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_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_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_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_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_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_div(z, w, prec, rnd=round_fast):
a, b = z
c, d = w
wp = prec + 10
# mag = c*c + d*d
mag = mpf_add(mpf_mul(c, c), mpf_mul(d, d), wp)
# (a*c+b*d)/mag, (b*c-a*d)/mag
t = mpf_add(mpf_mul(a, c), mpf_mul(b, d), wp)
u = mpf_sub(mpf_mul(b, c), mpf_mul(a, d), wp)
return mpf_div(t, mag, prec, rnd), mpf_div(u, mag, prec, rnd)
def mpc_div(z, w, prec, rnd=round_fast):
a, b = z
c, d = w
wp = prec + 10
# mag = c*c + d*d
mag = mpf_add(mpf_mul(c, c), mpf_mul(d, d), wp)
# (a*c+b*d)/mag, (b*c-a*d)/mag
t = mpf_add(mpf_mul(a,c), mpf_mul(b,d), wp)
u = mpf_sub(mpf_mul(b,c), mpf_mul(a,d), wp)
return mpf_div(t,mag,prec,rnd), mpf_div(u,mag,prec,rnd)
def mpf_zeta_int(s, prec, rnd=round_fast):
"""
Optimized computation of zeta(s) for an integer s.
"""
wp = prec + 20
s = int(s)
if s in zeta_int_cache and zeta_int_cache[s][0] >= wp:
return mpf_pos(zeta_int_cache[s][1], prec, rnd)
if s < 2:
if s == 1:
raise ValueError("zeta(1) pole")
if not s:
return mpf_neg(fhalf)
return mpf_div(mpf_bernoulli(-s + 1, wp), from_int(s - 1), prec, rnd)
# 2^-s term vanishes?
if s >= wp:
return mpf_perturb(fone, 0, prec, rnd)
# 5^-s term vanishes?
elif s >= wp * 0.431:
t = one = 1 << wp
t += 1 << (wp - s)
t += one // (MPZ_THREE**s)
t += 1 << max(0, wp - s * 2)
return from_man_exp(t, -wp, prec, rnd)
else:
# Fast enough to sum directly?
# Even better, we use the Euler product (idea stolen from pari)
m = (float(wp) / (s - 1) + 1)
if m < 30:
needed_terms = int(2.0**m + 1)
if needed_terms < int(wp / 2.54 + 5) / 10:
t = fone
for k in list_primes(needed_terms):
#print k, needed_terms
powprec = int(wp - s * math.log(k, 2))
if powprec < 2:
break
a = mpf_sub(fone, mpf_pow_int(from_int(k), -s, powprec),
wp)
t = mpf_mul(t, a, wp)
return mpf_div(fone, t, wp)
# Use Borwein's algorithm
n = int(wp / 2.54 + 5)
d = borwein_coefficients(n)
t = MPZ_ZERO
s = MPZ(s)
for k in xrange(n):
t += (((-1)**k * (d[k] - d[n])) << wp) // (k + 1)**s
t = (t << wp) // (-d[n])
t = (t << wp) // ((1 << wp) - (1 << (wp + 1 - s)))
if (s in zeta_int_cache
and zeta_int_cache[s][0] < wp) or (s not in zeta_int_cache):
zeta_int_cache[s] = (wp, from_man_exp(t, -wp - wp))
return from_man_exp(t, -wp - wp, prec, rnd)
def mpf_zeta_int(s, prec, rnd=round_fast):
"""
Optimized computation of zeta(s) for an integer s.
"""
wp = prec + 20
s = int(s)
if s in zeta_int_cache and zeta_int_cache[s][0] >= wp:
return mpf_pos(zeta_int_cache[s][1], prec, rnd)
if s < 2:
if s == 1:
raise ValueError("zeta(1) pole")
if not s:
return mpf_neg(fhalf)
return mpf_div(mpf_bernoulli(-s+1, wp), from_int(s-1), prec, rnd)
# 2^-s term vanishes?
if s >= wp:
return mpf_perturb(fone, 0, prec, rnd)
# 5^-s term vanishes?
elif s >= wp*0.431:
t = one = 1 << wp
t += 1 << (wp - s)
t += one // (MPZ_THREE ** s)
t += 1 << max(0, wp - s*2)
return from_man_exp(t, -wp, prec, rnd)
else:
# Fast enough to sum directly?
# Even better, we use the Euler product (idea stolen from pari)
m = (float(wp)/(s-1) + 1)
if m < 30:
needed_terms = int(2.0**m + 1)
if needed_terms < int(wp/2.54 + 5) / 10:
t = fone
for k in list_primes(needed_terms):
#print k, needed_terms
powprec = int(wp - s*math.log(k,2))
if powprec < 2:
break
a = mpf_sub(fone, mpf_pow_int(from_int(k), -s, powprec), wp)
t = mpf_mul(t, a, wp)
return mpf_div(fone, t, wp)
# Use Borwein's algorithm
n = int(wp/2.54 + 5)
d = borwein_coefficients(n)
t = MPZ_ZERO
s = MPZ(s)
for k in xrange(n):
t += (((-1)**k * (d[k] - d[n])) << wp) // (k+1)**s
t = (t << wp) // (-d[n])
t = (t << wp) // ((1 << wp) - (1 << (wp+1-s)))
if (s in zeta_int_cache and zeta_int_cache[s][0] < wp) or (s not in zeta_int_cache):
zeta_int_cache[s] = (wp, from_man_exp(t, -wp-wp))
return from_man_exp(t, -wp-wp, prec, rnd)
def mpf_zeta(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_div(s, t, prec):
sa, sb = s
ta, tb = t
sas = mpf_sign(sa)
sbs = mpf_sign(sb)
tas = mpf_sign(ta)
tbs = mpf_sign(tb)
# 0 / X
if sas == sbs == 0:
# 0 / <interval containing 0>
if (tas < 0 and tbs > 0) or (tas == 0 or tbs == 0):
return fninf, finf
return fzero, fzero
# Denominator contains both negative and positive numbers;
# this should properly be a multi-interval, but the closest
# match is the entire (extended) real line
if tas < 0 and tbs > 0:
return fninf, finf
# Assume denominator to be nonnegative
if tas < 0:
return mpi_div(mpi_neg(s), mpi_neg(t), prec)
# Division by zero
# XXX: make sure all results make sense
if tas == 0:
# Numerator contains both signs?
if sas < 0 and sbs > 0:
return fninf, finf
if tas == tbs:
return fninf, finf
# Numerator positive?
if sas >= 0:
a = mpf_div(sa, tb, prec, round_floor)
b = finf
if sbs <= 0:
a = fninf
b = mpf_div(sb, tb, prec, round_ceiling)
# Division with positive denominator
# We still have to handle nans resulting from inf/0 or inf/inf
else:
# Nonnegative numerator
if sas >= 0:
a = mpf_div(sa, tb, prec, round_floor)
b = mpf_div(sb, ta, prec, round_ceiling)
if a == fnan: a = fzero
if b == fnan: b = finf
# Nonpositive numerator
elif sbs <= 0:
a = mpf_div(sa, ta, prec, round_floor)
b = mpf_div(sb, tb, prec, round_ceiling)
if a == fnan: a = fninf
if b == fnan: b = fzero
# Numerator contains both signs?
else:
a = mpf_div(sa, ta, prec, round_floor)
b = mpf_div(sb, ta, prec, round_ceiling)
if a == fnan: a = fninf
if b == fnan: b = finf
return a, b
def mpi_div(s, t, prec):
sa, sb = s
ta, tb = t
sas = mpf_sign(sa)
sbs = mpf_sign(sb)
tas = mpf_sign(ta)
tbs = mpf_sign(tb)
# 0 / X
if sas == sbs == 0:
# 0 / <interval containing 0>
if (tas < 0 and tbs > 0) or (tas == 0 or tbs == 0):
return fninf, finf
return fzero, fzero
# Denominator contains both negative and positive numbers;
# this should properly be a multi-interval, but the closest
# match is the entire (extended) real line
if tas < 0 and tbs > 0:
return fninf, finf
# Assume denominator to be nonnegative
if tas < 0:
return mpi_div(mpi_neg(s), mpi_neg(t), prec)
# Division by zero
# XXX: make sure all results make sense
if tas == 0:
# Numerator contains both signs?
if sas < 0 and sbs > 0:
return fninf, finf
if tas == tbs:
return fninf, finf
# Numerator positive?
if sas >= 0:
a = mpf_div(sa, tb, prec, round_floor)
b = finf
if sbs <= 0:
a = fninf
b = mpf_div(sb, tb, prec, round_ceiling)
# Division with positive denominator
# We still have to handle nans resulting from inf/0 or inf/inf
else:
# Nonnegative numerator
if sas >= 0:
a = mpf_div(sa, tb, prec, round_floor)
b = mpf_div(sb, ta, prec, round_ceiling)
if a == fnan: a = fzero
if b == fnan: b = finf
# Nonpositive numerator
elif sbs <= 0:
a = mpf_div(sa, ta, prec, round_floor)
b = mpf_div(sb, tb, prec, round_ceiling)
if a == fnan: a = fninf
if b == fnan: b = fzero
# Numerator contains both signs?
else:
a = mpf_div(sa, ta, prec, round_floor)
b = mpf_div(sb, ta, prec, round_ceiling)
if a == fnan: a = fninf
if b == fnan: b = finf
return a, b
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 khinchin_fixed(prec):
wp = int(prec + prec**0.5 + 15)
s = MP_ZERO
fac = from_int(4)
t = ONE = MP_ONE << wp
pi = mpf_pi(wp)
pipow = twopi2 = mpf_shift(mpf_mul(pi, pi, wp), 2)
n = 1
while 1:
zeta2n = mpf_abs(mpf_bernoulli(2 * n, wp))
zeta2n = mpf_mul(zeta2n, pipow, wp)
zeta2n = mpf_div(zeta2n, fac, wp)
zeta2n = to_fixed(zeta2n, wp)
term = (((zeta2n - ONE) * t) // n) >> wp
if term < 100:
break
#if not n % 100:
# print n, nstr(ln(term))
s += term
t += ONE // (2 * n + 1) - ONE // (2 * n)
n += 1
fac = mpf_mul_int(fac, (2 * n) * (2 * n - 1), wp)
pipow = mpf_mul(pipow, twopi2, wp)
s = (s << wp) // ln2_fixed(wp)
K = mpf_exp(from_man_exp(s, -wp), wp)
K = to_fixed(K, prec)
return K
def 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 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 mpf_pow(s, t, prec, rnd=round_fast):
"""
Compute s**t. Raises ComplexResult if s is negative and t is
fractional.
"""
ssign, sman, sexp, sbc = s
tsign, tman, texp, tbc = t
if ssign and texp < 0:
raise ComplexResult("negative number raised to a fractional power")
if texp >= 0:
return mpf_pow_int(s, (-1)**tsign * (tman<<texp), prec, rnd)
# s**(n/2) = sqrt(s)**n
if texp == -1:
if tman == 1:
if tsign:
return mpf_div(fone, mpf_sqrt(s, prec+10,
reciprocal_rnd[rnd]), prec, rnd)
return mpf_sqrt(s, prec, rnd)
else:
if tsign:
return mpf_pow_int(mpf_sqrt(s, prec+10,
reciprocal_rnd[rnd]), -tman, prec, rnd)
return mpf_pow_int(mpf_sqrt(s, prec+10, rnd), tman, prec, rnd)
# General formula: s**t = exp(t*log(s))
# TODO: handle rnd direction of the logarithm carefully
c = mpf_log(s, prec+10, rnd)
return mpf_exp(mpf_mul(t, c), 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_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 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_pow(s, t, prec, rnd=round_fast):
"""
Compute s**t. Raises ComplexResult if s is negative and t is
fractional.
"""
ssign, sman, sexp, sbc = s
tsign, tman, texp, tbc = t
if ssign and texp < 0:
raise ComplexResult("negative number raised to a fractional power")
if texp >= 0:
return mpf_pow_int(s, (-1)**tsign * (tman<<texp), prec, rnd)
# s**(n/2) = sqrt(s)**n
if texp == -1:
if tman == 1:
if tsign:
return mpf_div(fone, mpf_sqrt(s, prec+10,
reciprocal_rnd[rnd]), prec, rnd)
return mpf_sqrt(s, prec, rnd)
else:
if tsign:
return mpf_pow_int(mpf_sqrt(s, prec+10,
reciprocal_rnd[rnd]), -tman, prec, rnd)
return mpf_pow_int(mpf_sqrt(s, prec+10, rnd), tman, prec, rnd)
# General formula: s**t = exp(t*log(s))
# TODO: handle rnd direction of the logarithm carefully
c = mpf_log(s, prec+10, rnd)
return mpf_exp(mpf_mul(t, c), prec, rnd)
def twinprime_fixed(prec):
def I(n):
return sum(
moebius(d) << (n // d) for d in xrange(1, n + 1) if not n % d) // n
wp = 2 * prec + 30
res = fone
primes = [from_rational(1, p, wp) for p in [2, 3, 5, 7]]
ppowers = [mpf_mul(p, p, wp) for p in primes]
n = 2
while 1:
a = mpf_zeta_int(n, wp)
for i in range(4):
a = mpf_mul(a, mpf_sub(fone, ppowers[i]), wp)
ppowers[i] = mpf_mul(ppowers[i], primes[i], wp)
a = mpf_pow_int(a, -I(n), wp)
if mpf_pos(a, prec + 10, 'n') == fone:
break
#from libmpf import to_str
#print n, to_str(mpf_sub(fone, a), 6)
res = mpf_mul(res, a, wp)
n += 1
res = mpf_mul(res, from_int(3 * 15 * 35), wp)
res = mpf_div(res, from_int(4 * 16 * 36), wp)
return to_fixed(res, prec)
def 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_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 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 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_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_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_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_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 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_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_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_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 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 mpi_from_str_a_b(x, y, percent, prec):
wp = prec + 20
xa = from_str(x, wp, round_floor)
xb = from_str(x, wp, round_ceiling)
#ya = from_str(y, wp, round_floor)
y = from_str(y, wp, round_ceiling)
assert mpf_ge(y, fzero)
if percent:
y = mpf_mul(MAX(mpf_abs(xa), mpf_abs(xb)), y, wp, round_ceiling)
y = mpf_div(y, from_int(100), wp, round_ceiling)
a = mpf_sub(xa, y, prec, round_floor)
b = mpf_add(xb, y, prec, round_ceiling)
return a, b
def mpi_from_str_a_b(x, y, percent, prec):
wp = prec + 20
xa = from_str(x, wp, round_floor)
xb = from_str(x, wp, round_ceiling)
#ya = from_str(y, wp, round_floor)
y = from_str(y, wp, round_ceiling)
assert mpf_ge(y, fzero)
if percent:
y = mpf_mul(MAX(mpf_abs(xa), mpf_abs(xb)), y, wp, round_ceiling)
y = mpf_div(y, from_int(100), wp, round_ceiling)
a = mpf_sub(xa, y, prec, round_floor)
b = mpf_add(xb, y, prec, round_ceiling)
return a, b
def glaisher_fixed(prec):
wp = prec + 30
# Number of direct terms to sum before applying the Euler-Maclaurin
# formula to the tail. TODO: choose more intelligently
N = int(0.33*prec + 5)
ONE = MPZ_ONE << wp
# Euler-Maclaurin, step 1: sum log(k)/k**2 for k from 2 to N-1
s = MPZ_ZERO
for k in range(2, N):
#print k, N
s += log_int_fixed(k, wp) // k**2
logN = log_int_fixed(N, wp)
#logN = to_fixed(mpf_log(from_int(N), wp+20), wp)
# E-M step 2: integral of log(x)/x**2 from N to inf
s += (ONE + logN) // N
# E-M step 3: endpoint correction term f(N)/2
s += logN // (N**2 * 2)
# E-M step 4: the series of derivatives
pN = N**3
a = 1
b = -2
j = 3
fac = from_int(2)
k = 1
while 1:
# D(2*k-1) * B(2*k) / fac(2*k) [D(n) = nth derivative]
D = ((a << wp) + b*logN) // pN
D = from_man_exp(D, -wp)
B = mpf_bernoulli(2*k, wp)
term = mpf_mul(B, D, wp)
term = mpf_div(term, fac, wp)
term = to_fixed(term, wp)
if abs(term) < 100:
break
#if not k % 10:
# print k, math.log(int(abs(term)), 10)
s -= term
# Advance derivative twice
a, b, pN, j = b-a*j, -j*b, pN*N, j+1
a, b, pN, j = b-a*j, -j*b, pN*N, j+1
k += 1
fac = mpf_mul_int(fac, (2*k)*(2*k-1), wp)
# A = exp((6*s/pi**2 + log(2*pi) + euler)/12)
pi = pi_fixed(wp)
s *= 6
s = (s << wp) // (pi**2 >> wp)
s += euler_fixed(wp)
s += to_fixed(mpf_log(from_man_exp(2*pi, -wp), wp), wp)
s //= 12
A = mpf_exp(from_man_exp(s, -wp), wp)
return to_fixed(A, prec)
def glaisher_fixed(prec):
wp = prec + 30
# Number of direct terms to sum before applying the Euler-Maclaurin
# formula to the tail. TODO: choose more intelligently
N = int(0.33 * prec + 5)
ONE = MP_ONE << wp
# Euler-Maclaurin, step 1: sum log(k)/k**2 for k from 2 to N-1
s = MP_ZERO
for k in range(2, N):
#print k, N
s += log_int_fixed(k, wp) // k**2
logN = log_int_fixed(N, wp)
#logN = to_fixed(mpf_log(from_int(N), wp+20), wp)
# E-M step 2: integral of log(x)/x**2 from N to inf
s += (ONE + logN) // N
# E-M step 3: endpoint correction term f(N)/2
s += logN // (N**2 * 2)
# E-M step 4: the series of derivatives
pN = N**3
a = 1
b = -2
j = 3
fac = from_int(2)
k = 1
while 1:
# D(2*k-1) * B(2*k) / fac(2*k) [D(n) = nth derivative]
D = ((a << wp) + b * logN) // pN
D = from_man_exp(D, -wp)
B = mpf_bernoulli(2 * k, wp)
term = mpf_mul(B, D, wp)
term = mpf_div(term, fac, wp)
term = to_fixed(term, wp)
if abs(term) < 100:
break
#if not k % 10:
# print k, math.log(int(abs(term)), 10)
s -= term
# Advance derivative twice
a, b, pN, j = b - a * j, -j * b, pN * N, j + 1
a, b, pN, j = b - a * j, -j * b, pN * N, j + 1
k += 1
fac = mpf_mul_int(fac, (2 * k) * (2 * k - 1), wp)
# A = exp((6*s/pi**2 + log(2*pi) + euler)/12)
pi = pi_fixed(wp)
s *= 6
s = (s << wp) // (pi**2 >> wp)
s += euler_fixed(wp)
s += to_fixed(mpf_log(from_man_exp(2 * pi, -wp), wp), wp)
s //= 12
A = mpf_exp(from_man_exp(s, -wp), wp)
return to_fixed(A, prec)
def mertens_fixed(prec):
wp = prec + 20
m = 2
s = mpf_euler(wp)
while 1:
t = mpf_zeta_int(m, wp)
if t == fone:
break
t = mpf_log(t, wp)
t = mpf_mul_int(t, moebius(m), wp)
t = mpf_div(t, from_int(m), wp)
s = mpf_add(s, t)
m += 1
return to_fixed(s, prec)
def mertens_fixed(prec):
wp = prec + 20
m = 2
s = mpf_euler(wp)
while 1:
t = mpf_zeta_int(m, wp)
if t == fone:
break
t = mpf_log(t, wp)
t = mpf_mul_int(t, moebius(m), wp)
t = mpf_div(t, from_int(m), wp)
s = mpf_add(s, t)
m += 1
return to_fixed(s, prec)
def mpf_gamma(x, prec, rounding=round_fast, p1=1):
"""
Computes the gamma function of a real floating-point argument.
With p1=0, computes a factorial instead.
"""
sign, man, exp, bc = x
if not man:
if x == finf:
return finf
if x == fninf or x == fnan:
return fnan
# More precision is needed for enormous x. TODO:
# use Stirling's formula + Euler-Maclaurin summation
size = exp + bc
if size > 5:
size = int(size * math.log(size, 2))
wp = prec + max(0, size) + 15
if exp >= 0:
if sign or (p1 and not man):
raise ValueError("gamma function pole")
# A direct factorial is fastest
if exp + bc <= 10:
return from_int(int_fac((man << exp) - p1), prec, rounding)
reflect = sign or exp + bc < -1
if p1:
# Should be done exactly!
x = mpf_sub(x, fone, bc - exp + 2)
# x < 0.25
if reflect:
# gamma = pi / (sin(pi*x) * gamma(1-x))
wp += 15
pix = mpf_mul(x, mpf_pi(wp), wp)
t = mpf_sin_pi(x, wp)
g = mpf_gamma(mpf_sub(fone, x, wp), wp)
return mpf_div(pix, mpf_mul(t, g, wp), prec, rounding)
sprec, a, c = get_spouge_coefficients(wp)
s = spouge_sum_real(x, sprec, a, c)
# gamma = exp(log(x+a)*(x+0.5) - xpa) * s
xpa = mpf_add(x, from_int(a), wp)
logxpa = mpf_log(xpa, wp)
xph = mpf_add(x, fhalf, wp)
t = mpf_sub(mpf_mul(logxpa, xph, wp), xpa, wp)
t = mpf_mul(mpf_exp(t, wp), s, prec, rounding)
return t
def mpf_gamma(x, prec, rounding=round_fast, p1=1):
"""
Computes the gamma function of a real floating-point argument.
With p1=0, computes a factorial instead.
"""
sign, man, exp, bc = x
if not man:
if x == finf:
return finf
if x == fninf or x == fnan:
return fnan
# More precision is needed for enormous x. TODO:
# use Stirling's formula + Euler-Maclaurin summation
size = exp + bc
if size > 5:
size = int(size * math.log(size,2))
wp = prec + max(0, size) + 15
if exp >= 0:
if sign or (p1 and not man):
raise ValueError("gamma function pole")
# A direct factorial is fastest
if exp + bc <= 10:
return from_int(ifac((man<<exp)-p1), prec, rounding)
reflect = sign or exp+bc < -1
if p1:
# Should be done exactly!
x = mpf_sub(x, fone)
# x < 0.25
if reflect:
# gamma = pi / (sin(pi*x) * gamma(1-x))
wp += 15
pix = mpf_mul(x, mpf_pi(wp), wp)
t = mpf_sin_pi(x, wp)
g = mpf_gamma(mpf_sub(fone, x), wp)
return mpf_div(pix, mpf_mul(t, g, wp), prec, rounding)
sprec, a, c = get_spouge_coefficients(wp)
s = spouge_sum_real(x, sprec, a, c)
# gamma = exp(log(x+a)*(x+0.5) - xpa) * s
xpa = mpf_add(x, from_int(a), wp)
logxpa = mpf_log(xpa, wp)
xph = mpf_add(x, fhalf, wp)
t = mpf_sub(mpf_mul(logxpa, xph, wp), xpa, wp)
t = mpf_mul(mpf_exp(t, wp), s, prec, rounding)
return t
def 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
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_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
