def mpc_zeta(s, prec, rnd):
re, im = s
wp = prec + 20
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 = MP_ZERO
tim = MP_ZERO
one = MP_ONE << wp
one_2wp = MP_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 = cos_sin(from_man_exp(-imf * log_int_fixed(k+1, wp), -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)
q = mpc_sub(mpc_one, mpc_pow(mpc_two, mpc_sub(mpc_one, s, wp), wp), wp)
return mpc_div((tre, tim), q, prec, rnd)
def mpc_zeta(s, prec, rnd):
re, im = s
wp = prec + 20
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 = MP_ZERO
tim = MP_ZERO
one = MP_ONE << wp
one_2wp = MP_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 = cos_sin(
from_man_exp(-imf * log_int_fixed(k + 1, wp), -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)
q = mpc_sub(mpc_one, mpc_pow(mpc_two, mpc_sub(mpc_one, s, wp), wp), wp)
return mpc_div((tre, tim), q, prec, rnd)
def mpc_zeta(s, prec, rnd=round_fast, alt=0):
re, im = s
if im == fzero:
return mpf_zeta(re, prec, rnd, alt), fzero
wp = prec + 20
# 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 = MP_ZERO
tim = MP_ZERO
one = MP_ONE << wp
one_2wp = MP_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 = cos_sin(from_man_exp(-imf * log_int_fixed(k+1, wp), -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, mpc_sub(mpc_one, s, wp), wp), wp)
return mpc_div((tre, tim), q, prec, rnd)
def pi_fixed(prec, verbose=False, verbose_base=None):
"""
Compute floor(pi * 2**prec) as a big integer.
This is done using Chudnovsky's series (see comments in
libelefun.py for details).
"""
# The Chudnovsky series gives 14.18 digits per term
N = int(prec/3.3219280948/14.181647462 + 2)
if verbose:
print "binary splitting with N =", N
g, p, q = bs_chudnovsky(0, N, 0, verbose)
sqrtC = sqrt_fixed(CHUD_C<<prec, prec)
v = p*CHUD_C*sqrtC//((q+CHUD_A*p)*CHUD_D)
return v
def pi_fixed(prec, verbose=False, verbose_base=None):
"""
Compute floor(pi * 2**prec) as a big integer.
This is done using Chudnovsky's series (see comments in
libelefun.py for details).
"""
# The Chudnovsky series gives 14.18 digits per term
N = int(prec / 3.3219280948 / 14.181647462 + 2)
if verbose:
print "binary splitting with N =", N
g, p, q = bs_chudnovsky(0, N, 0, verbose)
sqrtC = sqrt_fixed(CHUD_C << prec, prec)
v = p * CHUD_C * sqrtC // ((q + CHUD_A * p) * CHUD_D)
return v
def expi_series(x, prec, r):
x >>= r
one = 1 << prec
x2 = (x*x) >> prec
s = x
a = x
k = 2
while a:
a = ((a * x2) >> prec) // (-k*(k+1))
s += a
k += 2
c = sqrt_fixed(one - ((s*s)>>prec), prec)
# Calculate (c + j*s)**(2**r) by repeated squaring
for j in range(r):
c, s = (c*c-s*s) >> prec, (2*c*s ) >> prec
return c, s
def expi_series(x, prec, r):
x >>= r
one = 1 << prec
x2 = (x * x) >> prec
s = x
a = x
k = 2
while a:
a = ((a * x2) >> prec) // (-k * (k + 1))
s += a
k += 2
c = sqrt_fixed(one - ((s * s) >> prec), prec)
# Calculate (c + j*s)**(2**r) by repeated squaring
for j in range(r):
c, s = (c * c - s * s) >> prec, (2 * c * s) >> prec
return c, s
def mpf_erfc(x, prec, rnd=round_fast):
sign, man, exp, bc = x
if not man:
if x == fzero: return fone
if x == finf: return fzero
if x == fninf: return ftwo
return fnan
wp = prec + 20
mag = bc + exp
# Preserve full accuracy when exponent grows huge
wp += max(0, 2 * mag)
regular_erf = sign or mag < 2
if regular_erf or not erfc_check_series(x, wp):
if regular_erf:
return mpf_sub(fone, mpf_erf(x, prec + 10, negative_rnd[rnd]),
prec, rnd)
# 1-erf(x) ~ exp(-x^2), increase prec to deal with cancellation
n = to_int(x)
return mpf_sub(fone, mpf_erf(x, prec + int(n**2 * 1.44) + 10), prec,
rnd)
s = term = MP_ONE << wp
term_prev = 0
t = (2 * to_fixed(x, wp)**2) >> wp
k = 1
while 1:
term = ((term * (2 * k - 1)) << wp) // t
if k > 4 and term > term_prev or not term:
break
if k & 1:
s -= term
else:
s += term
term_prev = term
#print k, to_str(from_man_exp(term, -wp, 50), 10)
k += 1
s = (s << wp) // sqrt_fixed(pi_fixed(wp), wp)
s = from_man_exp(s, -wp, wp)
z = mpf_exp(mpf_neg(mpf_mul(x, x, wp), wp), wp)
y = mpf_div(mpf_mul(z, s, wp), x, prec, rnd)
return y
def mpf_erf(x, prec, rnd=round_fast):
sign, man, exp, bc = x
if not man:
if x == fzero: return fzero
if x == finf: return fone
if x== fninf: return fnone
return fnan
size = exp + bc
lg = math.log
# The approximation erf(x) = 1 is accurate to > x^2 * log(e,2) bits
if size > 3 and 2*(size-1) + 0.528766 > lg(prec,2):
if sign:
return mpf_perturb(fnone, 0, prec, rnd)
else:
return mpf_perturb(fone, 1, prec, rnd)
# erf(x) ~ 2*x/sqrt(pi) close to 0
if size < -prec:
# 2*x
x = mpf_shift(x,1)
c = mpf_sqrt(mpf_pi(prec+20), prec+20)
# TODO: interval rounding
return mpf_div(x, c, prec, rnd)
wp = prec + abs(size) + 25
# Taylor series for erf, fixed-point summation
t = abs(to_fixed(x, wp))
t2 = (t*t) >> wp
s, term, k = t, 12345, 1
while term:
t = ((t * t2) >> wp) // k
term = t // (2*k+1)
if k & 1:
s -= term
else:
s += term
k += 1
s = (s << (wp+1)) // sqrt_fixed(pi_fixed(wp), wp)
if sign:
s = -s
return from_man_exp(s, -wp, prec, rnd)
def mpf_erf(x, prec, rnd=round_fast):
sign, man, exp, bc = x
if not man:
if x == fzero: return fzero
if x == finf: return fone
if x == fninf: return fnone
return fnan
size = exp + bc
lg = math.log
# The approximation erf(x) = 1 is accurate to > x^2 * log(e,2) bits
if size > 3 and 2 * (size - 1) + 0.528766 > lg(prec, 2):
if sign:
return mpf_perturb(fnone, 0, prec, rnd)
else:
return mpf_perturb(fone, 1, prec, rnd)
# erf(x) ~ 2*x/sqrt(pi) close to 0
if size < -prec:
# 2*x
x = mpf_shift(x, 1)
c = mpf_sqrt(mpf_pi(prec + 20), prec + 20)
# TODO: interval rounding
return mpf_div(x, c, prec, rnd)
wp = prec + abs(size) + 20
# Taylor series for erf, fixed-point summation
t = abs(to_fixed(x, wp))
t2 = (t * t) >> wp
s, term, k = t, 12345, 1
while term:
t = ((t * t2) >> wp) // k
term = t // (2 * k + 1)
if k & 1:
s -= term
else:
s += term
k += 1
s = (s << (wp + 1)) // sqrt_fixed(pi_fixed(wp), wp)
if sign:
s = -s
return from_man_exp(s, -wp, wp, rnd)
def mpf_erfc(x, prec, rnd=round_fast):
sign, man, exp, bc = x
if not man:
if x == fzero: return fone
if x == finf: return fzero
if x == fninf: return ftwo
return fnan
wp = prec + 20
mag = bc+exp
# Preserve full accuracy when exponent grows huge
wp += max(0, 2*mag)
regular_erf = sign or mag < 2
if regular_erf or not erfc_check_series(x, wp):
if regular_erf:
return mpf_sub(fone, mpf_erf(x, prec+10, negative_rnd[rnd]), prec, rnd)
# 1-erf(x) ~ exp(-x^2), increase prec to deal with cancellation
n = to_int(x)+1
return mpf_sub(fone, mpf_erf(x, prec + int(n**2*1.44) + 10), prec, rnd)
s = term = MPZ_ONE << wp
term_prev = 0
t = (2 * to_fixed(x, wp) ** 2) >> wp
k = 1
while 1:
term = ((term * (2*k - 1)) << wp) // t
if k > 4 and term > term_prev or not term:
break
if k & 1:
s -= term
else:
s += term
term_prev = term
#print k, to_str(from_man_exp(term, -wp, 50), 10)
k += 1
s = (s << wp) // sqrt_fixed(pi_fixed(wp), wp)
s = from_man_exp(s, -wp, wp)
z = mpf_exp(mpf_neg(mpf_mul(x,x,wp),wp),wp)
y = mpf_div(mpf_mul(z, s, wp), x, prec, rnd)
return y
def exp_series2(x, prec, r):
x >>= r
sign = 0
if x < 0:
sign = 1
x = -x
x2 = (x * x) >> prec
s1 = a = x
k = 3
while a:
a = ((a * x2) >> prec) // (k * (k - 1))
s1 += a
k += 2
c1 = sqrt_fixed(((s1 * s1) >> prec) + (1 << prec), prec)
if sign:
s = c1 - s1
else:
s = c1 + s1
# Calculate s**(2**r) by repeated squaring
while r:
s = (s * s) >> prec
r -= 1
return s
def exp_series2(x, prec, r):
x >>= r
sign = 0
if x < 0:
sign = 1
x = -x
x2 = (x*x) >> prec
s1 = a = x
k = 3
while a:
a = ((a * x2) >> prec) // (k*(k-1))
s1 += a
k += 2
c1 = sqrt_fixed(((s1*s1) >> prec) + (1<<prec), prec)
if sign:
s = c1 - s1
else:
s = c1 + s1
# Calculate s**(2**r) by repeated squaring
while r:
s = (s*s) >> prec
r -= 1
return s
def calc_cos_sin(which, y, swaps, prec, cos_rnd, sin_rnd):
"""
Simultaneous computation of cos and sin (internal function).
"""
y, wp = y
swap_cos_sin, cos_sign, sin_sign = swaps
if swap_cos_sin:
which_compute = -which
else:
which_compute = which
# XXX: assumes no swaps
if not y:
return fone, fzero
# Tiny nonzero argument
if wp > prec*2 + 30:
y = from_man_exp(y, -wp)
if swap_cos_sin:
cos_rnd, sin_rnd = sin_rnd, cos_rnd
cos_sign, sin_sign = sin_sign, cos_sign
if cos_sign: cos = mpf_perturb(fnone, 0, prec, cos_rnd)
else: cos = mpf_perturb(fone, 1, prec, cos_rnd)
if sin_sign: sin = mpf_perturb(mpf_neg(y), 0, prec, sin_rnd)
else: sin = mpf_perturb(y, 1, prec, sin_rnd)
if swap_cos_sin:
cos, sin = sin, cos
return cos, sin
# Use standard Taylor series
if prec < 600:
if which_compute == 0:
sin = sin_taylor(y, wp)
# only need to evaluate one of the series
cos = sqrt_fixed((1<<wp) - ((sin*sin)>>wp), wp)
elif which_compute == 1:
sin = 0
cos = cos_taylor(y, wp)
elif which_compute == -1:
sin = sin_taylor(y, wp)
cos = 0
# Use exp(i*x) with Brent's trick
else:
r = int(0.137 * prec**0.579)
ep = r+20
cos, sin = expi_series(y<<ep, wp+ep, r)
cos >>= ep
sin >>= ep
if swap_cos_sin:
cos, sin = sin, cos
if cos_rnd is not round_nearest:
# Round and set correct signs
# XXX: this logic needs a second look
ONE = MP_ONE << wp
if cos_sign:
cos += (-1)**(cos_rnd in (round_ceiling, round_down))
cos = min(ONE, cos)
else:
cos += (-1)**(cos_rnd in (round_ceiling, round_up))
cos = min(ONE, cos)
if sin_sign:
sin += (-1)**(sin_rnd in (round_ceiling, round_down))
sin = min(ONE, sin)
else:
sin += (-1)**(sin_rnd in (round_ceiling, round_up))
sin = min(ONE, sin)
if which != -1:
cos = normalize(cos_sign, cos, -wp, bitcount(cos), prec, cos_rnd)
if which != 1:
sin = normalize(sin_sign, sin, -wp, bitcount(sin), prec, sin_rnd)
return cos, sin
def sqrtpi_fixed(prec):
return sqrt_fixed(pi_fixed(prec), prec)
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)
def sqrtpi_fixed(prec):
return sqrt_fixed(pi_fixed(prec), prec)
def mpc_zetasum(s, a, n, derivatives, reflect, prec):
"""
Fast version of mp._zetasum, assuming s = complex, a = integer.
"""
wp = prec + 10
have_derivatives = derivatives != [0]
have_one_derivative = len(derivatives) == 1
# parse s
sre, sim = s
critical_line = (sre == fhalf)
sre = to_fixed(sre, wp)
sim = to_fixed(sim, wp)
maxd = max(derivatives)
if not have_one_derivative:
derivatives = range(maxd + 1)
# x_d = 0, y_d = 0
xre = [MPZ_ZERO for d in derivatives]
xim = [MPZ_ZERO for d in derivatives]
if reflect:
yre = [MPZ_ZERO for d in derivatives]
yim = [MPZ_ZERO for d in derivatives]
else:
yre = yim = []
one = MPZ_ONE << wp
one_2wp = MPZ_ONE << (2 * wp)
for w in xrange(a, a + n + 1):
log = log_int_fixed(w, wp)
cos, sin = cos_sin_fixed_prod(-sim * log, wp)
if critical_line:
u = one_2wp // sqrt_fixed(w << wp, wp)
else:
u = exp_fixed_prod(-sre * log, wp)
xterm_re = (u * cos) >> wp
xterm_im = (u * sin) >> wp
if reflect:
reciprocal = (one_2wp // (u * w))
yterm_re = (reciprocal * cos) >> wp
yterm_im = (reciprocal * sin) >> wp
if have_derivatives:
if have_one_derivative:
log = pow_fixed(log, maxd, wp)
xre[0] += (xterm_re * log) >> wp
xim[0] += (xterm_im * log) >> wp
if reflect:
yre[0] += (yterm_re * log) >> wp
yim[0] += (yterm_im * log) >> wp
else:
t = MPZ_ONE << wp
for d in derivatives:
xre[d] += (xterm_re * t) >> wp
xim[d] += (xterm_im * t) >> wp
if reflect:
yre[d] += (yterm_re * t) >> wp
yim[d] += (yterm_im * t) >> wp
t = (t * log) >> wp
else:
xre[0] += xterm_re
xim[0] += xterm_im
if reflect:
yre[0] += yterm_re
yim[0] += yterm_im
if have_derivatives:
if have_one_derivative:
if maxd % 2:
xre[0] = -xre[0]
xim[0] = -xim[0]
if reflect:
yre[0] = -yre[0]
yim[0] = -yim[0]
else:
xre = [(-1)**d * xre[d] for d in derivatives]
xim = [(-1)**d * xim[d] for d in derivatives]
if reflect:
yre = [(-1)**d * yre[d] for d in derivatives]
yim = [(-1)**d * yim[d] for d in derivatives]
xs = [(from_man_exp(xa, -wp, prec, 'n'), from_man_exp(xb, -wp, prec, 'n'))
for (xa, xb) in zip(xre, xim)]
ys = [(from_man_exp(ya, -wp, prec, 'n'), from_man_exp(yb, -wp, prec, 'n'))
for (ya, yb) in zip(yre, yim)]
return xs, ys
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)
def mpc_zeta(s, prec, rnd=round_fast, alt=0):
re, im = s
if im == fzero:
return mpf_zeta(re, prec, rnd, alt), fzero
wp = prec + 20
# 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 = MP_ZERO
tim = MP_ZERO
one = MP_ONE << wp
one_2wp = MP_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 = cos_sin(
from_man_exp(-imf * log_int_fixed(k + 1, wp), -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, mpc_sub(mpc_one, s, wp), wp), wp)
return mpc_div((tre, tim), q, prec, rnd)
def mpc_zetasum(s, a, n, derivatives, reflect, prec):
"""
Fast version of mp._zetasum, assuming s = complex, a = integer.
"""
wp = prec + 10
have_derivatives = derivatives != [0]
have_one_derivative = len(derivatives) == 1
# parse s
sre, sim = s
critical_line = (sre == fhalf)
sre = to_fixed(sre, wp)
sim = to_fixed(sim, wp)
maxd = max(derivatives)
if not have_one_derivative:
derivatives = range(maxd+1)
# x_d = 0, y_d = 0
xre = [MPZ_ZERO for d in derivatives]
xim = [MPZ_ZERO for d in derivatives]
if reflect:
yre = [MPZ_ZERO for d in derivatives]
yim = [MPZ_ZERO for d in derivatives]
else:
yre = yim = []
one = MPZ_ONE << wp
one_2wp = MPZ_ONE << (2*wp)
for w in xrange(a, a+n+1):
log = log_int_fixed(w, wp)
cos, sin = cos_sin_fixed_prod(-sim*log, wp)
if critical_line:
u = one_2wp // sqrt_fixed(w << wp, wp)
else:
u = exp_fixed_prod(-sre*log, wp)
xterm_re = (u * cos) >> wp
xterm_im = (u * sin) >> wp
if reflect:
reciprocal = (one_2wp // (u*w))
yterm_re = (reciprocal * cos) >> wp
yterm_im = (reciprocal * sin) >> wp
if have_derivatives:
if have_one_derivative:
log = pow_fixed(log, maxd, wp)
xre[0] += (xterm_re * log) >> wp
xim[0] += (xterm_im * log) >> wp
if reflect:
yre[0] += (yterm_re * log) >> wp
yim[0] += (yterm_im * log) >> wp
else:
t = MPZ_ONE << wp
for d in derivatives:
xre[d] += (xterm_re * t) >> wp
xim[d] += (xterm_im * t) >> wp
if reflect:
yre[d] += (yterm_re * t) >> wp
yim[d] += (yterm_im * t) >> wp
t = (t * log) >> wp
else:
xre[0] += xterm_re
xim[0] += xterm_im
if reflect:
yre[0] += yterm_re
yim[0] += yterm_im
if have_derivatives:
if have_one_derivative:
if maxd % 2:
xre[0] = -xre[0]
xim[0] = -xim[0]
if reflect:
yre[0] = -yre[0]
yim[0] = -yim[0]
else:
xre = [(-1)**d * xre[d] for d in derivatives]
xim = [(-1)**d * xim[d] for d in derivatives]
if reflect:
yre = [(-1)**d * yre[d] for d in derivatives]
yim = [(-1)**d * yim[d] for d in derivatives]
xs = [(from_man_exp(xa, -wp, prec, 'n'), from_man_exp(xb, -wp, prec, 'n'))
for (xa, xb) in zip(xre, xim)]
ys = [(from_man_exp(ya, -wp, prec, 'n'), from_man_exp(yb, -wp, prec, 'n'))
for (ya, yb) in zip(yre, yim)]
return xs, ys
def calc_cos_sin(which, y, swaps, prec, cos_rnd, sin_rnd):
"""
Simultaneous computation of cos and sin (internal function).
"""
y, wp = y
swap_cos_sin, cos_sign, sin_sign = swaps
if swap_cos_sin:
which_compute = -which
else:
which_compute = which
# XXX: assumes no swaps
if not y:
return fone, fzero
# Tiny nonzero argument
if wp > prec * 2 + 30:
y = from_man_exp(y, -wp)
if swap_cos_sin:
cos_rnd, sin_rnd = sin_rnd, cos_rnd
cos_sign, sin_sign = sin_sign, cos_sign
if cos_sign: cos = mpf_perturb(fnone, 0, prec, cos_rnd)
else: cos = mpf_perturb(fone, 1, prec, cos_rnd)
if sin_sign: sin = mpf_perturb(mpf_neg(y), 0, prec, sin_rnd)
else: sin = mpf_perturb(y, 1, prec, sin_rnd)
if swap_cos_sin:
cos, sin = sin, cos
return cos, sin
# Use standard Taylor series
if prec < 600:
if which_compute == 0:
sin = sin_taylor(y, wp)
# only need to evaluate one of the series
cos = sqrt_fixed((1 << wp) - ((sin * sin) >> wp), wp)
elif which_compute == 1:
sin = 0
cos = cos_taylor(y, wp)
elif which_compute == -1:
sin = sin_taylor(y, wp)
cos = 0
# Use exp(i*x) with Brent's trick
else:
r = int(0.137 * prec**0.579)
ep = r + 20
cos, sin = expi_series(y << ep, wp + ep, r)
cos >>= ep
sin >>= ep
if swap_cos_sin:
cos, sin = sin, cos
if cos_rnd is not round_nearest:
# Round and set correct signs
# XXX: this logic needs a second look
ONE = MP_ONE << wp
if cos_sign:
cos += (-1)**(cos_rnd in (round_ceiling, round_down))
cos = min(ONE, cos)
else:
cos += (-1)**(cos_rnd in (round_ceiling, round_up))
cos = min(ONE, cos)
if sin_sign:
sin += (-1)**(sin_rnd in (round_ceiling, round_down))
sin = min(ONE, sin)
else:
sin += (-1)**(sin_rnd in (round_ceiling, round_up))
sin = min(ONE, sin)
if which != -1:
cos = normalize(cos_sign, cos, -wp, bitcount(cos), prec, cos_rnd)
if which != 1:
sin = normalize(sin_sign, sin, -wp, bitcount(sin), prec, sin_rnd)
return cos, sin
