def test_incomplete(x, y, z):
# TODO: sign of sqrt(x, y, z)
assume(distinct(x, y, z))
# https://dlmf.nist.gov/19.21.E7
assert nice_and_close(
abs(3 * elliprf(x, y, z) - ((x - y) * elliprd(y, z, x) +
(z - y) * elliprd(x, y, z))),
abs(3 * (y / x / z)**0.5))
# https://dlmf.nist.gov/19.21.E8
assert nice_and_close(
abs(elliprd(y, z, x) + elliprd(z, x, y) + elliprd(x, y, z)),
abs(3 * (x * y * z)**(-0.5)))
# https://dlmf.nist.gov/19.21.E9
assert nice_and_close(
x * elliprd(y, z, x) + y * elliprd(z, x, y) + z * elliprd(x, y, z),
3 * elliprf(x, y, z))
# https://dlmf.nist.gov/19.21.E10
assert nice_and_close(
abs(2 * elliprg(x, y, z) - (z * elliprf(x, y, z) - (x - z) *
(y - z) * elliprd(x, y, z) / 3)),
abs((x * y / z)**0.5))
# https://dlmf.nist.gov/19.21.E11
assert nice_and_close(
6 * elliprg(x, y, z), 3 * (x + y + z) * elliprf(x, y, z) -
sum(_x**2 * elliprd(_y, _z, _x)
for _x, _y, _z in circular_shifts((x, y, z))))
assert nice_and_close(
6 * elliprg(x, y, z),
sum(_x * (_y + _z) * elliprd(_y, _z, _x)
for _x, _y, _z in circular_shifts((x, y, z))))
def test_complete_3(y, z, p):
# https://dlmf.nist.gov/19.21.E6
assume(distinct(y, z, p))
if z < y < p or p < y < z:
y, z = z, y
r = (y - p) / (y - z)
assume(r > 0)
assert nice_and_close(
(r * p)**0.5 / z * elliprj(0, y, z, p),
(r - 1) * elliprf(0, y, z) * elliprd(p, r * z, z) +
elliprd(0, y, z) * elliprf(p, r * z, z))
def test_elliprj(x, y, z, p):
# TODO: properly analytically continue
assume(distinct(x, y, z, p))
# https://dlmf.nist.gov/19.21.E13
q = (y - x) * (z - x) / (p - x) + x
# # https://dlmf.nist.gov/19.21.E12
# TODO: elliprc sign issue
assert nice_and_close(
((p - x) * elliprj(x, y, z, p) +
(q - x) * elliprj(x, y, z, q)).real,
(3 * (elliprf(x, y, z) - elliprc(y * z / x, p * q / x))).real)
# https://dlmf.nist.gov/19.21.E15
# special case of above with x=0
q = y * z / p
assert nice_and_close(
(p * elliprj(0, y, z, p) + q * elliprj(0, y, z, q)).real,
3 * elliprf(0, y, z).real)
def test_elliprj(x, y, z, p, l):
assert nice_and_close(elliprj(x, x, x, x), x**(-1.5))
assert nice_and_close(elliprj(l * x, l * y, l * z, l * p),
elliprj(x, y, z, p) / l**1.5)
assert nice_and_close(elliprj(x, y, z, z), elliprd(x, y, z))
assert isinf(elliprj(0, 0, z, p))
assert nice_and_close(elliprj(x, x, x, p), elliprd(p, p, x))
assert nice_and_close(
elliprj(x, x, x, p).real,
3 * (elliprc(x, p) - x**(-0.5)) / (x - p))
assert nice_and_close(elliprj(x, y, y, y), elliprd(x, y, y))
# assert nice_and_close(elliprj(0, y, z, (y*z)**0.5), 1.5/(y*z)**0.5 * elliprf(0, y, z))
# assert nice_and_close(elliprj(0, y, z, -(y*z)**0.5), -1.5/(y*z)**0.5 * elliprf(0, y, z))
# TODO: elliprc sign issue
assume(distinct(x, y, z))
p = x + ((y - x) * (z - x))**0.5
assert nice_and_close(
(p - x) * elliprj(x, y, z, p),
1.5 * (elliprf(x, y, z) - x**0.5 * elliprc(y * z, p**2)))
def test_elliprj2(y, p):
assume(distinct(y, p) and y != 0)
assert nice_and_close(
elliprj(0, y, y, -p).real, -1.5 * pi / y**0.5 / (y + p))
def test_elliprj1(x, y, p):
assume(distinct(y, p))
assert nice_and_close(elliprj(0, y, y, p),
1.5 * pi / (y * p**0.5 + p * y**0.5))
assert nice_and_close(elliprj(x, y, y, p),
3 * (elliprc(x, y) - elliprc(x, p)) / (p - y))
def test_stress(self, m, args, z1, z2):
assume(distinct(*args, d_min=SMALL))
assume(abs(z1-z2) >= SMALL)
self._test(args, m, z1, z2, rtol=1e-2, atol=1e-2)