def mpi_pow_int(s, n, prec):
sa, sb = s
if n < 0:
return mpi_div((fone, fone), mpi_pow_int(s, -n, prec + 20), prec)
if n == 0:
return (fone, fone)
if n == 1:
return s
# Odd -- signs are preserved
if n & 1:
a = mpf_pow_int(sa, n, prec, round_floor)
b = mpf_pow_int(sb, n, prec, round_ceiling)
# Even -- important to ensure positivity
else:
sas = mpf_sign(sa)
sbs = mpf_sign(sb)
# Nonnegative?
if sas >= 0:
a = mpf_pow_int(sa, n, prec, round_floor)
b = mpf_pow_int(sb, n, prec, round_ceiling)
# Nonpositive?
elif sbs <= 0:
a = mpf_pow_int(sb, n, prec, round_floor)
b = mpf_pow_int(sa, n, prec, round_ceiling)
# Mixed signs?
else:
a = fzero
# max(-a,b)**n
sa = mpf_neg(sa)
if mpf_ge(sa, sb):
b = mpf_pow_int(sa, n, prec, round_ceiling)
else:
b = mpf_pow_int(sb, n, prec, round_ceiling)
return a, b
def mpi_pow_int(s, n, prec):
sa, sb = s
if n < 0:
return mpi_div((fone, fone), mpi_pow_int(s, -n, prec+20), prec)
if n == 0:
return (fone, fone)
if n == 1:
return s
# Odd -- signs are preserved
if n & 1:
a = mpf_pow_int(sa, n, prec, round_floor)
b = mpf_pow_int(sb, n, prec, round_ceiling)
# Even -- important to ensure positivity
else:
sas = mpf_sign(sa)
sbs = mpf_sign(sb)
# Nonnegative?
if sas >= 0:
a = mpf_pow_int(sa, n, prec, round_floor)
b = mpf_pow_int(sb, n, prec, round_ceiling)
# Nonpositive?
elif sbs <= 0:
a = mpf_pow_int(sb, n, prec, round_floor)
b = mpf_pow_int(sa, n, prec, round_ceiling)
# Mixed signs?
else:
a = fzero
# max(-a,b)**n
sa = mpf_neg(sa)
if mpf_ge(sa, sb):
b = mpf_pow_int(sa, n, prec, round_ceiling)
else:
b = mpf_pow_int(sb, n, prec, round_ceiling)
return a, b
def 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 mpi_square(s, prec=0):
sa, sb = s
if mpf_ge(sa, fzero):
a = mpf_mul(sa, sa, prec, round_floor)
b = mpf_mul(sb, sb, prec, round_ceiling)
elif mpf_le(sb, fzero):
a = mpf_mul(sb, sb, prec, round_floor)
b = mpf_mul(sa, sa, prec, round_ceiling)
else:
sa = mpf_neg(sa)
sa, sb = mpf_min_max([sa, sb])
a = fzero
b = mpf_mul(sb, sb, prec, round_ceiling)
return a, b
def mpi_square(s, prec=0):
sa, sb = s
if mpf_ge(sa, fzero):
a = mpf_mul(sa, sa, prec, round_floor)
b = mpf_mul(sb, sb, prec, round_ceiling)
elif mpf_le(sb, fzero):
a = mpf_mul(sb, sb, prec, round_floor)
b = mpf_mul(sa, sa, prec, round_ceiling)
else:
sa = mpf_neg(sa)
sa, sb = mpf_min_max([sa, sb])
a = fzero
b = mpf_mul(sb, sb, prec, round_ceiling)
return a, b
def mpi_atan2(y, x, prec):
ya, yb = y
xa, xb = x
# Constrained to the real line
if ya == yb == fzero:
if mpf_ge(xa, fzero):
return mpi_zero
return mpi_pi(prec)
# Right half-plane
if mpf_ge(xa, fzero):
if mpf_ge(ya, fzero):
a = mpf_atan2(ya, xb, prec, round_floor)
else:
a = mpf_atan2(ya, xa, prec, round_floor)
if mpf_ge(yb, fzero):
b = mpf_atan2(yb, xa, prec, round_ceiling)
else:
b = mpf_atan2(yb, xb, prec, round_ceiling)
# Upper half-plane
elif mpf_ge(ya, fzero):
b = mpf_atan2(ya, xa, prec, round_ceiling)
if mpf_le(xb, fzero):
a = mpf_atan2(yb, xb, prec, round_floor)
else:
a = mpf_atan2(ya, xb, prec, round_floor)
# Lower half-plane
elif mpf_le(yb, fzero):
a = mpf_atan2(yb, xa, prec, round_floor)
if mpf_le(xb, fzero):
b = mpf_atan2(ya, xb, prec, round_ceiling)
else:
b = mpf_atan2(yb, xb, prec, round_ceiling)
# Covering the origin
else:
b = mpf_pi(prec, round_ceiling)
a = mpf_neg(b)
return a, b
def mpi_atan2(y, x, prec):
ya, yb = y
xa, xb = x
# Constrained to the real line
if ya == yb == fzero:
if mpf_ge(xa, fzero):
return mpi_zero
return mpi_pi(prec)
# Right half-plane
if mpf_ge(xa, fzero):
if mpf_ge(ya, fzero):
a = mpf_atan2(ya, xb, prec, round_floor)
else:
a = mpf_atan2(ya, xa, prec, round_floor)
if mpf_ge(yb, fzero):
b = mpf_atan2(yb, xa, prec, round_ceiling)
else:
b = mpf_atan2(yb, xb, prec, round_ceiling)
# Upper half-plane
elif mpf_ge(ya, fzero):
b = mpf_atan2(ya, xa, prec, round_ceiling)
if mpf_le(xb, fzero):
a = mpf_atan2(yb, xb, prec, round_floor)
else:
a = mpf_atan2(ya, xb, prec, round_floor)
# Lower half-plane
elif mpf_le(yb, fzero):
a = mpf_atan2(yb, xa, prec, round_floor)
if mpf_le(xb, fzero):
b = mpf_atan2(ya, xb, prec, round_ceiling)
else:
b = mpf_atan2(yb, xb, prec, round_ceiling)
# Covering the origin
else:
b = mpf_pi(prec, round_ceiling)
a = mpf_neg(b)
return a, b
def MAX(x, y):
if mpf_ge(x, y):
return x
return y
def MAX(x, y):
if mpf_ge(x, y):
return x
return y
def mpci_gamma(z, prec, type=0):
(a1, a2), (b1, b2) = z
# Real case
if b1 == b2 == fzero and (type != 3 or mpf_gt(a1, fzero)):
return mpi_gamma(z, prec, type), mpi_zero
# Estimate precision
wp = prec + 20
if type != 3:
amag = a2[2] + a2[3]
bmag = b2[2] + b2[3]
if a2 != fzero:
mag = max(amag, bmag)
else:
mag = bmag
an = abs(to_int(a2))
bn = abs(to_int(b2))
absn = max(an, bn)
gamma_size = max(0, absn * mag)
wp += bitcount(gamma_size)
# Assume type != 1
if type == 1:
(a1, a2) = mpi_add((a1, a2), mpi_one, wp)
z = (a1, a2), (b1, b2)
type = 0
# Avoid non-monotonic region near the negative real axis
if mpf_lt(a1, gamma_min_b):
if mpi_overlap((b1, b2), (gamma_mono_imag_a, gamma_mono_imag_b)):
# TODO: reflection formula
#if mpf_lt(a2, mpf_shift(fone,-1)):
# znew = mpci_sub((mpi_one,mpi_zero),z,wp)
# ...
# Recurrence:
# gamma(z) = gamma(z+1)/z
znew = mpi_add((a1, a2), mpi_one, wp), (b1, b2)
if type == 0:
return mpci_div(mpci_gamma(znew, prec + 2, 0), z, prec)
if type == 2:
return mpci_mul(mpci_gamma(znew, prec + 2, 2), z, prec)
if type == 3:
return mpci_sub(mpci_gamma(znew, prec + 2, 3),
mpci_log(z, prec + 2), prec)
# Use monotonicity (except for a small region close to the
# origin and near poles)
# upper half-plane
if mpf_ge(b1, fzero):
minre = mpc_loggamma((a1, b2), wp, round_floor)
maxre = mpc_loggamma((a2, b1), wp, round_ceiling)
minim = mpc_loggamma((a1, b1), wp, round_floor)
maxim = mpc_loggamma((a2, b2), wp, round_ceiling)
# lower half-plane
elif mpf_le(b2, fzero):
minre = mpc_loggamma((a1, b1), wp, round_floor)
maxre = mpc_loggamma((a2, b2), wp, round_ceiling)
minim = mpc_loggamma((a2, b1), wp, round_floor)
maxim = mpc_loggamma((a1, b2), wp, round_ceiling)
# crosses real axis
else:
maxre = mpc_loggamma((a2, fzero), wp, round_ceiling)
# stretches more into the lower half-plane
if mpf_gt(mpf_neg(b1), b2):
minre = mpc_loggamma((a1, b1), wp, round_ceiling)
else:
minre = mpc_loggamma((a1, b2), wp, round_ceiling)
minim = mpc_loggamma((a2, b1), wp, round_floor)
maxim = mpc_loggamma((a2, b2), wp, round_floor)
w = (minre[0], maxre[0]), (minim[1], maxim[1])
if type == 3:
return mpi_pos(w[0], prec), mpi_pos(w[1], prec)
if type == 2:
w = mpci_neg(w)
return mpci_exp(w, prec)
def mpi_lt(s, t):
sa, sb = s
ta, tb = t
if mpf_lt(sb, ta): return True
if mpf_ge(sa, tb): return False
return None
def mpi_lt(s, t):
sa, sb = s
ta, tb = t
if mpf_lt(sb, ta): return True
if mpf_ge(sa, tb): return False
return None
def mpci_gamma(z, prec, type=0):
(a1,a2), (b1,b2) = z
# Real case
if b1 == b2 == fzero and (type != 3 or mpf_gt(a1,fzero)):
return mpi_gamma(z, prec, type), mpi_zero
# Estimate precision
wp = prec+20
if type != 3:
amag = a2[2]+a2[3]
bmag = b2[2]+b2[3]
if a2 != fzero:
mag = max(amag, bmag)
else:
mag = bmag
an = abs(to_int(a2))
bn = abs(to_int(b2))
absn = max(an, bn)
gamma_size = max(0,absn*mag)
wp += bitcount(gamma_size)
# Assume type != 1
if type == 1:
(a1,a2) = mpi_add((a1,a2), mpi_one, wp); z = (a1,a2), (b1,b2)
type = 0
# Avoid non-monotonic region near the negative real axis
if mpf_lt(a1, gamma_min_b):
if mpi_overlap((b1,b2), (gamma_mono_imag_a, gamma_mono_imag_b)):
# TODO: reflection formula
#if mpf_lt(a2, mpf_shift(fone,-1)):
# znew = mpci_sub((mpi_one,mpi_zero),z,wp)
# ...
# Recurrence:
# gamma(z) = gamma(z+1)/z
znew = mpi_add((a1,a2), mpi_one, wp), (b1,b2)
if type == 0: return mpci_div(mpci_gamma(znew, prec+2, 0), z, prec)
if type == 2: return mpci_mul(mpci_gamma(znew, prec+2, 2), z, prec)
if type == 3: return mpci_sub(mpci_gamma(znew, prec+2, 3), mpci_log(z,prec+2), prec)
# Use monotonicity (except for a small region close to the
# origin and near poles)
# upper half-plane
if mpf_ge(b1, fzero):
minre = mpc_loggamma((a1,b2), wp, round_floor)
maxre = mpc_loggamma((a2,b1), wp, round_ceiling)
minim = mpc_loggamma((a1,b1), wp, round_floor)
maxim = mpc_loggamma((a2,b2), wp, round_ceiling)
# lower half-plane
elif mpf_le(b2, fzero):
minre = mpc_loggamma((a1,b1), wp, round_floor)
maxre = mpc_loggamma((a2,b2), wp, round_ceiling)
minim = mpc_loggamma((a2,b1), wp, round_floor)
maxim = mpc_loggamma((a1,b2), wp, round_ceiling)
# crosses real axis
else:
maxre = mpc_loggamma((a2,fzero), wp, round_ceiling)
# stretches more into the lower half-plane
if mpf_gt(mpf_neg(b1), b2):
minre = mpc_loggamma((a1,b1), wp, round_ceiling)
else:
minre = mpc_loggamma((a1,b2), wp, round_ceiling)
minim = mpc_loggamma((a2,b1), wp, round_floor)
maxim = mpc_loggamma((a2,b2), wp, round_floor)
w = (minre[0], maxre[0]), (minim[1], maxim[1])
if type == 3:
return mpi_pos(w[0], prec), mpi_pos(w[1], prec)
if type == 2:
w = mpci_neg(w)
return mpci_exp(w, prec)