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Values of the error function $\operatorname{erf}(x)$
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Numbers
$x$ 
$\operatorname{erf}(x)$
0:
0
1/6:
0.1863362842332081025946067385261728344531269806380057173698633987032438746046701941572545600301753876
1/5:
0.2227025892104784541401390068001438163882690384302276056209350238883636742719121828703537237787407004
1/4:
0.2763263901682369329850682677648157120653539778923112540824719312626850838973022251851848058497413668
1/3:
0.3626481117660629334081786401478658796921415903725372392400185940574592693607102820066494286377752688
2/5:
0.4283923550466684551036038453201724441218629285225903834950863461133333685056718024227261581105386271
1/2:
0.5204998778130465376827466538919645287364515757579637000588057256471935217168535709147882187347877570
3/5:
0.6038560908479259225626224360567232065642733648000979055529703290184541051110738626355966614440960122
2/3:
0.6542214138488396842683835306000422920407217843858573007814022170904852801521010440196826358150469635
3/4:
0.7111556336535151315989378345914107773742059540965372322781333971250363687640495611107932538900989000
4/5:
0.7421009647076604861671105865029458773176895799147087268821357305984278671460148026602250357625748838
5/6:
0.7614071706835645381007505785993300168988199533012327799141161304825475837181980445871957258017537188
1:
0.8427007929497148693412206350826092592960669979663029084599378978347172540960108412619833253481448885
7/6:
0.9010398459805941781379158747151874394715451583805586397424495593103464643425655793530902742149436950
6/5:
0.9103139782296353802384057757153736772278970559690271452435026559277575470452890761863933867070550002
5/4:
0.9229001282564582301365234811972811404236014387022283297936684875922200565312010874754980022358767232
4/3:
0.9406535612080801226660221657882701788364365903481297488172133044771168313161588647824024283806506687
7/5:
0.9522851197626488105164826915334316745996767629463686396773095080459706081143015676380516639553442601
3/2:
0.9661051464753107270669762616459478586814104792576367804499678464421328544237507262448665492802771884
8/5:
0.9763483833446440077742834471420044624957100450770820473516959816819234758363746814744735293032479746
5/3:
0.9815778745459009899744560154669926556342173945126387675288949495438880782894135321763924740699133112
7/4:
0.9866716712191824437722111001286879766073023135644546218047564516294549210334122827410851459207148116
9/5:
0.9890905016357307141837328107558492270765546602110084134651163284821543373341128717384392425473548191
11/6:
0.9904781088159011572673063238155078863905411652071193760806835793600351244852911459312963349814630012
2:
0.9953222650189527341620692563672529286108917970400600767383523262004372807199951773676290080196806805
13/6:
0.9978169552626377469054647316363201725390096265708047939123765596565554536344519009607834035479135629
11/5:
0.9981371537020181085565482439713732889116475569401125001196517649800096639416487830190935657646775144
9/4:
0.9985372834133188483020892036270170460884832513334481682391335130809033188898760953955389868199542747
7/3:
0.9990325715497522541260363354086670041548132574654671715483539952955601588929867048085938453541124872
12/5:
0.9993114861033549214302550678293683197376639450608221724928957331085156092499299620570661811236618924
5/2:
0.9995930479825550410604357842600250872796513225962865798608792212309029939701503343580384559212367757
13/5:
0.9997639655834706507960089967924560353732809733547307250150722378661423977136381167150705416034954639
8/3:
0.9998375591548131989940281010389528643576498030438670595276947852202797882616311623606868736534268004
11/4:
0.9998993780778803631630956080249130432349352621422640655161095794654526422025908961447328296681056893
14/5:
0.9999249868053345409757767547519009270706825462567914949398229632208426998611688407492169887115839911
17/6:
0.9999384891096472719308729397535804940808324937724613317896486150529063745838523784576915837959796335
3:
0.9999779095030014145586272238704176796201522929126007503427610451570575433163798677321837453491846837
19/6:
0.9999924775316923673779967900251937639584517557957946390245669378897618860984643496744881250170922486
16/5:
0.9999939742388482379050282576371638928741063252930246327431644475983756730838374725051605649823360077
13/4:
0.9999956972205363248781695241048697634024328649303539628228726069878661418265167475664819434514467218
10/3:
0.9999975715325270241569836160018165222719502379116011291714689457642090952503471134051884916111470918
17/5:
0.9999984780066371377146382428623426431809172935335313392606654249259090141184460187978604729499308868
7/2:
0.9999992569016276585872544763162439043642793399078272025374088904300564999604094706007257706534207839
18/5:
0.9999996441370069923147011844435800829426042867274983085135069638299391086138318815251091170151050830
11/3:
0.9999997845057885714571188205094183211080913970458874730848288115487832777497421568847474572796486714
15/4:
0.9999998862727434302033467409234169933911427716655971337348456451182392704498881999029301096419874109
19/5:
0.9999999229960725430358713018021821040708884939423593619671970032924782564921796171544810636157255613
23/6:
0.9999999407838148957447698751272210689007793261668477817491264015503066392581218923655880665077058541
4:
0.9999999845827420997199811478403265131159514278547464108088316570950057869589731887459272349865438194
25/6:
0.9999999961973401127406219723692221571866243567932024794628361373210002719761987572193820170795765068
21/5:
0.9999999971445058204078113842507807718238409818713614291842739113338060453444921276423818642575719748
17/4:
0.9999999981494258626132574799441619284712732638753432187656736155793321398275583255304260193443539590
13/3:
0.9999999991115399582615338852090764387508212626858419764872088200706017659495619014402669904208182231
22/5:
0.9999999995108289729394111582042657233902155455807403514063601490634516988615707095340449478106505066
9/2:
0.9999999998033839558457112523720839632335667339422329400388353024318814039097978137963074179858858477
23/5:
0.9999999999225040040255816810813715905240964276185642326018250496788102103727194780789708236334850035
14/3:
0.9999999999587907364708121941710417661586032692086644653521555425115582228231024182155464248626738987
19/4:
0.9999999999815149522785146891125705838990215767568862450907238134990538529508647045828231318371336899
24/5:
0.9999999999886478564150780390451145547058777571686535729220168900917115511364908894714811017987858863
29/6:
0.9999999999918203818869933516162117874966684425311473463467201917483919103346694585465369255077360887
5:
0.9999999999984625402055719651498116565146166211098819496852766200693120859440796086354413085352818561
31/6:
0.9999999999997263597968244160048228400807464047231279100631591714710576300152997994176049543884404363
26/5:
0.9999999999998075093890002764030582650854062286165191850709451059939341637825925938063649177474631441
21/4:
0.9999999999998868968673311284611721319520088814074200230930580854036242583768285691373213346089793833
16/3:
0.9999999999999538856023767405499979201521612782138233902212583730261419439551428932562187739913262736
27/5:
0.9999999999999777232132053220521422854863313340849827377412739021198132215638883530036234910573470595
11/2:
0.9999999999999926421520820256019369316376014299097917769150305481239370072470717996800899167954693387
28/5:
0.9999999999999976171637154169816328539492647022492287815662444388785261119478071548577292508957877407
17/3:
0.9999999999999988885199842545525413353151812341876634373684610177034496998435435089343452351519491769
23/4:
0.9999999999999995767863382574262374053280147281028367560173431350225490531089875307277000986182573174
29/5:
0.9999999999999997644410624843563351099121566365516295672902040567001967798451027797832561380435595631
35/6:
0.9999999999999998410466499972315754007374754749537642447665430424651235956565391974395248242707290120
6:
0.9999999999999999784802632875010868834066496008126153695224859383114578994721079489436627615150721396
37/6:
0.9999999999999999972420469860528623791062353120412715934381698325779413161662894935395415734310131217
31/5:
0.9999999999999999981833243827618689276217596540931819900575698640541577115611854154771477758284081430
25/4:
0.9999999999999999990327795868123746008517966815936978545148095245237027710924321717133239006639956528
19/3:
0.9999999999999999996654168849607824435886246672010682834180236903443869179845389424165938938838423543
32/5:
0.9999999999999999998582919652331588466896920218886628492684665682844701135134379269095020971704898672
13/2:
0.9999999999999999999615785167287935253012419545623122337855071529851098654155166909643150899456462428
33/5:
0.9999999999999999999897867483214242544629539880057231452486381580000801716378313137436618638043556509
20/3:
0.9999999999999999999958237750807397597349893833894134676534921381879122806461039141215966273500730152
27/4:
0.9999999999999999999986512321106388699487801303254088507580590015304025229774554817169147950213095994
34/5:
0.9999999999999999999993199139434668766490290218074873259452261815238786546867038340653478701587713251
41/6:
0.9999999999999999999995703439956803514515013892262315683704169672714083558663056072587613474675157928
7:
0.9999999999999999999999581617439222058560138598977610006774997038258618753984795836564695045380719012
43/6:
0.9999999999999999999999961440522965670481732028174609336281434321940189005717711007083318123266290475
36/5:
0.9999999999999999999999976222054336736908091931717568221122657344002544729831935158498015436848981660
29/4:
0.9999999999999999999999988533099185184988383305278089390182955212250805112764649322679441964198742355
22/3:
0.9999999999999999999999996636561801676480426819441157185171140089789781075502900144680432526937012009
37/5:
0.9999999999999999999999998751614353646671793802038090815264754698913705493614398764931980430081032402
15/2:
0.9999999999999999999999999722335061396943089933603379067758741326031135910422636901867643389998303230
38/5:
0.9999999999999999999999999939454648195107230596532503880491819571876968201792022921139707752486641683
23/3:
0.9999999999999999999999999978306230860382840334550945581079247116061033794011158795922361702832459250
31/4:
0.9999999999999999999999999994060252140482853784724676046693857832605530618489743655540844479328049897
39/5:
0.9999999999999999999999999997287588670563397724441841793453429580662635944876663932115641881488253271
47/6:
0.9999999999999999999999999998395953309787605732558226783285450190109489397629191601652976493992538216
8:
0.9999999999999999999999999999887757028270170729200321115568297209065680708355210366140873247585743508
49/6:
0.9999999999999999999999999999992567172319465381978072794849511446587101627824575681683283134975330359
41/5:
0.9999999999999999999999999999995709787977237068334299488033934424469345635719998687248819139140922226
33/4:
0.9999999999999999999999999999998126433529449500292138346882297368549223439134618456225547687396327679
25/3:
0.9999999999999999999999999999999534205077226598957874279048012324099419846957854193913706944279502709
42/5:
0.9999999999999999999999999999999848538464720268901584092663573011553643009933978553269750008283467373
17/2:
0.9999999999999999999999999999999972376759286662285538654970699422177798604984104197393592590518063928
43/5:
0.9999999999999999999999999999999995061230429225821771587067888579860928309770303973380607828570041829
26/3:
0.9999999999999999999999999999999998449800919789086332506156251541143883401125087627697967825182175318
35/4:
0.9999999999999999999999999999999999640288427135292798434464639366079008384070499478131356122356352658
44/5:
0.9999999999999999999999999999999999851263511075573772530075353180523307884651439487983967493290478613
53/6:
0.9999999999999999999999999999999999917675821285010478616331197987374633851913175309391757894244995491
9:
0.9999999999999999999999999999999999995862968253486189761946096532637475404289808014052037540194090098
55/6:
0.9999999999999999999999999999999999999803271836560325629509065864741481187017047502447925937267260661
46/5:
0.9999999999999999999999999999999999999893726844045952513430570048000635629850731558517626184682898444
37/4:
0.9999999999999999999999999999999999999957979627850802888654675057655465062468367018942723711925498293
28/3:
0.9999999999999999999999999999999999999991147713650520059212139000694398766013176657800589076058952374
47/5:
0.9999999999999999999999999999999999999997478766360737283509924927320428702843629855260829811822144856
19/2:
0.9999999999999999999999999999999999999999623078551434512005832291267895267803037910691651099408680663
48/5:
0.9999999999999999999999999999999999999999944760554006249935313618617299189019586072456674217615320514
29/3:
0.9999999999999999999999999999999999999999984813831391368255403609315337539189924541618392148301039656
39/4:
0.9999999999999999999999999999999999999999997014299167199421677943348238956876929325588129885187003342
49/5:
0.9999999999999999999999999999999999999999998882301580942856792650873034833965282758363828478496932973
59/6:
0.9999999999999999999999999999999999999999999421047064157540671265851211491681107131218099075753092400
10:
0.9999999999999999999999999999999999999999999979115124162374552429992137050422113884391818806788362730
Definition
The error function is $\operatorname{erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}\,\mathrm{d}t$ [4]. This table gives its values at rational $x$ with $0\leq x\leq10$.
Parameters
$x$
—   argument ($x\geq0$)
Formulas
(1)
For real $x$, $\operatorname{erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}\,\mathrm{d}t$, $\operatorname{erfc}(x)=1-\operatorname{erf}(x)$, and $\operatorname{erfi}(x)=-i\operatorname{erf}(ix)$ [4].
(2)
The functions $\operatorname{erf}$, $\operatorname{erfi}$, $F$, $S$ and $C$ are odd, and $\operatorname{erfc}(-x)=2-\operatorname{erfc}(x)$.
(3)
The Fresnel integrals $S$ and $C$, with $S(x)=\int_0^x\sin(\pi t^2/2)\,\mathrm{d}t$ and $C(x)=\int_0^x\cos(\pi t^2/2)\,\mathrm{d}t$, satisfy $C(x)+iS(x)=\frac{1+i}{2}\operatorname{erf}\!\left(\frac{\sqrt{\pi}}{2}(1-i)x\right)$ [3].
Comments
(4)
The probability integral $\alpha(x)$ is $\operatorname{erf}(x/\sqrt{2})$ [6]. The standard-normal tail $Q(x)$ is $\tfrac12\operatorname{erfc}(x/\sqrt2)$ [1]. The curve $t\mapsto(C(t),S(t))$ is Cornu's spiral, also called the Euler spiral or clothoid, for this Fresnel normalisation [7] [3].
(5)
Negative arguments are not listed separately. Values at negative arguments follow from (2).
(6)
At $x=0$ the value is exact, $\operatorname{erf}(0)=0$, and is stored as the integer rather than as a hundred places of one.
Programs
(P1)
Sage
import numberdb.sage as numberdb
from sage.rings.rational_field import QQ
from sage.rings.real_arb import RealBallField

field = RealBallField(numberdb.bits(100, losing=64))
x = field(QQ(1))
x.erf()
(P2)
Python
import mpmath

mpmath.mp.dps = 100
mpmath.erf(1)
Links
Similar tables
Values of the complementary error function $\operatorname{erfc}(x)$ —   $\operatorname{erfc}=1-\operatorname{erf}$
Values of the imaginary error function $\operatorname{erfi}(x)$ —   $\operatorname{erfi}(x)=-i\operatorname{erf}(ix)$
Values of Dawson's integral $F(x)$ —   the same Gaussian integrand, with $e^{+t^2}$
Values of the Fresnel sine integral $S(x)$ —   $C(x)+iS(x)$ is an error function of a complex argument (3)
Values of the Fresnel cosine integral $C(x)$ —   $C(x)+iS(x)$ is an error function of a complex argument (3)
Values of the exponential integral —   stores $\operatorname{Ei}(x)$ and $E_1(x)$, with exponential kernels $e^t/t$ and $e^{-t}/t$ rather than the Gaussian kernel $e^{-t^2}$
Values of the sine integral $\operatorname{Si}(x)$ —   stores $\operatorname{Si}(x)=\int_0^x\sin(t)/t\,\mathrm{d}t$, with linear phase in place of the Fresnel quadratic phase
Values of the cosine integral $\operatorname{Ci}(x)$ —   stores $\operatorname{Ci}(x)=\gamma+\log x+\int_0^x(\cos(t)-1)/t\,\mathrm{d}t$, with linear phase in place of the Fresnel quadratic phase
Values of the hyperbolic sine integral $\operatorname{Shi}(x)$ —   stores $\operatorname{Shi}(x)=\int_0^x\sinh(t)/t\,\mathrm{d}t$ at rational arguments
Values of the hyperbolic cosine integral $\operatorname{Chi}(x)$ —   stores $\operatorname{Chi}(x)=\gamma+\log x+\int_0^x(\cosh(t)-1)/t\,\mathrm{d}t$ at rational arguments
Values of the Gamma function at rational numbers —   contains $\Gamma(1/2)=\sqrt{\pi}$, the normalising constant in $\operatorname{erf}$ and $\operatorname{erfc}$
Data properties
Entries are of type: real number
How well the digits are known: proven
Table is complete: no (it holds $\operatorname{erf}(x)$ at every rational $x=a/b$ in lowest terms with $b\leq6$ and $0\leq x\leq10$, which are the arguments a reader is likely to have written down rather than the ones base ten makes short)
How they were obtained:

Each value was computed as a ball with arb at numberdb.bits(digits, losing=64) bits, and checked against mpmath at 150 decimal digits, against the identities in (2) and against the OEIS value at $x=1$, $\operatorname{erf}(1)$ [5].

more

The value at $x=0$ is exact and is checked to lie in the computed ball rather than taken from it.



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