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