"""TEST-ONLY stand-in for python-flint (float based, radius 0). NOT rigorous. Used only to debug the
certificate scripts in an environment without python-flint. The real run must use python-flint."""
import math
class ctx: prec=53
class arb:
    __slots__=('v',)
    def __init__(s,v=0.0,r=None):
        if isinstance(v,arb): v=v.v
        s.v=float(v)
    @staticmethod
    def pi(): return arb(math.pi)
    def lower(s): return arb(s.v)
    def upper(s): return arb(s.v)
    def mid(s): return arb(s.v)
    def rad(s): return arb(0.0)
    def __float__(s): return s.v
    def log(s): return arb(math.log(s.v))
    def exp(s): return arb(math.exp(s.v))
    def sqrt(s): return arb(math.sqrt(s.v))
    def cos(s): return arb(math.cos(s.v))
    def sin(s): return arb(math.sin(s.v))
    def atan(s): return arb(math.atan(s.v))
    def __add__(s,o):
        if type(o).__name__=='acb': return NotImplemented
        return arb(s.v+arb(o).v)
    __radd__=__add__
    def __sub__(s,o):
        if type(o).__name__=='acb': return NotImplemented
        return arb(s.v-arb(o).v)
    def __rsub__(s,o):
        if type(o).__name__=='acb': return NotImplemented
        return arb(arb(o).v-s.v)
    def __mul__(s,o):
        if type(o).__name__=='acb': return NotImplemented
        return arb(s.v*arb(o).v)
    __rmul__=__mul__
    def __truediv__(s,o):
        if type(o).__name__=='acb': return NotImplemented
        return arb(s.v/arb(o).v)
    def __rtruediv__(s,o):
        if type(o).__name__=='acb': return NotImplemented
        return arb(arb(o).v/s.v)
    def __neg__(s): return arb(-s.v)
    def __pow__(s,o): return arb(s.v**arb(o).v)
    def __abs__(s): return arb(abs(s.v))
    def __lt__(s,o): return s.v<arb(o).v
    def __gt__(s,o): return s.v>arb(o).v
    def __repr__(s): return 'arb(%r)'%s.v
import cmath
class acb:
    __slots__=('v',)
    def __init__(s,re=0.0,im=0.0):
        if isinstance(re,acb): s.v=re.v; return
        s.v=complex(float(arb(re).v) if not isinstance(re,complex) else re)+1j*float(arb(im).v)
    @property
    def real(s): return arb(s.v.real)
    @property
    def imag(s): return arb(s.v.imag)
    def log(s): return acb._c(cmath.log(s.v))
    def exp(s): return acb._c(cmath.exp(s.v))
    @staticmethod
    def _c(z): a=acb(); a.v=z; return a
    @staticmethod
    def _v(o):
        if isinstance(o,acb): return o.v
        if isinstance(o,arb): return o.v
        return complex(o)
    def __add__(s,o): return acb._c(s.v+acb._v(o))
    __radd__=__add__
    def __sub__(s,o): return acb._c(s.v-acb._v(o))
    def __rsub__(s,o): return acb._c(acb._v(o)-s.v)
    def __mul__(s,o): return acb._c(s.v*acb._v(o))
    __rmul__=__mul__
    def __truediv__(s,o): return acb._c(s.v/acb._v(o))
    def __rtruediv__(s,o): return acb._c(acb._v(o)/s.v)
    def __neg__(s): return acb._c(-s.v)
    def __abs__(s): return arb(abs(s.v))