def mpf_asin(x, prec, rnd=round_fast):
sign, man, exp, bc = x
if bc+exp > 0 and x not in (fone, fnone):
raise ComplexResult("asin(x) is real only for -1 <= x <= 1")
flag_nr = True
if prec < 1000 or exp+bc < -13:
flag_nr = False
else:
ebc = exp + bc
if ebc < -13:
flag_nr = False
elif ebc < -3:
if prec < 3000:
flag_nr = False
if not flag_nr:
# asin(x) = 2*atan(x/(1+sqrt(1-x**2)))
wp = prec + 15
a = mpf_mul(x, x)
b = mpf_add(fone, mpf_sqrt(mpf_sub(fone, a, wp), wp), wp)
c = mpf_div(x, b, wp)
return mpf_shift(mpf_atan(c, prec, rnd), 1)
# use Newton's method
extra = 10
extra_p = 10
prec2 = prec + extra
r = math.asin(to_float(x))
r = from_float(r, 50, rnd)
for p in giant_steps(50, prec2):
wp = p + extra_p
c, s = cos_sin(r, wp, rnd)
tmp = mpf_sub(x, s, wp, rnd)
tmp = mpf_div(tmp, c, wp, rnd)
r = mpf_add(r, tmp, wp, rnd)
sign, man, exp, bc = r
return normalize(sign, man, exp, bc, prec, rnd)
def mpf_asin(x, prec, rnd=round_fast):
sign, man, exp, bc = x
if bc+exp > 0 and x not in (fone, fnone):
raise ComplexResult("asin(x) is real only for -1 <= x <= 1")
flag_nr = True
if prec < 1000 or exp+bc < -13:
flag_nr = False
else:
ebc = exp + bc
if ebc < -13:
flag_nr = False
elif ebc < -3:
if prec < 3000:
flag_nr = False
if not flag_nr:
# asin(x) = 2*atan(x/(1+sqrt(1-x**2)))
wp = prec + 15
a = mpf_mul(x, x)
b = mpf_add(fone, mpf_sqrt(mpf_sub(fone, a, wp), wp), wp)
c = mpf_div(x, b, wp)
return mpf_shift(mpf_atan(c, prec, rnd), 1)
# use Newton's method
extra = 10
extra_p = 10
prec2 = prec + extra
r = math.asin(to_float(x))
r = from_float(r, 50, rnd)
for p in giant_steps(50, prec2):
wp = p + extra_p
c, s = cos_sin(r, wp, rnd)
tmp = mpf_sub(x, s, wp, rnd)
tmp = mpf_div(tmp, c, wp, rnd)
r = mpf_add(r, tmp, wp, rnd)
sign, man, exp, bc = r
return normalize(sign, man, exp, bc, prec, rnd)
def exp_newton(x, prec):
extra = 10
r = mpf_exp(x, 60)
start = 50
prevp = start
for p in giant_steps(start, prec+extra, 4):
h = mpf_sub(x, mpf_log(r, p), p)
h2 = mpf_mul(h, h, p)
h3 = mpf_mul(h2, h, p)
h4 = mpf_mul(h2, h2, p)
t = mpf_add(h, mpf_shift(h2, -1), p)
t = mpf_add(t, mpf_div(h3, from_int(6, p), p), p)
t = mpf_add(t, mpf_div(h4, from_int(24, p), p), p)
t = mpf_mul(r, t, p)
r = mpf_add(r, t, p)
return r
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 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 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
