def atan_newton(x, prec):
if prec >= 100:
r = math.atan((x>>(prec-53))/2.0**53)
else:
r = math.atan(x/2.0**prec)
prevp = 50
r = int(r * 2.0**53) >> (53-prevp)
extra_p = 50
for wp in giant_steps(prevp, prec):
wp += extra_p
r = r << (wp-prevp)
cos, sin = cos_sin_fixed(r, wp)
tan = (sin << wp) // cos
a = ((tan-rshift(x, prec-wp)) << wp) // ((MPZ_ONE<<wp) + ((tan**2)>>wp))
r = r - a
prevp = wp
return rshift(r, prevp-prec)
def atan_newton(x, prec):
if prec >= 100:
r = math.atan((x >> (prec - 53)) / 2.0 ** 53)
else:
r = math.atan(x / 2.0 ** prec)
prevp = 50
r = MPZ(int(r * 2.0 ** 53) >> (53 - prevp))
extra_p = 50
for wp in giant_steps(prevp, prec):
wp += extra_p
r = r << (wp - prevp)
cos, sin = cos_sin_fixed(r, wp)
tan = (sin << wp) // cos
a = ((tan - rshift(x, prec - wp)) << wp) // ((MPZ_ONE << wp) + ((tan ** 2) >> wp))
r = r - a
prevp = wp
return rshift(r, prevp - prec)
def atan_newton(x, prec):
if prec >= 100:
r = math.atan((x>>(prec-53))/2.0**53)
else:
r = math.atan(x/2.0**prec)
prevp = 50
r = int(r * 2.0**53) >> (53-prevp)
extra_p = 100
for p in giant_steps(prevp, prec):
s = int(0.137 * p**0.579)
p += s + 50
r = r << (p-prevp)
cos, sin = expi_series(r, p, s)
tan = (sin << p) // cos
a = ((tan - rshift(x, prec-p)) << p) // ((MP_ONE<<p) + ((tan**2)>>p))
r = r - a
prevp = p
return rshift(r, prevp-prec)
def atan_newton(x, prec):
if prec >= 100:
r = math.atan((x>>(prec-53))/2.0**53)
else:
r = math.atan(x/2.0**prec)
prevp = 50
r = int(r * 2.0**53) >> (53-prevp)
extra_p = 100
for p in giant_steps(prevp, prec):
s = int(0.137 * p**0.579)
p += s + 50
r = r << (p-prevp)
cos, sin = expi_series(r, p, s)
tan = (sin << p) // cos
a = ((tan - rshift(x, prec-p)) << p) // ((MP_ONE<<p) + ((tan**2)>>p))
r = r - a
prevp = p
return rshift(r, prevp-prec)
def nthroot_fixed(y, n, prec, exp1):
start = 50
try:
y1 = rshift(y, prec - n*start)
r = MP_BASE(y1**(1.0/n))
except OverflowError:
y1 = from_int(y1, start)
fn = from_int(n)
fn = mpf_rdiv_int(1, fn, start)
r = mpf_pow(y1, fn, start)
r = to_int(r)
extra = 10
extra1 = n
prevp = start
for p in giant_steps(start, prec+extra):
pm, pe = int_pow_fixed(r, n-1, prevp)
r2 = rshift(pm, (n-1)*prevp - p - pe - extra1)
B = lshift(y, 2*p-prec+extra1)//r2
r = (B + (n-1) * lshift(r, p-prevp))//n
prevp = p
return r
def nthroot_fixed(y, n, prec, exp1):
start = 50
try:
y1 = rshift(y, prec - n*start)
r = MP_BASE(int(y1**(1.0/n)))
except OverflowError:
y1 = from_int(y1, start)
fn = from_int(n)
fn = mpf_rdiv_int(1, fn, start)
r = mpf_pow(y1, fn, start)
r = to_int(r)
extra = 10
extra1 = n
prevp = start
for p in giant_steps(start, prec+extra):
pm, pe = int_pow_fixed(r, n-1, prevp)
r2 = rshift(pm, (n-1)*prevp - p - pe - extra1)
B = lshift(y, 2*p-prec+extra1)//r2
r = (B + (n-1) * lshift(r, p-prevp))//n
prevp = p
return r
def log_newton(x, prec):
extra = 10
# 40-bit approximation
fx = math.log(long(x)) - 0.69314718055994529*prec
r = MP_BASE(fx * 2.0**40)
prevp = 40
for p in giant_steps2(40, prec+extra):
rb = lshift(r, p-prevp)
# Parameters for exponential series
if p < 300:
r = int(2 * p**0.4)
exp_func = exp_series
else:
r = int(0.7 * p**0.5)
exp_func = exp_series2
exp_extra = r + 10
e = exp_func((-rb) << exp_extra, p + exp_extra, r)
s = ((rshift(x, prec-p)*e)>>(p + exp_extra)) - (1 << p)
s1 = -((s*s)>>(p+1))
r = rb + s + s1
prevp = p
return r >> extra
def log_newton(x, prec):
extra = 10
# 40-bit approximation
fx = math.log(long(x)) - 0.69314718055994529 * prec
r = MP_BASE(fx * 2.0**40)
prevp = 40
for p in giant_steps2(40, prec + extra):
rb = lshift(r, p - prevp)
# Parameters for exponential series
if p < 300:
r = int(2 * p**0.4)
exp_func = exp_series
else:
r = int(0.7 * p**0.5)
exp_func = exp_series2
exp_extra = r + 10
e = exp_func((-rb) << exp_extra, p + exp_extra, r)
s = ((rshift(x, prec - p) * e) >> (p + exp_extra)) - (1 << p)
s1 = -((s * s) >> (p + 1))
r = rb + s + s1
prevp = p
return r >> extra
def mpc_nthroot_fixed(a, b, n, prec):
# a, b signed integers at fixed precision prec
start = 50
a1 = int(rshift(a, prec - n*start))
b1 = int(rshift(b, prec - n*start))
try:
r = (a1 + 1j * b1)**(1.0/n)
re = r.real
im = r.imag
# XXX: workaround bug in gmpy
if abs(re) < 0.1: re = 0
if abs(im) < 0.1: im = 0
re = MP_BASE(re)
im = MP_BASE(im)
except OverflowError:
a1 = from_int(a1, start)
b1 = from_int(b1, start)
fn = from_int(n)
nth = mpf_rdiv_int(1, fn, start)
re, im = mpc_pow((a1, b1), (nth, fzero), start)
re = to_int(re)
im = to_int(im)
extra = 10
prevp = start
extra1 = n
for p in giant_steps(start, prec+extra):
# this is slow for large n, unlike int_pow_fixed
re2, im2 = complex_int_pow(re, im, n-1)
re2 = rshift(re2, (n-1)*prevp - p - extra1)
im2 = rshift(im2, (n-1)*prevp - p - extra1)
r4 = (re2*re2 + im2*im2) >> (p + extra1)
ap = rshift(a, prec - p)
bp = rshift(b, prec - p)
rec = (ap * re2 + bp * im2) >> p
imc = (-ap * im2 + bp * re2) >> p
reb = (rec << p) // r4
imb = (imc << p) // r4
re = (reb + (n-1)*lshift(re, p-prevp))//n
im = (imb + (n-1)*lshift(im, p-prevp))//n
prevp = p
return re, im
def mpc_nthroot_fixed(a, b, n, prec):
# a, b signed integers at fixed precision prec
start = 50
a1 = int(rshift(a, prec - n * start))
b1 = int(rshift(b, prec - n * start))
try:
r = (a1 + 1j * b1)**(1.0 / n)
re = r.real
im = r.imag
# XXX: workaround bug in gmpy
if abs(re) < 0.1: re = 0
if abs(im) < 0.1: im = 0
re = MP_BASE(re)
im = MP_BASE(im)
except OverflowError:
a1 = from_int(a1, start)
b1 = from_int(b1, start)
fn = from_int(n)
nth = mpf_rdiv_int(1, fn, start)
re, im = mpc_pow((a1, b1), (nth, fzero), start)
re = to_int(re)
im = to_int(im)
extra = 10
prevp = start
extra1 = n
for p in giant_steps(start, prec + extra):
# this is slow for large n, unlike int_pow_fixed
re2, im2 = complex_int_pow(re, im, n - 1)
re2 = rshift(re2, (n - 1) * prevp - p - extra1)
im2 = rshift(im2, (n - 1) * prevp - p - extra1)
r4 = (re2 * re2 + im2 * im2) >> (p + extra1)
ap = rshift(a, prec - p)
bp = rshift(b, prec - p)
rec = (ap * re2 + bp * im2) >> p
imc = (-ap * im2 + bp * re2) >> p
reb = (rec << p) // r4
imb = (imc << p) // r4
re = (reb + (n - 1) * lshift(re, p - prevp)) // n
im = (imb + (n - 1) * lshift(im, p - prevp)) // n
prevp = p
return re, im
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_nthroot(s, n, prec, rnd=round_fast):
"""nth-root of a positive number
Use the Newton method when faster, otherwise use x**(1/n)
"""
sign, man, exp, bc = s
if sign:
raise ComplexResult("nth root of a negative number")
if not man:
if s == fnan:
return fnan
if s == fzero:
if n > 0:
return fzero
if n == 0:
return fone
return finf
# Infinity
if not n:
return fnan
if n < 0:
return fzero
return finf
flag_inverse = False
if n < 2:
if n == 0:
return fone
if n == 1:
return mpf_pos(s, prec, rnd)
if n == -1:
return mpf_div(fone, s, prec, rnd)
# n < 0
rnd = reciprocal_rnd[rnd]
flag_inverse = True
extra_inverse = 5
prec += extra_inverse
n = -n
if n > 20 and (n >= 20000 or prec < int(233 + 28.3 * n**0.62)):
prec2 = prec + 10
fn = from_int(n)
nth = mpf_rdiv_int(1, fn, prec2)
r = mpf_pow(s, nth, prec2, rnd)
s = normalize(r[0], r[1], r[2], r[3], prec, rnd)
if flag_inverse:
return mpf_div(fone, s, prec-extra_inverse, rnd)
else:
return s
# Convert to a fixed-point number with prec2 bits.
prec2 = prec + 2*n - (prec%n)
# a few tests indicate that
# for 10 < n < 10**4 a bit more precision is needed
if n > 10:
prec2 += prec2//10
prec2 = prec2 - prec2%n
# Mantissa may have more bits than we need. Trim it down.
shift = bc - prec2
# Adjust exponents to make prec2 and exp+shift multiples of n.
sign1 = 0
es = exp+shift
if es < 0:
sign1 = 1
es = -es
if sign1:
shift += es%n
else:
shift -= es%n
man = rshift(man, shift)
extra = 10
exp1 = ((exp+shift-(n-1)*prec2)//n) - extra
rnd_shift = 0
if flag_inverse:
if rnd == 'u' or rnd == 'c':
rnd_shift = 1
else:
if rnd == 'd' or rnd == 'f':
rnd_shift = 1
man = nthroot_fixed(man+rnd_shift, n, prec2, exp1)
s = from_man_exp(man, exp1, prec, rnd)
if flag_inverse:
return mpf_div(fone, s, prec-extra_inverse, rnd)
else:
return s
def mpf_nthroot(s, n, prec, rnd=round_fast):
"""nth-root of a positive number
Use the Newton method when faster, otherwise use x**(1/n)
"""
sign, man, exp, bc = s
if sign:
raise ComplexResult("nth root of a negative number")
if not man:
if s == fnan:
return fnan
if s == fzero:
if n > 0:
return fzero
if n == 0:
return fone
return finf
# Infinity
if not n:
return fnan
if n < 0:
return fzero
return finf
flag_inverse = False
if n < 2:
if n == 0:
return fone
if n == 1:
return mpf_pos(s, prec, rnd)
if n == -1:
return mpf_div(fone, s, prec, rnd)
# n < 0
rnd = reciprocal_rnd[rnd]
flag_inverse = True
extra_inverse = 5
prec += extra_inverse
n = -n
if n > 20 and (n >= 20000 or prec < int(233 + 28.3 * n**0.62)):
prec2 = prec + 10
fn = from_int(n)
nth = mpf_rdiv_int(1, fn, prec2)
r = mpf_pow(s, nth, prec2, rnd)
s = normalize(r[0], r[1], r[2], r[3], prec, rnd)
if flag_inverse:
return mpf_div(fone, s, prec-extra_inverse, rnd)
else:
return s
# Convert to a fixed-point number with prec2 bits.
prec2 = prec + 2*n - (prec%n)
# a few tests indicate that
# for 10 < n < 10**4 a bit more precision is needed
if n > 10:
prec2 += prec2//10
prec2 = prec2 - prec2%n
# Mantissa may have more bits than we need. Trim it down.
shift = bc - prec2
# Adjust exponents to make prec2 and exp+shift multiples of n.
sign1 = 0
es = exp+shift
if es < 0:
sign1 = 1
es = -es
if sign1:
shift += es%n
else:
shift -= es%n
man = rshift(man, shift)
extra = 10
exp1 = ((exp+shift-(n-1)*prec2)//n) - extra
rnd_shift = 0
if flag_inverse:
if rnd == 'u' or rnd == 'c':
rnd_shift = 1
else:
if rnd == 'd' or rnd == 'f':
rnd_shift = 1
man = nthroot_fixed(man+rnd_shift, n, prec2, exp1)
s = from_man_exp(man, exp1, prec, rnd)
if flag_inverse:
return mpf_div(fone, s, prec-extra_inverse, rnd)
else:
return s
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)