Incorrect results for comparison expression

edit

I found a work-around in a question I had myself asked about 5 years ago (but apparently forgotten its lesson): "Incorrect result for comparison (precision issues?)" at ask.sagemath.org/question/32371 — the solution is to avoid the "<" operator for comparison (what a gotcha!), and define one's own smaller function:

def smaller(a, b):
    '''Returns True if a < b (and False if b < a). Assumes that a != b.'''
    prec = 53
    while prec < 100000:
        R = RealIntervalField(prec)
        try:
            R(a).intersection(R(b))
        except ValueError:
            return R(a) < R(b)
        else:
            #print '{} bits of precision is not enough'.format(prec)
            prec += prec // 10
    raise Exception('100000 bits of precision was not enough for comparing: ', a, b, ' -- Possibly they are equal')

Then it works correctly:

sage: bool(smaller(1/47749, -5564456128*e + 15125759978))
True

Note that this function assumes that a ≠ b:

sage: smaller(pi/4, 4*arctan(1/5) - arctan(1/239))
...
Exception: ('100000 bits of precision was not enough for comparing: ', 1/4*pi, 4*arctan(1/5) - arctan(1/239), ' -- Possibly they are equal')