Laboratoire de l'Informatique et du Parallélisme » Pobyso

"""

Actual functions to use in Sage

ST 2012-11-13

Command line syntax:

  use from Sage (via the "load" or the "attach" commands)

pobyso functions come in five flavors: 

- the _so_so (arguments and returned objects are pointers to Sollya objects, includes

  the void function and the no arguments function that return a pointer to a Sollya

  object);

- the _so_sa (argument are pointers to Sollya objects, returned objects are

  Sage/Python objects or, more generally, information is transfered from the Sollya

  world to Sage/Python world);

- the _sa_so (arguments are Sage/Python objects, returned objects are 

  pointers to Sollya objects);

- the sa_sa (arguments and returned objects are all Sage/Python objects);

- a catch all flavor without suffix.

NOTES:

Reported errors in Eclipse come from the calls to

the Sollya library

ToDo (among other things): 

 -memory management.

"""

from ctypes import *

import re

from sage.symbolic.expression_conversions import polynomial

"""

Create the equivalent to an enum for the Sollya function types.

"""

(SOLLYA_BASE_FUNC_ABS,

SOLLYA_BASE_FUNC_ACOS,

    SOLLYA_BASE_FUNC_ACOSH,

    SOLLYA_BASE_FUNC_ADD,

    SOLLYA_BASE_FUNC_ASIN,

    SOLLYA_BASE_FUNC_ASINH,

    SOLLYA_BASE_FUNC_ATAN,

    SOLLYA_BASE_FUNC_ATANH,

    SOLLYA_BASE_FUNC_CEIL,

    SOLLYA_BASE_FUNC_CONSTANT,

    SOLLYA_BASE_FUNC_COS,

    SOLLYA_BASE_FUNC_COSH,

    SOLLYA_BASE_FUNC_DIV,

    SOLLYA_BASE_FUNC_DOUBLE,

    SOLLYA_BASE_FUNC_DOUBLEDOUBLE,

    SOLLYA_BASE_FUNC_DOUBLEEXTENDED,

    SOLLYA_BASE_FUNC_ERF,

    SOLLYA_BASE_FUNC_ERFC,

    SOLLYA_BASE_FUNC_EXP,

    SOLLYA_BASE_FUNC_EXP_M1,

    SOLLYA_BASE_FUNC_FLOOR,

    SOLLYA_BASE_FUNC_FREE_VARIABLE,

    SOLLYA_BASE_FUNC_HALFPRECISION,

    SOLLYA_BASE_FUNC_LIBRARYCONSTANT,

    SOLLYA_BASE_FUNC_LIBRARYFUNCTION,

    SOLLYA_BASE_FUNC_LOG,

    SOLLYA_BASE_FUNC_LOG_10,

    SOLLYA_BASE_FUNC_LOG_1P,

    SOLLYA_BASE_FUNC_LOG_2,

    SOLLYA_BASE_FUNC_MUL,

    SOLLYA_BASE_FUNC_NEARESTINT,

    SOLLYA_BASE_FUNC_NEG,

    SOLLYA_BASE_FUNC_PI,

    SOLLYA_BASE_FUNC_POW,

    SOLLYA_BASE_FUNC_PROCEDUREFUNCTION,

    SOLLYA_BASE_FUNC_QUAD,

    SOLLYA_BASE_FUNC_SIN,

    SOLLYA_BASE_FUNC_SINGLE,

    SOLLYA_BASE_FUNC_SINH,

    SOLLYA_BASE_FUNC_SQRT,

    SOLLYA_BASE_FUNC_SUB,

    SOLLYA_BASE_FUNC_TAN,

    SOLLYA_BASE_FUNC_TANH,

SOLLYA_BASE_FUNC_TRIPLEDOUBLE) = map(int,xrange(44))

print "First constant - SOLLYA_BASE_FUNC_ABS: ", SOLLYA_BASE_FUNC_ABS

print "Last constant  - SOLLYA_BASE_FUNC_TRIPLEDOUBLE: ", SOLLYA_BASE_FUNC_TRIPLEDOUBLE

pobyso_max_arity = 9

def pobyso_autoprint(arg):

    sollya_lib_autoprint(arg,None)

def pobyso_autoprint_so_so(arg):

    sollya_lib_autoprint(arg,None)

def pobyso_cmp(rnArg, soCte):

    precisionOfCte = c_int(0)

    # From the Sollya constant, create a local Sage RealNumber.

    sollya_lib_get_prec_of_constant(precisionOfCte, soCte) 

    #print "Precision of constant: ", precisionOfCte

    RRRR = RealField(precisionOfCte.value)

    rnLocal = RRRR(0)

    sollya_lib_get_constant(get_rn_value(rnLocal), soCte)

    #print "rnDummy: ", rnDummy

    # Compare the local Sage RealNumber with rnArg.

    return(cmp_rn_value(rnArg, rnLocal))

def pobyso_constant(rnArg):

    """ Legacy function. See pobyso_constant_sa_so. """

    return(pobyso_constant_sa_so(rnArg))

def pobyso_constant_sa_so(rnArg):

    return(sollya_lib_constant(get_rn_value(rnArg)))

def pobyso_constant_1():

    """ Legacy function. See pobyso_constant_so_so. """

    return(pobyso_constant_1_so_so())

def pobyso_constant_1_so_so():

    return(pobyso_constant_from_int_sa_so(1))

def pobyso_constant_from_int(anInt):

    """ Legacy function. See pobyso_constant_from_int_sa_so. """

    return(pobyso_constant_from_int_sa_so(anInt))

def pobyso_constant_from_int_sa_so(anInt):

    return(sollya_lib_constant_from_int(int(anInt)))

def pobyso_function_type_as_string(funcType):

    """ Legacy function. See pobyso_function_type_as_string_so_sa. """

    return(pobyso_function_type_as_string_so_sa(funcType))

def pobyso_function_type_as_string_so_sa(funcType):

    """

    Numeric Sollya function codes -> Sage mathematical function names.

    Notice that pow -> ^ (a la Sage, not a la Python).

    """

    if funcType == SOLLYA_BASE_FUNC_ABS:

        return "abs"

    elif funcType == SOLLYA_BASE_FUNC_ACOS:

        return "arccos"

    elif funcType == SOLLYA_BASE_FUNC_ACOSH:

        return "arccosh"

    elif funcType == SOLLYA_BASE_FUNC_ADD:

        return "+"

    elif funcType == SOLLYA_BASE_FUNC_ASIN:

        return "arcsin"

    elif funcType == SOLLYA_BASE_FUNC_ASINH:

        return "arcsinh"

    elif funcType == SOLLYA_BASE_FUNC_ATAN:

        return "arctan"

    elif funcType == SOLLYA_BASE_FUNC_ATANH:

        return "arctanh"

    elif funcType == SOLLYA_BASE_FUNC_CEIL:

        return "ceil"

    elif funcType == SOLLYA_BASE_FUNC_CONSTANT:

        return "cte"

    elif funcType == SOLLYA_BASE_FUNC_COS:

        return "cos"

    elif funcType == SOLLYA_BASE_FUNC_COSH:

        return "cosh"

    elif funcType == SOLLYA_BASE_FUNC_DIV:

        return "/"

    elif funcType == SOLLYA_BASE_FUNC_DOUBLE:

        return "double"

    elif funcType == SOLLYA_BASE_FUNC_DOUBLEDOUBLE:

        return "doubleDouble"

    elif funcType == SOLLYA_BASE_FUNC_DOUBLEEXTENDED:

        return "doubleDxtended"

    elif funcType == SOLLYA_BASE_FUNC_ERF:

        return "erf"

    elif funcType == SOLLYA_BASE_FUNC_ERFC:

        return "erfc"

    elif funcType == SOLLYA_BASE_FUNC_EXP:

        return "exp"

    elif funcType == SOLLYA_BASE_FUNC_EXP_M1:

        return "expm1"

    elif funcType == SOLLYA_BASE_FUNC_FLOOR:

        return "floor"

    elif funcType == SOLLYA_BASE_FUNC_FREE_VARIABLE:

        return "freeVariable"

    elif funcType == SOLLYA_BASE_FUNC_HALFPRECISION:

        return "halfPrecision"

    elif funcType == SOLLYA_BASE_FUNC_LIBRARYCONSTANT:

        return "libraryConstant"

    elif funcType == SOLLYA_BASE_FUNC_LIBRARYFUNCTION:

        return "libraryFunction"

    elif funcType == SOLLYA_BASE_FUNC_LOG:

        return "log"

    elif funcType == SOLLYA_BASE_FUNC_LOG_10:

        return "log10"

    elif funcType == SOLLYA_BASE_FUNC_LOG_1P:

        return "log1p"

    elif funcType == SOLLYA_BASE_FUNC_LOG_2:

        return "log2"

    elif funcType == SOLLYA_BASE_FUNC_MUL:

        return "*"

    elif funcType == SOLLYA_BASE_FUNC_NEARESTINT:

        return "round"

    elif funcType == SOLLYA_BASE_FUNC_NEG:

        return "__neg__"

    elif funcType == SOLLYA_BASE_FUNC_PI:

        return "pi"

    elif funcType == SOLLYA_BASE_FUNC_POW:

        return "^"

    elif funcType == SOLLYA_BASE_FUNC_PROCEDUREFUNCTION:

        return "procedureFunction"

    elif funcType == SOLLYA_BASE_FUNC_QUAD:

        return "quad"

    elif funcType == SOLLYA_BASE_FUNC_SIN:

        return "sin"

    elif funcType == SOLLYA_BASE_FUNC_SINGLE:

        return "single"

    elif funcType == SOLLYA_BASE_FUNC_SINH:

        return "sinh"

    elif funcType == SOLLYA_BASE_FUNC_SQRT:

        return "sqrt"

    elif funcType == SOLLYA_BASE_FUNC_SUB:

        return "-"

    elif funcType == SOLLYA_BASE_FUNC_TAN:

        return "tan"

    elif funcType == SOLLYA_BASE_FUNC_TANH:

        return "tanh"

    elif funcType == SOLLYA_BASE_FUNC_TRIPLEDOUBLE:

        return "tripleDouble"

    else:

        return None

def pobyso_get_constant(rnArg, soConst):

    """ Legacy function. See pobyso_get_constant_so_sa. """

    pobyso_get_constant_so_sa(rnArg, soConst)

def pobyso_get_constant_so_sa(rnArg, soConst):

    set_rn_value(rnArg, soConst)

def pobyso_get_constant_as_rn(ctExp):

    """ Legacy function. See pobyso_get_constant_as_rn_so_sa. """ 

    return(pobyso_get_constant_as_rn_so_sa(ctExp))

def pobyso_get_constant_as_rn_so_sa(ctExp):

    precision  = pobyso_get_prec_of_constant(ctExp) 

    RRRR = RealField(precision)

    rn = RRRR(0)

    sollya_lib_get_constant(get_rn_value(rn), ctExp)

    return(rn)

def pobyso_get_constant_as_rn_with_rf(ctExp, realField):

    """ Legacy function. See ."""

    return(pobyso_get_constant_as_rn_with_rf_so_sa(ctExp, realField))

def pobyso_get_constant_as_rn_with_rf_so_sa(ctExp, realField):

    rn = realField(0)

    sollya_lib_get_constant(get_rn_value(rn), ctExp)

    return(rn)

def pobyso_get_free_variable_name():

    """ Legacy function. See pobyso_get_free_variable_name_so_sa."""

    return(pobyso_get_free_variable_name_so_sa())

def pobyso_get_free_variable_name_so_sa():

    return(sollya_lib_get_free_variable_name())

def pobyso_get_function_arity(expressionSo):

    """ Legacy function. See pobyso_get_function_arity_so_sa."""

    return(pobyso_get_function_arity_so_sa(expressionSo))

def pobyso_get_function_arity_so_sa(expressionSo):

    arity = c_int(0)

    sollya_lib_get_function_arity(byref(arity),expressionSo)

    return(int(arity.value))

def pobyso_get_head_function(expressionSo):

    """ Legacy function. See pobyso_get_head_function_so_sa. """

    return(pobyso_get_head_function_so_sa(expressionSo)) 

def pobyso_get_head_function_so_sa(expressionSo):

    functionType = c_int(0)

    sollya_lib_get_head_function(byref(functionType), expressionSo, None)

    return(int(functionType.value))

def pobyso_get_list_elements(soObj):

    """ Legacy function. See pobyso_get_list_elements_so_so. """

    return(pobyso_get_list_elements_so_so(soObj))

def pobyso_get_list_elements_so_so(soObj):

    # Type for array of pointers to sollya_obj_t

    listAddress = POINTER(c_longlong)()

    numElements = c_int(0)

    isEndElliptic = c_int(0)

    listAsList = []

    result = sollya_lib_get_list_elements(byref(listAddress),\

                                        byref(numElements),\

                                        byref(isEndElliptic),\

                                        soObj)

    if result == 0 :

        return None

    for i in xrange(0, numElements.value, 1):

        #print "address ", i, " ->", listAddress[i]

        listAsList.append(listAddress[i])

    return(listAsList, numElements.value, isEndElliptic.value)

def pobyso_get_max_prec_of_exp(soExp):

    """ Legacy function. See pobyso_get_max_prec_of_exp_so_sa. """

    return(pobyso_get_max_prec_of_exp_so_sa(soExp))

def pobyso_get_max_prec_of_exp_so_sa(soExp):

    """

    Get the maximum precision used for the numbers in a Sollya expression.

    TODO: 

    - error management;

    - correctly deal with numerical type such as DOUBLEEXTENDED.

    """

    maxPrecision = 0

    operator = pobyso_get_head_function_so_sa(soExp)

    if (operator != SOLLYA_BASE_FUNC_CONSTANT) and \

    (operator != SOLLYA_BASE_FUNC_FREE_VARIABLE):

        (arity, subexpressions) = pobyso_get_subfunctions_so_sa(soExp)

        for i in xrange(arity):

            maxPrecisionCandidate = \

                pobyso_get_max_prec_of_exp_so_sa(subexpressions[i])

            if maxPrecisionCandidate > maxPrecision:

                maxPrecision = maxPrecisionCandidate

        return(maxPrecision)

    elif operator == SOLLYA_BASE_FUNC_CONSTANT:

        #print pobyso_get_prec_of_constant(soExp)

        return(pobyso_get_prec_of_constant_so_sa(soExp))

    elif operator == SOLLYA_BASE_FUNC_FREE_VARIABLE:

        return(0)

    else:

        print "pobyso_get_max_prec_of_exp_so_sa: unexepected operator."

        return(0)

def pobyso_get_sage_exp_from_sollya_exp(sollyaExp, realField = RR):

    """ Legacy function. See pobyso_get_sage_exp_from_sollya_exp_so_sa. """

    return(pobyso_get_sage_exp_from_sollya_exp_so_sa(sollyaExp, realField = RR))

def pobyso_get_sage_exp_from_sollya_exp_so_sa(sollyaExp, realField = RR):

    """

    Get a Sage expression from a Sollya expression. 

    Currently only tested with polynomials with floating-point coefficients.

    Notice that, in the returned polynomial, the exponents are RealNumbers.

    """

    #pobyso_autoprint(sollyaExp)

    operator = pobyso_get_head_function(sollyaExp)

    # Constants and the free variable are special cases.

    # All other operator are dealt with in the same way.

    if (operator != SOLLYA_BASE_FUNC_CONSTANT) and \

       (operator != SOLLYA_BASE_FUNC_FREE_VARIABLE):

        (arity, subexpressions) = pobyso_get_subfunctions_so_sa(sollyaExp)

        if arity == 1:

            sageExp = eval(pobyso_function_type_as_string_so_sa(operator) + \

            "(" + pobyso_get_sage_exp_from_sollya_exp_so_sa(subexpressions[0], realField)\

             + ")")

        elif arity == 2:

            if operator == SOLLYA_BASE_FUNC_POW:

                operatorAsString = "**"

            else:

                operatorAsString = pobyso_function_type_as_string_so_sa(operator)

            sageExp = \

              eval("pobyso_get_sage_exp_from_sollya_exp_so_sa(subexpressions[0], realField)"\

              + " " + operatorAsString + " " + \

                   "pobyso_get_sage_exp_from_sollya_exp_so_sa(subexpressions[1], realField)")

        # We do not know yet how to deal with arity > 3 (is there any in Sollya anyway?).

        else:

            sageExp = eval('None')

        return(sageExp)

    elif operator == SOLLYA_BASE_FUNC_CONSTANT:

        #print "This is a constant"

        return pobyso_get_constant_as_rn_with_rf_so_sa(sollyaExp, realField)

    elif operator == SOLLYA_BASE_FUNC_FREE_VARIABLE:

        #print "This is free variable"

        return(eval(sollya_lib_get_free_variable_name()))

    else:

        print "Unexpected"

        return eval('None')

# End pobyso_get_sage_poly_from_sollya_poly

def pobyso_get_subfunctions(expressionSo):

    """ Legacy function. See pobyso_get_subfunctions_so_sa. """

    return(pobyso_get_subfunctions_so_sa(expressionSo)) 

def pobyso_get_subfunctions_so_sa(expressionSo):

    """

    Get the subfunctions of an expression.

    Return the number of subfunctions and the list of subfunctions addresses.

    Could not figure out another way than that ugly list of declarations

    to recover the addresses of the subfunctions.

    Arity is limited to 9.

    """

    subf0 = c_int(0)

    subf1 = c_int(0)

    subf2 = c_int(0)

    subf3 = c_int(0)

    subf4 = c_int(0)

    subf5 = c_int(0)

    subf6 = c_int(0)

    subf7 = c_int(0)

    subf8 = c_int(0)

    arity = c_int(0)

    nullPtr = POINTER(c_int)()

    sollya_lib_get_subfunctions(expressionSo, byref(arity), \

    byref(subf0), byref(subf1), byref(subf2), byref(subf3), byref(subf4), byref(subf5),\

     byref(subf6), byref(subf7), byref(subf8), nullPtr, None) 

#    byref(cast(subfunctions[0], POINTER(c_int))), byref(cast(subfunctions[0], POINTER(c_int))), \

#    byref(cast(subfunctions[2], POINTER(c_int))), byref(cast(subfunctions[3], POINTER(c_int))), \

#    byref(cast(subfunctions[4], POINTER(c_int))), byref(cast(subfunctions[5], POINTER(c_int))), \

#    byref(cast(subfunctions[6], POINTER(c_int))), byref(cast(subfunctions[7], POINTER(c_int))), \

#    byref(cast(subfunctions[8], POINTER(c_int))), nullPtr)

    subfunctions = [subf0, subf1, subf2, subf3, subf4, subf5, subf6, subf7, subf8]

    subs = []

    if arity.value > pobyso_max_arity:

        return(0,[])

    for i in xrange(arity.value):

        subs.append(int(subfunctions[i].value))

        #print subs[i]

    return(int(arity.value), subs)

def pobyso_get_prec():

    """ Legacy function. See pobyso_get_prec_so_sa(). """

    return(pobyso_get_prec_so_sa())

def pobyso_get_prec_so_sa():

    """

    Get the current default precision in Sollya.

    The return value is Sage/Python int.

    """

    retc = sollya_lib_get_prec(None)

    a = c_int(0)

    sollya_lib_get_constant_as_int(byref(a), retc)

    return(int(a.value))

def pobyso_get_prec_of_constant(ctExpSo):

    """ Legacy function. See pobyso_get_prec_of_constant_so_sa. """

    return(pobyso_get_prec_of_constant_so_sa(ctExpSo))

def pobyso_get_prec_of_constant_so_sa(ctExpSo):

    prec = c_int(0)

    retc = sollya_lib_get_prec_of_constant(byref(prec), ctExpSo, None)

    return(int(prec.value))

def pobyso_lib_init():

    sollya_lib_init(None)

def pobyso_name_free_variable(freeVariableName):

    """ Legacy function. See pobyso_name_free_variable_sa_so. """

    pobyso_name_free_variable_sa_so(freeVariableName)

def pobyso_name_free_variable_sa_so(freeVariableName):

    sollya_lib_name_free_variable(freeVariableName)

def pobyso_parse_string(string):

    """ Legacy function. See pobyso_parse_string_sa_so. """

    return(pobyso_parse_string_sa_so(string))

def pobyso_parse_string_sa_so(string):

    return(sollya_lib_parse_string(string))

def pobyso_range(rnLowerBound, rnUpperBound):

    """ Legacy function. See pobyso_range_sa_so. """

    pobyso_range_sa_so(rnLowerBound, rnUpperBound) 

def pobyso_range_sa_so(rnLowerBound, rnUpperBound):

    lowerBoundSo = sollya_lib_constant(get_rn_value(rnLowerBound))

    upperBoundSo = sollya_lib_constant(get_rn_value(rnUpperBound))

    rangeSo = sollya_lib_range(lowerBoundSo, upperBoundSo)

    return(rangeSo)

def pobyso_remez_canonical(func, \

                           degree, \

                           lowerBound, \

                           upperBound, \

                           weight = "1", \

                           quality = None):

    """ Legacy function. See pobyso_remez_canonical_sa_so. """

def pobyso_remez_canonical_sa_so(func, \

                                 degree, \

                                 lowerBound, \

                                 upperBound, \

                                 weight = "1", \

                                 quality = None):

    """

    All arguments are Sage/Python.

    The weight function must be passed as string. A Sage function will fail.

    The return value is a pointer to a Sollya function.

    """

    if parent(func) == parent("string"):

        functionSo = sollya_lib_parse_string(func)

    else:

        return(None)

    degreeSo = pobyso_constant_from_int(degree)

    rangeSo = pobyso_range_sa_so(lowerBound, upperBound)

    weightSo = pobyso_parse_string_sa_so(weight)

    if not quality is None:

        qualitySo= pobyso_constant_sa_so(quality)

    return(sollya_lib_remez(functionSo, degreeSo, rangeSo, weightSo, qualitySo, None))

def pobyso_remez_canonical_so_so(funcSo, \

                                 degreeSo, \

                                 rangeSo, \

                                 weightSo = pobyso_constant_1_so_so(),\

                                 qualitySo = None):

    """

    All arguments are pointers to Sollya objects.

    The return value is a pointer to a Sollya function.

    """

    if not sollya_lib_obj_is_function(funcSo):

        return(None)

    return(sollya_lib_remez(funcSo, degreeSo, rangeSo, weightSo, qualitySo, None))

def pobyso_set_canonical_off():

    sollya_lib_set_canonical(sollya_lib_off())

def pobyso_set_canonical_on():

    sollya_lib_set_canonical(sollya_lib_on())

def pobyso_set_prec(p):

    """ Legacy function. See pobyso_set_prec_sa_so. """

    return( pobyso_set_prec_sa_so(p))

def pobyso_set_prec_sa_so(p):

    a = c_int(p)

    precSo = c_void_p(sollya_lib_constant_from_int(a))

    sollya_lib_set_prec(precSo)

def pobyso_taylor(function, degree, point):

    """ Legacy function. See pobysoTaylor_so_so. """

    return(pobyso_taylor_so_so(function, degree, point))

def pobyso_taylor_so_so(function, degree, point):

    return(sollya_lib_taylor(function, degree, point))

def pobyso_taylorform(function, degree, point = None, interval = None, errorType=None):

    """ Legacy function. See ;"""

def pobyso_taylorform_sa_sa(functionSa, \

                            degree, \

                            point = None, \

                            interval = None, \

                            errorType=None):

    """

    Compute the Taylor form of 'degree' for 'functionSa' at 'point' 

    for 'interval' with 'errorType'. 

    point: must be a Real or a Real interval.

    return the Taylor form as an array

    TODO: take care of the interval and of point when it is an interval;

          when errorType is not None;

          take care of the other elements of the Taylor form (coefficients errors and

          delta.

    """

    # Absolute as the default error.

    if errorType is None:

        errorTypeSo = sollya_lib_absolute()

    else:

        #TODO: deal with the other case.

        pass

    varSa = functionSa.variables()[0]

    pointBaseRingString = str(point.base_ring())

    if not re.search('Real', pointBaseRingString):

        return None

    # Call Sollya but first "sollyafy" the arguments.

    sollya_lib_init(None)

    pobyso_name_free_variable_sa_so(str(varSa))

    #pobyso_set_prec_sa_so(300)

    # Sollyafy the function.

    functionSo = pobyso_parse_string_sa_so(functionSa._assume_str())

    if sollya_lib_obj_is_error(functionSo):

        print "pobyso_tailorform: function string can't be parsed!"

        return None

    # Sollyafy the degree

    degreeSo = sollya_lib_constant_from_int(int(degree))

    # Sollyafy the point

    if not re.search('Interval', pointBaseRingString):

        pointSo  = pobyso_constant_sa_so(point)

    else:

        # TODO: deal with the interval case.

        pass

    # Call Sollya

    taylorFormSo = sollya_lib_taylorform(functionSo, degreeSo, pointSo, errorTypeSo,\

                                         None)

    (tfsAsList, numElements, isEndElliptic) = \

            pobyso_get_list_elements_so_so(taylorFormSo)

    polySo = tfsAsList[0]

    maxPrecision = pobyso_get_max_prec_of_exp_so_sa(polySo)

    polyRealField = RealField(maxPrecision)

    expSa = pobyso_get_sage_exp_from_sollya_exp_so_sa(polySo, polyRealField)

    sollya_lib_close()

    polynomialRing = polyRealField[str(varSa)]

    polySa = polynomial(expSa, polynomialRing)

    taylorFormSa = [polySa]

    return(taylorFormSa)                    

def pobyso_univar_polynomial_print_reverse(polySa):

    """ Legacy function. See pobyso_univar_polynomial_print_reverse_so_so. """

    return(pobyso_univar_polynomial_print_reverse_so_so(polySa))

def pobyso_univar_polynomial_print_reverse_so_so(polySa):

    """

    Return the string representation of a univariate polynomial with

    monomials ordered in the x^0..x^n order of the monomials.

    Remember: Sage

    """

    polynomialRing = polySa.base_ring()

    # A very expensive solution:

    # -create a fake multivariate polynomial field with only one variable,

    #   specifying a negative lexicographical order;

    mpolynomialRing = PolynomialRing(polynomialRing.base(), \

                                     polynomialRing.variable_name(), \

                                     1, order='neglex')

    # - convert the univariate argument polynomial into a multivariate

    #   version;

    p = mpolynomialRing(polySa)

    # - return the string representation of the converted form.

    # There is no simple str() method defined for p's class.

    return(p.__str__())

#

print "Superficial test of pobyso:"    

print pobyso_get_prec()  

pobyso_set_prec(165)

print pobyso_get_prec()  

a=100

print type(a)

id(a)

print "Max arity: ", pobyso_max_arity

print "Function tripleDouble (43) as a string: ", pobyso_function_type_as_string(43)

print "Function None (44) as a string: ", pobyso_function_type_as_string(44)