def __neg__(self):
return Real._new(mlib.mpf_neg(self._mpf_), self._prec)
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 = fpowi(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 = libmpc.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, xre_acc, yim_acc = evalf(base, prec+5, options)
# General complex square root
if xim:
re, im = libmpc.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, yre_acc, yim_acc = 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, yre_acc, yim_acc = evalf(exp, prec, options)
# Pure exponential function; no need to evalf the base
if base is S.Exp1:
if yim:
re, im = libmpc.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, xre_acc, yim_acc = 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 = libmpc.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 = libmpc.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 = libmpc.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):
args = v.args
# 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
target_prec = prec
# XXX: big overestimate
prec = prec + len(args) + 5
direction = 0
# Empty product is 1
man, exp, bc = MP_BASE(1), 0, 1
direction = 0
complex_factors = []
# First, we multiply all pure real or pure imaginary numbers.
# direction tells us that the result should be multiplied by
# i**direction
for arg in args:
re, im, a, aim = evalf(arg, prec, options)
if re and im:
complex_factors.append((re, im, a, aim))
continue
elif re:
s, m, e, b = re
elif im:
a = aim
direction += 1
s, m, e, b = im
else:
return None, None, None, None
direction += 2*s
man *= m
exp += e
bc += b
if bc > 3*prec:
man >>= prec
exp += prec
acc = min(acc, a)
sign = (direction & 2) >> 1
v = normalize(sign, man, exp, bitcount(man), prec, round_nearest)
if complex_factors:
# Multiply first complex number by the existing real scalar
re, im, re_acc, im_acc = complex_factors[0]
re = mpf_mul(re, v, prec)
im = mpf_mul(im, v, prec)
re_acc = min(re_acc, acc)
im_acc = min(im_acc, acc)
# Multiply consecutive complex factors
complex_factors = complex_factors[1:]
for wre, wim, wre_acc, wim_acc in complex_factors:
re, im, re_acc, im_acc = cmul((re, re_acc), (im,im_acc),
(wre,wre_acc), (wim,wim_acc), prec, target_prec)
if options.get('verbose'):
print "MUL: obtained accuracy", re_acc, im_acc, "expected", target_prec
# multiply by i
if direction & 1:
return mpf_neg(im), re, re_acc, im_acc
else:
return re, im, re_acc, im_acc
else:
# multiply by i
if direction & 1:
return None, v, None, acc
else:
return v, None, acc, None
return re, im, re_accuracy, im_accuracy
prec = prec + max(10+2**i, diff)
options['maxprec'] = min(oldmaxprec, 2*prec)
if options.get('verbose'):
print "ADD: restarting with prec", prec
i += 1
finally:
options['maxprec'] = oldmaxprec
# Helper for complex multiplication
# XXX: should be able to multiply directly, and use complex_accuracy
# to obtain the final accuracy
def cmul((a, aacc), (b, bacc), (c, cacc), (d, dacc), prec, target_prec):
A, Aacc = mpf_mul(a,c,prec), min(aacc, cacc)
B, Bacc = mpf_mul(mpf_neg(b),d,prec), min(bacc, dacc)
C, Cacc = mpf_mul(a,d,prec), min(aacc, dacc)
D, Dacc = mpf_mul(b,c,prec), min(bacc, cacc)
re, re_accuracy = add_terms([(A, Aacc), (B, Bacc)], prec, target_prec)
im, im_accuracy = add_terms([(C, Cacc), (D, Cacc)], prec, target_prec)
return re, im, re_accuracy, im_accuracy
def evalf_mul(v, prec, options):
args = v.args
# 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
target_prec = prec
# XXX: big overestimate
def __neg__(self):
return Real._new(mlib.mpf_neg(self._mpf_), self._prec)
def evalf_mul(v, prec, options):
args = v.args
# 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
target_prec = prec
# XXX: big overestimate
prec = prec + len(args) + 5
direction = 0
# Empty product is 1
man, exp, bc = MP_BASE(1), 0, 1
direction = 0
complex_factors = []
# First, we multiply all pure real or pure imaginary numbers.
# direction tells us that the result should be multiplied by
# i**direction
for arg in args:
re, im, re_acc, im_acc = evalf(arg, 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*prec:
man >>= prec
exp += prec
acc = min(acc, w_acc)
sign = (direction & 2) >> 1
v = normalize(sign, man, exp, bitcount(man), prec, round_nearest)
if complex_factors:
# make existing real scalar look like an imaginary and
# multiply by the remaining complex numbers
re, im = v, (0, MP_BASE(0), 0, 0)
for wre, wim, wre_acc, wim_acc in complex_factors:
# 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)))
A = mpf_mul(re, wre, prec)
B = mpf_mul(mpf_neg(im), wim, prec)
C = mpf_mul(re, wim, prec)
D = mpf_mul(im, wre, prec)
re, xre_acc = add_terms([(A, acc), (B, acc)], prec, target_prec)
im, xim_acc = add_terms([(C, acc), (D, acc)], prec, target_prec)
if options.get('verbose'):
print "MUL: wanted", target_prec, "accurate bits, got", acc
# multiply by i
if direction & 1:
return mpf_neg(im), re, acc, acc
else:
return re, im, acc, acc
else:
# multiply by i
if direction & 1:
return None, v, None, acc
else:
return v, None, acc, None
