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_ellipe(x, prec, rnd=round_fast):
# http://functions.wolfram.com/EllipticIntegrals/
# EllipticK/20/01/0001/
# E = (1-m)*(K'(m)*2*m + K(m))
sign, man, exp, bc = x
if not man:
if x == fzero:
return mpf_shift(mpf_pi(prec, rnd), -1)
if x == fninf:
return finf
if x == fnan:
return x
if x == finf:
raise ComplexResult
if x == fone:
return fone
wp = prec+20
mag = exp+bc
if mag < -wp:
return mpf_shift(mpf_pi(prec, rnd), -1)
# Compute a finite difference for K'
p = max(mag, 0) - wp
h = mpf_shift(fone, p)
K = mpf_ellipk(x, 2*wp)
Kh = mpf_ellipk(mpf_sub(x, h), 2*wp)
Kdiff = mpf_shift(mpf_sub(K, Kh), -p)
t = mpf_sub(fone, x)
b = mpf_mul(Kdiff, mpf_shift(x,1), wp)
return mpf_mul(t, mpf_add(K, b), 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_agm(a, b, prec, rnd=round_fast):
"""
Computes the arithmetic-geometric mean agm(a,b) for
nonnegative mpf values a, b.
"""
asign, aman, aexp, abc = a
bsign, bman, bexp, bbc = b
if asign or bsign:
raise ComplexResult("agm of a negative number")
# Handle inf, nan or zero in either operand
if not (aman and bman):
if a == fnan or b == fnan:
return fnan
if a == finf:
if b == fzero:
return fnan
return finf
if b == finf:
if a == fzero:
return fnan
return finf
# agm(0,x) = agm(x,0) = 0
return fzero
wp = prec + 20
amag = aexp+abc
bmag = bexp+bbc
mag_delta = amag - bmag
# Reduce to roughly the same magnitude using floating-point AGM
abs_mag_delta = abs(mag_delta)
if abs_mag_delta > 10:
while abs_mag_delta > 10:
a, b = mpf_shift(mpf_add(a,b,wp),-1), \
mpf_sqrt(mpf_mul(a,b,wp),wp)
abs_mag_delta //= 2
asign, aman, aexp, abc = a
bsign, bman, bexp, bbc = b
amag = aexp+abc
bmag = bexp+bbc
mag_delta = amag - bmag
#print to_float(a), to_float(b)
# Use agm(a,b) = agm(x*a,x*b)/x to obtain a, b ~= 1
min_mag = min(amag,bmag)
max_mag = max(amag,bmag)
n = 0
# If too small, we lose precision when going to fixed-point
if min_mag < -8:
n = -min_mag
# If too large, we waste time using fixed-point with large numbers
elif max_mag > 20:
n = -max_mag
if n:
a = mpf_shift(a, n)
b = mpf_shift(b, n)
#print to_float(a), to_float(b)
af = to_fixed(a, wp)
bf = to_fixed(b, wp)
g = agm_fixed(af, bf, wp)
return from_man_exp(g, -wp-n, 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 mpc_agm(a, b, prec, rnd=round_fast):
"""
Complex AGM.
TODO:
* check that convergence works as intended
* optimize
* select a nonarbitrary branch
"""
if mpc_is_infnan(a) or mpc_is_infnan(b):
return fnan, fnan
if mpc_zero in (a, b):
return fzero, fzero
if mpc_neg(a) == b:
return fzero, fzero
wp = prec+20
eps = mpf_shift(fone, -wp+10)
while 1:
a1 = mpc_shift(mpc_add(a, b, wp), -1)
b1 = mpc_sqrt(mpc_mul(a, b, wp), wp)
a, b = a1, b1
size = mpf_min_max([mpc_abs(a,10), mpc_abs(b,10)])[1]
err = mpc_abs(mpc_sub(a, b, 10), 10)
if size == fzero or mpf_lt(err, mpf_mul(eps, size)):
return a
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 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_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 mpf_asin(x, prec, rnd=round_fast):
sign, man, exp, bc = x
if bc+exp > 0 and x not in (fone, fnone):
raise ComplexResult("asin(x) is real only for -1 <= x <= 1")
flag_nr = True
if prec < 1000 or exp+bc < -13:
flag_nr = False
else:
ebc = exp + bc
if ebc < -13:
flag_nr = False
elif ebc < -3:
if prec < 3000:
flag_nr = False
if not flag_nr:
# asin(x) = 2*atan(x/(1+sqrt(1-x**2)))
wp = prec + 15
a = mpf_mul(x, x)
b = mpf_add(fone, mpf_sqrt(mpf_sub(fone, a, wp), wp), wp)
c = mpf_div(x, b, wp)
return mpf_shift(mpf_atan(c, prec, rnd), 1)
# use Newton's method
extra = 10
extra_p = 10
prec2 = prec + extra
r = math.asin(to_float(x))
r = from_float(r, 50, rnd)
for p in giant_steps(50, prec2):
wp = p + extra_p
c, s = cos_sin(r, wp, rnd)
tmp = mpf_sub(x, s, wp, rnd)
tmp = mpf_div(tmp, c, wp, rnd)
r = mpf_add(r, tmp, wp, rnd)
sign, man, exp, bc = r
return normalize(sign, man, exp, bc, prec, rnd)
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 cosh_sinh(x, prec, rnd=round_fast, tanh=0):
"""Simultaneously compute (cosh(x), sinh(x)) for real x"""
sign, man, exp, bc = x
if (not man) and exp:
if tanh:
if x == finf: return fone
if x == fninf: return fnone
return fnan
if x == finf: return (finf, finf)
if x == fninf: return (finf, fninf)
return fnan, fnan
if sign:
man = -man
mag = exp + bc
prec2 = prec + 20
if mag < -3:
# Extremely close to 0, sinh(x) ~= x and cosh(x) ~= 1
if mag < -prec-2:
if tanh:
return mpf_perturb(x, 1-sign, prec, rnd)
cosh = mpf_perturb(fone, 0, prec, rnd)
sinh = mpf_perturb(x, sign, prec, rnd)
return cosh, sinh
# Avoid cancellation when computing sinh
# TODO: might be faster to use sinh series directly
prec2 += (-mag) + 4
# In the general case, we use
# cosh(x) = (exp(x) + exp(-x))/2
# sinh(x) = (exp(x) - exp(-x))/2
# and note that the exponential only needs to be computed once.
ep = mpf_exp(x, prec2)
em = mpf_div(fone, ep, prec2)
if tanh:
ch = mpf_add(ep, em, prec2, rnd)
sh = mpf_sub(ep, em, prec2, rnd)
return mpf_div(sh, ch, prec, rnd)
else:
ch = mpf_shift(mpf_add(ep, em, prec, rnd), -1)
sh = mpf_shift(mpf_sub(ep, em, prec, rnd), -1)
return ch, sh
def mpc_square(z, prec, rnd=round_fast):
# (a+b*I)**2 == a**2 - b**2 + 2*I*a*b
a, b = z
p = mpf_mul(a, a)
q = mpf_mul(b, b)
r = mpf_mul(a, b, prec, rnd)
re = mpf_sub(p, q, prec, rnd)
im = mpf_shift(r, 1)
return re, im
def mpc_square(z, prec, rnd=round_fast):
# (a+b*I)**2 == a**2 - b**2 + 2*I*a*b
a, b = z
p = mpf_mul(a,a)
q = mpf_mul(b,b)
r = mpf_mul(a,b, prec, rnd)
re = mpf_sub(p, q, prec, rnd)
im = mpf_shift(r, 1)
return re, im
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 mpc_tan(z, prec, rnd=round_fast):
"""Complex tangent. Computed as tan(a+bi) = sin(2a)/M + sinh(2b)/M*i
where M = cos(2a) + cosh(2b)."""
a, b = z
asign, aman, aexp, abc = a
bsign, bman, bexp, bbc = b
if b == fzero: return mpf_tan(a, prec, rnd), fzero
if a == fzero: return fzero, mpf_tanh(b, prec, rnd)
wp = prec + 15
a = mpf_shift(a, 1)
b = mpf_shift(b, 1)
c, s = cos_sin(a, wp)
ch, sh = cosh_sinh(b, wp)
# TODO: handle cancellation when c ~= -1 and ch ~= 1
mag = mpf_add(c, ch, wp)
re = mpf_div(s, mag, prec, rnd)
im = mpf_div(sh, mag, prec, rnd)
return re, im
def mpc_tan(z, prec, rnd=round_fast):
"""Complex tangent. Computed as tan(a+bi) = sin(2a)/M + sinh(2b)/M*i
where M = cos(2a) + cosh(2b)."""
a, b = z
asign, aman, aexp, abc = a
bsign, bman, bexp, bbc = b
if b == fzero: return mpf_tan(a, prec, rnd), fzero
if a == fzero: return fzero, mpf_tanh(b, prec, rnd)
wp = prec + 15
a = mpf_shift(a, 1)
b = mpf_shift(b, 1)
c, s = cos_sin(a, wp)
ch, sh = cosh_sinh(b, wp)
# TODO: handle cancellation when c ~= -1 and ch ~= 1
mag = mpf_add(c, ch, wp)
re = mpf_div(s, mag, prec, rnd)
im = mpf_div(sh, mag, prec, rnd)
return re, im
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_asin(x, prec, rnd=round_fast):
sign, man, exp, bc = x
if bc + exp > 0 and x not in (fone, fnone):
raise ComplexResult("asin(x) is real only for -1 <= x <= 1")
# asin(x) = 2*atan(x/(1+sqrt(1-x**2)))
wp = prec + 15
a = mpf_mul(x, x)
b = mpf_add(fone, mpf_sqrt(mpf_sub(fone, a, wp), wp), wp)
c = mpf_div(x, b, wp)
return mpf_shift(mpf_atan(c, prec, rnd), 1)
def mpf_asin(x, prec, rnd=round_fast):
sign, man, exp, bc = x
if bc + exp > 0 and x not in (fone, fnone):
raise ComplexResult("asin(x) is real only for -1 <= x <= 1")
# asin(x) = 2*atan(x/(1+sqrt(1-x**2)))
wp = prec + 15
a = mpf_mul(x, x)
b = mpf_add(fone, mpf_sqrt(mpf_sub(fone, a, wp), wp), wp)
c = mpf_div(x, b, wp)
return mpf_shift(mpf_atan(c, prec, rnd), 1)
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_ellipk(x, prec, rnd=round_fast):
if not x[1]:
if x == fzero:
return mpf_shift(mpf_pi(prec, rnd), -1)
if x == fninf:
return fzero
if x == fnan:
return x
if x == fone:
return finf
# TODO: for |x| << 1/2, one could use fall back to
# pi/2 * hyp2f1_rat((1,2),(1,2),(1,1), x)
wp = prec + 15
# Use K(x) = pi/2/agm(1,a) where a = sqrt(1-x)
# The sqrt raises ComplexResult if x > 0
a = mpf_sqrt(mpf_sub(fone, x, wp), wp)
v = mpf_agm1(a, wp)
r = mpf_div(mpf_pi(wp), v, prec, rnd)
return mpf_shift(r, -1)
def mpf_ellipk(x, prec, rnd=round_fast):
if not x[1]:
if x == fzero:
return mpf_shift(mpf_pi(prec, rnd), -1)
if x == fninf:
return fzero
if x == fnan:
return x
if x == fone:
return finf
# TODO: for |x| << 1/2, one could use fall back to
# pi/2 * hyp2f1_rat((1,2),(1,2),(1,1), x)
wp = prec + 15
# Use K(x) = pi/2/agm(1,a) where a = sqrt(1-x)
# The sqrt raises ComplexResult if x > 0
a = mpf_sqrt(mpf_sub(fone, x, wp), wp)
v = mpf_agm1(a, wp)
r = mpf_div(mpf_pi(wp), v, prec, rnd)
return mpf_shift(r, -1)
def mpf_atanh(x, prec, rnd=round_fast):
# atanh(x) = log((1+x)/(1-x))/2
sign, man, exp, bc = x
mag = bc + exp
if mag > 0:
raise ComplexResult("atanh(x) is real only for -1 < x < 1")
wp = prec + 15
a = mpf_add(x, fone, wp)
b = mpf_sub(fone, x, wp)
return mpf_shift(mpf_log(mpf_div(a, b, wp), prec, rnd), -1)
def mpf_atanh(x, prec, rnd=round_fast):
# atanh(x) = log((1+x)/(1-x))/2
sign, man, exp, bc = x
mag = bc + exp
if mag > 0:
raise ComplexResult("atanh(x) is real only for -1 < x < 1")
wp = prec + 15
a = mpf_add(x, fone, wp)
b = mpf_sub(fone, x, wp)
return mpf_shift(mpf_log(mpf_div(a, b, wp), prec, rnd), -1)
def mpf_acos(x, prec, rnd=round_fast):
# acos(x) = 2*atan(sqrt(1-x**2)/(1+x))
sign, man, exp, bc = x
if bc + exp > 0:
if x not in (fone, fnone):
raise ComplexResult("acos(x) is real only for -1 <= x <= 1")
if x == fnone:
return mpf_pi(prec, rnd)
wp = prec + 15
a = mpf_mul(x, x)
b = mpf_sqrt(mpf_sub(fone, a, wp), wp)
c = mpf_div(b, mpf_add(fone, x, wp), wp)
return mpf_shift(mpf_atan(c, prec, rnd), 1)
def mpf_acos(x, prec, rnd=round_fast):
# acos(x) = 2*atan(sqrt(1-x**2)/(1+x))
sign, man, exp, bc = x
if bc + exp > 0:
if x not in (fone, fnone):
raise ComplexResult("acos(x) is real only for -1 <= x <= 1")
if x == fnone:
return mpf_pi(prec, rnd)
wp = prec + 15
a = mpf_mul(x, x)
b = mpf_sqrt(mpf_sub(fone, a, wp), wp)
c = mpf_div(b, mpf_add(fone, x, wp), wp)
return mpf_shift(mpf_atan(c, prec, rnd), 1)
def mpc_fibonacci(z, prec, rnd=round_fast):
re, im = z
if im == fzero:
return (mpf_fibonacci(re, prec, rnd), fzero)
size = max(abs(re[2] + re[3]), abs(re[2] + re[3]))
wp = prec + size + 20
a = mpf_phi(wp)
b = mpf_add(mpf_shift(a, 1), fnone, wp)
u = mpc_pow((a, fzero), z, wp)
v = mpc_cos_pi(z, wp)
v = mpc_div(v, u, wp)
u = mpc_sub(u, v, wp)
u = mpc_div_mpf(u, b, prec, rnd)
return u
def mpc_psi0(z, prec, rnd=round_fast):
"""
Computation of the digamma function (psi function of order 0)
of a complex argument.
"""
re, im = z
# Fall back to the real case
if im == fzero:
return (mpf_psi0(re, prec, rnd), fzero)
wp = prec + 20
sign, man, exp, bc = re
# Reflection formula
if sign and exp + bc > 3:
c = mpc_cos_pi(z, wp)
s = mpc_sin_pi(z, wp)
q = mpc_mul(mpc_div(c, s, wp), (mpf_pi(wp), fzero), wp)
p = mpc_psi0(mpc_sub(mpc_one, z, wp), wp)
return mpc_sub(p, q, prec, rnd)
# Just the logarithmic term
if (not sign) and bc + exp > wp:
return mpc_log(mpc_sub(z, mpc_one, wp), prec, rnd)
# Initial recurrence to obtain a large enough z
w = to_int(re)
n = int(0.11 * wp) + 2
s = mpc_zero
if w < n:
for k in xrange(w, n):
s = mpc_sub(s, mpc_div(mpc_one, z, wp), wp)
z = mpc_add_mpf(z, fone, wp)
z = mpc_sub(z, mpc_one, wp)
# Logarithmic and endpoint term
s = mpc_add(s, mpc_log(z, wp), wp)
s = mpc_add(s, mpc_div(mpc_half, z, wp), wp)
# Euler-Maclaurin remainder sum
z2 = mpc_mul(z, z, wp)
t = mpc_one
prev = mpc_zero
k = 1
eps = mpf_shift(fone, -wp + 2)
while 1:
t = mpc_mul(t, z2, wp)
bern = mpf_bernoulli(2 * k, wp)
term = mpc_div((bern, fzero), mpc_mul_int(t, 2 * k, wp), wp)
s = mpc_sub(s, term, wp)
szterm = mpc_abs(term, 10)
if k > 2 and mpf_le(szterm, eps):
break
prev = term
k += 1
return s
def mpc_fibonacci(z, prec, rnd=round_fast):
re, im = z
if im == fzero:
return (mpf_fibonacci(re, prec, rnd), fzero)
size = max(abs(re[2]+re[3]), abs(re[2]+re[3]))
wp = prec + size + 20
a = mpf_phi(wp)
b = mpf_add(mpf_shift(a, 1), fnone, wp)
u = mpc_pow((a, fzero), z, wp)
v = mpc_cos_pi(z, wp)
v = mpc_div(v, u, wp)
u = mpc_sub(u, v, wp)
u = mpc_div_mpf(u, b, prec, rnd)
return u
def mpc_psi0(z, prec, rnd=round_fast):
"""
Computation of the digamma function (psi function of order 0)
of a complex argument.
"""
re, im = z
# Fall back to the real case
if im == fzero:
return (mpf_psi0(re, prec, rnd), fzero)
wp = prec + 20
sign, man, exp, bc = re
# Reflection formula
if sign and exp+bc > 3:
c = mpc_cos_pi(z, wp)
s = mpc_sin_pi(z, wp)
q = mpc_mul_mpf(mpc_div(c, s, wp), mpf_pi(wp), wp)
p = mpc_psi0(mpc_sub(mpc_one, z, wp), wp)
return mpc_sub(p, q, prec, rnd)
# Just the logarithmic term
if (not sign) and bc + exp > wp:
return mpc_log(mpc_sub(z, mpc_one, wp), prec, rnd)
# Initial recurrence to obtain a large enough z
w = to_int(re)
n = int(0.11*wp) + 2
s = mpc_zero
if w < n:
for k in xrange(w, n):
s = mpc_sub(s, mpc_reciprocal(z, wp), wp)
z = mpc_add_mpf(z, fone, wp)
z = mpc_sub(z, mpc_one, wp)
# Logarithmic and endpoint term
s = mpc_add(s, mpc_log(z, wp), wp)
s = mpc_add(s, mpc_div(mpc_half, z, wp), wp)
# Euler-Maclaurin remainder sum
z2 = mpc_square(z, wp)
t = mpc_one
prev = mpc_zero
k = 1
eps = mpf_shift(fone, -wp+2)
while 1:
t = mpc_mul(t, z2, wp)
bern = mpf_bernoulli(2*k, wp)
term = mpc_mpf_div(bern, mpc_mul_int(t, 2*k, wp), wp)
s = mpc_sub(s, term, wp)
szterm = mpc_abs(term, 10)
if k > 2 and mpf_le(szterm, eps):
break
prev = term
k += 1
return s
def ldexp(ctx, x, n):
r"""
Computes `x 2^n` efficiently. No rounding is performed.
The argument `x` must be a real floating-point number (or
possible to convert into one) and `n` must be a Python ``int``.
>>> from mpmath import *
>>> ldexp(1, 10)
mpf('1024.0')
>>> ldexp(1, -3)
mpf('0.125')
"""
x = ctx.convert(x)
return ctx.make_mpf(libmpf.mpf_shift(x._mpf_, n))
def ldexp(ctx, x, n):
r"""
Computes `x 2^n` efficiently. No rounding is performed.
The argument `x` must be a real floating-point number (or
possible to convert into one) and `n` must be a Python ``int``.
>>> from mpmath import *
>>> ldexp(1, 10)
mpf('1024.0')
>>> ldexp(1, -3)
mpf('0.125')
"""
x = ctx.convert(x)
return ctx.make_mpf(libmpf.mpf_shift(x._mpf_, n))
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 mpc_ellipe(z, prec, rnd=round_fast):
re, im = z
if im == fzero:
if re == finf:
return (fzero, finf)
if mpf_le(re, fone):
return mpf_ellipe(re, prec, rnd), fzero
wp = prec + 15
mag = mpc_abs(z, 1)
p = max(mag[2]+mag[3], 0) - wp
h = mpf_shift(fone, p)
K = mpc_ellipk(z, 2*wp)
Kh = mpc_ellipk(mpc_add_mpf(z, h, 2*wp), 2*wp)
Kdiff = mpc_shift(mpc_sub(Kh, K, wp), -p)
t = mpc_sub(mpc_one, z, wp)
b = mpc_mul(Kdiff, mpc_shift(z,1), wp)
return mpc_mul(t, mpc_add(K, b, wp), prec, rnd)
def mpc_ellipe(z, prec, rnd=round_fast):
re, im = z
if im == fzero:
if re == finf:
return (fzero, finf)
if mpf_le(re, fone):
return mpf_ellipe(re, prec, rnd), fzero
wp = prec + 15
mag = mpc_abs(z, 1)
p = max(mag[2] + mag[3], 0) - wp
h = mpf_shift(fone, p)
K = mpc_ellipk(z, 2 * wp)
Kh = mpc_ellipk(mpc_add_mpf(z, h, 2 * wp), 2 * wp)
Kdiff = mpc_shift(mpc_sub(Kh, K, wp), -p)
t = mpc_sub(mpc_one, z, wp)
b = mpc_mul(Kdiff, mpc_shift(z, 1), wp)
return mpc_mul(t, mpc_add(K, b, wp), prec, rnd)
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_atanh(x, prec, rnd=round_fast):
# atanh(x) = log((1+x)/(1-x))/2
sign, man, exp, bc = x
if (not man) and exp:
if x in (fzero, fnan):
return x
raise ComplexResult("atanh(x) is real only for -1 <= x <= 1")
mag = bc + exp
if mag > 0:
if mag == 1 and man == 1:
return [finf, fninf][sign]
raise ComplexResult("atanh(x) is real only for -1 <= x <= 1")
wp = prec + 15
if mag < -8:
if mag < -wp:
return mpf_perturb(x, sign, prec, rnd)
wp += (-mag)
a = mpf_add(x, fone, wp)
b = mpf_sub(fone, x, wp)
return mpf_shift(mpf_log(mpf_div(a, b, wp), prec, rnd), -1)
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_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_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 mpf_log_hypot(a, b, prec, rnd):
"""
Computes log(sqrt(a^2+b^2)) accurately.
"""
# If either a or b is inf/nan/0, assume it to be a
if not b[1]:
a, b = b, a
# a is inf/nan/0
if not a[1]:
# both are inf/nan/0
if not b[1]:
if a == b == fzero:
return fninf
if fnan in (a, b):
return fnan
# at least one term is (+/- inf)^2
return finf
# only a is inf/nan/0
if a == fzero:
# log(sqrt(0+b^2)) = log(|b|)
return mpf_log(mpf_abs(b), prec, rnd)
if a == fnan:
return fnan
return finf
# Exact
a2 = mpf_mul(a,a)
b2 = mpf_mul(b,b)
extra = 20
# Not exact
h2 = mpf_add(a2, b2, prec+extra)
cancelled = mpf_add(h2, fnone, 10)
mag_cancelled = cancelled[2]+cancelled[3]
# Just redo the sum exactly if necessary (could be smarter
# and avoid memory allocation when a or b is precisely 1
# and the other is tiny...)
if cancelled == fzero or mag_cancelled < -extra//2:
h2 = mpf_add(a2, b2, prec+extra-min(a2[2],b2[2]))
return mpf_shift(mpf_log(h2, prec, rnd), -1)
def ln_sqrt2pi_fixed(prec):
wp = prec + 10
# ln(sqrt(2*pi)) = ln(2*pi)/2
return to_fixed(mpf_log(mpf_shift(mpf_pi(wp), 1), wp), prec - 1)
def mpf_log(x, prec, rnd=round_fast):
"""
Compute the natural logarithm of the mpf value x. If x is negative,
ComplexResult is raised.
"""
sign, man, exp, bc = x
#------------------------------------------------------------------
# Handle special values
if not man:
if x == fzero: return fninf
if x == finf: return finf
if x == fnan: return fnan
if sign:
raise ComplexResult("logarithm of a negative number")
wp = prec + 20
#------------------------------------------------------------------
# Handle log(2^n) = log(n)*2.
# Here we catch the only possible exact value, log(1) = 0
if man == 1:
if not exp:
return fzero
return from_man_exp(exp*ln2_fixed(wp), -wp, prec, rnd)
mag = exp+bc
abs_mag = abs(mag)
#------------------------------------------------------------------
# Handle x = 1+eps, where log(x) ~ x. We need to check for
# cancellation when moving to fixed-point math and compensate
# by increasing the precision. Note that abs_mag in (0, 1) <=>
# 0.5 < x < 2 and x != 1
if abs_mag <= 1:
# Calculate t = x-1 to measure distance from 1 in bits
tsign = 1-abs_mag
if tsign:
tman = (MP_ONE<<bc) - man
else:
tman = man - (MP_ONE<<(bc-1))
tbc = bitcount(tman)
cancellation = bc - tbc
if cancellation > wp:
t = normalize(tsign, tman, abs_mag-bc, tbc, tbc, 'n')
return mpf_perturb(t, tsign, prec, rnd)
else:
wp += cancellation
# TODO: if close enough to 1, we could use Taylor series
# even in the AGM precision range, since the Taylor series
# converges rapidly
#------------------------------------------------------------------
# Another special case:
# n*log(2) is a good enough approximation
if abs_mag > 10000:
if bitcount(abs_mag) > wp:
return from_man_exp(exp*ln2_fixed(wp), -wp, prec, rnd)
#------------------------------------------------------------------
# General case.
# Perform argument reduction using log(x) = log(x*2^n) - n*log(2):
# If we are in the Taylor precision range, choose magnitude 0 or 1.
# If we are in the AGM precision range, choose magnitude -m for
# some large m; benchmarking on one machine showed m = prec/20 to be
# optimal between 1000 and 100,000 digits.
if wp <= LOG_TAYLOR_PREC:
m = log_taylor_cached(lshift(man, wp-bc), wp)
if mag:
m += mag*ln2_fixed(wp)
else:
optimal_mag = -wp//LOG_AGM_MAG_PREC_RATIO
n = optimal_mag - mag
x = mpf_shift(x, n)
wp += (-optimal_mag)
m = -log_agm(to_fixed(x, wp), wp)
m -= n*ln2_fixed(wp)
return from_man_exp(m, -wp, prec, rnd)
def mpf_ci_si(x, prec, rnd=round_fast, which=2):
"""
Calculation of Ci(x), Si(x) for real x.
which = 0 -- returns (Ci(x), -)
which = 1 -- returns (Si(x), -)
which = 2 -- returns (Ci(x), Si(x))
Note: if x < 0, Ci(x) needs an additional imaginary term, pi*i.
"""
wp = prec + 20
sign, man, exp, bc = x
ci, si = None, None
if not man:
if x == fzero:
return (fninf, fzero)
if x == fnan:
return (x, x)
ci = fzero
if which != 0:
if x == finf:
si = mpf_shift(mpf_pi(prec, rnd), -1)
if x == fninf:
si = mpf_neg(mpf_shift(mpf_pi(prec, negative_rnd[rnd]), -1))
return (ci, si)
# For small x: Ci(x) ~ euler + log(x), Si(x) ~ x
mag = exp + bc
if mag < -wp:
if which != 0:
si = mpf_perturb(x, 1 - sign, prec, rnd)
if which != 1:
y = mpf_euler(wp)
xabs = mpf_abs(x)
ci = mpf_add(y, mpf_log(xabs, wp), prec, rnd)
return ci, si
# For huge x: Ci(x) ~ sin(x)/x, Si(x) ~ pi/2
elif mag > wp:
if which != 0:
if sign:
si = mpf_neg(mpf_pi(prec, negative_rnd[rnd]))
else:
si = mpf_pi(prec, rnd)
si = mpf_shift(si, -1)
if which != 1:
ci = mpf_div(mpf_sin(x, wp), x, prec, rnd)
return ci, si
else:
wp += abs(mag)
# Use an asymptotic series? The smallest value of n!/x^n
# occurs for n ~ x, where the magnitude is ~ exp(-x).
asymptotic = mag - 1 > math.log(wp, 2)
# Case 1: convergent series near 0
if not asymptotic:
if which != 0:
si = mpf_pos(mpf_ci_si_taylor(x, wp, 1), prec, rnd)
if which != 1:
ci = mpf_ci_si_taylor(x, wp, 0)
ci = mpf_add(ci, mpf_euler(wp), wp)
ci = mpf_add(ci, mpf_log(mpf_abs(x), wp), prec, rnd)
return ci, si
x = mpf_abs(x)
# Case 2: asymptotic series for x >> 1
xf = to_fixed(x, wp)
xr = (MP_ONE << (2 * wp)) // xf # 1/x
s1 = (MP_ONE << wp)
s2 = xr
t = xr
k = 2
while t:
t = -t
t = (t * xr * k) >> wp
k += 1
s1 += t
t = (t * xr * k) >> wp
k += 1
s2 += t
s1 = from_man_exp(s1, -wp)
s2 = from_man_exp(s2, -wp)
s1 = mpf_div(s1, x, wp)
s2 = mpf_div(s2, x, wp)
cos, sin = cos_sin(x, wp)
# Ci(x) = sin(x)*s1-cos(x)*s2
# Si(x) = pi/2-cos(x)*s1-sin(x)*s2
if which != 0:
si = mpf_add(mpf_mul(cos, s1), mpf_mul(sin, s2), wp)
si = mpf_sub(mpf_shift(mpf_pi(wp), -1), si, wp)
if sign:
si = mpf_neg(si)
si = mpf_pos(si, prec, rnd)
if which != 1:
ci = mpf_sub(mpf_mul(sin, s1), mpf_mul(cos, s2), prec, rnd)
return ci, si
def mpc_sub_mpf((a, b), p, prec, rnd=round_fast):
return mpf_sub(a, p, prec, rnd), b
def mpc_pos((a, b), prec, rnd=round_fast):
return mpf_pos(a, prec, rnd), mpf_pos(b, prec, rnd)
def mpc_neg((a, b), prec=None, rnd=round_fast):
return mpf_neg(a, prec, rnd), mpf_neg(b, prec, rnd)
def mpc_shift((a, b), n):
return mpf_shift(a, n), mpf_shift(b, n)
def mpc_abs((a, b), prec, rnd=round_fast):
"""Absolute value of a complex number, |a+bi|.
Returns an mpf value."""
return mpf_hypot(a, b, prec, rnd)
def mpc_arg((a, b), prec, rnd=round_fast):
"""Argument of a complex number. Returns an mpf value."""
return mpf_atan2(b, a, prec, rnd)
def mpc_floor((a, b), prec, rnd=round_fast):
return mpf_floor(a, prec, rnd), mpf_floor(b, prec, rnd)
def acos_asin(z, prec, rnd, n):
""" complex acos for n = 0, asin for n = 1
The algorithm is described in
T.E. Hull, T.F. Fairgrieve and P.T.P. Tang
'Implementing the Complex Arcsine and Arcosine Functions
using Exception Handling',
ACM Trans. on Math. Software Vol. 23 (1997), p299
The complex acos and asin can be defined as
acos(z) = acos(beta) - I*sign(a)* log(alpha + sqrt(alpha**2 -1))
asin(z) = asin(beta) + I*sign(a)* log(alpha + sqrt(alpha**2 -1))
where z = a + I*b
alpha = (1/2)*(r + s); beta = (1/2)*(r - s) = a/alpha
r = sqrt((a+1)**2 + y**2); s = sqrt((a-1)**2 + y**2)
These expressions are rewritten in different ways in different
regions, delimited by two crossovers alpha_crossover and beta_crossover,
and by abs(a) <= 1, in order to improve the numerical accuracy.
"""
a, b = z
wp = prec + 10
# special cases with real argument
if b == fzero:
am = mpf_sub(fone, mpf_abs(a), wp)
# case abs(a) <= 1
if not am[0]:
if n == 0:
return mpf_acos(a, prec, rnd), fzero
else:
return mpf_asin(a, prec, rnd), fzero
# cases abs(a) > 1
else:
# case a < -1
if a[0]:
pi = mpf_pi(prec, rnd)
c = mpf_acosh(mpf_neg(a), prec, rnd)
if n == 0:
return pi, mpf_neg(c)
else:
return mpf_neg(mpf_shift(pi, -1)), c
# case a > 1
else:
c = mpf_acosh(a, prec, rnd)
if n == 0:
return fzero, c
else:
pi = mpf_pi(prec, rnd)
return mpf_shift(pi, -1), mpf_neg(c)
asign = bsign = 0
if a[0]:
a = mpf_neg(a)
asign = 1
if b[0]:
b = mpf_neg(b)
bsign = 1
am = mpf_sub(fone, a, wp)
ap = mpf_add(fone, a, wp)
r = mpf_hypot(ap, b, wp)
s = mpf_hypot(am, b, wp)
alpha = mpf_shift(mpf_add(r, s, wp), -1)
beta = mpf_div(a, alpha, wp)
b2 = mpf_mul(b, b, wp)
# case beta <= beta_crossover
if not mpf_sub(beta_crossover, beta, wp)[0]:
if n == 0:
re = mpf_acos(beta, wp)
else:
re = mpf_asin(beta, wp)
else:
# to compute the real part in this region use the identity
# asin(beta) = atan(beta/sqrt(1-beta**2))
# beta/sqrt(1-beta**2) = (alpha + a) * (alpha - a)
# alpha + a is numerically accurate; alpha - a can have
# cancellations leading to numerical inaccuracies, so rewrite
# it in differente ways according to the region
Ax = mpf_add(alpha, a, wp)
# case a <= 1
if not am[0]:
# c = b*b/(r + (a+1)); d = (s + (1-a))
# alpha - a = (1/2)*(c + d)
# case n=0: re = atan(sqrt((1/2) * Ax * (c + d))/a)
# case n=1: re = atan(a/sqrt((1/2) * Ax * (c + d)))
c = mpf_div(b2, mpf_add(r, ap, wp), wp)
d = mpf_add(s, am, wp)
re = mpf_shift(mpf_mul(Ax, mpf_add(c, d, wp), wp), -1)
if n == 0:
re = mpf_atan(mpf_div(mpf_sqrt(re, wp), a, wp), wp)
else:
re = mpf_atan(mpf_div(a, mpf_sqrt(re, wp), wp), wp)
else:
# c = Ax/(r + (a+1)); d = Ax/(s - (1-a))
# alpha - a = (1/2)*(c + d)
# case n = 0: re = atan(b*sqrt(c + d)/2/a)
# case n = 1: re = atan(a/(b*sqrt(c + d)/2)
c = mpf_div(Ax, mpf_add(r, ap, wp), wp)
d = mpf_div(Ax, mpf_sub(s, am, wp), wp)
re = mpf_shift(mpf_add(c, d, wp), -1)
re = mpf_mul(b, mpf_sqrt(re, wp), wp)
if n == 0:
re = mpf_atan(mpf_div(re, a, wp), wp)
else:
re = mpf_atan(mpf_div(a, re, wp), wp)
# to compute alpha + sqrt(alpha**2 - 1), if alpha <= alpha_crossover
# replace it with 1 + Am1 + sqrt(Am1*(alpha+1)))
# where Am1 = alpha -1
# if alpha <= alpha_crossover:
if not mpf_sub(alpha_crossover, alpha, wp)[0]:
c1 = mpf_div(b2, mpf_add(r, ap, wp), wp)
# case a < 1
if mpf_neg(am)[0]:
# Am1 = (1/2) * (b*b/(r + (a+1)) + b*b/(s + (1-a))
c2 = mpf_add(s, am, wp)
c2 = mpf_div(b2, c2, wp)
Am1 = mpf_shift(mpf_add(c1, c2, wp), -1)
else:
# Am1 = (1/2) * (b*b/(r + (a+1)) + (s - (1-a)))
c2 = mpf_sub(s, am, wp)
Am1 = mpf_shift(mpf_add(c1, c2, wp), -1)
# im = log(1 + Am1 + sqrt(Am1*(alpha+1)))
im = mpf_mul(Am1, mpf_add(alpha, fone, wp), wp)
im = mpf_log(mpf_add(fone, mpf_add(Am1, mpf_sqrt(im, wp), wp), wp), wp)
else:
# im = log(alpha + sqrt(alpha*alpha - 1))
im = mpf_sqrt(mpf_sub(mpf_mul(alpha, alpha, wp), fone, wp), wp)
im = mpf_log(mpf_add(alpha, im, wp), wp)
if asign:
if n == 0:
re = mpf_sub(mpf_pi(wp), re, wp)
else:
re = mpf_neg(re)
if not bsign and n == 0:
im = mpf_neg(im)
if bsign and n == 1:
im = mpf_neg(im)
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 atan_inf(sign, prec, rnd):
if not sign:
return mpf_shift(mpf_pi(prec, rnd), -1)
return mpf_neg(mpf_shift(mpf_pi(prec, negative_rnd[rnd]), -1))
def _jacobi_theta3(z, q):
extra1 = 10
extra2 = 20
MIN = 2
if z == zero:
if isinstance(q, mpf):
wp = mp.prec + extra1
x = to_fixed(q._mpf_, wp)
s = x
a = b = x
x2 = (x*x) >> wp
while abs(a) > MIN:
b = (b*x2) >> wp
a = (a*b) >> wp
s += a
s = (1 << wp) + (s << 1)
s = mpf(from_man_exp(s, -wp, mp.prec, 'n'))
return s
else:
wp = mp.prec + extra1
xre, xim = q._mpc_
xre = to_fixed(xre, wp)
xim = to_fixed(xim, wp)
x2re = (xre*xre - xim*xim) >> wp
x2im = (xre*xim) >> (wp - 1)
sre = are = bre = xre
sim = aim = bim = xim
while are**2 + aim**2 > MIN:
bre, bim = (bre * x2re - bim * x2im) >> wp, \
(bre * x2im + bim * x2re) >> wp
are, aim = (are * bre - aim * bim) >> wp, \
(are * bim + aim * bre) >> wp
sre += are
sim += aim
sre = (1 << wp) + (sre << 1)
sim = (sim << 1)
sre = from_man_exp(sre, -wp, mp.prec, 'n')
sim = from_man_exp(sim, -wp, mp.prec, 'n')
s = mpc(sre, sim)
return s
else:
if isinstance(q, mpf) and isinstance(z, mpf):
s = MP_ZERO
wp = mp.prec + extra1
x = to_fixed(q._mpf_, wp)
a = b = x
x2 = (x*x) >> wp
c1, s1 = cos_sin(mpf_shift(z._mpf_, 1), wp)
c1 = to_fixed(c1, wp)
s1 = to_fixed(s1, wp)
cn = c1
sn = s1
s += (a * cn) >> wp
while abs(a) > MIN:
b = (b*x2) >> wp
a = (a*b) >> wp
cn, sn = (cn*c1 - sn*s1) >> wp, (sn*c1 + cn*s1) >> wp
s += (a * cn) >> wp
s = (1 << wp) + (s << 1)
s = mpf(from_man_exp(s, -wp, mp.prec, 'n'))
return s
# case z real, q complex
elif isinstance(z, mpf):
wp = mp.prec + extra2
xre, xim = q._mpc_
xre = to_fixed(xre, wp)
xim = to_fixed(xim, wp)
x2re = (xre*xre - xim*xim) >> wp
x2im = (xre*xim) >> (wp - 1)
are = bre = xre
aim = bim = xim
c1, s1 = cos_sin(mpf_shift(z._mpf_, 1), wp)
c1 = to_fixed(c1, wp)
s1 = to_fixed(s1, wp)
cn = c1
sn = s1
sre = (are * cn) >> wp
sim = (aim * cn) >> wp
while are**2 + aim**2 > MIN:
bre, bim = (bre * x2re - bim * x2im) >> wp, \
(bre * x2im + bim * x2re) >> wp
are, aim = (are * bre - aim * bim) >> wp, \
(are * bim + aim * bre) >> wp
cn, sn = (cn*c1 - sn*s1) >> wp, (sn*c1 + cn*s1) >> wp
sre += (are * cn) >> wp
sim += (aim * cn) >> wp
sre = (1 << wp) + (sre << 1)
sim = (sim << 1)
sre = from_man_exp(sre, -wp, mp.prec, 'n')
sim = from_man_exp(sim, -wp, mp.prec, 'n')
s = mpc(sre, sim)
return s
#case z complex, q real
elif isinstance(q, mpf):
wp = mp.prec + extra2
x = to_fixed(q._mpf_, wp)
a = b = x
x2 = (x*x) >> wp
prec0 = mp.prec
mp.prec = wp
c1 = cos(2*z)
s1 = sin(2*z)
mp.prec = prec0
cnre = c1re = to_fixed(c1.real._mpf_, wp)
cnim = c1im = to_fixed(c1.imag._mpf_, wp)
snre = s1re = to_fixed(s1.real._mpf_, wp)
snim = s1im = to_fixed(s1.imag._mpf_, wp)
sre = (a * cnre) >> wp
sim = (a * cnim) >> wp
while abs(a) > MIN:
b = (b*x2) >> wp
a = (a*b) >> wp
t1 = (cnre*c1re - cnim*c1im - snre*s1re + snim*s1im) >> wp
t2 = (cnre*c1im + cnim*c1re - snre*s1im - snim*s1re) >> wp
t3 = (snre*c1re - snim*c1im + cnre*s1re - cnim*s1im) >> wp
t4 = (snre*c1im + snim*c1re + cnre*s1im + cnim*s1re) >> wp
cnre = t1
cnim = t2
snre = t3
snim = t4
sre += (a * cnre) >> wp
sim += (a * cnim) >> wp
sre = (1 << wp) + (sre << 1)
sim = (sim << 1)
sre = from_man_exp(sre, -wp, mp.prec, 'n')
sim = from_man_exp(sim, -wp, mp.prec, 'n')
s = mpc(sre, sim)
return s
# case z and q complex
else:
wp = mp.prec + extra2
xre, xim = q._mpc_
xre = to_fixed(xre, wp)
xim = to_fixed(xim, wp)
x2re = (xre*xre - xim*xim) >> wp
x2im = (xre*xim) >> (wp - 1)
are = bre = xre
aim = bim = xim
prec0 = mp.prec
mp.prec = wp
# cos(2*z), sin(2*z) with z complex
c1 = cos(2*z)
s1 = sin(2*z)
mp.prec = prec0
cnre = c1re = to_fixed(c1.real._mpf_, wp)
cnim = c1im = to_fixed(c1.imag._mpf_, wp)
snre = s1re = to_fixed(s1.real._mpf_, wp)
snim = s1im = to_fixed(s1.imag._mpf_, wp)
sre = (are * cnre - aim * cnim) >> wp
sim = (aim * cnre + are * cnim) >> wp
while are**2 + aim**2 > MIN:
bre, bim = (bre * x2re - bim * x2im) >> wp, \
(bre * x2im + bim * x2re) >> wp
are, aim = (are * bre - aim * bim) >> wp, \
(are * bim + aim * bre) >> wp
t1 = (cnre*c1re - cnim*c1im - snre*s1re + snim*s1im) >> wp
t2 = (cnre*c1im + cnim*c1re - snre*s1im - snim*s1re) >> wp
t3 = (snre*c1re - snim*c1im + cnre*s1re - cnim*s1im) >> wp
t4 = (snre*c1im + snim*c1re + cnre*s1im + cnim*s1re) >> wp
cnre = t1
cnim = t2
snre = t3
snim = t4
sre += (are * cnre - aim * cnim) >> wp
sim += (aim * cnre + are * cnim) >> wp
sre = (1 << wp) + (sre << 1)
sim = (sim << 1)
sre = from_man_exp(sre, -wp, mp.prec, 'n')
sim = from_man_exp(sim, -wp, mp.prec, 'n')
s = mpc(sre, sim)
return s
def _djacobi_theta3(z, q, nd):
"""nd=1,2,3 order of the derivative with respect to z"""
MIN = 2
extra1 = 10
extra2 = 20
if isinstance(q, mpf) and isinstance(z, mpf):
s = MP_ZERO
wp = mp.prec + extra1
x = to_fixed(q._mpf_, wp)
a = b = x
x2 = (x*x) >> wp
c1, s1 = cos_sin(mpf_shift(z._mpf_, 1), wp)
c1 = to_fixed(c1, wp)
s1 = to_fixed(s1, wp)
cn = c1
sn = s1
if (nd&1):
s += (a * sn) >> wp
else:
s += (a * cn) >> wp
n = 2
while abs(a) > MIN:
b = (b*x2) >> wp
a = (a*b) >> wp
cn, sn = (cn*c1 - sn*s1) >> wp, (sn*c1 + cn*s1) >> wp
if nd&1:
s += (a * sn * n**nd) >> wp
else:
s += (a * cn * n**nd) >> wp
n += 1
s = -(s << (nd+1))
s = mpf(from_man_exp(s, -wp, mp.prec, 'n'))
# case z real, q complex
elif isinstance(z, mpf):
wp = mp.prec + extra2
xre, xim = q._mpc_
xre = to_fixed(xre, wp)
xim = to_fixed(xim, wp)
x2re = (xre*xre - xim*xim) >> wp
x2im = (xre*xim) >> (wp - 1)
are = bre = xre
aim = bim = xim
c1, s1 = cos_sin(mpf_shift(z._mpf_, 1), wp)
c1 = to_fixed(c1, wp)
s1 = to_fixed(s1, wp)
cn = c1
sn = s1
if (nd&1):
sre = (are * sn) >> wp
sim = (aim * sn) >> wp
else:
sre = (are * cn) >> wp
sim = (aim * cn) >> wp
n = 2
while are**2 + aim**2 > MIN:
bre, bim = (bre * x2re - bim * x2im) >> wp, \
(bre * x2im + bim * x2re) >> wp
are, aim = (are * bre - aim * bim) >> wp, \
(are * bim + aim * bre) >> wp
cn, sn = (cn*c1 - sn*s1) >> wp, (sn*c1 + cn*s1) >> wp
if nd&1:
sre += (are * sn * n**nd) >> wp
sim += (aim * sn * n**nd) >> wp
else:
sre += (are * cn * n**nd) >> wp
sim += (aim * cn * n**nd) >> wp
n += 1
sre = -(sre << (nd+1))
sim = -(sim << (nd+1))
sre = from_man_exp(sre, -wp, mp.prec, 'n')
sim = from_man_exp(sim, -wp, mp.prec, 'n')
s = mpc(sre, sim)
#case z complex, q real
elif isinstance(q, mpf):
wp = mp.prec + extra2
x = to_fixed(q._mpf_, wp)
a = b = x
x2 = (x*x) >> wp
prec0 = mp.prec
mp.prec = wp
c1 = cos(2*z)
s1 = sin(2*z)
mp.prec = prec0
cnre = c1re = to_fixed(c1.real._mpf_, wp)
cnim = c1im = to_fixed(c1.imag._mpf_, wp)
snre = s1re = to_fixed(s1.real._mpf_, wp)
snim = s1im = to_fixed(s1.imag._mpf_, wp)
if (nd&1):
sre = (a * snre) >> wp
sim = (a * snim) >> wp
else:
sre = (a * cnre) >> wp
sim = (a * cnim) >> wp
n = 2
while abs(a) > MIN:
b = (b*x2) >> wp
a = (a*b) >> wp
t1 = (cnre*c1re - cnim*c1im - snre*s1re + snim*s1im) >> wp
t2 = (cnre*c1im + cnim*c1re - snre*s1im - snim*s1re) >> wp
t3 = (snre*c1re - snim*c1im + cnre*s1re - cnim*s1im) >> wp
t4 = (snre*c1im + snim*c1re + cnre*s1im + cnim*s1re) >> wp
cnre = t1
cnim = t2
snre = t3
snim = t4
if (nd&1):
sre += (a * snre * n**nd) >> wp
sim += (a * snim * n**nd) >> wp
else:
sre += (a * cnre * n**nd) >> wp
sim += (a * cnim * n**nd) >> wp
n += 1
sre = -(sre << (nd+1))
sim = -(sim << (nd+1))
sre = from_man_exp(sre, -wp, mp.prec, 'n')
sim = from_man_exp(sim, -wp, mp.prec, 'n')
s = mpc(sre, sim)
# case z and q complex
else:
wp = mp.prec + extra2
xre, xim = q._mpc_
xre = to_fixed(xre, wp)
xim = to_fixed(xim, wp)
x2re = (xre*xre - xim*xim) >> wp
x2im = (xre*xim) >> (wp - 1)
are = bre = xre
aim = bim = xim
prec0 = mp.prec
mp.prec = wp
# cos(2*z), sin(2*z) with z complex
c1 = cos(2*z)
s1 = sin(2*z)
mp.prec = prec0
cnre = c1re = to_fixed(c1.real._mpf_, wp)
cnim = c1im = to_fixed(c1.imag._mpf_, wp)
snre = s1re = to_fixed(s1.real._mpf_, wp)
snim = s1im = to_fixed(s1.imag._mpf_, wp)
if (nd&1):
sre = (are * snre - aim * snim) >> wp
sim = (aim * snre + are * snim) >> wp
else:
sre = (are * cnre - aim * cnim) >> wp
sim = (aim * cnre + are * cnim) >> wp
n = 2
while are**2 + aim**2 > MIN:
bre, bim = (bre * x2re - bim * x2im) >> wp, \
(bre * x2im + bim * x2re) >> wp
are, aim = (are * bre - aim * bim) >> wp, \
(are * bim + aim * bre) >> wp
t1 = (cnre*c1re - cnim*c1im - snre*s1re + snim*s1im) >> wp
t2 = (cnre*c1im + cnim*c1re - snre*s1im - snim*s1re) >> wp
t3 = (snre*c1re - snim*c1im + cnre*s1re - cnim*s1im) >> wp
t4 = (snre*c1im + snim*c1re + cnre*s1im + cnim*s1re) >> wp
cnre = t1
cnim = t2
snre = t3
snim = t4
if(nd&1):
sre += ((are * snre - aim * snim) * n**nd) >> wp
sim += ((aim * snre + are * snim) * n**nd) >> wp
else:
sre += ((are * cnre - aim * cnim) * n**nd) >> wp
sim += ((aim * cnre + are * cnim) * n**nd) >> wp
n += 1
sre = -(sre << (nd+1))
sim = -(sim << (nd+1))
sre = from_man_exp(sre, -wp, mp.prec, 'n')
sim = from_man_exp(sim, -wp, mp.prec, 'n')
s = mpc(sre, sim)
if (nd&1):
return (-1)**(nd//2) * s
else:
return (-1)**(1 + nd//2) * s
def mpi_mid(s, prec):
sa, sb = s
return mpf_shift(mpf_add(sa, sb, prec, round_nearest), -1)
def mpf_cosh_sinh(x, prec, rnd=round_fast, tanh=0):
"""Simultaneously compute (cosh(x), sinh(x)) for real x"""
sign, man, exp, bc = x
if (not man) and exp:
if tanh:
if x == finf: return fone
if x == fninf: return fnone
return fnan
if x == finf: return (finf, finf)
if x == fninf: return (finf, fninf)
return fnan, fnan
mag = exp + bc
wp = prec + 14
if mag < -4:
# Extremely close to 0, sinh(x) ~= x and cosh(x) ~= 1
if mag < -wp:
if tanh:
return mpf_perturb(x, 1 - sign, prec, rnd)
cosh = mpf_perturb(fone, 0, prec, rnd)
sinh = mpf_perturb(x, sign, prec, rnd)
return cosh, sinh
# Fix for cancellation when computing sinh
wp += (-mag)
# Does exp(-2*x) vanish?
if mag > 10:
if 3 * (1 << (mag - 1)) > wp:
# XXX: rounding
if tanh:
return mpf_perturb([fone, fnone][sign], 1 - sign, prec, rnd)
c = s = mpf_shift(mpf_exp(mpf_abs(x), prec, rnd), -1)
if sign:
s = mpf_neg(s)
return c, s
# |x| > 1
if mag > 1:
wpmod = wp + mag
offset = exp + wpmod
if offset >= 0:
t = man << offset
else:
t = man >> (-offset)
lg2 = ln2_fixed(wpmod)
n, t = divmod(t, lg2)
n = int(n)
t >>= mag
else:
offset = exp + wp
if offset >= 0:
t = man << offset
else:
t = man >> (-offset)
n = 0
a, b = exp_expneg_basecase(t, wp)
# TODO: optimize division precision
cosh = a + (b >> (2 * n))
sinh = a - (b >> (2 * n))
if sign:
sinh = -sinh
if tanh:
man = (sinh << wp) // cosh
return from_man_exp(man, -wp, prec, rnd)
else:
cosh = from_man_exp(cosh, n - wp - 1, prec, rnd)
sinh = from_man_exp(sinh, n - wp - 1, prec, rnd)
return cosh, sinh
