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 try_convert_mpf_value(x, prec, rounding):
if isinstance(x, float): return from_float(x)
if hasattr(x, '_mpf_'): return x._mpf_
if hasattr(x, '_mpmath_'):
t = mpmathify(x._mpmath_(prec, rounding))
if isinstance(t, mpf):
return t._mpf_
return NotImplemented
def try_convert_mpf_value(x, prec, rounding):
if isinstance(x, float): return from_float(x)
if hasattr(x, '_mpf_'): return x._mpf_
if hasattr(x, '_mpmath_'):
t = convert_lossless(x._mpmath_(prec, rounding))
if isinstance(t, mpf):
return t._mpf_
return NotImplemented
def mpf_convert_rhs(x):
if isinstance(x, int_types): return from_int(x)
if isinstance(x, float): return from_float(x)
if isinstance(x, complex_types): return mpc(x)
if hasattr(x, '_mpf_'): return x._mpf_
if hasattr(x, '_mpmath_'):
t = mpmathify(x._mpmath_(*prec_rounding))
if isinstance(t, mpf):
return t._mpf_
return t
return NotImplemented
def mpf_convert_arg(x, prec, rounding):
if isinstance(x, int_types): return from_int(x)
if isinstance(x, float): return from_float(x)
if isinstance(x, basestring): return from_str(x, prec, rounding)
if isinstance(x, constant): return x.func(prec, rounding)
if hasattr(x, '_mpf_'): return x._mpf_
if hasattr(x, '_mpmath_'):
t = mpmathify(x._mpmath_(prec, rounding))
if isinstance(t, mpf):
return t._mpf_
raise TypeError("cannot create mpf from " + repr(x))
def mpf_convert_rhs(x):
if isinstance(x, int_types): return from_int(x)
if isinstance(x, float): return from_float(x)
if isinstance(x, complex_types): return mpc(x)
if hasattr(x, '_mpf_'): return x._mpf_
if hasattr(x, '_mpmath_'):
t = convert_lossless(x._mpmath_(*prec_rounding))
if isinstance(t, mpf):
return t._mpf_
return t
return NotImplemented
def mpf_convert_arg(x, prec, rounding):
if isinstance(x, int_types): return from_int(x)
if isinstance(x, float): return from_float(x)
if isinstance(x, basestring): return from_str(x, prec, rounding)
if isinstance(x, constant): return x.func(prec, rounding)
if hasattr(x, '_mpf_'): return x._mpf_
if hasattr(x, '_mpmath_'):
t = convert_lossless(x._mpmath_(prec, rounding))
if isinstance(t, mpf):
return t._mpf_
raise TypeError("cannot create mpf from " + repr(x))
def convert_lossless(x, strings=True):
"""Attempt to convert x to an mpf or mpc losslessly. If x is an
mpf or mpc, return it unchanged. If x is an int, create an mpf with
sufficient precision to represent it exactly. If x is a str, just
convert it to an mpf with the current working precision (perhaps
this should be done differently...)"""
if isinstance(x, mpnumeric): return x
if isinstance(x, int_types): return make_mpf(from_int(x))
if isinstance(x, float): return make_mpf(from_float(x))
if isinstance(x, complex): return mpc(x)
if strings and isinstance(x, basestring): return make_mpf(from_str(x, *prec_rounding))
if hasattr(x, '_mpf_'): return make_mpf(x._mpf_)
if hasattr(x, '_mpc_'): return make_mpc(x._mpc_)
if hasattr(x, '_mpmath_'): return convert_lossless(x._mpmath_(*prec_rounding))
raise TypeError("cannot create mpf from " + repr(x))
def mpmathify(x, strings=True):
"""
Converts *x* to an ``mpf`` or ``mpc``. If *x* is of type ``mpf``,
``mpc``, ``int``, ``float``, ``complex``, the conversion
will be performed losslessly.
If *x* is a string, the result will be rounded to the present
working precision. Strings representing fractions or complex
numbers are permitted.
>>> from sympy.mpmath import *
>>> mp.dps = 15
>>> mpmathify(3.5)
mpf('3.5')
>>> mpmathify('2.1')
mpf('2.1000000000000001')
>>> mpmathify('3/4')
mpf('0.75')
>>> mpmathify('2+3j')
mpc(real='2.0', imag='3.0')
"""
if isinstance(x, mpnumeric): return x
if isinstance(x, int_types): return make_mpf(from_int(x))
if isinstance(x, float): return make_mpf(from_float(x))
if isinstance(x, complex): return mpc(x)
if strings and isinstance(x, basestring):
try:
return make_mpf(from_str(x, *prec_rounding))
except Exception, e:
if '/' in x:
fract = x.split('/')
assert len(fract) == 2
return mpmathify(fract[0]) / mpmathify(fract[1])
if 'j' in x.lower():
x = x.lower().replace(' ', '')
match = get_complex.match(x)
re = match.group('re')
if not re:
re = 0
im = match.group('im').rstrip('j')
return mpc(mpmathify(re),
mpmathify(im))
raise e
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
beta_crossover = from_float(0.6417)
alpha_crossover = from_float(1.5)
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
# TODO: avoid loss of accuracy
def mpc_atan((a, b), prec, rnd=round_fast):
# 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)
return mpf_neg(mpf_shift(b, -1)), mpf_shift(a, -1)
beta_crossover = from_float(0.6417)
alpha_crossover = from_float(1.5)
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
if n == 1:
return mpci_pos(x, prec)
if n == 2:
return mpci_square(x, prec)
wp = prec + 20
result = (mpi_one, mpi_zero)
while n:
if n & 1:
result = mpci_mul(result, x, wp)
n -= 1
x = mpci_square(x, wp)
n >>= 1
return mpci_pos(result, prec)
gamma_min_a = from_float(1.46163214496)
gamma_min_b = from_float(1.46163214497)
gamma_min = (gamma_min_a, gamma_min_b)
gamma_mono_imag_a = from_float(-1.1)
gamma_mono_imag_b = from_float(1.1)
def mpi_overlap(x, y):
a, b = x
c, d = y
if mpf_lt(d, a): return False
if mpf_gt(c, b): return False
return True
# type = 0 -- gamma
return mpi_one, mpi_zero
if n == 1:
return mpci_pos(x, prec)
if n == 2:
return mpci_square(x, prec)
wp = prec + 20
result = (mpi_one, mpi_zero)
while n:
if n & 1:
result = mpci_mul(result, x, wp)
n -= 1
x = mpci_square(x, wp)
n >>= 1
return mpci_pos(result, prec)
gamma_min_a = from_float(1.46163214496)
gamma_min_b = from_float(1.46163214497)
gamma_min = (gamma_min_a, gamma_min_b)
gamma_mono_imag_a = from_float(-1.1)
gamma_mono_imag_b = from_float(1.1)
def mpi_overlap(x, y):
a, b = x
c, d = y
if mpf_lt(d, a): return False
if mpf_gt(c, b): return False
return True
# type = 0 -- gamma
# type = 1 -- factorial
# type = 2 -- 1/gamma