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 cos_sin(x):
"""
cos_sin(x) calculates both the cosine and the sine of x rounded
to the nearest Float value, and returns the tuple (cos(x), sin(x)).
"""
if not isinstance(x, Float):
x = Float(x)
bits_from_unit = abs(bitcount(x.man) + x.exp)
prec = Float._prec + bits_from_unit + 15
xf = make_fixed(x, prec)
n, rx = _trig_reduce(xf, prec)
case = n % 4
one = 1 << prec
if case == 0:
s = _sin_series(rx, prec)
c = _sqrt_fixed(one - ((s * s) >> prec), prec)
elif case == 1:
c = -_sin_series(rx, prec)
s = _sqrt_fixed(one - ((c * c) >> prec), prec)
elif case == 2:
s = -_sin_series(rx, prec)
c = -_sqrt_fixed(one - ((s * s) >> prec), prec)
elif case == 3:
c = _sin_series(rx, prec)
s = -_sqrt_fixed(one - ((c * c) >> prec), prec)
return Float((c, -prec)), Float((s, -prec))
def _lower_gamma_series(are, aim, zre, zim, prec):
are = make_fixed(are, prec)
aim = make_fixed(aim, prec)
zre = make_fixed(zre, prec)
zim = make_fixed(zim, prec)
one = 1 << prec
cre = sre = one
cim = sim = 0
while abs(cre) > 3 or abs(cim) > 3:
# c = (c * z) << prec
cre, cim = (cre*zre-cim*zim)>>prec, (cim*zre+cre*zim)>>prec
# c = c / (a+k)
are += one
mag = ((are**2 + aim**2) >> prec)
cre, cim = (cre*are + cim*aim)//mag, (cim*are - cre*aim)//mag
sre += cre
sim += cim
#k += 1
sre = Float((sre, -prec))
sim = Float((sim, -prec))
return ComplexFloat(sre, sim)
def _spouge_sum(x, prec, a, c):
if isinstance(x, Float):
# Regular fixed-point summation
x = make_fixed(x, prec)
s = c[0]
for k in xrange(1, a):
s += (c[k] << prec) // (x + (k << prec))
return Float((s, -prec))
elif isinstance(x, (Rational, int, long)):
# Here we can save some work
if isinstance(x, (int, long)):
p, q = x, 1
else:
p, q = x.p, x.q
s = c[0]
for k in xrange(1, a):
s += c[k] * q // (p+q*k)
return Float((s, -prec))
elif isinstance(x, ComplexFloat):
"""
For a complex number a + b*I, we have
c_k (a+k)*c_k b * c_k
------------- = --------- - ------- * I
(a + b*I) + k M M
2 2 2 2 2
where M = (a+k) + b = (a + b ) + (2*a*k + k )
"""
re = make_fixed(x.real, prec)
im = make_fixed(x.imag, prec)
sre, sim = c[0], 0
mag = ((re**2)>>prec) + ((im**2)>>prec)
for k in xrange(1, a):
M = mag + re*(2*k) + ((k**2) << prec)
sre += (c[k] * (re + (k << prec))) // M
sim -= (c[k] * im) // M
return ComplexFloat(Float((sre, -prec)), Float((sim, -prec)))
def _atan_series_2(x):
prec = Float._prec
prec = prec + 15
x = make_fixed(x, prec)
one = 1 << prec
x2 = (x * x) >> prec
y = (x2 << prec) // (one + x2)
s = a = one
for n in xrange(1, 1000000):
a = ((a * y) >> prec) * (2 * n) // (2 * n + 1)
if a < 100:
break
s += a
return Float(((y * s) // x, -prec))
def _atan_series_1(x):
prec = Float._prec
# Increase absolute precision when extremely close to 0
bc = bitcount(x.man)
diff = -(bc + x.exp)
if diff > 10:
if 3 * diff - 4 > prec: # x**3 term vanishes; atan(x) ~x
return +x
prec = prec + diff
prec += 15 # XXX: better estimate for number of guard bits
x = make_fixed(x, prec)
x2 = (x * x) >> prec
one = 1 << prec
s = a = x
for n in xrange(1, 1000000):
a = (a * x2) >> prec
s += a // ((-1) ** n * (n + n + 1))
if -100 < a < 100:
break
return Float((s, -prec))
def exp(x):
"""
exp(x) -- compute the exponential function of the real or complex
number x
"""
if isinstance(x, (ComplexFloat, complex)):
mag = exp(x.real)
re, im = cos_sin(x.imag)
return ComplexFloat(mag * re, mag * im)
else:
if not isinstance(x, Float):
x = Float(x)
# extra precision needs to be similar in magnitude to log_2(|x|)
prec = Float._prec + 4 + max(0, bitcount(x.man) + x.exp)
t = make_fixed(x, prec)
if abs(x) > 1:
lg2 = log2_fixed(prec)
n, t = divmod(t, lg2)
else:
n = 0
y = _exp_series(t, prec)
return Float((y, -prec + n))