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# Source code for skopt.sampler.sobol

    """
      Authors:
        Original FORTRAN77 version of i4_sobol by Bennett Fox.
        MATLAB version by John Burkardt.
        PYTHON version by Corrado Chisari

        Original Python version of is_prime by Corrado Chisari

        Original MATLAB versions of other functions by John Burkardt.
        PYTHON versions by Corrado Chisari

        Modified Python version by Holger Nahrstaedt

        Original code is available from
        http://people.sc.fsu.edu/~jburkardt/py_src/sobol/sobol.html
    """

    import warnings

    import numpy as np
    from sklearn.utils import check_random_state

    from ..space import Space
    from .base import InitialPointGenerator

    class Sobol(InitialPointGenerator):
        """Generates a new quasirandom Sobol' vector with each call.

        Parameters
        ----------
        skip : int
            Skipped seed number.

        randomize : bool, default=False
            When set to True, random shift is applied.

        Notes
        -----
        Sobol' sequences [1]_ provide :math:`n=2^m` low discrepancy points in
        :math:`[0,1)^{dim}`. Scrambling them makes them suitable for singular
        integrands, provides a means of error estimation, and can improve their
        rate of convergence.

        There are many versions of Sobol' sequences depending on their
        'direction numbers'. Here, the maximum number of dimension is 40.

        The routine adapts the ideas of Antonov and Saleev [2]_.

        .. warning::

           Sobol' sequences are a quadrature rule and they lose their balance
           properties if one uses a sample size that is not a power of 2, or skips
           the first point, or thins the sequence [5]_.

           If :math:`n=2^m` points are not enough then one should take :math:`2^M`
           points for :math:`M>m`. When scrambling, the number R of independent
           replicates does not have to be a power of 2.

           Sobol' sequences are generated to some number :math:`B` of bits. Then
           after :math:`2^B` points have been generated, the sequence will repeat.
           Currently :math:`B=30`.

        References
        ----------
        .. [1] I. M. Sobol. The distribution of points in a cube and the accurate
           evaluation of integrals. Zh. Vychisl. Mat. i Mat. Phys., 7:784-802,
           1967.

        .. [2] Antonov, Saleev,
           USSR Computational Mathematics and Mathematical Physics,
           Volume 19, 1980, pages 252 - 256.

        .. [3] Paul Bratley, Bennett Fox,
           Algorithm 659:
           Implementing Sobol's Quasirandom Sequence Generator,
           ACM Transactions on Mathematical Software,
           Volume 14, Number 1, pages 88-100, 1988.

        .. [4] Bennett Fox,
           Algorithm 647:
           Implementation and Relative Efficiency of Quasirandom
           Sequence Generators,

        .. [5] Art B. Owen. On dropping the first Sobol' point. arXiv 2008.08051,
           2020.
        """

        def __init__(self, skip=0, randomize=True):

            if not (skip & (skip - 1) == 0):
                raise ValueError(
                    "The balance properties of Sobol' points require"
                    " skipping a power of 2."
                )
            if skip != 0:
                warnings.warn(
                    f"{skip} points have been skipped: "
                    f"{skip} points can be generated before the "
                    f"sequence repeats."
                )
            self.skip = skip

            self.num_generated = 0
            self.randomize = randomize

            self.dim_max = 40
            self.log_max = 30
            self.atmost = 2**self.log_max - 1
            self.lastq = None
            self.maxcol = None
            self.poly = None
            self.recipd = None
            self.seed_save = -1
            self.v = np.zeros((self.dim_max, self.log_max))
            self.dim_num_save = -1

        def init(self, dim_num):
            self.dim_num_save = dim_num
            self.v = np.zeros((self.dim_max, self.log_max))
            self.v[0:40, 0] = np.transpose(
                [
                    1,
                    1,
                    1,
                    1,
                    1,
                    1,
                    1,
                    1,
                    1,
                    1,
                    1,
                    1,
                    1,
                    1,
                    1,
                    1,
                    1,
                    1,
                    1,
                    1,
                    1,
                    1,
                    1,
                    1,
                    1,
                    1,
                    1,
                    1,
                    1,
                    1,
                    1,
                    1,
                    1,
                    1,
                    1,
                    1,
                    1,
                    1,
                    1,
                    1,
                ]
            )

            self.v[2:40, 1] = np.transpose(
                [
                    1,
                    3,
                    1,
                    3,
                    1,
                    3,
                    3,
                    1,
                    3,
                    1,
                    3,
                    1,
                    3,
                    1,
                    1,
                    3,
                    1,
                    3,
                    1,
                    3,
                    1,
                    3,
                    3,
                    1,
                    3,
                    1,
                    3,
                    1,
                    3,
                    1,
                    1,
                    3,
                    1,
                    3,
                    1,
                    3,
                    1,
                    3,
                ]
            )

            self.v[3:40, 2] = np.transpose(
                [
                    7,
                    5,
                    1,
                    3,
                    3,
                    7,
                    5,
                    5,
                    7,
                    7,
                    1,
                    3,
                    3,
                    7,
                    5,
                    1,
                    1,
                    5,
                    3,
                    3,
                    1,
                    7,
                    5,
                    1,
                    3,
                    3,
                    7,
                    5,
                    1,
                    1,
                    5,
                    7,
                    7,
                    5,
                    1,
                    3,
                    3,
                ]
            )

            self.v[5:40, 3] = np.transpose(
                [
                    1,
                    7,
                    9,
                    13,
                    11,
                    1,
                    3,
                    7,
                    9,
                    5,
                    13,
                    13,
                    11,
                    3,
                    15,
                    5,
                    3,
                    15,
                    7,
                    9,
                    13,
                    9,
                    1,
                    11,
                    7,
                    5,
                    15,
                    1,
                    15,
                    11,
                    5,
                    3,
                    1,
                    7,
                    9,
                ]
            )

            self.v[7:40, 4] = np.transpose(
                [
                    9,
                    3,
                    27,
                    15,
                    29,
                    21,
                    23,
                    19,
                    11,
                    25,
                    7,
                    13,
                    17,
                    1,
                    25,
                    29,
                    3,
                    31,
                    11,
                    5,
                    23,
                    27,
                    19,
                    21,
                    5,
                    1,
                    17,
                    13,
                    7,
                    15,
                    9,
                    31,
                    9,
                ]
            )

            self.v[13:40, 5] = np.transpose(
                [
                    37,
                    33,
                    7,
                    5,
                    11,
                    39,
                    63,
                    27,
                    17,
                    15,
                    23,
                    29,
                    3,
                    21,
                    13,
                    31,
                    25,
                    9,
                    49,
                    33,
                    19,
                    29,
                    11,
                    19,
                    27,
                    15,
                    25,
                ]
            )

            self.v[19:40, 6] = np.transpose(
                [
                    13,
                    33,
                    115,
                    41,
                    79,
                    17,
                    29,
                    119,
                    75,
                    73,
                    105,
                    7,
                    59,
                    65,
                    21,
                    3,
                    113,
                    61,
                    89,
                    45,
                    107,
                ]
            )

            self.v[37:40, 7] = np.transpose([7, 23, 39])

            #  Set POLY.
            self.poly = [
                1,
                3,
                7,
                11,
                13,
                19,
                25,
                37,
                59,
                47,
                61,
                55,
                41,
                67,
                97,
                91,
                109,
                103,
                115,
                131,
                193,
                137,
                145,
                143,
                241,
                157,
                185,
                167,
                229,
                171,
                213,
                191,
                253,
                203,
                211,
                239,
                247,
                285,
                369,
                299,
            ]

            #  Find the number of bits in ATMOST.
            self.maxcol = _bit_hi1(self.atmost)

            #  Initialize row 1 of V.
            self.v[0, 0 : self.maxcol] = 1

            #  Check parameters.
            if dim_num < 1 or self.dim_max < dim_num:
                raise ValueError(
                    f'I4_SOBOL - Fatal error!\n'
                    f'  The spatial dimension DIM_NUM should '
                    f'satisfy:\n'
                    f'  1 <= DIM_NUM <= {self.dim_max}\n'
                    f'  But this input value is DIM_NUM = {dim_num}'
                )

            #  Initialize the remaining rows of V.
            for i in range(2, dim_num + 1):

                #  The bits of the integer POLY(I) gives the form of polynomial I.
                #  Find the degree of polynomial I from binary encoding.
                j = self.poly[i - 1]
                m = 0
                j //= 2
                while j > 0:
                    j //= 2
                    m += 1

                #  Expand this bit pattern to separate components
                #  of the logical array INCLUD.
                j = self.poly[i - 1]
                includ = np.zeros(m)
                for k in range(m, 0, -1):
                    j2 = j // 2
                    includ[k - 1] = j != 2 * j2
                    j = j2

                #  Calculate the remaining elements of row I as explained
                #  in Bratley and Fox, section 2.
                for j in range(m + 1, self.maxcol + 1):
                    newv = self.v[i - 1, j - m - 1]
                    p2 = 1
                    for k in range(1, m + 1):
                        p2 *= 2
                        if includ[k - 1]:
                            newv = np.bitwise_xor(
                                int(newv), int(p2 * self.v[i - 1, j - k - 1])
                            )
                    self.v[i - 1, j - 1] = newv
            #  Multiply columns of V by appropriate power of 2.
            p2 = 1
            for j in range(self.maxcol - 1, 0, -1):
                p2 *= 2
                self.v[0:dim_num, j - 1] = self.v[0:dim_num, j - 1] * p2

            #  RECIPD is 1/(common denominator of the elements in V).
            self.recipd = 1.0 / (2 * p2)
            self.lastq = np.zeros(dim_num)

        def generate(self, dimensions, n_samples, random_state=None):
            """Creates samples from Sobol' set.

            Parameters
            ----------
            dimensions : list, shape (n_dims,)
                List of search space dimensions.
                Each search dimension can be defined either as

                - a `(lower_bound, upper_bound)` tuple (for `Real` or `Integer`
                  dimensions),
                - a `(lower_bound, upper_bound, "prior")` tuple (for `Real`
                  dimensions),
                - as a list of categories (for `Categorical` dimensions), or
                - an instance of a `Dimension` object (`Real`, `Integer` or
                  `Categorical`).
            n_samples : int
                The order of the Sobol' sequence. Defines the number of samples.
            random_state : int, RandomState instance, or None (default)
                Set random state to something other than None for reproducible
                results.

            Returns
            -------
            sample : array_like (n_samples, dim)
                Sobol' set.
            """
            total_n_samples = self.num_generated + n_samples
            if not (total_n_samples & (total_n_samples - 1) == 0):
                warnings.warn(
                    "The balance properties of Sobol' points require "
                    "n to be a power of 2. {0} points have been "
                    "previously generated, then: n={0}+{1}={2}. ".format(
                        self.num_generated, n_samples, total_n_samples
                    )
                )
            if self.skip != 0 and total_n_samples > self.skip:
                raise ValueError(
                    f"{self.skip} points have been skipped: "
                    f"generating "
                    f"{n_samples} more points would cause the "
                    f"sequence to repeat."
                )

            rng = check_random_state(random_state)
            space = Space(dimensions)
            n_dim = space.n_dims
            transformer = space.get_transformer()
            space.set_transformer("normalize")
            r = np.full((n_samples, n_dim), np.nan)

            seed = self.skip
            for j in range(n_samples):
                r[j, 0:n_dim], seed = self._sobol(n_dim, seed)

            if self.randomize:
                r = _random_shift(r, rng)

            r = space.inverse_transform(r)
            space.set_transformer(transformer)

            self.num_generated += n_samples

            return r

        def _sobol(self, dim_num, seed):
            """Generates a new quasirandom Sobol' vector with each call.

            Parameters
            ----------
            dim_num : int
              Number of spatial dimensions.
              `dim_num` must satisfy 1 <= DIM_NUM <= 40.

            seed : int
              the `seed` for the sequence.
              This is essentially the index in the sequence of the quasirandom
              value to be generated. On output, `seed` has been set to the
              appropriate next value, usually simply `seed`+1.
              If `seed` is less than 0 on input, it is treated as though it were 0.
              An input value of 0 requests the first (0-th) element of
              the sequence.

            Returns
            -------
            vector, seed : np.array (n_dim,), int
                The next quasirandom vector and the seed of its next vector.
            """
            #  Things to do only if the dimension changed.
            if dim_num != self.dim_num_save:
                self.init(dim_num)

            seed = int(np.floor(seed))

            if seed < 0:
                seed = 0

            pos_lo0 = 1
            if seed == 0:
                self.lastq = np.zeros(dim_num)

            elif seed == self.seed_save + 1:

                #  Find the position of the right-hand zero in SEED.
                pos_lo0 = _bit_lo0(seed)

            elif seed <= self.seed_save:

                self.seed_save = 0
                self.lastq = np.zeros(dim_num)

                for seed_temp in range(int(self.seed_save), int(seed)):
                    pos_lo0 = _bit_lo0(seed_temp)
                    for i in range(1, dim_num + 1):
                        self.lastq[i - 1] = np.bitwise_xor(
                            int(self.lastq[i - 1]), int(self.v[i - 1, pos_lo0 - 1])
                        )

                pos_lo0 = _bit_lo0(seed)

            elif self.seed_save + 1 < seed:

                for seed_temp in range(int(self.seed_save + 1), int(seed)):
                    pos_lo0 = _bit_lo0(seed_temp)
                    for i in range(1, dim_num + 1):
                        self.lastq[i - 1] = np.bitwise_xor(
                            int(self.lastq[i - 1]), int(self.v[i - 1, pos_lo0 - 1])
                        )

                pos_lo0 = _bit_lo0(seed)

            #  Check that the user is not calling too many times!
            if self.maxcol < pos_lo0:
                raise ValueError(
                    f'I4_SOBOL - Fatal error!\n'
                    f' Too many calls!\n'
                    f' MAXCOL = {self.maxcol}\n'
                    f' L =      {pos_lo0}\n'
                )

            #  Calculate the new components of QUASI.
            quasi = np.zeros(dim_num)
            for i in range(1, dim_num + 1):
                quasi[i - 1] = self.lastq[i - 1] * self.recipd
                self.lastq[i - 1] = np.bitwise_xor(
                    int(self.lastq[i - 1]), int(self.v[i - 1, pos_lo0 - 1])
                )

            self.seed_save = seed
            seed += 1

            return [quasi, seed]

    def _bit_hi1(n):
        """Returns the position of the high 1 bit base 2 in an integer.

        Parameters
        ----------
        n : int
            Input, should be positive.
        """
        bin_repr = np.binary_repr(n)
        most_left_one = bin_repr.find('1')
        if most_left_one == -1:
            return 0
        else:
            return len(bin_repr) - most_left_one

    def _bit_lo0(n):
        """Returns the position of the low 0 bit base 2 in an integer.

        Parameters
        ----------
        n : int
            Input, should be positive.
        """
        bin_repr = np.binary_repr(n)
        most_right_zero = bin_repr[::-1].find('0')
        if most_right_zero == -1:
            most_right_zero = len(bin_repr)
        return most_right_zero + 1

    def _random_shift(dm, random_state=None):
        """Random shifting of a vector.

        Randomization of the quasi-MC samples can be achieved in the easiest manner
        by random shift (or the Cranley-Patterson rotation).

        References
        -----------
        .. [1] C. Lemieux, "Monte Carlo and Quasi-Monte Carlo Sampling," Springer
           Series in Statistics 692, Springer Science+Business Media, New York,
           2009

        Parameters
        ----------
        dm : array, shape(n, d)
            Input matrix.
        random_state : int, RandomState instance, or None (default)
            Set random state to something other than None for reproducible
            results.

        Returns
        -------
        dm :  array, shape(n, d)
            Randomized Sobol' design matrix.
        """
        rng = check_random_state(random_state)
        # Generate random shift matrix from uniform distribution
        shift = np.repeat(rng.rand(1, dm.shape[1]), dm.shape[0], axis=0)
        # Return the shifted Sobol' design
        return (dm + shift) % 1