def twinprime_fixed(prec):
def I(n):
return sum(
moebius(d) << (n // d) for d in xrange(1, n + 1) if not n % d) // n
wp = 2 * prec + 30
res = fone
primes = [from_rational(1, p, wp) for p in [2, 3, 5, 7]]
ppowers = [mpf_mul(p, p, wp) for p in primes]
n = 2
while 1:
a = mpf_zeta_int(n, wp)
for i in range(4):
a = mpf_mul(a, mpf_sub(fone, ppowers[i]), wp)
ppowers[i] = mpf_mul(ppowers[i], primes[i], wp)
a = mpf_pow_int(a, -I(n), wp)
if mpf_pos(a, prec + 10, 'n') == fone:
break
#from libmpf import to_str
#print n, to_str(mpf_sub(fone, a), 6)
res = mpf_mul(res, a, wp)
n += 1
res = mpf_mul(res, from_int(3 * 15 * 35), wp)
res = mpf_div(res, from_int(4 * 16 * 36), wp)
return to_fixed(res, prec)
def mpf_gamma_int(n, prec, rounding=round_fast):
if n < 1000:
return from_int(ifac(n-1), prec, rounding)
# XXX: choose the cutoff less arbitrarily
size = int(n*math.log(n,2))
if prec > size/20.0:
return from_int(ifac(n-1), prec, rounding)
return mpf_gamma(from_int(n), prec, rounding)
def mpf_gamma_int(n, prec, rounding=round_fast):
if n < 1000:
return from_int(int_fac(n - 1), prec, rounding)
# XXX: choose the cutoff less arbitrarily
size = int(n * math.log(n, 2))
if prec > size / 20.0:
return from_int(int_fac(n - 1), prec, rounding)
return mpf_gamma(from_int(n), prec, rounding)
def mpc_psi(m, z, prec, rnd=round_fast):
"""
Computation of the polygamma function of arbitrary integer order
m >= 0, for a complex argument z.
"""
if m == 0:
return mpc_psi0(z, prec, rnd)
re, im = z
wp = prec + 20
sign, man, exp, bc = re
if not man:
if re == finf and im == fzero:
return (fzero, fzero)
if re == fnan:
return fnan
# Recurrence
w = to_int(re)
n = int(0.4 * wp + 4 * m)
s = mpc_zero
if w < n:
for k in xrange(w, n):
t = mpc_pow_int(z, -m - 1, wp)
s = mpc_add(s, t, wp)
z = mpc_add_mpf(z, fone, wp)
zm = mpc_pow_int(z, -m, wp)
z2 = mpc_pow_int(z, -2, wp)
# 1/m*(z+N)^m
integral_term = mpc_div_mpf(zm, from_int(m), wp)
s = mpc_add(s, integral_term, wp)
# 1/2*(z+N)^(-(m+1))
s = mpc_add(s, mpc_mul_mpf(mpc_div(zm, z, wp), fhalf, wp), wp)
a = m + 1
b = 2
k = 1
# Important: we want to sum up to the *relative* error,
# not the absolute error, because psi^(m)(z) might be tiny
magn = mpc_abs(s, 10)
magn = magn[2] + magn[3]
eps = mpf_shift(fone, magn - wp + 2)
while 1:
zm = mpc_mul(zm, z2, wp)
bern = mpf_bernoulli(2 * k, wp)
scal = mpf_mul_int(bern, a, wp)
scal = mpf_div(scal, from_int(b), wp)
term = mpc_mul_mpf(zm, scal, wp)
s = mpc_add(s, term, wp)
szterm = mpc_abs(term, 10)
if k > 2 and mpf_le(szterm, eps):
break
#print k, to_str(szterm, 10), to_str(eps, 10)
a *= (m + 2 * k) * (m + 2 * k + 1)
b *= (2 * k + 1) * (2 * k + 2)
k += 1
# Scale and sign factor
v = mpc_mul_mpf(s, mpf_gamma(from_int(m + 1), wp), prec, rnd)
if not (m & 1):
v = mpf_neg(v[0]), mpf_neg(v[1])
return v
def mpc_psi(m, z, prec, rnd=round_fast):
"""
Computation of the polygamma function of arbitrary integer order
m >= 0, for a complex argument z.
"""
if m == 0:
return mpc_psi0(z, prec, rnd)
re, im = z
wp = prec + 20
sign, man, exp, bc = re
if not man:
if re == finf and im == fzero:
return (fzero, fzero)
if re == fnan:
return fnan
# Recurrence
w = to_int(re)
n = int(0.4*wp + 4*m)
s = mpc_zero
if w < n:
for k in xrange(w, n):
t = mpc_pow_int(z, -m-1, wp)
s = mpc_add(s, t, wp)
z = mpc_add_mpf(z, fone, wp)
zm = mpc_pow_int(z, -m, wp)
z2 = mpc_pow_int(z, -2, wp)
# 1/m*(z+N)^m
integral_term = mpc_div_mpf(zm, from_int(m), wp)
s = mpc_add(s, integral_term, wp)
# 1/2*(z+N)^(-(m+1))
s = mpc_add(s, mpc_mul_mpf(mpc_div(zm, z, wp), fhalf, wp), wp)
a = m + 1
b = 2
k = 1
# Important: we want to sum up to the *relative* error,
# not the absolute error, because psi^(m)(z) might be tiny
magn = mpc_abs(s, 10)
magn = magn[2]+magn[3]
eps = mpf_shift(fone, magn-wp+2)
while 1:
zm = mpc_mul(zm, z2, wp)
bern = mpf_bernoulli(2*k, wp)
scal = mpf_mul_int(bern, a, wp)
scal = mpf_div(scal, from_int(b), wp)
term = mpc_mul_mpf(zm, scal, wp)
s = mpc_add(s, term, wp)
szterm = mpc_abs(term, 10)
if k > 2 and mpf_le(szterm, eps):
break
#print k, to_str(szterm, 10), to_str(eps, 10)
a *= (m+2*k)*(m+2*k+1)
b *= (2*k+1)*(2*k+2)
k += 1
# Scale and sign factor
v = mpc_mul_mpf(s, mpf_gamma(from_int(m+1), wp), prec, rnd)
if not (m & 1):
v = mpf_neg(v[0]), mpf_neg(v[1])
return v
def mpf_zeta_int(s, prec, rnd=round_fast):
"""
Optimized computation of zeta(s) for an integer s.
"""
wp = prec + 20
s = int(s)
if s in zeta_int_cache and zeta_int_cache[s][0] >= wp:
return mpf_pos(zeta_int_cache[s][1], prec, rnd)
if s < 2:
if s == 1:
raise ValueError("zeta(1) pole")
if not s:
return mpf_neg(fhalf)
return mpf_div(mpf_bernoulli(-s + 1, wp), from_int(s - 1), prec, rnd)
# 2^-s term vanishes?
if s >= wp:
return mpf_perturb(fone, 0, prec, rnd)
# 5^-s term vanishes?
elif s >= wp * 0.431:
t = one = 1 << wp
t += 1 << (wp - s)
t += one // (MPZ_THREE**s)
t += 1 << max(0, wp - s * 2)
return from_man_exp(t, -wp, prec, rnd)
else:
# Fast enough to sum directly?
# Even better, we use the Euler product (idea stolen from pari)
m = (float(wp) / (s - 1) + 1)
if m < 30:
needed_terms = int(2.0**m + 1)
if needed_terms < int(wp / 2.54 + 5) / 10:
t = fone
for k in list_primes(needed_terms):
#print k, needed_terms
powprec = int(wp - s * math.log(k, 2))
if powprec < 2:
break
a = mpf_sub(fone, mpf_pow_int(from_int(k), -s, powprec),
wp)
t = mpf_mul(t, a, wp)
return mpf_div(fone, t, wp)
# Use Borwein's algorithm
n = int(wp / 2.54 + 5)
d = borwein_coefficients(n)
t = MPZ_ZERO
s = MPZ(s)
for k in xrange(n):
t += (((-1)**k * (d[k] - d[n])) << wp) // (k + 1)**s
t = (t << wp) // (-d[n])
t = (t << wp) // ((1 << wp) - (1 << (wp + 1 - s)))
if (s in zeta_int_cache
and zeta_int_cache[s][0] < wp) or (s not in zeta_int_cache):
zeta_int_cache[s] = (wp, from_man_exp(t, -wp - wp))
return from_man_exp(t, -wp - wp, prec, rnd)
def mpf_zeta_int(s, prec, rnd=round_fast):
"""
Optimized computation of zeta(s) for an integer s.
"""
wp = prec + 20
s = int(s)
if s in zeta_int_cache and zeta_int_cache[s][0] >= wp:
return mpf_pos(zeta_int_cache[s][1], prec, rnd)
if s < 2:
if s == 1:
raise ValueError("zeta(1) pole")
if not s:
return mpf_neg(fhalf)
return mpf_div(mpf_bernoulli(-s+1, wp), from_int(s-1), prec, rnd)
# 2^-s term vanishes?
if s >= wp:
return mpf_perturb(fone, 0, prec, rnd)
# 5^-s term vanishes?
elif s >= wp*0.431:
t = one = 1 << wp
t += 1 << (wp - s)
t += one // (MPZ_THREE ** s)
t += 1 << max(0, wp - s*2)
return from_man_exp(t, -wp, prec, rnd)
else:
# Fast enough to sum directly?
# Even better, we use the Euler product (idea stolen from pari)
m = (float(wp)/(s-1) + 1)
if m < 30:
needed_terms = int(2.0**m + 1)
if needed_terms < int(wp/2.54 + 5) / 10:
t = fone
for k in list_primes(needed_terms):
#print k, needed_terms
powprec = int(wp - s*math.log(k,2))
if powprec < 2:
break
a = mpf_sub(fone, mpf_pow_int(from_int(k), -s, powprec), wp)
t = mpf_mul(t, a, wp)
return mpf_div(fone, t, wp)
# Use Borwein's algorithm
n = int(wp/2.54 + 5)
d = borwein_coefficients(n)
t = MPZ_ZERO
s = MPZ(s)
for k in xrange(n):
t += (((-1)**k * (d[k] - d[n])) << wp) // (k+1)**s
t = (t << wp) // (-d[n])
t = (t << wp) // ((1 << wp) - (1 << (wp+1-s)))
if (s in zeta_int_cache and zeta_int_cache[s][0] < wp) or (s not in zeta_int_cache):
zeta_int_cache[s] = (wp, from_man_exp(t, -wp-wp))
return from_man_exp(t, -wp-wp, prec, rnd)
def exp_newton(x, prec):
extra = 10
r = mpf_exp(x, 60)
start = 50
prevp = start
for p in giant_steps(start, prec+extra, 4):
h = mpf_sub(x, mpf_log(r, p), p)
h2 = mpf_mul(h, h, p)
h3 = mpf_mul(h2, h, p)
h4 = mpf_mul(h2, h2, p)
t = mpf_add(h, mpf_shift(h2, -1), p)
t = mpf_add(t, mpf_div(h3, from_int(6, p), p), p)
t = mpf_add(t, mpf_div(h4, from_int(24, p), p), p)
t = mpf_mul(r, t, p)
r = mpf_add(r, t, p)
return r
def khinchin_fixed(prec):
wp = int(prec + prec**0.5 + 15)
s = MP_ZERO
fac = from_int(4)
t = ONE = MP_ONE << wp
pi = mpf_pi(wp)
pipow = twopi2 = mpf_shift(mpf_mul(pi, pi, wp), 2)
n = 1
while 1:
zeta2n = mpf_abs(mpf_bernoulli(2 * n, wp))
zeta2n = mpf_mul(zeta2n, pipow, wp)
zeta2n = mpf_div(zeta2n, fac, wp)
zeta2n = to_fixed(zeta2n, wp)
term = (((zeta2n - ONE) * t) // n) >> wp
if term < 100:
break
#if not n % 100:
# print n, nstr(ln(term))
s += term
t += ONE // (2 * n + 1) - ONE // (2 * n)
n += 1
fac = mpf_mul_int(fac, (2 * n) * (2 * n - 1), wp)
pipow = mpf_mul(pipow, twopi2, wp)
s = (s << wp) // ln2_fixed(wp)
K = mpf_exp(from_man_exp(s, -wp), wp)
K = to_fixed(K, prec)
return K
def khinchin_fixed(prec):
wp = int(prec + prec**0.5 + 15)
s = MPZ_ZERO
fac = from_int(4)
t = ONE = MPZ_ONE << wp
pi = mpf_pi(wp)
pipow = twopi2 = mpf_shift(mpf_mul(pi, pi, wp), 2)
n = 1
while 1:
zeta2n = mpf_abs(mpf_bernoulli(2*n, wp))
zeta2n = mpf_mul(zeta2n, pipow, wp)
zeta2n = mpf_div(zeta2n, fac, wp)
zeta2n = to_fixed(zeta2n, wp)
term = (((zeta2n - ONE) * t) // n) >> wp
if term < 100:
break
#if not n % 10:
# print n, math.log(int(abs(term)))
s += term
t += ONE//(2*n+1) - ONE//(2*n)
n += 1
fac = mpf_mul_int(fac, (2*n)*(2*n-1), wp)
pipow = mpf_mul(pipow, twopi2, wp)
s = (s << wp) // ln2_fixed(wp)
K = mpf_exp(from_man_exp(s, -wp), wp)
K = to_fixed(K, prec)
return K
def __rsub__(s, t):
prec, rounding = prec_rounding
if type(t) in int_types:
return make_mpf(mpf_sub(from_int(t), s._mpf_, prec, rounding))
t = mpf_convert_lhs(t)
if t is NotImplemented:
return t
return t - s
def __rsub__(s, t):
prec, rounding = prec_rounding
if type(t) in int_types:
return make_mpf(mpf_sub(from_int(t), s._mpf_, prec, rounding))
t = mpf_convert_lhs(t)
if t is NotImplemented:
return t
return t - s
def calc_spouge_coefficients(a, prec):
wp = prec + int(a * 1.4)
c = [0] * a
# b = exp(a-1)
b = mpf_exp(from_int(a - 1), wp)
# e = exp(1)
e = mpf_exp(fone, wp)
# sqrt(2*pi)
sq2pi = mpf_sqrt(mpf_shift(mpf_pi(wp), 1), wp)
c[0] = to_fixed(sq2pi, prec)
for k in xrange(1, a):
# c[k] = ((-1)**(k-1) * (a-k)**k) * b / sqrt(a-k)
term = mpf_mul_int(b, ((-1)**(k - 1) * (a - k)**k), wp)
term = mpf_div(term, mpf_sqrt(from_int(a - k), wp), wp)
c[k] = to_fixed(term, prec)
# b = b / (e * k)
b = mpf_div(b, mpf_mul(e, from_int(k), wp), wp)
return c
def calc_spouge_coefficients(a, prec):
wp = prec + int(a*1.4)
c = [0] * a
# b = exp(a-1)
b = mpf_exp(from_int(a-1), wp)
# e = exp(1)
e = mpf_exp(fone, wp)
# sqrt(2*pi)
sq2pi = mpf_sqrt(mpf_shift(mpf_pi(wp), 1), wp)
c[0] = to_fixed(sq2pi, prec)
for k in xrange(1, a):
# c[k] = ((-1)**(k-1) * (a-k)**k) * b / sqrt(a-k)
term = mpf_mul_int(b, ((-1)**(k-1) * (a-k)**k), wp)
term = mpf_div(term, mpf_sqrt(from_int(a-k), wp), wp)
c[k] = to_fixed(term, prec)
# b = b / (e * k)
b = mpf_div(b, mpf_mul(e, from_int(k), wp), wp)
return c
def mpf_gamma(x, prec, rounding=round_fast, p1=1):
"""
Computes the gamma function of a real floating-point argument.
With p1=0, computes a factorial instead.
"""
sign, man, exp, bc = x
if not man:
if x == finf:
return finf
if x == fninf or x == fnan:
return fnan
# More precision is needed for enormous x. TODO:
# use Stirling's formula + Euler-Maclaurin summation
size = exp + bc
if size > 5:
size = int(size * math.log(size, 2))
wp = prec + max(0, size) + 15
if exp >= 0:
if sign or (p1 and not man):
raise ValueError("gamma function pole")
# A direct factorial is fastest
if exp + bc <= 10:
return from_int(int_fac((man << exp) - p1), prec, rounding)
reflect = sign or exp + bc < -1
if p1:
# Should be done exactly!
x = mpf_sub(x, fone, bc - exp + 2)
# x < 0.25
if reflect:
# gamma = pi / (sin(pi*x) * gamma(1-x))
wp += 15
pix = mpf_mul(x, mpf_pi(wp), wp)
t = mpf_sin_pi(x, wp)
g = mpf_gamma(mpf_sub(fone, x, wp), wp)
return mpf_div(pix, mpf_mul(t, g, wp), prec, rounding)
sprec, a, c = get_spouge_coefficients(wp)
s = spouge_sum_real(x, sprec, a, c)
# gamma = exp(log(x+a)*(x+0.5) - xpa) * s
xpa = mpf_add(x, from_int(a), wp)
logxpa = mpf_log(xpa, wp)
xph = mpf_add(x, fhalf, wp)
t = mpf_sub(mpf_mul(logxpa, xph, wp), xpa, wp)
t = mpf_mul(mpf_exp(t, wp), s, prec, rounding)
return t
def mpf_gamma(x, prec, rounding=round_fast, p1=1):
"""
Computes the gamma function of a real floating-point argument.
With p1=0, computes a factorial instead.
"""
sign, man, exp, bc = x
if not man:
if x == finf:
return finf
if x == fninf or x == fnan:
return fnan
# More precision is needed for enormous x. TODO:
# use Stirling's formula + Euler-Maclaurin summation
size = exp + bc
if size > 5:
size = int(size * math.log(size,2))
wp = prec + max(0, size) + 15
if exp >= 0:
if sign or (p1 and not man):
raise ValueError("gamma function pole")
# A direct factorial is fastest
if exp + bc <= 10:
return from_int(ifac((man<<exp)-p1), prec, rounding)
reflect = sign or exp+bc < -1
if p1:
# Should be done exactly!
x = mpf_sub(x, fone)
# x < 0.25
if reflect:
# gamma = pi / (sin(pi*x) * gamma(1-x))
wp += 15
pix = mpf_mul(x, mpf_pi(wp), wp)
t = mpf_sin_pi(x, wp)
g = mpf_gamma(mpf_sub(fone, x), wp)
return mpf_div(pix, mpf_mul(t, g, wp), prec, rounding)
sprec, a, c = get_spouge_coefficients(wp)
s = spouge_sum_real(x, sprec, a, c)
# gamma = exp(log(x+a)*(x+0.5) - xpa) * s
xpa = mpf_add(x, from_int(a), wp)
logxpa = mpf_log(xpa, wp)
xph = mpf_add(x, fhalf, wp)
t = mpf_sub(mpf_mul(logxpa, xph, wp), xpa, wp)
t = mpf_mul(mpf_exp(t, wp), s, prec, rounding)
return t
def mpi_pow(s, t, prec):
ta, tb = t
if ta == tb and ta not in (finf, fninf):
if ta == from_int(to_int(ta)):
return mpi_pow_int(s, to_int(ta), prec)
if ta == fhalf:
return mpi_sqrt(s, prec)
u = mpi_log(s, prec + 20)
v = mpi_mul(u, t, prec + 20)
return mpi_exp(v, prec)
def mpi_pow(s, t, prec):
ta, tb = t
if ta == tb and ta not in (finf, fninf):
if ta == from_int(to_int(ta)):
return mpi_pow_int(s, to_int(ta), prec)
if ta == fhalf:
return mpi_sqrt(s, prec)
u = mpi_log(s, prec + 20)
v = mpi_mul(u, t, prec + 20)
return mpi_exp(v, prec)
def mpf_convert_arg(x, prec, rounding):
if isinstance(x, int_types): return from_int(x)
if isinstance(x, float): return from_float(x)
if isinstance(x, basestring): return from_str(x, prec, rounding)
if isinstance(x, constant): return x.func(prec, rounding)
if hasattr(x, '_mpf_'): return x._mpf_
if hasattr(x, '_mpmath_'):
t = convert_lossless(x._mpmath_(prec, rounding))
if isinstance(t, mpf):
return t._mpf_
raise TypeError("cannot create mpf from " + repr(x))
def mpf_convert_rhs(x):
if isinstance(x, int_types): return from_int(x)
if isinstance(x, float): return from_float(x)
if isinstance(x, complex_types): return mpc(x)
if hasattr(x, '_mpf_'): return x._mpf_
if hasattr(x, '_mpmath_'):
t = convert_lossless(x._mpmath_(*prec_rounding))
if isinstance(t, mpf):
return t._mpf_
return t
return NotImplemented
def mpc_gamma(x, prec, rounding=round_fast, p1=1):
re, im = x
if im == fzero:
return mpf_gamma(re, prec, rounding, p1), fzero
# More precision is needed for enormous x.
sign, man, exp, bc = re
isign, iman, iexp, ibc = im
if re == fzero:
size = iexp + ibc
else:
size = max(exp + bc, iexp + ibc)
if size > 5:
size = int(size * math.log(size, 2))
reflect = sign or (exp + bc < -1)
wp = prec + max(0, size) + 25
# Near x = 0 pole (TODO: other poles)
if p1:
if size < -prec - 5:
return mpc_add_mpf(mpc_div(mpc_one, x, 2*prec+10), \
mpf_neg(mpf_euler(2*prec+10)), prec, rounding)
elif size < -5:
wp += (-2 * size)
if p1:
# Should be done exactly!
re_orig = re
re = mpf_sub(re, fone, bc + abs(exp) + 2)
x = re, im
if reflect:
# Reflection formula
wp += 15
pi = mpf_pi(wp), fzero
pix = mpc_mul(x, pi, wp)
t = mpc_sin_pi(x, wp)
u = mpc_sub(mpc_one, x, wp)
g = mpc_gamma(u, wp)
w = mpc_mul(t, g, wp)
return mpc_div(pix, w, wp)
# Extremely close to the real line?
# XXX: reflection formula
if iexp + ibc < -wp:
a = mpf_gamma(re_orig, wp)
b = mpf_psi0(re_orig, wp)
gamma_diff = mpf_div(a, b, wp)
return mpf_pos(a, prec, rounding), mpf_mul(gamma_diff, im, prec,
rounding)
sprec, a, c = get_spouge_coefficients(wp)
s = spouge_sum_complex(re, im, sprec, a, c)
# gamma = exp(log(x+a)*(x+0.5) - xpa) * s
repa = mpf_add(re, from_int(a), wp)
logxpa = mpc_log((repa, im), wp)
reph = mpf_add(re, fhalf, wp)
t = mpc_sub(mpc_mul(logxpa, (reph, im), wp), (repa, im), wp)
t = mpc_mul(mpc_exp(t, wp), s, prec, rounding)
return t
def mpf_convert_arg(x, prec, rounding):
if isinstance(x, int_types): return from_int(x)
if isinstance(x, float): return from_float(x)
if isinstance(x, basestring): return from_str(x, prec, rounding)
if isinstance(x, constant): return x.func(prec, rounding)
if hasattr(x, '_mpf_'): return x._mpf_
if hasattr(x, '_mpmath_'):
t = mpmathify(x._mpmath_(prec, rounding))
if isinstance(t, mpf):
return t._mpf_
raise TypeError("cannot create mpf from " + repr(x))
def mpf_convert_rhs(x):
if isinstance(x, int_types): return from_int(x)
if isinstance(x, float): return from_float(x)
if isinstance(x, complex_types): return mpc(x)
if hasattr(x, '_mpf_'): return x._mpf_
if hasattr(x, '_mpmath_'):
t = mpmathify(x._mpmath_(*prec_rounding))
if isinstance(t, mpf):
return t._mpf_
return t
return NotImplemented
def log_int_fixed(n, prec):
if n in log_int_cache:
value, vprec = log_int_cache[n]
if vprec >= prec:
return value >> (vprec - prec)
extra = 30
vprec = prec + extra
v = to_fixed(mpf_log(from_int(n), vprec+5), vprec)
if n < MAX_LOG_INT_CACHE:
log_int_cache[n] = (v, vprec)
return v >> extra
def mpc_nthroot_fixed(a, b, n, prec):
# a, b signed integers at fixed precision prec
start = 50
a1 = int(rshift(a, prec - n * start))
b1 = int(rshift(b, prec - n * start))
try:
r = (a1 + 1j * b1)**(1.0 / n)
re = r.real
im = r.imag
# XXX: workaround bug in gmpy
if abs(re) < 0.1: re = 0
if abs(im) < 0.1: im = 0
re = MP_BASE(re)
im = MP_BASE(im)
except OverflowError:
a1 = from_int(a1, start)
b1 = from_int(b1, start)
fn = from_int(n)
nth = mpf_rdiv_int(1, fn, start)
re, im = mpc_pow((a1, b1), (nth, fzero), start)
re = to_int(re)
im = to_int(im)
extra = 10
prevp = start
extra1 = n
for p in giant_steps(start, prec + extra):
# this is slow for large n, unlike int_pow_fixed
re2, im2 = complex_int_pow(re, im, n - 1)
re2 = rshift(re2, (n - 1) * prevp - p - extra1)
im2 = rshift(im2, (n - 1) * prevp - p - extra1)
r4 = (re2 * re2 + im2 * im2) >> (p + extra1)
ap = rshift(a, prec - p)
bp = rshift(b, prec - p)
rec = (ap * re2 + bp * im2) >> p
imc = (-ap * im2 + bp * re2) >> p
reb = (rec << p) // r4
imb = (imc << p) // r4
re = (reb + (n - 1) * lshift(re, p - prevp)) // n
im = (imb + (n - 1) * lshift(im, p - prevp)) // n
prevp = p
return re, im
def mpc_nthroot_fixed(a, b, n, prec):
# a, b signed integers at fixed precision prec
start = 50
a1 = int(rshift(a, prec - n*start))
b1 = int(rshift(b, prec - n*start))
try:
r = (a1 + 1j * b1)**(1.0/n)
re = r.real
im = r.imag
# XXX: workaround bug in gmpy
if abs(re) < 0.1: re = 0
if abs(im) < 0.1: im = 0
re = MP_BASE(re)
im = MP_BASE(im)
except OverflowError:
a1 = from_int(a1, start)
b1 = from_int(b1, start)
fn = from_int(n)
nth = mpf_rdiv_int(1, fn, start)
re, im = mpc_pow((a1, b1), (nth, fzero), start)
re = to_int(re)
im = to_int(im)
extra = 10
prevp = start
extra1 = n
for p in giant_steps(start, prec+extra):
# this is slow for large n, unlike int_pow_fixed
re2, im2 = complex_int_pow(re, im, n-1)
re2 = rshift(re2, (n-1)*prevp - p - extra1)
im2 = rshift(im2, (n-1)*prevp - p - extra1)
r4 = (re2*re2 + im2*im2) >> (p + extra1)
ap = rshift(a, prec - p)
bp = rshift(b, prec - p)
rec = (ap * re2 + bp * im2) >> p
imc = (-ap * im2 + bp * re2) >> p
reb = (rec << p) // r4
imb = (imc << p) // r4
re = (reb + (n-1)*lshift(re, p-prevp))//n
im = (imb + (n-1)*lshift(im, p-prevp))//n
prevp = p
return re, im
def nthroot_fixed(y, n, prec, exp1):
start = 50
try:
y1 = rshift(y, prec - n*start)
r = MP_BASE(int(y1**(1.0/n)))
except OverflowError:
y1 = from_int(y1, start)
fn = from_int(n)
fn = mpf_rdiv_int(1, fn, start)
r = mpf_pow(y1, fn, start)
r = to_int(r)
extra = 10
extra1 = n
prevp = start
for p in giant_steps(start, prec+extra):
pm, pe = int_pow_fixed(r, n-1, prevp)
r2 = rshift(pm, (n-1)*prevp - p - pe - extra1)
B = lshift(y, 2*p-prec+extra1)//r2
r = (B + (n-1) * lshift(r, p-prevp))//n
prevp = p
return r
def mpc_gamma(x, prec, rounding=round_fast, p1=1):
re, im = x
if im == fzero:
return mpf_gamma(re, prec, rounding, p1), fzero
# More precision is needed for enormous x.
sign, man, exp, bc = re
isign, iman, iexp, ibc = im
if re == fzero:
size = iexp+ibc
else:
size = max(exp+bc, iexp+ibc)
if size > 5:
size = int(size * math.log(size,2))
reflect = sign or (exp+bc < -1)
wp = prec + max(0, size) + 25
# Near x = 0 pole (TODO: other poles)
if p1:
if size < -prec-5:
return mpc_add_mpf(mpc_div(mpc_one, x, 2*prec+10), \
mpf_neg(mpf_euler(2*prec+10)), prec, rounding)
elif size < -5:
wp += (-2*size)
if p1:
# Should be done exactly!
re_orig = re
re = mpf_sub(re, fone, bc+abs(exp)+2)
x = re, im
if reflect:
# Reflection formula
wp += 15
pi = mpf_pi(wp), fzero
pix = mpc_mul(x, pi, wp)
t = mpc_sin_pi(x, wp)
u = mpc_sub(mpc_one, x, wp)
g = mpc_gamma(u, wp)
w = mpc_mul(t, g, wp)
return mpc_div(pix, w, wp)
# Extremely close to the real line?
# XXX: reflection formula
if iexp+ibc < -wp:
a = mpf_gamma(re_orig, wp)
b = mpf_psi0(re_orig, wp)
gamma_diff = mpf_div(a, b, wp)
return mpf_pos(a, prec, rounding), mpf_mul(gamma_diff, im, prec, rounding)
sprec, a, c = get_spouge_coefficients(wp)
s = spouge_sum_complex(re, im, sprec, a, c)
# gamma = exp(log(x+a)*(x+0.5) - xpa) * s
repa = mpf_add(re, from_int(a), wp)
logxpa = mpc_log((repa, im), wp)
reph = mpf_add(re, fhalf, wp)
t = mpc_sub(mpc_mul(logxpa, (reph, im), wp), (repa, im), wp)
t = mpc_mul(mpc_exp(t, wp), s, prec, rounding)
return t
def nthroot_fixed(y, n, prec, exp1):
start = 50
try:
y1 = rshift(y, prec - n*start)
r = MP_BASE(y1**(1.0/n))
except OverflowError:
y1 = from_int(y1, start)
fn = from_int(n)
fn = mpf_rdiv_int(1, fn, start)
r = mpf_pow(y1, fn, start)
r = to_int(r)
extra = 10
extra1 = n
prevp = start
for p in giant_steps(start, prec+extra):
pm, pe = int_pow_fixed(r, n-1, prevp)
r2 = rshift(pm, (n-1)*prevp - p - pe - extra1)
B = lshift(y, 2*p-prec+extra1)//r2
r = (B + (n-1) * lshift(r, p-prevp))//n
prevp = p
return r
def mpi_from_str_a_b(x, y, percent, prec):
wp = prec + 20
xa = from_str(x, wp, round_floor)
xb = from_str(x, wp, round_ceiling)
#ya = from_str(y, wp, round_floor)
y = from_str(y, wp, round_ceiling)
assert mpf_ge(y, fzero)
if percent:
y = mpf_mul(MAX(mpf_abs(xa), mpf_abs(xb)), y, wp, round_ceiling)
y = mpf_div(y, from_int(100), wp, round_ceiling)
a = mpf_sub(xa, y, prec, round_floor)
b = mpf_add(xb, y, prec, round_ceiling)
return a, b
def mpi_from_str_a_b(x, y, percent, prec):
wp = prec + 20
xa = from_str(x, wp, round_floor)
xb = from_str(x, wp, round_ceiling)
#ya = from_str(y, wp, round_floor)
y = from_str(y, wp, round_ceiling)
assert mpf_ge(y, fzero)
if percent:
y = mpf_mul(MAX(mpf_abs(xa), mpf_abs(xb)), y, wp, round_ceiling)
y = mpf_div(y, from_int(100), wp, round_ceiling)
a = mpf_sub(xa, y, prec, round_floor)
b = mpf_add(xb, y, prec, round_ceiling)
return a, b
def glaisher_fixed(prec):
wp = prec + 30
# Number of direct terms to sum before applying the Euler-Maclaurin
# formula to the tail. TODO: choose more intelligently
N = int(0.33 * prec + 5)
ONE = MP_ONE << wp
# Euler-Maclaurin, step 1: sum log(k)/k**2 for k from 2 to N-1
s = MP_ZERO
for k in range(2, N):
#print k, N
s += log_int_fixed(k, wp) // k**2
logN = log_int_fixed(N, wp)
#logN = to_fixed(mpf_log(from_int(N), wp+20), wp)
# E-M step 2: integral of log(x)/x**2 from N to inf
s += (ONE + logN) // N
# E-M step 3: endpoint correction term f(N)/2
s += logN // (N**2 * 2)
# E-M step 4: the series of derivatives
pN = N**3
a = 1
b = -2
j = 3
fac = from_int(2)
k = 1
while 1:
# D(2*k-1) * B(2*k) / fac(2*k) [D(n) = nth derivative]
D = ((a << wp) + b * logN) // pN
D = from_man_exp(D, -wp)
B = mpf_bernoulli(2 * k, wp)
term = mpf_mul(B, D, wp)
term = mpf_div(term, fac, wp)
term = to_fixed(term, wp)
if abs(term) < 100:
break
#if not k % 10:
# print k, math.log(int(abs(term)), 10)
s -= term
# Advance derivative twice
a, b, pN, j = b - a * j, -j * b, pN * N, j + 1
a, b, pN, j = b - a * j, -j * b, pN * N, j + 1
k += 1
fac = mpf_mul_int(fac, (2 * k) * (2 * k - 1), wp)
# A = exp((6*s/pi**2 + log(2*pi) + euler)/12)
pi = pi_fixed(wp)
s *= 6
s = (s << wp) // (pi**2 >> wp)
s += euler_fixed(wp)
s += to_fixed(mpf_log(from_man_exp(2 * pi, -wp), wp), wp)
s //= 12
A = mpf_exp(from_man_exp(s, -wp), wp)
return to_fixed(A, prec)
def glaisher_fixed(prec):
wp = prec + 30
# Number of direct terms to sum before applying the Euler-Maclaurin
# formula to the tail. TODO: choose more intelligently
N = int(0.33*prec + 5)
ONE = MPZ_ONE << wp
# Euler-Maclaurin, step 1: sum log(k)/k**2 for k from 2 to N-1
s = MPZ_ZERO
for k in range(2, N):
#print k, N
s += log_int_fixed(k, wp) // k**2
logN = log_int_fixed(N, wp)
#logN = to_fixed(mpf_log(from_int(N), wp+20), wp)
# E-M step 2: integral of log(x)/x**2 from N to inf
s += (ONE + logN) // N
# E-M step 3: endpoint correction term f(N)/2
s += logN // (N**2 * 2)
# E-M step 4: the series of derivatives
pN = N**3
a = 1
b = -2
j = 3
fac = from_int(2)
k = 1
while 1:
# D(2*k-1) * B(2*k) / fac(2*k) [D(n) = nth derivative]
D = ((a << wp) + b*logN) // pN
D = from_man_exp(D, -wp)
B = mpf_bernoulli(2*k, wp)
term = mpf_mul(B, D, wp)
term = mpf_div(term, fac, wp)
term = to_fixed(term, wp)
if abs(term) < 100:
break
#if not k % 10:
# print k, math.log(int(abs(term)), 10)
s -= term
# Advance derivative twice
a, b, pN, j = b-a*j, -j*b, pN*N, j+1
a, b, pN, j = b-a*j, -j*b, pN*N, j+1
k += 1
fac = mpf_mul_int(fac, (2*k)*(2*k-1), wp)
# A = exp((6*s/pi**2 + log(2*pi) + euler)/12)
pi = pi_fixed(wp)
s *= 6
s = (s << wp) // (pi**2 >> wp)
s += euler_fixed(wp)
s += to_fixed(mpf_log(from_man_exp(2*pi, -wp), wp), wp)
s //= 12
A = mpf_exp(from_man_exp(s, -wp), wp)
return to_fixed(A, prec)
def mertens_fixed(prec):
wp = prec + 20
m = 2
s = mpf_euler(wp)
while 1:
t = mpf_zeta_int(m, wp)
if t == fone:
break
t = mpf_log(t, wp)
t = mpf_mul_int(t, moebius(m), wp)
t = mpf_div(t, from_int(m), wp)
s = mpf_add(s, t)
m += 1
return to_fixed(s, prec)
def twinprime_fixed(prec):
def I(n):
return sum(moebius(d)<<(n//d) for d in xrange(1,n+1) if not n%d)//n
wp = 2*prec + 30
res = fone
primes = [from_rational(1,p,wp) for p in [2,3,5,7]]
ppowers = [mpf_mul(p,p,wp) for p in primes]
n = 2
while 1:
a = mpf_zeta_int(n, wp)
for i in range(4):
a = mpf_mul(a, mpf_sub(fone, ppowers[i]), wp)
ppowers[i] = mpf_mul(ppowers[i], primes[i], wp)
a = mpf_pow_int(a, -I(n), wp)
if mpf_pos(a, prec+10, 'n') == fone:
break
#from libmpf import to_str
#print n, to_str(mpf_sub(fone, a), 6)
res = mpf_mul(res, a, wp)
n += 1
res = mpf_mul(res, from_int(3*15*35), wp)
res = mpf_div(res, from_int(4*16*36), wp)
return to_fixed(res, prec)
def mertens_fixed(prec):
wp = prec + 20
m = 2
s = mpf_euler(wp)
while 1:
t = mpf_zeta_int(m, wp)
if t == fone:
break
t = mpf_log(t, wp)
t = mpf_mul_int(t, moebius(m), wp)
t = mpf_div(t, from_int(m), wp)
s = mpf_add(s, t)
m += 1
return to_fixed(s, prec)
def convert_lossless(x, strings=True):
"""Attempt to convert x to an mpf or mpc losslessly. If x is an
mpf or mpc, return it unchanged. If x is an int, create an mpf with
sufficient precision to represent it exactly. If x is a str, just
convert it to an mpf with the current working precision (perhaps
this should be done differently...)"""
if isinstance(x, mpnumeric): return x
if isinstance(x, int_types): return make_mpf(from_int(x))
if isinstance(x, float): return make_mpf(from_float(x))
if isinstance(x, complex): return mpc(x)
if strings and isinstance(x, basestring): return make_mpf(from_str(x, *prec_rounding))
if hasattr(x, '_mpf_'): return make_mpf(x._mpf_)
if hasattr(x, '_mpc_'): return make_mpc(x._mpc_)
if hasattr(x, '_mpmath_'): return convert_lossless(x._mpmath_(*prec_rounding))
raise TypeError("cannot create mpf from " + repr(x))
def mpmathify(x, strings=True):
"""
Converts *x* to an ``mpf`` or ``mpc``. If *x* is of type ``mpf``,
``mpc``, ``int``, ``float``, ``complex``, the conversion
will be performed losslessly.
If *x* is a string, the result will be rounded to the present
working precision. Strings representing fractions or complex
numbers are permitted.
>>> from sympy.mpmath import *
>>> mp.dps = 15
>>> mpmathify(3.5)
mpf('3.5')
>>> mpmathify('2.1')
mpf('2.1000000000000001')
>>> mpmathify('3/4')
mpf('0.75')
>>> mpmathify('2+3j')
mpc(real='2.0', imag='3.0')
"""
if isinstance(x, mpnumeric): return x
if isinstance(x, int_types): return make_mpf(from_int(x))
if isinstance(x, float): return make_mpf(from_float(x))
if isinstance(x, complex): return mpc(x)
if strings and isinstance(x, basestring):
try:
return make_mpf(from_str(x, *prec_rounding))
except Exception, e:
if '/' in x:
fract = x.split('/')
assert len(fract) == 2
return mpmathify(fract[0]) / mpmathify(fract[1])
if 'j' in x.lower():
x = x.lower().replace(' ', '')
match = get_complex.match(x)
re = match.group('re')
if not re:
re = 0
im = match.group('im').rstrip('j')
return mpc(mpmathify(re),
mpmathify(im))
raise e
def log_int_fixed(n, prec, ln2=None):
"""
Fast computation of log(n), caching the value for small n,
intended for zeta sums.
"""
if n in log_int_cache:
value, vprec = log_int_cache[n]
if vprec >= prec:
return value >> (vprec - prec)
wp = prec + 10
if wp <= LOG_TAYLOR_SHIFT:
if ln2 is None:
ln2 = ln2_fixed(wp)
r = bitcount(n)
x = n << (wp - r)
v = log_taylor_cached(x, wp) + r * ln2
else:
v = to_fixed(mpf_log(from_int(n), wp + 5), wp)
if n < MAX_LOG_INT_CACHE:
log_int_cache[n] = (v, wp)
return v >> (wp - prec)
def log_int_fixed(n, prec, ln2=None):
"""
Fast computation of log(n), caching the value for small n,
intended for zeta sums.
"""
if n in log_int_cache:
value, vprec = log_int_cache[n]
if vprec >= prec:
return value >> (vprec - prec)
wp = prec + 10
if wp <= LOG_TAYLOR_SHIFT:
if ln2 is None:
ln2 = ln2_fixed(wp)
r = bitcount(n)
x = n << (wp - r)
v = log_taylor_cached(x, wp) + r * ln2
else:
v = to_fixed(mpf_log(from_int(n), wp + 5), wp)
if n < MAX_LOG_INT_CACHE:
log_int_cache[n] = (v, wp)
return v >> (wp - prec)
def mpc_nthroot(z, n, prec, rnd=round_fast):
"""
Complex n-th root.
Use Newton method as in the real case when it is faster,
otherwise use z**(1/n)
"""
a, b = z
if a[0] == 0 and b == fzero:
re = mpf_nthroot(a, n, prec, rnd)
return (re, fzero)
if n < 2:
if n == 0:
return mpc_one
if n == 1:
return mpc_pos((a, b), prec, rnd)
if n == -1:
return mpc_div(mpc_one, (a, b), prec, rnd)
inverse = mpc_nthroot((a, b), -n, prec + 5, reciprocal_rnd[rnd])
return mpc_div(mpc_one, inverse, prec, rnd)
if n <= 20:
prec2 = int(1.2 * (prec + 10))
asign, aman, aexp, abc = a
bsign, bman, bexp, bbc = b
pf = mpc_abs((a, b), prec)
if pf[-2] + pf[-1] > -10 and pf[-2] + pf[-1] < prec:
af = to_fixed(a, prec2)
bf = to_fixed(b, prec2)
re, im = mpc_nthroot_fixed(af, bf, n, prec2)
extra = 10
re = from_man_exp(re, -prec2 - extra, prec2, rnd)
im = from_man_exp(im, -prec2 - extra, prec2, rnd)
return re, im
fn = from_int(n)
prec2 = prec + 10 + 10
nth = mpf_rdiv_int(1, fn, prec2)
re, im = mpc_pow((a, b), (nth, fzero), prec2, rnd)
re = normalize(re[0], re[1], re[2], re[3], prec, rnd)
im = normalize(im[0], im[1], im[2], im[3], prec, rnd)
return re, im
def mpc_nthroot(z, n, prec, rnd=round_fast):
"""
Complex n-th root.
Use Newton method as in the real case when it is faster,
otherwise use z**(1/n)
"""
a, b = z
if a[0] == 0 and b == fzero:
re = mpf_nthroot(a, n, prec, rnd)
return (re, fzero)
if n < 2:
if n == 0:
return mpc_one
if n == 1:
return mpc_pos((a, b), prec, rnd)
if n == -1:
return mpc_div(mpc_one, (a, b), prec, rnd)
inverse = mpc_nthroot((a, b), -n, prec+5, reciprocal_rnd[rnd])
return mpc_div(mpc_one, inverse, prec, rnd)
if n <= 20:
prec2 = int(1.2 * (prec + 10))
asign, aman, aexp, abc = a
bsign, bman, bexp, bbc = b
pf = mpc_abs((a,b), prec)
if pf[-2] + pf[-1] > -10 and pf[-2] + pf[-1] < prec:
af = to_fixed(a, prec2)
bf = to_fixed(b, prec2)
re, im = mpc_nthroot_fixed(af, bf, n, prec2)
extra = 10
re = from_man_exp(re, -prec2-extra, prec2, rnd)
im = from_man_exp(im, -prec2-extra, prec2, rnd)
return re, im
fn = from_int(n)
prec2 = prec+10 + 10
nth = mpf_rdiv_int(1, fn, prec2)
re, im = mpc_pow((a, b), (nth, fzero), prec2, rnd)
re = normalize(re[0], re[1], re[2], re[3], prec, rnd)
im = normalize(im[0], im[1], im[2], im[3], prec, rnd)
return re, im
def bernfrac(n):
"""
Computes integers (p,q) such that p/q = B_n exactly, where
B_n denotes the nth Bernoulli number.
Use bernoulli(n) to get a floating-point approximation
instead of the exact fraction (much faster for large n).
"""
n = int(n)
if n < 3:
return [(1, 1), (-1, 2), (1, 6)][n]
if n & 1:
return (0, 1)
q = 1
for k in list_primes(n+1):
if not (n % (k-1)):
q *= k
prec = bernoulli_size(n) + int(math.log(q,2)) + 20
b = mpf_bernoulli(n, prec)
p = mpf_mul(b, from_int(q))
pint = to_int(p, round_nearest)
return (pint, q)
def mpf_fibonacci(x, prec, rnd=round_fast):
sign, man, exp, bc = x
if not man:
if x == fninf:
return fnan
return x
# F(2^n) ~= 2^(2^n)
size = abs(exp+bc)
if exp >= 0:
# Exact
if size < 10 or size <= bitcount(prec):
return from_int(ifib(to_int(x)), prec, rnd)
# Use the modified Binet formula
wp = prec + size + 20
a = mpf_phi(wp)
b = mpf_add(mpf_shift(a, 1), fnone, wp)
u = mpf_pow(a, x, wp)
v = mpf_cos_pi(x, wp)
v = mpf_div(v, u, wp)
u = mpf_sub(u, v, wp)
u = mpf_div(u, b, prec, rnd)
return u
def bernfrac(n):
"""
Computes integers (p,q) such that p/q = B_n exactly, where
B_n denotes the nth Bernoulli number.
Use bernoulli(n) to get a floating-point approximation
instead of the exact fraction (much faster for large n).
"""
n = int(n)
if n < 3:
return [(1, 1), (-1, 2), (1, 6)][n]
if n & 1:
return (0, 1)
q = 1
for k in list_primes(n + 1):
if not (n % (k - 1)):
q *= k
prec = bernoulli_size(n) + int(math.log(q, 2)) + 20
b = mpf_bernoulli(n, prec)
p = mpf_mul(b, from_int(q))
pint = to_int(p, round_nearest)
return (pint, q)
def mpf_nthroot(s, n, prec, rnd=round_fast):
"""nth-root of a positive number
Use the Newton method when faster, otherwise use x**(1/n)
"""
sign, man, exp, bc = s
if sign:
raise ComplexResult("nth root of a negative number")
if not man:
if s == fnan:
return fnan
if s == fzero:
if n > 0:
return fzero
if n == 0:
return fone
return finf
# Infinity
if not n:
return fnan
if n < 0:
return fzero
return finf
flag_inverse = False
if n < 2:
if n == 0:
return fone
if n == 1:
return mpf_pos(s, prec, rnd)
if n == -1:
return mpf_div(fone, s, prec, rnd)
# n < 0
rnd = reciprocal_rnd[rnd]
flag_inverse = True
extra_inverse = 5
prec += extra_inverse
n = -n
if n > 20 and (n >= 20000 or prec < int(233 + 28.3 * n**0.62)):
prec2 = prec + 10
fn = from_int(n)
nth = mpf_rdiv_int(1, fn, prec2)
r = mpf_pow(s, nth, prec2, rnd)
s = normalize(r[0], r[1], r[2], r[3], prec, rnd)
if flag_inverse:
return mpf_div(fone, s, prec-extra_inverse, rnd)
else:
return s
# Convert to a fixed-point number with prec2 bits.
prec2 = prec + 2*n - (prec%n)
# a few tests indicate that
# for 10 < n < 10**4 a bit more precision is needed
if n > 10:
prec2 += prec2//10
prec2 = prec2 - prec2%n
# Mantissa may have more bits than we need. Trim it down.
shift = bc - prec2
# Adjust exponents to make prec2 and exp+shift multiples of n.
sign1 = 0
es = exp+shift
if es < 0:
sign1 = 1
es = -es
if sign1:
shift += es%n
else:
shift -= es%n
man = rshift(man, shift)
extra = 10
exp1 = ((exp+shift-(n-1)*prec2)//n) - extra
rnd_shift = 0
if flag_inverse:
if rnd == 'u' or rnd == 'c':
rnd_shift = 1
else:
if rnd == 'd' or rnd == 'f':
rnd_shift = 1
man = nthroot_fixed(man+rnd_shift, n, prec2, exp1)
s = from_man_exp(man, exp1, prec, rnd)
if flag_inverse:
return mpf_div(fone, s, prec-extra_inverse, rnd)
else:
return s
def bernfrac(n):
r"""
Returns a tuple of integers `(p, q)` such that `p/q = B_n` exactly,
where `B_n` denotes the `n`-th Bernoulli number. The fraction is
always reduced to lowest terms. Note that for `n > 1` and `n` odd,
`B_n = 0`, and `(0, 1)` is returned.
**Examples**
The first few Bernoulli numbers are exactly::
>>> from mpmath import *
>>> for n in range(15):
... p, q = bernfrac(n)
... print n, "%s/%s" % (p, q)
...
0 1/1
1 -1/2
2 1/6
3 0/1
4 -1/30
5 0/1
6 1/42
7 0/1
8 -1/30
9 0/1
10 5/66
11 0/1
12 -691/2730
13 0/1
14 7/6
This function works for arbitrarily large `n`::
>>> p, q = bernfrac(10**4)
>>> print q
2338224387510
>>> print len(str(p))
27692
>>> mp.dps = 15
>>> print mpf(p) / q
-9.04942396360948e+27677
>>> print bernoulli(10**4)
-9.04942396360948e+27677
Note: :func:`bernoulli` computes a floating-point approximation
directly, without computing the exact fraction first.
This is much faster for large `n`.
**Algorithm**
:func:`bernfrac` works by computing the value of `B_n` numerically
and then using the von Staudt-Clausen theorem [1] to reconstruct
the exact fraction. For large `n`, this is significantly faster than
computing `B_1, B_2, \ldots, B_2` recursively with exact arithmetic.
The implementation has been tested for `n = 10^m` up to `m = 6`.
In practice, :func:`bernfrac` appears to be about three times
slower than the specialized program calcbn.exe [2]
**References**
1. MathWorld, von Staudt-Clausen Theorem:
http://mathworld.wolfram.com/vonStaudt-ClausenTheorem.html
2. The Bernoulli Number Page:
http://www.bernoulli.org/
"""
n = int(n)
if n < 3:
return [(1, 1), (-1, 2), (1, 6)][n]
if n & 1:
return (0, 1)
q = 1
for k in list_primes(n + 1):
if not (n % (k - 1)):
q *= k
prec = bernoulli_size(n) + int(math.log(q, 2)) + 20
b = mpf_bernoulli(n, prec)
p = mpf_mul(b, from_int(q))
pint = to_int(p, round_nearest)
return (pint, q)
[n/6]
___
\ / n + 3 \
S(n) = ) | | * B
/___ \ n - 6*k / n-6*k
k = 1
For isolated large Bernoulli numbers, we use the Riemann zeta function
to calculate a numerical value for B_n. The von Staudt-Clausen theorem
can then be used to optionally find the exact value of the
numerator and denominator.
"""
bernoulli_cache = {}
f3 = from_int(3)
f6 = from_int(6)
def bernoulli_size(n):
"""Accurately estimate the size of B_n (even n > 2 only)"""
lgn = math.log(n, 2)
return int(2.326 + 0.5 * lgn + n * (lgn - 4.094))
BERNOULLI_PREC_CUTOFF = bernoulli_size(MAX_BERNOULLI_CACHE)
def mpf_bernoulli(n, prec, rnd=None):
"""Computation of Bernoulli numbers (numerically)"""
if n < 2:
def mpf_exp(x, prec, rnd=round_fast):
sign, man, exp, bc = x
if not man:
if not exp:
return fone
if x == fninf:
return fzero
return x
mag = bc+exp
# Fast handling e**n. TODO: the best cutoff depends on both the
# size of n and the precision.
if prec > 600 and exp >= 0:
e = mpf_e(prec+10+int(1.45*mag))
return mpf_pow_int(e, (-1)**sign *(man<<exp), prec, rnd)
if mag < -prec-10:
return mpf_perturb(fone, sign, prec, rnd)
# extra precision needs to be similar in magnitude to log_2(|x|)
# for the modulo reduction, plus r for the error from squaring r times
wp = prec + max(0, mag)
if wp < 300:
r = int(2*wp**0.4)
if mag < 0:
r = max(1, r + mag)
wp += r + 20
t = to_fixed(x, wp)
# abs(x) > 1?
if mag > 1:
lg2 = ln2_fixed(wp)
n, t = divmod(t, lg2)
else:
n = 0
man = exp_series(t, wp, r)
else:
use_newton = False
# put a bound on exp to avoid infinite recursion in exp_newton
# TODO find a good bound
if wp > LIM_EXP_SERIES2 and exp < 1000:
if mag > 0:
use_newton = True
elif mag <= 0 and -mag <= ns_exp[-1]:
i = bisect(ns_exp, -mag-1)
if i < len(ns_exp):
wp0 = precs_exp[i]
if wp > wp0:
use_newton = True
if not use_newton:
r = int(0.7 * wp**0.5)
if mag < 0:
r = max(1, r + mag)
wp += r + 20
t = to_fixed(x, wp)
if mag > 1:
lg2 = ln2_fixed(wp)
n, t = divmod(t, lg2)
else:
n = 0
man = exp_series2(t, wp, r)
else:
# if x is very small or very large use
# exp(x + m) = exp(x) * e**m
if mag > LIM_MAG:
wp += mag*10 + 100
n = int(mag * math.log(2)) + 1
x = mpf_sub(x, from_int(n, wp), wp)
elif mag <= 0:
wp += -mag*10 + 100
if mag < 0:
n = int(-mag * math.log(2)) + 1
x = mpf_add(x, from_int(n, wp), wp)
res = exp_newton(x, wp)
sign, man, exp, bc = res
if mag < 0:
t = mpf_pow_int(mpf_e(wp), n, wp)
res = mpf_div(res, t, wp)
sign, man, exp, bc = res
if mag > LIM_MAG:
t = mpf_pow_int(mpf_e(wp), n, wp)
res = mpf_mul(res, t, wp)
sign, man, exp, bc = res
return normalize(sign, man, exp, bc, prec, rnd)
bc = bitcount(man)
return normalize(0, man, int(-wp+n), bc, prec, rnd)
def mpf_nthroot(s, n, prec, rnd=round_fast):
"""nth-root of a positive number
Use the Newton method when faster, otherwise use x**(1/n)
"""
sign, man, exp, bc = s
if sign:
raise ComplexResult("nth root of a negative number")
if not man:
if s == fnan:
return fnan
if s == fzero:
if n > 0:
return fzero
if n == 0:
return fone
return finf
# Infinity
if not n:
return fnan
if n < 0:
return fzero
return finf
flag_inverse = False
if n < 2:
if n == 0:
return fone
if n == 1:
return mpf_pos(s, prec, rnd)
if n == -1:
return mpf_div(fone, s, prec, rnd)
# n < 0
rnd = reciprocal_rnd[rnd]
flag_inverse = True
extra_inverse = 5
prec += extra_inverse
n = -n
if n > 20 and (n >= 20000 or prec < int(233 + 28.3 * n**0.62)):
prec2 = prec + 10
fn = from_int(n)
nth = mpf_rdiv_int(1, fn, prec2)
r = mpf_pow(s, nth, prec2, rnd)
s = normalize(r[0], r[1], r[2], r[3], prec, rnd)
if flag_inverse:
return mpf_div(fone, s, prec-extra_inverse, rnd)
else:
return s
# Convert to a fixed-point number with prec2 bits.
prec2 = prec + 2*n - (prec%n)
# a few tests indicate that
# for 10 < n < 10**4 a bit more precision is needed
if n > 10:
prec2 += prec2//10
prec2 = prec2 - prec2%n
# Mantissa may have more bits than we need. Trim it down.
shift = bc - prec2
# Adjust exponents to make prec2 and exp+shift multiples of n.
sign1 = 0
es = exp+shift
if es < 0:
sign1 = 1
es = -es
if sign1:
shift += es%n
else:
shift -= es%n
man = rshift(man, shift)
extra = 10
exp1 = ((exp+shift-(n-1)*prec2)//n) - extra
rnd_shift = 0
if flag_inverse:
if rnd == 'u' or rnd == 'c':
rnd_shift = 1
else:
if rnd == 'd' or rnd == 'f':
rnd_shift = 1
man = nthroot_fixed(man+rnd_shift, n, prec2, exp1)
s = from_man_exp(man, exp1, prec, rnd)
if flag_inverse:
return mpf_div(fone, s, prec-extra_inverse, rnd)
else:
return s
def bernfrac(n):
r"""
Returns a tuple of integers `(p, q)` such that `p/q = B_n` exactly,
where `B_n` denotes the `n`-th Bernoulli number. The fraction is
always reduced to lowest terms. Note that for `n > 1` and `n` odd,
`B_n = 0`, and `(0, 1)` is returned.
**Examples**
The first few Bernoulli numbers are exactly::
>>> from mpmath import *
>>> for n in range(15):
... p, q = bernfrac(n)
... print n, "%s/%s" % (p, q)
...
0 1/1
1 -1/2
2 1/6
3 0/1
4 -1/30
5 0/1
6 1/42
7 0/1
8 -1/30
9 0/1
10 5/66
11 0/1
12 -691/2730
13 0/1
14 7/6
This function works for arbitrarily large `n`::
>>> p, q = bernfrac(10**4)
>>> print q
2338224387510
>>> print len(str(p))
27692
>>> mp.dps = 15
>>> print mpf(p) / q
-9.04942396360948e+27677
>>> print bernoulli(10**4)
-9.04942396360948e+27677
Note: :func:`bernoulli` computes a floating-point approximation
directly, without computing the exact fraction first.
This is much faster for large `n`.
**Algorithm**
:func:`bernfrac` works by computing the value of `B_n` numerically
and then using the von Staudt-Clausen theorem [1] to reconstruct
the exact fraction. For large `n`, this is significantly faster than
computing `B_1, B_2, \ldots, B_2` recursively with exact arithmetic.
The implementation has been tested for `n = 10^m` up to `m = 6`.
In practice, :func:`bernfrac` appears to be about three times
slower than the specialized program calcbn.exe [2]
**References**
1. MathWorld, von Staudt-Clausen Theorem:
http://mathworld.wolfram.com/vonStaudt-ClausenTheorem.html
2. The Bernoulli Number Page:
http://www.bernoulli.org/
"""
n = int(n)
if n < 3:
return [(1, 1), (-1, 2), (1, 6)][n]
if n & 1:
return (0, 1)
q = 1
for k in list_primes(n+1):
if not (n % (k-1)):
q *= k
prec = bernoulli_size(n) + int(math.log(q,2)) + 20
b = mpf_bernoulli(n, prec)
p = mpf_mul(b, from_int(q))
pint = to_int(p, round_nearest)
return (pint, q)
"""
if a[0] == 0 and b == fzero:
re = mpf_nthroot(a, n, prec, rnd)
return (re, fzero)
if n < 2:
if n == 0:
return mpc_one
if n == 1:
return mpc_pos((a, b), prec, rnd)
if n == -1:
return mpc_div(mpc_one, (a, b), prec, rnd)
inverse = mpc_nthroot((a, b), -n, prec + 5, reciprocal_rnd[rnd])
return mpc_div(mpc_one, inverse, prec, rnd)
if n > 20:
fn = from_int(n)
prec2 = prec + 10
nth = mpf_rdiv_int(1, fn, prec2)
re, im = mpc_pow((a, b), (nth, fzero), prec2, rnd)
re = normalize(re[0], re[1], re[2], re[3], prec, rnd)
im = normalize(im[0], im[1], im[2], im[3], prec, rnd)
return re, im
prec2 = int(1.2 * (prec + 10))
asign, aman, aexp, abc = a
bsign, bman, bexp, bbc = b
af = to_fixed(a, prec2)
bf = to_fixed(b, prec2)
re, im = mpc_nthroot_fixed(af, bf, n, prec2)
extra = 10
re = from_man_exp(re, -prec2 - extra, prec2, rnd)
im = from_man_exp(im, -prec2 - extra, prec2, rnd)
def mpf_bernoulli(n, prec, rnd=None):
"""Computation of Bernoulli numbers (numerically)"""
if n < 2:
if n < 0:
raise ValueError("Bernoulli numbers only defined for n >= 0")
if n == 0:
return fone
if n == 1:
return mpf_neg(fhalf)
# For odd n > 1, the Bernoulli numbers are zero
if n & 1:
return fzero
# If precision is extremely high, we can save time by computing
# the Bernoulli number at a lower precision that is sufficient to
# obtain the exact fraction, round to the exact fraction, and
# convert the fraction back to an mpf value at the original precision
if prec > BERNOULLI_PREC_CUTOFF and prec > bernoulli_size(n)*1.1 + 1000:
p, q = bernfrac(n)
return from_rational(p, q, prec, rnd or round_floor)
if n > MAX_BERNOULLI_CACHE:
return mpf_bernoulli_huge(n, prec, rnd)
wp = prec + 30
# Reuse nearby precisions
wp += 32 - (prec & 31)
cached = bernoulli_cache.get(wp)
if cached:
numbers, state = cached
if n in numbers:
if not rnd:
return numbers[n]
return mpf_pos(numbers[n], prec, rnd)
m, bin, bin1 = state
if n - m > 10:
return mpf_bernoulli_huge(n, prec, rnd)
else:
if n > 10:
return mpf_bernoulli_huge(n, prec, rnd)
numbers = {0:fone}
m, bin, bin1 = state = [2, MPZ(10), MPZ_ONE]
bernoulli_cache[wp] = (numbers, state)
while m <= n:
#print m
case = m % 6
# Accurately estimate size of B_m so we can use
# fixed point math without using too much precision
szbm = bernoulli_size(m)
s = 0
sexp = max(0, szbm) - wp
if m < 6:
a = MPZ_ZERO
else:
a = bin1
for j in xrange(1, m//6+1):
usign, uman, uexp, ubc = u = numbers[m-6*j]
if usign:
uman = -uman
s += lshift(a*uman, uexp-sexp)
# Update inner binomial coefficient
j6 = 6*j
a *= ((m-5-j6)*(m-4-j6)*(m-3-j6)*(m-2-j6)*(m-1-j6)*(m-j6))
a //= ((4+j6)*(5+j6)*(6+j6)*(7+j6)*(8+j6)*(9+j6))
if case == 0: b = mpf_rdiv_int(m+3, f3, wp)
if case == 2: b = mpf_rdiv_int(m+3, f3, wp)
if case == 4: b = mpf_rdiv_int(-m-3, f6, wp)
s = from_man_exp(s, sexp, wp)
b = mpf_div(mpf_sub(b, s, wp), from_int(bin), wp)
numbers[m] = b
m += 2
# Update outer binomial coefficient
bin = bin * ((m+2)*(m+3)) // (m*(m-1))
if m > 6:
bin1 = bin1 * ((2+m)*(3+m)) // ((m-7)*(m-6))
state[:] = [m, bin, bin1]
return numbers[n]
def mpf_bernoulli(n, prec, rnd=None):
"""Computation of Bernoulli numbers (numerically)"""
if n < 2:
if n < 0:
raise ValueError("Bernoulli numbers only defined for n >= 0")
if n == 0:
return fone
if n == 1:
return mpf_neg(fhalf)
# For odd n > 1, the Bernoulli numbers are zero
if n & 1:
return fzero
# If precision is extremely high, we can save time by computing
# the Bernoulli number at a lower precision that is sufficient to
# obtain the exact fraction, round to the exact fraction, and
# convert the fraction back to an mpf value at the original precision
if prec > BERNOULLI_PREC_CUTOFF and prec > bernoulli_size(n) * 1.1 + 1000:
p, q = bernfrac(n)
return from_rational(p, q, prec, rnd or round_floor)
if n > MAX_BERNOULLI_CACHE:
return mpf_bernoulli_huge(n, prec, rnd)
wp = prec + 30
# Reuse nearby precisions
wp += 32 - (prec & 31)
cached = bernoulli_cache.get(wp)
if cached:
numbers, state = cached
if n in numbers:
if not rnd:
return numbers[n]
return mpf_pos(numbers[n], prec, rnd)
m, bin, bin1 = state
if n - m > 10:
return mpf_bernoulli_huge(n, prec, rnd)
else:
if n > 10:
return mpf_bernoulli_huge(n, prec, rnd)
numbers = {0: fone}
m, bin, bin1 = state = [2, MP_BASE(10), MP_ONE]
bernoulli_cache[wp] = (numbers, state)
while m <= n:
#print m
case = m % 6
# Accurately estimate size of B_m so we can use
# fixed point math without using too much precision
szbm = bernoulli_size(m)
s = 0
sexp = max(0, szbm) - wp
if m < 6:
a = MP_ZERO
else:
a = bin1
for j in xrange(1, m // 6 + 1):
usign, uman, uexp, ubc = u = numbers[m - 6 * j]
if usign:
uman = -uman
s += lshift(a * uman, uexp - sexp)
# Update inner binomial coefficient
j6 = 6 * j
a *= ((m - 5 - j6) * (m - 4 - j6) * (m - 3 - j6) * (m - 2 - j6) *
(m - 1 - j6) * (m - j6))
a //= ((4 + j6) * (5 + j6) * (6 + j6) * (7 + j6) * (8 + j6) *
(9 + j6))
if case == 0: b = mpf_rdiv_int(m + 3, f3, wp)
if case == 2: b = mpf_rdiv_int(m + 3, f3, wp)
if case == 4: b = mpf_rdiv_int(-m - 3, f6, wp)
s = from_man_exp(s, sexp, wp)
b = mpf_div(mpf_sub(b, s, wp), from_int(bin), wp)
numbers[m] = b
m += 2
# Update outer binomial coefficient
bin = bin * ((m + 2) * (m + 3)) // (m * (m - 1))
if m > 6:
bin1 = bin1 * ((2 + m) * (3 + m)) // ((m - 7) * (m - 6))
state[:] = [m, bin, bin1]
[n/6]
___
\ / n + 3 \
S(n) = ) | | * B
/___ \ n - 6*k / n-6*k
k = 1
For isolated large Bernoulli numbers, we use the Riemann zeta function
to calculate a numerical value for B_n. The von Staudt-Clausen theorem
can then be used to optionally find the exact value of the
numerator and denominator.
"""
bernoulli_cache = {}
f3 = from_int(3)
f6 = from_int(6)
def bernoulli_size(n):
"""Accurately estimate the size of B_n (even n > 2 only)"""
lgn = math.log(n,2)
return int(2.326 + 0.5*lgn + n*(lgn - 4.094))
BERNOULLI_PREC_CUTOFF = bernoulli_size(MAX_BERNOULLI_CACHE)
def mpf_bernoulli(n, prec, rnd=None):
"""Computation of Bernoulli numbers (numerically)"""
if n < 2:
if n < 0:
raise ValueError("Bernoulli numbers only defined for n >= 0")
if n == 0:
def mpf_expint(n, x, prec, rnd=round_fast, gamma=False):
"""
E_n(x), n an integer, x real
With gamma=True, computes Gamma(n,x) (upper incomplete gamma function)
Returns (real, None) if real, otherwise (real, imag)
The imaginary part is an optional branch cut term
"""
sign, man, exp, bc = x
if not man:
if gamma:
if x == fzero:
# Actually gamma function pole
if n <= 0:
return finf
return mpf_gamma_int(n, prec, rnd)
if x == finf:
return fzero, None
# TODO: could return finite imaginary value at -inf
return fnan, fnan
else:
if x == fzero:
if n > 1:
return from_rational(1, n - 1, prec, rnd), None
else:
return finf, None
if x == finf:
return fzero, None
return fnan, fnan
n_orig = n
if gamma:
n = 1 - n
wp = prec + 20
xmag = exp + bc
# Beware of near-poles
if xmag < -10:
raise NotImplementedError
nmag = bitcount(abs(n))
have_imag = n > 0 and sign
negx = mpf_neg(x)
# Skip series if direct convergence
if n == 0 or 2 * nmag - xmag < -wp:
if gamma:
v = mpf_exp(negx, wp)
re = mpf_mul(v, mpf_pow_int(x, n_orig - 1, wp), prec, rnd)
else:
v = mpf_exp(negx, wp)
re = mpf_div(v, x, prec, rnd)
else:
# Finite number of terms, or...
can_use_asymptotic_series = -3 * wp < n <= 0
# ...large enough?
if not can_use_asymptotic_series:
xi = abs(to_int(x))
m = min(max(1, xi - n), 2 * wp)
siz = -n * nmag + (m + n) * bitcount(abs(m + n)) - m * xmag - (
144 * m // 100)
tol = -wp - 10
can_use_asymptotic_series = siz < tol
if can_use_asymptotic_series:
r = ((-MP_ONE) << (wp + wp)) // to_fixed(x, wp)
m = n
t = r * m
s = MP_ONE << wp
while m and t:
s += t
m += 1
t = (m * r * t) >> wp
v = mpf_exp(negx, wp)
if gamma:
# ~ exp(-x) * x^(n-1) * (1 + ...)
v = mpf_mul(v, mpf_pow_int(x, n_orig - 1, wp), wp)
else:
# ~ exp(-x)/x * (1 + ...)
v = mpf_div(v, x, wp)
re = mpf_mul(v, from_man_exp(s, -wp), prec, rnd)
elif n == 1:
re = mpf_neg(mpf_ei(negx, prec, rnd))
elif n > 0 and n < 3 * wp:
T1 = mpf_neg(mpf_ei(negx, wp))
if gamma:
if n_orig & 1:
T1 = mpf_neg(T1)
else:
T1 = mpf_mul(T1, mpf_pow_int(negx, n - 1, wp), wp)
r = t = to_fixed(x, wp)
facs = [1] * (n - 1)
for k in range(1, n - 1):
facs[k] = facs[k - 1] * k
facs = facs[::-1]
s = facs[0] << wp
for k in range(1, n - 1):
if k & 1:
s -= facs[k] * t
else:
s += facs[k] * t
t = (t * r) >> wp
T2 = from_man_exp(s, -wp, wp)
T2 = mpf_mul(T2, mpf_exp(negx, wp))
if gamma:
T2 = mpf_mul(T2, mpf_pow_int(x, n_orig, wp), wp)
R = mpf_add(T1, T2)
re = mpf_div(R, from_int(int_fac(n - 1)), prec, rnd)
else:
raise NotImplementedError
if have_imag:
M = from_int(-int_fac(n - 1))
if gamma:
im = mpf_div(mpf_pi(wp), M, prec, rnd)
else:
im = mpf_div(
mpf_mul(mpf_pi(wp), mpf_pow_int(negx, n_orig - 1, wp), wp), M,
prec, rnd)
return re, im
else:
return re, None
def mpc_zeta(s, prec, rnd=round_fast, alt=0, force=False):
re, im = s
if im == fzero:
return mpf_zeta(re, prec, rnd, alt), fzero
# slow for large s
if (not force) and mpf_gt(mpc_abs(s, 10), from_int(prec)):
raise NotImplementedError
wp = prec + 20
# Near pole
r = mpc_sub(mpc_one, s, wp)
asign, aman, aexp, abc = mpc_abs(r, 10)
pole_dist = -2*(aexp+abc)
if pole_dist > wp:
if alt:
q = mpf_ln2(wp)
y = mpf_mul(q, mpf_euler(wp), wp)
g = mpf_shift(mpf_mul(q, q, wp), -1)
g = mpf_sub(y, g)
z = mpc_mul_mpf(r, mpf_neg(g), wp)
z = mpc_add_mpf(z, q, wp)
return mpc_pos(z, prec, rnd)
else:
q = mpc_neg(mpc_div(mpc_one, r, wp))
q = mpc_add_mpf(q, mpf_euler(wp), wp)
return mpc_pos(q, prec, rnd)
else:
wp += max(0, pole_dist)
# Reflection formula. To be rigorous, we should reflect to the left of
# re = 1/2 (see comments for mpf_zeta), but this leads to unnecessary
# slowdown for interesting values of s
if mpf_lt(re, fzero):
# XXX: could use the separate refl. formula for Dirichlet eta
if alt:
q = mpc_sub(mpc_one, mpc_pow(mpc_two, mpc_sub(mpc_one, s, wp),
wp), wp)
return mpc_mul(mpc_zeta(s, wp), q, prec, rnd)
# XXX: -1 should be done exactly
y = mpc_sub(mpc_one, s, 10*wp)
a = mpc_gamma(y, wp)
b = mpc_zeta(y, wp)
c = mpc_sin_pi(mpc_shift(s, -1), wp)
rsign, rman, rexp, rbc = re
isign, iman, iexp, ibc = im
mag = max(rexp+rbc, iexp+ibc)
wp2 = wp + mag
pi = mpf_pi(wp+wp2)
pi2 = (mpf_shift(pi, 1), fzero)
d = mpc_div_mpf(mpc_pow(pi2, s, wp2), pi, wp2)
return mpc_mul(a,mpc_mul(b,mpc_mul(c,d,wp),wp),prec,rnd)
n = int(wp/2.54 + 5)
n += int(0.9*abs(to_int(im)))
d = borwein_coefficients(n)
ref = to_fixed(re, wp)
imf = to_fixed(im, wp)
tre = MPZ_ZERO
tim = MPZ_ZERO
one = MPZ_ONE << wp
one_2wp = MPZ_ONE << (2*wp)
critical_line = re == fhalf
for k in xrange(n):
log = log_int_fixed(k+1, wp)
# A square root is much cheaper than an exp
if critical_line:
w = one_2wp // sqrt_fixed((k+1) << wp, wp)
else:
w = to_fixed(mpf_exp(from_man_exp(-ref*log, -2*wp), wp), wp)
if k & 1:
w *= (d[n] - d[k])
else:
w *= (d[k] - d[n])
wre, wim = mpf_cos_sin(from_man_exp(-imf * log, -2*wp), wp)
tre += (w * to_fixed(wre, wp)) >> wp
tim += (w * to_fixed(wim, wp)) >> wp
tre //= (-d[n])
tim //= (-d[n])
tre = from_man_exp(tre, -wp, wp)
tim = from_man_exp(tim, -wp, wp)
if alt:
return mpc_pos((tre, tim), prec, rnd)
else:
q = mpc_sub(mpc_one, mpc_pow(mpc_two, r, wp), wp)
return mpc_div((tre, tim), q, prec, rnd)
