_tanhsinh.py

# mypy: disable-error-code="attr-defined"
import numpy as np
from scipy import special
from scipy.optimize import OptimizeResult
from scipy.optimize._zeros_py import (  # noqa: F401
    _scalar_optimization_initialize,
    _scalar_optimization_loop,
    _ECONVERGED, _ESIGNERR, _ECONVERR,
    _EVALUEERR, _ECALLBACK, _EINPROGRESS)

# todo:
#  figure out warning situation
#  address https://github.com/scipy/scipy/pull/18650#discussion_r1233032521
#  without `minweight`, we are also suppressing infinities within the interval.
#    Is that OK? If so, we can probably get rid of `status=3`.
#  Add heuristic to stop when improvement is too slow / antithrashing
#  support singularities? interval subdivision? this feature will be added
#    eventually, but do we adjust the interface now?
#  When doing log-integration, should the tolerances control the error of the
#    log-integral or the error of the integral?  The trouble is that `log`
#    inherently looses some precision so it may not be possible to refine
#    the integral further. Example: 7th moment of stats.f(15, 20)
#  respect function evaluation limit?
#  make public?


def _tanhsinh(f, a, b, *, args=(), log=False, maxfun=None, maxlevel=None,
               minlevel=2, atol=None, rtol=None, callback=None):
    """Evaluate a convergent integral numerically using tanh-sinh quadrature.

    In practice, tanh-sinh quadrature achieves quadratic convergence for
    many integrands: the number of accurate *digits* scales roughly linearly
    with the number of function evaluations [1]_.

    Either or both of the limits of integration may be infinite, and
    singularities at the endpoints are acceptable. Divergent integrals and
    integrands with non-finite derivatives or singularities within an interval
    are out of scope, but the latter may be evaluated be calling `_tanhsinh` on
    each sub-interval separately.

    Parameters
    ----------
    f : callable
        The function to be integrated. The signature must be::
            func(x: ndarray, *args) -> ndarray
         where each element of ``x`` is a finite real and ``args`` is a tuple,
         which may contain an arbitrary number of arrays that are broadcastable
         with `x`. ``func`` must be an elementwise function: each element
         ``func(x)[i]`` must equal ``func(x[i])`` for all indices ``i``.
         If ``func`` returns a value with complex dtype when evaluated at
         either endpoint, subsequent arguments ``x`` will have complex dtype
         (but zero imaginary part).
    a, b : array_like
        Real lower and upper limits of integration. Must be broadcastable.
        Elements may be infinite.
    args : tuple, optional
        Additional positional arguments to be passed to `func`. Must be arrays
        broadcastable with `a` and `b`. If the callable to be integrated
        requires arguments that are not broadcastable with `a` and `b`, wrap
        that callable with `f`. See Examples.
    log : bool, default: False
        Setting to True indicates that `f` returns the log of the integrand
        and that `atol` and `rtol` are expressed as the logs of the absolute
        and relative errors. In this case, the result object will contain the
        log of the integral and error. This is useful for integrands for which
        numerical underflow or overflow would lead to inaccuracies.
        When ``log=True``, the integrand (the exponential of `f`) must be real,
        but it may be negative, in which case the log of the integrand is a
        complex number with an imaginary part that is an odd multiple of π.
    maxlevel : int, default: 10
        The maximum refinement level of the algorithm.

        At the zeroth level, `f` is called once, performing 16 function
        evaluations. At each subsequent level, `f` is called once more,
        approximately doubling the number of function evaluations that have
        been performed. Accordingly, for many integrands, each successive level
        will double the number of accurate digits in the result (up to the
        limits of floating point precision).

        The algorithm will terminate after completing level `maxlevel` or after
        another termination condition is satisfied, whichever comes first.
    minlevel : int, default: 2
        The level at which to begin iteration (default: 2). This does not
        change the total number of function evaluations or the abscissae at
        which the function is evaluated; it changes only the *number of times*
        `f` is called. If ``minlevel=k``, then the integrand is evaluated at
        all abscissae from levels ``0`` through ``k`` in a single call.
        Note that if `minlevel` exceeds `maxlevel`, the provided `minlevel` is
        ignored, and `minlevel` is set equal to `maxlevel`.
    atol, rtol : float, optional
        Absolute termination tolerance (default: 0) and relative termination
        tolerance (default: ``eps**0.75``, where ``eps`` is the precision of
        the result dtype), respectively. The error estimate is as
        described in [1]_ Section 5. While not theoretically rigorous or
        conservative, it is said to work well in practice. Must be non-negative
        and finite if `log` is False, and must be expressed as the log of a
        non-negative and finite number if `log` is True.
    callback : callable, optional
        An optional user-supplied function to be called before the first
        iteration and after each iteration.
        Called as ``callback(res)``, where ``res`` is an ``OptimizeResult``
        similar to that returned by `_differentiate` (but containing the
        current iterate's values of all variables). If `callback` raises a
        ``StopIteration``, the algorithm will terminate immediately and
        `_tanhsinh` will return a result object.

    Returns
    -------
    res : OptimizeResult
        An instance of `scipy.optimize.OptimizeResult` with the following
        attributes. (The descriptions are written as though the values will be
        scalars; however, if `func` returns an array, the outputs will be
        arrays of the same shape.)
        success : bool
            ``True`` when the algorithm terminated successfully (status ``0``).
        status : int
            An integer representing the exit status of the algorithm.
            ``0`` : The algorithm converged to the specified tolerances.
            ``-1`` : (unused)
            ``-2`` : The maximum number of iterations was reached.
            ``-3`` : A non-finite value was encountered.
            ``-4`` : Iteration was terminated by `callback`.
            ``1`` : The algorithm is proceeding normally (in `callback` only).
        integral : float
            An estimate of the integral
        error : float
            An estimate of the error. Only available if level two or higher
            has been completed; otherwise NaN.
        nit : int
            The number of iterations performed.
        nfev : int
            The number of points at which `func` was evaluated.

    See Also
    --------
    quad, quadrature

    Notes
    -----
    Implements the algorithm as described in [1]_ with minor adaptations for
    finite-precision arithmetic, including some described by [2]_ and [3]_. The
    tanh-sinh scheme was originally introduced in [4]_.

    Due floating-point error in the abscissae, the function may be evaluated
    at the endpoints of the interval during iterations. The values returned by
    the function at the endpoints will be ignored.

    References
    ----------
    [1] Bailey, David H., Karthik Jeyabalan, and Xiaoye S. Li. "A comparison of
        three high-precision quadrature schemes." Experimental Mathematics 14.3
        (2005): 317-329.
    [2] Vanherck, Joren, Bart Sorée, and Wim Magnus. "Tanh-sinh quadrature for
        single and multiple integration using floating-point arithmetic."
        arXiv preprint arXiv:2007.15057 (2020).
    [3] van Engelen, Robert A.  "Improving the Double Exponential Quadrature
        Tanh-Sinh, Sinh-Sinh and Exp-Sinh Formulas."
        https://www.genivia.com/files/qthsh.pdf
    [4] Takahasi, Hidetosi, and Masatake Mori. "Double exponential formulas for
        numerical integration." Publications of the Research Institute for
        Mathematical Sciences 9.3 (1974): 721-741.

    Example
    -------
    Evaluate the Gaussian integral:

    >>> import numpy as np
    >>> from scipy.integrate._tanhsinh import _tanhsinh
    >>> def f(x):
    ...     return np.exp(-x**2)
    >>> res = _tanhsinh(f, -np.inf, np.inf)
    >>> res.integral  # true value is np.sqrt(np.pi), 1.7724538509055159
     1.7724538509055159
    >>> res.error  # actual error is 0
    4.0007963937534104e-16

    The value of the Gaussian function (bell curve) is nearly zero for
    arguments sufficiently far from zero, so the value of the integral
    over a finite interval is nearly the same.

    >>> _tanhsinh(f, -20, 20).integral
    1.772453850905518

    However, with unfavorable integration limits, the integration scheme
    may not be able to find the important region.

    >>> _tanhsinh(f, -np.inf, 1000).integral
    4.500490856620352

    In such cases, or when there are singularities within the interval,
    break the integral into parts with endpoints at the important points.

    >>> _tanhsinh(f, -np.inf, 0).integral + _tanhsinh(f, 0, 1000).integral
    1.772453850905404

    For integration involving very large or very small magnitudes, use
    log-integration. (For illustrative purposes, the following example shows a
    case in which both regular and log-integration work, but for more extreme
    limits of integration, log-integration would avoid the underflow
    experienced when evaluating the integral normally.)

    >>> res = _tanhsinh(f, 20, 30, rtol=1e-10)
    >>> res.integral, res.error
    4.7819613911309014e-176, 4.670364401645202e-187
    >>> def log_f(x):
    ...     return -x**2
    >>> np.exp(res.integral), np.exp(res.error)
    4.7819613911306924e-176, 4.670364401645093e-187

    The limits of integration and elements of `args` may be broadcastable
    arrays, and integration is performed elementwise.

    >>> from scipy import stats
    >>> dist = stats.gausshyper(13.8, 3.12, 2.51, 5.18)
    >>> a, b = dist.support()
    >>> x = np.linspace(a, b, 100)
    >>> res = _tanhsinh(dist.pdf, a, x)
    >>> ref = dist.cdf(x)
    >>> np.allclose(res.integral, ref)

    """
    tmp = f, a, b, log, maxfun, maxlevel, minlevel, atol, rtol, args, callback
    tmp = _tanhsinh_iv(*tmp)
    f, a, b, log, maxfun, maxlevel, minlevel, atol, rtol, args, callback = tmp

    # Initialization
    # `_scalar_optimization_initialize` does several important jobs, including
    # ensuring that limits, each of the `args`, and the output of `f`
    # broadcast correctly and are of consistent types. To save a function
    # evaluation, I pass the midpoint of the integration interval. This comes
    # at a cost of some gymnastics to ensure that the midpoint has the right
    # shape and dtype. Did you know that 0d and >0d arrays follow different
    # type promotion rules?
    with np.errstate(over='ignore', invalid='ignore', divide='ignore'):
        c = ((a.ravel() + b.ravel())/2).reshape(a.shape)
        c[np.isinf(a)] = b[np.isinf(a)]  # takes care of infinite a
        c[np.isinf(b)] = a[np.isinf(b)]  # takes care of infinite b
        c[np.isnan(c)] = 0  # takes care of infinite a and b
        tmp = _scalar_optimization_initialize(f, (c,), args, complex_ok=True)
    xs, fs, args, shape, dtype = tmp
    a = np.broadcast_to(a, shape).astype(dtype).ravel()
    b = np.broadcast_to(b, shape).astype(dtype).ravel()

    # Transform improper integrals
    a, b, a0, negative, abinf, ainf, binf = _transform_integrals(a, b)

    # Define variables we'll need
    nit, nfev = 0, 1  # one function evaluation performed above
    zero = -np.inf if log else 0
    pi = dtype.type(np.pi)
    maxiter = maxlevel - minlevel + 1
    eps = np.finfo(dtype).eps
    if rtol is None:
        rtol = 0.75*np.log(eps) if log else eps**0.75

    Sn = np.full(shape, zero, dtype=dtype).ravel()  # latest integral estimate
    Sn[np.isnan(a) | np.isnan(b) | np.isnan(fs[0])] = np.nan
    Sk = np.empty_like(Sn).reshape(-1, 1)[:, 0:0]  # all integral estimates
    aerr = np.full(shape, np.nan, dtype=dtype).ravel()  # absolute error
    status = np.full(shape, _EINPROGRESS, dtype=int).ravel()
    h0 = np.real(_get_base_step(dtype=dtype))  # base step

    # For term `d4` of error estimate ([1] Section 5), we need to keep the
    # most extreme abscissae and corresponding `fj`s, `wj`s in Euler-Maclaurin
    # sum. Here, we initialize these variables.
    xr0 = np.full(shape, -np.inf, dtype=dtype).ravel()
    fr0 = np.full(shape, np.nan, dtype=dtype).ravel()
    wr0 = np.zeros(shape, dtype=dtype).ravel()
    xl0 = np.full(shape, np.inf, dtype=dtype).ravel()
    fl0 = np.full(shape, np.nan, dtype=dtype).ravel()
    wl0 = np.zeros(shape, dtype=dtype).ravel()
    d4 = np.zeros(shape, dtype=dtype).ravel()

    work = OptimizeResult(
        Sn=Sn, Sk=Sk, aerr=aerr, h=h0, log=log, dtype=dtype, pi=pi, eps=eps,
        a=a.reshape(-1, 1), b=b.reshape(-1, 1),  # integration limits
        n=minlevel, nit=nit, nfev=nfev, status=status,  # iter/eval counts
        xr0=xr0, fr0=fr0, wr0=wr0, xl0=xl0, fl0=fl0, wl0=wl0, d4=d4,  # err est
        ainf=ainf, binf=binf, abinf=abinf, a0=a0.reshape(-1, 1))  # transforms
    # Constant scalars don't need to be put in `work` unless they need to be
    # passed outside `tanhsinh`. Examples: atol, rtol, h0, minlevel.

    # Correspondence between terms in the `work` object and the result
    res_work_pairs = [('status', 'status'), ('integral', 'Sn'),
                      ('error', 'aerr'), ('nit', 'nit'), ('nfev', 'nfev')]

    def pre_func_eval(work):
        # Determine abscissae at which to evaluate `f`
        work.h = h0 / 2**work.n
        xjc, wj = _get_pairs(work.n, h0, dtype=work.dtype,
                             inclusive=(work.n == minlevel))
        work.xj, work.wj = _transform_to_limits(xjc, wj, work.a, work.b)

        # Perform abscissae substitutions for infinite limits of integration
        xj = work.xj.copy()
        xj[work.abinf] = xj[work.abinf] / (1 - xj[work.abinf]**2)
        xj[work.binf] = 1/xj[work.binf] - 1 + work.a0[work.binf]
        xj[work.ainf] *= -1
        return xj

    def post_func_eval(x, fj, work):
        # Weight integrand as required by substitutions for infinite limits
        if work.log:
            fj[work.abinf] += (np.log(1 + work.xj[work.abinf] ** 2)
                               - 2*np.log(1 - work.xj[work.abinf] ** 2))
            fj[work.binf] -= 2 * np.log(work.xj[work.binf])
        else:
            fj[work.abinf] *= ((1 + work.xj[work.abinf]**2) /
                               (1 - work.xj[work.abinf]**2)**2)
            fj[work.binf] *= work.xj[work.binf]**-2.

        # Estimate integral with Euler-Maclaurin Sum
        fjwj, Sn = _euler_maclaurin_sum(fj, work)
        if work.Sk.shape[-1]:
            Snm1 = work.Sk[:, -1]
            Sn = (special.logsumexp([Snm1 - np.log(2), Sn], axis=0) if log
                  else Snm1 / 2 + Sn)

        work.fjwj = fjwj
        work.Sn = Sn

    def check_termination(work):
        """Terminate due to convergence or encountering non-finite values"""
        stop = np.zeros(work.Sn.shape, dtype=bool)

        # Terminate before first iteration if integration limits are equal
        if work.nit == 0:
            i = (work.a == work.b).ravel()  # ravel singleton dimension
            zero = -np.inf if log else 0
            work.Sn[i] = zero
            work.aerr[i] = zero
            work.status[i] = _ECONVERGED
            stop[i] = True
        else:
            # Terminate if convergence criterion is met
            work.rerr, work.aerr = _estimate_error(work)
            i = ((work.rerr < rtol) | (work.rerr + np.real(work.Sn) < atol) if log
                 else (work.rerr < rtol) | (work.rerr * abs(work.Sn) < atol))
            work.status[i] = _ECONVERGED
            stop[i] = True

        # Terminate if integral estimate becomes invalid
        if log:
            i = (np.isposinf(np.real(work.Sn)) | np.isnan(work.Sn)) & ~stop
        else:
            i = ~np.isfinite(work.Sn) & ~stop
        work.status[i] = _EVALUEERR
        stop[i] = True

        return stop

    def post_termination_check(work):
        work.n += 1
        work.Sk = np.concatenate((work.Sk, work.Sn[:, np.newaxis]), axis=-1)
        return

    def customize_result(res, shape):
        # If the integration limits were such that b < a, we reversed them
        # to perform the calculation, and the final result needs to be negated.
        if log and np.any(negative):
            pi = res['integral'].dtype.type(np.pi)
            j = np.complex64(1j)  # minimum complex type
            res['integral'] = res['integral'] + negative*pi*j
        else:
            res['integral'][negative] *= -1

        # For this algorithm, it seems more appropriate to report the maximum
        # level rather than the number of iterations in which it was performed.
        res['maxlevel'] = minlevel + res['nit'] - 1
        res['maxlevel'][res['nit'] == 0] = -1
        del res['nit']
        return shape

    # Suppress all warnings initially, since there are many places in the code
    # for which this is expected behavior.
    with np.errstate(over='ignore', invalid='ignore', divide='ignore'):
        res = _scalar_optimization_loop(work, callback, shape, maxiter, f,
                                        args, dtype, pre_func_eval,
                                        post_func_eval, check_termination,
                                        post_termination_check,
                                        customize_result, res_work_pairs)
    return res


def _get_base_step(dtype=np.float64):
    # Compute the base step length for the provided dtype. Theoretically, the
    # Euler-Maclaurin sum is infinite, but it gets cut off when either the
    # weights underflow or the abscissae cannot be distinguished from the
    # limits of integration. The latter happens to occur first for float32 and
    # float64, and it occurs when `xjc` (the abscissa complement)
    # in `_compute_pair` underflows. We can solve for the argument `tmax` at
    # which it will underflow using [2] Eq. 13.
    fmin = 4*np.finfo(dtype).tiny  # stay a little away from the limit
    tmax = np.arcsinh(np.log(2/fmin - 1) / np.pi)

    # Based on this, we can choose a base step size `h` for level 0.
    # The number of function evaluations will be `2 + m*2^(k+1)`, where `k` is
    # the level and `m` is an integer we get to choose. I choose
    # m = _N_BASE_STEPS = `8` somewhat arbitrarily, but a rationale is that a
    # power of 2 makes floating point arithmetic more predictable. It also
    # results in a base step size close to `1`, which is what [1] uses (and I
    # used here until I found [2] and these ideas settled).
    h0 = tmax / _N_BASE_STEPS
    return h0.astype(dtype)


_N_BASE_STEPS = 8


def _compute_pair(k, h0):
    # Compute the abscissa-weight pairs for each level k. See [1] page 9.

    # For now, we compute and store in 64-bit precision. If higher-precision
    # data types become better supported, it would be good to compute these
    # using the highest precision available. Or, once there is an Array API-
    # compatible arbitrary precision array, we can compute at the required
    # precision.

    # "....each level k of abscissa-weight pairs uses h = 2 **-k"
    # We adapt to floating point arithmetic using ideas of [2].
    h = h0 / 2**k
    max = _N_BASE_STEPS * 2**k

    # For iterations after the first, "....the integrand function needs to be
    # evaluated only at the odd-indexed abscissas at each level."
    j = np.arange(max+1) if k == 0 else np.arange(1, max+1, 2)
    jh = j * h

    # "In this case... the weights wj = u1/cosh(u2)^2, where..."
    pi_2 = np.pi / 2
    u1 = pi_2*np.cosh(jh)
    u2 = pi_2*np.sinh(jh)
    # Denominators get big here. Overflow then underflow doesn't need warning.
    # with np.errstate(under='ignore', over='ignore'):
    wj = u1 / np.cosh(u2)**2
    # "We actually store 1-xj = 1/(...)."
    xjc = 1 / (np.exp(u2) * np.cosh(u2))  # complement of xj = np.tanh(u2)

    # When level k == 0, the zeroth xj corresponds with xj = 0. To simplify
    # code, the function will be evaluated there twice; each gets half weight.
    wj[0] = wj[0] / 2 if k == 0 else wj[0]

    return xjc, wj  # store at full precision


def _pair_cache(k, h0):
    # Cache the abscissa-weight pairs up to a specified level.
    # Abscissae and weights of consecutive levels are concatenated.
    # `index` records the indices that correspond with each level:
    # `xjc[index[k]:index[k+1]` extracts the level `k` abscissae.
    if h0 != _pair_cache.h0:
        _pair_cache.xjc = np.empty(0)
        _pair_cache.wj = np.empty(0)
        _pair_cache.indices = [0]

    xjcs = [_pair_cache.xjc]
    wjs = [_pair_cache.wj]

    for i in range(len(_pair_cache.indices)-1, k + 1):
        xjc, wj = _compute_pair(i, h0)
        xjcs.append(xjc)
        wjs.append(wj)
        _pair_cache.indices.append(_pair_cache.indices[-1] + len(xjc))

    _pair_cache.xjc = np.concatenate(xjcs)
    _pair_cache.wj = np.concatenate(wjs)
    _pair_cache.h0 = h0

_pair_cache.xjc = np.empty(0)
_pair_cache.wj = np.empty(0)
_pair_cache.indices = [0]
_pair_cache.h0 = None


def _get_pairs(k, h0, inclusive=False, dtype=np.float64):
    # Retrieve the specified abscissa-weight pairs from the cache
    # If `inclusive`, return all up to and including the specified level
    if len(_pair_cache.indices) <= k+2 or h0 != _pair_cache.h0:
        _pair_cache(k, h0)

    xjc = _pair_cache.xjc
    wj = _pair_cache.wj
    indices = _pair_cache.indices

    start = 0 if inclusive else indices[k]
    end = indices[k+1]

    return xjc[start:end].astype(dtype), wj[start:end].astype(dtype)


def _transform_to_limits(xjc, wj, a, b):
    # Transform integral according to user-specified limits. This is just
    # math that follows from the fact that the standard limits are (-1, 1).
    # Note: If we had stored xj instead of xjc, we would have
    # xj = alpha * xj + beta, where beta = (a + b)/2
    alpha = (b - a) / 2
    xj = np.concatenate((-alpha * xjc + b, alpha * xjc + a), axis=-1)
    wj = wj*alpha  # arguments get broadcasted, so we can't use *=
    wj = np.concatenate((wj, wj), axis=-1)

    # Points at the boundaries can be generated due to finite precision
    # arithmetic, but these function values aren't supposed to be included in
    # the Euler-Maclaurin sum. Ideally we wouldn't evaluate the function at
    # these points; however, we can't easily filter out points since this
    # function is vectorized. Instead, zero the weights.
    invalid = (xj <= a) | (xj >= b)
    wj[invalid] = 0
    return xj, wj


def _euler_maclaurin_sum(fj, work):
    # Perform the Euler-Maclaurin Sum, [1] Section 4

    # The error estimate needs to know the magnitude of the last term
    # omitted from the Euler-Maclaurin sum. This is a bit involved because
    # it may have been computed at a previous level. I sure hope it's worth
    # all the trouble.
    xr0, fr0, wr0 = work.xr0, work.fr0, work.wr0
    xl0, fl0, wl0 = work.xl0, work.fl0, work.wl0

    # It is much more convenient to work with the transposes of our work
    # variables here.
    xj, fj, wj = work.xj.T, fj.T, work.wj.T
    n_x, n_active = xj.shape  # number of abscissae, number of active elements

    # We'll work with the left and right sides separately
    xr, xl = xj.reshape(2, n_x // 2, n_active).copy()  # this gets modified
    fr, fl = fj.reshape(2, n_x // 2, n_active)
    wr, wl = wj.reshape(2, n_x // 2, n_active)

    invalid_r = ~np.isfinite(fr) | (wr == 0)
    invalid_l = ~np.isfinite(fl) | (wl == 0)

    # integer index of the maximum abscissa at this level
    xr[invalid_r] = -np.inf
    ir = np.argmax(xr, axis=0, keepdims=True)
    # abscissa, function value, and weight at this index
    xr_max = np.take_along_axis(xr, ir, axis=0)[0]
    fr_max = np.take_along_axis(fr, ir, axis=0)[0]
    wr_max = np.take_along_axis(wr, ir, axis=0)[0]
    # boolean indices at which maximum abscissa at this level exceeds
    # the incumbent maximum abscissa (from all previous levels)
    j = xr_max > xr0
    # Update record of the incumbent abscissa, function value, and weight
    xr0[j] = xr_max[j]
    fr0[j] = fr_max[j]
    wr0[j] = wr_max[j]

    # integer index of the minimum abscissa at this level
    xl[invalid_l] = np.inf
    il = np.argmin(xl, axis=0, keepdims=True)
    # abscissa, function value, and weight at this index
    xl_min = np.take_along_axis(xl, il, axis=0)[0]
    fl_min = np.take_along_axis(fl, il, axis=0)[0]
    wl_min = np.take_along_axis(wl, il, axis=0)[0]
    # boolean indices at which minimum abscissa at this level is less than
    # the incumbent minimum abscissa (from all previous levels)
    j = xl_min < xl0
    # Update record of the incumbent abscissa, function value, and weight
    xl0[j] = xl_min[j]
    fl0[j] = fl_min[j]
    wl0[j] = wl_min[j]
    fj = fj.T

    # Compute the error estimate `d4` - the magnitude of the leftmost or
    # rightmost term, whichever is greater.
    flwl0 = fl0 + np.log(wl0) if work.log else fl0 * wl0  # leftmost term
    frwr0 = fr0 + np.log(wr0) if work.log else fr0 * wr0  # rightmost term
    magnitude = np.real if work.log else np.abs
    work.d4 = np.maximum(magnitude(flwl0), magnitude(frwr0))

    # There are two approaches to dealing with function values that are
    # numerically infinite due to approaching a singularity - zero them, or
    # replace them with the function value at the nearest non-infinite point.
    # [3] pg. 22 suggests the latter, so let's do that given that we have the
    # information.
    fr0b = np.broadcast_to(fr0[np.newaxis, :], fr.shape)
    fl0b = np.broadcast_to(fl0[np.newaxis, :], fl.shape)
    fr[invalid_r] = fr0b[invalid_r]
    fl[invalid_l] = fl0b[invalid_l]

    # When wj is zero, log emits a warning
    # with np.errstate(divide='ignore'):
    fjwj = fj + np.log(work.wj) if work.log else fj * work.wj

    # update integral estimate
    Sn = (special.logsumexp(fjwj + np.log(work.h), axis=-1) if work.log
          else np.sum(fjwj, axis=-1) * work.h)

    work.xr0, work.fr0, work.wr0 = xr0, fr0, wr0
    work.xl0, work.fl0, work.wl0 = xl0, fl0, wl0

    return fjwj, Sn


def _estimate_error(work):
    # Estimate the error according to [1] Section 5

    if work.n == 0 or work.nit == 0:
        # The paper says to use "one" as the error before it can be calculated.
        # NaN seems to be more appropriate.
        nan = np.full_like(work.Sn, np.nan)
        return nan, nan

    indices = _pair_cache.indices

    n_active = len(work.Sn)  # number of active elements
    axis_kwargs = dict(axis=-1, keepdims=True)

    # With a jump start (starting at level higher than 0), we haven't
    # explicitly calculated the integral estimate at lower levels. But we have
    # all the function value-weight products, so we can compute the
    # lower-level estimates.
    if work.Sk.shape[-1] == 0:
        h = 2 * work.h  # step size at this level
        n_x = indices[work.n]  # number of abscissa up to this level
        # The right and left fjwj terms from all levels are concatenated along
        # the last axis. Get out only the terms up to this level.
        fjwj_rl = work.fjwj.reshape(n_active, 2, -1)
        fjwj = fjwj_rl[:, :, :n_x].reshape(n_active, 2*n_x)
        # Compute the Euler-Maclaurin sum at this level
        Snm1 = (special.logsumexp(fjwj, **axis_kwargs) + np.log(h) if work.log
                else np.sum(fjwj, **axis_kwargs) * h)
        work.Sk = np.concatenate((Snm1, work.Sk), axis=-1)

    if work.n == 1:
        nan = np.full_like(work.Sn, np.nan)
        return nan, nan

    # The paper says not to calculate the error for n<=2, but it's not clear
    # about whether it starts at level 0 or level 1. We start at level 0, so
    # why not compute the error beginning in level 2?
    if work.Sk.shape[-1] < 2:
        h = 4 * work.h  # step size at this level
        n_x = indices[work.n-1]  # number of abscissa up to this level
        # The right and left fjwj terms from all levels are concatenated along
        # the last axis. Get out only the terms up to this level.
        fjwj_rl = work.fjwj.reshape(len(work.Sn), 2, -1)
        fjwj = fjwj_rl[..., :n_x].reshape(n_active, 2*n_x)
        # Compute the Euler-Maclaurin sum at this level
        Snm2 = (special.logsumexp(fjwj, **axis_kwargs) + np.log(h) if work.log
                else np.sum(fjwj, **axis_kwargs) * h)
        work.Sk = np.concatenate((Snm2, work.Sk), axis=-1)

    Snm2 = work.Sk[..., -2]
    Snm1 = work.Sk[..., -1]

    e1 = work.eps

    if work.log:
        log_e1 = np.log(e1)
        # Currently, only real integrals are supported in log-scale. All
        # complex values have imaginary part in increments of pi*j, which just
        # carries sign information of the original integral, so use of
        # `np.real` here is equivalent to absolute value in real scale.
        d1 = np.real(special.logsumexp([work.Sn, Snm1 + work.pi*1j], axis=0))
        d2 = np.real(special.logsumexp([work.Sn, Snm2 + work.pi*1j], axis=0))
        d3 = log_e1 + np.max(np.real(work.fjwj), axis=-1)
        d4 = work.d4
        aerr = np.max([d1 ** 2 / d2, 2 * d1, d3, d4], axis=0)
        rerr = np.maximum(log_e1, aerr - np.real(work.Sn))
    else:
        # Note: explicit computation of log10 of each of these is unnecessary.
        d1 = np.abs(work.Sn - Snm1)
        d2 = np.abs(work.Sn - Snm2)
        d3 = e1 * np.max(np.abs(work.fjwj), axis=-1)
        d4 = work.d4
        # If `d1` is 0, no need to warn. This does the right thing.
        # with np.errstate(divide='ignore'):
        aerr = np.max([d1**(np.log(d1)/np.log(d2)), d1**2, d3, d4], axis=0)
        rerr = np.maximum(e1, aerr/np.abs(work.Sn))
    return rerr, aerr.reshape(work.Sn.shape)


def _transform_integrals(a, b):
    # Transform integrals to a form with finite a < b
    # For b < a, we reverse the limits and will multiply the final result by -1
    # For infinite limit on the right, we use the substitution x = 1/t - 1 + a
    # For infinite limit on the left, we substitute x = -x and treat as above
    # For infinite limits, we substitute x = t / (1-t**2)

    negative = b < a
    a[negative], b[negative] = b[negative], a[negative]

    abinf = np.isinf(a) & np.isinf(b)
    a[abinf], b[abinf] = -1, 1

    ainf = np.isinf(a)
    a[ainf], b[ainf] = -b[ainf], -a[ainf]

    binf = np.isinf(b)
    a0 = a.copy()
    a[binf], b[binf] = 0, 1

    return a, b, a0, negative, abinf, ainf, binf


def _tanhsinh_iv(f, a, b, log, maxfun, maxlevel, minlevel,
                 atol, rtol, args, callback):
    # Input validation and standardization

    message = '`f` must be callable.'
    if not callable(f):
        raise ValueError(message)

    message = 'All elements of `a` and `b` must be real numbers.'
    a, b = np.broadcast_arrays(a, b)
    if np.any(np.iscomplex(a)) or np.any(np.iscomplex(b)):
        raise ValueError(message)

    message = '`log` must be True or False.'
    if log not in {True, False}:
        raise ValueError(message)
    log = bool(log)

    if atol is None:
        atol = -np.inf if log else 0

    rtol_temp = rtol if rtol is not None else 0.

    params = np.asarray([atol, rtol_temp, 0.])
    message = "`atol` and `rtol` must be real numbers."
    if not np.issubdtype(params.dtype, np.floating):
        raise ValueError(message)

    if log:
        message = '`atol` and `rtol` may not be positive infinity.'
        if np.any(np.isposinf(params)):
            raise ValueError(message)
    else:
        message = '`atol` and `rtol` must be non-negative and finite.'
        if np.any(params < 0) or np.any(np.isinf(params)):
            raise ValueError(message)
    atol = params[0]
    rtol = rtol if rtol is None else params[1]

    BIGINT = float(2**62)
    if maxfun is None and maxlevel is None:
        maxlevel = 10

    maxfun = BIGINT if maxfun is None else maxfun
    maxlevel = BIGINT if maxlevel is None else maxlevel

    message = '`maxfun`, `maxlevel`, and `minlevel` must be integers.'
    params = np.asarray([maxfun, maxlevel, minlevel])
    if not (np.issubdtype(params.dtype, np.number)
            and np.all(np.isreal(params))
            and np.all(params.astype(np.int64) == params)):
        raise ValueError(message)
    message = '`maxfun`, `maxlevel`, and `minlevel` must be non-negative.'
    if np.any(params < 0):
        raise ValueError(message)
    maxfun, maxlevel, minlevel = params.astype(np.int64)
    minlevel = min(minlevel, maxlevel)

    if not np.iterable(args):
        args = (args,)

    if callback is not None and not callable(callback):
        raise ValueError('`callback` must be callable.')

    return f, a, b, log, maxfun, maxlevel, minlevel, atol, rtol, args, callback