xref: /netbsd/external/gpl3/gcc/dist/libquadmath/math/erfq.c (revision 81418a27)

/*
 * ====================================================
 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
 *
 * Developed at SunPro, a Sun Microsystems, Inc. business.
 * Permission to use, copy, modify, and distribute this
 * software is freely granted, provided that this notice
 * is preserved.
 * ====================================================
 */

/* Modifications and expansions for 128-bit long double are
   Copyright (C) 2001 Stephen L. Moshier <moshier@na-net.ornl.gov>
   and are incorporated herein by permission of the author.  The author
   reserves the right to distribute this material elsewhere under different
   copying permissions.  These modifications are distributed here under
   the following terms:

    This library is free software; you can redistribute it and/or
    modify it under the terms of the GNU Lesser General Public
    License as published by the Free Software Foundation; either
    version 2.1 of the License, or (at your option) any later version.

    This library is distributed in the hope that it will be useful,
    but WITHOUT ANY WARRANTY; without even the implied warranty of
    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
    Lesser General Public License for more details.

    You should have received a copy of the GNU Lesser General Public
    License along with this library; if not, see
    <http://www.gnu.org/licenses/>.  */

/* double erf(double x)
 * double erfc(double x)
 *			     x
 *		      2      |\
 *     erf(x)  =  ---------  | exp(-t*t)dt
 *		   sqrt(pi) \|
 *			     0
 *
 *     erfc(x) =  1-erf(x)
 *  Note that
 *		erf(-x) = -erf(x)
 *		erfc(-x) = 2 - erfc(x)
 *
 * Method:
 *	1.  erf(x)  = x + x*R(x^2) for |x| in [0, 7/8]
 *	   Remark. The formula is derived by noting
 *          erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....)
 *	   and that
 *          2/sqrt(pi) = 1.128379167095512573896158903121545171688
 *	   is close to one.
 *
 *      1a. erf(x)  = 1 - erfc(x), for |x| > 1.0
 *          erfc(x) = 1 - erf(x)  if |x| < 1/4
 *
 *      2. For |x| in [7/8, 1], let s = |x| - 1, and
 *         c = 0.84506291151 rounded to single (24 bits)
 *	erf(s + c)  = sign(x) * (c  + P1(s)/Q1(s))
 *	   Remark: here we use the taylor series expansion at x=1.
 *		erf(1+s) = erf(1) + s*Poly(s)
 *			 = 0.845.. + P1(s)/Q1(s)
 *	   Note that |P1/Q1|< 0.078 for x in [0.84375,1.25]
 *
 *      3. For x in [1/4, 5/4],
 *	erfc(s + const)  = erfc(const)  + s P1(s)/Q1(s)
 *              for const = 1/4, 3/8, ..., 9/8
 *              and 0 <= s <= 1/8 .
 *
 *      4. For x in [5/4, 107],
 *	erfc(x) = (1/x)*exp(-x*x-0.5625 + R(z))
 *              z=1/x^2
 *         The interval is partitioned into several segments
 *         of width 1/8 in 1/x.
 *
 *      Note1:
 *	   To compute exp(-x*x-0.5625+R/S), let s be a single
 *	   precision number and s := x; then
 *		-x*x = -s*s + (s-x)*(s+x)
 *	        exp(-x*x-0.5626+R/S) =
 *			exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S);
 *      Note2:
 *	   Here 4 and 5 make use of the asymptotic series
 *			  exp(-x*x)
 *		erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) )
 *			  x*sqrt(pi)
 *
 *      5. For inf > x >= 107
 *	erf(x)  = sign(x) *(1 - tiny)  (raise inexact)
 *	erfc(x) = tiny*tiny (raise underflow) if x > 0
 *			= 2 - tiny if x<0
 *
 *      7. Special case:
 *	erf(0)  = 0, erf(inf)  = 1, erf(-inf) = -1,
 *	erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2,
 *		erfc/erf(NaN) is NaN
 */

#include "quadmath-imp.h"

/* Evaluate P[n] x^n  +  P[n-1] x^(n-1)  +  ...  +  P[0] */

static __float128
neval (__float128 x, const __float128 *p, int n)
{
  __float128 y;

  p += n;
  y = *p--;
  do
    {
      y = y * x + *p--;
    }
  while (--n > 0);
  return y;
}


/* Evaluate x^n+1  +  P[n] x^(n)  +  P[n-1] x^(n-1)  +  ...  +  P[0] */

static __float128
deval (__float128 x, const __float128 *p, int n)
{
  __float128 y;

  p += n;
  y = x + *p--;
  do
    {
      y = y * x + *p--;
    }
  while (--n > 0);
  return y;
}



static const __float128
tiny = 1e-4931Q,
  one = 1,
  two = 2,
  /* 2/sqrt(pi) - 1 */
  efx = 1.2837916709551257389615890312154517168810E-1Q;


/* erf(x)  = x  + x R(x^2)
   0 <= x <= 7/8
   Peak relative error 1.8e-35  */
#define NTN1 8
static const __float128 TN1[NTN1 + 1] =
{
 -3.858252324254637124543172907442106422373E10Q,
  9.580319248590464682316366876952214879858E10Q,
  1.302170519734879977595901236693040544854E10Q,
  2.922956950426397417800321486727032845006E9Q,
  1.764317520783319397868923218385468729799E8Q,
  1.573436014601118630105796794840834145120E7Q,
  4.028077380105721388745632295157816229289E5Q,
  1.644056806467289066852135096352853491530E4Q,
  3.390868480059991640235675479463287886081E1Q
};
#define NTD1 8
static const __float128 TD1[NTD1 + 1] =
{
  -3.005357030696532927149885530689529032152E11Q,
  -1.342602283126282827411658673839982164042E11Q,
  -2.777153893355340961288511024443668743399E10Q,
  -3.483826391033531996955620074072768276974E9Q,
  -2.906321047071299585682722511260895227921E8Q,
  -1.653347985722154162439387878512427542691E7Q,
  -6.245520581562848778466500301865173123136E5Q,
  -1.402124304177498828590239373389110545142E4Q,
  -1.209368072473510674493129989468348633579E2Q
/* 1.0E0 */
};


/* erf(z+1)  = erf_const + P(z)/Q(z)
   -.125 <= z <= 0
   Peak relative error 7.3e-36  */
static const __float128 erf_const = 0.845062911510467529296875Q;
#define NTN2 8
static const __float128 TN2[NTN2 + 1] =
{
 -4.088889697077485301010486931817357000235E1Q,
  7.157046430681808553842307502826960051036E3Q,
 -2.191561912574409865550015485451373731780E3Q,
  2.180174916555316874988981177654057337219E3Q,
  2.848578658049670668231333682379720943455E2Q,
  1.630362490952512836762810462174798925274E2Q,
  6.317712353961866974143739396865293596895E0Q,
  2.450441034183492434655586496522857578066E1Q,
  5.127662277706787664956025545897050896203E-1Q
};
#define NTD2 8
static const __float128 TD2[NTD2 + 1] =
{
  1.731026445926834008273768924015161048885E4Q,
  1.209682239007990370796112604286048173750E4Q,
  1.160950290217993641320602282462976163857E4Q,
  5.394294645127126577825507169061355698157E3Q,
  2.791239340533632669442158497532521776093E3Q,
  8.989365571337319032943005387378993827684E2Q,
  2.974016493766349409725385710897298069677E2Q,
  6.148192754590376378740261072533527271947E1Q,
  1.178502892490738445655468927408440847480E1Q
 /* 1.0E0 */
};


/* erfc(x + 0.25) = erfc(0.25) + x R(x)
   0 <= x < 0.125
   Peak relative error 1.4e-35  */
#define NRNr13 8
static const __float128 RNr13[NRNr13 + 1] =
{
 -2.353707097641280550282633036456457014829E3Q,
  3.871159656228743599994116143079870279866E2Q,
 -3.888105134258266192210485617504098426679E2Q,
 -2.129998539120061668038806696199343094971E1Q,
 -8.125462263594034672468446317145384108734E1Q,
  8.151549093983505810118308635926270319660E0Q,
 -5.033362032729207310462422357772568553670E0Q,
 -4.253956621135136090295893547735851168471E-2Q,
 -8.098602878463854789780108161581050357814E-2Q
};
#define NRDr13 7
static const __float128 RDr13[NRDr13 + 1] =
{
  2.220448796306693503549505450626652881752E3Q,
  1.899133258779578688791041599040951431383E2Q,
  1.061906712284961110196427571557149268454E3Q,
  7.497086072306967965180978101974566760042E1Q,
  2.146796115662672795876463568170441327274E2Q,
  1.120156008362573736664338015952284925592E1Q,
  2.211014952075052616409845051695042741074E1Q,
  6.469655675326150785692908453094054988938E-1Q
 /* 1.0E0 */
};
/* erfc(0.25) = C13a + C13b to extra precision.  */
static const __float128 C13a = 0.723663330078125Q;
static const __float128 C13b = 1.0279753638067014931732235184287934646022E-5Q;


/* erfc(x + 0.375) = erfc(0.375) + x R(x)
   0 <= x < 0.125
   Peak relative error 1.2e-35  */
#define NRNr14 8
static const __float128 RNr14[NRNr14 + 1] =
{
 -2.446164016404426277577283038988918202456E3Q,
  6.718753324496563913392217011618096698140E2Q,
 -4.581631138049836157425391886957389240794E2Q,
 -2.382844088987092233033215402335026078208E1Q,
 -7.119237852400600507927038680970936336458E1Q,
  1.313609646108420136332418282286454287146E1Q,
 -6.188608702082264389155862490056401365834E0Q,
 -2.787116601106678287277373011101132659279E-2Q,
 -2.230395570574153963203348263549700967918E-2Q
};
#define NRDr14 7
static const __float128 RDr14[NRDr14 + 1] =
{
  2.495187439241869732696223349840963702875E3Q,
  2.503549449872925580011284635695738412162E2Q,
  1.159033560988895481698051531263861842461E3Q,
  9.493751466542304491261487998684383688622E1Q,
  2.276214929562354328261422263078480321204E2Q,
  1.367697521219069280358984081407807931847E1Q,
  2.276988395995528495055594829206582732682E1Q,
  7.647745753648996559837591812375456641163E-1Q
 /* 1.0E0 */
};
/* erfc(0.375) = C14a + C14b to extra precision.  */
static const __float128 C14a = 0.5958709716796875Q;
static const __float128 C14b = 1.2118885490201676174914080878232469565953E-5Q;

/* erfc(x + 0.5) = erfc(0.5) + x R(x)
   0 <= x < 0.125
   Peak relative error 4.7e-36  */
#define NRNr15 8
static const __float128 RNr15[NRNr15 + 1] =
{
 -2.624212418011181487924855581955853461925E3Q,
  8.473828904647825181073831556439301342756E2Q,
 -5.286207458628380765099405359607331669027E2Q,
 -3.895781234155315729088407259045269652318E1Q,
 -6.200857908065163618041240848728398496256E1Q,
  1.469324610346924001393137895116129204737E1Q,
 -6.961356525370658572800674953305625578903E0Q,
  5.145724386641163809595512876629030548495E-3Q,
  1.990253655948179713415957791776180406812E-2Q
};
#define NRDr15 7
static const __float128 RDr15[NRDr15 + 1] =
{
  2.986190760847974943034021764693341524962E3Q,
  5.288262758961073066335410218650047725985E2Q,
  1.363649178071006978355113026427856008978E3Q,
  1.921707975649915894241864988942255320833E2Q,
  2.588651100651029023069013885900085533226E2Q,
  2.628752920321455606558942309396855629459E1Q,
  2.455649035885114308978333741080991380610E1Q,
  1.378826653595128464383127836412100939126E0Q
  /* 1.0E0 */
};
/* erfc(0.5) = C15a + C15b to extra precision.  */
static const __float128 C15a = 0.4794921875Q;
static const __float128 C15b = 7.9346869534623172533461080354712635484242E-6Q;

/* erfc(x + 0.625) = erfc(0.625) + x R(x)
   0 <= x < 0.125
   Peak relative error 5.1e-36  */
#define NRNr16 8
static const __float128 RNr16[NRNr16 + 1] =
{
 -2.347887943200680563784690094002722906820E3Q,
  8.008590660692105004780722726421020136482E2Q,
 -5.257363310384119728760181252132311447963E2Q,
 -4.471737717857801230450290232600243795637E1Q,
 -4.849540386452573306708795324759300320304E1Q,
  1.140885264677134679275986782978655952843E1Q,
 -6.731591085460269447926746876983786152300E0Q,
  1.370831653033047440345050025876085121231E-1Q,
  2.022958279982138755020825717073966576670E-2Q,
};
#define NRDr16 7
static const __float128 RDr16[NRDr16 + 1] =
{
  3.075166170024837215399323264868308087281E3Q,
  8.730468942160798031608053127270430036627E2Q,
  1.458472799166340479742581949088453244767E3Q,
  3.230423687568019709453130785873540386217E2Q,
  2.804009872719893612081109617983169474655E2Q,
  4.465334221323222943418085830026979293091E1Q,
  2.612723259683205928103787842214809134746E1Q,
  2.341526751185244109722204018543276124997E0Q,
  /* 1.0E0 */
};
/* erfc(0.625) = C16a + C16b to extra precision.  */
static const __float128 C16a = 0.3767547607421875Q;
static const __float128 C16b = 4.3570693945275513594941232097252997287766E-6Q;

/* erfc(x + 0.75) = erfc(0.75) + x R(x)
   0 <= x < 0.125
   Peak relative error 1.7e-35  */
#define NRNr17 8
static const __float128 RNr17[NRNr17 + 1] =
{
  -1.767068734220277728233364375724380366826E3Q,
  6.693746645665242832426891888805363898707E2Q,
  -4.746224241837275958126060307406616817753E2Q,
  -2.274160637728782675145666064841883803196E1Q,
  -3.541232266140939050094370552538987982637E1Q,
  6.988950514747052676394491563585179503865E0Q,
  -5.807687216836540830881352383529281215100E0Q,
  3.631915988567346438830283503729569443642E-1Q,
  -1.488945487149634820537348176770282391202E-2Q
};
#define NRDr17 7
static const __float128 RDr17[NRDr17 + 1] =
{
  2.748457523498150741964464942246913394647E3Q,
  1.020213390713477686776037331757871252652E3Q,
  1.388857635935432621972601695296561952738E3Q,
  3.903363681143817750895999579637315491087E2Q,
  2.784568344378139499217928969529219886578E2Q,
  5.555800830216764702779238020065345401144E1Q,
  2.646215470959050279430447295801291168941E1Q,
  2.984905282103517497081766758550112011265E0Q,
  /* 1.0E0 */
};
/* erfc(0.75) = C17a + C17b to extra precision.  */
static const __float128 C17a = 0.2888336181640625Q;
static const __float128 C17b = 1.0748182422368401062165408589222625794046E-5Q;


/* erfc(x + 0.875) = erfc(0.875) + x R(x)
   0 <= x < 0.125
   Peak relative error 2.2e-35  */
#define NRNr18 8
static const __float128 RNr18[NRNr18 + 1] =
{
 -1.342044899087593397419622771847219619588E3Q,
  6.127221294229172997509252330961641850598E2Q,
 -4.519821356522291185621206350470820610727E2Q,
  1.223275177825128732497510264197915160235E1Q,
 -2.730789571382971355625020710543532867692E1Q,
  4.045181204921538886880171727755445395862E0Q,
 -4.925146477876592723401384464691452700539E0Q,
  5.933878036611279244654299924101068088582E-1Q,
 -5.557645435858916025452563379795159124753E-2Q
};
#define NRDr18 7
static const __float128 RDr18[NRDr18 + 1] =
{
  2.557518000661700588758505116291983092951E3Q,
  1.070171433382888994954602511991940418588E3Q,
  1.344842834423493081054489613250688918709E3Q,
  4.161144478449381901208660598266288188426E2Q,
  2.763670252219855198052378138756906980422E2Q,
  5.998153487868943708236273854747564557632E1Q,
  2.657695108438628847733050476209037025318E1Q,
  3.252140524394421868923289114410336976512E0Q,
  /* 1.0E0 */
};
/* erfc(0.875) = C18a + C18b to extra precision.  */
static const __float128 C18a = 0.215911865234375Q;
static const __float128 C18b = 1.3073705765341685464282101150637224028267E-5Q;

/* erfc(x + 1.0) = erfc(1.0) + x R(x)
   0 <= x < 0.125
   Peak relative error 1.6e-35  */
#define NRNr19 8
static const __float128 RNr19[NRNr19 + 1] =
{
 -1.139180936454157193495882956565663294826E3Q,
  6.134903129086899737514712477207945973616E2Q,
 -4.628909024715329562325555164720732868263E2Q,
  4.165702387210732352564932347500364010833E1Q,
 -2.286979913515229747204101330405771801610E1Q,
  1.870695256449872743066783202326943667722E0Q,
 -4.177486601273105752879868187237000032364E0Q,
  7.533980372789646140112424811291782526263E-1Q,
 -8.629945436917752003058064731308767664446E-2Q
};
#define NRDr19 7
static const __float128 RDr19[NRDr19 + 1] =
{
  2.744303447981132701432716278363418643778E3Q,
  1.266396359526187065222528050591302171471E3Q,
  1.466739461422073351497972255511919814273E3Q,
  4.868710570759693955597496520298058147162E2Q,
  2.993694301559756046478189634131722579643E2Q,
  6.868976819510254139741559102693828237440E1Q,
  2.801505816247677193480190483913753613630E1Q,
  3.604439909194350263552750347742663954481E0Q,
  /* 1.0E0 */
};
/* erfc(1.0) = C19a + C19b to extra precision.  */
static const __float128 C19a = 0.15728759765625Q;
static const __float128 C19b = 1.1609394035130658779364917390740703933002E-5Q;

/* erfc(x + 1.125) = erfc(1.125) + x R(x)
   0 <= x < 0.125
   Peak relative error 3.6e-36  */
#define NRNr20 8
static const __float128 RNr20[NRNr20 + 1] =
{
 -9.652706916457973956366721379612508047640E2Q,
  5.577066396050932776683469951773643880634E2Q,
 -4.406335508848496713572223098693575485978E2Q,
  5.202893466490242733570232680736966655434E1Q,
 -1.931311847665757913322495948705563937159E1Q,
 -9.364318268748287664267341457164918090611E-2Q,
 -3.306390351286352764891355375882586201069E0Q,
  7.573806045289044647727613003096916516475E-1Q,
 -9.611744011489092894027478899545635991213E-2Q
};
#define NRDr20 7
static const __float128 RDr20[NRDr20 + 1] =
{
  3.032829629520142564106649167182428189014E3Q,
  1.659648470721967719961167083684972196891E3Q,
  1.703545128657284619402511356932569292535E3Q,
  6.393465677731598872500200253155257708763E2Q,
  3.489131397281030947405287112726059221934E2Q,
  8.848641738570783406484348434387611713070E1Q,
  3.132269062552392974833215844236160958502E1Q,
  4.430131663290563523933419966185230513168E0Q
 /* 1.0E0 */
};
/* erfc(1.125) = C20a + C20b to extra precision.  */
static const __float128 C20a = 0.111602783203125Q;
static const __float128 C20b = 8.9850951672359304215530728365232161564636E-6Q;

/* erfc(1/x) = 1/x exp (-1/x^2 - 0.5625 + R(1/x^2))
   7/8 <= 1/x < 1
   Peak relative error 1.4e-35  */
#define NRNr8 9
static const __float128 RNr8[NRNr8 + 1] =
{
  3.587451489255356250759834295199296936784E1Q,
  5.406249749087340431871378009874875889602E2Q,
  2.931301290625250886238822286506381194157E3Q,
  7.359254185241795584113047248898753470923E3Q,
  9.201031849810636104112101947312492532314E3Q,
  5.749697096193191467751650366613289284777E3Q,
  1.710415234419860825710780802678697889231E3Q,
  2.150753982543378580859546706243022719599E2Q,
  8.740953582272147335100537849981160931197E0Q,
  4.876422978828717219629814794707963640913E-2Q
};
#define NRDr8 8
static const __float128 RDr8[NRDr8 + 1] =
{
  6.358593134096908350929496535931630140282E1Q,
  9.900253816552450073757174323424051765523E2Q,
  5.642928777856801020545245437089490805186E3Q,
  1.524195375199570868195152698617273739609E4Q,
  2.113829644500006749947332935305800887345E4Q,
  1.526438562626465706267943737310282977138E4Q,
  5.561370922149241457131421914140039411782E3Q,
  9.394035530179705051609070428036834496942E2Q,
  6.147019596150394577984175188032707343615E1Q
  /* 1.0E0 */
};

/* erfc(1/x) = 1/x exp (-1/x^2 - 0.5625 + R(1/x^2))
   0.75 <= 1/x <= 0.875
   Peak relative error 2.0e-36  */
#define NRNr7 9
static const __float128 RNr7[NRNr7 + 1] =
{
 1.686222193385987690785945787708644476545E1Q,
 1.178224543567604215602418571310612066594E3Q,
 1.764550584290149466653899886088166091093E4Q,
 1.073758321890334822002849369898232811561E5Q,
 3.132840749205943137619839114451290324371E5Q,
 4.607864939974100224615527007793867585915E5Q,
 3.389781820105852303125270837910972384510E5Q,
 1.174042187110565202875011358512564753399E5Q,
 1.660013606011167144046604892622504338313E4Q,
 6.700393957480661937695573729183733234400E2Q
};
#define NRDr7 9
static const __float128 RDr7[NRDr7 + 1] =
{
-1.709305024718358874701575813642933561169E3Q,
-3.280033887481333199580464617020514788369E4Q,
-2.345284228022521885093072363418750835214E5Q,
-8.086758123097763971926711729242327554917E5Q,
-1.456900414510108718402423999575992450138E6Q,
-1.391654264881255068392389037292702041855E6Q,
-6.842360801869939983674527468509852583855E5Q,
-1.597430214446573566179675395199807533371E5Q,
-1.488876130609876681421645314851760773480E4Q,
-3.511762950935060301403599443436465645703E2Q
 /* 1.0E0 */
};

/* erfc(1/x) = 1/x exp(-1/x^2 - 0.5625 + R(1/x^2))
   5/8 <= 1/x < 3/4
   Peak relative error 1.9e-35  */
#define NRNr6 9
static const __float128 RNr6[NRNr6 + 1] =
{
 1.642076876176834390623842732352935761108E0Q,
 1.207150003611117689000664385596211076662E2Q,
 2.119260779316389904742873816462800103939E3Q,
 1.562942227734663441801452930916044224174E4Q,
 5.656779189549710079988084081145693580479E4Q,
 1.052166241021481691922831746350942786299E5Q,
 9.949798524786000595621602790068349165758E4Q,
 4.491790734080265043407035220188849562856E4Q,
 8.377074098301530326270432059434791287601E3Q,
 4.506934806567986810091824791963991057083E2Q
};
#define NRDr6 9
static const __float128 RDr6[NRDr6 + 1] =
{
-1.664557643928263091879301304019826629067E2Q,
-3.800035902507656624590531122291160668452E3Q,
-3.277028191591734928360050685359277076056E4Q,
-1.381359471502885446400589109566587443987E5Q,
-3.082204287382581873532528989283748656546E5Q,
-3.691071488256738343008271448234631037095E5Q,
-2.300482443038349815750714219117566715043E5Q,
-6.873955300927636236692803579555752171530E4Q,
-8.262158817978334142081581542749986845399E3Q,
-2.517122254384430859629423488157361983661E2Q
 /* 1.00 */
};

/* erfc(1/x) = 1/x exp(-1/x^2 - 0.5625 + R(1/x^2))
   1/2 <= 1/x < 5/8
   Peak relative error 4.6e-36  */
#define NRNr5 10
static const __float128 RNr5[NRNr5 + 1] =
{
-3.332258927455285458355550878136506961608E-3Q,
-2.697100758900280402659586595884478660721E-1Q,
-6.083328551139621521416618424949137195536E0Q,
-6.119863528983308012970821226810162441263E1Q,
-3.176535282475593173248810678636522589861E2Q,
-8.933395175080560925809992467187963260693E2Q,
-1.360019508488475978060917477620199499560E3Q,
-1.075075579828188621541398761300910213280E3Q,
-4.017346561586014822824459436695197089916E2Q,
-5.857581368145266249509589726077645791341E1Q,
-2.077715925587834606379119585995758954399E0Q
};
#define NRDr5 9
static const __float128 RDr5[NRDr5 + 1] =
{
 3.377879570417399341550710467744693125385E-1Q,
 1.021963322742390735430008860602594456187E1Q,
 1.200847646592942095192766255154827011939E2Q,
 7.118915528142927104078182863387116942836E2Q,
 2.318159380062066469386544552429625026238E3Q,
 4.238729853534009221025582008928765281620E3Q,
 4.279114907284825886266493994833515580782E3Q,
 2.257277186663261531053293222591851737504E3Q,
 5.570475501285054293371908382916063822957E2Q,
 5.142189243856288981145786492585432443560E1Q
 /* 1.0E0 */
};

/* erfc(1/x) = 1/x exp(-1/x^2 - 0.5625 + R(1/x^2))
   3/8 <= 1/x < 1/2
   Peak relative error 2.0e-36  */
#define NRNr4 10
static const __float128 RNr4[NRNr4 + 1] =
{
 3.258530712024527835089319075288494524465E-3Q,
 2.987056016877277929720231688689431056567E-1Q,
 8.738729089340199750734409156830371528862E0Q,
 1.207211160148647782396337792426311125923E2Q,
 8.997558632489032902250523945248208224445E2Q,
 3.798025197699757225978410230530640879762E3Q,
 9.113203668683080975637043118209210146846E3Q,
 1.203285891339933238608683715194034900149E4Q,
 8.100647057919140328536743641735339740855E3Q,
 2.383888249907144945837976899822927411769E3Q,
 2.127493573166454249221983582495245662319E2Q
};
#define NRDr4 10
static const __float128 RDr4[NRDr4 + 1] =
{
-3.303141981514540274165450687270180479586E-1Q,
-1.353768629363605300707949368917687066724E1Q,
-2.206127630303621521950193783894598987033E2Q,
-1.861800338758066696514480386180875607204E3Q,
-8.889048775872605708249140016201753255599E3Q,
-2.465888106627948210478692168261494857089E4Q,
-3.934642211710774494879042116768390014289E4Q,
-3.455077258242252974937480623730228841003E4Q,
-1.524083977439690284820586063729912653196E4Q,
-2.810541887397984804237552337349093953857E3Q,
-1.343929553541159933824901621702567066156E2Q
 /* 1.0E0 */
};

/* erfc(1/x) = 1/x exp(-1/x^2 - 0.5625 + R(1/x^2))
   1/4 <= 1/x < 3/8
   Peak relative error 8.4e-37  */
#define NRNr3 11
static const __float128 RNr3[NRNr3 + 1] =
{
-1.952401126551202208698629992497306292987E-6Q,
-2.130881743066372952515162564941682716125E-4Q,
-8.376493958090190943737529486107282224387E-3Q,
-1.650592646560987700661598877522831234791E-1Q,
-1.839290818933317338111364667708678163199E0Q,
-1.216278715570882422410442318517814388470E1Q,
-4.818759344462360427612133632533779091386E1Q,
-1.120994661297476876804405329172164436784E2Q,
-1.452850765662319264191141091859300126931E2Q,
-9.485207851128957108648038238656777241333E1Q,
-2.563663855025796641216191848818620020073E1Q,
-1.787995944187565676837847610706317833247E0Q
};
#define NRDr3 10
static const __float128 RDr3[NRDr3 + 1] =
{
 1.979130686770349481460559711878399476903E-4Q,
 1.156941716128488266238105813374635099057E-2Q,
 2.752657634309886336431266395637285974292E-1Q,
 3.482245457248318787349778336603569327521E0Q,
 2.569347069372696358578399521203959253162E1Q,
 1.142279000180457419740314694631879921561E2Q,
 3.056503977190564294341422623108332700840E2Q,
 4.780844020923794821656358157128719184422E2Q,
 4.105972727212554277496256802312730410518E2Q,
 1.724072188063746970865027817017067646246E2Q,
 2.815939183464818198705278118326590370435E1Q
 /* 1.0E0 */
};

/* erfc(1/x) = 1/x exp(-1/x^2 - 0.5625 + R(1/x^2))
   1/8 <= 1/x < 1/4
   Peak relative error 1.5e-36  */
#define NRNr2 11
static const __float128 RNr2[NRNr2 + 1] =
{
-2.638914383420287212401687401284326363787E-8Q,
-3.479198370260633977258201271399116766619E-6Q,
-1.783985295335697686382487087502222519983E-4Q,
-4.777876933122576014266349277217559356276E-3Q,
-7.450634738987325004070761301045014986520E-2Q,
-7.068318854874733315971973707247467326619E-1Q,
-4.113919921935944795764071670806867038732E0Q,
-1.440447573226906222417767283691888875082E1Q,
-2.883484031530718428417168042141288943905E1Q,
-2.990886974328476387277797361464279931446E1Q,
-1.325283914915104866248279787536128997331E1Q,
-1.572436106228070195510230310658206154374E0Q
};
#define NRDr2 10
static const __float128 RDr2[NRDr2 + 1] =
{
 2.675042728136731923554119302571867799673E-6Q,
 2.170997868451812708585443282998329996268E-4Q,
 7.249969752687540289422684951196241427445E-3Q,
 1.302040375859768674620410563307838448508E-1Q,
 1.380202483082910888897654537144485285549E0Q,
 8.926594113174165352623847870299170069350E0Q,
 3.521089584782616472372909095331572607185E1Q,
 8.233547427533181375185259050330809105570E1Q,
 1.072971579885803033079469639073292840135E2Q,
 6.943803113337964469736022094105143158033E1Q,
 1.775695341031607738233608307835017282662E1Q
 /* 1.0E0 */
};

/* erfc(1/x) = 1/x exp(-1/x^2 - 0.5625 + R(1/x^2))
   1/128 <= 1/x < 1/8
   Peak relative error 2.2e-36  */
#define NRNr1 9
static const __float128 RNr1[NRNr1 + 1] =
{
-4.250780883202361946697751475473042685782E-8Q,
-5.375777053288612282487696975623206383019E-6Q,
-2.573645949220896816208565944117382460452E-4Q,
-6.199032928113542080263152610799113086319E-3Q,
-8.262721198693404060380104048479916247786E-2Q,
-6.242615227257324746371284637695778043982E-1Q,
-2.609874739199595400225113299437099626386E0Q,
-5.581967563336676737146358534602770006970E0Q,
-5.124398923356022609707490956634280573882E0Q,
-1.290865243944292370661544030414667556649E0Q
};
#define NRDr1 8
static const __float128 RDr1[NRDr1 + 1] =
{
 4.308976661749509034845251315983612976224E-6Q,
 3.265390126432780184125233455960049294580E-4Q,
 9.811328839187040701901866531796570418691E-3Q,
 1.511222515036021033410078631914783519649E-1Q,
 1.289264341917429958858379585970225092274E0Q,
 6.147640356182230769548007536914983522270E0Q,
 1.573966871337739784518246317003956180750E1Q,
 1.955534123435095067199574045529218238263E1Q,
 9.472613121363135472247929109615785855865E0Q
  /* 1.0E0 */
};


__float128
erfq (__float128 x)
{
  __float128 a, y, z;
  int32_t i, ix, sign;
  ieee854_float128 u;

  u.value = x;
  sign = u.words32.w0;
  ix = sign & 0x7fffffff;

  if (ix >= 0x7fff0000)
    {				/* erf(nan)=nan */
      i = ((sign & 0xffff0000) >> 31) << 1;
      return (__float128) (1 - i) + one / x;	/* erf(+-inf)=+-1 */
    }

  if (ix >= 0x3fff0000) /* |x| >= 1.0 */
    {
      if (ix >= 0x40030000 && sign > 0)
	return one; /* x >= 16, avoid spurious underflow from erfc.  */
      y = erfcq (x);
      return (one - y);
      /*    return (one - erfcq (x)); */
    }
  u.words32.w0 = ix;
  a = u.value;
  z = x * x;
  if (ix < 0x3ffec000)  /* a < 0.875 */
    {
      if (ix < 0x3fc60000) /* |x|<2**-57 */
	{
	  if (ix < 0x00080000)
	    {
	      /* Avoid spurious underflow.  */
	      __float128 ret =  0.0625 * (16.0 * x + (16.0 * efx) * x);
	      math_check_force_underflow (ret);
	      return ret;
	    }
	  return x + efx * x;
	}
      y = a + a * neval (z, TN1, NTN1) / deval (z, TD1, NTD1);
    }
  else
    {
      a = a - one;
      y = erf_const + neval (a, TN2, NTN2) / deval (a, TD2, NTD2);
    }

  if (sign & 0x80000000) /* x < 0 */
    y = -y;
  return( y );
}


__float128
erfcq (__float128 x)
{
  __float128 y, z, p, r;
  int32_t i, ix, sign;
  ieee854_float128 u;

  u.value = x;
  sign = u.words32.w0;
  ix = sign & 0x7fffffff;
  u.words32.w0 = ix;

  if (ix >= 0x7fff0000)
    {				/* erfc(nan)=nan */
      /* erfc(+-inf)=0,2 */
      return (__float128) (((uint32_t) sign >> 31) << 1) + one / x;
    }

  if (ix < 0x3ffd0000) /* |x| <1/4 */
    {
      if (ix < 0x3f8d0000) /* |x|<2**-114 */
	return one - x;
      return one - erfq (x);
    }
  if (ix < 0x3fff4000) /* 1.25 */
    {
      x = u.value;
      i = 8.0 * x;
      switch (i)
	{
	case 2:
	  z = x - 0.25Q;
	  y = C13b + z * neval (z, RNr13, NRNr13) / deval (z, RDr13, NRDr13);
	  y += C13a;
	  break;
	case 3:
	  z = x - 0.375Q;
	  y = C14b + z * neval (z, RNr14, NRNr14) / deval (z, RDr14, NRDr14);
	  y += C14a;
	  break;
	case 4:
	  z = x - 0.5Q;
	  y = C15b + z * neval (z, RNr15, NRNr15) / deval (z, RDr15, NRDr15);
	  y += C15a;
	  break;
	case 5:
	  z = x - 0.625Q;
	  y = C16b + z * neval (z, RNr16, NRNr16) / deval (z, RDr16, NRDr16);
	  y += C16a;
	  break;
	case 6:
	  z = x - 0.75Q;
	  y = C17b + z * neval (z, RNr17, NRNr17) / deval (z, RDr17, NRDr17);
	  y += C17a;
	  break;
	case 7:
	  z = x - 0.875Q;
	  y = C18b + z * neval (z, RNr18, NRNr18) / deval (z, RDr18, NRDr18);
	  y += C18a;
	  break;
	case 8:
	  z = x - 1;
	  y = C19b + z * neval (z, RNr19, NRNr19) / deval (z, RDr19, NRDr19);
	  y += C19a;
	  break;
	default: /* i == 9.  */
	  z = x - 1.125Q;
	  y = C20b + z * neval (z, RNr20, NRNr20) / deval (z, RDr20, NRDr20);
	  y += C20a;
	  break;
	}
      if (sign & 0x80000000)
	y = 2 - y;
      return y;
    }
  /* 1.25 < |x| < 107 */
  if (ix < 0x4005ac00)
    {
      /* x < -9 */
      if ((ix >= 0x40022000) && (sign & 0x80000000))
	return two - tiny;

      x = fabsq (x);
      z = one / (x * x);
      i = 8.0 / x;
      switch (i)
	{
	default:
	case 0:
	  p = neval (z, RNr1, NRNr1) / deval (z, RDr1, NRDr1);
	  break;
	case 1:
	  p = neval (z, RNr2, NRNr2) / deval (z, RDr2, NRDr2);
	  break;
	case 2:
	  p = neval (z, RNr3, NRNr3) / deval (z, RDr3, NRDr3);
	  break;
	case 3:
	  p = neval (z, RNr4, NRNr4) / deval (z, RDr4, NRDr4);
	  break;
	case 4:
	  p = neval (z, RNr5, NRNr5) / deval (z, RDr5, NRDr5);
	  break;
	case 5:
	  p = neval (z, RNr6, NRNr6) / deval (z, RDr6, NRDr6);
	  break;
	case 6:
	  p = neval (z, RNr7, NRNr7) / deval (z, RDr7, NRDr7);
	  break;
	case 7:
	  p = neval (z, RNr8, NRNr8) / deval (z, RDr8, NRDr8);
	  break;
	}
      u.value = x;
      u.words32.w3 = 0;
      u.words32.w2 &= 0xfe000000;
      z = u.value;
      r = expq (-z * z - 0.5625) *
	expq ((z - x) * (z + x) + p);
      if ((sign & 0x80000000) == 0)
	{
	  __float128 ret = r / x;
	  if (ret == 0)
	    errno = ERANGE;
	  return ret;
	}
      else
	return two - r / x;
    }
  else
    {
      if ((sign & 0x80000000) == 0)
	{
	  errno = ERANGE;
	  return tiny * tiny;
	}
      else
	return two - tiny;
    }
}