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
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
def evalf_mul(v, prec, options):
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 = []
from sympy.core.numbers import Float
for arg in args:
arg = evalf(arg, prec, options)
if arg[0] is None:
continue
arg = Float._new(arg[0], 1)
if arg is S.NaN or arg.is_infinite:
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)
re = mpf_add(A, B, use_prec)
im = mpf_add(C, D, 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_mul(v, prec, options):
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 = []
from sympy.core.numbers import Float
for arg in args:
arg = evalf(arg, prec, options)
if arg[0] is None:
continue
arg = Float._new(arg[0], 1)
if arg is S.NaN or arg.is_infinite:
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)
re = mpf_add(A, B, use_prec)
im = mpf_add(C, D, 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