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