def add_terms(terms, prec, target_prec):
"""
Helper for evalf_add. Adds a list of (mpfval, accuracy) terms.
"""
if len(terms) == 1:
if not terms[0]:
# XXX: this is supposed to represent a scaled zero
return mpf_shift(fone, target_prec), -1
return terms[0]
max_extra_prec = 2 * prec
sum_man, sum_exp, absolute_error = 0, 0, MINUS_INF
for x, accuracy in terms:
if not x:
continue
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 > max_extra_prec) and \
((not sum_man) or delta-bitcount(abs(sum_man)) > max_extra_prec):
sum_man = man
sum_exp = exp
else:
sum_man += (man << delta)
else:
delta = -delta
# x much smaller than existing sum?
if delta - bc > max_extra_prec:
if not sum_man:
sum_man, sum_exp = man, exp
else:
sum_man = (sum_man << delta) + man
sum_exp = exp
if absolute_error == MINUS_INF:
return None, None
if not sum_man:
# XXX: this is supposed to represent a scaled zero
return mpf_shift(fone, absolute_error), -1
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,
round_nearest), sum_accuracy
#print "returning", to_str(r[0],50), r[1]
return r
def add_terms(terms, prec, target_prec):
"""
Helper for evalf_add. Adds a list of (mpfval, accuracy) terms.
"""
if len(terms) == 1:
if not terms[0]:
# XXX: this is supposed to represent a scaled zero
return mpf_shift(fone, target_prec), -1
return terms[0]
max_extra_prec = 2*prec
sum_man, sum_exp, absolute_error = 0, 0, MINUS_INF
for x, accuracy in terms:
if not x:
continue
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 > max_extra_prec) and \
((not sum_man) or delta-bitcount(abs(sum_man)) > max_extra_prec):
sum_man = man
sum_exp = exp
else:
sum_man += (man << delta)
else:
delta = -delta
# x much smaller than existing sum?
if delta-bc > max_extra_prec:
if not sum_man:
sum_man, sum_exp = man, exp
else:
sum_man = (sum_man << delta) + man
sum_exp = exp
if absolute_error == MINUS_INF:
return None, None
if not sum_man:
# XXX: this is supposed to represent a scaled zero
return mpf_shift(fone, absolute_error), -1
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,
round_nearest), sum_accuracy
#print "returning", to_str(r[0],50), r[1]
return r
def npartitions(n, verbose=False):
"""
Calculate the partition function P(n), i.e. the number of ways that
n can be written as a sum of positive integers.
P(n) is computed using the Hardy-Ramanujan-Rademacher formula,
described e.g. at http://mathworld.wolfram.com/PartitionFunctionP.html
The correctness of this implementation has been tested for 10**n
up to n = 8.
"""
n = int(n)
if n < 0: return 0
if n <= 5: return [1, 1, 2, 3, 5, 7][n]
# Estimate number of bits in p(n). This formula could be tidied
pbits = int((math.pi*(2*n/3.)**0.5-math.log(4*n))/math.log(10)+1)*\
math.log(10,2)
prec = p = int(pbits*1.1 + 100)
s = fzero
M = max(6, int(0.24*n**0.5+4))
sq23pi = mpf_mul(mpf_sqrt(from_rational(2,3,p), p), mpf_pi(p), p)
sqrt8 = mpf_sqrt(from_int(8), p)
for q in xrange(1, M):
a = A(n,q,p)
d = D(n,q,p, sq23pi, sqrt8)
s = mpf_add(s, mpf_mul(a, d), prec)
if verbose:
print "step", q, "of", M, to_str(a, 10), to_str(d, 10)
# On average, the terms decrease rapidly in magnitude. Dynamically
# reducing the precision greatly improves performance.
p = bitcount(abs(to_int(d))) + 50
np = to_int(mpf_add(s, fhalf, prec))
return int(np)
def npartitions(n, verbose=False):
"""
Calculate the partition function P(n), i.e. the number of ways that
n can be written as a sum of positive integers.
P(n) is computed using the Hardy-Ramanujan-Rademacher formula,
described e.g. at http://mathworld.wolfram.com/PartitionFunctionP.html
The correctness of this implementation has been tested for 10**n
up to n = 8.
"""
n = int(n)
if n < 0: return 0
if n <= 5: return [1, 1, 2, 3, 5, 7][n]
# Estimate number of bits in p(n). This formula could be tidied
pbits = int((math.pi*(2*n/3.)**0.5-math.log(4*n))/math.log(10)+1)*\
math.log(10,2)
prec = p = int(pbits * 1.1 + 100)
s = fzero
M = max(6, int(0.24 * n**0.5 + 4))
sq23pi = mpf_mul(mpf_sqrt(from_rational(2, 3, p), p), mpf_pi(p), p)
sqrt8 = mpf_sqrt(from_int(8), p)
for q in xrange(1, M):
a = A(n, q, p)
d = D(n, q, p, sq23pi, sqrt8)
s = mpf_add(s, mpf_mul(a, d), prec)
if verbose:
print "step", q, "of", M, to_str(a, 10), to_str(d, 10)
# On average, the terms decrease rapidly in magnitude. Dynamically
# reducing the precision greatly improves performance.
p = bitcount(abs(to_int(d))) + 50
np = to_int(mpf_add(s, fhalf, prec))
return int(np)
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)
# 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:
# intialize 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 add_terms(terms, prec, target_prec):
"""
Helper for evalf_add. Adds a list of (mpfval, accuracy) terms.
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
"""
terms = [t for t in terms if not iszero(t)]
if not terms:
return None, None
elif len(terms) == 1:
return terms[0]
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
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 = MPZ(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, MPZ(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
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 = MPZ(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, MPZ(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
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)
# 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:
# intialize 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 add_terms(terms, prec, target_prec):
"""
Helper for evalf_add. Adds a list of (mpfval, accuracy) terms.
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
"""
terms = [t for t in terms if not iszero(t)]
if not terms:
return None, None
elif len(terms) == 1:
return terms[0]
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
