> restart:
> Digits:=30:
> a:=0.1;
> b:=beta;
> f:=a*x^2+b;
> k:=subs(x=0,diff(f,x$2)/sqrt(1+diff(f,x)^2));
> g:=1/(2*Pi)*exp(-(x^2+y^2)/2);
> PHI:=proc(x)
>   int(1/sqrt(2*Pi)*exp(-t^2/2),t=-infinity..x)
> end:

                               a := 0.1


                              b := beta


                                    2
                          f := 0.1 x  + beta


                               k := 0.2


                                      2      2
                                     x      y
                              exp(- ---- - ----)
                                     2      2
                     g := 1/2 ------------------
                                      Pi

> Pdef:=evalf(Int(int(g,y=f..infinity),x=-infinity..infinity));
> for i from 0 to 32 do
>   beta:=8.0*i/32:
>   Pdef:=evalf(Int(int(g,y=f..infinity),x=-infinity..infinity)):
>   PFORM:=PHI(-beta):
>   PSORM:=PHI(-beta)/sqrt(1+beta*k):
> lbeta[i]:=beta:
> lPFORM[i]:=PFORM:
> lPSORM[i]:=PSORM:
> lPDEF[i]:=Pdef:
>   print([beta,PFORM,PSORM,Pdef,PFORM/Pdef,PSORM/Pdef]);
> od:
> plot([seq([lbeta[i],lPSORM[i]/lPDEF[i]],i=0..100)]);
> 

             Float(infinity)
            /
           |
  Pdef :=  |                 -0.199471140200716338969973029967
           |
          /
            Float(-infinity)

                                               2
        exp(-0.500000000000000000000000000000 x ) (erf(

                                           2
        0.0707106781186547524400844362105 x

         + 0.707106781186547524400844362105 beta) - 1.) dx


  [0., 0.500000000000000000000000000000,

        0.500000000000000000000000000000,

        0.461021882779949487433112608223,

        1.08454721711909534461306306089,

        1.08454721711909534461306306089]


  [0.250000000000000000000000000000, 0.401293674317076275759146208419,

        0.391622526039819653097621966085,

        0.364805645459419223555872433619,

        1.10002046106415302219756008185,

        1.07351004819738603626967362214]


  [0.500000000000000000000000000000, 0.308537538725986896362295389392,

        0.294179000553141676323633259680,

        0.276406871510791646914025035263,

        1.11624409711514999074964755074,

        1.06429698706551931371779049750]


  [0.750000000000000000000000000000, 0.226627352376868199327062169383,

        0.211331095770201496300592876080,

        0.200014849865906181092409100087,

        1.13305263348598143575089714016,

        1.05657702871502763445327315386]


  [1.00000000000000000000000000000, 0.158655253931457051414767454368,

        0.144831769074948694907766136143,

        0.137924758985751001750425892628,

        1.15030292674172963974068160358,

        1.05007810156776315946958656739]


  [1.25000000000000000000000000000, 0.105649773666855257688772764026,

        0.0944960302906222305274413240302,

        0.0904633017477591072436722948851,

        1.16787439354624638114727178974,

        1.04457861326029943601203090438]


  [1.50000000000000000000000000000, 0.0668072012688580660044940409799,

        0.0585937916203107043052937877069,

        0.0563456501745595511902624715678,

        1.18566741286840234433831348534,

        1.03989911268725060251244139959]


  [1.75000000000000000000000000000, 0.0400591568638170904187573498856,

        0.0344774327548164726048053139994,

        0.0332827583308622774323121525445,

        1.20360086942286951225603913360,

        1.03589469394567586574078439676]


  [2.00000000000000000000000000000, 0.0227501319481792072002826371665,

        0.0192273708116388127844460424066,

        0.0186230812426850280522151598594,

        1.22160944538193678473053440478,

        1.03244842038108726057716500219]


  [2.25000000000000000000000000000, 0.0122244726550447031526239312997,

        0.0101518719759711004318482092473,

        0.00986130064758826832158090273531,

        1.23964100597971167592507379122,

        1.02946582187958094212703538420]


  [2.50000000000000000000000000000,

        0.00620966532577613516697810457419,

        0.00507017050720166838673376542834,

        0.00493749799035946477343316631781,

        1.25765424875120865971096676028,

        1.02687039409458972826342913077]


  [2.75000000000000000000000000000,

        0.00297976323505455675429424698643,

        0.00239340342594886005084812775183,

        0.00233593938149220078077614026092,

        1.27561667852488558612695934074,

        1.02459997246159323362138212620]


  [3.00000000000000000000000000000,

        0.00134989803163009452665181476759,

        0.00106718809723227905853795125583,

        0.00104359875879428088410498887472,

        1.29350291024655463561677147824,

        1.02260383910886599303657409708]


  [3.25000000000000000000000000000,

        0.000577025042390767042916919314251,

        0.000449213386255970177400813261343, %1,

        0.000577025042390767042916919314251
        -----------------------------------,
                        %1

        0.000449213386255970177400813261343
        -----------------------------------]
                        %1

           Float(infinity)
          /
         |
  %1 :=  |                 -0.199471140200716338969973029967
         |
        /
          Float(-infinity)

                                               2
        exp(-0.500000000000000000000000000000 x ) (erf(

                                           2
        0.0707106781186547524400844362105 x

         + 2.29809703885627945430274417684) - 1.) dx


  [3.50000000000000000000000000000,

        0.000232629079035525036349925886728,

        0.000178418359008055515441331636208,

        0.000175044291391449743990991382019,

        1.32897266849622091071827960009,

        1.01927550787166419971206581107]


  [3.75000000000000000000000000000,

        0.0000884172852008038678177546690265,

        0.0000668371852116567589126756623144,

        0.0000656630801047212575450968414543,

        1.34652966415516281090757712817,

        1.01788074980739567443225474185]


  [4.00000000000000000000000000000,

        0.0000316712418331199212537707567222,

        0.0000236063498902303980856650019982,

        0.0000232201388695715321954939137217,

        1.36395574595908230146107243660,

        1.01663258875531403826433404533]


  [4.25000000000000000000000000000,

        0.0000106885257749344204692005578127,

                                           -5
        0.785836043835833355650050719700 10  ,

                                           -5
        0.773832870969607301569131268853 10  ,

        1.38124473331583481100139212482,

        1.01551132462386375557891731162]

Warning, computation interrupted

> plot({[t,subs(x=t,f),t=-2..2],[1/k*cos(t),1/k*sin(t)+1/k+b,t=0..2*Pi]}
> ,scaling=constrained);

> PSORM:=PHI(-beta)/sqrt(1+beta*k);
> PSORMc:=1-PHI(-beta)/sqrt(1+beta*k);

                                                -23
                   PSORM := 0.166278825165857 10


                             PSORMc := 1.

> 
