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# Copyright (c) 2020, nicolas deutschmann
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import numpy as np
import torch
import torch.nn.functional as F


def piecewise_linear_transform(x, q_tilde, compute_jacobian=True, outlier_passthru=True):
    """Apply an element-wise piecewise-linear transformation to some variables

    Args:
        x : torch.Tensor
            a tensor with shape (N,k) where N is the batch dimension while k is the dimension of the variable space. This variable span the k-dimensional unit
        hypercube

        q_tilde: torch.Tensor
                is a tensor with shape (N,k,b) where b is the number of bins.
                This contains the un-normalized heights of the bins of the piecewise-constant PDF for dimension k,
                i.e. q_tilde lives in all of R and we don't impose a constraint on their sum yet.
                Normalization is imposed in this function using softmax.

        compute_jacobian : bool, optional
                            determines whether the jacobian should be compute or None is returned

    Returns:
        tuple of torch.Tensor
            pair `(y,h)`.
            - `y` is a tensor with shape (N,k) living in the k-dimensional unit hypercube
            - `j` is the jacobian of the transformation with shape (N,) if compute_jacobian==True, else None.
    """
    logj = None

    third_dimension_softmax = torch.nn.Softmax(dim=2)

    # Compute the bin width w
    N, k, b = q_tilde.shape
    Nx, kx = x.shape
    assert N == Nx and k == kx, "Shape mismatch"

    w = 1.0 / b

    # Compute normalized bin heights with softmax function on bin dimension
    q = 1.0 / w * third_dimension_softmax(q_tilde)
    # x is in the mx-th bin: x \in [0,1],
    # mx \in [[0,b-1]], so we clamp away the case x == 1
    mx = torch.clamp(torch.floor(b * x), 0, b - 1).to(torch.long)
    # Need special error handling because trying to index with mx
    # if it contains nans will lock the GPU. (device-side assert triggered)
    if torch.any(torch.isnan(mx)).item() or torch.any(mx < 0) or torch.any(mx >= b):
        raise AvertedCUDARuntimeError("NaN detected in PWLinear bin indexing")

    # We compute the output variable in-place
    out = x - mx * w  # alpha (element of [0.,w], the position of x in its bin

    # Multiply by the slope
    # q has shape (N,k,b), mxu = mx.unsqueeze(-1) has shape (N,k) with entries that are a b-index
    # gather defines slope[i, j, k] = q[i, j, mxu[i, j, k]] with k taking only 0 as a value
    # i.e. we say slope[i, j] = q[i, j, mx [i, j]]
    slopes = torch.gather(q, 2, mx.unsqueeze(-1)).squeeze(-1)
    out = out * slopes
    # The jacobian is the product of the slopes in all dimensions

    # Compute the integral over the left-bins.
    # 1. Compute all integrals: cumulative sum of bin height * bin weight.
    # We want that index i contains the cumsum *strictly to the left* so we shift by 1
    # leaving the first entry null, which is achieved with a roll and assignment
    q_left_integrals = torch.roll(torch.cumsum(q, 2) * w, 1, 2)
    q_left_integrals[:, :, 0] = 0

    # 2. Access the correct index to get the left integral of each point and add it to our transformation
    out = out + torch.gather(q_left_integrals, 2, mx.unsqueeze(-1)).squeeze(-1)

    # Regularization: points must be strictly within the unit hypercube
    # Use the dtype information from pytorch
    eps = torch.finfo(out.dtype).eps
    out = out.clamp(min=eps, max=1.0 - eps)
    oob_mask = torch.logical_or(x < 0.0, x > 1.0).detach().float()
    if outlier_passthru:
        out = out * (1 - oob_mask) + x * oob_mask
        slopes = slopes * (1 - oob_mask) + oob_mask

    if compute_jacobian:
        # logj = torch.log(torch.prod(slopes.float(), 1))
        logj = torch.sum(torch.log(slopes), 1)
    del slopes

    return out, logj


def piecewise_linear_inverse_transform(y, q_tilde, compute_jacobian=True, outlier_passthru=True):
    """
    Apply inverse of an element-wise piecewise-linear transformation to some
    variables

    Args:
        y : torch.Tensor
            a tensor with shape (N,k) where N is the batch dimension while k is the dimension of the variable space. This variable span the k-dimensional unit hypercube

        q_tilde: torch.Tensor
                is a tensor with shape (N,k,b) where b is the number of bins. This contains the un-normalized heights of the bins of the piecewise-constant PDF for dimension k, i.e. q_tilde lives in all of R and we don't impose a constraint on their sum yet. Normalization is imposed in this function using softmax.

        compute_jacobian : bool, optional
                            determines whether the jacobian should be compute or None is returned

    Returns:
        tuple of torch.Tensor
            pair `(x,h)`.
            - `x` is a tensor with shape (N,k) living in the k-dimensional unit hypercube
            - `j` is the jacobian of the transformation with shape (N,) if compute_jacobian==True, else None.
    """

    third_dimension_softmax = torch.nn.Softmax(dim=2)
    # Compute the bin width w
    N, k, b = q_tilde.shape
    Ny, ky = y.shape
    assert N == Ny and k == ky, "Shape mismatch"

    w = 1.0 / b

    # Compute normalized bin heights with softmax function on the bin dimension
    q = 1.0 / w * third_dimension_softmax(q_tilde)

    # Compute the integral over the left-bins in the forward transform.
    # 1. Compute all integrals: cumulative sum of bin height * bin weight.
    # We want that index i contains the cumsum *strictly to the left*,
    # so we shift by 1 leaving the first entry null,
    # which is achieved with a roll and assignment
    q_left_integrals = torch.roll(torch.cumsum(q.float(), 2) * w, 1, 2)
    q_left_integrals[:, :, 0] = 0

    # Find which bin each y belongs to by finding the smallest bin such that
    # y - q_left_integral is positive

    edges = (y.unsqueeze(-1) - q_left_integrals).detach()
    # y and q_left_integrals are between 0 and 1,
    # so that their difference is at most 1.
    # By setting the negative values to 2., we know that the
    # smallest value left is the smallest positive
    edges[edges < 0] = 2.0
    edges = torch.clamp(torch.argmin(edges, dim=2), 0, b - 1).to(torch.long)

    # Need special error handling because trying to index with mx
    # if it contains nans will lock the GPU. (device-side assert triggered)
    if torch.any(torch.isnan(edges)).item() or torch.any(edges < 0) or torch.any(edges >= b):
        raise AvertedCUDARuntimeError("NaN detected in PWLinear bin indexing")

    # Gather the left integrals at each edge. See comment about gathering in q_left_integrals
    # for the unsqueeze
    q_left_integrals = q_left_integrals.gather(2, edges.unsqueeze(-1)).squeeze(-1)

    # Gather the slope at each edge.
    q = q.gather(2, edges.unsqueeze(-1)).squeeze(-1)

    # Build the output
    x = (y - q_left_integrals) / q + edges * w

    # Regularization: points must be strictly within the unit hypercube
    # Use the dtype information from pytorch
    eps = torch.finfo(x.dtype).eps
    x = x.clamp(min=eps, max=1.0 - eps)
    oob_mask = torch.logical_or(y < 0.0, y > 1.0).detach().float()
    if outlier_passthru:
        x = x * (1 - oob_mask) + y * oob_mask
        q = q * (1 - oob_mask) + oob_mask

    # Prepare the jacobian
    logj = None
    if compute_jacobian:
        # logj = - torch.log(torch.prod(q, 1))
        logj = -torch.sum(torch.log(q.float()), 1)
    return x.detach(), logj


def unbounded_piecewise_quadratic_transform(x, w_tilde, v_tilde, upper=1, lower=0, inverse=False):
    assert upper > lower
    _range = upper - lower
    inside_interval_mask = (x >= lower) & (x < upper)
    outside_interval_mask = ~inside_interval_mask

    outputs = torch.zeros_like(x)
    log_j = torch.zeros_like(x)

    outputs[outside_interval_mask] = x[outside_interval_mask]
    log_j[outside_interval_mask] = 0

    output, _log_j = piecewise_quadratic_transform(
        (x[inside_interval_mask] - lower) / _range,
        w_tilde[inside_interval_mask, :],
        v_tilde[inside_interval_mask, :],
        inverse=inverse,
    )
    outputs[inside_interval_mask] = output * _range + lower
    if not inverse:
        # the before and after transformation cancel out, so the log_j would be just as it is.
        log_j[inside_interval_mask] = _log_j
    else:
        log_j = None
    return outputs, log_j


def weighted_softmax(v, w):
    # to avoid NaN...
    v = v - torch.max(v, dim=-1, keepdim=True)[0]
    v = torch.exp(v) + 1e-8  # to avoid NaN...
    v_sum = torch.sum((v[..., :-1] + v[..., 1:]) / 2 * w, dim=-1, keepdim=True)
    return v / v_sum


def piecewise_quadratic_transform(x, w_tilde, v_tilde, inverse=False):
    """Element-wise piecewise-quadratic transformation
    Args:
        x : torch.Tensor
            *, The variable spans the D-dim unit hypercube ([0,1))
        w_tilde : torch.Tensor
            * x K defined in the paper
        v_tilde : torch.Tensor
            * x (K+1) defined in the paper
        inverse : bool
            forward or inverse
    Returns:
        c : torch.Tensor
            *, transformed value
        log_j : torch.Tensor
            *, log determinant of the Jacobian matrix
    """
    w = torch.softmax(w_tilde, dim=-1)
    v = weighted_softmax(v_tilde, w)
    w_cumsum = torch.cumsum(w, dim=-1)
    # force sum = 1
    w_cumsum[..., -1] = 1.0
    w_cumsum_shift = F.pad(w_cumsum, (1, 0), 'constant', 0)
    cdf = torch.cumsum((v[..., 1:] + v[..., :-1]) / 2 * w, dim=-1)
    # force sum = 1
    cdf[..., -1] = 1.0
    cdf_shift = F.pad(cdf, (1, 0), 'constant', 0)

    if not inverse:
        # * x D x 1, (w_cumsum[idx-1] < x <= w_cumsum[idx])
        bin_index = torch.searchsorted(w_cumsum, x.unsqueeze(-1))
    else:
        # * x D x 1, (cdf[idx-1] < x <= cdf[idx])
        bin_index = torch.searchsorted(cdf, x.unsqueeze(-1))

    w_b = torch.gather(w, -1, bin_index).squeeze(-1)
    w_bn1 = torch.gather(w_cumsum_shift, -1, bin_index).squeeze(-1)
    v_b = torch.gather(v, -1, bin_index).squeeze(-1)
    v_bp1 = torch.gather(v, -1, bin_index + 1).squeeze(-1)
    cdf_bn1 = torch.gather(cdf_shift, -1, bin_index).squeeze(-1)

    if not inverse:
        alpha = (x - w_bn1) / w_b.clamp(min=torch.finfo(w_b.dtype).eps)
        c = (alpha ** 2) / 2 * (v_bp1 - v_b) * w_b + alpha * v_b * w_b + cdf_bn1

        # just sum of log pdfs
        log_j = torch.lerp(v_b, v_bp1, alpha).clamp(min=torch.finfo(c.dtype).eps).log()

        # make sure it falls into [0,1)
        c = c.clamp(min=torch.finfo(c.dtype).eps, max=1.0 - torch.finfo(c.dtype).eps)
        return c, log_j
    else:
        # quadratic equation for alpha
        # alpha should fall into (0, 1]. Since a, b > 0, the symmetry axis -b/2a < 0 and we should pick the larger root
        # skip calculating the log_j in inverse since we don't need it
        a = (v_bp1 - v_b) * w_b / 2
        b = v_b * w_b
        c = cdf_bn1 - x
        alpha = (-b + torch.sqrt((b ** 2) - 4 * a * c)) / (2 * a)
        inv = alpha * w_b + w_bn1

        # make sure it falls into [0,1)
        inv = inv.clamp(min=torch.finfo(c.dtype).eps, max=1.0 - torch.finfo(inv.dtype).eps)
        return inv, None


def piecewise_rational_quadratic_transform(
    inputs,
    unnormalized_widths,
    unnormalized_heights,
    unnormalized_derivatives,
    inverse=False,
    tails=None,
    tail_bound=1.0,
    min_bin_width=1e-3,
    min_bin_height=1e-3,
    min_derivative=1e-3,
):

    if tails is None:
        spline_fn = rational_quadratic_spline
        spline_kwargs = {}
    else:
        spline_fn = unconstrained_rational_quadratic_spline
        spline_kwargs = {'tails': tails, 'tail_bound': tail_bound}

    outputs, logabsdet = spline_fn(
        inputs=inputs,
        unnormalized_widths=unnormalized_widths,
        unnormalized_heights=unnormalized_heights,
        unnormalized_derivatives=unnormalized_derivatives,
        inverse=inverse,
        min_bin_width=min_bin_width,
        min_bin_height=min_bin_height,
        min_derivative=min_derivative,
        **spline_kwargs
    )
    return outputs, logabsdet


def searchsorted(bin_locations, inputs, eps=1e-6):
    bin_locations[..., -1] += eps
    return torch.sum(inputs[..., None] >= bin_locations, dim=-1) - 1


def unconstrained_rational_quadratic_spline(
    inputs,
    unnormalized_widths,
    unnormalized_heights,
    unnormalized_derivatives,
    inverse=False,
    tails='linear',
    tail_bound=1.0,
    min_bin_width=1e-3,
    min_bin_height=1e-3,
    min_derivative=1e-3,
):
    inside_interval_mask = (inputs >= -tail_bound) & (inputs <= tail_bound)
    outside_interval_mask = ~inside_interval_mask

    outputs = torch.zeros_like(inputs)
    logabsdet = torch.zeros_like(inputs)

    if tails == 'linear':
        unnormalized_derivatives = F.pad(unnormalized_derivatives, pad=(1, 1))
        constant = np.log(np.exp(1 - min_derivative) - 1)
        unnormalized_derivatives[..., 0] = constant
        unnormalized_derivatives[..., -1] = constant

        outputs[outside_interval_mask] = inputs[outside_interval_mask]
        logabsdet[outside_interval_mask] = 0
    else:
        raise RuntimeError('{} tails are not implemented.'.format(tails))

    outputs[inside_interval_mask], logabsdet[inside_interval_mask] = rational_quadratic_spline(
        inputs=inputs[inside_interval_mask],
        unnormalized_widths=unnormalized_widths[inside_interval_mask, :],
        unnormalized_heights=unnormalized_heights[inside_interval_mask, :],
        unnormalized_derivatives=unnormalized_derivatives[inside_interval_mask, :],
        inverse=inverse,
        left=-tail_bound,
        right=tail_bound,
        bottom=-tail_bound,
        top=tail_bound,
        min_bin_width=min_bin_width,
        min_bin_height=min_bin_height,
        min_derivative=min_derivative,
    )

    return outputs, logabsdet


def rational_quadratic_spline(
    inputs,
    unnormalized_widths,
    unnormalized_heights,
    unnormalized_derivatives,
    inverse=False,
    left=0.0,
    right=1.0,
    bottom=0.0,
    top=1.0,
    min_bin_width=1e-3,
    min_bin_height=1e-3,
    min_derivative=1e-3,
):

    if torch.min(inputs) < left or torch.max(inputs) > right:
        raise ValueError('Input to a transform is not within its domain')

    num_bins = unnormalized_widths.shape[-1]

    if min_bin_width * num_bins > 1.0:
        raise ValueError('Minimal bin width too large for the number of bins')
    if min_bin_height * num_bins > 1.0:
        raise ValueError('Minimal bin height too large for the number of bins')

    widths = F.softmax(unnormalized_widths, dim=-1)
    widths = min_bin_width + (1 - min_bin_width * num_bins) * widths
    cumwidths = torch.cumsum(widths, dim=-1)
    cumwidths = F.pad(cumwidths, pad=(1, 0), mode='constant', value=0.0)
    cumwidths = (right - left) * cumwidths + left
    cumwidths[..., 0] = left
    cumwidths[..., -1] = right
    widths = cumwidths[..., 1:] - cumwidths[..., :-1]

    derivatives = min_derivative + F.softplus(unnormalized_derivatives)

    heights = F.softmax(unnormalized_heights, dim=-1)
    heights = min_bin_height + (1 - min_bin_height * num_bins) * heights
    cumheights = torch.cumsum(heights, dim=-1)
    cumheights = F.pad(cumheights, pad=(1, 0), mode='constant', value=0.0)
    cumheights = (top - bottom) * cumheights + bottom
    cumheights[..., 0] = bottom
    cumheights[..., -1] = top
    heights = cumheights[..., 1:] - cumheights[..., :-1]

    if inverse:
        bin_idx = searchsorted(cumheights, inputs)[..., None]
    else:
        bin_idx = searchsorted(cumwidths, inputs)[..., None]

    input_cumwidths = cumwidths.gather(-1, bin_idx)[..., 0]
    input_bin_widths = widths.gather(-1, bin_idx)[..., 0]

    input_cumheights = cumheights.gather(-1, bin_idx)[..., 0]
    delta = heights / widths
    input_delta = delta.gather(-1, bin_idx)[..., 0]

    input_derivatives = derivatives.gather(-1, bin_idx)[..., 0]
    input_derivatives_plus_one = derivatives[..., 1:].gather(-1, bin_idx)[..., 0]

    input_heights = heights.gather(-1, bin_idx)[..., 0]

    if inverse:
        a = (inputs - input_cumheights) * (
            input_derivatives + input_derivatives_plus_one - 2 * input_delta
        ) + input_heights * (input_delta - input_derivatives)
        b = input_heights * input_derivatives - (inputs - input_cumheights) * (
            input_derivatives + input_derivatives_plus_one - 2 * input_delta
        )
        c = -input_delta * (inputs - input_cumheights)

        discriminant = b.pow(2) - 4 * a * c
        assert (discriminant >= 0).all()

        root = (2 * c) / (-b - torch.sqrt(discriminant))
        outputs = root * input_bin_widths + input_cumwidths

        theta_one_minus_theta = root * (1 - root)
        denominator = input_delta + (
            (input_derivatives + input_derivatives_plus_one - 2 * input_delta) * theta_one_minus_theta
        )
        derivative_numerator = input_delta.pow(2) * (
            input_derivatives_plus_one * root.pow(2)
            + 2 * input_delta * theta_one_minus_theta
            + input_derivatives * (1 - root).pow(2)
        )
        logabsdet = torch.log(derivative_numerator) - 2 * torch.log(denominator)

        return outputs, -logabsdet
    else:
        theta = (inputs - input_cumwidths) / input_bin_widths
        theta_one_minus_theta = theta * (1 - theta)

        numerator = input_heights * (input_delta * theta.pow(2) + input_derivatives * theta_one_minus_theta)
        denominator = input_delta + (
            (input_derivatives + input_derivatives_plus_one - 2 * input_delta) * theta_one_minus_theta
        )
        outputs = input_cumheights + numerator / denominator

        derivative_numerator = input_delta.pow(2) * (
            input_derivatives_plus_one * theta.pow(2)
            + 2 * input_delta * theta_one_minus_theta
            + input_derivatives * (1 - theta).pow(2)
        )
        logabsdet = torch.log(derivative_numerator) - 2 * torch.log(denominator)

        return outputs, logabsdet