def _calc_spouge_coefficients(a, prec):
"""
Calculate Spouge coefficients for approximation with parameter a.
Return a list of big integers representing the coefficients in
fixed-point form with a precision of prec bits.
"""
# We'll store the coefficients as fixed-point numbers but calculate
# them as Floats for convenience. The initial terms are huge, so we
# need to allocate extra bits to ensure full accuracy. The integer
# part of the largest term has size ~= exp(a) or 2**(1.4*a)
floatprec = prec + int(a*1.4)
Float.store()
Float.setprec(floatprec)
c = [0] * a
b = exp(a-1)
e = exp(1)
c[0] = make_fixed(sqrt(2*pi_float()), prec)
for k in range(1, a):
# print "%02f" % (100.0 * k / a), "% done"
c[k] = make_fixed(((-1)**(k-1) * (a-k)**k) * b / sqrt(a-k), prec)
# Divide off e and k instead of computing exp and k! from scratch
b = b / (e * k)
Float.revert()
return c
def evalf(expr):
"""
evalf(expr) attempts to evaluate a SymPy expression to a Float or
ComplexFloat with an error smaller than 10**(-Float.getdps())
"""
if isinstance(expr, (Float, ComplexFloat)):
return expr
elif isinstance(expr, (int, float)):
return Float(expr)
elif isinstance(expr, complex):
return ComplexFloat(expr)
expr = Basic.sympify(expr)
if isinstance(expr, (Rational)):
y = Float(expr)
elif isinstance(expr, Real):
y = Float(str(expr))
elif expr is I:
y = ComplexFloat(0,1)
elif expr is pi:
y = constants.pi_float()
elif expr is E:
y = functions.exp(1)
elif isinstance(expr, Mul):
factors = expr[:]
workprec = Float.getprec() + 1 + len(factors)
Float.store()
Float.setprec(workprec)
y = Float(1)
for f in factors:
y *= evalf(f)
Float.revert()
elif isinstance(expr, Pow):
base, expt = expr[:]
workprec = Float.getprec() + 8 # may need more
Float.store()
Float.setprec(workprec)
base = evalf(base)
expt = evalf(expt)
if expt == 0.5:
y = functions.sqrt(base)
else:
y = functions.exp(functions.log(base) * expt)
Float.revert()
elif isinstance(expr, Basic.exp):
Float.store()
Float.setprec(Float.getprec() + 3)
#XXX: how is it possible, that this works:
x = evalf(expr[0])
#and this too:
#x = evalf(expr[1])
#?? (Try to uncomment it and you'll see)
y = functions.exp(x)
Float.revert()
elif isinstance(expr, Add):
# TODO: this doesn't yet work as it should.
# We need some way to handle sums whose results are
# very close to 0, and when necessary, repeat the
# summation with higher precision
reqprec = Float.getprec()
Float.store()
Float.setprec(10)
terms = expr[:]
approxterms = [abs(evalf(x)) for x in terms]
min_mag = min(x.exp for x in approxterms)
max_mag = max(x.exp+bitcount(x.man) for x in approxterms)
Float.setprec(reqprec - 10 + max_mag - min_mag + 1 + len(terms))
workprec = Float.getdps()
y = 0
for t in terms:
y += evalf(t)
Float.revert()
else:
# print expr, expr.__class__
raise NotImplementedError
# print expr, y
return +y