<SYSTEM_TASK:>
Solve the following problem using Python, implementing the functions described below, one line at a time
<END_TASK>
<USER_TASK:>
Description:
 def encode(cls, public_key, scalar, precision=None, max_exponent=None):
 """Return an encoding of an int or float. This encoding is carefully chosen so that it supports the same operations as the Paillier cryptosystem. If *scalar* is a float, first approximate it as an int, `int_rep`: scalar = int_rep * (:attr:`BASE` ** :attr:`exponent`), for some (typically negative) integer exponent, which can be tuned using *precision* and *max_exponent*. Specifically, :attr:`exponent` is chosen to be equal to or less than *max_exponent*, and such that the number *precision* is not rounded to zero. Having found an integer representation for the float (or having been given an int `scalar`), we then represent this integer as a non-negative integer < :attr:`~PaillierPublicKey.n`. Paillier homomorphic arithemetic works modulo :attr:`~PaillierPublicKey.n`. We take the convention that a number x < n/3 is positive, and that a number x > 2n/3 is negative. The range n/3 < x < 2n/3 allows for overflow detection. Args: public_key (PaillierPublicKey):
 public key for which to encode (this is necessary because :attr:`~PaillierPublicKey.n` varies). scalar: an int or float to be encrypted. If int, it must satisfy abs(*value*) < :attr:`~PaillierPublicKey.n`/3. If float, it must satisfy abs(*value* / *precision*) << :attr:`~PaillierPublicKey.n`/3 (i.e. if a float is near the limit then detectable overflow may still occur) precision (float):
 Choose exponent (i.e. fix the precision) so that this number is distinguishable from zero. If `scalar` is a float, then this is set so that minimal precision is lost. Lower precision leads to smaller encodings, which might yield faster computation. max_exponent (int):
 Ensure that the exponent of the returned `EncryptedNumber` is at most this. Returns: EncodedNumber: Encoded form of *scalar*, ready for encryption against *public_key*. """

        # Calculate the maximum exponent for desired precision
        if precision is None:
            if isinstance(scalar, int):
                prec_exponent = 0
            elif isinstance(scalar, float):
                # Encode with *at least* as much precision as the python float
                # What's the base-2 exponent on the float?
                bin_flt_exponent = math.frexp(scalar)[1]

                # What's the base-2 exponent of the least significant bit?
                # The least significant bit has value 2 ** bin_lsb_exponent
                bin_lsb_exponent = bin_flt_exponent - cls.FLOAT_MANTISSA_BITS

                # What's the corresponding base BASE exponent? Round that down.
                prec_exponent = math.floor(bin_lsb_exponent / cls.LOG2_BASE)
            else:
                raise TypeError("Don't know the precision of type %s."
                                % type(scalar))
        else:
            prec_exponent = math.floor(math.log(precision, cls.BASE))

        # Remember exponents are negative for numbers < 1.
        # If we're going to store numbers with a more negative
        # exponent than demanded by the precision, then we may
        # as well bump up the actual precision.
        if max_exponent is None:
            exponent = prec_exponent
        else:
            exponent = min(max_exponent, prec_exponent)

        # Use rationals instead of floats to avoid overflow.
        int_rep = round(fractions.Fraction(scalar)
                        * fractions.Fraction(cls.BASE) ** -exponent)

        if abs(int_rep) > public_key.max_int:
            raise ValueError('Integer needs to be within +/- %d but got %d'
                             % (public_key.max_int, int_rep))

        # Wrap negative numbers by adding n
        return cls(public_key, int_rep % public_key.n, exponent)