Python cmath.pi() Examples

def show(self):
        super().show()  # Clear working area
        ssd = self.device
        x0 = self.x0
        y0 = self.y0
        radius = self.radius
        adivs = self.adivs
        rdivs = self.rdivs
        diam = 2 * radius
        if rdivs > 0:
            for r in range(1, rdivs + 1):
                circle(ssd, self.xp_origin, self.yp_origin, round(radius * r / rdivs), self.gridcolor)
        if adivs > 0:
            v = complex(1)
            m = rect(1, pi / adivs)
            for _ in range(adivs):
                self.cline(-v, v, self.gridcolor)
                v *= m
        ssd.vline(x0 + radius, y0, diam, self.fgcolor)
        ssd.hline(x0, y0 + radius, diam, self.fgcolor) 

def test_tauegamma_implementation(self):
        input_dict_list=[{
        'Cgamma_taue':np.random.random()*1e-8*exp(1j*2*pi*np.random.random()),
        'Cgamma_etau':np.random.random()*1e-8*exp(1j*2*pi*np.random.random()),
        } for i in range(10)]
        BRs = np.array([
            flavio.np_prediction(
                'BR(tau->egamma)',
                Wilson(input_dict, 100,  'WET', 'flavio')
            )
            for input_dict in input_dict_list
        ])
        compare_BRs = np.array([
            compare_BR(
                Wilson(input_dict, 100,  'WET', 'flavio'),
                'tau', 'e',
            )
            for input_dict in input_dict_list
        ])
        self.assertAlmostEqual(np.max(np.abs(1-BRs/compare_BRs)), 0, delta=0.002) 

def test_tauegamma_implementation(self):
        input_dict_list=[{
        'Cgamma_mue':np.random.random()*1e-8*exp(1j*2*pi*np.random.random()),
        'Cgamma_emu':np.random.random()*1e-8*exp(1j*2*pi*np.random.random()),
        } for i in range(10)]
        BRs = np.array([
            flavio.np_prediction(
                'BR(mu->egamma)',
                Wilson(input_dict, 100,  'WET', 'flavio')
            )
            for input_dict in input_dict_list
        ])
        compare_BRs = np.array([
            compare_BR(
                Wilson(input_dict, 100,  'WET', 'flavio'),
                'mu', 'e',
            )
            for input_dict in input_dict_list
        ])
        self.assertAlmostEqual(np.max(np.abs(1-BRs/compare_BRs)), 0, delta=0.005) 

def test_taumugamma_implementation(self):
        input_dict_list=[{
        'Cgamma_taumu':np.random.random()*1e-8*exp(1j*2*pi*np.random.random()),
        'Cgamma_mutau':np.random.random()*1e-8*exp(1j*2*pi*np.random.random()),
        } for i in range(10)]
        BRs = np.array([
            flavio.np_prediction(
                'BR(tau->mugamma)',
                Wilson(input_dict, 100,  'WET', 'flavio')
            )
            for input_dict in input_dict_list
        ])
        compare_BRs = np.array([
            compare_BR(
                Wilson(input_dict, 100,  'WET', 'flavio'),
                'tau', 'mu',
            )
            for input_dict in input_dict_list
        ])
        self.assertAlmostEqual(np.max(np.abs(1-BRs/compare_BRs)), 0, delta=0.02) 

def arrow(tft, origin, vec, lc, color):
    ccw = cmath.exp(3j * cmath.pi/4)  # Unit vectors
    cw = cmath.exp(-3j * cmath.pi/4)
    length, theta = cmath.polar(vec)
    uv = cmath.rect(1, theta)  # Unit rotation vector
    start = -vec
    if length > 3 * lc:  # If line is long
        ds = cmath.rect(lc, theta)
        start += ds  # shorten to allow for length of tail chevrons
    chev = lc + 0j
    pline(tft, origin, vec, color)  # Origin to tip
    pline(tft, origin, start, color)  # Origin to tail
    pline(tft, origin + conj(vec), chev*ccw*uv, color)  # Tip chevron
    pline(tft, origin + conj(vec), chev*cw*uv, color)
    if length > lc:  # Confusing appearance of very short vectors with tail chevron
        pline(tft, origin + conj(start), chev*ccw*uv, color)  # Tail chevron
        pline(tft, origin + conj(start), chev*cw*uv, color)

# Vector display 

def aclock(dial, lbldate, lbltim):
    uv = lambda phi : rect(1, phi)  # Return a unit vector of phase phi
    days = ('Monday', 'Tuesday', 'Wednesday', 'Thursday', 'Friday', 'Saturday',
            'Sunday')
    months = ('January', 'February', 'March', 'April', 'May', 'June', 'July',
              'August', 'September', 'October', 'November', 'December')

    hrs = Pointer(dial)
    mins = Pointer(dial)
    secs = Pointer(dial)

    hstart =  0 + 0.7j  # Pointer lengths. Position at top.
    mstart = 0 + 1j
    sstart = 0 + 1j 

    while True:
        t = time.localtime()
        hrs.value(hstart * uv(-t[3] * pi/6 - t[4] * pi / 360), CYAN)
        mins.value(mstart * uv(-t[4] * pi/30), CYAN)
        secs.value(sstart * uv(-t[5] * pi/30), RED)
        lbltim.value('{:02d}.{:02d}.{:02d}'.format(t[3], t[4], t[5]))
        lbldate.value('{} {} {} {}'.format(days[t[6]], t[2], months[t[1] - 1], t[0]))
        await asyncio.sleep(1) 

def compare_BR(wc_wilson, l1, l2):
    scale = flavio.config['renormalization scale'][l1+'decays']
    ll = wcxf_sector_names[l1, l2]
    par = flavio.default_parameters.get_central_all()
    wc_obj = flavio.WilsonCoefficients.from_wilson(wc_wilson, par)
    par = flavio.parameters.default_parameters.get_central_all()
    wc = wc_obj.get_wc(ll, scale, par, nf_out=4)
    alpha = flavio.physics.running.running.get_alpha_e(par, scale, nf_out=4)
    e = sqrt(4 * pi * alpha)
    ml = par['m_' + l1]
    # cf. (18) of hep-ph/0404211
    pre = 48 * pi**3 * alpha / par['GF']**2
    DL = 2 / (e * ml) * wc['Cgamma_' + l1 + l2]
    DR = 2 / (e * ml) * wc['Cgamma_' + l2 + l1].conjugate()
    if l1 == 'tau':
        BR_SL = par['BR(tau->{}nunu)'.format(l2)]
    else:
        BR_SL = 1  # BR(mu->enunu) = 1
    return pre * (abs(DL)**2 + abs(DR)**2) * BR_SL 

def arrow(dev, origin, vec, lc, color, ccw=cmath.exp(3j * cmath.pi/4), cw=cmath.exp(-3j * cmath.pi/4)):
    length, theta = cmath.polar(vec)
    uv = cmath.rect(1, theta)  # Unit rotation vector
    start = -vec
    if length > 3 * lc:  # If line is long
        ds = cmath.rect(lc, theta)
        start += ds  # shorten to allow for length of tail chevrons
    chev = lc + 0j
    polar(dev, origin, vec, color)  # Origin to tip
    polar(dev, origin, start, color)  # Origin to tail
    polar(dev, origin + conj(vec), chev*ccw*uv, color)  # Tip chevron
    polar(dev, origin + conj(vec), chev*cw*uv, color)
    if length > lc:  # Confusing appearance of very short vectors with tail chevron
        polar(dev, origin + conj(start), chev*ccw*uv, color)  # Tail chevron
        polar(dev, origin + conj(start), chev*cw*uv, color)

# If a (framebuf based) device is passed to refresh, the screen is cleared.
# None causes pending widgets to be drawn and the result to be copied to hardware.
# The pend mechanism enables a displayable object to postpone its renedering
# until it is complete: efficient for e.g. Dial which may have multiple Pointers 

def clock(x):
    print('Clock test.')
    refresh(ssd, True)  # Clear any prior image
    lbl = Label(wri, 5, 85, 'Clock')
    dial = Dial(wri, 5, 5, height = 75, ticks = 12, bdcolor=None, label=50)  # Border in fg color
    hrs = Pointer(dial)
    mins = Pointer(dial)
    hrs.value(0 + 0.7j, RED)
    mins.value(0 + 0.9j, YELLOW)
    dm = cmath.rect(1, -cmath.pi/30)  # Rotate by 1 minute (CW)
    dh = cmath.rect(1, -cmath.pi/1800)  # Rotate hours by 1 minute
    for n in range(x):
        refresh(ssd)
        utime.sleep_ms(200)
        mins.value(mins.value() * dm, YELLOW)
        hrs.value(hrs.value() * dh, RED)
        dial.text('ticks: {}'.format(n))
    lbl.value('Done') 

def test_polar(self):
        self.assertCISEqual(polar(0), (0., 0.))
        self.assertCISEqual(polar(1.), (1., 0.))
        self.assertCISEqual(polar(-1.), (1., pi))
        self.assertCISEqual(polar(1j), (1., pi/2))
        self.assertCISEqual(polar(-1j), (1., -pi/2)) 

def test_rect(self):
        self.assertCEqual(rect(0, 0), (0, 0))
        self.assertCEqual(rect(1, 0), (1., 0))
        self.assertCEqual(rect(1, -pi), (-1., 0))
        self.assertCEqual(rect(1, pi/2), (0, 1.))
        self.assertCEqual(rect(1, -pi/2), (0, -1.)) 

def test_phase(self):
        self.assertAlmostEqual(phase(0), 0.)
        self.assertAlmostEqual(phase(1.), 0.)
        self.assertAlmostEqual(phase(-1.), pi)
        self.assertAlmostEqual(phase(-1.+1E-300j), pi)
        self.assertAlmostEqual(phase(-1.-1E-300j), -pi)
        self.assertAlmostEqual(phase(1j), pi/2)
        self.assertAlmostEqual(phase(-1j), -pi/2)

        # zeros
        self.assertEqual(phase(complex(0.0, 0.0)), 0.0)
        self.assertEqual(phase(complex(0.0, -0.0)), -0.0)
        self.assertEqual(phase(complex(-0.0, 0.0)), pi)
        self.assertEqual(phase(complex(-0.0, -0.0)), -pi)

        # infinities
        self.assertAlmostEqual(phase(complex(-INF, -0.0)), -pi)
        self.assertAlmostEqual(phase(complex(-INF, -2.3)), -pi)
        self.assertAlmostEqual(phase(complex(-INF, -INF)), -0.75*pi)
        self.assertAlmostEqual(phase(complex(-2.3, -INF)), -pi/2)
        self.assertAlmostEqual(phase(complex(-0.0, -INF)), -pi/2)
        self.assertAlmostEqual(phase(complex(0.0, -INF)), -pi/2)
        self.assertAlmostEqual(phase(complex(2.3, -INF)), -pi/2)
        self.assertAlmostEqual(phase(complex(INF, -INF)), -pi/4)
        self.assertEqual(phase(complex(INF, -2.3)), -0.0)
        self.assertEqual(phase(complex(INF, -0.0)), -0.0)
        self.assertEqual(phase(complex(INF, 0.0)), 0.0)
        self.assertEqual(phase(complex(INF, 2.3)), 0.0)
        self.assertAlmostEqual(phase(complex(INF, INF)), pi/4)
        self.assertAlmostEqual(phase(complex(2.3, INF)), pi/2)
        self.assertAlmostEqual(phase(complex(0.0, INF)), pi/2)
        self.assertAlmostEqual(phase(complex(-0.0, INF)), pi/2)
        self.assertAlmostEqual(phase(complex(-2.3, INF)), pi/2)
        self.assertAlmostEqual(phase(complex(-INF, INF)), 0.75*pi)
        self.assertAlmostEqual(phase(complex(-INF, 2.3)), pi)
        self.assertAlmostEqual(phase(complex(-INF, 0.0)), pi)

        # real or imaginary part NaN
        for z in complex_nans:
            self.assertTrue(math.isnan(phase(z))) 

def test_rect(self):
        self.assertCEqual(rect(0, 0), (0, 0))
        self.assertCEqual(rect(1, 0), (1., 0))
        self.assertCEqual(rect(1, -pi), (-1., 0))
        self.assertCEqual(rect(1, pi/2), (0, 1.))
        self.assertCEqual(rect(1, -pi/2), (0, -1.)) 

def test_rect(self):
        self.assertCEqual(rect(0, 0), (0, 0))
        self.assertCEqual(rect(1, 0), (1., 0))
        self.assertCEqual(rect(1, -pi), (-1., 0))
        self.assertCEqual(rect(1, pi/2), (0, 1.))
        self.assertCEqual(rect(1, -pi/2), (0, -1.)) 

def test_constants(self):
        e_expected = 2.71828182845904523536
        pi_expected = 3.14159265358979323846
        self.assertAlmostEqual(cmath.pi, pi_expected, places=9,
            msg="cmath.pi is {}; should be {}".format(cmath.pi, pi_expected))
        self.assertAlmostEqual(cmath.e, e_expected, places=9,
            msg="cmath.e is {}; should be {}".format(cmath.e, e_expected)) 

def _draw_background(self, divisions: int = 10):
        """
        Draws the background of the dial

        :param divisions: the number of divisions
        between 'ticks' shown on the dial
        :return: None
        """
        self.canvas.create_arc(2, 2, self.size-2, self.size-2,
                               style=tk.PIESLICE, start=-60, extent=30,
                               fill='red')
        self.canvas.create_arc(2, 2, self.size-2, self.size-2,
                               style=tk.PIESLICE, start=-30, extent=60,
                               fill='yellow')
        self.canvas.create_arc(2, 2, self.size-2, self.size-2,
                               style=tk.PIESLICE, start=30, extent=210,
                               fill='green')

        # find the distance between the center and the inner tick radius
        inner_tick_radius = int(self.size * 0.4)
        outer_tick_radius = int(self.size * 0.5)

        for tick in range(divisions):
            angle_in_radians = (2.0 * cmath.pi / 3.0) \
                               + tick/divisions * (5.0 * cmath.pi / 3.0)
            inner_point = cmath.rect(inner_tick_radius, angle_in_radians)
            outer_point = cmath.rect(outer_tick_radius, angle_in_radians)

            self.canvas.create_line(
                *self.to_absolute(inner_point.real, inner_point.imag),
                *self.to_absolute(outer_point.real, outer_point.imag),
                width=1
            ) 

def set_value(self, number: (float, int)):
        """
        Sets the value of the graphic

        :param number: the number (must be between 0 and \
        'max_range' or the scale will peg the limits
        :return: None
        """
        self.canvas.delete('all')
        self.canvas.create_image(0, 0, image=self.image, anchor='nw')

        number = number if number <= self.max_value else self.max_value
        number = 0.0 if number < 0.0 else number

        radius = 0.9 * self.size/2.0
        angle_in_radians = (2.0 * cmath.pi / 3.0) \
            + number / self.max_value * (5.0 * cmath.pi / 3.0)

        center = cmath.rect(0, 0)
        outer = cmath.rect(radius, angle_in_radians)
        if self.needle_thickness == 0:
            line_width = int(5 * self.size / 200)
            line_width = 1 if line_width < 1 else line_width
        else:
            line_width = self.needle_thickness

        self.canvas.create_line(
            *self.to_absolute(center.real, center.imag),
            *self.to_absolute(outer.real, outer.imag),
            width=line_width,
            fill=self.needle_color
        )

        self.readout['text'] = '{}{}'.format(number, self.unit) 

def test_phase(self):
        self.assertAlmostEqual(phase(0), 0.)
        self.assertAlmostEqual(phase(1.), 0.)
        self.assertAlmostEqual(phase(-1.), pi)
        self.assertAlmostEqual(phase(-1.+1E-300j), pi)
        self.assertAlmostEqual(phase(-1.-1E-300j), -pi)
        self.assertAlmostEqual(phase(1j), pi/2)
        self.assertAlmostEqual(phase(-1j), -pi/2)

        # zeros
        self.assertEqual(phase(complex(0.0, 0.0)), 0.0)
        self.assertEqual(phase(complex(0.0, -0.0)), -0.0)
        self.assertEqual(phase(complex(-0.0, 0.0)), pi)
        self.assertEqual(phase(complex(-0.0, -0.0)), -pi)

        # infinities
        self.assertAlmostEqual(phase(complex(-INF, -0.0)), -pi)
        self.assertAlmostEqual(phase(complex(-INF, -2.3)), -pi)
        self.assertAlmostEqual(phase(complex(-INF, -INF)), -0.75*pi)
        self.assertAlmostEqual(phase(complex(-2.3, -INF)), -pi/2)
        self.assertAlmostEqual(phase(complex(-0.0, -INF)), -pi/2)
        self.assertAlmostEqual(phase(complex(0.0, -INF)), -pi/2)
        self.assertAlmostEqual(phase(complex(2.3, -INF)), -pi/2)
        self.assertAlmostEqual(phase(complex(INF, -INF)), -pi/4)
        self.assertEqual(phase(complex(INF, -2.3)), -0.0)
        self.assertEqual(phase(complex(INF, -0.0)), -0.0)
        self.assertEqual(phase(complex(INF, 0.0)), 0.0)
        self.assertEqual(phase(complex(INF, 2.3)), 0.0)
        self.assertAlmostEqual(phase(complex(INF, INF)), pi/4)
        self.assertAlmostEqual(phase(complex(2.3, INF)), pi/2)
        self.assertAlmostEqual(phase(complex(0.0, INF)), pi/2)
        self.assertAlmostEqual(phase(complex(-0.0, INF)), pi/2)
        self.assertAlmostEqual(phase(complex(-2.3, INF)), pi/2)
        self.assertAlmostEqual(phase(complex(-INF, INF)), 0.75*pi)
        self.assertAlmostEqual(phase(complex(-INF, 2.3)), pi)
        self.assertAlmostEqual(phase(complex(-INF, 0.0)), pi)

        # real or imaginary part NaN
        for z in complex_nans:
            self.assertTrue(math.isnan(phase(z))) 

def test_constants(self):
        e_expected = 2.71828182845904523536
        pi_expected = 3.14159265358979323846
        self.assertAlmostEqual(cmath.pi, pi_expected, places=9,
            msg="cmath.pi is {}; should be {}".format(cmath.pi, pi_expected))
        self.assertAlmostEqual(cmath.e, e_expected, places=9,
            msg="cmath.e is {}; should be {}".format(cmath.e, e_expected)) 

def show(self):
        # cache bound variables
        tft = self.tft
        ticks = self.ticks
        radius = self.radius
        xo = self.xorigin
        yo = self.yorigin
        vor = self.vor
        if self.redraw:  # An overlaying screen has closed. Force redraw.
            self.redraw = False
            self.drawn = False
        if not self.drawn:
            self.drawn = True
            vtstart = (1 - self.TICKLEN) * radius + 0j  # start of tick
            vtick = self.TICKLEN * radius + 0j  # tick
            vrot = cmath.exp(2j * cmath.pi/ticks)  # unit rotation
            for _ in range(ticks):
                pline(tft, vor + conj(vtstart), vtick, self.fgcolor)
                vtick *= vrot
                vtstart *= vrot
            tft.draw_circle(xo, yo, radius, self.fgcolor)

        vshort = 1000  # Length of shortest vector
        for v in self.vectors:
            val = v.value() * radius  # val is complex
            vshort = min(vshort, cmath.polar(val)[0])
            v.show()
        if isinstance(self.pip, tuple) and vshort > 9:
            tft.fill_circle(xo, yo, 3, self.pip) 

def test_rect(self):
        self.assertCEqual(rect(0, 0), (0, 0))
        self.assertCEqual(rect(1, 0), (1., 0))
        self.assertCEqual(rect(1, -pi), (-1., 0))
        self.assertCEqual(rect(1, pi/2), (0, 1.))
        self.assertCEqual(rect(1, -pi/2), (0, -1.)) 

def test_phase(self):
        self.assertAlmostEqual(phase(0), 0.)
        self.assertAlmostEqual(phase(1.), 0.)
        self.assertAlmostEqual(phase(-1.), pi)
        self.assertAlmostEqual(phase(-1.+1E-300j), pi)
        self.assertAlmostEqual(phase(-1.-1E-300j), -pi)
        self.assertAlmostEqual(phase(1j), pi/2)
        self.assertAlmostEqual(phase(-1j), -pi/2)

        # zeros
        self.assertEqual(phase(complex(0.0, 0.0)), 0.0)
        self.assertEqual(phase(complex(0.0, -0.0)), -0.0)
        self.assertEqual(phase(complex(-0.0, 0.0)), pi)
        self.assertEqual(phase(complex(-0.0, -0.0)), -pi)

        # infinities
        self.assertAlmostEqual(phase(complex(-INF, -0.0)), -pi)
        self.assertAlmostEqual(phase(complex(-INF, -2.3)), -pi)
        self.assertAlmostEqual(phase(complex(-INF, -INF)), -0.75*pi)
        self.assertAlmostEqual(phase(complex(-2.3, -INF)), -pi/2)
        self.assertAlmostEqual(phase(complex(-0.0, -INF)), -pi/2)
        self.assertAlmostEqual(phase(complex(0.0, -INF)), -pi/2)
        self.assertAlmostEqual(phase(complex(2.3, -INF)), -pi/2)
        self.assertAlmostEqual(phase(complex(INF, -INF)), -pi/4)
        self.assertEqual(phase(complex(INF, -2.3)), -0.0)
        self.assertEqual(phase(complex(INF, -0.0)), -0.0)
        self.assertEqual(phase(complex(INF, 0.0)), 0.0)
        self.assertEqual(phase(complex(INF, 2.3)), 0.0)
        self.assertAlmostEqual(phase(complex(INF, INF)), pi/4)
        self.assertAlmostEqual(phase(complex(2.3, INF)), pi/2)
        self.assertAlmostEqual(phase(complex(0.0, INF)), pi/2)
        self.assertAlmostEqual(phase(complex(-0.0, INF)), pi/2)
        self.assertAlmostEqual(phase(complex(-2.3, INF)), pi/2)
        self.assertAlmostEqual(phase(complex(-INF, INF)), 0.75*pi)
        self.assertAlmostEqual(phase(complex(-INF, 2.3)), pi)
        self.assertAlmostEqual(phase(complex(-INF, 0.0)), pi)

        # real or imaginary part NaN
        for z in complex_nans:
            self.assertTrue(math.isnan(phase(z))) 

def test_polar(self):
        self.assertCISEqual(polar(0), (0., 0.))
        self.assertCISEqual(polar(1.), (1., 0.))
        self.assertCISEqual(polar(-1.), (1., pi))
        self.assertCISEqual(polar(1j), (1., pi/2))
        self.assertCISEqual(polar(-1j), (1., -pi/2)) 

def test_constants(self):
        e_expected = 2.71828182845904523536
        pi_expected = 3.14159265358979323846
        self.assertAlmostEqual(cmath.pi, pi_expected, places=9,
            msg="cmath.pi is {}; should be {}".format(cmath.pi, pi_expected))
        self.assertAlmostEqual(cmath.e, e_expected, places=9,
            msg="cmath.e is {}; should be {}".format(cmath.e, e_expected)) 

def get_sequence_impedance_matrix(self, phase_impedance_matrix):
        """Get sequence impedance matrix from phase impedance matrix."""

        phase_impedance_matrix = np.array(phase_impedance_matrix)
        # If we have a 3 by 3 phase impedance matrix
        if phase_impedance_matrix.shape == (3, 3):
            a = cmath.exp(complex(0, 2.0 / 3 * cmath.pi))
            A = np.array(
                [
                    [complex(1.0, 0), complex(1.0, 0), complex(1.0, 0)],
                    [complex(1.0, 0), a ** 2, a],
                    [complex(1.0, 0), a, a ** 2],
                ]
            )
            A_inv = (
                1.0
                / 3.0
                * np.array(
                    [
                        [complex(1.0, 0), complex(1.0, 0), complex(1.0, 0)],
                        [complex(1.0, 0), a, a ** 2],
                        [complex(1.0, 0), a ** 2, a],
                    ]
                )
            )
        # if we have a 2 by 2 phase impedance matrix
        elif phase_impedance_matrix.shape == (2, 2):
            A = np.array([[1.0, 1.0], [1.0, -1.0]])
            A_inv = np.array([[0.5, 0.5], [0.5, -0.5]])
        else:
            return []
        return np.dot(A_inv, np.dot(phase_impedance_matrix, A)) 

def fft(a, inv=False):
    n = len(a)
    w = [cmath.rect(1, (-2 if inv else 2) * cmath.pi * i / n) for i in range(n >> 1)]
    rev = [0] * n
    for i in range(n):
        rev[i] = rev[i >> 1] >> 1
        if i & 1:
            rev[i] |= n >> 1
        if i < rev[i]:
            a[i], a[rev[i]] = a[rev[i]], a[i]

    step = 2
    while step <= n:
        half, diff = step >> 1, n // step
        for i in range(0, n, step):
            pw = 0
            for j in range(i, i + half):
                v = a[j + half] * w[pw]
                a[j + half] = a[j] - v
                a[j] += v
                pw += diff
        step <<= 1

    if inv:
        for i in range(n):
            a[i] /= n 

def test_rect(self):
        self.assertCEqual(rect(0, 0), (0, 0))
        self.assertCEqual(rect(1, 0), (1., 0))
        self.assertCEqual(rect(1, -pi), (-1., 0))
        self.assertCEqual(rect(1, pi/2), (0, 1.))
        self.assertCEqual(rect(1, -pi/2), (0, -1.)) 

def test_phase(self):
        self.assertAlmostEqual(phase(0), 0.)
        self.assertAlmostEqual(phase(1.), 0.)
        self.assertAlmostEqual(phase(-1.), pi)
        self.assertAlmostEqual(phase(-1.+1E-300j), pi)
        self.assertAlmostEqual(phase(-1.-1E-300j), -pi)
        self.assertAlmostEqual(phase(1j), pi/2)
        self.assertAlmostEqual(phase(-1j), -pi/2)

        # zeros
        self.assertEqual(phase(complex(0.0, 0.0)), 0.0)
        self.assertEqual(phase(complex(0.0, -0.0)), -0.0)
        self.assertEqual(phase(complex(-0.0, 0.0)), pi)
        self.assertEqual(phase(complex(-0.0, -0.0)), -pi)

        # infinities
        self.assertAlmostEqual(phase(complex(-INF, -0.0)), -pi)
        self.assertAlmostEqual(phase(complex(-INF, -2.3)), -pi)
        self.assertAlmostEqual(phase(complex(-INF, -INF)), -0.75*pi)
        self.assertAlmostEqual(phase(complex(-2.3, -INF)), -pi/2)
        self.assertAlmostEqual(phase(complex(-0.0, -INF)), -pi/2)
        self.assertAlmostEqual(phase(complex(0.0, -INF)), -pi/2)
        self.assertAlmostEqual(phase(complex(2.3, -INF)), -pi/2)
        self.assertAlmostEqual(phase(complex(INF, -INF)), -pi/4)
        self.assertEqual(phase(complex(INF, -2.3)), -0.0)
        self.assertEqual(phase(complex(INF, -0.0)), -0.0)
        self.assertEqual(phase(complex(INF, 0.0)), 0.0)
        self.assertEqual(phase(complex(INF, 2.3)), 0.0)
        self.assertAlmostEqual(phase(complex(INF, INF)), pi/4)
        self.assertAlmostEqual(phase(complex(2.3, INF)), pi/2)
        self.assertAlmostEqual(phase(complex(0.0, INF)), pi/2)
        self.assertAlmostEqual(phase(complex(-0.0, INF)), pi/2)
        self.assertAlmostEqual(phase(complex(-2.3, INF)), pi/2)
        self.assertAlmostEqual(phase(complex(-INF, INF)), 0.75*pi)
        self.assertAlmostEqual(phase(complex(-INF, 2.3)), pi)
        self.assertAlmostEqual(phase(complex(-INF, 0.0)), pi)

        # real or imaginary part NaN
        for z in complex_nans:
            self.assertTrue(math.isnan(phase(z))) 

def check_polar(self, func):
        def check(arg, expected):
            got = func(arg)
            for e, g in zip(expected, got):
                self.rAssertAlmostEqual(e, g)
        check(0, (0., 0.))
        check(1, (1., 0.))
        check(-1, (1., pi))
        check(1j, (1., pi / 2))
        check(-3j, (3., -pi / 2))
        inf = float('inf')
        check(complex(inf, 0), (inf, 0.))
        check(complex(-inf, 0), (inf, pi))
        check(complex(3, inf), (inf, pi / 2))
        check(complex(5, -inf), (inf, -pi / 2))
        check(complex(inf, inf), (inf, pi / 4))
        check(complex(inf, -inf), (inf, -pi / 4))
        check(complex(-inf, inf), (inf, 3 * pi / 4))
        check(complex(-inf, -inf), (inf, -3 * pi / 4))
        nan = float('nan')
        check(complex(nan, 0), (nan, nan))
        check(complex(0, nan), (nan, nan))
        check(complex(nan, nan), (nan, nan))
        check(complex(inf, nan), (inf, nan))
        check(complex(-inf, nan), (inf, nan))
        check(complex(nan, inf), (inf, nan))
        check(complex(nan, -inf), (inf, nan)) 

def test_constants(self):
        e_expected = 2.71828182845904523536
        pi_expected = 3.14159265358979323846
        self.assertAlmostEqual(cmath.pi, pi_expected, places=9,
            msg="cmath.pi is {}; should be {}".format(cmath.pi, pi_expected))
        self.assertAlmostEqual(cmath.e, e_expected, places=9,
            msg="cmath.e is {}; should be {}".format(cmath.e, e_expected))