def _jacobi_theta2(z, q):
extra1 = 10
extra2 = 20
# the loops below break when the fixed precision quantities
# a and b go to zero;
# right shifting small negative numbers by wp one obtains -1, not zero,
# so the condition a**2 + b**2 > MIN is used to break the loops.
MIN = 2
if z == zero:
if isinstance(q, mpf):
wp = mp.prec + extra1
x = to_fixed(q._mpf_, wp)
x2 = (x*x) >> wp
a = b = x2
s = x2
while abs(a) > MIN:
b = (b*x2) >> wp
a = (a*b) >> wp
s += a
s = (1 << (wp+1)) + (s << 1)
s = mpf(from_man_exp(s, -wp, mp.prec, 'n'))
else:
wp = mp.prec + extra1
xre, xim = q._mpc_
xre = to_fixed(xre, wp)
xim = to_fixed(xim, wp)
x2re = (xre*xre - xim*xim) >> wp
x2im = (xre*xim) >> (wp - 1)
are = bre = x2re
aim = bim = x2im
sre = (1<<wp) + are
sim = aim
while are**2 + aim**2 > MIN:
bre, bim = (bre * x2re - bim * x2im) >> wp, \
(bre * x2im + bim * x2re) >> wp
are, aim = (are * bre - aim * bim) >> wp, \
(are * bim + aim * bre) >> wp
sre += are
sim += aim
sre = (sre << 1)
sim = (sim << 1)
sre = from_man_exp(sre, -wp, mp.prec, 'n')
sim = from_man_exp(sim, -wp, mp.prec, 'n')
s = mpc(sre, sim)
else:
if isinstance(q, mpf) and isinstance(z, mpf):
wp = mp.prec + extra1
x = to_fixed(q._mpf_, wp)
x2 = (x*x) >> wp
a = b = x2
c1, s1 = cos_sin(z._mpf_, wp)
cn = c1 = to_fixed(c1, wp)
sn = s1 = to_fixed(s1, wp)
c2 = (c1*c1 - s1*s1) >> wp
s2 = (c1 * s1) >> (wp - 1)
cn, sn = (cn*c2 - sn*s2) >> wp, (sn*c2 + cn*s2) >> wp
s = c1 + ((a * cn) >> wp)
while abs(a) > MIN:
b = (b*x2) >> wp
a = (a*b) >> wp
cn, sn = (cn*c2 - sn*s2) >> wp, (sn*c2 + cn*s2) >> wp
s += (a * cn) >> wp
s = (s << 1)
s = mpf(from_man_exp(s, -wp, mp.prec, 'n'))
s *= nthroot(q, 4)
return s
# case z real, q complex
elif isinstance(z, mpf):
wp = mp.prec + extra2
xre, xim = q._mpc_
xre = to_fixed(xre, wp)
xim = to_fixed(xim, wp)
x2re = (xre*xre - xim*xim) >> wp
x2im = (xre*xim) >> (wp - 1)
are = bre = x2re
aim = bim = x2im
c1, s1 = cos_sin(z._mpf_, wp)
cn = c1 = to_fixed(c1, wp)
sn = s1 = to_fixed(s1, wp)
c2 = (c1*c1 - s1*s1) >> wp
s2 = (c1 * s1) >> (wp - 1)
cn, sn = (cn*c2 - sn*s2) >> wp, (sn*c2 + cn*s2) >> wp
sre = c1 + ((are * cn) >> wp)
sim = ((aim * cn) >> wp)
while are**2 + aim**2 > MIN:
bre, bim = (bre * x2re - bim * x2im) >> wp, \
(bre * x2im + bim * x2re) >> wp
are, aim = (are * bre - aim * bim) >> wp, \
(are * bim + aim * bre) >> wp
cn, sn = (cn*c2 - sn*s2) >> wp, (sn*c2 + cn*s2) >> wp
sre += ((are * cn) >> wp)
sim += ((aim * cn) >> wp)
sre = (sre << 1)
sim = (sim << 1)
sre = from_man_exp(sre, -wp, mp.prec, 'n')
sim = from_man_exp(sim, -wp, mp.prec, 'n')
s = mpc(sre, sim)
#case z complex, q real
elif isinstance(q, mpf):
wp = mp.prec + extra2
x = to_fixed(q._mpf_, wp)
x2 = (x*x) >> wp
a = b = x2
prec0 = mp.prec
mp.prec = wp
c1 = cos(z)
s1 = sin(z)
mp.prec = prec0
cnre = c1re = to_fixed(c1.real._mpf_, wp)
cnim = c1im = to_fixed(c1.imag._mpf_, wp)
snre = s1re = to_fixed(s1.real._mpf_, wp)
snim = s1im = to_fixed(s1.imag._mpf_, wp)
#c2 = (c1*c1 - s1*s1) >> wp
c2re = (c1re*c1re - c1im*c1im - s1re*s1re + s1im*s1im) >> wp
c2im = (c1re*c1im - s1re*s1im) >> (wp - 1)
#s2 = (c1 * s1) >> (wp - 1)
s2re = (c1re*s1re - c1im*s1im) >> (wp - 1)
s2im = (c1re*s1im + c1im*s1re) >> (wp - 1)
#cn, sn = (cn*c2 - sn*s2) >> wp, (sn*c2 + cn*s2) >> wp
t1 = (cnre*c2re - cnim*c2im - snre*s2re + snim*s2im) >> wp
t2 = (cnre*c2im + cnim*c2re - snre*s2im - snim*s2re) >> wp
t3 = (snre*c2re - snim*c2im + cnre*s2re - cnim*s2im) >> wp
t4 = (snre*c2im + snim*c2re + cnre*s2im + cnim*s2re) >> wp
cnre = t1
cnim = t2
snre = t3
snim = t4
sre = c1re + ((a * cnre) >> wp)
sim = c1im + ((a * cnim) >> wp)
while abs(a) > MIN:
b = (b*x2) >> wp
a = (a*b) >> wp
t1 = (cnre*c2re - cnim*c2im - snre*s2re + snim*s2im) >> wp
t2 = (cnre*c2im + cnim*c2re - snre*s2im - snim*s2re) >> wp
t3 = (snre*c2re - snim*c2im + cnre*s2re - cnim*s2im) >> wp
t4 = (snre*c2im + snim*c2re + cnre*s2im + cnim*s2re) >> wp
cnre = t1
cnim = t2
snre = t3
snim = t4
sre += ((a * cnre) >> wp)
sim += ((a * cnim) >> wp)
sre = (sre << 1)
sim = (sim << 1)
sre = from_man_exp(sre, -wp, mp.prec, 'n')
sim = from_man_exp(sim, -wp, mp.prec, 'n')
s = mpc(sre, sim)
# case z and q complex
else:
wp = mp.prec + extra2
xre, xim = q._mpc_
xre = to_fixed(xre, wp)
xim = to_fixed(xim, wp)
x2re = (xre*xre - xim*xim) >> wp
x2im = (xre*xim) >> (wp - 1)
are = bre = x2re
aim = bim = x2im
prec0 = mp.prec
mp.prec = wp
# cos(z), siz(z) with z complex
c1 = cos(z)
s1 = sin(z)
mp.prec = prec0
cnre = c1re = to_fixed(c1.real._mpf_, wp)
cnim = c1im = to_fixed(c1.imag._mpf_, wp)
snre = s1re = to_fixed(s1.real._mpf_, wp)
snim = s1im = to_fixed(s1.imag._mpf_, wp)
c2re = (c1re*c1re - c1im*c1im - s1re*s1re + s1im*s1im) >> wp
c2im = (c1re*c1im - s1re*s1im) >> (wp - 1)
s2re = (c1re*s1re - c1im*s1im) >> (wp - 1)
s2im = (c1re*s1im + c1im*s1re) >> (wp - 1)
t1 = (cnre*c2re - cnim*c2im - snre*s2re + snim*s2im) >> wp
t2 = (cnre*c2im + cnim*c2re - snre*s2im - snim*s2re) >> wp
t3 = (snre*c2re - snim*c2im + cnre*s2re - cnim*s2im) >> wp
t4 = (snre*c2im + snim*c2re + cnre*s2im + cnim*s2re) >> wp
cnre = t1
cnim = t2
snre = t3
snim = t4
n = 1
termre = c1re
termim = c1im
sre = c1re + ((are * cnre - aim * cnim) >> wp)
sim = c1im + ((are * cnim + aim * cnre) >> wp)
n = 3
termre = ((are * cnre - aim * cnim) >> wp)
termim = ((are * cnim + aim * cnre) >> wp)
sre = c1re + ((are * cnre - aim * cnim) >> wp)
sim = c1im + ((are * cnim + aim * cnre) >> wp)
n = 5
while are**2 + aim**2 > MIN:
bre, bim = (bre * x2re - bim * x2im) >> wp, \
(bre * x2im + bim * x2re) >> wp
are, aim = (are * bre - aim * bim) >> wp, \
(are * bim + aim * bre) >> wp
#cn, sn = (cn*c1 - sn*s1) >> wp, (sn*c1 + cn*s1) >> wp
t1 = (cnre*c2re - cnim*c2im - snre*s2re + snim*s2im) >> wp
t2 = (cnre*c2im + cnim*c2re - snre*s2im - snim*s2re) >> wp
t3 = (snre*c2re - snim*c2im + cnre*s2re - cnim*s2im) >> wp
t4 = (snre*c2im + snim*c2re + cnre*s2im + cnim*s2re) >> wp
cnre = t1
cnim = t2
snre = t3
snim = t4
termre = ((are * cnre - aim * cnim) >> wp)
termim = ((aim * cnre + are * cnim) >> wp)
sre += ((are * cnre - aim * cnim) >> wp)
sim += ((aim * cnre + are * cnim) >> wp)
n += 2
sre = (sre << 1)
sim = (sim << 1)
sre = from_man_exp(sre, -wp, mp.prec, 'n')
sim = from_man_exp(sim, -wp, mp.prec, 'n')
s = mpc(sre, sim)
s *= nthroot(q, 4)
return s
def _jacobi_theta3(z, q):
extra1 = 10
extra2 = 20
MIN = 2
if z == zero:
if isinstance(q, mpf):
wp = mp.prec + extra1
x = to_fixed(q._mpf_, wp)
s = x
a = b = x
x2 = (x*x) >> wp
while abs(a) > MIN:
b = (b*x2) >> wp
a = (a*b) >> wp
s += a
s = (1 << wp) + (s << 1)
s = mpf(from_man_exp(s, -wp, mp.prec, 'n'))
return s
else:
wp = mp.prec + extra1
xre, xim = q._mpc_
xre = to_fixed(xre, wp)
xim = to_fixed(xim, wp)
x2re = (xre*xre - xim*xim) >> wp
x2im = (xre*xim) >> (wp - 1)
sre = are = bre = xre
sim = aim = bim = xim
while are**2 + aim**2 > MIN:
bre, bim = (bre * x2re - bim * x2im) >> wp, \
(bre * x2im + bim * x2re) >> wp
are, aim = (are * bre - aim * bim) >> wp, \
(are * bim + aim * bre) >> wp
sre += are
sim += aim
sre = (1 << wp) + (sre << 1)
sim = (sim << 1)
sre = from_man_exp(sre, -wp, mp.prec, 'n')
sim = from_man_exp(sim, -wp, mp.prec, 'n')
s = mpc(sre, sim)
return s
else:
if isinstance(q, mpf) and isinstance(z, mpf):
s = MP_ZERO
wp = mp.prec + extra1
x = to_fixed(q._mpf_, wp)
a = b = x
x2 = (x*x) >> wp
c1, s1 = cos_sin(mpf_shift(z._mpf_, 1), wp)
c1 = to_fixed(c1, wp)
s1 = to_fixed(s1, wp)
cn = c1
sn = s1
s += (a * cn) >> wp
while abs(a) > MIN:
b = (b*x2) >> wp
a = (a*b) >> wp
cn, sn = (cn*c1 - sn*s1) >> wp, (sn*c1 + cn*s1) >> wp
s += (a * cn) >> wp
s = (1 << wp) + (s << 1)
s = mpf(from_man_exp(s, -wp, mp.prec, 'n'))
return s
# case z real, q complex
elif isinstance(z, mpf):
wp = mp.prec + extra2
xre, xim = q._mpc_
xre = to_fixed(xre, wp)
xim = to_fixed(xim, wp)
x2re = (xre*xre - xim*xim) >> wp
x2im = (xre*xim) >> (wp - 1)
are = bre = xre
aim = bim = xim
c1, s1 = cos_sin(mpf_shift(z._mpf_, 1), wp)
c1 = to_fixed(c1, wp)
s1 = to_fixed(s1, wp)
cn = c1
sn = s1
sre = (are * cn) >> wp
sim = (aim * cn) >> wp
while are**2 + aim**2 > MIN:
bre, bim = (bre * x2re - bim * x2im) >> wp, \
(bre * x2im + bim * x2re) >> wp
are, aim = (are * bre - aim * bim) >> wp, \
(are * bim + aim * bre) >> wp
cn, sn = (cn*c1 - sn*s1) >> wp, (sn*c1 + cn*s1) >> wp
sre += (are * cn) >> wp
sim += (aim * cn) >> wp
sre = (1 << wp) + (sre << 1)
sim = (sim << 1)
sre = from_man_exp(sre, -wp, mp.prec, 'n')
sim = from_man_exp(sim, -wp, mp.prec, 'n')
s = mpc(sre, sim)
return s
#case z complex, q real
elif isinstance(q, mpf):
wp = mp.prec + extra2
x = to_fixed(q._mpf_, wp)
a = b = x
x2 = (x*x) >> wp
prec0 = mp.prec
mp.prec = wp
c1 = cos(2*z)
s1 = sin(2*z)
mp.prec = prec0
cnre = c1re = to_fixed(c1.real._mpf_, wp)
cnim = c1im = to_fixed(c1.imag._mpf_, wp)
snre = s1re = to_fixed(s1.real._mpf_, wp)
snim = s1im = to_fixed(s1.imag._mpf_, wp)
sre = (a * cnre) >> wp
sim = (a * cnim) >> wp
while abs(a) > MIN:
b = (b*x2) >> wp
a = (a*b) >> wp
t1 = (cnre*c1re - cnim*c1im - snre*s1re + snim*s1im) >> wp
t2 = (cnre*c1im + cnim*c1re - snre*s1im - snim*s1re) >> wp
t3 = (snre*c1re - snim*c1im + cnre*s1re - cnim*s1im) >> wp
t4 = (snre*c1im + snim*c1re + cnre*s1im + cnim*s1re) >> wp
cnre = t1
cnim = t2
snre = t3
snim = t4
sre += (a * cnre) >> wp
sim += (a * cnim) >> wp
sre = (1 << wp) + (sre << 1)
sim = (sim << 1)
sre = from_man_exp(sre, -wp, mp.prec, 'n')
sim = from_man_exp(sim, -wp, mp.prec, 'n')
s = mpc(sre, sim)
return s
# case z and q complex
else:
wp = mp.prec + extra2
xre, xim = q._mpc_
xre = to_fixed(xre, wp)
xim = to_fixed(xim, wp)
x2re = (xre*xre - xim*xim) >> wp
x2im = (xre*xim) >> (wp - 1)
are = bre = xre
aim = bim = xim
prec0 = mp.prec
mp.prec = wp
# cos(2*z), sin(2*z) with z complex
c1 = cos(2*z)
s1 = sin(2*z)
mp.prec = prec0
cnre = c1re = to_fixed(c1.real._mpf_, wp)
cnim = c1im = to_fixed(c1.imag._mpf_, wp)
snre = s1re = to_fixed(s1.real._mpf_, wp)
snim = s1im = to_fixed(s1.imag._mpf_, wp)
sre = (are * cnre - aim * cnim) >> wp
sim = (aim * cnre + are * cnim) >> wp
while are**2 + aim**2 > MIN:
bre, bim = (bre * x2re - bim * x2im) >> wp, \
(bre * x2im + bim * x2re) >> wp
are, aim = (are * bre - aim * bim) >> wp, \
(are * bim + aim * bre) >> wp
t1 = (cnre*c1re - cnim*c1im - snre*s1re + snim*s1im) >> wp
t2 = (cnre*c1im + cnim*c1re - snre*s1im - snim*s1re) >> wp
t3 = (snre*c1re - snim*c1im + cnre*s1re - cnim*s1im) >> wp
t4 = (snre*c1im + snim*c1re + cnre*s1im + cnim*s1re) >> wp
cnre = t1
cnim = t2
snre = t3
snim = t4
sre += (are * cnre - aim * cnim) >> wp
sim += (aim * cnre + are * cnim) >> wp
sre = (1 << wp) + (sre << 1)
sim = (sim << 1)
sre = from_man_exp(sre, -wp, mp.prec, 'n')
sim = from_man_exp(sim, -wp, mp.prec, 'n')
s = mpc(sre, sim)
return s
def _djacobi_theta3(z, q, nd):
"""nd=1,2,3 order of the derivative with respect to z"""
MIN = 2
extra1 = 10
extra2 = 20
if isinstance(q, mpf) and isinstance(z, mpf):
s = MP_ZERO
wp = mp.prec + extra1
x = to_fixed(q._mpf_, wp)
a = b = x
x2 = (x*x) >> wp
c1, s1 = cos_sin(mpf_shift(z._mpf_, 1), wp)
c1 = to_fixed(c1, wp)
s1 = to_fixed(s1, wp)
cn = c1
sn = s1
if (nd&1):
s += (a * sn) >> wp
else:
s += (a * cn) >> wp
n = 2
while abs(a) > MIN:
b = (b*x2) >> wp
a = (a*b) >> wp
cn, sn = (cn*c1 - sn*s1) >> wp, (sn*c1 + cn*s1) >> wp
if nd&1:
s += (a * sn * n**nd) >> wp
else:
s += (a * cn * n**nd) >> wp
n += 1
s = -(s << (nd+1))
s = mpf(from_man_exp(s, -wp, mp.prec, 'n'))
# case z real, q complex
elif isinstance(z, mpf):
wp = mp.prec + extra2
xre, xim = q._mpc_
xre = to_fixed(xre, wp)
xim = to_fixed(xim, wp)
x2re = (xre*xre - xim*xim) >> wp
x2im = (xre*xim) >> (wp - 1)
are = bre = xre
aim = bim = xim
c1, s1 = cos_sin(mpf_shift(z._mpf_, 1), wp)
c1 = to_fixed(c1, wp)
s1 = to_fixed(s1, wp)
cn = c1
sn = s1
if (nd&1):
sre = (are * sn) >> wp
sim = (aim * sn) >> wp
else:
sre = (are * cn) >> wp
sim = (aim * cn) >> wp
n = 2
while are**2 + aim**2 > MIN:
bre, bim = (bre * x2re - bim * x2im) >> wp, \
(bre * x2im + bim * x2re) >> wp
are, aim = (are * bre - aim * bim) >> wp, \
(are * bim + aim * bre) >> wp
cn, sn = (cn*c1 - sn*s1) >> wp, (sn*c1 + cn*s1) >> wp
if nd&1:
sre += (are * sn * n**nd) >> wp
sim += (aim * sn * n**nd) >> wp
else:
sre += (are * cn * n**nd) >> wp
sim += (aim * cn * n**nd) >> wp
n += 1
sre = -(sre << (nd+1))
sim = -(sim << (nd+1))
sre = from_man_exp(sre, -wp, mp.prec, 'n')
sim = from_man_exp(sim, -wp, mp.prec, 'n')
s = mpc(sre, sim)
#case z complex, q real
elif isinstance(q, mpf):
wp = mp.prec + extra2
x = to_fixed(q._mpf_, wp)
a = b = x
x2 = (x*x) >> wp
prec0 = mp.prec
mp.prec = wp
c1 = cos(2*z)
s1 = sin(2*z)
mp.prec = prec0
cnre = c1re = to_fixed(c1.real._mpf_, wp)
cnim = c1im = to_fixed(c1.imag._mpf_, wp)
snre = s1re = to_fixed(s1.real._mpf_, wp)
snim = s1im = to_fixed(s1.imag._mpf_, wp)
if (nd&1):
sre = (a * snre) >> wp
sim = (a * snim) >> wp
else:
sre = (a * cnre) >> wp
sim = (a * cnim) >> wp
n = 2
while abs(a) > MIN:
b = (b*x2) >> wp
a = (a*b) >> wp
t1 = (cnre*c1re - cnim*c1im - snre*s1re + snim*s1im) >> wp
t2 = (cnre*c1im + cnim*c1re - snre*s1im - snim*s1re) >> wp
t3 = (snre*c1re - snim*c1im + cnre*s1re - cnim*s1im) >> wp
t4 = (snre*c1im + snim*c1re + cnre*s1im + cnim*s1re) >> wp
cnre = t1
cnim = t2
snre = t3
snim = t4
if (nd&1):
sre += (a * snre * n**nd) >> wp
sim += (a * snim * n**nd) >> wp
else:
sre += (a * cnre * n**nd) >> wp
sim += (a * cnim * n**nd) >> wp
n += 1
sre = -(sre << (nd+1))
sim = -(sim << (nd+1))
sre = from_man_exp(sre, -wp, mp.prec, 'n')
sim = from_man_exp(sim, -wp, mp.prec, 'n')
s = mpc(sre, sim)
# case z and q complex
else:
wp = mp.prec + extra2
xre, xim = q._mpc_
xre = to_fixed(xre, wp)
xim = to_fixed(xim, wp)
x2re = (xre*xre - xim*xim) >> wp
x2im = (xre*xim) >> (wp - 1)
are = bre = xre
aim = bim = xim
prec0 = mp.prec
mp.prec = wp
# cos(2*z), sin(2*z) with z complex
c1 = cos(2*z)
s1 = sin(2*z)
mp.prec = prec0
cnre = c1re = to_fixed(c1.real._mpf_, wp)
cnim = c1im = to_fixed(c1.imag._mpf_, wp)
snre = s1re = to_fixed(s1.real._mpf_, wp)
snim = s1im = to_fixed(s1.imag._mpf_, wp)
if (nd&1):
sre = (are * snre - aim * snim) >> wp
sim = (aim * snre + are * snim) >> wp
else:
sre = (are * cnre - aim * cnim) >> wp
sim = (aim * cnre + are * cnim) >> wp
n = 2
while are**2 + aim**2 > MIN:
bre, bim = (bre * x2re - bim * x2im) >> wp, \
(bre * x2im + bim * x2re) >> wp
are, aim = (are * bre - aim * bim) >> wp, \
(are * bim + aim * bre) >> wp
t1 = (cnre*c1re - cnim*c1im - snre*s1re + snim*s1im) >> wp
t2 = (cnre*c1im + cnim*c1re - snre*s1im - snim*s1re) >> wp
t3 = (snre*c1re - snim*c1im + cnre*s1re - cnim*s1im) >> wp
t4 = (snre*c1im + snim*c1re + cnre*s1im + cnim*s1re) >> wp
cnre = t1
cnim = t2
snre = t3
snim = t4
if(nd&1):
sre += ((are * snre - aim * snim) * n**nd) >> wp
sim += ((aim * snre + are * snim) * n**nd) >> wp
else:
sre += ((are * cnre - aim * cnim) * n**nd) >> wp
sim += ((aim * cnre + are * cnim) * n**nd) >> wp
n += 1
sre = -(sre << (nd+1))
sim = -(sim << (nd+1))
sre = from_man_exp(sre, -wp, mp.prec, 'n')
sim = from_man_exp(sim, -wp, mp.prec, 'n')
s = mpc(sre, sim)
if (nd&1):
return (-1)**(nd//2) * s
else:
return (-1)**(1 + nd//2) * s
def _djacobi_theta2(z, q, nd):
MIN = 2
extra1 = 10
extra2 = 20
if isinstance(q, mpf) and isinstance(z, mpf):
wp = mp.prec + extra1
x = to_fixed(q._mpf_, wp)
x2 = (x*x) >> wp
a = b = x2
c1, s1 = cos_sin(z._mpf_, wp)
cn = c1 = to_fixed(c1, wp)
sn = s1 = to_fixed(s1, wp)
c2 = (c1*c1 - s1*s1) >> wp
s2 = (c1 * s1) >> (wp - 1)
cn, sn = (cn*c2 - sn*s2) >> wp, (sn*c2 + cn*s2) >> wp
if (nd&1):
s = s1 + ((a * sn * 3**nd) >> wp)
else:
s = c1 + ((a * cn * 3**nd) >> wp)
n = 2
while abs(a) > MIN:
b = (b*x2) >> wp
a = (a*b) >> wp
cn, sn = (cn*c2 - sn*s2) >> wp, (sn*c2 + cn*s2) >> wp
if nd&1:
s += (a * sn * (2*n+1)**nd) >> wp
else:
s += (a * cn * (2*n+1)**nd) >> wp
n += 1
s = -(s << 1)
s = mpf(from_man_exp(s, -wp, mp.prec, 'n'))
# case z real, q complex
elif isinstance(z, mpf):
wp = mp.prec + extra2
xre, xim = q._mpc_
xre = to_fixed(xre, wp)
xim = to_fixed(xim, wp)
x2re = (xre*xre - xim*xim) >> wp
x2im = (xre*xim) >> (wp - 1)
are = bre = x2re
aim = bim = x2im
c1, s1 = cos_sin(z._mpf_, wp)
cn = c1 = to_fixed(c1, wp)
sn = s1 = to_fixed(s1, wp)
c2 = (c1*c1 - s1*s1) >> wp
s2 = (c1 * s1) >> (wp - 1)
cn, sn = (cn*c2 - sn*s2) >> wp, (sn*c2 + cn*s2) >> wp
if (nd&1):
sre = s1 + ((are * sn * 3**nd) >> wp)
sim = ((aim * sn * 3**nd) >> wp)
else:
sre = c1 + ((are * cn * 3**nd) >> wp)
sim = ((aim * cn * 3**nd) >> wp)
n = 5
while are**2 + aim**2 > MIN:
bre, bim = (bre * x2re - bim * x2im) >> wp, \
(bre * x2im + bim * x2re) >> wp
are, aim = (are * bre - aim * bim) >> wp, \
(are * bim + aim * bre) >> wp
cn, sn = (cn*c2 - sn*s2) >> wp, (sn*c2 + cn*s2) >> wp
if (nd&1):
sre += ((are * sn * n**nd) >> wp)
sim += ((aim * sn * n**nd) >> wp)
else:
sre += ((are * cn * n**nd) >> wp)
sim += ((aim * cn * n**nd) >> wp)
n += 2
sre = -(sre << 1)
sim = -(sim << 1)
sre = from_man_exp(sre, -wp, mp.prec, 'n')
sim = from_man_exp(sim, -wp, mp.prec, 'n')
s = mpc(sre, sim)
#case z complex, q real
elif isinstance(q, mpf):
wp = mp.prec + extra2
x = to_fixed(q._mpf_, wp)
x2 = (x*x) >> wp
a = b = x2
prec0 = mp.prec
mp.prec = wp
c1 = cos(z)
s1 = sin(z)
mp.prec = prec0
cnre = c1re = to_fixed(c1.real._mpf_, wp)
cnim = c1im = to_fixed(c1.imag._mpf_, wp)
snre = s1re = to_fixed(s1.real._mpf_, wp)
snim = s1im = to_fixed(s1.imag._mpf_, wp)
#c2 = (c1*c1 - s1*s1) >> wp
c2re = (c1re*c1re - c1im*c1im - s1re*s1re + s1im*s1im) >> wp
c2im = (c1re*c1im - s1re*s1im) >> (wp - 1)
#s2 = (c1 * s1) >> (wp - 1)
s2re = (c1re*s1re - c1im*s1im) >> (wp - 1)
s2im = (c1re*s1im + c1im*s1re) >> (wp - 1)
#cn, sn = (cn*c2 - sn*s2) >> wp, (sn*c2 + cn*s2) >> wp
t1 = (cnre*c2re - cnim*c2im - snre*s2re + snim*s2im) >> wp
t2 = (cnre*c2im + cnim*c2re - snre*s2im - snim*s2re) >> wp
t3 = (snre*c2re - snim*c2im + cnre*s2re - cnim*s2im) >> wp
t4 = (snre*c2im + snim*c2re + cnre*s2im + cnim*s2re) >> wp
cnre = t1
cnim = t2
snre = t3
snim = t4
if (nd&1):
sre = s1re + ((a * snre * 3**nd) >> wp)
sim = s1im + ((a * snim * 3**nd) >> wp)
else:
sre = c1re + ((a * cnre * 3**nd) >> wp)
sim = c1im + ((a * cnim * 3**nd) >> wp)
n = 5
while abs(a) > MIN:
b = (b*x2) >> wp
a = (a*b) >> wp
t1 = (cnre*c2re - cnim*c2im - snre*s2re + snim*s2im) >> wp
t2 = (cnre*c2im + cnim*c2re - snre*s2im - snim*s2re) >> wp
t3 = (snre*c2re - snim*c2im + cnre*s2re - cnim*s2im) >> wp
t4 = (snre*c2im + snim*c2re + cnre*s2im + cnim*s2re) >> wp
cnre = t1
cnim = t2
snre = t3
snim = t4
if (nd&1):
sre += ((a * snre * n**nd) >> wp)
sim += ((a * snim * n**nd) >> wp)
else:
sre += ((a * cnre * n**nd) >> wp)
sim += ((a * cnim * n**nd) >> wp)
n += 2
sre = -(sre << 1)
sim = -(sim << 1)
sre = from_man_exp(sre, -wp, mp.prec, 'n')
sim = from_man_exp(sim, -wp, mp.prec, 'n')
s = mpc(sre, sim)
# case z and q complex
else:
wp = mp.prec + extra2
xre, xim = q._mpc_
xre = to_fixed(xre, wp)
xim = to_fixed(xim, wp)
x2re = (xre*xre - xim*xim) >> wp
x2im = (xre*xim) >> (wp - 1)
are = bre = x2re
aim = bim = x2im
prec0 = mp.prec
mp.prec = wp
# cos(2*z), siz(2*z) with z complex
c1 = cos(z)
s1 = sin(z)
mp.prec = prec0
cnre = c1re = to_fixed(c1.real._mpf_, wp)
cnim = c1im = to_fixed(c1.imag._mpf_, wp)
snre = s1re = to_fixed(s1.real._mpf_, wp)
snim = s1im = to_fixed(s1.imag._mpf_, wp)
c2re = (c1re*c1re - c1im*c1im - s1re*s1re + s1im*s1im) >> wp
c2im = (c1re*c1im - s1re*s1im) >> (wp - 1)
s2re = (c1re*s1re - c1im*s1im) >> (wp - 1)
s2im = (c1re*s1im + c1im*s1re) >> (wp - 1)
t1 = (cnre*c2re - cnim*c2im - snre*s2re + snim*s2im) >> wp
t2 = (cnre*c2im + cnim*c2re - snre*s2im - snim*s2re) >> wp
t3 = (snre*c2re - snim*c2im + cnre*s2re - cnim*s2im) >> wp
t4 = (snre*c2im + snim*c2re + cnre*s2im + cnim*s2re) >> wp
cnre = t1
cnim = t2
snre = t3
snim = t4
if (nd&1):
sre = s1re + (((are * snre - aim * snim) * 3**nd) >> wp)
sim = s1im + (((are * snim + aim * snre)* 3**nd) >> wp)
else:
sre = c1re + (((are * cnre - aim * cnim) * 3**nd) >> wp)
sim = c1im + (((are * cnim + aim * cnre)* 3**nd) >> wp)
n = 5
while are**2 + aim**2 > MIN:
bre, bim = (bre * x2re - bim * x2im) >> wp, \
(bre * x2im + bim * x2re) >> wp
are, aim = (are * bre - aim * bim) >> wp, \
(are * bim + aim * bre) >> wp
#cn, sn = (cn*c1 - sn*s1) >> wp, (sn*c1 + cn*s1) >> wp
t1 = (cnre*c2re - cnim*c2im - snre*s2re + snim*s2im) >> wp
t2 = (cnre*c2im + cnim*c2re - snre*s2im - snim*s2re) >> wp
t3 = (snre*c2re - snim*c2im + cnre*s2re - cnim*s2im) >> wp
t4 = (snre*c2im + snim*c2re + cnre*s2im + cnim*s2re) >> wp
cnre = t1
cnim = t2
snre = t3
snim = t4
if (nd&1):
sre += (((are * snre - aim * snim) * n**nd) >> wp)
sim += (((aim * snre + are * snim) * n**nd) >> wp)
else:
sre += (((are * cnre - aim * cnim) * n**nd) >> wp)
sim += (((aim * cnre + are * cnim) * n**nd) >> wp)
n += 2
sre = -(sre << 1)
sim = -(sim << 1)
sre = from_man_exp(sre, -wp, mp.prec, 'n')
sim = from_man_exp(sim, -wp, mp.prec, 'n')
s = mpc(sre, sim)
s *= nthroot(q, 4)
if (nd&1):
return (-1)**(nd//2) * s
else:
return (-1)**(1 + nd//2) * s
