Good afternoon, everyone!
Please, tell me if the community knows anything about the strange behavior of NVidia graphics cards in relation to the product of two double precision numbers in OpenGL computational shader?
Here’s the deal - I created a computational shader that processes geodetic data, and quickly realized that float precision is not enough for most trigonometric operations.
I used the GL_ARB_gpu_shader_fp64 extension to enable support for double numbers, and implemented my sine, cosine, and arctangent functions with double precision using Taylor expansions and the Padé polynomial.
However, nothing has changed in my calculations, even though I am absolutely sure that my functions are correct. I did some debugging of the shader to find out what the problem was and discovered that the product of two different double numbers is rounded to float, even though the result of the product is of type double.
Especially interesting is the fact that I encounter this behavior only on NVidia graphics cards (RTX 4070 and 4080), but on AMD graphics cards (RX 580, RX 6700) everything is correct (for the same code).
I implemented a test program on Qt 5.15 comparing the results of sine calculation for some data set on CPU using <math.h> library and the results of calculation by my function inside the shader on GPU. This program outputs the original data set for sine calculation, as well as the sets of processed (sine) data obtained on CPU and GPU, then compares the results and outputs the maximum and average calculation error (relative to math.h).
I’ll attach the shader code with some comments, and say that the control program is exactly right.
Thank you for any help!
P.S. It is worth noting that I have tried a lot of things. I tried splitting the data into small parts, checking intermediate calculations, using the “precise” modifier, but EVERYTHING points to the fact that the NVidia compiler apparently optimizes the product of two double numbers to float precision.
P.P.S. I apologize for the long table in the code. I have shortened it considerably, but it is still large. Without it, I’m afraid nobody will believe me that my algorithm works correctly at all.
#version 430 core
#extension GL_ARB_gpu_shader_fp64 : enable
#extension GL_ARB_shader_precision : enable
layout (local_size_x = 1024, local_size_y = 1, local_size_z = 1) in;
layout (std430, binding = 0) buffer debugInfoBH {
double debugInfo[];
};
double PI = 3.141592653589793238462643383279lf;
double TAU = 6.283185307179586477lf;
// =============================================
double DsinTaylorTable[][6] =
{
{0.000000000000000005lf, 0.999999999999997002lf, 0.000000000000843494lf, -0.166666666787488260lf, 0.000000009230097997lf, 0.008333016025534760lf},
{0.000000000000010030lf, 0.999999999997764344lf, 0.000000000204954024lf, -0.166666676452553619lf, 0.000000249197463313lf, 0.008330477708129645lf},
{0.000000000000358366lf, 0.999999999952088658lf, 0.000000002635343280lf, -0.166666742164831294lf, 0.000001153551389282lf, 0.008325401846515482lf},
{0.000000000003776854lf, 0.999999999639332060lf, 0.000000014170424681lf, -0.166666956641516290lf, 0.000003164766486946lf, 0.008317789986848892lf},
{0.000000000021928181lf, 0.999999998371296939lf, 0.000000049771323972lf, -0.166667458840684707lf, 0.000006724642939041lf, 0.008307644447776066lf},
{0.000000000089306340lf, 0.999999994572779105lf, 0.000000135697713448lf, -0.166668433820118034lf, 0.000012274037160153lf, 0.008294968319726490lf},
{0.000000000287429289lf, 0.999999985219709231lf, 0.000000312706413340lf, -0.166670112549209631lf, 0.000020252592888140lf, 0.008279765463971562lf},
{0.000000000782214904lf, 0.999999965139087954lf, 0.000000639227861574lf, -0.166672771674073517lf, 0.000031098472829729lf, 0.008262040511448420lf},
{0.000000001877392519lf, 0.999999926172204900lf, 0.000001194515572567lf, -0.166676733236006042lf, 0.000045248090983012lf, 0.008241798861349308lf},
{0.000000004086756766lf, 0.999999856202947845lf, 0.000002081763728415lf, -0.166682364343480105lf, 0.000063135845759211lf, 0.008219046679476929lf},
{0.000000008228016685lf, 0.999999738053285347lf, 0.000003431188072986lf, -0.166690076797882808lf, 0.000085193854025860lf, 0.008193790896366288lf},
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{0.000000027832342708lf, 0.999999255653487462lf, 0.000008190726251800lf, -0.166713613850164782lf, 0.000143536102464266lf, 0.008135800059332779lf},
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{0.000000078492317312lf, 0.999998190211353277lf, 0.000017169590296906lf, -0.166751515550308815lf, 0.000223676103719489lf, 0.008067897003150897lf},
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{0.000000620248090388lf, 0.999989363351067806lf, 0.000075059490306173lf, -0.166942600866746477lf, 0.000541208515298265lf, 0.007855345759101486lf},
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{0.000001224552577062lf, 0.999980951358492187lf, 0.000121934477460228lf, -0.167073306537060401lf, 0.000723581340045019lf, 0.007753479733183538lf},
{0.000001680451189645lf, 0.999975019802682774lf, 0.000152809230264006lf, -0.167153674502075550lf, 0.000828199514485871lf, 0.007698996104260723lf},
{0.000002274198133700lf, 0.999967630077316483lf, 0.000189603772741548lf, -0.167245291656012707lf, 0.000942279254793323lf, 0.007642167286542700lf},
{0.000003038685110535lf, 0.999958511198620337lf, 0.000233118416619688lf, -0.167349131050946953lf, 0.001066193613198381lf, 0.007583010590637860lf},
{0.000004012721296132lf, 0.999947356900559581lf, 0.000284219151764770lf, -0.167466199499387874lf, 0.001200309563060838lf, 0.007521544036248839lf},
{0.000005241772663811lf, 0.999933822916915238lf, 0.000343839521083300lf, -0.167597536512179096lf, 0.001344987763804342lf, 0.007457786346683542lf},
{0.000006778753990633lf, 0.999917524207646902lf, 0.000412982384693034lf, -0.167744213198104869lf, 0.001500582328751474lf, 0.007391756943151848lf},
{0.000008684874062914lf, 0.999898032136234183lf, 0.000492721569702398lf, -0.167907331125916381lf, 0.001667440595966789lf, 0.007323475938849712lf},
{0.000011030534409651lf, 0.999874871604883086lf, 0.000584203402036659lf, -0.168088021149515932lf, 0.001845902902214550lf, 0.007252964132832497lf},
{0.000013896281704118lf, 0.999847518154685977lf, 0.000688648116851764lf, -0.168287442197061932lf, 0.002036302360136726lf, 0.007180243003679382lf},
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{0.000160463734056759lf, 0.998763685126346501lf, 0.003922086905081834lf, -0.173154725088832118lf, 0.005734880564614381lf, 0.006044786425102396lf},
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{0.044405983288408224lf, 0.852483046473811568lf, 0.202648466740645572lf, -0.312743758916803838lf, 0.056812431514744882lf, -0.001803663449484191lf},
{0.047066797411352002lf, 0.845082603437648183lf, 0.210881584087554669lf, -0.317323535649823796lf, 0.058086221185735587lf, -0.001945378032132546lf},
{0.049847625556333097lf, 0.837422730274222649lf, 0.219321387748838081lf, -0.321973144925080557lf, 0.059366999379265151lf, -0.002086500033787012lf},
{0.052751597970352504lf, 0.829499827872358320lf, 0.227967868906223808lf, -0.326691251855874110lf, 0.060654268781659706lf, -0.002226986467318808lf},
{0.055781803427652266lf, 0.821310511056549064lf, 0.236820787721035264lf, -0.331476434463233438lf, 0.061947522616185746lf, -0.002366794539199356lf},
{0.058941280710039012lf, 0.812851623399495793lf, 0.245879666106955996lf, -0.336327183061661039lf, 0.063246244832346366lf, -0.002505881662535609lf},
{0.062233009725744637lf, 0.804120252307703565lf, 0.255143780547669397lf, -0.341241899697122542lf, 0.064549910300272115lf, -0.002644205470042434lf},
{0.065659902264529224lf, 0.795113744371116038lf, 0.264612154966364532lf, -0.346218897638077339lf, 0.065857985010135511lf, -0.002781723826948091lf},
{0.069224792387135192lf, 0.785829720967342160lf, 0.274283553654143586lf, -0.351256400920325951lf, 0.067169926276516967lf, -0.002918394843828895lf},
{0.072930426447610386lf, 0.776266094110505955lf, 0.284156474264415759lf, -0.356352543946418465lf, 0.068485182947646855lf, -0.003054176889369144lf},
{0.076779452747455554lf, 0.766421082534321840lf, 0.294229140880395312lf, -0.361505371140344756lf, 0.069803195619446645lf, -0.003189028603042414lf},
{0.080774410820971476lf, 0.756293227998504181lf, 0.304499497162862287lf, -0.366712836658199204lf, 0.071123396854289950lf, -0.003322908907710385lf},
{0.084917720351658343lf, 0.745881411807165051lf, 0.314965199585365729lf, -0.371972804155483727lf, 0.072445211404401896lf, -0.003455777022135324lf},
{0.089211669719959530lf, 0.735184871527385875lf, 0.325623610764080862lf, -0.377283046611686934lf, 0.073768056439813809lf, -0.003587592473402461lf},
{0.093658404183129984lf, 0.724203217895687423lf, 0.336471792889549703lf, -0.382641246212745312lf, 0.075091341780786969lf, -0.003718315109248406lf},
{0.098259913688499845lf, 0.712936451899674584lf, 0.347506501267542289lf, -0.388044994291967482lf, 0.076414470134618898lf, -0.003847905110291949lf},
{0.103018020321880566lf, 0.701384982021666881lf, 0.358724177976299963lf, -0.393491791329968255lf, 0.077736837336741735lf, -0.003976323002163404lf},
{0.107934365393394294lf, 0.689549641630688104lf, 0.370120945647412047lf, -0.398979047014133503lf, 0.079057832596021810lf, -0.004103529667528892lf},
{0.113010396163514049lf, 0.677431706508740517lf, 0.381692601377584317lf, -0.404504080358103257lf, 0.080376838744166984lf, -0.004229486358005868lf},
{0.118247352212620505lf, 0.665032912496849704lf, 0.393434610778555016lf, -0.410064119881731692lf, 0.081693232489146117lf, -0.004354154705966241lf},
{0.123646251457948608lf, 0.652355473245946671lf, 0.405342102172394081lf, -0.415656303851949849lf, 0.083006384672523997lf, -0.004477496736223533lf},
{0.129207875822307194lf, 0.639402098057201940lf, 0.417409860939419008lf, -0.421277680584927761lf, 0.084315660530612596lf, -0.004599474877600485lf},
{0.134932756559547951lf, 0.626176009796040489lf, 0.429632324025926327lf, -0.426925208809898094lf, 0.085620419959337835lf, -0.004720051974373607lf},
{0.140821159242302762lf, 0.612680962863626277lf, 0.442003574618920370lf, -0.432595758094972138lf, 0.086920017782719461lf, -0.004839191297591166lf},
{0.146873068418083674lf, 0.598921261209221845lf, 0.454517336994991417lf, -0.438286109335248641lf, 0.088213804024860093lf, -0.004956856556261178lf},
{0.153088171940439555lf, 0.584901776366409498lf, 0.467166971550447729lf, -0.443992955303478487lf, 0.089501124185337164lf, -0.005073011908406005lf},
{0.159465844982420890lf, 0.570627965495801526lf, 0.479945470019780718lf, -0.449712901263520481lf, 0.090781319517890660lf, -0.005187621971980164lf},
{0.166005133740237287lf, 0.556105889416457222lf, 0.492845450889479064lf, -0.455442465646785144lf, 0.092053727312297495lf, -0.005300651835648033lf},
{0.172704738835556704lf, 0.541342230607878783lf, 0.505859155014169226lf, -0.461178080791833422lf, 0.093317681179321793lf, -0.005412067069418198lf},
{0.179562998425537801lf, 0.526344311164082868lf, 0.518978441441994121lf, -0.466916093747262118lf, 0.094572511338629164lf, -0.005521833735131145lf},
{0.186577871030267461lf, 0.511120110680904904lf, 0.532194783456081733lf, -0.472652767137972107lf, 0.095817544909551328lf, -0.005629918396797170lf},
{0.193746918087925657lf, 0.495678284057337715lf, 0.545499264838888331lf, -0.478384280094883152lf, 0.097052106204585972lf, -0.005736288130781283lf},
{0.201067286248598487lf, 0.480028179191400506lf, 0.558882576366125483lf, -0.484106729248125311lf, 0.098275517025515763lf, -0.005840910535832091lf},
{0.208535689418296144lf, 0.464179854550690330lf, 0.572335012536908883lf, -0.489816129783697429lf, 0.099487096962028468lf, -0.005943753742951513lf},
{0.216148390565362503lf, 0.448144096597491215lf, 0.585846468546668664lf, -0.495508416563555654lf, 0.100686163692719363lf, -0.006044786425102396lf},
{0.223901183302098516lf, 0.431932437047984674lf, 0.599406437509290346lf, -0.501179445309053007lf, 0.101872033288355826lf, -0.006143977806751034lf},
{0.231789373255045561lf, 0.415557169944867244lf, 0.613004007934847994lf, -0.506824993847618366lf, 0.103044020517282259lf, -0.006241297673241684lf},
{0.239807759238030771lf, 0.399031368522377416lf, 0.626627861469194580lf, -0.512440763422532863lf, 0.104201439152843287lf, -0.006336716380000258lf},
{0.247950614242683498lf, 0.382368901842496944lf, 0.640266270901580614lf, -0.518022380065615629lf, 0.105343602282700927lf, -0.006430204861564333lf},
{0.256211666261806581lf, 0.365584451180838599lf, 0.653907098446344648lf, -0.523565396032605213lf, 0.106469822619921509lf, -0.006521734640436783lf},
{0.264584078961601188lf, 0.348693526140521737lf, 0.667537794304618681lf, -0.529065291300983676lf, 0.107579412815706210lf, -0.006611277835760293lf},
{0.273060432219373705lf, 0.331712480472103888lf, 0.681145395511868257lf, -0.534517475129953956lf, 0.108671685773638629lf, -0.006698807171810145lf},
{0.281632702544017288lf, 0.314658527577445835lf, 0.694716525076961799lf, -0.539917287682246227lf, 0.109745954965321496lf, -0.006784295986302660lf},
{0.290292243397170191lf, 0.297549755675212468lf, 0.708237391418327089lf, -0.545260001707395192lf, 0.110801534747274430lf, -0.006867718238516798lf},
{0.299029765433585837lf, 0.280405142605530955lf, 0.721693788102637734lf, -0.550540824286091746lf, 0.111837740678962835lf, -0.006949048517226402lf},
{0.307835316679894166lf, 0.263244570251172982lf, 0.735071093891306315lf, -0.555754898635177907lf, 0.112853889841828273lf, -0.007028262048440715lf},
{0.316698262671531339lf, 0.246088838552513511lf, 0.748354273099941514lf, -0.560897305972820948lf, 0.113849301159189037lf, -0.007105334702950769lf},
{0.325607266568269216lf, 0.228959679093367452lf, 0.761527876275755577lf, -0.565963067443361445lf, 0.114823295716880006lf, -0.007180243003679382lf},
{0.334550269269340728lf, 0.211879768234719545lf, 0.774576041197776011lf, -0.570947146101302150lf, 0.115775197084498738lf, -0.007252964132832497lf},
{0.343514469549814683lf, 0.194872739773287895lf, 0.787482494204538641lf, -0.575844448953861443lf, 0.116704331637126124lf, -0.007323475938849712lf},
{0.352486304240439363lf, 0.177963197101751236lf, 0.800230551853782668lf, -0.580649829061487210lf, 0.117610028877387432lf, -0.007391756943151848lf},
{0.361451428473787750lf, 0.161176724847469810lf, 0.812803122918507537lf, -0.585358087695684537lf, 0.118491621757720739lf, -0.007457786346683542lf},
{0.370394696020109215lf, 0.144539899966451779lf, 0.825182710723560309lf, -0.589963976553483649lf, 0.119348447002718197lf, -0.007521544036248839lf},
{0.379300139736874709lf, 0.128080302269329316lf, 0.837351415826777412lf, -0.594462200027833010lf, 0.120179845431405877lf, -0.007583010590637860lf},
{0.388150952156572693lf, 0.111826524356096302lf, 0.849290939048477078lf, -0.598847417533171944lf, 0.120985162279327277lf, -0.007642167286542700lf},
{0.396929466237862183lf, 0.095808180936379994lf, 0.860982584852967081lf, -0.603114245885400746lf, 0.121763747520295493lf, -0.007698996104260723lf},
{0.405617136305739034lf, 0.080055917512067643lf, 0.872407265085493488lf, -0.607257261735433818lf, 0.122514956187678770lf, -0.007753479733183538lf},
{0.414194519206916745lf, 0.064601418399162669lf, 0.883545503067900029lf, -0.611271004055485179lf, 0.123238148695084107lf, -0.007805601577069981lf},
{0.422641255707107977lf, 0.049477414065814185lf, 0.894377438056048302lf, -0.615149976677205279lf, 0.123932691156303093lf, -0.007855345759101486lf},
{0.430936052157463223lf, 0.034717687763571869lf, 0.904882830061847221lf, -0.618888650880749736lf, 0.124597955704384936lf, -0.007902697126718327lf},
{0.439056662457850244lf, 0.020357081429032142lf, 0.915041065042556445lf, -0.622481468033835084lf, 0.125233320809700455lf, -0.007947641256235224lf},
{0.446979870345194819lf, 0.006431500833169013lf, 0.924831160459786306lf, -0.625922842279793668lf, 0.125838171596862169lf, -0.007990164457234942lf},
{0.454681472035526113lf, -0.007022080044185475lf, 0.934231771210436213lf, -0.629207163273618031lf, 0.126411900160364465lf, -0.008030253776738525lf},
{0.462136259248857595lf, -0.019965615437089190lf, 0.943221195931569056lf, -0.632328798964943539lf, 0.126953905878809337lf, -0.008067897003150897lf},
{0.469318002646428756lf, -0.032359985035637306lf, 0.951777383681018607lf, -0.635282098426894537lf, 0.127463595727581702lf, -0.008103082669980638lf},
{0.476199435710288110lf, -0.044164992069454345lf, 0.959877940995299328lf, -0.638061394729680265lf, 0.127940384589840556lf, -0.008135800059332779lf},
{0.482752239095556757lf, -0.055339362254267610lf, 0.967500139326147091lf, -0.640661007857802112lf, 0.128383695565690820lf, -0.008166039205173579lf},
{0.488947025486123898lf, -0.065840743623018283lf, 0.974620922856818339lf, -0.643075247669698147lf, 0.128792960279402124lf, -0.008193790896366288lf},
{0.494753324984868570lf, -0.075625707262681696lf, 0.981216916699006347lf, -0.645298416898624883lf, 0.129167619184540700lf, -0.008219046679476929lf},
{0.500139571069837530lf, -0.084649748977713102lf, 0.987264435471032487lf, -0.647324814193542375lf, 0.129507121866881542lf, -0.008241798861349308lf},
{0.505073087148136146lf, -0.092867291900650517lf, 0.992739492257706413lf, -0.649148737198744441lf, 0.129810927344967769lf, -0.008262040511448420lf},
{0.509520073739611945lf, -0.100231690070145374lf, 0.997617807952016378lf, -0.650764485670938586lf, 0.130078504368185854lf, -0.008279765463971562lf},
{0.513445596322596565lf, -0.106695232996243061lf, 1.001874820978577052lf, -0.652166364632464468lf, 0.130309331712224225lf, -0.008294968319726490lf},
{0.516813573874371990lf, -0.112209151232440074lf, 1.005485697398495581lf, -0.653348687559298757lf, 0.130502898471785672lf, -0.008307644447776066lf},
{0.519586768139094435lf, -0.116723622973579327lf, 1.008425341395084773lf, -0.654305779602474713lf, 0.130658704350423038lf, -0.008317789986848892lf},
{0.521726773656238763lf, -0.120187781698261775lf, 1.010668406139599629lf, -0.655031980841516037lf, 0.130776259947368995lf, -0.008325401846515482lf},
{0.523194008582743120lf, -0.122549724873972934lf, 1.012189305035905873lf, -0.655521649568457820lf, 0.130855087041231444lf, -0.008330477708129645lf},
{0.523947706342212882lf, -0.123756523742706015lf, 1.012962223342756429lf, -0.655769165601004289lf, 0.130894718870428084lf, -0.008333016025534760lf}
};
double k = 79.5774715459476595lf; // y = floor(k*x + b)
double dsin(double x);
// =============================================
int testsLength = 17; // change simultaneously with main program (manually)
void main()
{
// TaylorDOUBLEfma
// ====================================================
uint index = gl_GlobalInvocationID.x;
if (index < testsLength)
{
debugInfo[index] = dsin(debugInfo[index]);
}
// ====================================================
}
double dsin(double x)
{
x = mod(x+PI, TAU)-PI;
double signum = +1.00000000000000000lf;
if (x < 0)
{
x = abs(x);
signum = -1.000000000000000000lf;
}
double x2 = x*x, x3 = x2*x, x4 = x2*x2, x5 = x2*x3;
double TaylorValue;
int num = int(k*x);
TaylorValue = fma(x5, DsinTaylorTable[num][5], fma(x4, DsinTaylorTable[num][4], fma(x3, DsinTaylorTable[num][3], fma(x2, DsinTaylorTable[num][2], fma(x, DsinTaylorTable[num][1], DsinTaylorTable[num][0])))));
// ^ This works correctly on AMD, but not on NVidia ^
// "precise" modificator changes nothing.
// Deletion of "fma"s changes nothing.
// Tests show that "x5 * DsinTaylorTable[num][5]" (and similar) gives incorrect (float) result.
return TaylorValue*signum;
}