Method _power_exact

Lib/_pydecimal.py:2005–2251

Attempt to compute self**other exactly. Given Decimals self and other and an integer p, attempt to compute an exact result for the power self**other, with p digits of precision. Return None if self**other is not exactly representable in p digits. Assumes th

(self, other, p)

        return _dec_from_triple(sign, str(base), 0)

    def _power_exact(self, other, p):
        """Attempt to compute self**other exactly.

        Given Decimals self and other and an integer p, attempt to
        compute an exact result for the power self**other, with p
        digits of precision.  Return None if self**other is not
        exactly representable in p digits.

        Assumes that elimination of special cases has already been
        performed: self and other must both be nonspecial; self must
        be positive and not numerically equal to 1; other must be
        nonzero.  For efficiency, other._exp should not be too large,
        so that 10**abs(other._exp) is a feasible calculation."""

        # In the comments below, we write x for the value of self and y for the
        # value of other.  Write x = xc*10**xe and abs(y) = yc*10**ye, with xc
        # and yc positive integers not divisible by 10.

        # The main purpose of this method is to identify the *failure*
        # of x**y to be exactly representable with as little effort as
        # possible.  So we look for cheap and easy tests that
        # eliminate the possibility of x**y being exact.  Only if all
        # these tests are passed do we go on to actually compute x**y.

        # Here's the main idea.  Express y as a rational number m/n, with m and
        # n relatively prime and n>0.  Then for x**y to be exactly
        # representable (at *any* precision), xc must be the nth power of a
        # positive integer and xe must be divisible by n.  If y is negative
        # then additionally xc must be a power of either 2 or 5, hence a power
        # of 2**n or 5**n.
        #
        # There's a limit to how small |y| can be: if y=m/n as above
        # then:
        #
        #  (1) if xc != 1 then for the result to be representable we
        #      need xc**(1/n) >= 2, and hence also xc**|y| >= 2.  So
        #      if |y| <= 1/nbits(xc) then xc < 2**nbits(xc) <=
        #      2**(1/|y|), hence xc**|y| < 2 and the result is not
        #      representable.
        #
        #  (2) if xe != 0, |xe|*(1/n) >= 1, so |xe|*|y| >= 1.  Hence if
        #      |y| < 1/|xe| then the result is not representable.
        #
        # Note that since x is not equal to 1, at least one of (1) and
        # (2) must apply.  Now |y| < 1/nbits(xc) iff |yc|*nbits(xc) <
        # 10**-ye iff len(str(|yc|*nbits(xc)) <= -ye.
        #
        # There's also a limit to how large y can be, at least if it's
        # positive: the normalized result will have coefficient xc**y,
        # so if it's representable then xc**y < 10**p, and y <
        # p/log10(xc).  Hence if y*log10(xc) >= p then the result is
        # not exactly representable.

        # if len(str(abs(yc*xe)) <= -ye then abs(yc*xe) < 10**-ye,
        # so |y| < 1/xe and the result is not representable.
        # Similarly, len(str(abs(yc)*xc_bits)) <= -ye implies |y|
        # < 1/nbits(xc).