def mpf_atan2(y, x, prec, rnd=round_fast):
xsign, xman, xexp, xbc = x
ysign, yman, yexp, ybc = y
if not yman:
if y == fzero and x != fnan:
if mpf_sign(x) >= 0:
return fzero
return mpf_pi(prec, rnd)
if y in (finf, fninf):
if x in (finf, fninf):
return fnan
# pi/2
if y == finf:
return mpf_shift(mpf_pi(prec, rnd), -1)
# -pi/2
return mpf_neg(mpf_shift(mpf_pi(prec, negative_rnd[rnd]), -1))
return fnan
if ysign:
return mpf_neg(mpf_atan2(mpf_neg(y), x, prec, negative_rnd[rnd]))
if not xman:
if x == fnan:
return fnan
if x == finf:
return fzero
if x == fninf:
return mpf_pi(prec, rnd)
if y == fzero:
return fzero
return mpf_shift(mpf_pi(prec, rnd), -1)
tquo = mpf_atan(mpf_div(y, x, prec+4), prec+4)
if xsign:
return mpf_add(mpf_pi(prec+4), tquo, prec, rnd)
else:
return mpf_pos(tquo, prec, rnd)
def mpc_sqrt(z, prec, rnd=round_fast):
"""Complex square root (principal branch).
We have sqrt(a+bi) = sqrt((r+a)/2) + b/sqrt(2*(r+a))*i where
r = abs(a+bi), when a+bi is not a negative real number."""
a, b = z
if b == fzero:
if a == fzero:
return (a, b)
# When a+bi is a negative real number, we get a real sqrt times i
if a[0]:
im = mpf_sqrt(mpf_neg(a), prec, rnd)
return (fzero, im)
else:
re = mpf_sqrt(a, prec, rnd)
return (re, fzero)
wp = prec+20
if not a[0]: # case a positive
t = mpf_add(mpc_abs((a, b), wp), a, wp) # t = abs(a+bi) + a
u = mpf_shift(t, -1) # u = t/2
re = mpf_sqrt(u, prec, rnd) # re = sqrt(u)
v = mpf_shift(t, 1) # v = 2*t
w = mpf_sqrt(v, wp) # w = sqrt(v)
im = mpf_div(b, w, prec, rnd) # im = b / w
else: # case a negative
t = mpf_sub(mpc_abs((a, b), wp), a, wp) # t = abs(a+bi) - a
u = mpf_shift(t, -1) # u = t/2
im = mpf_sqrt(u, prec, rnd) # im = sqrt(u)
v = mpf_shift(t, 1) # v = 2*t
w = mpf_sqrt(v, wp) # w = sqrt(v)
re = mpf_div(b, w, prec, rnd) # re = b/w
if b[0]:
re = mpf_neg(re)
im = mpf_neg(im)
return re, im
def mpf_atan2(y, x, prec, rnd=round_fast):
xsign, xman, xexp, xbc = x
ysign, yman, yexp, ybc = y
if not yman:
if y == fnan or x == fnan:
return fnan
if mpf_sign(x) >= 0:
return fzero
return mpf_pi(prec, rnd)
if ysign:
return mpf_neg(mpf_atan2(mpf_neg(y), x, prec, rnd))
if not xman:
if x == fnan:
return fnan
if x == finf:
return fzero
if x == fninf:
return mpf_pi(prec, rnd)
if not yman:
return fzero
return mpf_shift(mpf_pi(prec, rnd), -1)
tquo = mpf_atan(mpf_div(y, x, prec+4), prec+4)
if xsign:
return mpf_add(mpf_pi(prec+4), tquo, prec, rnd)
else:
return mpf_pos(tquo, prec, rnd)
def mpc_sqrt(z, prec, rnd=round_fast):
"""Complex square root (principal branch).
We have sqrt(a+bi) = sqrt((r+a)/2) + b/sqrt(2*(r+a))*i where
r = abs(a+bi), when a+bi is not a negative real number."""
a, b = z
if b == fzero:
if a == fzero:
return (a, b)
# When a+bi is a negative real number, we get a real sqrt times i
if a[0]:
im = mpf_sqrt(mpf_neg(a), prec, rnd)
return (fzero, im)
else:
re = mpf_sqrt(a, prec, rnd)
return (re, fzero)
wp = prec + 20
if not a[0]: # case a positive
t = mpf_add(mpc_abs((a, b), wp), a, wp) # t = abs(a+bi) + a
u = mpf_shift(t, -1) # u = t/2
re = mpf_sqrt(u, prec, rnd) # re = sqrt(u)
v = mpf_shift(t, 1) # v = 2*t
w = mpf_sqrt(v, wp) # w = sqrt(v)
im = mpf_div(b, w, prec, rnd) # im = b / w
else: # case a negative
t = mpf_sub(mpc_abs((a, b), wp), a, wp) # t = abs(a+bi) - a
u = mpf_shift(t, -1) # u = t/2
im = mpf_sqrt(u, prec, rnd) # im = sqrt(u)
v = mpf_shift(t, 1) # v = 2*t
w = mpf_sqrt(v, wp) # w = sqrt(v)
re = mpf_div(b, w, prec, rnd) # re = b/w
if b[0]:
re = mpf_neg(re)
im = mpf_neg(im)
return re, im
def mpf_cos_sin_pi(x, prec, rnd=round_fast):
"""Accurate computation of (cos(pi*x), sin(pi*x))
for x close to an integer"""
sign, man, exp, bc = x
if not man:
return cos_sin(x, prec, rnd)
# Exactly an integer or half-integer?
if exp >= -1:
if exp == -1:
c = fzero
s = (fone, fnone)[bool(man & 2) ^ sign]
elif exp == 0:
c, s = (fnone, fzero)
else:
c, s = (fone, fzero)
return c, s
# Close to 0 ?
size = exp + bc
if size < -(prec+5):
return (fone, mpf_mul(x, mpf_pi(wp), prec, rnd))
if sign:
man = -man
# Subtract nearest integer (= modulo pi)
nint = ((man >> (-exp-1)) + 1) >> 1
man = man - (nint << (-exp))
x = from_man_exp(man, exp, prec)
x = mpf_mul(x, mpf_pi(prec), prec)
# Shifted an odd multiple of pi ?
if nint & 1:
c, s = cos_sin(x, prec, negative_rnd[rnd])
return mpf_neg(c), mpf_neg(s)
else:
return cos_sin(x, prec, rnd)
def mpf_cos_sin_pi(x, prec, rnd=round_fast):
"""Accurate computation of (cos(pi*x), sin(pi*x))
for x close to an integer"""
sign, man, exp, bc = x
if not man:
return cos_sin(x, prec, rnd)
# Exactly an integer or half-integer?
if exp >= -1:
if exp == -1:
c = fzero
s = (fone, fnone)[bool(man & 2) ^ sign]
elif exp == 0:
c, s = (fnone, fzero)
else:
c, s = (fone, fzero)
return c, s
# Close to 0 ?
size = exp + bc
if size < -(prec + 5):
return (fone, mpf_mul(x, mpf_pi(wp), prec, rnd))
if sign:
man = -man
# Subtract nearest integer (= modulo pi)
nint = ((man >> (-exp - 1)) + 1) >> 1
man = man - (nint << (-exp))
x = from_man_exp(man, exp, prec)
x = mpf_mul(x, mpf_pi(prec), prec)
# Shifted an odd multiple of pi ?
if nint & 1:
c, s = cos_sin(x, prec, negative_rnd[rnd])
return mpf_neg(c), mpf_neg(s)
else:
return cos_sin(x, prec, rnd)
def mpf_atan2(y, x, prec, rnd=round_fast):
xsign, xman, xexp, xbc = x
ysign, yman, yexp, ybc = y
if not yman:
if y == fzero and x != fnan:
if mpf_sign(x) >= 0:
return fzero
return mpf_pi(prec, rnd)
if y in (finf, fninf):
if x in (finf, fninf):
return fnan
# pi/2
if y == finf:
return mpf_shift(mpf_pi(prec, rnd), -1)
# -pi/2
return mpf_neg(mpf_shift(mpf_pi(prec, negative_rnd[rnd]), -1))
return fnan
if ysign:
return mpf_neg(mpf_atan2(mpf_neg(y), x, prec, negative_rnd[rnd]))
if not xman:
if x == fnan:
return fnan
if x == finf:
return fzero
if x == fninf:
return mpf_pi(prec, rnd)
if y == fzero:
return fzero
return mpf_shift(mpf_pi(prec, rnd), -1)
tquo = mpf_atan(mpf_div(y, x, prec+4), prec+4)
if xsign:
return mpf_add(mpf_pi(prec+4), tquo, prec, rnd)
else:
return mpf_pos(tquo, prec, rnd)
def mpf_atan2(y, x, prec, rnd=round_fast):
xsign, xman, xexp, xbc = x
ysign, yman, yexp, ybc = y
if not yman:
if y == fnan or x == fnan:
return fnan
if mpf_sign(x) >= 0:
return fzero
return mpf_pi(prec, rnd)
if ysign:
return mpf_neg(mpf_atan2(mpf_neg(y), x, prec, rnd))
if not xman:
if x == fnan:
return fnan
if x == finf:
return fzero
if x == fninf:
return mpf_pi(prec, rnd)
if not yman:
return fzero
return mpf_shift(mpf_pi(prec, rnd), -1)
tquo = mpf_atan(mpf_div(y, x, prec + 4), prec + 4)
if xsign:
return mpf_add(mpf_pi(prec + 4), tquo, prec, rnd)
else:
return mpf_pos(tquo, prec, rnd)
def mpc_acosh(z, prec, rnd=round_fast):
# acosh(z) = -I * acos(z) for Im(acos(z)) <= 0
# +I * acos(z) otherwise
a, b = mpc_acos(z, prec, rnd)
if b[0] or b == fzero:
return mpf_neg(b), a
else:
return b, mpf_neg(a)
def mpc_acosh(z, prec, rnd=round_fast):
# acosh(z) = -I * acos(z) for Im(acos(z)) <= 0
# +I * acos(z) otherwise
a, b = mpc_acos(z, prec, rnd)
if b[0] or b == fzero:
return mpf_neg(b), a
else:
return b, mpf_neg(a)
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 mpc_expj(z, prec, rnd='f'):
re, im = z
if im == fzero:
return mpf_cos_sin(re, prec, rnd)
if re == fzero:
return mpf_exp(mpf_neg(im), prec, rnd), fzero
ey = mpf_exp(mpf_neg(im), prec + 10)
c, s = mpf_cos_sin(re, prec + 10)
re = mpf_mul(ey, c, prec, rnd)
im = mpf_mul(ey, s, prec, rnd)
return re, im
def mpc_expj(z, prec, rnd="f"):
re, im = z
if im == fzero:
return mpf_cos_sin(re, prec, rnd)
if re == fzero:
return mpf_exp(mpf_neg(im), prec, rnd), fzero
ey = mpf_exp(mpf_neg(im), prec + 10)
c, s = mpf_cos_sin(re, prec + 10)
re = mpf_mul(ey, c, prec, rnd)
im = mpf_mul(ey, s, prec, rnd)
return re, im
def mpc_reciprocal(z, prec, rnd=round_fast):
"""Calculate 1/z efficiently"""
a, b = z
m = mpf_add(mpf_mul(a, a), mpf_mul(b, b), prec + 10)
re = mpf_div(a, m, prec, rnd)
im = mpf_neg(mpf_div(b, m, prec, rnd))
return re, im
def mpi_pow_int(s, n, prec):
sa, sb = s
if n < 0:
return mpi_div((fone, fone), mpi_pow_int(s, -n, prec+20), prec)
if n == 0:
return (fone, fone)
if n == 1:
return s
# Odd -- signs are preserved
if n & 1:
a = mpf_pow_int(sa, n, prec, round_floor)
b = mpf_pow_int(sb, n, prec, round_ceiling)
# Even -- important to ensure positivity
else:
sas = mpf_sign(sa)
sbs = mpf_sign(sb)
# Nonnegative?
if sas >= 0:
a = mpf_pow_int(sa, n, prec, round_floor)
b = mpf_pow_int(sb, n, prec, round_ceiling)
# Nonpositive?
elif sbs <= 0:
a = mpf_pow_int(sb, n, prec, round_floor)
b = mpf_pow_int(sa, n, prec, round_ceiling)
# Mixed signs?
else:
a = fzero
# max(-a,b)**n
sa = mpf_neg(sa)
if mpf_ge(sa, sb):
b = mpf_pow_int(sa, n, prec, round_ceiling)
else:
b = mpf_pow_int(sb, n, prec, round_ceiling)
return a, b
def mpc_to_str(z, dps, **kwargs):
re, im = z
rs = to_str(re, dps)
if im[0]:
return rs + " - " + to_str(mpf_neg(im), dps, **kwargs) + "j"
else:
return rs + " + " + to_str(im, dps, **kwargs) + "j"
def mpc_to_str(z, dps, **kwargs):
re, im = z
rs = to_str(re, dps)
if im[0]:
return rs + " - " + to_str(mpf_neg(im), dps, **kwargs) + "j"
else:
return rs + " + " + to_str(im, dps, **kwargs) + "j"
def mpi_pow_int(s, n, prec):
sa, sb = s
if n < 0:
return mpi_div((fone, fone), mpi_pow_int(s, -n, prec + 20), prec)
if n == 0:
return (fone, fone)
if n == 1:
return s
# Odd -- signs are preserved
if n & 1:
a = mpf_pow_int(sa, n, prec, round_floor)
b = mpf_pow_int(sb, n, prec, round_ceiling)
# Even -- important to ensure positivity
else:
sas = mpf_sign(sa)
sbs = mpf_sign(sb)
# Nonnegative?
if sas >= 0:
a = mpf_pow_int(sa, n, prec, round_floor)
b = mpf_pow_int(sb, n, prec, round_ceiling)
# Nonpositive?
elif sbs <= 0:
a = mpf_pow_int(sb, n, prec, round_floor)
b = mpf_pow_int(sa, n, prec, round_ceiling)
# Mixed signs?
else:
a = fzero
# max(-a,b)**n
sa = mpf_neg(sa)
if mpf_ge(sa, sb):
b = mpf_pow_int(sa, n, prec, round_ceiling)
else:
b = mpf_pow_int(sb, n, prec, round_ceiling)
return a, b
def mpc_mpf_div(p, z, prec, rnd=round_fast):
"""Calculate p/z where p is real efficiently"""
a, b = z
m = mpf_add(mpf_mul(a,a),mpf_mul(b,b), prec+10)
re = mpf_div(mpf_mul(a,p), m, prec, rnd)
im = mpf_div(mpf_neg(mpf_mul(b,p)), m, prec, rnd)
return re, im
def mpc_reciprocal(z, prec, rnd=round_fast):
"""Calculate 1/z efficiently"""
a, b = z
m = mpf_add(mpf_mul(a,a),mpf_mul(b,b),prec+10)
re = mpf_div(a, m, prec, rnd)
im = mpf_neg(mpf_div(b, m, prec, rnd))
return re, im
def mpc_mpf_div(p, z, prec, rnd=round_fast):
"""Calculate p/z where p is real efficiently"""
a, b = z
m = mpf_add(mpf_mul(a, a), mpf_mul(b, b), prec + 10)
re = mpf_div(mpf_mul(a, p), m, prec, rnd)
im = mpf_div(mpf_neg(mpf_mul(b, p)), m, prec, rnd)
return re, im
def mpc_pow_int(z, n, prec, rnd=round_fast):
a, b = z
if b == fzero:
return mpf_pow_int(a, n, prec, rnd), fzero
if a == fzero:
v = mpf_pow_int(b, n, prec, rnd)
n %= 4
if n == 0:
return v, fzero
elif n == 1:
return fzero, v
elif n == 2:
return mpf_neg(v), fzero
elif n == 3:
return fzero, mpf_neg(v)
if n == 0:
return mpc_one
if n == 1:
return mpc_pos(z, prec, rnd)
if n == 2:
return mpc_square(z, prec, rnd)
if n == -1:
return mpc_reciprocal(z, prec, rnd)
if n < 0:
return mpc_reciprocal(mpc_pow_int(z, -n, prec + 4), prec, rnd)
asign, aman, aexp, abc = a
bsign, bman, bexp, bbc = b
if asign:
aman = -aman
if bsign:
bman = -bman
de = aexp - bexp
abs_de = abs(de)
exact_size = n * (abs_de + max(abc, bbc))
if exact_size < 10000:
if de > 0:
aman <<= de
aexp = bexp
else:
bman <<= -de
bexp = aexp
re, im = complex_int_pow(aman, bman, n)
re = from_man_exp(re, int(n * aexp), prec, rnd)
im = from_man_exp(im, int(n * bexp), prec, rnd)
return re, im
return mpc_exp(mpc_mul_int(mpc_log(z, prec + 10), n, prec + 10), prec, rnd)
def mpc_mul_imag_mpf(z, x, prec, rnd=round_fast):
"""
Multiply the mpc value z by I*x where x is an mpf value.
"""
a, b = z
re = mpf_neg(mpf_mul(b, x, prec, rnd))
im = mpf_mul(a, x, prec, rnd)
return re, im
def mpc_mul_imag_mpf(z, x, prec, rnd=round_fast):
"""
Multiply the mpc value z by I*x where x is an mpf value.
"""
a, b = z
re = mpf_neg(mpf_mul(b, x, prec, rnd))
im = mpf_mul(a, x, prec, rnd)
return re, im
def mpf_cos_sin_pi(x, prec, rnd=round_fast):
"""Accurate computation of (cos(pi*x), sin(pi*x))
for x close to an integer"""
sign, man, exp, bc = x
if not man:
return cos_sin(x, prec, rnd)
# Exactly an integer or half-integer?
if exp >= -1:
if exp == -1:
c = fzero
s = (fone, fnone)[bool(man & 2) ^ sign]
elif exp == 0:
c, s = (fnone, fzero)
else:
c, s = (fone, fzero)
return c, s
# Close to 0 ?
size = exp + bc
if size < -(prec+5):
c = mpf_perturb(fone, 1, prec, rnd)
s = mpf_perturb(mpf_mul(x, mpf_pi(prec)), sign, prec, rnd)
return c, s
if sign:
man = -man
# Subtract nearest half-integer (= modulo pi/2)
nhint = ((man >> (-exp-2)) + 1) >> 1
man = man - (nhint << (-exp-1))
x = from_man_exp(man, exp, prec)
x = mpf_mul(x, mpf_pi(prec), prec)
# XXX: with some more work, could call calc_cos_sin,
# to save some time and to get rounding right
case = nhint % 4
if case == 0:
c, s = cos_sin(x, prec, rnd)
elif case == 1:
s, c = cos_sin(x, prec, rnd)
c = mpf_neg(c)
elif case == 2:
c, s = cos_sin(x, prec, rnd)
c = mpf_neg(c)
s = mpf_neg(s)
else:
s, c = cos_sin(x, prec, rnd)
s = mpf_neg(s)
return c, s
def mpc_atan(z, prec, rnd=round_fast):
a, b = z
# atan(z) = (I/2)*(log(1-I*z) - log(1+I*z))
# x = 1-I*z = 1 + b - I*a
# y = 1+I*z = 1 - b + I*a
wp = prec + 15
x = mpf_add(fone, b, wp), mpf_neg(a)
y = mpf_sub(fone, b, wp), a
l1 = mpc_log(x, wp)
l2 = mpc_log(y, wp)
a, b = mpc_sub(l1, l2, prec, rnd)
# (I/2) * (a+b*I) = (-b/2 + a/2*I)
v = mpf_neg(mpf_shift(b,-1)), mpf_shift(a,-1)
# Subtraction at infinity gives correct real part but
# wrong imaginary part (should be zero)
if v[1] == fnan and mpc_is_inf(z):
v = (v[0], fzero)
return v
def mpc_atan(z, prec, rnd=round_fast):
a, b = z
# atan(z) = (I/2)*(log(1-I*z) - log(1+I*z))
# x = 1-I*z = 1 + b - I*a
# y = 1+I*z = 1 - b + I*a
wp = prec + 15
x = mpf_add(fone, b, wp), mpf_neg(a)
y = mpf_sub(fone, b, wp), a
l1 = mpc_log(x, wp)
l2 = mpc_log(y, wp)
a, b = mpc_sub(l1, l2, prec, rnd)
# (I/2) * (a+b*I) = (-b/2 + a/2*I)
v = mpf_neg(mpf_shift(b, -1)), mpf_shift(a, -1)
# Subtraction at infinity gives correct real part but
# wrong imaginary part (should be zero)
if v[1] == fnan and mpc_is_inf(z):
v = (v[0], fzero)
return v
def mpf_bernoulli_huge(n, prec, rnd=None):
wp = prec + 10
piprec = wp + int(math.log(n, 2))
v = mpf_gamma_int(n + 1, wp)
v = mpf_mul(v, mpf_zeta_int(n, wp), wp)
v = mpf_mul(v, mpf_pow_int(mpf_pi(piprec), -n, wp))
v = mpf_shift(v, 1 - n)
if not n & 3:
v = mpf_neg(v)
return mpf_pos(v, prec, rnd or round_fast)
def mpf_bernoulli_huge(n, prec, rnd=None):
wp = prec + 10
piprec = wp + int(math.log(n,2))
v = mpf_gamma_int(n+1, wp)
v = mpf_mul(v, mpf_zeta_int(n, wp), wp)
v = mpf_mul(v, mpf_pow_int(mpf_pi(piprec), -n, wp))
v = mpf_shift(v, 1-n)
if not n & 3:
v = mpf_neg(v)
return mpf_pos(v, prec, rnd or round_fast)
def mpc_cos_pi(z, prec, rnd=round_fast):
a, b = z
b = mpf_mul(b, mpf_pi(prec + 5), prec + 5)
if a == fzero:
return mpf_cosh(b, prec, rnd), fzero
wp = prec + 6
c, s = mpf_cos_sin_pi(a, wp)
ch, sh = mpf_cosh_sinh(b, wp)
re = mpf_mul(c, ch, prec, rnd)
im = mpf_mul(s, sh, prec, rnd)
return re, mpf_neg(im)
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 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 mpc_cos_pi(z, prec, rnd=round_fast):
a, b = z
b = mpf_mul(b, mpf_pi(prec+5), prec+5)
if a == fzero:
return mpf_cosh(b, prec, rnd), fzero
wp = prec + 6
c, s = mpf_cos_sin_pi(a, wp)
ch, sh = mpf_cosh_sinh(b, wp)
re = mpf_mul(c, ch, prec, rnd)
im = mpf_mul(s, sh, prec, rnd)
return re, mpf_neg(im)
def mpf_ei(x, prec, rnd=round_fast, e1=False):
if e1:
x = mpf_neg(x)
sign, man, exp, bc = x
if e1 and not sign:
if x == fzero:
return finf
raise ComplexResult("E1(x) for x < 0")
if man:
xabs = 0, man, exp, bc
xmag = exp + bc
wp = prec + 20
can_use_asymp = xmag > wp
if not can_use_asymp:
if exp >= 0:
xabsint = man << exp
else:
xabsint = man >> (-exp)
can_use_asymp = xabsint > int(wp * 0.693) + 10
if can_use_asymp:
if xmag > wp:
v = fone
else:
v = from_man_exp(ei_asymptotic(to_fixed(x, wp), wp), -wp)
v = mpf_mul(v, mpf_exp(x, wp), wp)
v = mpf_div(v, x, prec, rnd)
else:
wp += 2 * int(to_int(xabs))
u = to_fixed(x, wp)
v = ei_taylor(u, wp) + euler_fixed(wp)
t1 = from_man_exp(v, -wp)
t2 = mpf_log(xabs, wp)
v = mpf_add(t1, t2, prec, rnd)
else:
if x == fzero: v = fninf
elif x == finf: v = finf
elif x == fninf: v = fzero
else: v = fnan
if e1:
v = mpf_neg(v)
return v
def mpf_ei(x, prec, rnd=round_fast, e1=False):
if e1:
x = mpf_neg(x)
sign, man, exp, bc = x
if e1 and not sign:
if x == fzero:
return finf
raise ComplexResult("E1(x) for x < 0")
if man:
xabs = 0, man, exp, bc
xmag = exp+bc
wp = prec + 20
can_use_asymp = xmag > wp
if not can_use_asymp:
if exp >= 0:
xabsint = man << exp
else:
xabsint = man >> (-exp)
can_use_asymp = xabsint > int(wp*0.693) + 10
if can_use_asymp:
if xmag > wp:
v = fone
else:
v = from_man_exp(ei_asymptotic(to_fixed(x, wp), wp), -wp)
v = mpf_mul(v, mpf_exp(x, wp), wp)
v = mpf_div(v, x, prec, rnd)
else:
wp += 2*int(to_int(xabs))
u = to_fixed(x, wp)
v = ei_taylor(u, wp) + euler_fixed(wp)
t1 = from_man_exp(v,-wp)
t2 = mpf_log(xabs,wp)
v = mpf_add(t1, t2, prec, rnd)
else:
if x == fzero: v = fninf
elif x == finf: v = finf
elif x == fninf: v = fzero
else: v = fnan
if e1:
v = mpf_neg(v)
return v
def mpi_abs(s, prec):
sa, sb = s
sas = mpf_sign(sa)
sbs = mpf_sign(sb)
# Both points nonnegative?
if sas >= 0:
a = mpf_pos(sa, prec, round_floor)
b = mpf_pos(sb, prec, round_ceiling)
# Upper point nonnegative?
elif sbs >= 0:
a = fzero
negsa = mpf_neg(sa)
if mpf_lt(negsa, sb):
b = mpf_pos(sb, prec, round_ceiling)
else:
b = mpf_pos(negsa, prec, round_ceiling)
# Both negative?
else:
a = mpf_neg(sb, prec, round_floor)
b = mpf_neg(sa, prec, round_ceiling)
return a, b
def mpi_abs(s, prec):
sa, sb = s
sas = mpf_sign(sa)
sbs = mpf_sign(sb)
# Both points nonnegative?
if sas >= 0:
a = mpf_pos(sa, prec, round_floor)
b = mpf_pos(sb, prec, round_ceiling)
# Upper point nonnegative?
elif sbs >= 0:
a = fzero
negsa = mpf_neg(sa)
if mpf_lt(negsa, sb):
b = mpf_pos(sb, prec, round_ceiling)
else:
b = mpf_pos(negsa, prec, round_ceiling)
# Both negative?
else:
a = mpf_neg(sb, prec, round_floor)
b = mpf_neg(sa, prec, round_ceiling)
return a, b
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_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 mpc_cos_sin(z, prec, rnd=round_fast):
a, b = z
if a == fzero:
ch, sh = mpf_cosh_sinh(b, prec, rnd)
return (ch, fzero), (sh, fzero)
wp = prec + 6
c, s = mpf_cos_sin(a, wp)
ch, sh = mpf_cosh_sinh(b, wp)
cre = mpf_mul(c, ch, prec, rnd)
cim = mpf_mul(s, sh, prec, rnd)
sre = mpf_mul(s, ch, prec, rnd)
sim = mpf_mul(c, sh, prec, rnd)
return (cre, mpf_neg(cim)), (sre, sim)
def mpc_pow_int(z, n, prec, rnd=round_fast):
a, b = z
if b == fzero:
return mpf_pow_int(a, n, prec, rnd), fzero
if a == fzero:
v = mpf_pow_int(b, n, prec, rnd)
n %= 4
if n == 0:
return v, fzero
elif n == 1:
return fzero, v
elif n == 2:
return mpf_neg(v), fzero
elif n == 3:
return fzero, mpf_neg(v)
if n == 0: return mpc_one
if n == 1: return mpc_pos(z, prec, rnd)
if n == 2: return mpc_square(z, prec, rnd)
if n == -1: return mpc_reciprocal(z, prec, rnd)
if n < 0: return mpc_reciprocal(mpc_pow_int(z, -n, prec+4), prec, rnd)
asign, aman, aexp, abc = a
bsign, bman, bexp, bbc = b
if asign: aman = -aman
if bsign: bman = -bman
de = aexp - bexp
abs_de = abs(de)
exact_size = n*(abs_de + max(abc, bbc))
if exact_size < 10000:
if de > 0:
aman <<= de
aexp = bexp
else:
bman <<= (-de)
bexp = aexp
re, im = complex_int_pow(aman, bman, n)
re = from_man_exp(re, int(n*aexp), prec, rnd)
im = from_man_exp(im, int(n*bexp), prec, rnd)
return re, im
return mpc_exp(mpc_mul_int(mpc_log(z, prec+10), n, prec+10), prec, rnd)
def mpc_cos_sin(z, prec, rnd=round_fast):
a, b = z
if a == fzero:
ch, sh = mpf_cosh_sinh(b, prec, rnd)
return (ch, fzero), (sh, fzero)
wp = prec + 6
c, s = mpf_cos_sin(a, wp)
ch, sh = mpf_cosh_sinh(b, wp)
cre = mpf_mul(c, ch, prec, rnd)
cim = mpf_mul(s, sh, prec, rnd)
sre = mpf_mul(s, ch, prec, rnd)
sim = mpf_mul(c, sh, prec, rnd)
return (cre, mpf_neg(cim)), (sre, sim)
def mpi_square(s, prec=0):
sa, sb = s
if mpf_ge(sa, fzero):
a = mpf_mul(sa, sa, prec, round_floor)
b = mpf_mul(sb, sb, prec, round_ceiling)
elif mpf_le(sb, fzero):
a = mpf_mul(sb, sb, prec, round_floor)
b = mpf_mul(sa, sa, prec, round_ceiling)
else:
sa = mpf_neg(sa)
sa, sb = mpf_min_max([sa, sb])
a = fzero
b = mpf_mul(sb, sb, prec, round_ceiling)
return a, b
def mpf_zeta(s, prec, rnd=round_fast):
sign, man, exp, bc = s
if not man:
if s == fzero:
return mpf_neg(fhalf)
if s == finf:
return fone
return fnan
wp = prec + 20
# First term vanishes?
if (not sign) and (exp + bc > (math.log(wp,2) + 2)):
if rnd in (round_up, round_ceiling):
return mpf_add(fone, mpf_shift(fone,-wp-10), prec, rnd)
return fone
elif exp >= 0:
return mpf_zeta_int(to_int(s), prec, rnd)
# Less than 0.5?
if sign or (exp+bc) < 0:
# XXX: -1 should be done exactly
y = mpf_sub(fone, s, 10*wp)
a = mpf_gamma(y, wp)
b = mpf_zeta(y, wp)
c = mpf_sin_pi(mpf_shift(s, -1), wp)
wp2 = wp + (exp+bc)
pi = mpf_pi(wp+wp2)
d = mpf_div(mpf_pow(mpf_shift(pi, 1), s, wp2), pi, wp2)
return mpf_mul(a,mpf_mul(b,mpf_mul(c,d,wp),wp),prec,rnd)
t = MP_ZERO
#wp += 16 - (prec & 15)
# Use Borwein's algorithm
n = int(wp/2.54 + 5)
d = borwein_coefficients(n)
t = MP_ZERO
sf = to_fixed(s, wp)
for k in xrange(n):
u = from_man_exp(-sf*log_int_fixed(k+1, wp), -2*wp, wp)
esign, eman, eexp, ebc = mpf_exp(u, wp)
offset = eexp + wp
if offset >= 0:
w = ((d[k] - d[n]) * eman) << offset
else:
w = ((d[k] - d[n]) * eman) >> (-offset)
if k & 1:
t -= w
else:
t += w
t = t // (-d[n])
t = from_man_exp(t, -wp, wp)
q = mpf_sub(fone, mpf_pow(ftwo, mpf_sub(fone, s, wp), wp), wp)
return mpf_div(t, q, prec, rnd)
def mpf_zeta(s, prec, rnd=round_fast):
sign, man, exp, bc = s
if not man:
if s == fzero:
return mpf_neg(fhalf)
if s == finf:
return fone
return fnan
wp = prec + 20
# First term vanishes?
if (not sign) and (exp + bc > (math.log(wp, 2) + 2)):
if rnd in (round_up, round_ceiling):
return mpf_add(fone, mpf_shift(fone, -wp - 10), prec, rnd)
return fone
elif exp >= 0:
return mpf_zeta_int(to_int(s), prec, rnd)
# Less than 0.5?
if sign or (exp + bc) < 0:
# XXX: -1 should be done exactly
y = mpf_sub(fone, s, 10 * wp)
a = mpf_gamma(y, wp)
b = mpf_zeta(y, wp)
c = mpf_sin_pi(mpf_shift(s, -1), wp)
wp2 = wp + (exp + bc)
pi = mpf_pi(wp + wp2)
d = mpf_div(mpf_pow(mpf_shift(pi, 1), s, wp2), pi, wp2)
return mpf_mul(a, mpf_mul(b, mpf_mul(c, d, wp), wp), prec, rnd)
t = MP_ZERO
#wp += 16 - (prec & 15)
# Use Borwein's algorithm
n = int(wp / 2.54 + 5)
d = borwein_coefficients(n)
t = MP_ZERO
sf = to_fixed(s, wp)
for k in xrange(n):
u = from_man_exp(-sf * log_int_fixed(k + 1, wp), -2 * wp, wp)
esign, eman, eexp, ebc = mpf_exp(u, wp)
offset = eexp + wp
if offset >= 0:
w = ((d[k] - d[n]) * eman) << offset
else:
w = ((d[k] - d[n]) * eman) >> (-offset)
if k & 1:
t -= w
else:
t += w
t = t // (-d[n])
t = from_man_exp(t, -wp, wp)
q = mpf_sub(fone, mpf_pow(ftwo, mpf_sub(fone, s, wp), wp), wp)
return mpf_div(t, q, prec, rnd)
def mpi_square(s, prec=0):
sa, sb = s
if mpf_ge(sa, fzero):
a = mpf_mul(sa, sa, prec, round_floor)
b = mpf_mul(sb, sb, prec, round_ceiling)
elif mpf_le(sb, fzero):
a = mpf_mul(sb, sb, prec, round_floor)
b = mpf_mul(sa, sa, prec, round_ceiling)
else:
sa = mpf_neg(sa)
sa, sb = mpf_min_max([sa, sb])
a = fzero
b = mpf_mul(sb, sb, prec, round_ceiling)
return a, b
def mpc_expjpi(z, prec, rnd='f'):
re, im = z
if im == fzero:
return mpf_cos_sin_pi(re, prec, rnd)
sign, man, exp, bc = im
wp = prec + 10
if man:
wp += max(0, exp + bc)
im = mpf_neg(mpf_mul(mpf_pi(wp), im, wp))
if re == fzero:
return mpf_exp(im, prec, rnd), fzero
ey = mpf_exp(im, prec + 10)
c, s = mpf_cos_sin_pi(re, prec + 10)
re = mpf_mul(ey, c, prec, rnd)
im = mpf_mul(ey, s, prec, rnd)
return re, im
def mpc_expjpi(z, prec, rnd="f"):
re, im = z
if im == fzero:
return mpf_cos_sin_pi(re, prec, rnd)
sign, man, exp, bc = im
wp = prec + 10
if man:
wp += max(0, exp + bc)
im = mpf_neg(mpf_mul(mpf_pi(wp), im, wp))
if re == fzero:
return mpf_exp(im, prec, rnd), fzero
ey = mpf_exp(im, prec + 10)
c, s = mpf_cos_sin_pi(re, prec + 10)
re = mpf_mul(ey, c, prec, rnd)
im = mpf_mul(ey, s, prec, rnd)
return re, im
def mpf_asinh(x, prec, rnd=round_fast):
wp = prec + 20
sign, man, exp, bc = x
mag = exp+bc
if mag < -8:
if mag < -wp:
return mpf_perturb(x, 1-sign, prec, rnd)
wp += (-mag)
# asinh(x) = log(x+sqrt(x**2+1))
# use reflection symmetry to avoid cancellation
q = mpf_sqrt(mpf_add(mpf_mul(x, x), fone, wp), wp)
q = mpf_add(mpf_abs(x), q, wp)
if sign:
return mpf_neg(mpf_log(q, prec, negative_rnd[rnd]))
else:
return mpf_log(q, prec, rnd)
def mpc_cos(z, prec, rnd=round_fast):
"""Complex cosine. The formula used is cos(a+bi) = cos(a)*cosh(b) -
sin(a)*sinh(b)*i.
The same comments apply as for the complex exp: only real
multiplications are pewrormed, so no cancellation errors are
possible. The formula is also efficient since we can compute both
pairs (cos, sin) and (cosh, sinh) in single stwps."""
a, b = z
if a == fzero:
return mpf_cosh(b, prec, rnd), fzero
wp = prec + 6
c, s = mpf_cos_sin(a, wp)
ch, sh = mpf_cosh_sinh(b, wp)
re = mpf_mul(c, ch, prec, rnd)
im = mpf_mul(s, sh, prec, rnd)
return re, mpf_neg(im)
def mpc_cos_sin_pi(z, prec, rnd=round_fast):
a, b = z
if b == fzero:
c, s = mpf_cos_sin_pi(a, prec, rnd)
return (c, fzero), (s, fzero)
b = mpf_mul(b, mpf_pi(prec + 5), prec + 5)
if a == fzero:
ch, sh = mpf_cosh_sinh(b, prec, rnd)
return (ch, fzero), (fzero, sh)
wp = prec + 6
c, s = mpf_cos_sin_pi(a, wp)
ch, sh = mpf_cosh_sinh(b, wp)
cre = mpf_mul(c, ch, prec, rnd)
cim = mpf_mul(s, sh, prec, rnd)
sre = mpf_mul(s, ch, prec, rnd)
sim = mpf_mul(c, sh, prec, rnd)
return (cre, mpf_neg(cim)), (sre, sim)
def mpc_cos(z, prec, rnd=round_fast):
"""Complex cosine. The formula used is cos(a+bi) = cos(a)*cosh(b) -
sin(a)*sinh(b)*i.
The same comments apply as for the complex exp: only real
multiplications are pewrormed, so no cancellation errors are
possible. The formula is also efficient since we can compute both
pairs (cos, sin) and (cosh, sinh) in single stwps."""
a, b = z
if a == fzero:
return mpf_cosh(b, prec, rnd), fzero
wp = prec + 6
c, s = mpf_cos_sin(a, wp)
ch, sh = mpf_cosh_sinh(b, wp)
re = mpf_mul(c, ch, prec, rnd)
im = mpf_mul(s, sh, prec, rnd)
return re, mpf_neg(im)
def mpc_cos_sin_pi(z, prec, rnd=round_fast):
a, b = z
if b == fzero:
c, s = mpf_cos_sin_pi(a, prec, rnd)
return (c, fzero), (s, fzero)
b = mpf_mul(b, mpf_pi(prec + 5), prec + 5)
if a == fzero:
ch, sh = mpf_cosh_sinh(b, prec, rnd)
return (ch, fzero), (fzero, sh)
wp = prec + 6
c, s = mpf_cos_sin_pi(a, wp)
ch, sh = mpf_cosh_sinh(b, wp)
cre = mpf_mul(c, ch, prec, rnd)
cim = mpf_mul(s, sh, prec, rnd)
sre = mpf_mul(s, ch, prec, rnd)
sim = mpf_mul(c, sh, prec, rnd)
return (cre, mpf_neg(cim)), (sre, sim)
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_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 mpi_atan2(y, x, prec):
ya, yb = y
xa, xb = x
# Constrained to the real line
if ya == yb == fzero:
if mpf_ge(xa, fzero):
return mpi_zero
return mpi_pi(prec)
# Right half-plane
if mpf_ge(xa, fzero):
if mpf_ge(ya, fzero):
a = mpf_atan2(ya, xb, prec, round_floor)
else:
a = mpf_atan2(ya, xa, prec, round_floor)
if mpf_ge(yb, fzero):
b = mpf_atan2(yb, xa, prec, round_ceiling)
else:
b = mpf_atan2(yb, xb, prec, round_ceiling)
# Upper half-plane
elif mpf_ge(ya, fzero):
b = mpf_atan2(ya, xa, prec, round_ceiling)
if mpf_le(xb, fzero):
a = mpf_atan2(yb, xb, prec, round_floor)
else:
a = mpf_atan2(ya, xb, prec, round_floor)
# Lower half-plane
elif mpf_le(yb, fzero):
a = mpf_atan2(yb, xa, prec, round_floor)
if mpf_le(xb, fzero):
b = mpf_atan2(ya, xb, prec, round_ceiling)
else:
b = mpf_atan2(yb, xb, prec, round_ceiling)
# Covering the origin
else:
b = mpf_pi(prec, round_ceiling)
a = mpf_neg(b)
return a, b
def mpi_atan2(y, x, prec):
ya, yb = y
xa, xb = x
# Constrained to the real line
if ya == yb == fzero:
if mpf_ge(xa, fzero):
return mpi_zero
return mpi_pi(prec)
# Right half-plane
if mpf_ge(xa, fzero):
if mpf_ge(ya, fzero):
a = mpf_atan2(ya, xb, prec, round_floor)
else:
a = mpf_atan2(ya, xa, prec, round_floor)
if mpf_ge(yb, fzero):
b = mpf_atan2(yb, xa, prec, round_ceiling)
else:
b = mpf_atan2(yb, xb, prec, round_ceiling)
# Upper half-plane
elif mpf_ge(ya, fzero):
b = mpf_atan2(ya, xa, prec, round_ceiling)
if mpf_le(xb, fzero):
a = mpf_atan2(yb, xb, prec, round_floor)
else:
a = mpf_atan2(ya, xb, prec, round_floor)
# Lower half-plane
elif mpf_le(yb, fzero):
a = mpf_atan2(yb, xa, prec, round_floor)
if mpf_le(xb, fzero):
b = mpf_atan2(ya, xb, prec, round_ceiling)
else:
b = mpf_atan2(yb, xb, prec, round_ceiling)
# Covering the origin
else:
b = mpf_pi(prec, round_ceiling)
a = mpf_neg(b)
return a, b
