Function _ilog

tools/python-3.11.9-amd64/Lib/_pydecimal.py:5738–5784

Integer approximation to M*log(x/M), with absolute error boundable in terms only of x/M. Given positive integers x and M, return an integer approximation to M * log(x/M). For L = 8 and 0.1 <= x/M <= 10 the difference between the approximation and the exact result is at most 22

(x, M, L = 8)

    return q + (2*r + (q&1) > b)

def _ilog(x, M, L = 8):
    """Integer approximation to M*log(x/M), with absolute error boundable
    in terms only of x/M.

    Given positive integers x and M, return an integer approximation to
    M * log(x/M).  For L = 8 and 0.1 <= x/M <= 10 the difference
    between the approximation and the exact result is at most 22.  For
    L = 8 and 1.0 <= x/M <= 10.0 the difference is at most 15.  In
    both cases these are upper bounds on the error; it will usually be
    much smaller."""

    # The basic algorithm is the following: let log1p be the function
    # log1p(x) = log(1+x).  Then log(x/M) = log1p((x-M)/M).  We use
    # the reduction
    #
    #    log1p(y) = 2*log1p(y/(1+sqrt(1+y)))
    #
    # repeatedly until the argument to log1p is small (< 2**-L in
    # absolute value).  For small y we can use the Taylor series
    # expansion
    #
    #    log1p(y) ~ y - y**2/2 + y**3/3 - ... - (-y)**T/T
    #
    # truncating at T such that y**T is small enough.  The whole
    # computation is carried out in a form of fixed-point arithmetic,
    # with a real number z being represented by an integer
    # approximation to z*M.  To avoid loss of precision, the y below
    # is actually an integer approximation to 2**R*y*M, where R is the
    # number of reductions performed so far.

    y = x-M
    # argument reduction; R = number of reductions performed
    R = 0
    while (R <= L and abs(y) << L-R >= M or
           R > L and abs(y) >> R-L >= M):
        y = _div_nearest((M*y) << 1,
                         M + _sqrt_nearest(M*(M+_rshift_nearest(y, R)), M))
        R += 1

    # Taylor series with T terms
    T = -int(-10*len(str(M))//(3*L))
    yshift = _rshift_nearest(y, R)
    w = _div_nearest(M, T)
    for k in range(T-1, 0, -1):
        w = _div_nearest(M, k) - _div_nearest(yshift*w, M)

    return _div_nearest(w*y, M)

def _dlog10(c, e, p):
    """Given integers c, e and p with c > 0, p >= 0, compute an integer