#BEGIN OUTFILE1
# before write_ruby top matter
ALWAYS = 1
INFO = 2
DEBUGL = 3
DEBUGMASSIVE = 4
MAX_UNCHANGED = 10
ATS_MAX_TERMS = 20
# before generate init omniout const
$glob_iolevel = INFO
# before write_ats library and user def block
#BEGIN ATS LIBRARY BLOCK
# Begin Function number 2
def omniout_str(iolevel,str)
    if ($glob_iolevel >= iolevel)  then # if number 1
        puts str
    end# end if 1
end # End Function number 2
# Begin Function number 3
def omniout_str_noeol(iolevel,str)
    if ($glob_iolevel >= iolevel) then # if number 1
        printf("%s", str)
    end# end if 1
end # End Function number 3
# Begin Function number 4
def omniout_labstr(iolevel,label,str)
    if ($glob_iolevel >= iolevel)  then # if number 1
        puts label + str
    end# end if 1
end # End Function number 4
# Begin Function number 5
def omniout_float(iolevel,prelabel,prelen,value,vallen,postlabel)
    if ($glob_iolevel >= iolevel)  then # if number 1
        puts prelabel.ljust(30) + value.to_s + postlabel
    end# end if 1
end # End Function number 5
# Begin Function number 6
def omniout_int(iolevel,prelabel,prelen,value,vallen,postlabel)
    if ($glob_iolevel >= iolevel)  then # if number 1
        puts prelabel.ljust(32) + value.to_s + postlabel
    end# end if 1
end # End Function number 6
# Begin Function number 7
def omniout_float_arr(iolevel,prelabel,elemnt,prelen,value,vallen,postlabel)
    if ($glob_iolevel >= iolevel)  then # if number 1
        puts (prelabel.ljust(12) + "  " + elemnt.to_s + "  "  + value.to_s + " " + postlabel)
    end# end if 1
end # End Function number 7
# Begin Function number 8
def dump_series(iolevel,dump_label,series_name,arr_series,numb)
    if ($glob_iolevel >= iolevel)  then # if number 1
        i = 1
        while (i <= numb) do
        puts dump_label + series_name  + "[" + i.to_s + "]" + arr_series[i].to_s
        i += 1
        end
        end
    end # End Function number 8
    # Begin Function number 9
    def dump_series_2(iolevel,dump_label,series_name2,arr_series2,numb,subnum,arr_x)
        if ($glob_iolevel >= iolevel)  then # if number 2
            sub = 1;
            while (sub <= subnum) do
            i =  1;
            while (i <= numb) do
            puts dump_label + series_name2 + "[" + sub.to_s + "," + i.to_s + "]" + arr_series2[sub,i].to_s
        end# end do number 0
        sub += 1;
    end# end do number -1
end# end if 2
end # End Function number 9
# Begin Function number 10
def cs_info(iolevel,str)
    if ($glob_iolevel >= iolevel)  then # if number 2
        puts "cs_info " + str  + " $glob_correct_start_flag = " , $glob_correct_start_flag.to_s + "$glob_h := " + $glob_h + "$glob_reached_optimal_h := "  +  $glob_reached_optimal_h
    end# end if 2
end # End Function number 10
# Begin Function number 11
def logitem_time(fd,secs_in)
    fd.printf("<td>")
    if (secs_in >= 0)  then # if number 2
        years_int  =  int_trunc(secs_in / $glob_sec_in_year)
        sec_temp  =  (secs_in % $glob_sec_in_year)
        days_int  =  int_trunc(sec_temp / $glob_sec_in_day)
        sec_temp  =  (sec_temp % $glob_sec_in_day)
        hours_int  =  int_trunc(sec_temp / $glob_sec_in_hour)
        sec_temp  =  (sec_temp % $glob_sec_in_hour)
        minutes_int  = int_trunc(sec_temp / $glob_sec_in_minute)
        sec_int  = (sec_temp % $glob_sec_in_minute)
        if  (years_int > 0)  then # if number 3
            fd.printf(years_int.to_s + " Years " + days_int.to_s + " Days " + hours_int.to_s + " Hours " + minutes_int.to_s + " Minutes " + sec_int.to_s + " Seconds")
        elsif
        (days_int > 0)  then # if number 4
            fd.printf(days_int.to_s + " Days " + hours_int.to_s + " Hours " + minutes_int.to_s + " Minutes " + sec_int.to_s + " Seconds")
        elsif
        (hours_int > 0)  then # if number 5
            fd.printf(hours_int.to_s + " Hours " + minutes_int.to_s + " Minutes " + sec_int.to_s + " Seconds")
        elsif
        (minutes_int > 0)  then # if number 6
            fd.printf(minutes_int.to_s + " Minutes " + sec_int.to_s + " Seconds")
        else
            fd.printf(sec_int.to_s + " Seconds")
        end# end if 6
    else
        fd.printf(" 0.0 Seconds")
    end# end if 5
    fd.printf("</td>")
end # End Function number 11
# Begin Function number 12
def omniout_timestr(secs_in)
    if (secs_in >= 0)  then # if number 5
        years_int  =  int_trunc(secs_in / $glob_sec_in_year)
        sec_temp  =  (secs_in % $glob_sec_in_year)
        days_int  =  int_trunc(sec_temp / $glob_sec_in_day)
        sec_temp  =  (sec_temp % $glob_sec_in_day)
        hours_int  =  int_trunc(sec_temp / $glob_sec_in_hour)
        sec_temp  =  (sec_temp % $glob_sec_in_hour)
        minutes_int  = int_trunc(sec_temp / $glob_sec_in_minute)
        sec_int  = (sec_temp % $glob_sec_in_minute)
        if  (years_int > 0)  then # if number 6
            puts years_int.to_s + " Years " + days_int.to_s + " Days " + hours_int.to_s + " Hours " + minutes_int.to_s + " Minutes " + sec_int.to_s + " Seconds"
        elsif
        (days_int > 0)  then # if number 7
            puts days_int.to_s + " Days " + hours_int.to_s + " Hours " + minutes_int.to_s + " Minutes " + sec_int.to_s + " Seconds"
        elsif
        (hours_int > 0)  then # if number 8
            puts hours_int.to_s + " Hours " + minutes_int.to_s + " Minutes " + sec_int.to_s + " Seconds"
        elsif
        (minutes_int > 0)  then # if number 9
            puts minutes_int.to_s + " Minutes " + sec_int.to_s + " Seconds"
        else
            puts sec_int.to_s + " Seconds"
        end# end if 9
    else
        puts " 0.0 Seconds"
    end# end if 8
end # End Function number 12
# Begin Function number 13
def zero_ats_ar(arr_a)
    iii  = 1
    while (iii <= ATS_MAX_TERMS) do # do number -1
        arr_a [iii] = $glob__0
        iii  =  iii + 1
    end# end do number -1
end # End Function number 13
# Begin Function number 14
def ats(mmm_ats,arr_a,arr_b,jjj_ats)
    ret_ats  = $glob__0
    if (jjj_ats <=  mmm_ats)  then # if number 8
        ma_ats  =  mmm_ats + 1
        iii_ats  =  jjj_ats
        while (iii_ats <= mmm_ats) do # do number -1
            lll_ats  =  ma_ats - iii_ats
            if ((lll_ats <= ATS_MAX_TERMS  and (iii_ats <= ATS_MAX_TERMS) ))  then # if number 9
                ret_ats  =  ret_ats + c(arr_a[iii_ats])*c(arr_b[lll_ats])
            end# end if 9
            iii_ats  =  iii_ats + 1
        end# end do number -1
    end# end if 8
    return ( ret_ats)
end # End Function number 14
# Begin Function number 15
def att(mmm_att,arr_aa,arr_bb,jjj_att)
    ret_att  = $glob__0
    if (jjj_att < mmm_att)  then # if number 8
        ma_att  =  mmm_att + 2
        iii_att  =  jjj_att
        while ((iii_att < mmm_att) and (iii_att <= ATS_MAX_TERMS) )   do # do number -1
            lll_att  =  ma_att - iii_att
            al_att  =  (lll_att - 1)
            if ((lll_att <= ATS_MAX_TERMS  and (iii_att <= ATS_MAX_TERMS) ))  then # if number 9
                ret_att  =   ret_att + c(arr_aa[iii_att])*c(arr_bb[lll_att])* c(al_att)
            end# end if 9
            iii_att  =  iii_att + 1
        end# end do number -1
        ret_att  =  ret_att / c(mmm_att)
    end# end if 8
    return ( ret_att)
end # End Function number 15
# Begin Function number 16
def logditto(file)
    file.printf("<td>")
    file.printf("ditto")
    file.printf("</td>")
end # End Function number 16
# Begin Function number 17
def logitem_integer(file,n)
    file.printf("<td>")
    file.printf("%d",n)
    file.printf("</td>")
end # End Function number 17
# Begin Function number 18
def logitem_str(file,str)
    file.printf("<td>")
    file.printf(str.to_s)
    file.printf("</td>")
end # End Function number 18
# Begin Function number 19
def logitem_good_digits(file,rel_error)
    file.printf("<td>")
    file.printf("%d",$glob_min_good_digits)
    file.printf("</td>")
end # End Function number 19
# Begin Function number 20
def log_revs(file,revs)
    file.printf(revs.to_s)
end # End Function number 20
# Begin Function number 21
def logitem_float(file,x)
    file.printf("<td>")
    file.printf(x.to_s)
    file.printf("</td>")
end # End Function number 21
# Begin Function number 22
def logitem_h_reason(file)
    file.printf("<td>")
    if ($glob_h_reason  == 1) then # if number 8
        file.printf("Max H")
    elsif
    ($glob_h_reason == 2) then # if number 9
        file.printf("Display Interval")
    elsif
    ($glob_h_reason == 3) then # if number 10
        file.printf("Optimal")
    elsif
    ($glob_h_reason == 4) then # if number 11
        file.printf("Pole Accuracy")
    elsif
    ($glob_h_reason == 5) then # if number 12
        file.printf("Min H (Pole)")
    elsif
    ($glob_h_reason == 6) then # if number 13
        file.printf("Pole")
    elsif
    ($glob_h_reason == 7) then # if number 14
        file.printf("Opt Iter")
    else
        file.printf("Impossible")
    end# end if 14
    file.printf("</td>")
end # End Function number 22
# Begin Function number 23
def logstart(file)
    file.printf("<tr>")
end # End Function number 23
# Begin Function number 24
def logend(file)
    file.printf("</tr>")
end # End Function number 24
# Begin Function number 25
def chk_data()
    errflag  =  false
    if ($glob_max_iter < 2)  then # if number 14
        omniout_str(ALWAYS,"Illegal max_iter")
        errflag  =  true
    end# end if 14
    if (errflag)   then # if number 14
    end# end if 14
end # End Function number 25
# Begin Function number 26
def comp_expect_sec(t_end2,t_start2,t2,clock_sec2)
    ms2 = c(clock_sec2)
    sub1 = c(t_end2-t_start2)
    sub2 = c(t2-t_start2)
    if (sub1 == $glob__0)  then # if number 14
        sec_left = $glob__0
    else
        if (sub2 > $glob__0)  then # if number 15
            rrr = (sub1/sub2)
            sec_left  = rrr * c(ms2) - c(ms2)
        else
            sec_left  = $glob__0
        end# end if 15
    end# end if 14
    return ( sec_left)
end # End Function number 26
# Begin Function number 27
def comp_percent(t_end2,t_start2, t2)
    sub1  =  (t_end2-t_start2)
    sub2  =  (t2-t_start2)
    if (sub2 > $glob_small_float)  then # if number 14
        rrr  =  ($glob__100*sub2)/sub1
    else
        rrr  =  0.0
    end# end if 14
    return (   rrr)
end # End Function number 27
# Begin Function number 28
def comp_rad_from_ratio(term1,term2,last_no)
    #TOP TWO TERM RADIUS ANALYSIS
    if (float_abs(term2) > $glob__0)  then # if number 14
        ret =  float_abs(term1 * $glob_h / term2)
    else
        ret = $glob_larger_float
    end# end if 14
    return (   ret)
    #BOTTOM TWO TERM RADIUS ANALYSIS
end # End Function number 28
# Begin Function number 29
def comp_ord_from_ratio(term1,term2,last_no)
    #TOP TWO TERM ORDER ANALYSIS
    if (float_abs(term2) > $glob__0)  then # if number 14
        ret = $glob__1 +  float_abs(term2) * c(last_no) * ln(float_abs(term1 * $glob_h / term2))/ln(c(last_no))
    else
        ret = $glob_larger_float
    end# end if 14
    return (   ret)
    #BOTTOM TWO TERM ORDER ANALYSIS
end # End Function number 29
# Begin Function number 30
def c(in_val)
    #To Force Conversion when needed
    ret = in_val
    return (   ret)
    #End Conversion
end # End Function number 30
# Begin Function number 31
def comp_rad_from_three_terms(term1,term2,term3,last_no)
    #TOP THREE TERM RADIUS ANALYSIS
    temp = float_abs(term2*term2*c(last_no)+$glob__m2*term2*term2-term1*term3*c(last_no)+term1*term3)
    if (float_abs(temp) > $glob__0)  then # if number 14
        ret = float_abs((term2*$glob_h*term1)/(temp))
    else
        ret = $glob_larger_float
    end# end if 14
    return ( ret)
    #BOTTOM THREE TERM RADIUS ANALYSIS
end # End Function number 31
# Begin Function number 32
def comp_ord_from_three_terms(term1,term2,term3,last_no)
    #TOP THREE TERM ORDER ANALYSIS
    ret = float_abs(($glob__4*term1*term3*c(last_no)-$glob__3*term1*term3-$glob__4*term2*term2*c(last_no)+$glob__4*term2*term2+term2*term2*c(last_no*last_no)-term1*term3*c(last_no*last_no))/(term2*term2*c(last_no)-$glob__2*term2*term2-term1*term3*c(last_no)+term1*term3))
    return ( ret)
    #TOP THREE TERM ORDER ANALYSIS
end # End Function number 32
# Begin Function number 33
def comp_rad_from_six_terms(term1,term2,term3,term4,term5,term6,last_no)
    #TOP SIX TERM RADIUS ANALYSIS
    if ((term5  != $glob__0) and (term4  != $glob__0) and (term3  != $glob__0) and  (term2  != $glob__0) and (term1  != $glob__0)) then # if number 14
        rm0 = term6/term5
        rm1 = term5/term4
        rm2 = term4/term3
        rm3 = term3/term2
        rm4 = term2/term1
        nr1 =  c(last_no-1)*rm0 - $glob__2*c(last_no-2)*rm1 + c(last_no-3)*rm2
        nr2 =  c(last_no-2)*rm1 - $glob__2*c(last_no-3)*rm2 + c(last_no-4)*rm3
        dr1 =  $glob__m1/rm1 + $glob__2/rm2 - $glob__1/rm3
        dr2 =  $glob__m1/rm2 + $glob__2/rm3 - $glob__1/rm4
        ds1 =  $glob__3/rm1 - $glob__8/rm2 + $glob__5/rm3
        ds2 =  $glob__3/rm2 - $glob__8/rm3 + $glob__5/rm4
        if ((float_abs(nr1 * dr2 - nr2 * dr1)  ==  $glob__0)  or  (float_abs(dr1) == $glob__0))  then # if number 15
            rad_c =  $glob_larger_float
            ord_no =  $glob_larger_float
        else
            if (float_abs(nr1*dr2 - nr2 * dr1) > $glob__0)  then # if number 16
                rcs =  ((ds1*dr2 - ds2*dr1 +dr1*dr2)/(nr1*dr2 - nr2 * dr1))
                #(Manuels)  rcs  =  (ds1*dr2 - ds2*dr1)/(nr1*dr2 - nr2 * dr1)
                ord_no =  (rcs*nr1 - ds1)/($glob__2*dr1) -c(last_no)/$glob__2
                if (float_abs(rcs)  != $glob__0)  then # if number 17
                    if (rcs > $glob__0)  then # if number 18
                        rad_c  =  sqrt(rcs) * float_abs($glob_h)
                    else
                        rad_c  =  $glob_larger_float
                        ord_no =  $glob_larger_float
                    end# end if 18
                else
                    rad_c =  $glob_larger_float
                    ord_no =  $glob_larger_float
                end# end if 17
            else
                rad_c =  $glob_larger_float
                ord_no =  $glob_larger_float
            end# end if 16
        end# end if 15
    else
        rad_c =  $glob_larger_float
        ord_no =  $glob_larger_float
    end# end if 14
    $glob_six_term_ord_save =  ord_no
    return ( rad_c)
    #BOTTOM SIX TERM RADIUS ANALYSIS
end # End Function number 33
# Begin Function number 34
def comp_ord_from_six_terms(term1,term2,term3,term4,term5,term6,last_no)
    #TOP SIX TERM ORDER ANALYSIS
    #TOP SAVED FROM SIX TERM RADIUS ANALYSIS
    return ( $glob_six_term_ord_save)
    #BOTTOM SIX TERM ORDER ANALYSIS
end # End Function number 34
# Begin Function number 35
def factorial_2(nnn)
    if (nnn > 0)  then # if number 14
        return ( nnn * factorial_2(nnn - 1))
    else
        return ( 1.0)
    end# end if 14
end # End Function number 35
# Begin Function number 36
def factorial_1(nnn)
    if (nnn <= ATS_MAX_TERMS) then # if number 14
        if ($array_fact_1[nnn] == 0) then # if number 15
            ret = factorial_2(nnn)
            $array_fact_1[nnn] = ret
        else
            ret = $array_fact_1[nnn]
        end# end if 15
    else
        ret = factorial_2(nnn)
    end# end if 14
    return ( ret)
end # End Function number 36
# Begin Function number 37
def factorial_3(mmm,nnn)
    if ((nnn <= ATS_MAX_TERMS)  and  (mmm <= ATS_MAX_TERMS)) then # if number 14
        if ($array_fact_2[mmm][nnn] == 0) then # if number 15
            ret = factorial_1(mmm)/factorial_1(nnn)
            $array_fact_2[mmm][nnn] = ret
        else
            ret = $array_fact_2[mmm][nnn]
        end# end if 15
    else
        ret = factorial_2(mmm)/factorial_2(nnn)
    end# end if 14
    return ( ret)
end # End Function number 37
# Begin Function number 38
def convfloat(mmm)
    return(mmm.to_f)
end # End Function number 38
# Begin Function number 39
def elapsed_time_seconds()
    t = Time.now
    return(t.to_i)
end # End Function number 39
# Begin Function number 40
def expt(x,y)
    if ((x <= 0.0) and (y < 0.0)) then # if number 14
        puts "expt error x = " + x.to_s + "y = " + y.to_s
    end# end if 14
    return(x**y)
end # End Function number 40
# Begin Function number 41
def Si(x)
    return(0.0)
end # End Function number 41
# Begin Function number 42
def Ci(x)
    return(0.0)
end # End Function number 42
# Begin Function number 43
def float_abs(x)
    return(x.abs)
end # End Function number 43
# Begin Function number 44
def int_abs(x)
    return(x.abs)
end # End Function number 44
# Begin Function number 45
def exp(x)
    return(Math.exp(x))
end # End Function number 45
# Begin Function number 46
def int_trunc(x)
    return(x.to_i)
end # End Function number 46
# Begin Function number 47
def floor(x)
    return(x.floor)
end # End Function number 47
# Begin Function number 48
def sin(x)
    return(Math.sin(x))
end # End Function number 48
# Begin Function number 49
def neg(x)
    return(-x)
end # End Function number 49
# Begin Function number 50
def cos(x)
    return(Math.cos(x))
end # End Function number 50
# Begin Function number 51
def tan(x)
    return(Math.tan(x))
end # End Function number 51
# Begin Function number 52
def arccos(x)
    return(Math.acos(x))
end # End Function number 52
# Begin Function number 53
def arccosh(x)
    return(Math.acosh(x))
end # End Function number 53
# Begin Function number 54
def arcsin(x)
    return(Math.asin(x))
end # End Function number 54
# Begin Function number 55
def arcsinh(x)
    return(Math.asinh(x))
end # End Function number 55
# Begin Function number 56
def arctan(x)
    return(Math.atan(x))
end # End Function number 56
# Begin Function number 57
def arctanh(x)
    return(Math.atanh(x))
end # End Function number 57
# Begin Function number 58
def cosh(x)
    return(Math.cosh(x))
end # End Function number 58
# Begin Function number 59
def erf(x)
    return(Math.erf(x))
end # End Function number 59
# Begin Function number 60
def log(x)
    return(Math.log(x))
end # End Function number 60
# Begin Function number 61
def ln(x)
    return(Math.log(x))
end # End Function number 61
# Begin Function number 62
def log10(x)
    return(Math.log10(x))
end # End Function number 62
# Begin Function number 63
def sinh(x)
    return(Math.sinh(x))
end # End Function number 63
# Begin Function number 64
def tanh(x)
    return(Math.tanh(x))
end # End Function number 64
# Begin Function number 65
def sqrt(x)
    return(Math.sqrt(x))
end # End Function number 65
# Begin Function number 66
def array2d(op3,op4)
    i = 0
    x1 = Array.new(op3.to_i + 1)
    while i <= op3.to_i + 1  do # do number -1
        x1[i] = Array.new(op4.to_i + 1)
        i += 1
    end# end do number -1
    return x1
end # End Function number 66
# Begin Function number 67
def estimated_needed_step_error(x_start,x_end,estimated_h,estimated_answer)
    omniout_float(ALWAYS,"$glob_desired_digits_correct",32,$glob_desired_digits_correct,32,"")
    desired_abs_gbl_error = expt($glob__10,c( -$glob_desired_digits_correct)) * c(float_abs(c(estimated_answer)))
    omniout_float(ALWAYS,"estimated_h",32,estimated_h,32,"")
    omniout_float(ALWAYS,"estimated_answer",32,estimated_answer,32,"")
    omniout_float(ALWAYS,"desired_abs_gbl_error",32,desired_abs_gbl_error,32,"")
    range = (x_end - x_start)
    omniout_float(ALWAYS,"range",32,range,32,"")
    estimated_steps = range / estimated_h
    omniout_float(ALWAYS,"estimated_steps",32,estimated_steps,32,"")
    step_error = (c(float_abs(desired_abs_gbl_error) /sqrt(c( estimated_steps))/c(ATS_MAX_TERMS)))
    omniout_float(ALWAYS,"step_error",32,step_error,32,"")
    return ( (step_error))
end # End Function number 67
#END ATS LIBRARY BLOCK
#BEGIN USER FUNCTION BLOCK
#BEGIN BLOCK 3
#BEGIN USER DEF BLOCK
def exact_soln_y (x)
x = c(x)
return(arctan(c(x)))
end
#END USER DEF BLOCK
#END BLOCK 3
#END USER FUNCTION BLOCK
# before write_aux functions
# Begin Function number 2
def display_poles()
    if (($glob_type_given_pole == 1) or ($glob_type_given_pole == 2))  then # if number 1
        rad_given = sqrt(($array_x[1] - $array_given_rad_poles[1][1]) * ($array_x[1] - $array_given_rad_poles[1][1]) +  $array_given_rad_poles[1][2] * $array_given_rad_poles[1][2])
        omniout_float(ALWAYS,"Radius of convergence (given) for eq  1           ",4,rad_given,4," ")
        omniout_float(ALWAYS,"Order of pole (given)                             ",4,$array_given_ord_poles[1][1],4," ")
        if (rad_given < $glob_least_given_sing)  then # if number 2
            $glob_least_given_sing =  rad_given
        end# end if 2
    elsif
    ($glob_type_given_pole == 3) then # if number 2
        omniout_str(ALWAYS,"NO POLE (given) for Equation 1")
    elsif
    ($glob_type_given_pole == 5) then # if number 3
        omniout_str(ALWAYS,"SOME POLE (given) for Equation 1")
    else
        omniout_str(ALWAYS,"NO INFO (given) for Equation 1")
    end# end if 3
    if ($array_rad_test_poles[1][1] < $glob_large_float)  then # if number 3
        omniout_float(ALWAYS,"Radius of convergence (ratio test) for eq  1      ",4,$array_rad_test_poles[1][1],4," ")
        if ($array_rad_test_poles[1][1]< $glob_least_ratio_sing) then # if number 4
            $glob_least_ratio_sing = $array_rad_test_poles[1][1]
        end# end if 4
        omniout_float(ALWAYS,"Order of pole (ratio test)                        ",4, $array_ord_test_poles[1][1],4," ")
    else
        omniout_str(ALWAYS,"NO POLE (ratio test) for Equation 1")
    end# end if 3
    if (($array_rad_test_poles[1][2] >  $glob__small) and ($array_rad_test_poles[1][2] < $glob_large_float))  then # if number 3
        omniout_float(ALWAYS,"Radius of convergence (three term test) for eq 1  ",4,$array_rad_test_poles[1][2],4," ")
        if ($array_rad_test_poles[1][2]< $glob_least_3_sing) then # if number 4
            $glob_least_3_sing = $array_rad_test_poles[1][2]
        end# end if 4
        omniout_float(ALWAYS,"Order of pole (three term test)                   ",4, $array_ord_test_poles[1][2],4," ")
    else
        omniout_str(ALWAYS,"NO REAL POLE (three term test) for Equation 1")
    end# end if 3
    if (($array_rad_test_poles[1][3] >  $glob__small) and ($array_rad_test_poles[1][3] < $glob_large_float))  then # if number 3
        omniout_float(ALWAYS,"Radius of convergence (six term test) for eq 1    ",4,$array_rad_test_poles[1][3],4," ")
        if ($array_rad_test_poles[1][3]< $glob_least_6_sing) then # if number 4
            $glob_least_6_sing = $array_rad_test_poles[1][3]
        end# end if 4
        omniout_float(ALWAYS,"Order of pole (six term test)                     ",4, $array_ord_test_poles[1][3],4," ")
    else
        omniout_str(ALWAYS,"NO COMPLEX POLE (six term test) for Equation 1")
    end# end if 3
end # End Function number 2
# Begin Function number 3
def my_check_sign( x0 ,xf)
    if (xf > x0) then # if number 3
        ret = $glob__1
    else
        ret = $glob__m1
    end# end if 3
    return ( ret)
end # End Function number 3
# Begin Function number 4
def est_size_answer()
    min_size = $glob_estimated_size_answer
    if (float_abs($array_y[1]) < min_size) then # if number 3
        min_size = float_abs($array_y[1])
        omniout_float(ALWAYS,"min_size",32,min_size,32,"")
    end# end if 3
    if (min_size < $glob__1) then # if number 3
        min_size = $glob__1
        omniout_float(ALWAYS,"min_size",32,min_size,32,"")
    end# end if 3
    return ( min_size)
end # End Function number 4
# Begin Function number 5
def test_suggested_h()
    max_estimated_step_error  = $glob__small
    no_terms = ATS_MAX_TERMS
    hn_div_ho = $glob__0_5
    hn_div_ho_2 = $glob__0_25
    hn_div_ho_3 = $glob__0_125
    omniout_float(ALWAYS,"hn_div_ho",32,hn_div_ho,32,"")
    omniout_float(ALWAYS,"hn_div_ho_2",32,hn_div_ho_2,32,"")
    omniout_float(ALWAYS,"hn_div_ho_3",32,hn_div_ho_3,32,"")
    est_tmp = float_abs($array_y[no_terms-3] + $array_y[no_terms - 2] * hn_div_ho + $array_y[no_terms - 1] * hn_div_ho_2 + $array_y[no_terms] * hn_div_ho_3)
    if (est_tmp >= max_estimated_step_error) then # if number 3
        max_estimated_step_error = est_tmp
    end# end if 3
    omniout_float(ALWAYS,"max_estimated_step_error",32,max_estimated_step_error,32,"")
    return ( max_estimated_step_error)
end # End Function number 5
# Begin Function number 6
def track_estimated_error()
    no_terms = ATS_MAX_TERMS
    hn_div_ho = $glob__0_5
    hn_div_ho_2 = $glob__0_25
    hn_div_ho_3 = $glob__0_125
    est_tmp = c(float_abs($array_y[no_terms-3])) + c(float_abs($array_y[no_terms - 2])) * c(hn_div_ho) + c(float_abs($array_y[no_terms - 1])) * c(hn_div_ho_2) + c(float_abs($array_y[no_terms])) * c(hn_div_ho_3)
    if ($glob_prec * c(float_abs($array_y[1])) > c(est_tmp)) then # if number 3
        est_tmp = c($glob_prec) * c(float_abs($array_y[1]))
    end# end if 3
    if (c(est_tmp) >= c($array_max_est_error[1])) then # if number 3
        $array_max_est_error[1] = c(est_tmp)
    end# end if 3
end # End Function number 6
# Begin Function number 7
def reached_interval()
    if (($glob_check_sign * $array_x[1]) >= ($glob_check_sign * $glob_next_display - $glob_h/$glob__10)) then # if number 3
        ret = true
    else
        ret = false
    end# end if 3
    return(ret)
end # End Function number 7
# Begin Function number 8
def display_alot(iter)
    #TOP DISPLAY ALOT
    if (reached_interval())  then # if number 3
        if (iter >= 0)  then # if number 4
            ind_var = $array_x[1]
            omniout_float(ALWAYS,"x[1]                             ",33,ind_var,20," ")
            closed_form_val_y = exact_soln_y(ind_var)
            omniout_float(ALWAYS,"y[1] (closed_form)               ",33,closed_form_val_y,20," ")
            term_no   =  1
            numeric_val = $array_y[term_no]
            abserr = float_abs(numeric_val - closed_form_val_y)
            omniout_float(ALWAYS,"y[1] (numeric)                   ",33,numeric_val,20," ")
            if (c(float_abs(closed_form_val_y)) > c($glob_prec))  then # if number 5
                relerr = abserr*$glob__100/float_abs(closed_form_val_y)
                if (c(relerr) > c($glob_prec)) then # if number 6
                    $glob_good_digits = -int_trunc(log10(c(relerr))) + 3
                else
                    $glob_good_digits = 16
                end# end if 6
            else
                relerr =  $glob__m1
                $glob_good_digits = -16
            end# end if 5
            if ($glob_good_digits < $glob_min_good_digits) then # if number 5
                $glob_min_good_digits  = $glob_good_digits
            end# end if 5
            if ($glob_apfp_est_good_digits < $glob_min_apfp_est_good_digits) then # if number 5
                $glob_min_apfp_est_good_digits  = $glob_apfp_est_good_digits
            end# end if 5
            if (c(float_abs(numeric_val)) > c($glob_prec))  then # if number 5
                est_rel_err = c($array_max_est_error[1])*$glob__100 * c(sqrt($glob_iter))*c(22)*c(ATS_MAX_TERMS)/c(float_abs(numeric_val))
                if (est_rel_err > $glob_prec) then # if number 6
                    $glob_est_digits = -int_trunc(log10(est_rel_err)) + 3
                else
                    $glob_est_digits = 16
                end# end if 6
            else
                relerr =  $glob__m1
                $glob_est_digits = -16
            end# end if 5
            $array_est_digits[1] = $glob_est_digits
            if ($glob_iter  == 1) then # if number 5
                $array_1st_rel_error[1] = relerr
            else
                $array_last_rel_error[1] = relerr
            end# end if 5
            $array_est_rel_error[1] = est_rel_err
            omniout_float(ALWAYS,"absolute error                   ",4,abserr,20," ")
            omniout_float(ALWAYS,"relative error                   ",4,relerr,20,"%")
            omniout_int(INFO,"Desired digits                   ",32,$glob_desired_digits_correct,4," ")
            omniout_int(INFO,"Estimated correct digits         ",32,$glob_est_digits,4," ")
            omniout_int(INFO,"Correct digits                   ",32,$glob_good_digits,4," ")
            omniout_float(ALWAYS,"h                                ",4,$glob_h,20," ")
        end# end if 4
        #BOTTOM DISPLAY ALOT
    end# end if 3
end # End Function number 8
# Begin Function number 9
def prog_report(x_start,x_end)
    #TOP PROGRESS REPORT
    clock_sec1   = elapsed_time_seconds()
    total_clock_sec   =  (clock_sec1) - ($glob_orig_start_sec)
    $glob_clock_sec   =  (clock_sec1) - ($glob_clock_start_sec)
    left_sec   =  ($glob_max_sec) + ($glob_orig_start_sec) - (clock_sec1)
    expect_sec   =  comp_expect_sec((x_end),(x_start),($array_x[1]) + ($glob_h)  ,( clock_sec1) - ($glob_orig_start_sec))
    opt_clock_sec   = ( clock_sec1) - ($glob_optimal_clock_start_sec)
    $glob_optimal_expect_sec   =  comp_expect_sec((x_end),(x_start),($array_x[1]) +( $glob_h) ,( opt_clock_sec))
    $glob_total_exp_sec   = $glob_optimal_expect_sec + c(total_clock_sec)
    percent_done   =  comp_percent((x_end),(x_start),($array_x[1]) + ($glob_h))
    $glob_percent_done = percent_done
    omniout_str_noeol(INFO,"Total Elapsed Time              ")
    omniout_timestr((total_clock_sec))
    omniout_str_noeol(INFO,"Elapsed Time(since restart)     ")
    omniout_timestr(($glob_clock_sec))
    if (c(percent_done) < $glob__100)  then # if number 3
        omniout_str_noeol(INFO,"Expected Time Remaining         ")
        omniout_timestr((expect_sec))
        omniout_str_noeol(INFO,"Optimized Time Remaining        ")
        omniout_timestr(($glob_optimal_expect_sec))
        omniout_str_noeol(INFO,"Expected Total Time             ")
        omniout_timestr(($glob_total_exp_sec))
    end# end if 3
    omniout_str_noeol(INFO,"Time to Timeout                 ")
    omniout_timestr((left_sec))
    omniout_float(INFO,    "Percent Done                    ",33,percent_done,4,"%")
    #BOTTOM PROGRESS REPORT
end # End Function number 9
# Begin Function number 10
def check_for_pole()
    #TOP CHECK FOR POLE
    tmp_rad =  $glob_larger_float
    prev_tmp_rad =  $glob_larger_float
    tmp_ratio =  $glob_larger_float
    rad_c =  $glob_larger_float
    $array_rad_test_poles[1][1] =  $glob_larger_float
    $array_ord_test_poles[1][1] =  $glob_larger_float
    found_sing = 1
    last_no =  ATS_MAX_TERMS - 1 - 10
    cnt =  0
    while (last_no < ATS_MAX_TERMS-3 and found_sing == 1) do # do number 1
        tmp_rad = comp_rad_from_ratio($array_y_higher[1][last_no-1],$array_y_higher[1][last_no],last_no)
        if (float_abs(prev_tmp_rad) > $glob__0) then # if number 3
            tmp_ratio =  tmp_rad / prev_tmp_rad
        else
            tmp_ratio =  $glob_large_float
        end# end if 3
        if ((cnt > 0 ) and (tmp_ratio < $glob_upper_ratio_limit) and (tmp_ratio > $glob_lower_ratio_limit)) then # if number 3
            rad_c = tmp_rad
        elsif
        (cnt == 0) then # if number 4
            rad_c = tmp_rad
        elsif
        (cnt > 0) then # if number 5
            found_sing = 0
        end# end if 5
        prev_tmp_rad = tmp_rad
        cnt =  cnt + 1
        last_no = last_no + 1
    end# end do number 1
    if (found_sing == 1) then # if number 5
        if (rad_c  < $array_rad_test_poles[1][1]) then # if number 6
            $array_rad_test_poles[1][1] = rad_c
            last_no = last_no - 1
            tmp_ord = comp_ord_from_ratio($array_y_higher[1][last_no-1],$array_y_higher[1][last_no],last_no)
            $array_rad_test_poles[1][1] = rad_c
            $array_ord_test_poles[1][1] = tmp_ord
        end# end if 6
    end# end if 5
    #BOTTOM general radius test1
    tmp_rad =  $glob_larger_float
    prev_tmp_rad =  $glob_larger_float
    tmp_ratio =  $glob_larger_float
    rad_c =  $glob_larger_float
    $array_rad_test_poles[1][2] =  $glob_larger_float
    $array_ord_test_poles[1][2] =  $glob_larger_float
    found_sing = 1
    last_no =  ATS_MAX_TERMS - 1 - 10
    cnt =  0
    while (last_no < ATS_MAX_TERMS-4 and found_sing == 1) do # do number 1
        tmp_rad = comp_rad_from_three_terms($array_y_higher[1][last_no-2],$array_y_higher[1][last_no-1],$array_y_higher[1][last_no],last_no)
        if (float_abs(prev_tmp_rad) > $glob__0) then # if number 5
            tmp_ratio =  tmp_rad / prev_tmp_rad
        else
            tmp_ratio =  $glob_large_float
        end# end if 5
        if ((cnt > 0 ) and (tmp_ratio < $glob_upper_ratio_limit) and (tmp_ratio > $glob_lower_ratio_limit)) then # if number 5
            rad_c = tmp_rad
        elsif
        (cnt == 0) then # if number 6
            rad_c = tmp_rad
        elsif
        (cnt > 0) then # if number 7
            found_sing = 0
        end# end if 7
        prev_tmp_rad = tmp_rad
        cnt =  cnt + 1
        last_no = last_no + 1
    end# end do number 1
    if (found_sing == 1) then # if number 7
        if (rad_c  < $array_rad_test_poles[1][2]) then # if number 8
            $array_rad_test_poles[1][2] = rad_c
            last_no = last_no - 1
            tmp_ord = comp_ord_from_three_terms($array_y_higher[1][last_no-2],$array_y_higher[1][last_no-1],$array_y_higher[1][last_no],last_no)
            $array_rad_test_poles[1][2] = rad_c
            if (rad_c  <  $glob_min_pole_est) then # if number 9
                $glob_min_pole_est = rad_c
            end# end if 9
            $array_ord_test_poles[1][2] = tmp_ord
        end# end if 8
    end# end if 7
    #BOTTOM general radius test1
    tmp_rad =  $glob_larger_float
    prev_tmp_rad =  $glob_larger_float
    tmp_ratio =  $glob_larger_float
    rad_c =  $glob_larger_float
    $array_rad_test_poles[1][3] =  $glob_larger_float
    $array_ord_test_poles[1][3] =  $glob_larger_float
    found_sing = 1
    last_no =  ATS_MAX_TERMS - 1 - 10
    cnt =  0
    while (last_no < ATS_MAX_TERMS-7 and found_sing == 1) do # do number 1
        tmp_rad = comp_rad_from_six_terms($array_y_higher[1][last_no-5],$array_y_higher[1][last_no-4],$array_y_higher[1][last_no-3],$array_y_higher[1][last_no-2],$array_y_higher[1][last_no-1],$array_y_higher[1][last_no],last_no)
        if (float_abs(prev_tmp_rad) > $glob__0) then # if number 7
            tmp_ratio =  tmp_rad / prev_tmp_rad
        else
            tmp_ratio =  $glob_large_float
        end# end if 7
        if ((cnt > 0 ) and (tmp_ratio < $glob_upper_ratio_limit) and (tmp_ratio > $glob_lower_ratio_limit)) then # if number 7
            rad_c = tmp_rad
        elsif
        (cnt == 0) then # if number 8
            rad_c = tmp_rad
        elsif
        (cnt > 0) then # if number 9
            found_sing = 0
        end# end if 9
        prev_tmp_rad = tmp_rad
        cnt =  cnt + 1
        last_no = last_no + 1
    end# end do number 1
    if (found_sing == 1) then # if number 9
        if (rad_c  < $array_rad_test_poles[1][3]) then # if number 10
            $array_rad_test_poles[1][3] = rad_c
            last_no = last_no - 1
            tmp_ord = comp_ord_from_six_terms($array_y_higher[1][last_no-5],$array_y_higher[1][last_no-4],$array_y_higher[1][last_no-3],$array_y_higher[1][last_no-2],$array_y_higher[1][last_no-1],$array_y_higher[1][last_no],last_no)
            $array_rad_test_poles[1][3] = rad_c
            if (rad_c  <  $glob_min_pole_est) then # if number 11
                $glob_min_pole_est = rad_c
            end# end if 11
            $array_ord_test_poles[1][3] = tmp_ord
        end# end if 10
    end# end if 9
    #BOTTOM general radius test1
    #START ADJUST ALL SERIES
    if (float_abs($glob_min_pole_est) * $glob_ratio_of_radius < float_abs($glob_h)) then # if number 9
        h_new = $glob_check_sign * $glob_min_pole_est * $glob_ratio_of_radius
        omniout_str(ALWAYS,"SETTING H FOR POLE")
        $glob_h_reason = 6
        if ($glob_check_sign * $glob_min_h > $glob_check_sign * h_new) then # if number 10
            omniout_str(ALWAYS,"SETTING H FOR MIN H")
            h_new = $glob_min_h
            $glob_h_reason = 5
        end# end if 10
        term = 1
        ratio = 1.0
        while (term <= ATS_MAX_TERMS)  do # do number 1
            $array_y[term] = $array_y[term]* ratio
            $array_y_higher[1][term] = $array_y_higher[1][term]* ratio
            $array_x[term] = $array_x[term]* ratio
            ratio = ratio * h_new / float_abs($glob_h)
            term = term + 1
        end# end do number 1
        $glob_h = h_new
    end# end if 9
    #BOTTOM ADJUST ALL SERIES
    if (reached_interval())  then # if number 9
        display_poles()
    end# end if 9
end # End Function number 10
# Begin Function number 11
def atomall()
    #TOP ATOMALL
    #END OUTFILE1
    #BEGIN OUTFILE2
    #END OUTFILE2
    #BEGIN ATOMHDR1
    #emit pre mult LINEAR - LINEAR $eq_no = 1 i = 1
    $array_tmp1[1] = $array_x[1] * $array_x[1]
    #emit pre add FULL - CONST $eq_no = 1 i = 1
    $array_tmp2[1] = $array_tmp1[1] + $array_const_1D0[1]
    #emit pre div CONST FULL $eq_no = 1 i = 1
    $array_tmp3[1] = $array_const_1D0[1] / $array_tmp2[1]
    #emit pre add CONST FULL $eq_no = 1 i = 1
    $array_tmp4[1] = $array_const_0D0[1] + $array_tmp3[1]
    #emit pre assign xxx $eq_no = 1 i = 1 $min_hdrs = 5
    if ( not  $array_y_set_initial[1][2]) then # if number 1
        if (1 <= ATS_MAX_TERMS) then # if number 2
            temporary = c($array_tmp4[1]) * (expt(($glob_h) , c(1))) * c(factorial_3(0,1))
            if (2 <= ATS_MAX_TERMS) then # if number 3
                $array_y[2] = temporary
                $array_y_higher[1][2] = temporary
            end# end if 3
            temporary = c(temporary) / c($glob_h) * c(1)
            $array_y_higher[2][1] = c(temporary)
        end# end if 2
    end# end if 1
    kkk = 2
    #END ATOMHDR1
    #BEGIN ATOMHDR2
    #emit pre mult LINEAR - LINEAR $eq_no = 1 i = 2
    $array_tmp1[2] = $array_x[1] * $array_x[2] + $array_x[2] * $array_x[1]
    #emit pre add FULL CONST $eq_no = 1 i = 2
    $array_tmp2[2] = $array_tmp1[2]
    #emit pre div CONST FULL $eq_no = 1 i = 2
    $array_tmp3[2] = neg(ats(2,$array_tmp2,$array_tmp3,2)) / $array_tmp2[1]
    #emit pre add CONST FULL $eq_no = 1 i = 2
    $array_tmp4[2] = $array_tmp3[2]
    #emit pre assign xxx $eq_no = 1 i = 2 $min_hdrs = 5
    if ( not  $array_y_set_initial[1][3]) then # if number 1
        if (2 <= ATS_MAX_TERMS) then # if number 2
            temporary = c($array_tmp4[2]) * (expt(($glob_h) , c(1))) * c(factorial_3(1,2))
            if (3 <= ATS_MAX_TERMS) then # if number 3
                $array_y[3] = temporary
                $array_y_higher[1][3] = temporary
            end# end if 3
            temporary = c(temporary) / c($glob_h) * c(2)
            $array_y_higher[2][2] = c(temporary)
        end# end if 2
    end# end if 1
    kkk = 3
    #END ATOMHDR2
    #BEGIN ATOMHDR3
    #emit pre mult LINEAR - LINEAR $eq_no = 1 i = 3
    $array_tmp1[3] = $array_x[2] * $array_x[2]
    #emit pre add FULL CONST $eq_no = 1 i = 3
    $array_tmp2[3] = $array_tmp1[3]
    #emit pre div CONST FULL $eq_no = 1 i = 3
    $array_tmp3[3] = neg(ats(3,$array_tmp2,$array_tmp3,2)) / $array_tmp2[1]
    #emit pre add CONST FULL $eq_no = 1 i = 3
    $array_tmp4[3] = $array_tmp3[3]
    #emit pre assign xxx $eq_no = 1 i = 3 $min_hdrs = 5
    if ( not  $array_y_set_initial[1][4]) then # if number 1
        if (3 <= ATS_MAX_TERMS) then # if number 2
            temporary = c($array_tmp4[3]) * (expt(($glob_h) , c(1))) * c(factorial_3(2,3))
            if (4 <= ATS_MAX_TERMS) then # if number 3
                $array_y[4] = temporary
                $array_y_higher[1][4] = temporary
            end# end if 3
            temporary = c(temporary) / c($glob_h) * c(3)
            $array_y_higher[2][3] = c(temporary)
        end# end if 2
    end# end if 1
    kkk = 4
    #END ATOMHDR3
    #BEGIN ATOMHDR4
    #emit pre add FULL CONST $eq_no = 1 i = 4
    $array_tmp2[4] = $array_tmp1[4]
    #emit pre div CONST FULL $eq_no = 1 i = 4
    $array_tmp3[4] = neg(ats(4,$array_tmp2,$array_tmp3,2)) / $array_tmp2[1]
    #emit pre add CONST FULL $eq_no = 1 i = 4
    $array_tmp4[4] = $array_tmp3[4]
    #emit pre assign xxx $eq_no = 1 i = 4 $min_hdrs = 5
    if ( not  $array_y_set_initial[1][5]) then # if number 1
        if (4 <= ATS_MAX_TERMS) then # if number 2
            temporary = c($array_tmp4[4]) * (expt(($glob_h) , c(1))) * c(factorial_3(3,4))
            if (5 <= ATS_MAX_TERMS) then # if number 3
                $array_y[5] = temporary
                $array_y_higher[1][5] = temporary
            end# end if 3
            temporary = c(temporary) / c($glob_h) * c(4)
            $array_y_higher[2][4] = c(temporary)
        end# end if 2
    end# end if 1
    kkk = 5
    #END ATOMHDR4
    #BEGIN ATOMHDR5
    #emit pre add FULL CONST $eq_no = 1 i = 5
    $array_tmp2[5] = $array_tmp1[5]
    #emit pre div CONST FULL $eq_no = 1 i = 5
    $array_tmp3[5] = neg(ats(5,$array_tmp2,$array_tmp3,2)) / $array_tmp2[1]
    #emit pre add CONST FULL $eq_no = 1 i = 5
    $array_tmp4[5] = $array_tmp3[5]
    #emit pre assign xxx $eq_no = 1 i = 5 $min_hdrs = 5
    if ( not  $array_y_set_initial[1][6]) then # if number 1
        if (5 <= ATS_MAX_TERMS) then # if number 2
            temporary = c($array_tmp4[5]) * (expt(($glob_h) , c(1))) * c(factorial_3(4,5))
            if (6 <= ATS_MAX_TERMS) then # if number 3
                $array_y[6] = temporary
                $array_y_higher[1][6] = temporary
            end# end if 3
            temporary = c(temporary) / c($glob_h) * c(5)
            $array_y_higher[2][5] = c(temporary)
        end# end if 2
    end# end if 1
    kkk = 6
    #END ATOMHDR5
    #BEGIN OUTFILE3
    #Top Atomall While Loop-- outfile3
    while (kkk <= ATS_MAX_TERMS) do # do number 1
        #END OUTFILE3
        #BEGIN OUTFILE4
        #emit mult LINEAR - LINEAR $eq_no = 1 i = 1
        #emit FULL - NOT FULL add $eq_no = 1
        $array_tmp2[kkk] = $array_tmp1[kkk]
        #emit div CONST FULL $eq_no = 1 i = 1
        $array_tmp3[kkk] = neg(ats(kkk,$array_tmp2,$array_tmp3,2)) / $array_tmp2[1]
        #emit NOT FULL - FULL add $eq_no = 1
        $array_tmp4[kkk] = $array_tmp3[kkk]
        #emit assign $eq_no = 1
        order_d  = 1
        if (kkk + order_d <= ATS_MAX_TERMS) then # if number 1
            if ( not  $array_y_set_initial[1][kkk + order_d]) then # if number 2
                temporary = c($array_tmp4[kkk]) * expt(($glob_h) , c(order_d))  * c(factorial_3((kkk - 1),(kkk + order_d - 1)))
                $array_y[kkk + order_d] = c(temporary)
                $array_y_higher[1][kkk + order_d] = c(temporary)
                term = kkk + order_d - 1
                adj2 = kkk + order_d - 1
                adj3 = 2
                while ((term >= 1) and (term <= ATS_MAX_TERMS) and (adj3 < order_d + 1)) do # do number 1
                    if (adj3 <= order_d + 1) then # if number 3
                        if (adj2 > 0) then # if number 4
                            temporary = c(temporary) / c($glob_h) * c(adj2)
                        else
                            temporary = c(temporary)
                        end# end if 4
                        $array_y_higher[adj3][term] = c(temporary)
                    end# end if 3
                    term  = term - 1
                    adj2 = adj2 - 1
                    adj3 = adj3 + 1
                end# end do number 1
            end# end if 2
        end# end if 1
        kkk   =  kkk + 1
    end# end do number 1
    #BOTTOM ATOMALL
    #END OUTFILE4
    #BEGIN OUTFILE5
    #BOTTOM ATOMALL ???
end # End Function number 12
#END OUTFILE5
# before write_ruby_main_top
# Begin Function number 12
def main()
    #BEGIN OUTFILEMAIN
    # before first input block
    #BEGIN FIRST INPUT BLOCK
    #BEGIN BLOCK 1
    #BEGIN FIRST INPUT BLOCK
    # Digits:=32; ELIMINATED in preodein.rb
    max_terms=20
    #END BLOCK 1
    #END FIRST INPUT BLOCK
    #START OF INITS AFTER INPUT BLOCK
    $glob_html_log = true
    #END OF INITS AFTER INPUT BLOCK
    # before generate arrays
    $array_y_init= Array.new(ATS_MAX_TERMS + 1)
    $array_norms= Array.new(ATS_MAX_TERMS + 1)
    $array_fact_1= Array.new(ATS_MAX_TERMS + 1)
    $array_1st_rel_error= Array.new(2 + 1)
    $array_last_rel_error= Array.new(2 + 1)
    $array_est_rel_error= Array.new(2 + 1)
    $array_max_est_error= Array.new(2 + 1)
    $array_type_pole= Array.new(2 + 1)
    $array_type_real_pole= Array.new(2 + 1)
    $array_type_complex_pole= Array.new(2 + 1)
    $array_est_digits= Array.new(2 + 1)
    $array_y= Array.new(ATS_MAX_TERMS + 1)
    $array_x= Array.new(ATS_MAX_TERMS + 1)
    $array_tmp0= Array.new(ATS_MAX_TERMS + 1)
    $array_tmp1= Array.new(ATS_MAX_TERMS + 1)
    $array_tmp2= Array.new(ATS_MAX_TERMS + 1)
    $array_tmp3= Array.new(ATS_MAX_TERMS + 1)
    $array_tmp4= Array.new(ATS_MAX_TERMS + 1)
    $array_m1= Array.new(ATS_MAX_TERMS + 1)
    $array_y_higher = array2d(2 + 1 ,ATS_MAX_TERMS + 1)
    $array_y_higher_work = array2d(2 + 1 ,ATS_MAX_TERMS + 1)
    $array_y_higher_work2 = array2d(2 + 1 ,ATS_MAX_TERMS + 1)
    $array_y_set_initial = array2d(2 + 1 ,ATS_MAX_TERMS + 1)
    $array_given_rad_poles = array2d(2 + 1 ,3 + 1)
    $array_given_ord_poles = array2d(2 + 1 ,3 + 1)
    $array_rad_test_poles = array2d(2 + 1 ,4 + 1)
    $array_ord_test_poles = array2d(2 + 1 ,4 + 1)
    $array_fact_2 = array2d(ATS_MAX_TERMS + 1 ,ATS_MAX_TERMS + 1)
    # before generate constants
    # before generate globals definition
    #Top Generate Globals Definition
    $glob__small=c(0.1e-50)
    $glob_small_float=c(0.1e-50)
    $glob_smallish_float=c(0.1e-60)
    $glob_large_float=c(1.0e100)
    $glob_larger_float=c(1.1e100)
    $glob__m2=c(-2)
    $glob__m1=c(-1)
    $glob__0=c(0)
    $glob__1=c(1)
    $glob__2=c(2)
    $glob__3=c(3)
    $glob__4=c(4)
    $glob__5=c(5)
    $glob__8=c(8)
    $glob__10=c(10)
    $glob__100=c(100)
    $glob__pi=c(0.0)
    $glob__0_5=c(0.5)
    $glob__0_8=c(0.8)
    $glob__m0_8=c(-0.8)
    $glob__0_25=c(0.25)
    $glob__0_125=c(0.125)
    $glob_prec=c(1.0e-16)
    $glob_est_digits=1
    $glob_check_sign=c(1.0)
    $glob_desired_digits_correct=c(8.0)
    $glob_max_estimated_step_error=c(0.0)
    $glob_ratio_of_radius=c(0.1)
    $glob_percent_done=c(0.0)
    $glob_subiter_method=3
    $glob_total_exp_sec=c(0.1)
    $glob_optimal_expect_sec=c(0.1)
    $glob_estimated_size_answer=c(100.0)
    $glob_html_log=true
    $glob_min_good_digits=99999
    $glob_good_digits=0
    $glob_min_apfp_est_good_digits=99999
    $glob_apfp_est_good_digits=0
    $glob_max_opt_iter=10
    $glob_dump=false
    $glob_djd_debug=true
    $glob_display_flag=true
    $glob_djd_debug2=true
    $glob_h_reason=0
    $glob_sec_in_minute=60
    $glob_min_in_hour=60
    $glob_hours_in_day=24
    $glob_days_in_year=365
    $glob_sec_in_hour=3600
    $glob_sec_in_day=86400
    $glob_sec_in_year=31536000
    $glob_almost_1=c(0.9990)
    $glob_clock_sec=c(0.0)
    $glob_clock_start_sec=c(0.0)
    $glob_not_yet_finished=true
    $glob_initial_pass=true
    $glob_not_yet_start_msg=true
    $glob_reached_optimal_h=false
    $glob_optimal_done=false
    $glob_disp_incr=c(0.1)
    $glob_h=c(0.1)
    $glob_diff_rc_fm=c(0.1)
    $glob_diff_rc_fmm1=c(0.1)
    $glob_diff_rc_fmm2=c(0.1)
    $glob_diff_ord_fm=c(0.1)
    $glob_diff_ord_fmm1=c(0.1)
    $glob_diff_ord_fmm2=c(0.1)
    $glob_six_term_ord_save=c(0.1)
    $glob_guess_error_rc=c(0.1)
    $glob_guess_error_ord=c(0.1)
    $glob_max_h=c(0.1)
    $glob_min_h=c(0.000001)
    $glob_type_given_pole=0
    $glob_least_given_sing=c(9.9e200)
    $glob_least_ratio_sing=c(9.9e200)
    $glob_least_3_sing=c(9.9e100)
    $glob_least_6_sing=c(9.9e100)
    $glob_last_good_h=c(0.1)
    $glob_optimize=false
    $glob_look_poles=false
    $glob_display_interval=c(0.1)
    $glob_next_display=c(0.0)
    $glob_dump_closed_form=false
    $glob_abserr=c(0.1e-10)
    $glob_relerr=c(0.1e-10)
    $glob_min_pole_est=c(0.1e+10)
    $glob_max_hours=c(0.0)
    $glob_max_iter=1000
    $glob_max_rel_trunc_err=c(0.1e-10)
    $glob_max_trunc_err=c(0.1e-10)
    $glob_no_eqs=0
    $glob_optimal_clock_start_sec=c(0.0)
    $glob_optimal_start=c(0.0)
    $glob_upper_ratio_limit=c(1.0001)
    $glob_lower_ratio_limit=c(0.9999)
    $glob_unchanged_h_cnt=0
    $glob_warned=false
    $glob_warned2=false
    $glob_max_sec=c(10000.0)
    $glob_orig_start_sec=c(0.0)
    $glob_start=0
    $glob_iter=0
    $glob_normmax=c(0.0)
    $glob_max_minutes=c(0.0)
    #Bottom Generate Globals Deninition
    # before generate const definition
    $array_const_1= Array.new(ATS_MAX_TERMS + 1)
    $array_const_0D0= Array.new(ATS_MAX_TERMS + 1)
    $array_const_1D0= Array.new(ATS_MAX_TERMS + 1)
    # before arrays initialized
    term = 1
    while (term <= ATS_MAX_TERMS) do # do number 1
        $array_y_init[term] = c(0.0)
        term = term + 1
    end# end do number 1
    term = 1
    while (term <= ATS_MAX_TERMS) do # do number 1
        $array_norms[term] = c(0.0)
        term = term + 1
    end# end do number 1
    term = 1
    while (term <= ATS_MAX_TERMS) do # do number 1
        $array_fact_1[term] = c(0.0)
        term = term + 1
    end# end do number 1
    term = 1
    while (term <= 2) do # do number 1
        $array_1st_rel_error[term] = c(0.0)
        term = term + 1
    end# end do number 1
    term = 1
    while (term <= 2) do # do number 1
        $array_last_rel_error[term] = c(0.0)
        term = term + 1
    end# end do number 1
    term = 1
    while (term <= 2) do # do number 1
        $array_est_rel_error[term] = c(0.0)
        term = term + 1
    end# end do number 1
    term = 1
    while (term <= 2) do # do number 1
        $array_max_est_error[term] = c(0.0)
        term = term + 1
    end# end do number 1
    term = 1
    while (term <= 2) do # do number 1
        $array_type_pole[term] = 0
        term = term + 1
    end# end do number 1
    term = 1
    while (term <= 2) do # do number 1
        $array_type_real_pole[term] = 0
        term = term + 1
    end# end do number 1
    term = 1
    while (term <= 2) do # do number 1
        $array_type_complex_pole[term] = 0
        term = term + 1
    end# end do number 1
    term = 1
    while (term <= 2) do # do number 1
        $array_est_digits[term] = 0
        term = term + 1
    end# end do number 1
    term = 1
    while (term <= ATS_MAX_TERMS) do # do number 1
        $array_y[term] = c(0.0)
        term = term + 1
    end# end do number 1
    term = 1
    while (term <= ATS_MAX_TERMS) do # do number 1
        $array_x[term] = c(0.0)
        term = term + 1
    end# end do number 1
    term = 1
    while (term <= ATS_MAX_TERMS) do # do number 1
        $array_tmp0[term] = c(0.0)
        term = term + 1
    end# end do number 1
    term = 1
    while (term <= ATS_MAX_TERMS) do # do number 1
        $array_tmp1[term] = c(0.0)
        term = term + 1
    end# end do number 1
    term = 1
    while (term <= ATS_MAX_TERMS) do # do number 1
        $array_tmp2[term] = c(0.0)
        term = term + 1
    end# end do number 1
    term = 1
    while (term <= ATS_MAX_TERMS) do # do number 1
        $array_tmp3[term] = c(0.0)
        term = term + 1
    end# end do number 1
    term = 1
    while (term <= ATS_MAX_TERMS) do # do number 1
        $array_tmp4[term] = c(0.0)
        term = term + 1
    end# end do number 1
    term = 1
    while (term <= ATS_MAX_TERMS) do # do number 1
        $array_m1[term] = c(0.0)
        term = term + 1
    end# end do number 1
    ord = 1
    while (ord <=2) do # do number 1
        term = 1
        while (term <= ATS_MAX_TERMS) do # do number 2
            $array_y_higher[ord][term] = c(0.0)
            term = term + 1
        end# end do number 2
        ord = ord + 1
    end# end do number 1
    ord = 1
    while (ord <=2) do # do number 1
        term = 1
        while (term <= ATS_MAX_TERMS) do # do number 2
            $array_y_higher_work[ord][term] = c(0.0)
            term = term + 1
        end# end do number 2
        ord = ord + 1
    end# end do number 1
    ord = 1
    while (ord <=2) do # do number 1
        term = 1
        while (term <= ATS_MAX_TERMS) do # do number 2
            $array_y_higher_work2[ord][term] = c(0.0)
            term = term + 1
        end# end do number 2
        ord = ord + 1
    end# end do number 1
    ord = 1
    while (ord <=2) do # do number 1
        term = 1
        while (term <= ATS_MAX_TERMS) do # do number 2
            $array_y_set_initial[ord][term] = c(0.0)
            term = term + 1
        end# end do number 2
        ord = ord + 1
    end# end do number 1
    ord = 1
    while (ord <=2) do # do number 1
        term = 1
        while (term <= 3) do # do number 2
            $array_given_rad_poles[ord][term] = c(0.0)
            term = term + 1
        end# end do number 2
        ord = ord + 1
    end# end do number 1
    ord = 1
    while (ord <=2) do # do number 1
        term = 1
        while (term <= 3) do # do number 2
            $array_given_ord_poles[ord][term] = c(0.0)
            term = term + 1
        end# end do number 2
        ord = ord + 1
    end# end do number 1
    ord = 1
    while (ord <=2) do # do number 1
        term = 1
        while (term <= 4) do # do number 2
            $array_rad_test_poles[ord][term] = c(0.0)
            term = term + 1
        end# end do number 2
        ord = ord + 1
    end# end do number 1
    ord = 1
    while (ord <=2) do # do number 1
        term = 1
        while (term <= 4) do # do number 2
            $array_ord_test_poles[ord][term] = c(0.0)
            term = term + 1
        end# end do number 2
        ord = ord + 1
    end# end do number 1
    ord = 1
    while (ord <=ATS_MAX_TERMS) do # do number 1
        term = 1
        while (term <= ATS_MAX_TERMS) do # do number 2
            $array_fact_2[ord][term] = c(0.0)
            term = term + 1
        end# end do number 2
        ord = ord + 1
    end# end do number 1
    # before symbols initialized
    #BEGIN SYMBOLS INITIALIZATED
    zero_ats_ar($array_y)
    zero_ats_ar($array_x)
    zero_ats_ar($array_tmp0)
    zero_ats_ar($array_tmp1)
    zero_ats_ar($array_tmp2)
    zero_ats_ar($array_tmp3)
    zero_ats_ar($array_tmp4)
    zero_ats_ar($array_m1)
    zero_ats_ar($array_const_1)
    $array_const_1[1]   = c(1)
    zero_ats_ar($array_const_0D0)
    $array_const_0D0[1]   = c(0.0)
    zero_ats_ar($array_const_1D0)
    $array_const_1D0[1]   = c(1.0)
    zero_ats_ar($array_m1)
    $array_m1[1]   = $glob__m1
    #END SYMBOLS INITIALIZATED
    # before generate factorials init
    #Initing Factorial Tables
    iiif = 0
    while (iiif <= ATS_MAX_TERMS) do # do number 1
        jjjf = 0
        while (jjjf <= ATS_MAX_TERMS) do # do number 2
            $array_fact_1[iiif] = 0
            $array_fact_2[iiif][jjjf] = 0
            jjjf = jjjf + 1
        end# end do number 2
        iiif = iiif + 1
    end# end do number 1
    #Done Initing Factorial Table
    # before generate constants assign
    $glob_iolevel = 5
    $glob_yes_pole = 4
    $glob_no_pole = 3
    $glob_not_given = 0
    $glob_no_sing_tests = 4
    $glob_ratio_test = 1
    $glob_three_term_test = 2
    $glob_six_term_test = 3
    $glob_log_10 = log(c(10.0))
    # before generate globals assign
    $glob__small = c(0.1e-50)
    $glob_small_float = c(0.1e-50)
    $glob_smallish_float = c(0.1e-60)
    $glob_large_float = c(1.0e100)
    $glob_larger_float = c(1.1e100)
    $glob__m2 = c(-2)
    $glob__m1 = c(-1)
    $glob__0 = c(0)
    $glob__1 = c(1)
    $glob__2 = c(2)
    $glob__3 = c(3)
    $glob__4 = c(4)
    $glob__5 = c(5)
    $glob__8 = c(8)
    $glob__10 = c(10)
    $glob__100 = c(100)
    $glob__pi = c(0.0)
    $glob__0_5 = c(0.5)
    $glob__0_8 = c(0.8)
    $glob__m0_8 = c(-0.8)
    $glob__0_25 = c(0.25)
    $glob__0_125 = c(0.125)
    $glob_prec = c(1.0e-16)
    $glob_est_digits = 1
    $glob_check_sign = c(1.0)
    $glob_desired_digits_correct = c(8.0)
    $glob_max_estimated_step_error = c(0.0)
    $glob_ratio_of_radius = c(0.1)
    $glob_percent_done = c(0.0)
    $glob_subiter_method = 3
    $glob_total_exp_sec = c(0.1)
    $glob_optimal_expect_sec = c(0.1)
    $glob_estimated_size_answer = c(100.0)
    $glob_html_log = true
    $glob_min_good_digits = 99999
    $glob_good_digits = 0
    $glob_min_apfp_est_good_digits = 99999
    $glob_apfp_est_good_digits = 0
    $glob_max_opt_iter = 10
    $glob_dump = false
    $glob_djd_debug = true
    $glob_display_flag = true
    $glob_djd_debug2 = true
    $glob_h_reason = 0
    $glob_sec_in_minute = 60
    $glob_min_in_hour = 60
    $glob_hours_in_day = 24
    $glob_days_in_year = 365
    $glob_sec_in_hour = 3600
    $glob_sec_in_day = 86400
    $glob_sec_in_year = 31536000
    $glob_almost_1 = c(0.9990)
    $glob_clock_sec = c(0.0)
    $glob_clock_start_sec = c(0.0)
    $glob_not_yet_finished = true
    $glob_initial_pass = true
    $glob_not_yet_start_msg = true
    $glob_reached_optimal_h = false
    $glob_optimal_done = false
    $glob_disp_incr = c(0.1)
    $glob_h = c(0.1)
    $glob_diff_rc_fm = c(0.1)
    $glob_diff_rc_fmm1 = c(0.1)
    $glob_diff_rc_fmm2 = c(0.1)
    $glob_diff_ord_fm = c(0.1)
    $glob_diff_ord_fmm1 = c(0.1)
    $glob_diff_ord_fmm2 = c(0.1)
    $glob_six_term_ord_save = c(0.1)
    $glob_guess_error_rc = c(0.1)
    $glob_guess_error_ord = c(0.1)
    $glob_max_h = c(0.1)
    $glob_min_h = c(0.000001)
    $glob_type_given_pole = 0
    $glob_least_given_sing = c(9.9e200)
    $glob_least_ratio_sing = c(9.9e200)
    $glob_least_3_sing = c(9.9e100)
    $glob_least_6_sing = c(9.9e100)
    $glob_last_good_h = c(0.1)
    $glob_optimize = false
    $glob_look_poles = false
    $glob_display_interval = c(0.1)
    $glob_next_display = c(0.0)
    $glob_dump_closed_form = false
    $glob_abserr = c(0.1e-10)
    $glob_relerr = c(0.1e-10)
    $glob_min_pole_est = c(0.1e+10)
    $glob_max_hours = c(0.0)
    $glob_max_iter = 1000
    $glob_max_rel_trunc_err = c(0.1e-10)
    $glob_max_trunc_err = c(0.1e-10)
    $glob_no_eqs = 0
    $glob_optimal_clock_start_sec = c(0.0)
    $glob_optimal_start = c(0.0)
    $glob_upper_ratio_limit = c(1.0001)
    $glob_lower_ratio_limit = c(0.9999)
    $glob_unchanged_h_cnt = 0
    $glob_warned = false
    $glob_warned2 = false
    $glob_max_sec = c(10000.0)
    $glob_orig_start_sec = c(0.0)
    $glob_start = 0
    $glob_iter = 0
    $glob_normmax = c(0.0)
    $glob_max_minutes = c(0.0)
    # before generate set diff initial
    $array_y_set_initial[1][1] = true
    $array_y_set_initial[1][2] = false
    $array_y_set_initial[1][3] = false
    $array_y_set_initial[1][4] = false
    $array_y_set_initial[1][5] = false
    $array_y_set_initial[1][6] = false
    $array_y_set_initial[1][7] = false
    $array_y_set_initial[1][8] = false
    $array_y_set_initial[1][9] = false
    $array_y_set_initial[1][10] = false
    $array_y_set_initial[1][11] = false
    $array_y_set_initial[1][12] = false
    $array_y_set_initial[1][13] = false
    $array_y_set_initial[1][14] = false
    $array_y_set_initial[1][15] = false
    $array_y_set_initial[1][16] = false
    $array_y_set_initial[1][17] = false
    $array_y_set_initial[1][18] = false
    $array_y_set_initial[1][19] = false
    $array_y_set_initial[1][20] = false
    # set default block
    #Write Set Defaults
    $glob_orig_start_sec   =  elapsed_time_seconds()
    $glob_display_flag   =   true
    $glob_no_eqs   = 1
    $glob_iter   =  -1
    opt_iter   =  -1
    $glob_max_iter   =  50000
    $glob_max_hours   =  (0.0)
    $glob_max_minutes   =  (15.0)
    omniout_str(ALWAYS,"##############ECHO OF PROBLEM#################")
    omniout_str(ALWAYS,"##############temp/sing2_backpostode.ode#################")
    omniout_str(ALWAYS,"diff  (  y  ,  x  ,  1  )   =  1.0 /   ( x  *   x  +  1.0 )   ; ")
    omniout_str(ALWAYS,"!")
    omniout_str(ALWAYS,"#BEGIN FIRST INPUT BLOCK")
    omniout_str(ALWAYS,"# Digits:=32; ELIMINATED in preodein.rb")
    omniout_str(ALWAYS,"max_terms=20 ")
    omniout_str(ALWAYS,"!")
    omniout_str(ALWAYS,"#END FIRST INPUT BLOCK")
    omniout_str(ALWAYS,"#BEGIN SECOND INPUT BLOCK")
    omniout_str(ALWAYS,"x_start=c(-1.5) ")
    omniout_str(ALWAYS,"x_end=c(-2.0) ")
    omniout_str(ALWAYS,"$array_y_init[0 + 1] = exact_soln_y(x_start)")
    omniout_str(ALWAYS,"$glob_look_poles=true ")
    omniout_str(ALWAYS,"#  ELIMINATED in preodein.rb")
    omniout_str(ALWAYS,"#  ELIMINATED in preodein.rb")
    omniout_str(ALWAYS,"#  ELIMINATED in preodein.rb")
    omniout_str(ALWAYS,"#  ELIMINATED in preodein.rb")
    omniout_str(ALWAYS,"#  ELIMINATED in preodein.rb")
    omniout_str(ALWAYS,"#  ELIMINATED in preodein.rb")
    omniout_str(ALWAYS,"#  ELIMINATED in preodein.rb")
    omniout_str(ALWAYS,"$glob_type_given_pole=2 ")
    omniout_str(ALWAYS,"#  ELIMINATED in preodein.rb")
    omniout_str(ALWAYS,"$array_given_rad_poles[1][1]=c(0.0) ")
    omniout_str(ALWAYS,"#  ELIMINATED in preodein.rb")
    omniout_str(ALWAYS,"$array_given_rad_poles[1][2]=c(1.0) ")
    omniout_str(ALWAYS,"#  ELIMINATED in preodein.rb")
    omniout_str(ALWAYS,"$array_given_ord_poles[1][1]=c(1.0) ")
    omniout_str(ALWAYS,"#  ELIMINATED in preodein.rb")
    omniout_str(ALWAYS,"$array_given_ord_poles[1][2]=c(0.0) ")
    omniout_str(ALWAYS,"#END SECOND INPUT BLOCK")
    omniout_str(ALWAYS,"#BEGIN OVERRIDE  BLOCK")
    omniout_str(ALWAYS,"$glob_desired_digits_correct=8 ")
    omniout_str(ALWAYS,"$glob_max_minutes=(3.0) ")
    omniout_str(ALWAYS,"$glob_subiter_method=3 ")
    omniout_str(ALWAYS,"$glob_max_iter=100000 ")
    omniout_str(ALWAYS,"$glob_upper_ratio_limit=c(1.000001) ")
    omniout_str(ALWAYS,"$glob_lower_ratio_limit=c(0.999999) ")
    omniout_str(ALWAYS,"$glob_look_poles=false ")
    omniout_str(ALWAYS,"$glob_h=c(0.001) ")
    omniout_str(ALWAYS,"$glob_display_interval=c(0.01) ")
    omniout_str(ALWAYS,"#END OVERRIDE BLOCK")
    omniout_str(ALWAYS,"!")
    omniout_str(ALWAYS,"#BEGIN USER DEF BLOCK")
    omniout_str(ALWAYS,"def exact_soln_y (x)")
    omniout_str(ALWAYS,"x = c(x)")
    omniout_str(ALWAYS,"return(arctan(c(x))) ")
    omniout_str(ALWAYS,"end")
    omniout_str(ALWAYS,"")
    omniout_str(ALWAYS,"")
    omniout_str(ALWAYS,"#END USER DEF BLOCK")
    omniout_str(ALWAYS,"#######END OF ECHO OF PROBLEM#################")
    $glob_unchanged_h_cnt   =  0
    $glob_warned   =  false
    $glob_warned2   =  false
    $glob_small_float   = $glob__0
    $glob_smallish_float   = $glob__0
    $glob_large_float   =  c(1.0e100)
    $glob_larger_float   = c( 1.1e100)
    $glob_almost_1   = c( 0.99)
    # before second block
    #TOP SECOND INPUT BLOCK
    #BEGIN SECOND INPUT BLOCK
    #BEGIN BLOCK 2
    #END FIRST INPUT BLOCK
    #BEGIN SECOND INPUT BLOCK
    x_start=c(-1.5)
    x_end=c(-2.0)
    $array_y_init[0 + 1] = exact_soln_y(x_start)
    $glob_look_poles=true
    #  ELIMINATED in preodein.rb
    #  ELIMINATED in preodein.rb
    #  ELIMINATED in preodein.rb
    #  ELIMINATED in preodein.rb
    #  ELIMINATED in preodein.rb
    #  ELIMINATED in preodein.rb
    #  ELIMINATED in preodein.rb
    $glob_type_given_pole=2
    #  ELIMINATED in preodein.rb
    $array_given_rad_poles[1][1]=c(0.0)
    #  ELIMINATED in preodein.rb
    $array_given_rad_poles[1][2]=c(1.0)
    #  ELIMINATED in preodein.rb
    $array_given_ord_poles[1][1]=c(1.0)
    #  ELIMINATED in preodein.rb
    $array_given_ord_poles[1][2]=c(0.0)
    #END SECOND INPUT BLOCK
    #BEGIN OVERRIDE  BLOCK
    $glob_desired_digits_correct=8
    $glob_max_minutes=(3.0)
    $glob_subiter_method=3
    $glob_max_iter=100000
    $glob_upper_ratio_limit=c(1.000001)
    $glob_lower_ratio_limit=c(0.999999)
    $glob_look_poles=false
    $glob_h=c(0.001)
    $glob_display_interval=c(0.01)
    #END OVERRIDE BLOCK
    #END BLOCK 2
    #END SECOND INPUT BLOCK
    #BEGIN INITS AFTER SECOND INPUT BLOCK
    $glob_last_good_h = $glob_h
    $glob_max_sec   = (60.0) * ($glob_max_minutes) + (3600.0) * ($glob_max_hours)
    #  after second input block
    $glob_check_sign = c(my_check_sign(x_start,x_end))
    $glob__pi = arccos($glob__m1)
    $glob_prec = 1.0e-16
    if ($glob_optimize)  then # if number 9
        #BEGIN OPTIMIZE CODE
        omniout_str(ALWAYS,"START of Optimize")
        #Start Series -- INITIALIZE FOR OPTIMIZE
        found_h = false
        $glob_min_pole_est =  $glob_larger_float
        last_min_pole_est =  $glob_larger_float
        $glob_least_given_sing = $glob_larger_float
        $glob_least_ratio_sing = $glob_larger_float
        $glob_least_3_sing = $glob_larger_float
        $glob_least_6_sing = $glob_larger_float
        $glob_min_h = float_abs($glob_min_h) * $glob_check_sign
        $glob_max_h = float_abs($glob_max_h) * $glob_check_sign
        $glob_h = float_abs($glob_min_h) * $glob_check_sign
        $glob_display_interval = c((float_abs(c($glob_display_interval))) * ($glob_check_sign))
        display_max = c(x_end) - c(x_start)/$glob__10
        if (($glob_display_interval) > (display_max)) then # if number 10
            $glob_display_interval = c(display_max)
        end# end if 10
        chk_data()
        min_value = $glob_larger_float
        est_answer = est_size_answer()
        opt_iter = 1
        est_needed_step_err = estimated_needed_step_error(x_start,x_end,$glob_h,est_answer)
        omniout_float(ALWAYS,"est_needed_step_err",32,est_needed_step_err,16,"")
        estimated_step_error  = $glob_small_float
        while ((opt_iter <= 100) and ( not found_h)) do # do number 1
            omniout_int(ALWAYS,"opt_iter",32,opt_iter,4,"")
            $array_x[1]   = c(x_start)
            $array_x[2]   =  c($glob_h)
            $glob_next_display = c(x_start)
            order_diff = 1
            #Start Series $array_y
            term_no   =  1
            while (term_no <= order_diff) do # do number 2
                $array_y[term_no] = $array_y_init[term_no] * expt($glob_h , c(term_no - 1)) / c(factorial_1(term_no - 1))
                term_no   =  term_no + 1
            end# end do number 2
            rows   =  order_diff
            r_order   =  1
            while (r_order <= rows) do # do number 2
                term_no   =  1
                while (term_no <= (rows - r_order + 1)) do # do number 3
                    it  = term_no + r_order - 1
                    if (term_no < ATS_MAX_TERMS) then # if number 10
                        $array_y_higher[r_order][term_no] = $array_y_init[it]* expt($glob_h , c(term_no - 1)) / (c(factorial_1(term_no - 1)))
                    end# end if 10
                    term_no   =  term_no + 1
                end# end do number 3
                r_order   =  r_order + 1
            end# end do number 2
            atomall()
            if ($glob_check_sign * $glob_min_h >= $glob_check_sign * $glob_h) then # if number 10
                omniout_str(ALWAYS,"SETTING H FOR MIN H")
                $glob_h = $glob_check_sign * float_abs($glob_min_h)
                $glob_h_reason = 1
                found_h = true
            end# end if 10
            if ($glob_check_sign * $glob_display_interval <= $glob_check_sign * $glob_h) then # if number 10
                omniout_str(ALWAYS,"SETTING H FOR DISPLAY INTERVAL")
                $glob_h_reason = 2
                $glob_h = $glob_display_interval
                found_h = true
            end# end if 10
            if ($glob_look_poles)  then # if number 10
                check_for_pole()
            end# end if 10
            if ( not found_h) then # if number 10
                est_answer = est_size_answer()
                est_needed_step_err = estimated_needed_step_error(x_start,x_end,$glob_h,est_answer)
                omniout_float(ALWAYS,"est_needed_step_err",32,est_needed_step_err,16,"")
                estimated_step_error = test_suggested_h()
                omniout_float(ALWAYS,"estimated_step_error",32,estimated_step_error,32,"")
                if (estimated_step_error < est_needed_step_err) then # if number 11
                    omniout_str(ALWAYS,"Double H and LOOP")
                    $glob_h = $glob_h*$glob__2
                else
                    omniout_str(ALWAYS,"Found H for OPTIMAL")
                    found_h = true
                    $glob_h_reason = 3
                    $glob_h = $glob_h/$glob__2
                end# end if 11
            end# end if 10
            opt_iter = opt_iter + 1
        end# end do number 1
        if (( not found_h) and (opt_iter == 1)) then # if number 10
            omniout_str(ALWAYS,"Beginning $glob_h too large.")
            found_h = false
        end# end if 10
        if ($glob_check_sign * $glob_max_h <= $glob_check_sign * $glob_h) then # if number 10
            omniout_str(ALWAYS,"SETTING H FOR MAX H")
            $glob_h = $glob_check_sign * float_abs($glob_max_h)
            $glob_h_reason = 1
            found_h = true
        end# end if 10
    else
        found_h = true
        $glob_h = $glob_h * $glob_check_sign
    end# end if 9
    #END OPTIMIZE CODE
    if ($glob_html_log)  then # if number 9
        html_log_file = File.new("entry.html","w")
    end# end if 9
    #BEGIN SOLUTION CODE
    if (found_h) then # if number 9
        omniout_str(ALWAYS,"START of Soultion")
        #Start Series -- INITIALIZE FOR SOLUTION
        $array_x[1]   = c(x_start)
        $array_x[2]   =  c($glob_h)
        $glob_next_display = c(x_start)
        $glob_min_pole_est =  $glob_larger_float
        $glob_least_given_sing = $glob_larger_float
        $glob_least_ratio_sing = $glob_larger_float
        $glob_least_3_sing = $glob_larger_float
        $glob_least_6_sing = $glob_larger_float
        order_diff = 1
        #Start Series $array_y
        term_no   =  1
        while (term_no <= order_diff) do # do number 1
            $array_y[term_no] = $array_y_init[term_no] * expt($glob_h , c(term_no - 1)) / c(factorial_1(term_no - 1))
            term_no   =  term_no + 1
        end# end do number 1
        rows   =  order_diff
        r_order   =  1
        while (r_order <= rows) do # do number 1
            term_no   =  1
            while (term_no <= (rows - r_order + 1)) do # do number 2
                it  = term_no + r_order - 1
                if (term_no < ATS_MAX_TERMS) then # if number 10
                    $array_y_higher[r_order][term_no] = $array_y_init[it]* expt($glob_h , c(term_no - 1)) / (c(factorial_1(term_no - 1)))
                end# end if 10
                term_no   =  term_no + 1
            end# end do number 2
            r_order   =  r_order + 1
        end# end do number 1
        current_iter   =  1
        $glob_clock_start_sec   = elapsed_time_seconds()
        $glob_clock_sec   =  elapsed_time_seconds()
        $glob_iter   =  0
        omniout_str(DEBUGL," ")
        $glob_reached_optimal_h   =  true
        $glob_optimal_clock_start_sec = elapsed_time_seconds()
        while (($glob_iter < $glob_max_iter)  and ($glob_check_sign * $array_x[1] < $glob_check_sign * x_end )  and  ((($glob_clock_sec) - ($glob_orig_start_sec)) < ($glob_max_sec))) do # do number 1
            #left paren 0001C
            if (reached_interval()) then # if number 10
                omniout_str(INFO," ")
                omniout_str(INFO,"TOP MAIN SOLVE Loop")
            end# end if 10
            $glob_iter   =  $glob_iter + 1
            $glob_clock_sec   = elapsed_time_seconds()
            track_estimated_error()
            atomall()
            track_estimated_error()
            display_alot(current_iter)
            if ($glob_look_poles)  then # if number 10
                check_for_pole()
            end# end if 10
            if (reached_interval()) then # if number 10
                $glob_next_display  = $glob_next_display + $glob_display_interval
            end# end if 10
            $array_x[1]   = $array_x[1] + $glob_h
            $array_x[2]   =  $glob_h
            #Jump Series $array_y
            order_diff  = 2
            #START PART 1  SUM AND ADJUST
            #START SUM AND ADJUST EQ =1
            #sum_and_adjust $array_y
            #BEFORE ADJUST SUBSERIES EQ =1
            ord = 2
            calc_term = 1
            #adjust_subseries$array_y
            iii  = ATS_MAX_TERMS
            while (iii >= calc_term)   do # do number 2
                $array_y_higher_work[2][iii] = $array_y_higher[2][iii] / expt($glob_h , c(calc_term - 1)) / c(factorial_3(iii - calc_term , iii  - 1))
                iii  = iii - 1
            end# end do number 2
            #AFTER ADJUST SUBSERIES EQ =1
            #BEFORE SUM SUBSERIES EQ =1
            temp_sum = $glob__0
            ord = 2
            calc_term = 1
            #sum_subseries$array_y
            iii  = ATS_MAX_TERMS
            while (iii >= calc_term)   do # do number 2
                temp_sum = temp_sum + $array_y_higher_work[ord][iii]
                iii  = iii - 1
            end# end do number 2
            $array_y_higher_work2[ord][calc_term] = temp_sum * expt($glob_h , c(calc_term - 1)) / c(factorial_1(calc_term - 1))
            #AFTER SUM SUBSERIES EQ =1
            #BEFORE ADJUST SUBSERIES EQ =1
            ord = 1
            calc_term = 2
            #adjust_subseries$array_y
            iii  = ATS_MAX_TERMS
            while (iii >= calc_term)   do # do number 2
                $array_y_higher_work[1][iii] = $array_y_higher[1][iii] / expt($glob_h , c(calc_term - 1)) / c(factorial_3(iii - calc_term , iii  - 1))
                iii  = iii - 1
            end# end do number 2
            #AFTER ADJUST SUBSERIES EQ =1
            #BEFORE SUM SUBSERIES EQ =1
            temp_sum = $glob__0
            ord = 1
            calc_term = 2
            #sum_subseries$array_y
            iii  = ATS_MAX_TERMS
            while (iii >= calc_term)   do # do number 2
                temp_sum = temp_sum + $array_y_higher_work[ord][iii]
                iii  = iii - 1
            end# end do number 2
            $array_y_higher_work2[ord][calc_term] = temp_sum * expt($glob_h , c(calc_term - 1)) / c(factorial_1(calc_term - 1))
            #AFTER SUM SUBSERIES EQ =1
            #BEFORE ADJUST SUBSERIES EQ =1
            ord = 1
            calc_term = 1
            #adjust_subseries$array_y
            iii  = ATS_MAX_TERMS
            while (iii >= calc_term)   do # do number 2
                $array_y_higher_work[1][iii] = $array_y_higher[1][iii] / expt($glob_h , c(calc_term - 1)) / c(factorial_3(iii - calc_term , iii  - 1))
                iii  = iii - 1
            end# end do number 2
            #AFTER ADJUST SUBSERIES EQ =1
            #BEFORE SUM SUBSERIES EQ =1
            temp_sum = $glob__0
            ord = 1
            calc_term = 1
            #sum_subseries$array_y
            iii  = ATS_MAX_TERMS
            while (iii >= calc_term)   do # do number 2
                temp_sum = temp_sum + $array_y_higher_work[ord][iii]
                iii  = iii - 1
            end# end do number 2
            $array_y_higher_work2[ord][calc_term] = temp_sum * expt($glob_h , c(calc_term - 1)) / c(factorial_1(calc_term - 1))
            #AFTER SUM SUBSERIES EQ =1
            #END SUM AND ADJUST EQ =1
            #END PART 1
            #START PART 2  MOVE TERMS to REGULAR Array
            term_no = ATS_MAX_TERMS
            while (term_no >= 1) do # do number 2
                $array_y[term_no] = $array_y_higher_work2[1][term_no]
                ord = 1
                while (ord <= order_diff)  do # do number 3
                    $array_y_higher[ord][term_no] = $array_y_higher_work2[ord][term_no]
                    ord = ord + 1
                end# end do number 3
                term_no   =  term_no - 1
            end# end do number 2
            #END PART 2 HEVE MOVED TERMS to REGULAR Array
        end# end do number 1#right paren 0001C
        omniout_str(ALWAYS,"Finished!")
        if ($glob_iter >= $glob_max_iter)  then # if number 10
            omniout_str(ALWAYS,"Maximum Iterations Reached before Solution Completed!")
        end# end if 10
        if (elapsed_time_seconds() - ($glob_orig_start_sec) >= ($glob_max_sec  )) then # if number 10
            omniout_str(ALWAYS,"Maximum Time Reached before Solution Completed!")
        end# end if 10
        $glob_clock_sec   = elapsed_time_seconds()
        omniout_str(INFO,"diff  (  y  ,  x  ,  1  )   =  1.0 /   ( x  *   x  +  1.0 )   ; ")
        omniout_int(INFO,"Iterations                      ",32,$glob_iter,4," ")
        prog_report(x_start,x_end)
        if ($glob_html_log)  then # if number 10
            logstart(html_log_file)
            logitem_str(html_log_file,"2015-04-08T21:35:09-05:00")
            logitem_str(html_log_file,"Ruby")
            logitem_str(html_log_file,"<a href=\"sing2_back.ode.txt\">sing2_back</a>")
            logitem_str(html_log_file,"diff  (  y  ,  x  ,  1  )   =  1.0 /   ( x  *   x  +  1.0 )   ; ")
            logitem_float(html_log_file,x_start)
            logitem_float(html_log_file,x_end)
            logitem_float(html_log_file,$array_x[1])
            logitem_float(html_log_file,$glob_h)
            logitem_h_reason(html_log_file)
            logitem_str(html_log_file,"16")
            logitem_float(html_log_file,$glob_desired_digits_correct)
            if ($array_est_digits[1] != -16) then # if number 11
                logitem_integer(html_log_file,$array_est_digits[1])
            else
                logitem_str(html_log_file,"Unknown")
            end# end if 11
            if ($glob_min_good_digits != -16) then # if number 11
                logitem_integer(html_log_file,$glob_min_good_digits)
            else
                logitem_str(html_log_file,"Unknown")
            end# end if 11
            if ($glob_good_digits != -16) then # if number 11
                logitem_integer(html_log_file,$glob_good_digits)
            else
                logitem_str(html_log_file,"Unknown")
            end# end if 11
            logitem_str(html_log_file,"NA")
            logitem_str(html_log_file,"NA")
            logitem_integer(html_log_file,ATS_MAX_TERMS)
            if ($glob_type_given_pole  == 0) then # if number 11
                logitem_str(html_log_file,"Not Given")
                logitem_str(html_log_file,"NA")
            elsif
            ($glob_type_given_pole  == 4) then # if number 12
                logitem_str(html_log_file,"No Solution")
                logitem_str(html_log_file,"NA")
            elsif
            ($glob_type_given_pole  == 5) then # if number 13
                logitem_str(html_log_file,"Some Pole")
                logitem_str(html_log_file,"????")
            elsif
            ($glob_type_given_pole  == 3) then # if number 14
                logitem_str(html_log_file,"No Pole")
                logitem_str(html_log_file,"NA")
            elsif
            ($glob_type_given_pole  == 1) then # if number 15
                logitem_str(html_log_file,"Real Sing")
                logitem_float(html_log_file,$glob_least_given_sing)
            elsif
            ($glob_type_given_pole  == 2) then # if number 16
                logitem_str(html_log_file,"Complex Sing")
                logitem_float(html_log_file,$glob_least_given_sing)
            end# end if 16
            if ($glob_least_ratio_sing < $glob_large_float) then # if number 16
                logitem_float(html_log_file,$glob_least_ratio_sing)
            else
                logitem_str(html_log_file,"NONE")
            end# end if 16
            if ($glob_least_3_sing < $glob_large_float) then # if number 16
                logitem_float(html_log_file,$glob_least_3_sing)
            else
                logitem_str(html_log_file,"NONE")
            end# end if 16
            if ($glob_least_6_sing < $glob_large_float) then # if number 16
                logitem_float(html_log_file,$glob_least_6_sing)
            else
                logitem_str(html_log_file,"NONE")
            end# end if 16
            logitem_integer(html_log_file,$glob_iter)
            logitem_time(html_log_file,($glob_clock_sec))
            if (c($glob_percent_done) < $glob__100)  then # if number 16
                logitem_time(html_log_file,($glob_total_exp_sec))
                0
            else
                logitem_str(html_log_file,"Done")
                0
            end# end if 16
            log_revs(html_log_file,"<td> 305.fix.digits.div.zero </td>")
            logitem_str(html_log_file,"<a href=\"diffeq.sing2_back.rb.txt\">sing2_back diffeq.rb</a>")
            logitem_str(html_log_file,"<a href=\"results.sing2_back.rb.txt\">sing2_back Ruby results</a>")
            logitem_str(html_log_file,"OK")
            logend(html_log_file)
        end# end if 15
        if ($glob_html_log)  then # if number 15
            html_log_file.close;
        end# end if 15
    end# end if 14
    #END OUTFILEMAIN
end # End Function number 12
main()