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_psi0(x, prec, rnd=round_fast):
"""
Computation of the digamma function (psi function of order 0)
of a real argument.
"""
sign, man, exp, bc = x
wp = prec + 10
if not man:
if x == finf:
return x
if x == fninf or x == fnan:
return fnan
if x == fzero or (exp >= 0 and sign):
raise ValueError("polygamma pole")
# Reflection formula
if sign and exp + bc > 3:
c, s = mpf_cos_sin_pi(x, wp)
q = mpf_mul(mpf_div(c, s, wp), mpf_pi(wp), wp)
p = mpf_psi0(mpf_sub(fone, x, wp), wp)
return mpf_sub(p, q, prec, rnd)
# The logarithmic term is accurate enough
if (not sign) and bc + exp > wp:
return mpf_log(mpf_sub(x, fone, wp), prec, rnd)
# Initial recurrence to obtain a large enough x
m = to_int(x)
n = int(0.11 * wp) + 2
s = MP_ZERO
x = to_fixed(x, wp)
one = MP_ONE << wp
if m < n:
for k in xrange(m, n):
s -= (one << wp) // x
x += one
x -= one
# Logarithmic term
s += to_fixed(mpf_log(from_man_exp(x, -wp, wp), wp), wp)
# Endpoint term in Euler-Maclaurin expansion
s += (one << wp) // (2 * x)
# Euler-Maclaurin remainder sum
x2 = (x * x) >> wp
t = one
prev = 0
k = 1
while 1:
t = (t * x2) >> wp
bsign, bman, bexp, bbc = mpf_bernoulli(2 * k, wp)
offset = bexp + 2 * wp
if offset >= 0:
term = (bman << offset) // (t * (2 * k))
else:
term = (bman >> (-offset)) // (t * (2 * k))
if k & 1:
s -= term
else:
s += term
if k > 2 and term >= prev:
break
prev = term
k += 1
return from_man_exp(s, -wp, wp, rnd)
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_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_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_sin_pi(z, prec, rnd=round_fast):
a, b = z
b = mpf_mul(b, mpf_pi(prec+5), prec+5)
if a == fzero:
return fzero, mpf_sinh(b, prec, rnd)
wp = prec + 6
c, s = mpf_cos_sin_pi(a, wp)
ch, sh = mpf_cosh_sinh(b, wp)
re = mpf_mul(s, ch, prec, rnd)
im = mpf_mul(c, sh, prec, rnd)
return re, im
def mpf_psi0(x, prec, rnd=round_fast):
"""
Computation of the digamma function (psi function of order 0)
of a real argument.
"""
sign, man, exp, bc = x
wp = prec + 10
if not man:
if x == finf: return x
if x == fninf or x == fnan: return fnan
if x == fzero or (exp >= 0 and sign):
raise ValueError("polygamma pole")
# Reflection formula
if sign and exp + bc > 3:
c, s = mpf_cos_sin_pi(x, wp)
q = mpf_mul(mpf_div(c, s, wp), mpf_pi(wp), wp)
p = mpf_psi0(mpf_sub(fone, x, wp), wp)
return mpf_sub(p, q, prec, rnd)
# The logarithmic term is accurate enough
if (not sign) and bc + exp > wp:
return mpf_log(mpf_sub(x, fone, wp), prec, rnd)
# Initial recurrence to obtain a large enough x
m = to_int(x)
n = int(0.11 * wp) + 2
s = MP_ZERO
x = to_fixed(x, wp)
one = MP_ONE << wp
if m < n:
for k in xrange(m, n):
s -= (one << wp) // x
x += one
x -= one
# Logarithmic term
s += to_fixed(mpf_log(from_man_exp(x, -wp, wp), wp), wp)
# Endpoint term in Euler-Maclaurin expansion
s += (one << wp) // (2 * x)
# Euler-Maclaurin remainder sum
x2 = (x * x) >> wp
t = one
prev = 0
k = 1
while 1:
t = (t * x2) >> wp
bsign, bman, bexp, bbc = mpf_bernoulli(2 * k, wp)
offset = (bexp + 2 * wp)
if offset >= 0: term = (bman << offset) // (t * (2 * k))
else: term = (bman >> (-offset)) // (t * (2 * k))
if k & 1: s -= term
else: s += term
if k > 2 and term >= prev:
break
prev = term
k += 1
return from_man_exp(s, -wp, wp, rnd)
def mpc_sin_pi(z, prec, rnd=round_fast):
a, b = z
b = mpf_mul(b, mpf_pi(prec + 5), prec + 5)
if a == fzero:
return fzero, mpf_sinh(b, prec, rnd)
wp = prec + 6
c, s = mpf_cos_sin_pi(a, wp)
ch, sh = mpf_cosh_sinh(b, wp)
re = mpf_mul(s, ch, prec, rnd)
im = mpf_mul(c, sh, prec, rnd)
return re, im
def mpc_cos_pi(z, prec, rnd=round_fast):
a, b = z
if b == fzero:
return mpf_cos_pi(a, prec, rnd), fzero
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 mpc_cos_pi(z, prec, rnd=round_fast):
a, b = z
if b == fzero:
return mpf_cos_pi(a, prec, rnd), fzero
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)
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_cos_pi((a, b), prec, rnd=round_fast):
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 = 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_sin_pi((a, b), prec, rnd=round_fast):
b = mpf_mul(b, mpf_pi(prec+5), prec+5)
if a == fzero:
return fzero, mpf_sinh(b, prec, rnd)
wp = prec + 6
c, s = mpf_cos_sin_pi(a, wp)
ch, sh = cosh_sinh(b, wp)
re = mpf_mul(s, ch, prec, rnd)
im = mpf_mul(c, sh, prec, rnd)
return re, im
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_cos_pi((a, b), prec, rnd=round_fast):
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 = 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_sin_pi((a, b), prec, rnd=round_fast):
b = mpf_mul(b, mpf_pi(prec+5), prec+5)
if a == fzero:
return fzero, mpf_sinh(b, prec, rnd)
wp = prec + 6
c, s = mpf_cos_sin_pi(a, wp)
ch, sh = cosh_sinh(b, wp)
re = mpf_mul(s, ch, prec, rnd)
im = mpf_mul(c, sh, prec, rnd)
return re, im