def glaisher_fixed(prec):
wp = prec + 30
# Number of direct terms to sum before applying the Euler-Maclaurin
# formula to the tail. TODO: choose more intelligently
N = int(0.33*prec + 5)
ONE = MPZ_ONE << wp
# Euler-Maclaurin, step 1: sum log(k)/k**2 for k from 2 to N-1
s = MPZ_ZERO
for k in range(2, N):
#print k, N
s += log_int_fixed(k, wp) // k**2
logN = log_int_fixed(N, wp)
#logN = to_fixed(mpf_log(from_int(N), wp+20), wp)
# E-M step 2: integral of log(x)/x**2 from N to inf
s += (ONE + logN) // N
# E-M step 3: endpoint correction term f(N)/2
s += logN // (N**2 * 2)
# E-M step 4: the series of derivatives
pN = N**3
a = 1
b = -2
j = 3
fac = from_int(2)
k = 1
while 1:
# D(2*k-1) * B(2*k) / fac(2*k) [D(n) = nth derivative]
D = ((a << wp) + b*logN) // pN
D = from_man_exp(D, -wp)
B = mpf_bernoulli(2*k, wp)
term = mpf_mul(B, D, wp)
term = mpf_div(term, fac, wp)
term = to_fixed(term, wp)
if abs(term) < 100:
break
#if not k % 10:
# print k, math.log(int(abs(term)), 10)
s -= term
# Advance derivative twice
a, b, pN, j = b-a*j, -j*b, pN*N, j+1
a, b, pN, j = b-a*j, -j*b, pN*N, j+1
k += 1
fac = mpf_mul_int(fac, (2*k)*(2*k-1), wp)
# A = exp((6*s/pi**2 + log(2*pi) + euler)/12)
pi = pi_fixed(wp)
s *= 6
s = (s << wp) // (pi**2 >> wp)
s += euler_fixed(wp)
s += to_fixed(mpf_log(from_man_exp(2*pi, -wp), wp), wp)
s //= 12
A = mpf_exp(from_man_exp(s, -wp), wp)
return to_fixed(A, prec)
def glaisher_fixed(prec):
wp = prec + 30
# Number of direct terms to sum before applying the Euler-Maclaurin
# formula to the tail. TODO: choose more intelligently
N = int(0.33 * prec + 5)
ONE = MP_ONE << wp
# Euler-Maclaurin, step 1: sum log(k)/k**2 for k from 2 to N-1
s = MP_ZERO
for k in range(2, N):
#print k, N
s += log_int_fixed(k, wp) // k**2
logN = log_int_fixed(N, wp)
#logN = to_fixed(mpf_log(from_int(N), wp+20), wp)
# E-M step 2: integral of log(x)/x**2 from N to inf
s += (ONE + logN) // N
# E-M step 3: endpoint correction term f(N)/2
s += logN // (N**2 * 2)
# E-M step 4: the series of derivatives
pN = N**3
a = 1
b = -2
j = 3
fac = from_int(2)
k = 1
while 1:
# D(2*k-1) * B(2*k) / fac(2*k) [D(n) = nth derivative]
D = ((a << wp) + b * logN) // pN
D = from_man_exp(D, -wp)
B = mpf_bernoulli(2 * k, wp)
term = mpf_mul(B, D, wp)
term = mpf_div(term, fac, wp)
term = to_fixed(term, wp)
if abs(term) < 100:
break
#if not k % 10:
# print k, math.log(int(abs(term)), 10)
s -= term
# Advance derivative twice
a, b, pN, j = b - a * j, -j * b, pN * N, j + 1
a, b, pN, j = b - a * j, -j * b, pN * N, j + 1
k += 1
fac = mpf_mul_int(fac, (2 * k) * (2 * k - 1), wp)
# A = exp((6*s/pi**2 + log(2*pi) + euler)/12)
pi = pi_fixed(wp)
s *= 6
s = (s << wp) // (pi**2 >> wp)
s += euler_fixed(wp)
s += to_fixed(mpf_log(from_man_exp(2 * pi, -wp), wp), wp)
s //= 12
A = mpf_exp(from_man_exp(s, -wp), wp)
return to_fixed(A, prec)
def mpf_erfc(x, prec, rnd=round_fast):
sign, man, exp, bc = x
if not man:
if x == fzero: return fone
if x == finf: return fzero
if x == fninf: return ftwo
return fnan
wp = prec + 20
mag = bc + exp
# Preserve full accuracy when exponent grows huge
wp += max(0, 2 * mag)
regular_erf = sign or mag < 2
if regular_erf or not erfc_check_series(x, wp):
if regular_erf:
return mpf_sub(fone, mpf_erf(x, prec + 10, negative_rnd[rnd]),
prec, rnd)
# 1-erf(x) ~ exp(-x^2), increase prec to deal with cancellation
n = to_int(x)
return mpf_sub(fone, mpf_erf(x, prec + int(n**2 * 1.44) + 10), prec,
rnd)
s = term = MP_ONE << wp
term_prev = 0
t = (2 * to_fixed(x, wp)**2) >> wp
k = 1
while 1:
term = ((term * (2 * k - 1)) << wp) // t
if k > 4 and term > term_prev or not term:
break
if k & 1:
s -= term
else:
s += term
term_prev = term
#print k, to_str(from_man_exp(term, -wp, 50), 10)
k += 1
s = (s << wp) // sqrt_fixed(pi_fixed(wp), wp)
s = from_man_exp(s, -wp, wp)
z = mpf_exp(mpf_neg(mpf_mul(x, x, wp), wp), wp)
y = mpf_div(mpf_mul(z, s, wp), x, prec, rnd)
return y
def mpf_erf(x, prec, rnd=round_fast):
sign, man, exp, bc = x
if not man:
if x == fzero: return fzero
if x == finf: return fone
if x== fninf: return fnone
return fnan
size = exp + bc
lg = math.log
# The approximation erf(x) = 1 is accurate to > x^2 * log(e,2) bits
if size > 3 and 2*(size-1) + 0.528766 > lg(prec,2):
if sign:
return mpf_perturb(fnone, 0, prec, rnd)
else:
return mpf_perturb(fone, 1, prec, rnd)
# erf(x) ~ 2*x/sqrt(pi) close to 0
if size < -prec:
# 2*x
x = mpf_shift(x,1)
c = mpf_sqrt(mpf_pi(prec+20), prec+20)
# TODO: interval rounding
return mpf_div(x, c, prec, rnd)
wp = prec + abs(size) + 25
# Taylor series for erf, fixed-point summation
t = abs(to_fixed(x, wp))
t2 = (t*t) >> wp
s, term, k = t, 12345, 1
while term:
t = ((t * t2) >> wp) // k
term = t // (2*k+1)
if k & 1:
s -= term
else:
s += term
k += 1
s = (s << (wp+1)) // sqrt_fixed(pi_fixed(wp), wp)
if sign:
s = -s
return from_man_exp(s, -wp, prec, rnd)
def mpf_erf(x, prec, rnd=round_fast):
sign, man, exp, bc = x
if not man:
if x == fzero: return fzero
if x == finf: return fone
if x == fninf: return fnone
return fnan
size = exp + bc
lg = math.log
# The approximation erf(x) = 1 is accurate to > x^2 * log(e,2) bits
if size > 3 and 2 * (size - 1) + 0.528766 > lg(prec, 2):
if sign:
return mpf_perturb(fnone, 0, prec, rnd)
else:
return mpf_perturb(fone, 1, prec, rnd)
# erf(x) ~ 2*x/sqrt(pi) close to 0
if size < -prec:
# 2*x
x = mpf_shift(x, 1)
c = mpf_sqrt(mpf_pi(prec + 20), prec + 20)
# TODO: interval rounding
return mpf_div(x, c, prec, rnd)
wp = prec + abs(size) + 20
# Taylor series for erf, fixed-point summation
t = abs(to_fixed(x, wp))
t2 = (t * t) >> wp
s, term, k = t, 12345, 1
while term:
t = ((t * t2) >> wp) // k
term = t // (2 * k + 1)
if k & 1:
s -= term
else:
s += term
k += 1
s = (s << (wp + 1)) // sqrt_fixed(pi_fixed(wp), wp)
if sign:
s = -s
return from_man_exp(s, -wp, wp, rnd)
def mpf_erfc(x, prec, rnd=round_fast):
sign, man, exp, bc = x
if not man:
if x == fzero: return fone
if x == finf: return fzero
if x == fninf: return ftwo
return fnan
wp = prec + 20
mag = bc+exp
# Preserve full accuracy when exponent grows huge
wp += max(0, 2*mag)
regular_erf = sign or mag < 2
if regular_erf or not erfc_check_series(x, wp):
if regular_erf:
return mpf_sub(fone, mpf_erf(x, prec+10, negative_rnd[rnd]), prec, rnd)
# 1-erf(x) ~ exp(-x^2), increase prec to deal with cancellation
n = to_int(x)+1
return mpf_sub(fone, mpf_erf(x, prec + int(n**2*1.44) + 10), prec, rnd)
s = term = MPZ_ONE << wp
term_prev = 0
t = (2 * to_fixed(x, wp) ** 2) >> wp
k = 1
while 1:
term = ((term * (2*k - 1)) << wp) // t
if k > 4 and term > term_prev or not term:
break
if k & 1:
s -= term
else:
s += term
term_prev = term
#print k, to_str(from_man_exp(term, -wp, 50), 10)
k += 1
s = (s << wp) // sqrt_fixed(pi_fixed(wp), wp)
s = from_man_exp(s, -wp, wp)
z = mpf_exp(mpf_neg(mpf_mul(x,x,wp),wp),wp)
y = mpf_div(mpf_mul(z, s, wp), x, prec, rnd)
return y
