def mpf_atan(x, prec, rnd=round_fast):
sign, man, exp, bc = x
if not man:
if x == fzero: return fzero
if x == finf: return atan_inf(0, prec, rnd)
if x == fninf: return atan_inf(1, prec, rnd)
return fnan
mag = exp + bc
# Essentially infinity
if mag > prec+20:
return atan_inf(sign, prec, rnd)
# Essentially ~ x
if -mag > prec+20:
return mpf_perturb(x, 1-sign, prec, rnd)
wp = prec + 30 + abs(mag)
# For large x, use atan(x) = pi/2 - atan(1/x)
if mag >= 2:
x = mpf_rdiv_int(1, x, wp)
reciprocal = True
else:
reciprocal = False
t = to_fixed(x, wp)
if sign:
t = -t
if wp < ATAN_TAYLOR_PREC:
a = atan_taylor(t, wp)
else:
a = atan_newton(t, wp)
if reciprocal:
a = ((pi_fixed(wp)>>1)+1) - a
if sign:
a = -a
return from_man_exp(a, -wp, prec, rnd)
def mpf_atan(x, prec, rnd=round_fast):
sign, man, exp, bc = x
if not man:
if x == fzero: return fzero
if x == finf: return atan_inf(0, prec, rnd)
if x == fninf: return atan_inf(1, prec, rnd)
return fnan
mag = exp + bc
# Essentially infinity
if mag > prec+20:
return atan_inf(sign, prec, rnd)
# Essentially ~ x
if -mag > prec+20:
return mpf_perturb(x, 1-sign, prec, rnd)
wp = prec + 30 + abs(mag)
# For large x, use atan(x) = pi/2 - atan(1/x)
if mag >= 2:
x = mpf_rdiv_int(1, x, wp)
reciprocal = True
else:
reciprocal = False
t = to_fixed(x, wp)
if sign:
t = -t
if wp < ATAN_TAYLOR_PREC:
a = atan_taylor(t, wp)
else:
a = atan_newton(t, wp)
if reciprocal:
a = ((pi_fixed(wp)>>1)+1) - a
if sign:
a = -a
return from_man_exp(a, -wp, prec, rnd)
def mpf_cos_sin_pi(x, prec, rnd=round_fast):
"""Accurate computation of (cos(pi*x), sin(pi*x))
for x close to an integer"""
sign, man, exp, bc = x
if not man:
return cos_sin(x, prec, rnd)
# Exactly an integer or half-integer?
if exp >= -1:
if exp == -1:
c = fzero
s = (fone, fnone)[bool(man & 2) ^ sign]
elif exp == 0:
c, s = (fnone, fzero)
else:
c, s = (fone, fzero)
return c, s
# Close to 0 ?
size = exp + bc
if size < -(prec+5):
c = mpf_perturb(fone, 1, prec, rnd)
s = mpf_perturb(mpf_mul(x, mpf_pi(prec)), sign, prec, rnd)
return c, s
if sign:
man = -man
# Subtract nearest half-integer (= modulo pi/2)
nhint = ((man >> (-exp-2)) + 1) >> 1
man = man - (nhint << (-exp-1))
x = from_man_exp(man, exp, prec)
x = mpf_mul(x, mpf_pi(prec), prec)
# XXX: with some more work, could call calc_cos_sin,
# to save some time and to get rounding right
case = nhint % 4
if case == 0:
c, s = cos_sin(x, prec, rnd)
elif case == 1:
s, c = cos_sin(x, prec, rnd)
c = mpf_neg(c)
elif case == 2:
c, s = cos_sin(x, prec, rnd)
c = mpf_neg(c)
s = mpf_neg(s)
else:
s, c = cos_sin(x, prec, rnd)
s = mpf_neg(s)
return c, s
def cosh_sinh(x, prec, rnd=round_fast, tanh=0):
"""Simultaneously compute (cosh(x), sinh(x)) for real x"""
sign, man, exp, bc = x
if (not man) and exp:
if tanh:
if x == finf: return fone
if x == fninf: return fnone
return fnan
if x == finf: return (finf, finf)
if x == fninf: return (finf, fninf)
return fnan, fnan
if sign:
man = -man
mag = exp + bc
prec2 = prec + 20
if mag < -3:
# Extremely close to 0, sinh(x) ~= x and cosh(x) ~= 1
if mag < -prec-2:
if tanh:
return mpf_perturb(x, 1-sign, prec, rnd)
cosh = mpf_perturb(fone, 0, prec, rnd)
sinh = mpf_perturb(x, sign, prec, rnd)
return cosh, sinh
# Avoid cancellation when computing sinh
# TODO: might be faster to use sinh series directly
prec2 += (-mag) + 4
# In the general case, we use
# cosh(x) = (exp(x) + exp(-x))/2
# sinh(x) = (exp(x) - exp(-x))/2
# and note that the exponential only needs to be computed once.
ep = mpf_exp(x, prec2)
em = mpf_div(fone, ep, prec2)
if tanh:
ch = mpf_add(ep, em, prec2, rnd)
sh = mpf_sub(ep, em, prec2, rnd)
return mpf_div(sh, ch, prec, rnd)
else:
ch = mpf_shift(mpf_add(ep, em, prec, rnd), -1)
sh = mpf_shift(mpf_sub(ep, em, prec, rnd), -1)
return ch, sh
def mpf_zeta_int(s, prec, rnd=round_fast):
"""
Optimized computation of zeta(s) for an integer s.
"""
wp = prec + 20
s = int(s)
if s in zeta_int_cache and zeta_int_cache[s][0] >= wp:
return mpf_pos(zeta_int_cache[s][1], prec, rnd)
if s < 2:
if s == 1:
raise ValueError("zeta(1) pole")
if not s:
return mpf_neg(fhalf)
return mpf_div(mpf_bernoulli(-s + 1, wp), from_int(s - 1), prec, rnd)
# 2^-s term vanishes?
if s >= wp:
return mpf_perturb(fone, 0, prec, rnd)
# 5^-s term vanishes?
elif s >= wp * 0.431:
t = one = 1 << wp
t += 1 << (wp - s)
t += one // (MPZ_THREE**s)
t += 1 << max(0, wp - s * 2)
return from_man_exp(t, -wp, prec, rnd)
else:
# Fast enough to sum directly?
# Even better, we use the Euler product (idea stolen from pari)
m = (float(wp) / (s - 1) + 1)
if m < 30:
needed_terms = int(2.0**m + 1)
if needed_terms < int(wp / 2.54 + 5) / 10:
t = fone
for k in list_primes(needed_terms):
#print k, needed_terms
powprec = int(wp - s * math.log(k, 2))
if powprec < 2:
break
a = mpf_sub(fone, mpf_pow_int(from_int(k), -s, powprec),
wp)
t = mpf_mul(t, a, wp)
return mpf_div(fone, t, wp)
# Use Borwein's algorithm
n = int(wp / 2.54 + 5)
d = borwein_coefficients(n)
t = MPZ_ZERO
s = MPZ(s)
for k in xrange(n):
t += (((-1)**k * (d[k] - d[n])) << wp) // (k + 1)**s
t = (t << wp) // (-d[n])
t = (t << wp) // ((1 << wp) - (1 << (wp + 1 - s)))
if (s in zeta_int_cache
and zeta_int_cache[s][0] < wp) or (s not in zeta_int_cache):
zeta_int_cache[s] = (wp, from_man_exp(t, -wp - wp))
return from_man_exp(t, -wp - wp, prec, rnd)
def mpf_zeta_int(s, prec, rnd=round_fast):
"""
Optimized computation of zeta(s) for an integer s.
"""
wp = prec + 20
s = int(s)
if s in zeta_int_cache and zeta_int_cache[s][0] >= wp:
return mpf_pos(zeta_int_cache[s][1], prec, rnd)
if s < 2:
if s == 1:
raise ValueError("zeta(1) pole")
if not s:
return mpf_neg(fhalf)
return mpf_div(mpf_bernoulli(-s+1, wp), from_int(s-1), prec, rnd)
# 2^-s term vanishes?
if s >= wp:
return mpf_perturb(fone, 0, prec, rnd)
# 5^-s term vanishes?
elif s >= wp*0.431:
t = one = 1 << wp
t += 1 << (wp - s)
t += one // (MPZ_THREE ** s)
t += 1 << max(0, wp - s*2)
return from_man_exp(t, -wp, prec, rnd)
else:
# Fast enough to sum directly?
# Even better, we use the Euler product (idea stolen from pari)
m = (float(wp)/(s-1) + 1)
if m < 30:
needed_terms = int(2.0**m + 1)
if needed_terms < int(wp/2.54 + 5) / 10:
t = fone
for k in list_primes(needed_terms):
#print k, needed_terms
powprec = int(wp - s*math.log(k,2))
if powprec < 2:
break
a = mpf_sub(fone, mpf_pow_int(from_int(k), -s, powprec), wp)
t = mpf_mul(t, a, wp)
return mpf_div(fone, t, wp)
# Use Borwein's algorithm
n = int(wp/2.54 + 5)
d = borwein_coefficients(n)
t = MPZ_ZERO
s = MPZ(s)
for k in xrange(n):
t += (((-1)**k * (d[k] - d[n])) << wp) // (k+1)**s
t = (t << wp) // (-d[n])
t = (t << wp) // ((1 << wp) - (1 << (wp+1-s)))
if (s in zeta_int_cache and zeta_int_cache[s][0] < wp) or (s not in zeta_int_cache):
zeta_int_cache[s] = (wp, from_man_exp(t, -wp-wp))
return from_man_exp(t, -wp-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) + 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_asinh(x, prec, rnd=round_fast):
wp = prec + 20
sign, man, exp, bc = x
mag = exp+bc
if mag < -8:
if mag < -wp:
return mpf_perturb(x, 1-sign, prec, rnd)
wp += (-mag)
# asinh(x) = log(x+sqrt(x**2+1))
# use reflection symmetry to avoid cancellation
q = mpf_sqrt(mpf_add(mpf_mul(x, x), fone, wp), wp)
q = mpf_add(mpf_abs(x), q, wp)
if sign:
return mpf_neg(mpf_log(q, prec, negative_rnd[rnd]))
else:
return mpf_log(q, prec, rnd)
def mpf_exp(x, prec, rnd=round_fast):
sign, man, exp, bc = x
if not man:
if not exp:
return fone
if x == fninf:
return fzero
return x
# Fast handling e**n. TODO: the best cutoff depends on both the
# size of n and the precision.
if prec > 600 and exp >= 0:
return mpf_pow_int(mpf_e(prec + 10), (-1)**sign * (man << exp), prec,
rnd)
mag = bc + exp
if mag < -prec - 10:
return mpf_perturb(fone, sign, prec, rnd)
# extra precision needs to be similar in magnitude to log_2(|x|)
# for the modulo reduction, plus r for the error from squaring r times
wp = prec + max(0, mag)
if wp < 300:
r = int(2 * wp**0.4)
if mag < 0:
r = max(1, r + mag)
wp += r + 20
t = to_fixed(x, wp)
# abs(x) > 1?
if mag > 1:
lg2 = ln2_fixed(wp)
n, t = divmod(t, lg2)
else:
n = 0
man = exp_series(t, wp, r)
else:
r = int(0.7 * wp**0.5)
if mag < 0:
r = max(1, r + mag)
wp += r + 20
t = to_fixed(x, wp)
if mag > 1:
lg2 = ln2_fixed(wp)
n, t = divmod(t, lg2)
else:
n = 0
man = exp_series2(t, wp, r)
bc = wp - 2 + bctable[int(man >> (wp - 2))]
return normalize(0, man, int(-wp + n), bc, prec, rnd)
def mpf_exp(x, prec, rnd=round_fast):
sign, man, exp, bc = x
if not man:
if not exp:
return fone
if x == fninf:
return fzero
return x
# Fast handling e**n. TODO: the best cutoff depends on both the
# size of n and the precision.
if prec > 600 and exp >= 0:
return mpf_pow_int(mpf_e(prec+10), (-1)**sign *(man<<exp), prec, rnd)
mag = bc+exp
if mag < -prec-10:
return mpf_perturb(fone, sign, prec, rnd)
# extra precision needs to be similar in magnitude to log_2(|x|)
# for the modulo reduction, plus r for the error from squaring r times
wp = prec + max(0, mag)
if wp < 300:
r = int(2*wp**0.4)
if mag < 0:
r = max(1, r + mag)
wp += r + 20
t = to_fixed(x, wp)
# abs(x) > 1?
if mag > 1:
lg2 = ln2_fixed(wp)
n, t = divmod(t, lg2)
else:
n = 0
man = exp_series(t, wp, r)
else:
r = int(0.7 * wp**0.5)
if mag < 0:
r = max(1, r + mag)
wp += r + 20
t = to_fixed(x, wp)
if mag > 1:
lg2 = ln2_fixed(wp)
n, t = divmod(t, lg2)
else:
n = 0
man = exp_series2(t, wp, r)
bc = wp - 2 + bctable[int(man >> (wp - 2))]
return normalize(0, man, int(-wp+n), bc, prec, rnd)
def mpf_exp(x, prec, rnd=round_fast):
sign, man, exp, bc = x
if man:
mag = bc + exp
wp = prec + 14
if sign:
man = -man
# TODO: the best cutoff depends on both x and the precision.
if prec > 600 and exp >= 0:
# Need about log2(exp(n)) ~= 1.45*mag extra precision
e = mpf_e(wp + int(1.45 * mag))
return mpf_pow_int(e, man << exp, prec, rnd)
if mag < -wp:
return mpf_perturb(fone, sign, prec, rnd)
# |x| >= 2
if mag > 1:
# For large arguments: exp(2^mag*(1+eps)) =
# exp(2^mag)*exp(2^mag*eps) = exp(2^mag)*(1 + 2^mag*eps + ...)
# so about mag extra bits is required.
wpmod = wp + mag
offset = exp + wpmod
if offset >= 0:
t = man << offset
else:
t = man >> (-offset)
lg2 = ln2_fixed(wpmod)
n, t = divmod(t, lg2)
n = int(n)
t >>= mag
else:
offset = exp + wp
if offset >= 0:
t = man << offset
else:
t = man >> (-offset)
n = 0
man = exp_basecase(t, wp)
return from_man_exp(man, n - wp, prec, rnd)
if not exp:
return fone
if x == fninf:
return fzero
return x
def mpf_exp(x, prec, rnd=round_fast):
sign, man, exp, bc = x
if man:
mag = bc + exp
wp = prec + 14
if sign:
man = -man
# TODO: the best cutoff depends on both x and the precision.
if prec > 600 and exp >= 0:
# Need about log2(exp(n)) ~= 1.45*mag extra precision
e = mpf_e(wp + int(1.45 * mag))
return mpf_pow_int(e, man << exp, prec, rnd)
if mag < -wp:
return mpf_perturb(fone, sign, prec, rnd)
# |x| >= 2
if mag > 1:
# For large arguments: exp(2^mag*(1+eps)) =
# exp(2^mag)*exp(2^mag*eps) = exp(2^mag)*(1 + 2^mag*eps + ...)
# so about mag extra bits is required.
wpmod = wp + mag
offset = exp + wpmod
if offset >= 0:
t = man << offset
else:
t = man >> (-offset)
lg2 = ln2_fixed(wpmod)
n, t = divmod(t, lg2)
n = int(n)
t >>= mag
else:
offset = exp + wp
if offset >= 0:
t = man << offset
else:
t = man >> (-offset)
n = 0
man = exp_basecase(t, wp)
return from_man_exp(man, n - wp, prec, rnd)
if not exp:
return fone
if x == fninf:
return fzero
return x
def mpf_atanh(x, prec, rnd=round_fast):
# atanh(x) = log((1+x)/(1-x))/2
sign, man, exp, bc = x
if (not man) and exp:
if x in (fzero, fnan):
return x
raise ComplexResult("atanh(x) is real only for -1 <= x <= 1")
mag = bc + exp
if mag > 0:
if mag == 1 and man == 1:
return [finf, fninf][sign]
raise ComplexResult("atanh(x) is real only for -1 <= x <= 1")
wp = prec + 15
if mag < -8:
if mag < -wp:
return mpf_perturb(x, sign, prec, rnd)
wp += (-mag)
a = mpf_add(x, fone, wp)
b = mpf_sub(fone, x, wp)
return mpf_shift(mpf_log(mpf_div(a, b, wp), prec, rnd), -1)
def mpf_cosh_sinh(x, prec, rnd=round_fast, tanh=0):
"""Simultaneously compute (cosh(x), sinh(x)) for real x"""
sign, man, exp, bc = x
if (not man) and exp:
if tanh:
if x == finf: return fone
if x == fninf: return fnone
return fnan
if x == finf: return (finf, finf)
if x == fninf: return (finf, fninf)
return fnan, fnan
mag = exp + bc
wp = prec + 14
if mag < -4:
# Extremely close to 0, sinh(x) ~= x and cosh(x) ~= 1
if mag < -wp:
if tanh:
return mpf_perturb(x, 1 - sign, prec, rnd)
cosh = mpf_perturb(fone, 0, prec, rnd)
sinh = mpf_perturb(x, sign, prec, rnd)
return cosh, sinh
# Fix for cancellation when computing sinh
wp += (-mag)
# Does exp(-2*x) vanish?
if mag > 10:
if 3 * (1 << (mag - 1)) > wp:
# XXX: rounding
if tanh:
return mpf_perturb([fone, fnone][sign], 1 - sign, prec, rnd)
c = s = mpf_shift(mpf_exp(mpf_abs(x), prec, rnd), -1)
if sign:
s = mpf_neg(s)
return c, s
# |x| > 1
if mag > 1:
wpmod = wp + mag
offset = exp + wpmod
if offset >= 0:
t = man << offset
else:
t = man >> (-offset)
lg2 = ln2_fixed(wpmod)
n, t = divmod(t, lg2)
n = int(n)
t >>= mag
else:
offset = exp + wp
if offset >= 0:
t = man << offset
else:
t = man >> (-offset)
n = 0
a, b = exp_expneg_basecase(t, wp)
# TODO: optimize division precision
cosh = a + (b >> (2 * n))
sinh = a - (b >> (2 * n))
if sign:
sinh = -sinh
if tanh:
man = (sinh << wp) // cosh
return from_man_exp(man, -wp, prec, rnd)
else:
cosh = from_man_exp(cosh, n - wp - 1, prec, rnd)
sinh = from_man_exp(sinh, n - wp - 1, prec, rnd)
return cosh, sinh
def mpf_exp(x, prec, rnd=round_fast):
sign, man, exp, bc = x
if not man:
if not exp:
return fone
if x == fninf:
return fzero
return x
mag = bc+exp
# Fast handling e**n. TODO: the best cutoff depends on both the
# size of n and the precision.
if prec > 600 and exp >= 0:
e = mpf_e(prec+10+int(1.45*mag))
return mpf_pow_int(e, (-1)**sign *(man<<exp), prec, rnd)
if mag < -prec-10:
return mpf_perturb(fone, sign, prec, rnd)
# extra precision needs to be similar in magnitude to log_2(|x|)
# for the modulo reduction, plus r for the error from squaring r times
wp = prec + max(0, mag)
if wp < 300:
r = int(2*wp**0.4)
if mag < 0:
r = max(1, r + mag)
wp += r + 20
t = to_fixed(x, wp)
# abs(x) > 1?
if mag > 1:
lg2 = ln2_fixed(wp)
n, t = divmod(t, lg2)
else:
n = 0
man = exp_series(t, wp, r)
else:
use_newton = False
# put a bound on exp to avoid infinite recursion in exp_newton
# TODO find a good bound
if wp > LIM_EXP_SERIES2 and exp < 1000:
if mag > 0:
use_newton = True
elif mag <= 0 and -mag <= ns_exp[-1]:
i = bisect(ns_exp, -mag-1)
if i < len(ns_exp):
wp0 = precs_exp[i]
if wp > wp0:
use_newton = True
if not use_newton:
r = int(0.7 * wp**0.5)
if mag < 0:
r = max(1, r + mag)
wp += r + 20
t = to_fixed(x, wp)
if mag > 1:
lg2 = ln2_fixed(wp)
n, t = divmod(t, lg2)
else:
n = 0
man = exp_series2(t, wp, r)
else:
# if x is very small or very large use
# exp(x + m) = exp(x) * e**m
if mag > LIM_MAG:
wp += mag*10 + 100
n = int(mag * math.log(2)) + 1
x = mpf_sub(x, from_int(n, wp), wp)
elif mag <= 0:
wp += -mag*10 + 100
if mag < 0:
n = int(-mag * math.log(2)) + 1
x = mpf_add(x, from_int(n, wp), wp)
res = exp_newton(x, wp)
sign, man, exp, bc = res
if mag < 0:
t = mpf_pow_int(mpf_e(wp), n, wp)
res = mpf_div(res, t, wp)
sign, man, exp, bc = res
if mag > LIM_MAG:
t = mpf_pow_int(mpf_e(wp), n, wp)
res = mpf_mul(res, t, wp)
sign, man, exp, bc = res
return normalize(sign, man, exp, bc, prec, rnd)
bc = bitcount(man)
return normalize(0, man, int(-wp+n), bc, prec, rnd)
def mpf_log(x, prec, rnd=round_fast):
"""
Compute the natural logarithm of the mpf value x. If x is negative,
ComplexResult is raised.
"""
sign, man, exp, bc = x
#------------------------------------------------------------------
# Handle special values
if not man:
if x == fzero: return fninf
if x == finf: return finf
if x == fnan: return fnan
if sign:
raise ComplexResult("logarithm of a negative number")
wp = prec + 20
#------------------------------------------------------------------
# Handle log(2^n) = log(n)*2.
# Here we catch the only possible exact value, log(1) = 0
if man == 1:
if not exp:
return fzero
return from_man_exp(exp*ln2_fixed(wp), -wp, prec, rnd)
mag = exp+bc
abs_mag = abs(mag)
#------------------------------------------------------------------
# Handle x = 1+eps, where log(x) ~ x. We need to check for
# cancellation when moving to fixed-point math and compensate
# by increasing the precision. Note that abs_mag in (0, 1) <=>
# 0.5 < x < 2 and x != 1
if abs_mag <= 1:
# Calculate t = x-1 to measure distance from 1 in bits
tsign = 1-abs_mag
if tsign:
tman = (MP_ONE<<bc) - man
else:
tman = man - (MP_ONE<<(bc-1))
tbc = bitcount(tman)
cancellation = bc - tbc
if cancellation > wp:
t = normalize(tsign, tman, abs_mag-bc, tbc, tbc, 'n')
return mpf_perturb(t, tsign, prec, rnd)
else:
wp += cancellation
# TODO: if close enough to 1, we could use Taylor series
# even in the AGM precision range, since the Taylor series
# converges rapidly
#------------------------------------------------------------------
# Another special case:
# n*log(2) is a good enough approximation
if abs_mag > 10000:
if bitcount(abs_mag) > wp:
return from_man_exp(exp*ln2_fixed(wp), -wp, prec, rnd)
#------------------------------------------------------------------
# General case.
# Perform argument reduction using log(x) = log(x*2^n) - n*log(2):
# If we are in the Taylor precision range, choose magnitude 0 or 1.
# If we are in the AGM precision range, choose magnitude -m for
# some large m; benchmarking on one machine showed m = prec/20 to be
# optimal between 1000 and 100,000 digits.
if wp <= LOG_TAYLOR_PREC:
m = log_taylor_cached(lshift(man, wp-bc), wp)
if mag:
m += mag*ln2_fixed(wp)
else:
optimal_mag = -wp//LOG_AGM_MAG_PREC_RATIO
n = optimal_mag - mag
x = mpf_shift(x, n)
wp += (-optimal_mag)
m = -log_agm(to_fixed(x, wp), wp)
m -= n*ln2_fixed(wp)
return from_man_exp(m, -wp, prec, rnd)
def mpf_cosh_sinh(x, prec, rnd=round_fast, tanh=0):
"""Simultaneously compute (cosh(x), sinh(x)) for real x"""
sign, man, exp, bc = x
if (not man) and exp:
if tanh:
if x == finf:
return fone
if x == fninf:
return fnone
return fnan
if x == finf:
return (finf, finf)
if x == fninf:
return (finf, fninf)
return fnan, fnan
mag = exp + bc
wp = prec + 14
if mag < -4:
# Extremely close to 0, sinh(x) ~= x and cosh(x) ~= 1
if mag < -wp:
if tanh:
return mpf_perturb(x, 1 - sign, prec, rnd)
cosh = mpf_perturb(fone, 0, prec, rnd)
sinh = mpf_perturb(x, sign, prec, rnd)
return cosh, sinh
# Fix for cancellation when computing sinh
wp += -mag
# Does exp(-2*x) vanish?
if mag > 10:
if 3 * (1 << (mag - 1)) > wp:
# XXX: rounding
if tanh:
return mpf_perturb([fone, fnone][sign], 1 - sign, prec, rnd)
c = s = mpf_shift(mpf_exp(mpf_abs(x), prec, rnd), -1)
if sign:
s = mpf_neg(s)
return c, s
# |x| > 1
if mag > 1:
wpmod = wp + mag
offset = exp + wpmod
if offset >= 0:
t = man << offset
else:
t = man >> (-offset)
lg2 = ln2_fixed(wpmod)
n, t = divmod(t, lg2)
n = int(n)
t >>= mag
else:
offset = exp + wp
if offset >= 0:
t = man << offset
else:
t = man >> (-offset)
n = 0
a, b = exp_expneg_basecase(t, wp)
# TODO: optimize division precision
cosh = a + (b >> (2 * n))
sinh = a - (b >> (2 * n))
if sign:
sinh = -sinh
if tanh:
man = (sinh << wp) // cosh
return from_man_exp(man, -wp, prec, rnd)
else:
cosh = from_man_exp(cosh, n - wp - 1, prec, rnd)
sinh = from_man_exp(sinh, n - wp - 1, prec, rnd)
return cosh, sinh
def mpf_cos_sin(x, prec, rnd=round_fast, which=0, pi=False):
"""
which:
0 -- return cos(x), sin(x)
1 -- return cos(x)
2 -- return sin(x)
3 -- return tan(x)
if pi=True, compute for pi*x
"""
sign, man, exp, bc = x
if not man:
if exp:
c, s = fnan, fnan
else:
c, s = fone, fzero
if which == 0:
return c, s
if which == 1:
return c
if which == 2:
return s
if which == 3:
return s
mag = bc + exp
wp = prec + 10
# Extremely small?
if mag < 0:
if mag < -wp:
if pi:
x = mpf_mul(x, mpf_pi(wp))
c = mpf_perturb(fone, 1, prec, rnd)
s = mpf_perturb(x, 1 - sign, prec, rnd)
if which == 0:
return c, s
if which == 1:
return c
if which == 2:
return s
if which == 3:
return mpf_perturb(x, sign, prec, rnd)
if pi:
if exp >= -1:
if exp == -1:
c = fzero
s = (fone, fnone)[bool(man & 2) ^ sign]
elif exp == 0:
c, s = (fnone, fzero)
else:
c, s = (fone, fzero)
if which == 0:
return c, s
if which == 1:
return c
if which == 2:
return s
if which == 3:
return mpf_div(s, c, prec, rnd)
# Subtract nearest half-integer (= mod by pi/2)
n = ((man >> (-exp - 2)) + 1) >> 1
man = man - (n << (-exp - 1))
mag2 = bitcount(man) + exp
wp = prec + 10 - mag2
offset = exp + wp
if offset >= 0:
t = man << offset
else:
t = man >> (-offset)
t = (t * pi_fixed(wp)) >> wp
else:
t, n, wp = mod_pi2(man, exp, mag, wp)
c, s = cos_sin_basecase(t, wp)
m = n & 3
if m == 1:
c, s = -s, c
elif m == 2:
c, s = -c, -s
elif m == 3:
c, s = s, -c
if sign:
s = -s
if which == 0:
c = from_man_exp(c, -wp, prec, rnd)
s = from_man_exp(s, -wp, prec, rnd)
return c, s
if which == 1:
return from_man_exp(c, -wp, prec, rnd)
if which == 2:
return from_man_exp(s, -wp, prec, rnd)
if which == 3:
return from_rational(s, c, prec, rnd)
def mpf_zeta(s, prec, rnd=round_fast, alt=0):
sign, man, exp, bc = s
if not man:
if s == fzero:
if alt:
return fhalf
else:
return mpf_neg(fhalf)
if s == finf:
return fone
return fnan
wp = prec + 20
# First term vanishes?
if (not sign) and (exp + bc > (math.log(wp, 2) + 2)):
return mpf_perturb(fone, alt, prec, rnd)
# Optimize for integer arguments
elif exp >= 0:
if alt:
if s == fone:
return mpf_ln2(prec, rnd)
z = mpf_zeta_int(to_int(s), wp, negative_rnd[rnd])
q = mpf_sub(fone, mpf_pow(ftwo, mpf_sub(fone, s, wp), wp), wp)
return mpf_mul(z, q, prec, rnd)
else:
return mpf_zeta_int(to_int(s), prec, rnd)
# Negative: use the reflection formula
# Borwein only proves the accuracy bound for x >= 1/2. However, based on
# tests, the accuracy without reflection is quite good even some distance
# to the left of 1/2. XXX: verify this.
if sign:
# XXX: could use the separate refl. formula for Dirichlet eta
if alt:
q = mpf_sub(fone, mpf_pow(ftwo, mpf_sub(fone, s, wp), wp), wp)
return mpf_mul(mpf_zeta(s, wp), q, prec, rnd)
# XXX: -1 should be done exactly
y = mpf_sub(fone, s, 10 * wp)
a = mpf_gamma(y, wp)
b = mpf_zeta(y, wp)
c = mpf_sin_pi(mpf_shift(s, -1), wp)
wp2 = wp + (exp + bc)
pi = mpf_pi(wp + wp2)
d = mpf_div(mpf_pow(mpf_shift(pi, 1), s, wp2), pi, wp2)
return mpf_mul(a, mpf_mul(b, mpf_mul(c, d, wp), wp), prec, rnd)
t = MP_ZERO
#wp += 16 - (prec & 15)
# Use Borwein's algorithm
n = int(wp / 2.54 + 5)
d = borwein_coefficients(n)
t = MP_ZERO
sf = to_fixed(s, wp)
for k in xrange(n):
u = from_man_exp(-sf * log_int_fixed(k + 1, wp), -2 * wp, wp)
esign, eman, eexp, ebc = mpf_exp(u, wp)
offset = eexp + wp
if offset >= 0:
w = ((d[k] - d[n]) * eman) << offset
else:
w = ((d[k] - d[n]) * eman) >> (-offset)
if k & 1:
t -= w
else:
t += w
t = t // (-d[n])
t = from_man_exp(t, -wp, wp)
if alt:
return mpf_pos(t, prec, rnd)
else:
q = mpf_sub(fone, mpf_pow(ftwo, mpf_sub(fone, s, wp), wp), wp)
return mpf_div(t, q, prec, rnd)
def mpf_ci_si(x, prec, rnd=round_fast, which=2):
"""
Calculation of Ci(x), Si(x) for real x.
which = 0 -- returns (Ci(x), -)
which = 1 -- returns (Si(x), -)
which = 2 -- returns (Ci(x), Si(x))
Note: if x < 0, Ci(x) needs an additional imaginary term, pi*i.
"""
wp = prec + 20
sign, man, exp, bc = x
ci, si = None, None
if not man:
if x == fzero:
return (fninf, fzero)
if x == fnan:
return (x, x)
ci = fzero
if which != 0:
if x == finf:
si = mpf_shift(mpf_pi(prec, rnd), -1)
if x == fninf:
si = mpf_neg(mpf_shift(mpf_pi(prec, negative_rnd[rnd]), -1))
return (ci, si)
# For small x: Ci(x) ~ euler + log(x), Si(x) ~ x
mag = exp + bc
if mag < -wp:
if which != 0:
si = mpf_perturb(x, 1 - sign, prec, rnd)
if which != 1:
y = mpf_euler(wp)
xabs = mpf_abs(x)
ci = mpf_add(y, mpf_log(xabs, wp), prec, rnd)
return ci, si
# For huge x: Ci(x) ~ sin(x)/x, Si(x) ~ pi/2
elif mag > wp:
if which != 0:
if sign:
si = mpf_neg(mpf_pi(prec, negative_rnd[rnd]))
else:
si = mpf_pi(prec, rnd)
si = mpf_shift(si, -1)
if which != 1:
ci = mpf_div(mpf_sin(x, wp), x, prec, rnd)
return ci, si
else:
wp += abs(mag)
# Use an asymptotic series? The smallest value of n!/x^n
# occurs for n ~ x, where the magnitude is ~ exp(-x).
asymptotic = mag - 1 > math.log(wp, 2)
# Case 1: convergent series near 0
if not asymptotic:
if which != 0:
si = mpf_pos(mpf_ci_si_taylor(x, wp, 1), prec, rnd)
if which != 1:
ci = mpf_ci_si_taylor(x, wp, 0)
ci = mpf_add(ci, mpf_euler(wp), wp)
ci = mpf_add(ci, mpf_log(mpf_abs(x), wp), prec, rnd)
return ci, si
x = mpf_abs(x)
# Case 2: asymptotic series for x >> 1
xf = to_fixed(x, wp)
xr = (MP_ONE << (2 * wp)) // xf # 1/x
s1 = (MP_ONE << wp)
s2 = xr
t = xr
k = 2
while t:
t = -t
t = (t * xr * k) >> wp
k += 1
s1 += t
t = (t * xr * k) >> wp
k += 1
s2 += t
s1 = from_man_exp(s1, -wp)
s2 = from_man_exp(s2, -wp)
s1 = mpf_div(s1, x, wp)
s2 = mpf_div(s2, x, wp)
cos, sin = cos_sin(x, wp)
# Ci(x) = sin(x)*s1-cos(x)*s2
# Si(x) = pi/2-cos(x)*s1-sin(x)*s2
if which != 0:
si = mpf_add(mpf_mul(cos, s1), mpf_mul(sin, s2), wp)
si = mpf_sub(mpf_shift(mpf_pi(wp), -1), si, wp)
if sign:
si = mpf_neg(si)
si = mpf_pos(si, prec, rnd)
if which != 1:
ci = mpf_sub(mpf_mul(sin, s1), mpf_mul(cos, s2), prec, rnd)
return ci, si
def mpf_log(x, prec, rnd=round_fast):
"""
Compute the natural logarithm of the mpf value x. If x is negative,
ComplexResult is raised.
"""
sign, man, exp, bc = x
if not man:
if x == fzero: return fninf
if x == finf: return finf
if x == fnan: return fnan
if sign:
raise ComplexResult("logarithm of a negative number")
# log(2^n)
if man == 1:
if not exp:
return fzero
return from_man_exp(exp * ln2_fixed(prec + 20), -prec - 20, prec, rnd)
# Assume 20 bits to be sufficient for cancelling rounding errors
wp = prec + 20
mag = bc + exp
# Already on the standard interval, (0.5, 1)
if not mag:
t = rshift(man, bc - wp)
log2n = 0
# Watch out for "x = 0.9999"
# Proceed only if 1-x lost at most 15 bits of accuracy
res = (MP_ONE << wp) - t
if not (res >> (wp - 15)):
# Find out extra precision needed
delta = mpf_sub(x, fone, 10)
delta_bits = -(delta[2] + delta[3])
# O(x^2) term vanishes relatively
if delta_bits > wp + 10:
xm1 = mpf_sub(x, fone, prec, rnd)
return mpf_perturb(xm1, 1, prec, rnd)
else:
wp += delta_bits
t = rshift(man, bc - wp)
# Already on the standard interval, (1, 2)
elif mag == 1:
t = rshift(man, bc - wp - 1)
log2n = 0
# Watch out for "x = 1.0001"
# Similar to above; note that we flip signs
# to obtain a positive residual, ensuring that
# the following shift rounds down
res = t - (MP_ONE << wp)
if not (res >> (wp - 15)):
# Find out extra precision needed
delta = mpf_sub(x, fone, 10)
delta_bits = -(delta[2] + delta[3])
# O(x^2) term vanishes relatively
if delta_bits > wp + 10:
xm1 = mpf_sub(x, fone, prec, rnd)
return mpf_perturb(xm1, 1, prec, rnd)
else:
wp += delta_bits
t = rshift(man, bc - wp - 1)
# Rescale
else:
# Estimated precision needed for n*log(2) to
# be accurate relatively
wp += int(math.log(1 + abs(mag), 2))
log2n = mag * ln2_fixed(wp)
# Rescaled argument as a fixed-point number
t = rshift(man, bc - wp)
# Use the faster method
if wp < LOG_TAYLOR_PREC:
a = log_taylor(t, wp)
else:
a = log_newton(t, wp)
return from_man_exp(a + log2n, -wp, prec, rnd)
def mpf_ci_si(x, prec, rnd=round_fast, which=2):
"""
Calculation of Ci(x), Si(x) for real x.
which = 0 -- returns (Ci(x), -)
which = 1 -- returns (Si(x), -)
which = 2 -- returns (Ci(x), Si(x))
Note: if x < 0, Ci(x) needs an additional imaginary term, pi*i.
"""
wp = prec + 20
sign, man, exp, bc = x
ci, si = None, None
if not man:
if x == fzero:
return (fninf, fzero)
if x == fnan:
return (x, x)
ci = fzero
if which != 0:
if x == finf:
si = mpf_shift(mpf_pi(prec, rnd), -1)
if x == fninf:
si = mpf_neg(mpf_shift(mpf_pi(prec, negative_rnd[rnd]), -1))
return (ci, si)
# For small x: Ci(x) ~ euler + log(x), Si(x) ~ x
mag = exp+bc
if mag < -wp:
if which != 0:
si = mpf_perturb(x, 1-sign, prec, rnd)
if which != 1:
y = mpf_euler(wp)
xabs = mpf_abs(x)
ci = mpf_add(y, mpf_log(xabs, wp), prec, rnd)
return ci, si
# For huge x: Ci(x) ~ sin(x)/x, Si(x) ~ pi/2
elif mag > wp:
if which != 0:
if sign:
si = mpf_neg(mpf_pi(prec, negative_rnd[rnd]))
else:
si = mpf_pi(prec, rnd)
si = mpf_shift(si, -1)
if which != 1:
ci = mpf_div(mpf_sin(x, wp), x, prec, rnd)
return ci, si
else:
wp += abs(mag)
# Use an asymptotic series? The smallest value of n!/x^n
# occurs for n ~ x, where the magnitude is ~ exp(-x).
asymptotic = mag-1 > math.log(wp, 2)
# Case 1: convergent series near 0
if not asymptotic:
if which != 0:
si = mpf_pos(mpf_ci_si_taylor(x, wp, 1), prec, rnd)
if which != 1:
ci = mpf_ci_si_taylor(x, wp, 0)
ci = mpf_add(ci, mpf_euler(wp), wp)
ci = mpf_add(ci, mpf_log(mpf_abs(x), wp), prec, rnd)
return ci, si
x = mpf_abs(x)
# Case 2: asymptotic series for x >> 1
xf = to_fixed(x, wp)
xr = (MPZ_ONE<<(2*wp)) // xf # 1/x
s1 = (MPZ_ONE << wp)
s2 = xr
t = xr
k = 2
while t:
t = -t
t = (t*xr*k)>>wp
k += 1
s1 += t
t = (t*xr*k)>>wp
k += 1
s2 += t
s1 = from_man_exp(s1, -wp)
s2 = from_man_exp(s2, -wp)
s1 = mpf_div(s1, x, wp)
s2 = mpf_div(s2, x, wp)
cos, sin = mpf_cos_sin(x, wp)
# Ci(x) = sin(x)*s1-cos(x)*s2
# Si(x) = pi/2-cos(x)*s1-sin(x)*s2
if which != 0:
si = mpf_add(mpf_mul(cos, s1), mpf_mul(sin, s2), wp)
si = mpf_sub(mpf_shift(mpf_pi(wp), -1), si, wp)
if sign:
si = mpf_neg(si)
si = mpf_pos(si, prec, rnd)
if which != 1:
ci = mpf_sub(mpf_mul(sin, s1), mpf_mul(cos, s2), prec, rnd)
return ci, si
def mpf_log(x, prec, rnd=round_fast):
"""
Compute the natural logarithm of the mpf value x. If x is negative,
ComplexResult is raised.
"""
sign, man, exp, bc = x
if not man:
if x == fzero: return fninf
if x == finf: return finf
if x == fnan: return fnan
if sign:
raise ComplexResult("logarithm of a negative number")
# log(2^n)
if man == 1:
if not exp:
return fzero
return from_man_exp(exp*ln2_fixed(prec+20), -prec-20, prec, rnd)
# Assume 20 bits to be sufficient for cancelling rounding errors
wp = prec + 20
mag = bc + exp
# Already on the standard interval, (0.5, 1)
if not mag:
t = rshift(man, bc-wp)
log2n = 0
# Watch out for "x = 0.9999"
# Proceed only if 1-x lost at most 15 bits of accuracy
res = (MP_ONE << wp) - t
if not (res >> (wp - 15)):
# Find out extra precision needed
delta = mpf_sub(x, fone, 10)
delta_bits = -(delta[2] + delta[3])
# O(x^2) term vanishes relatively
if delta_bits > wp + 10:
xm1 = mpf_sub(x, fone, prec, rnd)
return mpf_perturb(xm1, 1, prec, rnd)
else:
wp += delta_bits
t = rshift(man, bc-wp)
# Already on the standard interval, (1, 2)
elif mag == 1:
t = rshift(man, bc-wp-1)
log2n = 0
# Watch out for "x = 1.0001"
# Similar to above; note that we flip signs
# to obtain a positive residual, ensuring that
# the following shift rounds down
res = t - (MP_ONE << wp)
if not (res >> (wp - 15)):
# Find out extra precision needed
delta = mpf_sub(x, fone, 10)
delta_bits = -(delta[2] + delta[3])
# O(x^2) term vanishes relatively
if delta_bits > wp + 10:
xm1 = mpf_sub(x, fone, prec, rnd)
return mpf_perturb(xm1, 1, prec, rnd)
else:
wp += delta_bits
t = rshift(man, bc-wp-1)
# Rescale
else:
# Estimated precision needed for n*log(2) to
# be accurate relatively
wp += int(math.log(1+abs(mag),2))
log2n = mag * ln2_fixed(wp)
# Rescaled argument as a fixed-point number
t = rshift(man, bc-wp)
# Use the faster method
if wp < LOG_TAYLOR_PREC:
a = log_taylor(t, wp)
else:
a = log_newton(t, wp)
return from_man_exp(a + log2n, -wp, prec, rnd)
def calc_cos_sin(which, y, swaps, prec, cos_rnd, sin_rnd):
"""
Simultaneous computation of cos and sin (internal function).
"""
y, wp = y
swap_cos_sin, cos_sign, sin_sign = swaps
if swap_cos_sin:
which_compute = -which
else:
which_compute = which
# XXX: assumes no swaps
if not y:
return fone, fzero
# Tiny nonzero argument
if wp > prec*2 + 30:
y = from_man_exp(y, -wp)
if swap_cos_sin:
cos_rnd, sin_rnd = sin_rnd, cos_rnd
cos_sign, sin_sign = sin_sign, cos_sign
if cos_sign: cos = mpf_perturb(fnone, 0, prec, cos_rnd)
else: cos = mpf_perturb(fone, 1, prec, cos_rnd)
if sin_sign: sin = mpf_perturb(mpf_neg(y), 0, prec, sin_rnd)
else: sin = mpf_perturb(y, 1, prec, sin_rnd)
if swap_cos_sin:
cos, sin = sin, cos
return cos, sin
# Use standard Taylor series
if prec < 600:
if which_compute == 0:
sin = sin_taylor(y, wp)
# only need to evaluate one of the series
cos = sqrt_fixed((1<<wp) - ((sin*sin)>>wp), wp)
elif which_compute == 1:
sin = 0
cos = cos_taylor(y, wp)
elif which_compute == -1:
sin = sin_taylor(y, wp)
cos = 0
# Use exp(i*x) with Brent's trick
else:
r = int(0.137 * prec**0.579)
ep = r+20
cos, sin = expi_series(y<<ep, wp+ep, r)
cos >>= ep
sin >>= ep
if swap_cos_sin:
cos, sin = sin, cos
if cos_rnd is not round_nearest:
# Round and set correct signs
# XXX: this logic needs a second look
ONE = MP_ONE << wp
if cos_sign:
cos += (-1)**(cos_rnd in (round_ceiling, round_down))
cos = min(ONE, cos)
else:
cos += (-1)**(cos_rnd in (round_ceiling, round_up))
cos = min(ONE, cos)
if sin_sign:
sin += (-1)**(sin_rnd in (round_ceiling, round_down))
sin = min(ONE, sin)
else:
sin += (-1)**(sin_rnd in (round_ceiling, round_up))
sin = min(ONE, sin)
if which != -1:
cos = normalize(cos_sign, cos, -wp, bitcount(cos), prec, cos_rnd)
if which != 1:
sin = normalize(sin_sign, sin, -wp, bitcount(sin), prec, sin_rnd)
return cos, sin
def mpf_cos_sin(x, prec, rnd=round_fast, which=0, pi=False):
"""
which:
0 -- return cos(x), sin(x)
1 -- return cos(x)
2 -- return sin(x)
3 -- return tan(x)
if pi=True, compute for pi*x
"""
sign, man, exp, bc = x
if not man:
if exp:
c, s = fnan, fnan
else:
c, s = fone, fzero
if which == 0: return c, s
if which == 1: return c
if which == 2: return s
if which == 3: return s
mag = bc + exp
wp = prec + 10
# Extremely small?
if mag < 0:
if mag < -wp:
if pi:
x = mpf_mul(x, mpf_pi(wp))
c = mpf_perturb(fone, 1, prec, rnd)
s = mpf_perturb(x, 1 - sign, prec, rnd)
if which == 0: return c, s
if which == 1: return c
if which == 2: return s
if which == 3: return mpf_perturb(x, sign, prec, rnd)
if pi:
if exp >= -1:
if exp == -1:
c = fzero
s = (fone, fnone)[bool(man & 2) ^ sign]
elif exp == 0:
c, s = (fnone, fzero)
else:
c, s = (fone, fzero)
if which == 0: return c, s
if which == 1: return c
if which == 2: return s
if which == 3: return mpf_div(s, c, prec, rnd)
# Subtract nearest half-integer (= mod by pi/2)
n = ((man >> (-exp - 2)) + 1) >> 1
man = man - (n << (-exp - 1))
mag2 = bitcount(man) + exp
wp = prec + 10 - mag2
offset = exp + wp
if offset >= 0:
t = man << offset
else:
t = man >> (-offset)
t = (t * pi_fixed(wp)) >> wp
else:
t, n, wp = mod_pi2(man, exp, mag, wp)
c, s = cos_sin_basecase(t, wp)
m = n & 3
if m == 1: c, s = -s, c
elif m == 2: c, s = -c, -s
elif m == 3: c, s = s, -c
if sign:
s = -s
if which == 0:
c = from_man_exp(c, -wp, prec, rnd)
s = from_man_exp(s, -wp, prec, rnd)
return c, s
if which == 1:
return from_man_exp(c, -wp, prec, rnd)
if which == 2:
return from_man_exp(s, -wp, prec, rnd)
if which == 3:
return from_rational(s, c, prec, rnd)
def mpf_zeta(s, prec, rnd=round_fast, alt=0):
sign, man, exp, bc = s
if not man:
if s == fzero:
if alt:
return fhalf
else:
return mpf_neg(fhalf)
if s == finf:
return fone
return fnan
wp = prec + 20
# First term vanishes?
if (not sign) and (exp + bc > (math.log(wp,2) + 2)):
return mpf_perturb(fone, alt, prec, rnd)
# Optimize for integer arguments
elif exp >= 0:
if alt:
if s == fone:
return mpf_ln2(prec, rnd)
z = mpf_zeta_int(to_int(s), wp, negative_rnd[rnd])
q = mpf_sub(fone, mpf_pow(ftwo, mpf_sub(fone, s, wp), wp), wp)
return mpf_mul(z, q, prec, rnd)
else:
return mpf_zeta_int(to_int(s), prec, rnd)
# Negative: use the reflection formula
# Borwein only proves the accuracy bound for x >= 1/2. However, based on
# tests, the accuracy without reflection is quite good even some distance
# to the left of 1/2. XXX: verify this.
if sign:
# XXX: could use the separate refl. formula for Dirichlet eta
if alt:
q = mpf_sub(fone, mpf_pow(ftwo, mpf_sub(fone, s, wp), wp), wp)
return mpf_mul(mpf_zeta(s, wp), q, prec, rnd)
# XXX: -1 should be done exactly
y = mpf_sub(fone, s, 10*wp)
a = mpf_gamma(y, wp)
b = mpf_zeta(y, wp)
c = mpf_sin_pi(mpf_shift(s, -1), wp)
wp2 = wp + (exp+bc)
pi = mpf_pi(wp+wp2)
d = mpf_div(mpf_pow(mpf_shift(pi, 1), s, wp2), pi, wp2)
return mpf_mul(a,mpf_mul(b,mpf_mul(c,d,wp),wp),prec,rnd)
# Near pole
r = mpf_sub(fone, s, wp)
asign, aman, aexp, abc = mpf_abs(r)
pole_dist = -2*(aexp+abc)
if pole_dist > wp:
if alt:
return mpf_ln2(prec, rnd)
else:
q = mpf_neg(mpf_div(fone, r, wp))
return mpf_add(q, mpf_euler(wp), prec, rnd)
else:
wp += max(0, pole_dist)
t = MPZ_ZERO
#wp += 16 - (prec & 15)
# Use Borwein's algorithm
n = int(wp/2.54 + 5)
d = borwein_coefficients(n)
t = MPZ_ZERO
sf = to_fixed(s, wp)
for k in xrange(n):
u = from_man_exp(-sf*log_int_fixed(k+1, wp), -2*wp, wp)
esign, eman, eexp, ebc = mpf_exp(u, wp)
offset = eexp + wp
if offset >= 0:
w = ((d[k] - d[n]) * eman) << offset
else:
w = ((d[k] - d[n]) * eman) >> (-offset)
if k & 1:
t -= w
else:
t += w
t = t // (-d[n])
t = from_man_exp(t, -wp, wp)
if alt:
return mpf_pos(t, prec, rnd)
else:
q = mpf_sub(fone, mpf_pow(ftwo, mpf_sub(fone, s, wp), wp), wp)
return mpf_div(t, q, prec, rnd)
def calc_cos_sin(which, y, swaps, prec, cos_rnd, sin_rnd):
"""
Simultaneous computation of cos and sin (internal function).
"""
y, wp = y
swap_cos_sin, cos_sign, sin_sign = swaps
if swap_cos_sin:
which_compute = -which
else:
which_compute = which
# XXX: assumes no swaps
if not y:
return fone, fzero
# Tiny nonzero argument
if wp > prec*2 + 30:
y = from_man_exp(y, -wp)
if swap_cos_sin:
cos_rnd, sin_rnd = sin_rnd, cos_rnd
cos_sign, sin_sign = sin_sign, cos_sign
if cos_sign: cos = mpf_perturb(fnone, 0, prec, cos_rnd)
else: cos = mpf_perturb(fone, 1, prec, cos_rnd)
if sin_sign: sin = mpf_perturb(mpf_neg(y), 0, prec, sin_rnd)
else: sin = mpf_perturb(y, 1, prec, sin_rnd)
if swap_cos_sin:
cos, sin = sin, cos
return cos, sin
# Use standard Taylor series
if prec < 600:
if which_compute == 0:
sin = sin_taylor(y, wp)
# only need to evaluate one of the series
cos = isqrt_fast((MP_ONE<<(2*wp)) - sin*sin)
elif which_compute == 1:
sin = 0
cos = cos_taylor(y, wp)
elif which_compute == -1:
sin = sin_taylor(y, wp)
cos = 0
# Use exp(i*x) with Brent's trick
else:
r = int(0.137 * prec**0.579)
ep = r+20
cos, sin = expi_series(y<<ep, wp+ep, r)
cos >>= ep
sin >>= ep
if swap_cos_sin:
cos, sin = sin, cos
if cos_rnd is not round_nearest:
# Round and set correct signs
# XXX: this logic needs a second look
ONE = MP_ONE << wp
if cos_sign:
cos += (-1)**(cos_rnd in (round_ceiling, round_down))
cos = min(ONE, cos)
else:
cos += (-1)**(cos_rnd in (round_ceiling, round_up))
cos = min(ONE, cos)
if sin_sign:
sin += (-1)**(sin_rnd in (round_ceiling, round_down))
sin = min(ONE, sin)
else:
sin += (-1)**(sin_rnd in (round_ceiling, round_up))
sin = min(ONE, sin)
if which != -1:
cos = normalize(cos_sign, cos, -wp, bitcount(cos), prec, cos_rnd)
if which != 1:
sin = normalize(sin_sign, sin, -wp, bitcount(sin), prec, sin_rnd)
return cos, sin
