# Source code for sympy.core.evalf

"""
Adaptive numerical evaluation of SymPy expressions, using mpmath
for mathematical functions.
"""
from __future__ import print_function, division

import math

import sympy.mpmath.libmp as libmp
from sympy.mpmath import inf as mpmath_inf
from sympy.mpmath.libmp import (from_int, from_man_exp, from_rational, fhalf,
fnan, fnone, fone, fzero, mpf_abs, mpf_add,
mpf_atan, mpf_atan2, mpf_cmp, mpf_cos, mpf_e, mpf_exp, mpf_log, mpf_lt,
mpf_mul, mpf_neg, mpf_pi, mpf_pow, mpf_pow_int, mpf_shift, mpf_sin,
mpf_sqrt, normalize, round_nearest, to_int, to_str)
from sympy.mpmath.libmp import bitcount as mpmath_bitcount
from sympy.mpmath.libmp.backend import MPZ
from sympy.mpmath.libmp.libmpc import _infs_nan
from sympy.mpmath.libmp.libmpf import dps_to_prec
from sympy.mpmath.libmp.gammazeta import mpf_bernoulli

from .compatibility import SYMPY_INTS
from .sympify import sympify
from .core import C
from .singleton import S
from .containers import Tuple

LG10 = math.log(10, 2)
rnd = round_nearest

def bitcount(n):
return mpmath_bitcount(int(n))

# Used in a few places as placeholder values to denote exponents and
# precision levels, e.g. of exact numbers. Must be careful to avoid
# passing these to mpmath functions or returning them in final results.
INF = float(mpmath_inf)
MINUS_INF = float(-mpmath_inf)

# ~= 100 digits. Real men set this to INF.
DEFAULT_MAXPREC = 333

[docs]class PrecisionExhausted(ArithmeticError):
pass

#----------------------------------------------------------------------------#
#                                                                            #
#              Helper functions for arithmetic and complex parts             #
#                                                                            #
#----------------------------------------------------------------------------#

"""
An mpf value tuple is a tuple of integers (sign, man, exp, bc)
representing a floating-point number: [1, -1][sign]*man*2**exp where
sign is 0 or 1 and bc should correspond to the number of bits used to
represent the mantissa (man) in binary notation, e.g.

>>> from sympy.core.evalf import bitcount
>>> sign, man, exp, bc = 0, 5, 1, 3
>>> n = [1, -1][sign]*man*2**exp
>>> n, bitcount(man)
(10, 3)

A temporary result is a tuple (re, im, re_acc, im_acc) where
re and im are nonzero mpf value tuples representing approximate
numbers, or None to denote exact zeros.

re_acc, im_acc are integers denoting log2(e) where e is the estimated
relative accuracy of the respective complex part, but may be anything
if the corresponding complex part is None.

"""

def fastlog(x):
"""Fast approximation of log2(x) for an mpf value tuple x.

Notes: Calculated as exponent + width of mantissa. This is an
approximation for two reasons: 1) it gives the ceil(log2(abs(x)))
value and 2) it is too high by 1 in the case that x is an exact
power of 2. Although this is easy to remedy by testing to see if
the odd mpf mantissa is 1 (indicating that one was dealing with
an exact power of 2) that would decrease the speed and is not
necessary as this is only being used as an approximation for the
number of bits in x. The correct return value could be written as
"x[2] + (x[3] if x[1] != 1 else 0)".
Since mpf tuples always have an odd mantissa, no check is done
to see if the mantissa is a multiple of 2 (in which case the
result would be too large by 1).

Examples
========

>>> from sympy import log
>>> from sympy.core.evalf import fastlog, bitcount
>>> s, m, e = 0, 5, 1
>>> bc = bitcount(m)
>>> n = [1, -1][s]*m*2**e
>>> n, (log(n)/log(2)).evalf(2), fastlog((s, m, e, bc))
(10, 3.3, 4)
"""

if not x or x == fzero:
return MINUS_INF
return x[2] + x[3]

def pure_complex(v):
"""Return a and b if v matches a + I*b where b is not zero and
a and b are Numbers, else None.

>>> from sympy.core.evalf import pure_complex
>>> from sympy import Tuple, I
>>> a, b = Tuple(2, 3)
>>> pure_complex(a)
>>> pure_complex(a + b*I)
(2, 3)
>>> pure_complex(I)
(0, 1)
"""
c, i = t.as_coeff_Mul()
if i is S.ImaginaryUnit:
return h, c

def scaled_zero(mag, sign=1):
"""Return an mpf representing a power of two with magnitude mag
and -1 for precision. Or, if mag is a scaled_zero tuple, then just
remove the sign from within the list that it was initially wrapped
in.

Examples
========

>>> from sympy.core.evalf import scaled_zero
>>> from sympy import Float
>>> z, p = scaled_zero(100)
>>> z, p
(([0], 1, 100, 1), -1)
>>> ok = scaled_zero(z)
>>> ok
(0, 1, 100, 1)
>>> Float(ok)
1.26765060022823e+30
>>> Float(ok, p)
0.e+30
>>> ok, p = scaled_zero(100, -1)
>>> Float(scaled_zero(ok), p)
-0.e+30
"""
if type(mag) is tuple and len(mag) == 4 and iszero(mag, scaled=True):
return (mag[0][0],) + mag[1:]
elif isinstance(mag, SYMPY_INTS):
if sign not in [-1, 1]:
raise ValueError('sign must be +/-1')
rv, p = mpf_shift(fone, mag), -1
s = 0 if sign == 1 else 1
rv = ([s],) + rv[1:]
return rv, p
else:
raise ValueError('scaled zero expects int or scaled_zero tuple.')

def iszero(mpf, scaled=False):
if not scaled:
return not mpf or not mpf[1] and not mpf[-1]
return mpf and type(mpf[0]) is list and mpf[1] == mpf[-1] == 1

def complex_accuracy(result):
"""
Returns relative accuracy of a complex number with given accuracies
for the real and imaginary parts. The relative accuracy is defined
in the complex norm sense as ||z|+|error|| / |z| where error
is equal to (real absolute error) + (imag absolute error)*i.

The full expression for the (logarithmic) error can be approximated
easily by using the max norm to approximate the complex norm.

In the worst case (re and im equal), this is wrong by a factor
sqrt(2), or by log2(sqrt(2)) = 0.5 bit.
"""
re, im, re_acc, im_acc = result
if not im:
if not re:
return INF
return re_acc
if not re:
return im_acc
re_size = fastlog(re)
im_size = fastlog(im)
absolute_error = max(re_size - re_acc, im_size - im_acc)
relative_error = absolute_error - max(re_size, im_size)
return -relative_error

def get_abs(expr, prec, options):
re, im, re_acc, im_acc = evalf(expr, prec + 2, options)
if not re:
re, re_acc, im, im_acc = im, im_acc, re, re_acc
if im:
return libmp.mpc_abs((re, im), prec), None, re_acc, None
elif re:
return mpf_abs(re), None, re_acc, None
else:
return None, None, None, None

def get_complex_part(expr, no, prec, options):
"""no = 0 for real part, no = 1 for imaginary part"""
workprec = prec
i = 0
while 1:
res = evalf(expr, workprec, options)
value, accuracy = res[no::2]
# XXX is the last one correct? Consider re((1+I)**2).n()
if (not value) or accuracy >= prec or -value[2] > prec:
return value, None, accuracy, None
workprec += max(30, 2**i)
i += 1

def evalf_abs(expr, prec, options):
return get_abs(expr.args[0], prec, options)

def evalf_re(expr, prec, options):
return get_complex_part(expr.args[0], 0, prec, options)

def evalf_im(expr, prec, options):
return get_complex_part(expr.args[0], 1, prec, options)

def finalize_complex(re, im, prec):
if re == fzero and im == fzero:
raise ValueError("got complex zero with unknown accuracy")
elif re == fzero:
return None, im, None, prec
elif im == fzero:
return re, None, prec, None

size_re = fastlog(re)
size_im = fastlog(im)
if size_re > size_im:
re_acc = prec
im_acc = prec + min(-(size_re - size_im), 0)
else:
im_acc = prec
re_acc = prec + min(-(size_im - size_re), 0)
return re, im, re_acc, im_acc

def chop_parts(value, prec):
"""
Chop off tiny real or complex parts.
"""
re, im, re_acc, im_acc = value
# Method 1: chop based on absolute value
if re and re not in _infs_nan and (fastlog(re) < -prec + 4):
re, re_acc = None, None
if im and im not in _infs_nan and (fastlog(im) < -prec + 4):
im, im_acc = None, None
# Method 2: chop if inaccurate and relatively small
if re and im:
delta = fastlog(re) - fastlog(im)
if re_acc < 2 and (delta - re_acc <= -prec + 4):
re, re_acc = None, None
if im_acc < 2 and (delta - im_acc >= prec - 4):
im, im_acc = None, None
return re, im, re_acc, im_acc

def check_target(expr, result, prec):
a = complex_accuracy(result)
if a < prec:
raise PrecisionExhausted("Failed to distinguish the expression: \n\n%s\n\n"
"from zero. Try simplifying the input, using chop=True, or providing "
"a higher maxn for evalf" % (expr))

def get_integer_part(expr, no, options, return_ints=False):
"""
With no = 1, computes ceiling(expr)
With no = -1, computes floor(expr)

Note: this function either gives the exact result or signals failure.
"""

# The expression is likely less than 2^30 or so
assumed_size = 30
ire, iim, ire_acc, iim_acc = evalf(expr, assumed_size, options)

# We now know the size, so we can calculate how much extra precision
# (if any) is needed to get within the nearest integer
if ire and iim:
gap = max(fastlog(ire) - ire_acc, fastlog(iim) - iim_acc)
elif ire:
gap = fastlog(ire) - ire_acc
elif iim:
gap = fastlog(iim) - iim_acc
else:
# ... or maybe the expression was exactly zero
return None, None, None, None

margin = 10

if gap >= -margin:
ire, iim, ire_acc, iim_acc = \
evalf(expr, margin + assumed_size + gap, options)

# We can now easily find the nearest integer, but to find floor/ceil, we
# must also calculate whether the difference to the nearest integer is
# positive or negative (which may fail if very close).
def calc_part(expr, nexpr):
nint = int(to_int(nexpr, rnd))
x, _, x_acc, _ = evalf(expr, 10, options)
try:
check_target(expr, (x, None, x_acc, None), 3)
except PrecisionExhausted:
if not expr.equals(0):
raise PrecisionExhausted
x = fzero
nint += int(no*(mpf_cmp(x or fzero, fzero) == no))
nint = from_int(nint)
return nint, fastlog(nint) + 10

re, im, re_acc, im_acc = None, None, None, None

if ire:
re, re_acc = calc_part(C.re(expr, evaluate=False), ire)
if iim:
im, im_acc = calc_part(C.im(expr, evaluate=False), iim)

if return_ints:
return int(to_int(re or fzero)), int(to_int(im or fzero))
return re, im, re_acc, im_acc

def evalf_ceiling(expr, prec, options):
return get_integer_part(expr.args[0], 1, options)

def evalf_floor(expr, prec, options):
return get_integer_part(expr.args[0], -1, options)

#----------------------------------------------------------------------------#
#                                                                            #
#                            Arithmetic operations                           #
#                                                                            #
#----------------------------------------------------------------------------#

"""

Returns
-------

- None, None if there are no non-zero terms;
- terms[0] if there is only 1 term;
- scaled_zero if the sum of the terms produces a zero by cancellation
e.g. mpfs representing 1 and -1 would produce a scaled zero which need
special handling since they are not actually zero and they are purposely
malformed to ensure that they can't be used in anything but accuracy
calculations;
- a tuple that is scaled to target_prec that corresponds to the
sum of the terms.

The returned mpf tuple will be normalized to target_prec; the input
prec is used to define the working precision.

XXX explain why this is needed and why one can't just loop using mpf_add
"""
from sympy.core.core import C

terms = [t for t in terms if not iszero(t)]
if not terms:
return None, None
elif len(terms) == 1:
return terms[0]

# see if any argument is NaN or oo and thus warrants a special return
special = []
for t in terms:
arg = C.Float._new(t[0], 1)
if arg is S.NaN or arg.is_unbounded:
special.append(arg)
if special:
rv = evalf(Add(*special), prec + 4, {})
return rv[0], rv[2]

working_prec = 2*prec
sum_man, sum_exp, absolute_error = 0, 0, MINUS_INF

for x, accuracy in terms:
sign, man, exp, bc = x
if sign:
man = -man
absolute_error = max(absolute_error, bc + exp - accuracy)
delta = exp - sum_exp
if exp >= sum_exp:
# x much larger than existing sum?
# first: quick test
if ((delta > working_prec) and
((not sum_man) or
delta - bitcount(abs(sum_man)) > working_prec)):
sum_man = man
sum_exp = exp
else:
sum_man += (man << delta)
else:
delta = -delta
# x much smaller than existing sum?
if delta - bc > working_prec:
if not sum_man:
sum_man, sum_exp = man, exp
else:
sum_man = (sum_man << delta) + man
sum_exp = exp
if not sum_man:
return scaled_zero(absolute_error)
if sum_man < 0:
sum_sign = 1
sum_man = -sum_man
else:
sum_sign = 0
sum_bc = bitcount(sum_man)
sum_accuracy = sum_exp + sum_bc - absolute_error
r = normalize(sum_sign, sum_man, sum_exp, sum_bc, target_prec,
rnd), sum_accuracy
#print "returning", to_str(r[0],50), r[1]
return r

res = pure_complex(v)
if res:
h, c = res
re, _, re_acc, _ = evalf(h, prec, options)
im, _, im_acc, _ = evalf(c, prec, options)
return re, im, re_acc, im_acc

oldmaxprec = options.get('maxprec', DEFAULT_MAXPREC)

i = 0
target_prec = prec
while 1:
options['maxprec'] = min(oldmaxprec, 2*prec)

terms = [evalf(arg, prec + 10, options) for arg in v.args]
[a[0::2] for a in terms if a[0]], prec, target_prec)
[a[1::2] for a in terms if a[1]], prec, target_prec)
acc = complex_accuracy((re, im, re_acc, im_acc))
if acc >= target_prec:
if options.get('verbose'):
print("ADD: wanted", target_prec, "accurate bits, got", re_acc, im_acc)
break
else:
if (prec - target_prec) > options['maxprec']:
break

prec = prec + max(10 + 2**i, target_prec - acc)
i += 1
if options.get('verbose'):

options['maxprec'] = oldmaxprec
if iszero(re, scaled=True):
re = scaled_zero(re)
if iszero(im, scaled=True):
im = scaled_zero(im)
return re, im, re_acc, im_acc

def evalf_mul(v, prec, options):
from sympy.core.core import C

res = pure_complex(v)
if res:
# the only pure complex that is a mul is h*I
_, h = res
im, _, im_acc, _ = evalf(h, prec, options)
return None, im, None, im_acc
args = list(v.args)

# see if any argument is NaN or oo and thus warrants a special return
special = []
for arg in args:
arg = evalf(arg, prec, options)
if arg[0] is None:
continue
arg = C.Float._new(arg[0], 1)
if arg is S.NaN or arg.is_unbounded:
special.append(arg)
if special:
from sympy.core.mul import Mul
special = Mul(*special)
return evalf(special, prec + 4, {})

# With guard digits, multiplication in the real case does not destroy
# accuracy. This is also true in the complex case when considering the
# total accuracy; however accuracy for the real or imaginary parts
# separately may be lower.
acc = prec

# XXX: big overestimate
working_prec = prec + len(args) + 5

# Empty product is 1
start = man, exp, bc = MPZ(1), 0, 1

# First, we multiply all pure real or pure imaginary numbers.
# direction tells us that the result should be multiplied by
# I**direction; all other numbers get put into complex_factors
# to be multiplied out after the first phase.
last = len(args)
direction = 0
args.append(S.One)
complex_factors = []

for i, arg in enumerate(args):
if i != last and pure_complex(arg):
args[-1] = (args[-1]*arg).expand()
continue
elif i == last and arg is S.One:
continue
re, im, re_acc, im_acc = evalf(arg, working_prec, options)
if re and im:
complex_factors.append((re, im, re_acc, im_acc))
continue
elif re:
(s, m, e, b), w_acc = re, re_acc
elif im:
(s, m, e, b), w_acc = im, im_acc
direction += 1
else:
return None, None, None, None
direction += 2*s
man *= m
exp += e
bc += b
if bc > 3*working_prec:
man >>= working_prec
exp += working_prec
acc = min(acc, w_acc)
sign = (direction & 2) >> 1
if not complex_factors:
v = normalize(sign, man, exp, bitcount(man), prec, rnd)
# multiply by i
if direction & 1:
return None, v, None, acc
else:
return v, None, acc, None
else:
# initialize with the first term
if (man, exp, bc) != start:
# there was a real part; give it an imaginary part
re, im = (sign, man, exp, bitcount(man)), (0, MPZ(0), 0, 0)
i0 = 0
else:
# there is no real part to start (other than the starting 1)
wre, wim, wre_acc, wim_acc = complex_factors[0]
acc = min(acc,
complex_accuracy((wre, wim, wre_acc, wim_acc)))
re = wre
im = wim
i0 = 1

for wre, wim, wre_acc, wim_acc in complex_factors[i0:]:
# acc is the overall accuracy of the product; we aren't
# computing exact accuracies of the product.
acc = min(acc,
complex_accuracy((wre, wim, wre_acc, wim_acc)))

use_prec = working_prec
A = mpf_mul(re, wre, use_prec)
B = mpf_mul(mpf_neg(im), wim, use_prec)
C = mpf_mul(re, wim, use_prec)
D = mpf_mul(im, wre, use_prec)
if options.get('verbose'):
print("MUL: wanted", prec, "accurate bits, got", acc)
# multiply by I
if direction & 1:
re, im = mpf_neg(im), re
return re, im, acc, acc

def evalf_pow(v, prec, options):

target_prec = prec
base, exp = v.args

# We handle x**n separately. This has two purposes: 1) it is much
# faster, because we avoid calling evalf on the exponent, and 2) it
# allows better handling of real/imaginary parts that are exactly zero
if exp.is_Integer:
p = exp.p
# Exact
if not p:
return fone, None, prec, None
# Exponentiation by p magnifies relative error by |p|, so the
# base must be evaluated with increased precision if p is large
prec += int(math.log(abs(p), 2))
re, im, re_acc, im_acc = evalf(base, prec + 5, options)
# Real to integer power
if re and not im:
return mpf_pow_int(re, p, target_prec), None, target_prec, None
# (x*I)**n = I**n * x**n
if im and not re:
z = mpf_pow_int(im, p, target_prec)
case = p % 4
if case == 0:
return z, None, target_prec, None
if case == 1:
return None, z, None, target_prec
if case == 2:
return mpf_neg(z), None, target_prec, None
if case == 3:
return None, mpf_neg(z), None, target_prec
# Zero raised to an integer power
if not re:
return None, None, None, None
# General complex number to arbitrary integer power
re, im = libmp.mpc_pow_int((re, im), p, prec)
# Assumes full accuracy in input
return finalize_complex(re, im, target_prec)

# Pure square root
if exp is S.Half:
xre, xim, _, _ = evalf(base, prec + 5, options)
# General complex square root
if xim:
re, im = libmp.mpc_sqrt((xre or fzero, xim), prec)
return finalize_complex(re, im, prec)
if not xre:
return None, None, None, None
# Square root of a negative real number
if mpf_lt(xre, fzero):
return None, mpf_sqrt(mpf_neg(xre), prec), None, prec
# Positive square root
return mpf_sqrt(xre, prec), None, prec, None

# We first evaluate the exponent to find its magnitude
# This determines the working precision that must be used
prec += 10
yre, yim, _, _ = evalf(exp, prec, options)
# Special cases: x**0
if not (yre or yim):
return fone, None, prec, None

ysize = fastlog(yre)
# Restart if too big
# XXX: prec + ysize might exceed maxprec
if ysize > 5:
prec += ysize
yre, yim, _, _ = evalf(exp, prec, options)

# Pure exponential function; no need to evalf the base
if base is S.Exp1:
if yim:
re, im = libmp.mpc_exp((yre or fzero, yim), prec)
return finalize_complex(re, im, target_prec)
return mpf_exp(yre, target_prec), None, target_prec, None

xre, xim, _, _ = evalf(base, prec + 5, options)
# 0**y
if not (xre or xim):
return None, None, None, None

# (real ** complex) or (complex ** complex)
if yim:
re, im = libmp.mpc_pow(
(xre or fzero, xim or fzero), (yre or fzero, yim),
target_prec)
return finalize_complex(re, im, target_prec)
# complex ** real
if xim:
re, im = libmp.mpc_pow_mpf((xre or fzero, xim), yre, target_prec)
return finalize_complex(re, im, target_prec)
# negative ** real
elif mpf_lt(xre, fzero):
re, im = libmp.mpc_pow_mpf((xre, fzero), yre, target_prec)
return finalize_complex(re, im, target_prec)
# positive ** real
else:
return mpf_pow(xre, yre, target_prec), None, target_prec, None

#----------------------------------------------------------------------------#
#                                                                            #
#                            Special functions                               #
#                                                                            #
#----------------------------------------------------------------------------#
def evalf_trig(v, prec, options):
"""
This function handles sin and cos of complex arguments.

TODO: should also handle tan of complex arguments.
"""
if v.func is C.cos:
func = mpf_cos
elif v.func is C.sin:
func = mpf_sin
else:
raise NotImplementedError
arg = v.args[0]
# 20 extra bits is possibly overkill. It does make the need
# to restart very unlikely
xprec = prec + 20
re, im, re_acc, im_acc = evalf(arg, xprec, options)
if im:
if 'subs' in options:
v = v.subs(options['subs'])
return evalf(v._eval_evalf(prec), prec, options)
if not re:
if v.func is C.cos:
return fone, None, prec, None
elif v.func is C.sin:
return None, None, None, None
else:
raise NotImplementedError
# For trigonometric functions, we are interested in the
# fixed-point (absolute) accuracy of the argument.
xsize = fastlog(re)
# Magnitude <= 1.0. OK to compute directly, because there is no
# danger of hitting the first root of cos (with sin, magnitude
# <= 2.0 would actually be ok)
if xsize < 1:
return func(re, prec, rnd), None, prec, None
# Very large
if xsize >= 10:
xprec = prec + xsize
re, im, re_acc, im_acc = evalf(arg, xprec, options)
# Need to repeat in case the argument is very close to a
# multiple of pi (or pi/2), hitting close to a root
while 1:
y = func(re, prec, rnd)
ysize = fastlog(y)
gap = -ysize
accuracy = (xprec - xsize) - gap
if accuracy < prec:
if options.get('verbose'):
print("SIN/COS", accuracy, "wanted", prec, "gap", gap)
print(to_str(y, 10))
if xprec > options.get('maxprec', DEFAULT_MAXPREC):
return y, None, accuracy, None
xprec += gap
re, im, re_acc, im_acc = evalf(arg, xprec, options)
continue
else:
return y, None, prec, None

def evalf_log(expr, prec, options):
arg = expr.args[0]
workprec = prec + 10
xre, xim, xacc, _ = evalf(arg, workprec, options)

if xim:
# XXX: use get_abs etc instead
re = evalf_log(
C.log(C.Abs(arg, evaluate=False), evaluate=False), prec, options)
im = mpf_atan2(xim, xre or fzero, prec)
return re[0], im, re[2], prec

imaginary_term = (mpf_cmp(xre, fzero) < 0)

re = mpf_log(mpf_abs(xre), prec, rnd)
size = fastlog(re)
if prec - size > workprec:
# We actually need to compute 1+x accurately, not x
xre, xim, _, _ = evalf_add(arg, prec, options)
prec2 = workprec - fastlog(xre)
re = mpf_log(mpf_add(xre, fone, prec2), prec, rnd)

re_acc = prec

if imaginary_term:
return re, mpf_pi(prec), re_acc, prec
else:
return re, None, re_acc, None

def evalf_atan(v, prec, options):
arg = v.args[0]
xre, xim, reacc, imacc = evalf(arg, prec + 5, options)
if xre is xim is None:
return (None,)*4
if xim:
raise NotImplementedError
return mpf_atan(xre, prec, rnd), None, prec, None

def evalf_subs(prec, subs):
""" Change all Float entries in subs to have precision prec. """
newsubs = {}
for a, b in subs.items():
b = S(b)
if b.is_Float:
b = b._eval_evalf(prec)
newsubs[a] = b
return newsubs

def evalf_piecewise(expr, prec, options):
if 'subs' in options:
expr = expr.subs(evalf_subs(prec, options['subs']))
newopts = options.copy()
del newopts['subs']
if hasattr(expr, 'func'):
return evalf(expr, prec, newopts)
if type(expr) == float:
return evalf(C.Float(expr), prec, newopts)
if type(expr) == int:
return evalf(C.Integer(expr), prec, newopts)

# We still have undefined symbols
raise NotImplementedError

def evalf_bernoulli(expr, prec, options):
arg = expr.args[0]
if not arg.is_Integer:
raise ValueError("Bernoulli number index must be an integer")
n = int(arg)
b = mpf_bernoulli(n, prec, rnd)
if b == fzero:
return None, None, None, None
return b, None, prec, None

#----------------------------------------------------------------------------#
#                                                                            #
#                            High-level operations                           #
#                                                                            #
#----------------------------------------------------------------------------#

def as_mpmath(x, prec, options):
x = sympify(x)
if isinstance(x, C.Zero):
return mpf(0)
if isinstance(x, C.Infinity):
return mpf('inf')
if isinstance(x, C.NegativeInfinity):
return mpf('-inf')
# XXX
re, im, _, _ = evalf(x, prec, options)
if im:
return mpc(re or fzero, im)
return mpf(re)

def do_integral(expr, prec, options):
func = expr.args[0]
x, xlow, xhigh = expr.args[1]
orig = mp.prec

oldmaxprec = options.get('maxprec', DEFAULT_MAXPREC)
options['maxprec'] = min(oldmaxprec, 2*prec)

try:
mp.prec = prec + 5
xlow = as_mpmath(xlow, prec + 15, options)
xhigh = as_mpmath(xhigh, prec + 15, options)

# Integration is like summation, and we can phone home from
# the integrand function to update accuracy summation style
# Note that this accuracy is inaccurate, since it fails
# to account for the variable quadrature weights,
# but it is better than nothing

have_part = [False, False]
max_real_term = [MINUS_INF]
max_imag_term = [MINUS_INF]

def f(t):
re, im, re_acc, im_acc = evalf(func, mp.prec, {'subs': {x: t}})

have_part[0] = re or have_part[0]
have_part[1] = im or have_part[1]

max_real_term[0] = max(max_real_term[0], fastlog(re))
max_imag_term[0] = max(max_imag_term[0], fastlog(im))

if im:
return mpc(re or fzero, im)
return mpf(re or fzero)

A = C.Wild('A', exclude=[x])
B = C.Wild('B', exclude=[x])
D = C.Wild('D')
m = func.match(C.cos(A*x + B)*D)
if not m:
m = func.match(C.sin(A*x + B)*D)
if not m:
raise ValueError("An integrand of the form sin(A*x+B)*f(x) "
"or cos(A*x+B)*f(x) is required for oscillatory quadrature")
period = as_mpmath(2*S.Pi/m[A], prec + 15, options)
result = quadosc(f, [xlow, xhigh], period=period)
# XXX: quadosc does not do error detection yet
else:

finally:
options['maxprec'] = oldmaxprec
mp.prec = orig

if have_part[0]:
re = result.real._mpf_
if re == fzero:
re, re_acc = scaled_zero(
re = scaled_zero(re)  # handled ok in evalf_integral
else:
re_acc = -max(max_real_term[0] - fastlog(re) -
else:
re, re_acc = None, None

if have_part[1]:
im = result.imag._mpf_
if im == fzero:
im, im_acc = scaled_zero(
im = scaled_zero(im)  # handled ok in evalf_integral
else:
im_acc = -max(max_imag_term[0] - fastlog(im) -
else:
im, im_acc = None, None

result = re, im, re_acc, im_acc
return result

def evalf_integral(expr, prec, options):
limits = expr.limits
if len(limits) != 1 or len(limits[0]) != 3:
raise NotImplementedError
workprec = prec
i = 0
maxprec = options.get('maxprec', INF)
while 1:
result = do_integral(expr, workprec, options)
# if a scaled_zero comes back accuracy will compute to -1
# which will cause workprec to increment by 1
accuracy = complex_accuracy(result)
if accuracy >= prec or workprec >= maxprec:
return result
workprec += prec - max(-2**i, accuracy)
i += 1

def check_convergence(numer, denom, n):
"""
Returns (h, g, p) where
-- h is:
> 0 for convergence of rate 1/factorial(n)**h
< 0 for divergence of rate factorial(n)**(-h)
= 0 for geometric or polynomial convergence or divergence

-- abs(g) is:
> 1 for geometric convergence of rate 1/h**n
< 1 for geometric divergence of rate h**n
= 1 for polynomial convergence or divergence

(g < 0 indicates an alternating series)

-- p is:
> 1 for polynomial convergence of rate 1/n**h
<= 1 for polynomial divergence of rate n**(-h)

"""
npol = C.Poly(numer, n)
dpol = C.Poly(denom, n)
p = npol.degree()
q = dpol.degree()
rate = q - p
if rate:
return rate, None, None
constant = dpol.LC() / npol.LC()
if abs(constant) != 1:
return rate, constant, None
if npol.degree() == dpol.degree() == 0:
return rate, constant, 0
pc = npol.all_coeffs()[1]
qc = dpol.all_coeffs()[1]
return rate, constant, (qc - pc)/dpol.LC()

def hypsum(expr, n, start, prec):
"""
Sum a rapidly convergent infinite hypergeometric series with
given general term, e.g. e = hypsum(1/factorial(n), n). The
quotient between successive terms must be a quotient of integer
polynomials.
"""
from sympy import hypersimp, lambdify

if start:
expr = expr.subs(n, n + start)
hs = hypersimp(expr, n)
if hs is None:
raise NotImplementedError("a hypergeometric series is required")
num, den = hs.as_numer_denom()

func1 = lambdify(n, num)
func2 = lambdify(n, den)

h, g, p = check_convergence(num, den, n)

if h < 0:
raise ValueError("Sum diverges like (n!)^%i" % (-h))

# Direct summation if geometric or faster
if h > 0 or (h == 0 and abs(g) > 1):
term = expr.subs(n, 0)
term = (MPZ(term.p) << prec) // term.q
s = term
k = 1
while abs(term) > 5:
term *= MPZ(func1(k - 1))
term //= MPZ(func2(k - 1))
s += term
k += 1
return from_man_exp(s, -prec)
else:
alt = g < 0
if abs(g) < 1:
raise ValueError("Sum diverges like (%i)^n" % abs(1/g))
if p < 1 or (p == 1 and not alt):
raise ValueError("Sum diverges like n^%i" % (-p))
# We have polynomial convergence: use Richardson extrapolation
# Need to use at least quad precision because a lot of cancellation
# might occur in the extrapolation process
prec2 = 4*prec
term = expr.subs(n, 0)
term = (MPZ(term.p) << prec2) // term.q

def summand(k, _term=[term]):
if k:
k = int(k)
_term[0] *= MPZ(func1(k - 1))
_term[0] //= MPZ(func2(k - 1))
return make_mpf(from_man_exp(_term[0], -prec2))

orig = mp.prec
try:
mp.prec = prec
v = nsum(summand, [0, mpmath_inf], method='richardson')
finally:
mp.prec = orig
return v._mpf_

def evalf_sum(expr, prec, options):
if 'subs' in options:
expr = expr.subs(options['subs'])
func = expr.function
limits = expr.limits
if len(limits) != 1 or len(limits[0]) != 3:
raise NotImplementedError
if func is S.Zero:
return mpf(0), None, None, None
prec2 = prec + 10
try:
n, a, b = limits[0]
if b != S.Infinity or a != int(a):
raise NotImplementedError
# Use fast hypergeometric summation if possible
v = hypsum(func, n, int(a), prec2)
delta = prec - fastlog(v)
if fastlog(v) < -10:
v = hypsum(func, n, int(a), delta)
return v, None, min(prec, delta), None
except NotImplementedError:
# Euler-Maclaurin summation for general series
eps = C.Float(2.0)**(-prec)
for i in range(1, 5):
m = n = 2**i * prec
s, err = expr.euler_maclaurin(m=m, n=n, eps=eps,
eval_integral=False)
err = err.evalf()
if err <= eps:
break
err = fastlog(evalf(abs(err), 20, options)[0])
re, im, re_acc, im_acc = evalf(s, prec2, options)
if re_acc is None:
re_acc = -err
if im_acc is None:
im_acc = -err
return re, im, re_acc, im_acc

#----------------------------------------------------------------------------#
#                                                                            #
#                            Symbolic interface                              #
#                                                                            #
#----------------------------------------------------------------------------#

def evalf_symbol(x, prec, options):
val = options['subs'][x]
if isinstance(val, mpf):
if not val:
return None, None, None, None
return val._mpf_, None, prec, None
else:
if not '_cache' in options:
options['_cache'] = {}
cache = options['_cache']
cached, cached_prec = cache.get(x.name, (None, MINUS_INF))
if cached_prec >= prec:
return cached
v = evalf(sympify(val), prec, options)
cache[x.name] = (v, prec)
return v

evalf_table = None

def _create_evalf_table():
global evalf_table
evalf_table = {
C.Symbol: evalf_symbol,
C.Dummy: evalf_symbol,
C.Float: lambda x, prec, options: (x._mpf_, None, prec, None),
C.Rational: lambda x, prec, options: (from_rational(x.p, x.q, prec), None, prec, None),
C.Integer: lambda x, prec, options: (from_int(x.p, prec), None, prec, None),
C.Zero: lambda x, prec, options: (None, None, prec, None),
C.One: lambda x, prec, options: (fone, None, prec, None),
C.Half: lambda x, prec, options: (fhalf, None, prec, None),
C.Pi: lambda x, prec, options: (mpf_pi(prec), None, prec, None),
C.Exp1: lambda x, prec, options: (mpf_e(prec), None, prec, None),
C.ImaginaryUnit: lambda x, prec, options: (None, fone, None, prec),
C.NegativeOne: lambda x, prec, options: (fnone, None, prec, None),
C.NaN : lambda x, prec, options: (fnan, None, prec, None),

C.exp: lambda x, prec, options: evalf_pow(C.Pow(S.Exp1, x.args[0],
evaluate=False), prec, options),

C.cos: evalf_trig,
C.sin: evalf_trig,

C.Mul: evalf_mul,
C.Pow: evalf_pow,

C.log: evalf_log,
C.atan: evalf_atan,
C.Abs: evalf_abs,

C.re: evalf_re,
C.im: evalf_im,
C.floor: evalf_floor,
C.ceiling: evalf_ceiling,

C.Integral: evalf_integral,
C.Sum: evalf_sum,
C.Piecewise: evalf_piecewise,

C.bernoulli: evalf_bernoulli,
}

def evalf(x, prec, options):
from sympy import re as re_, im as im_
try:
rf = evalf_table[x.func]
r = rf(x, prec, options)
except KeyError:
try:
# Fall back to ordinary evalf if possible
if 'subs' in options:
x = x.subs(evalf_subs(prec, options['subs']))
re, im = x._eval_evalf(prec).as_real_imag()
if re.has(re_) or im.has(im_):
raise NotImplementedError
if re == 0:
re = None
reprec = None
else:
re = re._to_mpmath(prec, allow_ints=False)._mpf_
reprec = prec
if im == 0:
im = None
imprec = None
else:
im = im._to_mpmath(prec, allow_ints=False)._mpf_
imprec = prec
r = re, im, reprec, imprec
except AttributeError:
raise NotImplementedError
if options.get("verbose"):
print("### input", x)
print("### output", to_str(r[0] or fzero, 50))
print("### raw", r ) # r[0], r[2]
print()
chop = options.get('chop', False)
if chop:
if chop is True:
chop_prec = prec
else:
# convert (approximately) from given tolerance;
# the formula here will will make 1e-i rounds to 0 for
# i in the range +/-27 while 2e-i will not be chopped
chop_prec = int(round(-3.321*math.log10(chop) + 2.5))
if chop_prec == 3:
chop_prec -= 1
r = chop_parts(r, chop_prec)
if options.get("strict"):
check_target(x, r, prec)
return r

class EvalfMixin(object):

__slots__ = []

def evalf(self, n=15, subs=None, maxn=100, chop=False, strict=False, quad=None, verbose=False):
"""
Evaluate the given formula to an accuracy of n digits.
Optional keyword arguments:

subs=<dict>
Substitute numerical values for symbols, e.g.
subs={x:3, y:1+pi}.

maxn=<integer>
Allow a maximum temporary working precision of maxn digits
(default=100)

chop=<bool>
Replace tiny real or imaginary parts in subresults
by exact zeros (default=False)

strict=<bool>
Raise PrecisionExhausted if any subresult fails to evaluate
to full accuracy, given the available maxprec
(default=False)

Choose algorithm for numerical quadrature. By default,
tanh-sinh quadrature is used. For oscillatory
integrals on an infinite interval, try quad='osc'.

verbose=<bool>
Print debug information (default=False)

"""
# for sake of sage that doesn't like evalf(1)
if n == 1 and isinstance(self, C.Number):
from sympy.core.expr import _mag
rv = self.evalf(2, subs, maxn, chop, strict, quad, verbose)
m = _mag(rv)
rv = rv.round(1 - m)
return rv

if not evalf_table:
_create_evalf_table()
prec = dps_to_prec(n)
options = {'maxprec': max(prec, int(maxn*LG10)), 'chop': chop,
'strict': strict, 'verbose': verbose}
if subs is not None:
options['subs'] = subs
try:
result = evalf(self, prec + 4, options)
except NotImplementedError:
# Fall back to the ordinary evalf
v = self._eval_evalf(prec)
if v is None:
return self
try:
# If the result is numerical, normalize it
result = evalf(v, prec, options)
except NotImplementedError:
# Probably contains symbols or unknown functions
return v
re, im, re_acc, im_acc = result
if re:
p = max(min(prec, re_acc), 1)
#re = mpf_pos(re, p, rnd)
re = C.Float._new(re, p)
else:
re = S.Zero
if im:
p = max(min(prec, im_acc), 1)
#im = mpf_pos(im, p, rnd)
im = C.Float._new(im, p)
return re + im*S.ImaginaryUnit
else:
return re

n = evalf

def _evalf(self, prec):
"""Helper for evalf. Does the same thing but takes binary precision"""
r = self._eval_evalf(prec)
if r is None:
r = self
return r

def _eval_evalf(self, prec):
return

def _to_mpmath(self, prec, allow_ints=True):
# mpmath functions accept ints as input
errmsg = "cannot convert to mpmath number"
if allow_ints and self.is_Integer:
return self.p
if hasattr(self, '_as_mpf_val'):
return make_mpf(self._as_mpf_val(prec))
try:
re, im, _, _ = evalf(self, prec, {})
if im:
if not re:
re = fzero
return make_mpc((re, im))
elif re:
return make_mpf(re)
else:
return make_mpf(fzero)
except NotImplementedError:
v = self._eval_evalf(prec)
if v is None:
raise ValueError(errmsg)
if v.is_Float:
return make_mpf(v._mpf_)
# Number + Number*I is also fine
re, im = v.as_real_imag()
if allow_ints and re.is_Integer:
re = from_int(re.p)
elif re.is_Float:
re = re._mpf_
else:
raise ValueError(errmsg)
if allow_ints and im.is_Integer:
im = from_int(im.p)
elif im.is_Float:
im = im._mpf_
else:
raise ValueError(errmsg)
return make_mpc((re, im))

[docs]def N(x, n=15, **options):
"""
Calls x.evalf(n, \*\*options).

Both .n() and N() are equivalent to .evalf(); use the one that you like better.

Examples
========

>>> from sympy import Sum, oo, N
>>> from sympy.abc import k
>>> Sum(1/k**k, (k, 1, oo))
Sum(k**(-k), (k, 1, oo))
>>> N(_, 4)
1.291

"""
return sympify(x).evalf(n, **options)