744140625,
                        R$ 0.998565673828125,   0.998443603515625,  0.998321533203125,  0.998199462890625,
                        R$ 0.998077392578125,   0.997955322265625,  0.997833251953125,  0.997711181640625,
                        R$ 0.997589111328125,   0.997467041015625,  0.997344970703125,  0.997222900390625,
                        R$ 0.997100830078125,   0.996978759765625,  0.996856689453125,  0.996734619140625,
                        R$ 0.996612548828125,   0.996490478515625,  0.996368408203125,  0.996246337890625,
                        R$ 0.996124267578125,   0.996002197265625,  0.995880126953125,  0.995758056640625,
                        R$ 0.995635986328125,   0.995513916015625,  0.995391845703125,  0.995269775390625,
                        R$ 0.995147705078125,   0.995025634765625,  0.994903564453125,  0.994781494140625,
                        R$ 0.994659423828125,   0.994537353515625,  0.99444580078125,   0.99432373046875,
                        R$ 0.99420166015625,    0.99407958984375,   0.99395751953125,   0.99383544921875,
                        R$ 0.99371337890625,    0.99359130859375,   0.99346923828125,   0.99334716796875,
                        R$ 0.99322509765625,    0.99310302734375,   0.99298095703125,   0.99285888671875,
                        R$ 0.99273681640625,    0.99261474609375,   0.99249267578125,   0.99237060546875,
                        R$ 1.00390625,          1.00384521484375,   1.0037841796875,    1.00372314453125,
                        R$ 1.003662109375,      1.00360107421875,   1.0035400390625,    1.00347900390625,
                        R$ 1.00341796875,       1.00335693359375,   1.0032958984375,    1.00323486328125,
                        R$ 1.003173828125,      1.00311279296875,   1.0030517578125,    1.00299072265625,
                        R$ 1.0029296875,        1.00286865234375,   1.0028076171875,    1.00274658203125,
                        R$ 1.002685546875,      1.00262451171875,   1.0025634765625,    1.00250244140625,
                        R$ 1.00244140625,       1.00238037109375,   1.0023193359375,    1.00225830078125,
                        R$ 1.002197265625,      1.00213623046875,   1.0020751953125,    1.00201416015625,
                        R$ 1.001953125,         1.00189208984375,   1.0018310546875,    1.00177001953125,
                        R$ 1.001708984375,      1.00164794921875,   1.0015869140625,    1.00152587890625,
                        R$ 1.00146484375,       1.00140380859375,   1.0013427734375,    1.00128173828125,
                        R$ 1.001220703125,      1.00115966796875,   1.0010986328125,    1.00103759765625,
                        R$ 1.0009765625,        1.00091552734375,   1.0008544921875,    1.00079345703125,
                        R$ 1.000732421875,      1.00067138671875,   1.0006103515625,    1.00054931640625,
                        R$ 1.00048828125,       1.00042724609375,   1.0003662109375,    1.00030517578125,
                        R$ 1.000244140625,      1.00018310546875,   1.0001220703125,    1.00006103515625,
                        R$ 1.0,                 0.0]



Btw, the tables seems to contains 2 or 4 extra Real8 used for align

[guga]

4th table

Code Select Expand


; array of 258 Real8 = 258*8 = 2064 bytes  (In fact, it is 256, but it contains 2 extra Real8 seems to be used for alignment)
[acrt_pow_log2_t2: R$ 0, 0, 0.0001320887224665057, 3.026078632670094e-14
R$ 0.0003082258335780352, 1.124369745920572e-14, 0.0004843844515107776
R$ 1.903404018633784e-13, 0.0006605645819490746, 1.360301172389496e-13
R$ 0.0008367662298951473, 1.008377266663287e-13, 0.00101298940057859
R$ 1.118400172014953e-13, 0.001189234099228997, 1.980402962614453e-13
R$ 0.001365500331303338, 1.629956576979862e-13, 0.001541788102031205
R$ 3.95652747764643e-14, 0.001718097416414821, 8.991134284368603e-14
R$ 0.001894428279911153, 1.233789962422219e-13, 0.002070780697749797
R$ 1.786182784606054e-13, 0.002247154675387719, 6.883773564429666e-14
R$ 0.002423550217827142, 6.392638790015439e-14, 0.002599967330297659
R$ 2.083336485192877e-13, 0.002776406018483613, 9.369659466456748e-14
R$ 0.00295286628715985, 2.230829667393829e-13, 0.003129348142010713
R$ 1.920027121301932e-13, 0.003305851588265796, 5.265098573379892e-14
R$ 0.00348237663092732, 8.65354229618834e-14, 0.003658923275452253
R$ 1.223560624394118e-13, 0.003835491527070189, 2.181273219431695e-13
R$ 0.004012081391238098, 2.064315977053137e-13, 0.004188692873185573
R$ 1.491675660714541e-13, 0.004365325978142209, 1.101774594110659e-13
R$ 0.004541980711337601, 1.552480180535887e-13, 0.004718657078228716
R$ 1.247377673641787e-13, 0.00489535508404515, 8.832532173246751e-14
R$ 0.005072074734016496, 1.176366628153659e-13, 0.005248816033599724
R$ 5.887241836403713e-14, 0.005425578987797053, 2.149298434083969e-13
R$ 0.005602363602292826, 2.085353984807866e-13, 0.005779169882316637
R$ 1.191140828574256e-13, 0.005955997833098081, 2.804304028312898e-14
R$ 0.006132847459866753, 1.865251616880913e-14, 0.006309718767852246
R$ 1.762268153757303e-13, 0.006486611762738903, 1.332579097004816e-13
R$ 0.006663526449528945, 2.0631477405019e-13, 0.006840462833906713
R$ 3.180229333022904e-14, 0.007017420920647055, 1.575779506824242e-13
R$ 0.007194400715434313, 2.239633846636919e-13, 0.007371402223725454
R$ 1.00613727776042e-13, 0.0075484254505227, 1.138922438134334e-13
R$ 0.007725470401283019, 1.373765885826852e-13, 0.007902537081463379
R$ 4.660712387442989e-14, 0.008035351354465092, 1.779540657512234e-13
R$ 0.008212456073806607, 4.84136748459322e-14, 0.008389582536892704
R$ 1.546045680667371e-13, 0.008566730749407725, 1.520621667666882e-13
R$ 0.008743900716581265, 1.530365318998788e-13, 0.008921092443870293
R$ 4.437230418092991e-14, 0.009098305936277029, 1.696306974974602e-13
R$ 0.009275541199485815, 1.922220893707314e-13, 0.009452798238953619
R$ 4.901690999851026e-15, 0.009630077059455289, 1.845178670144181e-13
R$ 0.009807377666902539, 1.730237013901949e-13, 0.009984700066524965
R$ 9.646737105227633e-14, 0.01016204426355216, 8.287209019893124e-14
R$ 0.0103394102634411, 3.486340610857252e-14, 0.01051679807142136
R$ 8.441752187352747e-14, 0.01069420769294993, 1.381149179584253e-13
R$ 0.01087163913348377, 1.045150006862668e-13, 0.01104909239825247
R$ 1.215307513229208e-13, -0.005624549193726125, -1.51981979419044e-13
R$ -0.005536834037002336, -8.39554315012231e-14, -0.005449113546774242
R$ -1.570959534331267e-13, -0.005361387722587097, -1.775402557762978e-13
R$ -0.005273656563758777, -1.786803973183034e-13, -0.005185920069607164
R$ -1.937900808455538e-13, -0.005098178239677509, -2.865094890736807e-14
R$ -0.005010431072832944, -1.710472567874437e-13, -0.004922678568846095
R$ -1.991501155910151e-13, -0.004834920727034842, -1.457595160852659e-13
R$ -0.004747157546717062, -4.355694898511842e-14, -0.004659389026983263
R$ -1.524790515599519e-13, -0.004571615167606069, -5.022287701538693e-14
R$ -0.004483835967448613, -2.23861593741848e-13, -0.004396051426510894
R$ -2.348137746263453e-14, -0.004308261543428671, -1.632918122378509e-13
R$ -0.004220466317974569, -2.20515431708447e-13, -0.004132665749466469
R$ -2.270034183096471e-13, -0.004044859837222248, -2.144882234864991e-13
R$ -0.003957048580559785, -2.145835387870935e-13, -0.003869231979024335
R$ -3.14105915037523e-14, -0.003781410031479027, -1.510928175215772e-13
R$ -0.003693582737696488, -1.501401042826415e-13, -0.003605750096994598
R$ -5.96908145033904e-14, -0.003517912108463861, -1.381380817806073e-13
R$ -0.00343006877164953, -1.890087553021017e-13, -0.003342220086096859
R$ -1.571072176473979e-14, -0.003254366050668978, -1.036539027268881e-13
R$ -0.003166506665138513, -2.863449751971163e-14, -0.003078641928595971
R$ -4.845068235489542e-14, -0.00299077184035923, -1.934078795479823e-13
R$ -0.002902896400200916, -3.894505306257473e-14, -0.002815015606984161
R$ -6.987673214832359e-14, -0.002727129460254218, -8.87772542080616e-14
R$ -0.002639237959328966, -1.254754375316458e-13, -0.002551341103526283
R$ -2.096808768047939e-13, -0.002463438892391423, -1.436102386090714e-13
R$ -0.00237553132501489, -1.841082586859539e-13, -0.002287618400941938
R$ -1.331530110894102e-13, -0.002199700119490444, -1.997690543224918e-14
R$ -0.002111776479750915, -1.010666577939101e-13, -0.002023847481268604
R$ -1.780422352896898e-13, -0.001935913123588762, -5.240417784726179e-14
R$ -0.001847973405574521, -2.076545954175557e-13, -0.001760028326998508
R$ -2.176814107444407e-13, -0.001672077887178602, -1.11000382887547e-13
R$ -0.001584122085205308, -1.433814026335814e-13, -0.001496160920623879
R$ -1.157274370122938e-13, -0.001408194392752193, -5.619552646294779e-14
R$ -0.001320222500680757, -2.20196755662203e-13, -0.001232245244182195
R$ -1.809015225701754e-13, -0.001144262622347014, -1.934815618999196e-13
R$ -0.001056274634720467, -5.82415387024573e-14, -0.0009682812803930574
R$ -3.01137209319898e-14, -0.0008802825586826657, -1.365369238995784e-13
R$ -0.0007922784691345441, -1.774564810476099e-13, -0.0007042690110665717
R$ -1.800715656018116e-13, -0.0006162541837966273, -1.714614858846534e-13
R$ -0.00052823398664259, -1.785856560630599e-13, -0.0004402084191497124
R$ -9.098914439675769e-16, -0.0003521774801811262, -1.19901080977114e-13
R$ -0.0002641411695094575, -1.074114298733444e-13, -0.0001760994862252119
R$ -2.172941584192324e-13, -0.00008805243010101549, -2.116142002074782e-14
R$ 0, 0]



5th table

Code Select Expand

; array of 258 Real8
[acrt_pow_rcp_t3: R$ 1.442694902420044, 1.442693471908569, 1.442692041397095
R$ 1.442690849304199, 1.442689418792725, 1.44268798828125
R$ 1.442686557769775, 1.44268536567688, 1.442683935165405
R$ 1.442682504653931, 1.442681074142456, 1.442679882049561
R$ 1.442678451538086, 1.442677021026611, 1.442675590515137
R$ 1.442674160003662, 1.442672967910767, 1.442671537399292
R$ 1.442670106887817, 1.442668676376343, 1.442667484283447
R$ 1.442666053771973, 1.442664623260498, 1.442663192749023
R$ 1.442662000656128, 1.442660570144653, 1.442659139633179
R$ 1.442657709121704, 1.442656278610229, 1.442655086517334
R$ 1.442653656005859, 1.442652225494385, 1.44265079498291
R$ 1.442649602890015, 1.44264817237854, 1.442646741867065
R$ 1.442645311355591, 1.442643880844116, 1.442642688751221
R$ 1.442641258239746, 1.442639827728271, 1.442638397216797
R$ 1.442637205123901, 1.442635774612427, 1.442634344100952
R$ 1.442632913589478, 1.442631721496582, 1.442630290985107
R$ 1.442628860473633, 1.442627429962158, 1.442625999450684
R$ 1.442624807357788, 1.442623376846313, 1.442621946334839
R$ 1.442620515823364, 1.442619323730469, 1.442617893218994
R$ 1.44261646270752, 1.442615032196045, 1.442613840103149
R$ 1.442612409591675, 1.4426109790802, 1.442609548568726
R$ 1.442608118057251, 1.442606925964355, 1.442605495452881
R$ 1.442604064941406, 1.442602634429932, 1.442601442337036
R$ 1.442600011825562, 1.442598581314087, 1.442597150802612
R$ 1.442595958709717, 1.442594528198242, 1.442593097686768
R$ 1.442591667175293, 1.442590236663818, 1.442589044570923
R$ 1.442587614059448, 1.442586183547974, 1.442584753036499
R$ 1.442583560943604, 1.442582130432129, 1.442580699920654
R$ 1.44257926940918, 1.442578077316284, 1.44257664680481
R$ 1.442575216293335, 1.44257378578186, 1.442572355270386
R$ 1.44257116317749, 1.442569732666016, 1.442568302154541
R$ 1.442566871643066, 1.442565679550171, 1.442564249038696
R$ 1.442562818527222, 1.442561388015747, 1.442560195922852
R$ 1.442558765411377, 1.442557334899902, 1.442555904388428
R$ 1.442554473876953, 1.442553281784058, 1.442551851272583
R$ 1.442550420761108, 1.442548990249634, 1.442547798156738
R$ 1.442546367645264, 1.442544937133789, 1.442543506622314
R$ 1.442542314529419, 1.442540884017944, 1.44253945350647
R$ 1.442538022994995, 1.442536592483521, 1.442535400390625
R$ 1.44253396987915, 1.442532539367676, 1.442531108856201
R$ 1.442529916763306, 1.442528486251831, 1.442527055740356
R$ 1.442525625228882, 1.442524433135986, 1.442523002624512
R$ 1.442521572113037, 1.442520141601562, 1.442782878875732
R$ 1.442782163619995, 1.442781686782837, 1.4427809715271
R$ 1.442780256271362, 1.442779541015625, 1.442778825759888
R$ 1.44277811050415, 1.442777395248413, 1.442776679992676
R$ 1.442775964736938, 1.44277548789978, 1.442774772644043
R$ 1.442774057388306, 1.442773342132568, 1.442772626876831
R$ 1.442771911621094, 1.442771196365356, 1.442770481109619
R$ 1.442770004272461, 1.442769289016724, 1.442768573760986
R$ 1.442767858505249, 1.442767143249512, 1.442766427993774
R$ 1.442765712738037, 1.4427649974823, 1.442764282226562
R$ 1.442763805389404, 1.442763090133667, 1.44276237487793
R$ 1.442761659622192, 1.442760944366455, 1.442760229110718
R$ 1.44275951385498, 1.442758798599243, 1.442758083343506
R$ 1.442757606506348, 1.44275689125061, 1.442756175994873
R$ 1.442755460739136, 1.442754745483398, 1.442754030227661
R$ 1.442753314971924, 1.442752599716187, 1.442751884460449
R$ 1.442751407623291, 1.442750692367554, 1.442749977111816
R$ 1.442749261856079, 1.442748546600342, 1.442747831344604
R$ 1.442747116088867, 1.44274640083313, 1.442745923995972
R$ 1.442745208740234, 1.442744493484497, 1.44274377822876
R$ 1.442743062973022, 1.442742347717285, 1.442741632461548
R$ 1.442740917205811, 1.442740201950073, 1.442739725112915
R$ 1.442739009857178, 1.44273829460144, 1.442737579345703
R$ 1.442736864089966, 1.442736148834229, 1.442735433578491
R$ 1.442734718322754, 1.442734003067017, 1.442733526229858
R$ 1.442732810974121, 1.442732095718384, 1.442731380462646
R$ 1.442730665206909, 1.442729949951172, 1.442729234695435
R$ 1.442728519439697, 1.442728042602539, 1.442727327346802
R$ 1.442726612091064, 1.442725896835327, 1.44272518157959
R$ 1.442724466323853, 1.442723751068115, 1.442723035812378
R$ 1.442722320556641, 1.442721843719482, 1.442721128463745
R$ 1.442720413208008, 1.442719697952271, 1.442718982696533
R$ 1.442718267440796, 1.442717552185059, 1.442716836929321
R$ 1.442716121673584, 1.442715644836426, 1.442714929580688
R$ 1.442714214324951, 1.442713499069214, 1.442712783813477
R$ 1.442712068557739, 1.442711353302002, 1.442710638046265
R$ 1.442709922790527, 1.442709445953369, 1.442708730697632
R$ 1.442708015441895, 1.442707300186157, 1.44270658493042
R$ 1.442705869674683, 1.442705154418945, 1.442704439163208
R$ 1.44270396232605, 1.442703247070312, 1.442702531814575
R$ 1.442701816558838, 1.442701101303101, 1.442700386047363
R$ 1.442699670791626, 1.442698955535889, 1.442698240280151
R$ 1.442697763442993, 1.442697048187256, 1.442696332931519
R$ 1.442695617675781, 1.442694902420044, 0.0]




Btw, the tables seems to contains 2 or 4 extra Real8 used for align

[guga]

6th table

Code Select Expand

; Array of 514 Real8
[acrt_pow_log2_t3: R$ 0, 0, 0.000001430512156730401, 1.64394539905828e-13
R$ 0.000002861025905076531, 1.556072355024121e-13, 0.000004053121756442124
R$ 1.70735242598682e-13, 0.000005483638233272359, 3.393325894260967e-14
R$ 0.000006914155846970971, 1.787047518188288e-13, 0.000008344675052285311
R$ 1.503051832822325e-13, 0.000009536775451124413, 1.490978738920983e-13
R$ 0.00001096729715754918, 2.200770702728377e-13, 0.00001239782045558968
R$ 1.178931752867144e-13, 0.00001382834511787223, 6.992267735039363e-14
R$ 0.00001502045006418484, 5.241449717899461e-14, 0.00001645097722757782
R$ 1.038425331121927e-13, 0.00001788150575521286, 2.094919362320289e-13
R$ 0.00001931203587446362, 1.419918441007808e-13, 0.00002074256735795643
R$ 1.28718745175558e-13, 0.00002193467798861093, 1.401354501619227e-13
R$ 0.00002336521197321417, 2.262858114976707e-13, 0.00002479574754943314
R$ 1.392974608476407e-13, 0.00002622628448989417, 1.065468867010195e-13
R$ 0.00002741839966802218, 1.017405996004179e-13, 0.00002884893910959363
R$ 1.684332551766157e-13, 0.00003027948014278081, 6.199798215530495e-14
R$ 0.00003171002254021005, 9.811269057749889e-15, 0.00003290214203843789
R$ 2.161901122993986e-13, 0.00003433268716435123, 3.609272316871314e-14
R$ 0.00003576323342713295, 1.376255398382174e-13, 0.00003719378128153039
R$ 6.604402453155766e-14, 0.00003862433050016989, 4.872466581044466e-14
R$ 0.00003981645591011329, 5.67963486239447e-14, 0.00004124700762986322
R$ 1.389649166312656e-13, 0.00004267756094122888, 4.802993632870214e-14
R$ 0.00004410811561683659, 1.136789631004368e-14, 0.00004530024557425349
R$ 3.328915612756188e-15, 0.00004673080275097163, 6.617457251950451e-14
R$ 0.00004816136129193183, 1.833011403488492e-13, 0.00004959192142450775
R$ 1.273377568401947e-13, 0.00005102248292132572, 1.256609106277783e-13
R$ 0.00005221461856308451, 1.467866447381837e-13, 0.00005364518278838659
R$ 1.726874796893837e-14, 0.00005507574815055705, 1.694190346924574e-13
R$ 0.00005650631510434323, 1.484929672453615e-13, 0.00005769845529357553
R$ 1.535859214430789e-13, 0.00005912902497584582, 4.838574145834865e-15
R$ 0.00006055959579498449, 1.377701944086278e-13, 0.00006199016820573888
R$ 9.763624460015895e-14, 0.0000631823129424447, 8.673086804021568e-14
R$ 0.00006461288785430952, 1.461690847569645e-13, 0.00006604346435779007
R$ 3.254970233753423e-14, 0.000067474041998139, 2.006228849556349e-13
R$ 0.00006890462123010366, 1.956440950204176e-13, 0.00007009677165115136
R$ 2.140450720435735e-13, 0.00007152735361160012, 8.128970207829634e-14
R$ 0.00007295793693629093, 2.864006052624349e-15, 0.00007438852139785013
R$ 2.061444727374861e-13, 0.00007558067636637134, 2.086250054972199e-13
R$ 0.00007701126378378831, 5.677498815065671e-14, 0.00007844185233807366
R$ 1.866391046552165e-13, 0.00007987244248397474, 1.434728174842135e-13
R$ 0.00008106460199996945, 1.300673564165687e-13, 0.00008249519464698096
R$ 1.865377077679364e-13, 0.0000839257888856082, 6.998562667522999e-14
R$ 0.00008535638448847749, 7.787601973542197e-15, 0.00008678698122821515
R$ 2.273201225063353e-13, 0.00008797914665592543, 1.598898362090234e-14
R$ 0.0000894097461241472, 1.078093970477927e-13, 0.0000908403471839847
R$ 2.662097616880261e-14, 0.00009227094938069058, 2.271738853027094e-13
R$ 0.00009346311935587437, 3.464198041886814e-17, 0.00009489372428106435
R$ 7.289521618551281e-14, 0.00009632433057049639, 2.00131416397277e-13
R$ 0.00009775493845154415, 1.54372380636876e-13, 0.00009894711274682777
R$ 1.388331600380832e-13, 0.000100377723128986, 1.927752371946668e-13
R$ 0.0001018083351027599, 7.373004990699465e-14, 0.0001032389484407759
R$ 9.074087114779278e-15, 0.0001046695629156602, 2.261838377661032e-13
R$ 0.0001058617431226594, 1.286058865906658e-14, 0.0001072923603260278
R$ 1.023227075261518e-13, 0.000108722979121012, 1.881116059284149e-14
R$ 0.0001101535990528646, 2.170761122825348e-13, 0.0001113457835799636
R$ 2.15430778185943e-13, 0.000112776406467674, 5.869419556051461e-14
R$ 0.0001142070304922527, 1.837420832915262e-13, 0.0001156376561084471
R$ 1.358299040615336e-13, 0.0001168298451830196, 1.185232625304213e-13
R$ 0.0001182604733003245, 1.70376673009607e-13, 0.0001196911030092451
R$ 4.927798835213344e-14, 0.0001211217338550341, 2.099773730455703e-13
R$ 0.0001225523662924388, 1.977302898129293e-13, 0.0001237445610513532
R$ 2.1015494979381e-13, 0.000125175196217242, 7.032471249622007e-14
R$ 0.0001266058325199992, 2.123033300979868e-13, 0.0001280364704143722
R$ 1.813462653542081e-13, 0.0001292286697207601, 1.781875130840851e-13
R$ 0.0001306593103436171, 1.966706792300232e-14, 0.0001320899521033425
R$ 1.429662633327201e-13, 0.0001335205954546836, 9.334056210042326e-14
R$ 0.0001347127993085451, 7.463285185876999e-14, 0.0001361434451609966
R$ 1.248372195255438e-13, 0.0001375740926050639, 2.124662671097775e-15
R$ 0.0001390047411859996, 1.612453458876318e-13, 0.000140435391358551
R$ 1.474547320027884e-13, 0.0001416276008967543, 1.586199831178549e-13
R$ 0.0001430582537977898, 1.731069528509697e-14, 0.0001444889078356937
R$ 1.578454334728518e-13, 0.0001459195634652133, 1.254796605408497e-13
R$ 0.0001471117775508901, 1.21173851764616e-13, 0.0001485424356815201
R$ 1.886828547109084e-13, 0.0001499730954037659, 8.329931886378599e-14
R$ 0.0001514037564902537, 3.239973344476338e-14, 0.0001525959751234041
R$ 1.265732033797346e-14, 0.0001540266387110023, 6.165228540573155e-14
R$ 0.0001554573036628426, 1.651391733188491e-13, 0.0001568879702062986
R$ 9.574712244446532e-14, 0.0001583186381139967, 8.085262204455472e-14
R$ 0.0001595108624314889, 9.112483540719435e-14, 0.0001609415328402974
R$ 1.761498190174993e-13, 0.0001623722048407217, 8.830665019094188e-14
R$ 0.000163802878205388, 5.497181822158605e-14, 0.0001649951070703537
R$ 4.988532823075644e-14, 0.0001664257829361304, 1.164897553899617e-13
R$ 0.0001678564603935229, 1.023681658595369e-14, 0.0001692871389877837
R$ 1.85876676588118e-13, 0.0001704793724002229, 1.654659400858969e-13
R$ 0.0001719100537229679, 2.136911591172563e-13, 0.0001733407366373285
R$ 8.906979878206305e-14, 0.0001747714209159312, 1.897834843856117e-14
R$ -0.00008797378177405335, -3.693324294747513e-14, -0.000087258569237747
R$ -2.04098621796641e-13, -0.00008678176095600065, -4.277051824350725e-14
R$ -0.00008606654796494695, -7.374159465758639e-14, -0.00008535133451914589
R$ -2.048945612320832e-13, -0.00008463612084597116, -2.088553909732898e-13
R$ -0.00008392090694542276, -8.562373233020722e-14, -0.000083205692590127
R$ -6.257290919454598e-14, -0.00008249047778008389, -1.397025700142615e-13
R$ -0.00008177526274266711, -8.963868779355458e-14, -0.00008106004725050298
R$ -1.397545864225669e-13, -0.00008058323669501988, -2.035567039913591e-13
R$ -0.00007986802074810839, -1.174732222150849e-13, -0.00007915280434644956
R$ -1.315685838110667e-13, -0.00007843758771741705, -1.846876178106521e-14
R$ -0.00007772237063363718, -5.547080012782316e-15, -0.00007700715309510997
R$ -9.280318695016504e-14, -0.00007629193532920908, -5.286305559340549e-14
R$ -0.00007557671710856084, -1.13100009828637e-13, -0.00007509990473408834
R$ -1.836480004341912e-13, -0.00007438468605869275, -1.076810821590146e-13
R$ -0.00007366946692854981, -1.318903119876763e-13, -0.00007295424757103319
R$ -2.890166291792855e-14, -0.00007223902775876923, -2.608845883346459e-14
R$ -0.00007152380749175791, -1.234503481742229e-13, -0.00007080858699737291
R$ -9.3613303936387e-14, -0.00007009336604824057, -1.639506500020817e-13
R$ -0.00006937814487173455, -1.070883593664447e-13, -0.00006890133045089897
R$ -1.753597191757648e-13, -0.00006818610859227192, -2.096621533793974e-13
R$ -0.0000674708865062712, -1.167640133814618e-13, -0.00006675566396552313
R$ -1.240386230616428e-13, -0.00006604044119740138, -4.111955412637959e-15
R$ -0.0000653252177471586, -2.117310097563184e-13, -0.00006460999429691583
R$ -6.47744081978722e-14, -0.0000638947703919257, -1.79891500581245e-14
R$ -0.00006317954603218823, -7.137488377091368e-14, -0.00006270272956498957
R$ -1.373520086128896e-13, -0.00006198750475050474, -5.452371356270698e-14
R$ -0.00006127227948127256, -7.186547285274053e-14, -0.00006055705375729303
R$ -1.893769349149114e-13, -0.00005984182780593983, -1.796840727373857e-13
R$ -0.00005912660162721295, -4.278653475103897e-14, -0.00005841137499373872
R$ -6.057644829455613e-15, -0.00005769614790551714, -6.949705140246527e-14
R$ -0.00005698092058992188, -5.730727456142644e-15, -0.00005650410207636014
R$ -6.939601293275178e-14, -0.00005578887407864386, -9.678425744222398e-14
R$ -0.00005507364562618022, -2.243395093511909e-13, -0.00005435841694634291
R$ -2.2468774164381e-13, -0.00005364318803913193, -9.782860274694785e-14
R$ -0.00005292795867717359, -7.113541653018028e-14, -0.00005221272886046791
R$ -1.446078314193281e-13, -0.00005149749881638854, -9.087182039645719e-14
R$ -0.0000510206784838374, -1.612224837130065e-13, -0.00005030544775763701
R$ -1.986365488158779e-13, -0.00004959021680406295, -1.088412504716354e-13
R$ -0.00004887498539574153, -1.192099125468918e-13, -0.00004815975376004644
R$ -2.368508021272758e-15, -0.00004744452144223033, -2.130640362035772e-13
R$ -0.00004672928912441421, -6.917511918592114e-14, -0.00004601405635185074
R$ -2.544875627605762e-14, -0.00004529882312453992, -8.188459589475289e-14
R$ -0.0000448220007456257, -1.498902417097785e-13, -0.00004410676706356753
R$ -7.009740366315591e-14, -0.00004339153292676201, -9.046583059791222e-14
R$ -0.00004267629833520914, -2.109951709328963e-13, -0.00004196106351628259
R$ -2.043113976432022e-13, -0.00004124582846998237, -7.041415914663291e-14
R$ -0.0000405305929689348, -3.667677930370069e-14, -0.00003981535701313987
R$ -1.030989065311628e-13, -0.00003910012082997127, -4.230651380202161e-14
R$ -0.00003862329640469397, -1.079495370629234e-13, -0.00003790805953940435
R$ -1.382970636984089e-13, -0.00003719282244674105, -4.142913281802618e-14
R$ -0.0000364775848993304, -4.471906827932448e-14, -0.00003576234689717239
R$ -1.481665184960979e-13, -0.00003504710866764071, -1.243974564383855e-13
R$ -0.00003433186998336168, -2.007852059621678e-13, -0.00003361663107170898
R$ -1.49955740036438e-13, -0.00003313980482744228, -2.222392170015739e-13
R$ -0.00003242456546104222, -3.517150250014857e-14, -0.00003170932541252114
R$ -1.756329858657198e-13, -0.00003099408513662638, -1.888759646221316e-13
R$ -0.00003027884463335795, -7.490008717916897e-14, -0.00002956360367534216
R$ -6.107867738932616e-14, -0.00002884836226257903, -1.474113836613424e-13
R$ -0.00002813312062244222, -1.065241789602021e-13, -0.00002741787852755806
R$ -1.657903871368305e-13, -0.00002694105046430195, -8.304386734796152e-15
R$ -0.00002622580768729676, -1.587009432423048e-13, -0.0000255105646829179
R$ -1.818762996023095e-13, -0.00002479532145116536, -7.783010422058627e-14
R$ -0.00002408007776466548, -7.393568094562017e-14, -0.00002336483362341824
R$ -1.701926781821414e-13, -0.00002264958925479732, -1.392270688911251e-13
R$ -0.00002193434443142905, -2.084121769194877e-13, -0.00002121909938068711
R$ -1.503739752271584e-13, -0.00002074226904369425, -2.178482426372076e-13
R$ -0.00002002702353820496, -2.356163523040752e-14, -0.00001931177735059464
R$ -1.567981313952534e-13, -0.00001859653093561064, -1.628100286465251e-13
R$ -0.00001788128429325297, -4.159697538494361e-14, -0.00001716603719614795
R$ -2.053229545393903e-14, -0.00001645078964429558, -9.961563725318647e-14
R$ -0.00001573554186506954, -5.147297373860604e-14, -0.00001502029363109614
R$ -1.03477628752059e-13, -0.0000145434612477402, -1.685208825968106e-13
R$ -0.00001382821255901945, -8.427205322821007e-14, -0.00001311296341555135
R$ -1.00169604781431e-13, -0.0000123977138173359, -2.162131856531851e-13
R$ -0.00001168246399174677, -2.050287687964293e-13, -0.00001096721393878397
R$ -6.661600260682923e-14, -0.00001025196343107382, -2.834821092276003e-14
R$ -0.000009536712468616315, -9.022504213884163e-14, -0.000009059878266270971
R$ -1.618482492123198e-13, -0.000008344626849066117, -8.74671033243007e-14
R$ -0.000007629374977113912, -1.13229642719529e-13, -0.000006914122877788031
R$ -1.176184034747526e-14, -0.0000061988703237148, -1.043702004355092e-14
R$ -0.000005483617314894218, -1.092548301994126e-13, -0.00000476836407869996
R$ -8.084124376296187e-14, -0.000004053110387758352, -1.525695845680414e-13
R$ -0.000003337856469443068, -9.706582556150695e-14, -0.000002861020220734645
R$ -1.662248380384509e-13, -0.000002145765620298334, -2.01831698580088e-13
R$ -0.000001430510792488349, -1.102055216811216e-13, -7.152555099310121e-7
R$ -1.187196311729539e-13, 0, 0]



7th table

Code Select Expand


; Array of 256 Real8 (Added 2 more t avoid errors. Now we have 258)
; This is to 2^(x/128) = a+b
[acrt_pow_exp2_tbl: R$ 1.0, 0.0, 1.005429901112803, 9.499186535455032e-17
R$ 1.0108892860517, -1.523477860336858e-17, 1.016378314910953
R$ -5.77217007319966e-17, 1.021897148654117, 5.109225028973444e-17
R$ 1.027445949118764, -4.95607417464537e-17, 1.033024879021228
R$ 7.600838874027088e-18, 1.038634101961379, 5.996273788852509e-17
R$ 1.044273782427414, 8.551889705537964e-17, 1.049944085800687
R$ 5.592937848127003e-17, 1.055645178360557, 1.759325738772092e-18
R$ 1.061377227289262, -1.197353708536566e-17, 1.067140400676824
R$ -7.899853966841581e-17, 1.072934867525976, -3.839668843358824e-18
R$ 1.07876079775712, -6.656660436056593e-17, 1.084618362213309
R$ 3.166152845816345e-17, 1.090507732665258, -3.046782079812471e-17
R$ 1.096429081816377, -5.919933484449315e-17, 1.102382583307841
R$ 5.266036871570694e-17, 1.108368411723679, -8.786813845180527e-17
R$ 1.114386742595892, 1.041027845684557e-16, 1.120437752409607
R$ -6.201085906554178e-17, 1.126521618608242, 5.165856758795456e-17
R$ 1.132638519598719, 3.237356166738e-17, 1.138788634756692
R$ 8.912812676025407e-17, 1.144972144431804, 4.641289892170011e-17
R$ 1.151189229952983, 3.250710218863827e-17, 1.157440073633751
R$ -9.123871231134399e-17, 1.163724858777577, 3.829204836924093e-17
R$ 1.17004376968325, -1.847744201790004e-18, 1.176396991650281
R$ 5.554203254218078e-17, 1.182784710984341, 1.542975430079076e-17
R$ 1.189207115002721, 3.982015231465646e-17, 1.195664392039827
R$ 4.616603670481481e-17, 1.202156731452703, 6.6449814992523e-17
R$ 1.208684323626582, -4.746725945228984e-17, 1.215247359980469
R$ -7.712630692681487e-17, 1.221846032972758, -1.061102121140269e-16
R$ 1.22848053610687, -1.89878163130253e-17, 1.235151063936933
R$ -1.075524434430784e-16, 1.241857812073484, 4.658027591836936e-17
R$ 1.248600977189205, -8.261810999021964e-17, 1.255380757024691
R$ -6.711389821296878e-18, 1.262197350394251, -3.084464887473846e-17
R$ 1.269050957191733, 2.667932131342186e-18, 1.275941778396392
R$ 9.91543024421429e-17, 1.282870016078778, 1.713594918243561e-17
R$ 1.289835873406666, 8.949257530897592e-17, 1.29683955465101
R$ 2.538250279488831e-17, 1.303881265191936, 8.647675598267871e-17
R$ 1.310961211524764, -7.181536135519454e-17, 1.318079601266064
R$ -5.457955827149153e-17, 1.325236643159741, -2.858731210038861e-17
R$ 1.332432547083162, -5.101586630916743e-17, 1.339667524053303
R$ 8.927282594831731e-17, 1.346941786232946, 3.224065101254679e-17
R$ 1.354255546936893, 7.700948379802988e-17, 1.361609020638225
R$ 1.533787661270668e-18, 1.369002422974591, 9.593797919118848e-17
R$ 1.37643597075453, -6.898588935871801e-17, 1.383909881963832
R$ -6.770511658794785e-17, 1.391424375771926, -4.906174865288989e-17
R$ 1.398979672538311, -9.614213209051322e-17, 1.406575993819015
R$ 7.034914812136421e-18, 1.414213562373095, -9.667293313452912e-17
R$ 1.421892602169166, -1.607782891589024e-17, 1.42961333839197
R$ -1.203164248905365e-17, 1.437375997448982, -4.204034016467557e-17
R$ 1.445180806977047, -3.023758134993987e-17, 1.453027995849053
R$ -5.779948609396105e-17, 1.460917794180647, -5.600377186075216e-17
R$ 1.468850433336982, 8.465882756533626e-17, 1.476826145939499
R$ -3.483994556892796e-17, 1.484845165872752, 1.078008676440748e-16
R$ 1.492907728291265, 1.419292015428404e-17, 1.501014069626426
R$ -6.413767275790234e-17, 1.509164427593423, -1.016455327754295e-16
R$ 1.517359041198215, -4.308699472043341e-17, 1.525598150744538
R$ -1.102494171234256e-16, 1.533881997840956, 8.875226844438446e-17
R$ 1.542210825407941, 7.94983480969762e-17, 1.550584877685
R$ -1.460070659068939e-17, 1.559004400237837, 3.781207053357528e-17
R$ 1.567469639965553, -1.035206176884972e-16, 1.575980845107886
R$ -1.01369164712783e-17, 1.584538265252494, -1.93377170345857e-17
R$ 1.593142151342267, -1.009440654231196e-16, 1.601792755682693
R$ -6.054917453527784e-17, 1.610490331949254, 2.470719256979789e-17
R$ 1.619235135194864, 2.094133415422909e-17, 1.628027421857348
R$ -6.712955084707083e-17, 1.636867449766964, 7.698325071319876e-17
R$ 1.645755478153965, -1.012567991367477e-16, 1.654691767656194
R$ 9.643294303196027e-17, 1.663676580326736, 5.8909926967131e-17
R$ 1.672710179641597, -5.476715964599563e-17, 1.681792830507429
R$ 8.199010020581497e-17, 1.690924799269305, -9.669671474394879e-17
R$ 1.700106353718523, -8.023719370397699e-18, 1.709337763100463
R$ -9.868779456632931e-17, 1.718619298122478, -1.851380418263111e-17
R$ 1.727951230961838, -1.075098186120464e-16, 1.737333835273706
R$ 3.164389299292957e-17, 1.746767386199169, -1.075229048350751e-16
R$ 1.756252160373299, 2.960140695448873e-17, 1.765788435933273
R$ 9.461315018083268e-17, 1.775376492526521, 6.429731796556571e-17
R$ 1.785016611318935, 1.533040012103131e-17, 1.794709075003107
R$ 1.822745842791209e-17, 1.804454167806624, -5.177222408793318e-17
R$ 1.814252175500399, -9.969531538920349e-17, 1.824103385407053
R$ -1.015962786227708e-16, 1.834008086409342, 3.283107224245627e-17
R$ 1.843966568958626, -5.939742026949965e-17, 1.853979125083385
R$ 9.761887490727594e-17, 1.864046048397789, 6.54091268062057e-17
R$ 1.8741676341103, -6.122763413004141e-17, 1.884344179032335
R$ -8.226593125533711e-17, 1.894575981586966, 3.40340353521653e-17
R$ 1.904863341817674, 6.533857514718277e-17, 1.915206561397147
R$ -1.061994605619596e-16, 1.925605943636125, -9.914963769693741e-17
R$ 1.936061793492294, 1.033238596067633e-16, 1.946574417579233
R$ 6.811022349533877e-17, 1.9571441241754, 8.960767791036667e-17
R$ 1.967771223233176, -1.031492801153113e-16, 1.978456026387951
R$ 4.038875310927817e-17, 1.989198846967266, 8.205132638369199e-18,
R$ 0, 0]


Btw, the tables seems to contains 2 or 4 extra Real8 used for align

[guga]

On the 1st table, it seems to b a sequence like this:

1st value = 16384/16384 = 16384/(256*64) = ((128*128)+0)/((128)*128) = ((128*128)+0/2)/((128)*128)
2nd value = 16257/16384 = 16257/(256*64) = ((127*128)+1)/((128)*128) = ((127*128)+2/2)/((128)*128) = ((127*128)+(128*(1/64))/2)/((128)*128)
3rd value = 16132/16384 = 16132/(256*64) = ((126*128)+4)/((128)*128) = ((126*128)+8/2)/((128)*128) = ((126*128)+(128*(4/64))/2)/((128)*128)
4th value = 16009/16384 = 16009/(256*64) = ((125*128)+9)/((128)*128) = ((125*128)+18/2)/((128)*128) = ((125*128)+(128*(9/64))/2)/((128)*128)
5th value = 15887.5/16384 = 15887.5/(256*64) = ((124*128)+15.5)/((128)*128) = ((124*128)+31/2)/((128)*128) = ((124*128)+(128*(15.5/64))/2)/((128)*128)
6th value = 15768/16384 = 15768/(256*64) = ((123*128)+24)/((128)*128) = ((123*128)+48/2)/((128)*128) = ((123*128)+(128*(24/64))/2)/((128)*128)
128th value = ((0*128)+16384/2)/((128)*128) ?

So, it seems to be a formula ion this format:

(((n-1)*128)+x/2)/((128)*128)

where n = Index f the serie varying from 0 to 128
x some unknown value that seems to be: 0/2, 2/2, 8/2, 18/2, 31/2, 48/2....16384/2
So, x seems also to be a sequence divided by 2 of: 0; 2; 8; 18; 31; 48.....16384. But how these numbers where calculated ?

[jj2007]

fixed mathematical series tables to accurately calculate the pow of a number

Sounds a bit like FastMath: calculate once y=f(x) using whatever slow function for a given range, then get y=f(x) really fast a Million times. Is that what you intend? If yes, what would be the slow function to generate the tables?

What would be the application of such an algo?

[guga]

Hi JJ

I´m using it to replace a pow function in RosAsm library (but it can be used in other apps and not only RosAsm). I succeeded to rebuild the libm_sse2_pow_precise from ucrtbased.dll and it has a high level of precision. But, although i succeeded to rebuild it, i have no idea how those tables where created. I´ll post the code soon, but i wanted to understand how those tables where generated, to see if it could be possible to achieve a even higher precision.

The internal code in ucrtbased.dll is a complete mess, highly optimized and thus, hard to maintain. That´s why i gave a try rebuilding it.

Did you used the libm_sse2_pow_precise before ? Take a look at it. It´s really precise, but the internal code is a mess.

[jj2007]

I'm afraid my math is not sufficient to understand the role of the tables

[daydreamer]

Guga
Using Taylor series or similar ?

[guga]

Hi Daydreamer,

yeah, but how those values were created ? I mena, how to reate a function to generate those exact values?

[daydreamer]

yeah, but how those values were created ? I mena, how to reate a function to generate those exact values?

I have concentrated on trigo functions since the day I discovered use cosine and sine in basic to draw circle

I have never coded a pow function or created pow LUT before, but someone else knows pow by heart
For others that LUT makes no sense,but those that used trigo LUT many years recognise it

[jack]

Hi guga, have a look at this pdf Hardware Implementation of Number Inversion Function
in the appendix there are tables that have at least some of the numbers that you posted, so maybe there's a hint in there somewhere as to how they were calculated

[jack]

Harmonized Parabolic Synthesis is a hardware-efficient methodology for approximating mathematical functions, including the power function. It combines Parabolic Synthesis with Second-Degree Interpolation to achieve high precision and low complexity. By using multiple intervals to approximate a mathematical expression, it can calculate functions like the inverse square root within a unit in the last place (ULP), according to DiVA portal.

[jack]

guga
ask your favorite AI can you explain how the LUT are computed using the "Harmonized Parabolic Synthesis" for the power function?

ChatGPT session

[InfiniteLoop]

Have you come full circle? Weren't these approximations done almost 10 years ago on this very forum?

It is straight forward enough to generate the Chebyshev coefficients using simultaneous linear equations. I have no idea what those integer numbers are for.  Pow needs log2(x) and 2^x.

[guga]

Hi guga, have a look at this pdf Hardware Implementation of Number Inversion Function
in the appendix there are tables that have at least some of the numbers that you posted, so maybe there's a hint in there somewhere as to how they were calculated

Tks, Jack, i´ll take a look