Upload libmpc.py with huggingface_hub

libmpc.py

@@ -0,0 +1,835 @@
+"""
+Low-level functions for complex arithmetic.
+"""
+
+import sys
+
+from .backend import MPZ, MPZ_ZERO, MPZ_ONE, MPZ_TWO, BACKEND
+
+from .libmpf import (\
+    round_floor, round_ceiling, round_down, round_up,
+    round_nearest, round_fast, bitcount,
+    bctable, normalize, normalize1, reciprocal_rnd, rshift, lshift, giant_steps,
+    negative_rnd,
+    to_str, to_fixed, from_man_exp, from_float, to_float, from_int, to_int,
+    fzero, fone, ftwo, fhalf, finf, fninf, fnan, fnone,
+    mpf_abs, mpf_pos, mpf_neg, mpf_add, mpf_sub, mpf_mul,
+    mpf_div, mpf_mul_int, mpf_shift, mpf_sqrt, mpf_hypot,
+    mpf_rdiv_int, mpf_floor, mpf_ceil, mpf_nint, mpf_frac,
+    mpf_sign, mpf_hash,
+    ComplexResult
+)
+
+from .libelefun import (\
+    mpf_pi, mpf_exp, mpf_log, mpf_cos_sin, mpf_cosh_sinh, mpf_tan, mpf_pow_int,
+    mpf_log_hypot,
+    mpf_cos_sin_pi, mpf_phi,
+    mpf_cos, mpf_sin, mpf_cos_pi, mpf_sin_pi,
+    mpf_atan, mpf_atan2, mpf_cosh, mpf_sinh, mpf_tanh,
+    mpf_asin, mpf_acos, mpf_acosh, mpf_nthroot, mpf_fibonacci
+)
+
+# An mpc value is a (real, imag) tuple
+mpc_one = fone, fzero
+mpc_zero = fzero, fzero
+mpc_two = ftwo, fzero
+mpc_half = (fhalf, fzero)
+
+_infs = (finf, fninf)
+_infs_nan = (finf, fninf, fnan)
+
+def mpc_is_inf(z):
+    """Check if either real or imaginary part is infinite"""
+    re, im = z
+    if re in _infs: return True
+    if im in _infs: return True
+    return False
+
+def mpc_is_infnan(z):
+    """Check if either real or imaginary part is infinite or nan"""
+    re, im = z
+    if re in _infs_nan: return True
+    if im in _infs_nan: return True
+    return False
+
+def mpc_to_str(z, dps, **kwargs):
+    re, im = z
+    rs = to_str(re, dps)
+    if im[0]:
+        return rs + " - " + to_str(mpf_neg(im), dps, **kwargs) + "j"
+    else:
+        return rs + " + " + to_str(im, dps, **kwargs) + "j"
+
+def mpc_to_complex(z, strict=False, rnd=round_fast):
+    re, im = z
+    return complex(to_float(re, strict, rnd), to_float(im, strict, rnd))
+
+def mpc_hash(z):
+    if sys.version_info >= (3, 2):
+        re, im = z
+        h = mpf_hash(re) + sys.hash_info.imag * mpf_hash(im)
+        # Need to reduce either module 2^32 or 2^64
+        h = h % (2**sys.hash_info.width)
+        return int(h)
+    else:
+        try:
+            return hash(mpc_to_complex(z, strict=True))
+        except OverflowError:
+            return hash(z)
+
+def mpc_conjugate(z, prec, rnd=round_fast):
+    re, im = z
+    return re, mpf_neg(im, prec, rnd)
+
+def mpc_is_nonzero(z):
+    return z != mpc_zero
+
+def mpc_add(z, w, prec, rnd=round_fast):
+    a, b = z
+    c, d = w
+    return mpf_add(a, c, prec, rnd), mpf_add(b, d, prec, rnd)
+
+def mpc_add_mpf(z, x, prec, rnd=round_fast):
+    a, b = z
+    return mpf_add(a, x, prec, rnd), b
+
+def mpc_sub(z, w, prec=0, rnd=round_fast):
+    a, b = z
+    c, d = w
+    return mpf_sub(a, c, prec, rnd), mpf_sub(b, d, prec, rnd)
+
+def mpc_sub_mpf(z, p, prec=0, rnd=round_fast):
+    a, b = z
+    return mpf_sub(a, p, prec, rnd), b
+
+def mpc_pos(z, prec, rnd=round_fast):
+    a, b = z
+    return mpf_pos(a, prec, rnd), mpf_pos(b, prec, rnd)
+
+def mpc_neg(z, prec=None, rnd=round_fast):
+    a, b = z
+    return mpf_neg(a, prec, rnd), mpf_neg(b, prec, rnd)
+
+def mpc_shift(z, n):
+    a, b = z
+    return mpf_shift(a, n), mpf_shift(b, n)
+
+def mpc_abs(z, prec, rnd=round_fast):
+    """Absolute value of a complex number, |a+bi|.
+    Returns an mpf value."""
+    a, b = z
+    return mpf_hypot(a, b, prec, rnd)
+
+def mpc_arg(z, prec, rnd=round_fast):
+    """Argument of a complex number. Returns an mpf value."""
+    a, b = z
+    return mpf_atan2(b, a, prec, rnd)
+
+def mpc_floor(z, prec, rnd=round_fast):
+    a, b = z
+    return mpf_floor(a, prec, rnd), mpf_floor(b, prec, rnd)
+
+def mpc_ceil(z, prec, rnd=round_fast):
+    a, b = z
+    return mpf_ceil(a, prec, rnd), mpf_ceil(b, prec, rnd)
+
+def mpc_nint(z, prec, rnd=round_fast):
+    a, b = z
+    return mpf_nint(a, prec, rnd), mpf_nint(b, prec, rnd)
+
+def mpc_frac(z, prec, rnd=round_fast):
+    a, b = z
+    return mpf_frac(a, prec, rnd), mpf_frac(b, prec, rnd)
+
+
+def mpc_mul(z, w, prec, rnd=round_fast):
+    """
+    Complex multiplication.
+
+    Returns the real and imaginary part of (a+bi)*(c+di), rounded to
+    the specified precision. The rounding mode applies to the real and
+    imaginary parts separately.
+    """
+    a, b = z
+    c, d = w
+    p = mpf_mul(a, c)
+    q = mpf_mul(b, d)
+    r = mpf_mul(a, d)
+    s = mpf_mul(b, c)
+    re = mpf_sub(p, q, prec, rnd)
+    im = mpf_add(r, s, prec, rnd)
+    return re, im
+
+def mpc_square(z, prec, rnd=round_fast):
+    # (a+b*I)**2 == a**2 - b**2 + 2*I*a*b
+    a, b = z
+    p = mpf_mul(a,a)
+    q = mpf_mul(b,b)
+    r = mpf_mul(a,b, prec, rnd)
+    re = mpf_sub(p, q, prec, rnd)
+    im = mpf_shift(r, 1)
+    return re, im
+
+def mpc_mul_mpf(z, p, prec, rnd=round_fast):
+    a, b = z
+    re = mpf_mul(a, p, prec, rnd)
+    im = mpf_mul(b, p, prec, rnd)
+    return re, im
+
+def mpc_mul_imag_mpf(z, x, prec, rnd=round_fast):
+    """
+    Multiply the mpc value z by I*x where x is an mpf value.
+    """
+    a, b = z
+    re = mpf_neg(mpf_mul(b, x, prec, rnd))
+    im = mpf_mul(a, x, prec, rnd)
+    return re, im
+
+def mpc_mul_int(z, n, prec, rnd=round_fast):
+    a, b = z
+    re = mpf_mul_int(a, n, prec, rnd)
+    im = mpf_mul_int(b, n, prec, rnd)
+    return re, im
+
+def mpc_div(z, w, prec, rnd=round_fast):
+    a, b = z
+    c, d = w
+    wp = prec + 10
+    # mag = c*c + d*d
+    mag = mpf_add(mpf_mul(c, c), mpf_mul(d, d), wp)
+    # (a*c+b*d)/mag, (b*c-a*d)/mag
+    t = mpf_add(mpf_mul(a,c), mpf_mul(b,d), wp)
+    u = mpf_sub(mpf_mul(b,c), mpf_mul(a,d), wp)
+    return mpf_div(t,mag,prec,rnd), mpf_div(u,mag,prec,rnd)
+
+def mpc_div_mpf(z, p, prec, rnd=round_fast):
+    """Calculate z/p where p is real"""
+    a, b = z
+    re = mpf_div(a, p, prec, rnd)
+    im = mpf_div(b, p, prec, rnd)
+    return re, im
+
+def mpc_reciprocal(z, prec, rnd=round_fast):
+    """Calculate 1/z efficiently"""
+    a, b = z
+    m = mpf_add(mpf_mul(a,a),mpf_mul(b,b),prec+10)
+    re = mpf_div(a, m, prec, rnd)
+    im = mpf_neg(mpf_div(b, m, prec, rnd))
+    return re, im
+
+def mpc_mpf_div(p, z, prec, rnd=round_fast):
+    """Calculate p/z where p is real efficiently"""
+    a, b = z
+    m = mpf_add(mpf_mul(a,a),mpf_mul(b,b), prec+10)
+    re = mpf_div(mpf_mul(a,p), m, prec, rnd)
+    im = mpf_div(mpf_neg(mpf_mul(b,p)), m, prec, rnd)
+    return re, im
+
+def complex_int_pow(a, b, n):
+    """Complex integer power: computes (a+b*I)**n exactly for
+    nonnegative n (a and b must be Python ints)."""
+    wre = 1
+    wim = 0
+    while n:
+        if n & 1:
+            wre, wim = wre*a - wim*b, wim*a + wre*b
+            n -= 1
+        a, b = a*a - b*b, 2*a*b
+        n //= 2
+    return wre, wim
+
+def mpc_pow(z, w, prec, rnd=round_fast):
+    if w[1] == fzero:
+        return mpc_pow_mpf(z, w[0], prec, rnd)
+    return mpc_exp(mpc_mul(mpc_log(z, prec+10), w, prec+10), prec, rnd)
+
+def mpc_pow_mpf(z, p, prec, rnd=round_fast):
+    psign, pman, pexp, pbc = p
+    if pexp >= 0:
+        return mpc_pow_int(z, (-1)**psign * (pman<<pexp), prec, rnd)
+    if pexp == -1:
+        sqrtz = mpc_sqrt(z, prec+10)
+        return mpc_pow_int(sqrtz, (-1)**psign * pman, prec, rnd)
+    return mpc_exp(mpc_mul_mpf(mpc_log(z, prec+10), p, prec+10), prec, rnd)
+
+def mpc_pow_int(z, n, prec, rnd=round_fast):
+    a, b = z
+    if b == fzero:
+        return mpf_pow_int(a, n, prec, rnd), fzero
+    if a == fzero:
+        v = mpf_pow_int(b, n, prec, rnd)
+        n %= 4
+        if n == 0:
+            return v, fzero
+        elif n == 1:
+            return fzero, v
+        elif n == 2:
+            return mpf_neg(v), fzero
+        elif n == 3:
+            return fzero, mpf_neg(v)
+    if n == 0: return mpc_one
+    if n == 1: return mpc_pos(z, prec, rnd)
+    if n == 2: return mpc_square(z, prec, rnd)
+    if n == -1: return mpc_reciprocal(z, prec, rnd)
+    if n < 0: return mpc_reciprocal(mpc_pow_int(z, -n, prec+4), prec, rnd)
+    asign, aman, aexp, abc = a
+    bsign, bman, bexp, bbc = b
+    if asign: aman = -aman
+    if bsign: bman = -bman
+    de = aexp - bexp
+    abs_de = abs(de)
+    exact_size = n*(abs_de + max(abc, bbc))
+    if exact_size < 10000:
+        if de > 0:
+            aman <<= de
+            aexp = bexp
+        else:
+            bman <<= (-de)
+            bexp = aexp
+        re, im = complex_int_pow(aman, bman, n)
+        re = from_man_exp(re, int(n*aexp), prec, rnd)
+        im = from_man_exp(im, int(n*bexp), prec, rnd)
+        return re, im
+    return mpc_exp(mpc_mul_int(mpc_log(z, prec+10), n, prec+10), prec, rnd)
+
+def mpc_sqrt(z, prec, rnd=round_fast):
+    """Complex square root (principal branch).
+
+    We have sqrt(a+bi) = sqrt((r+a)/2) + b/sqrt(2*(r+a))*i where
+    r = abs(a+bi), when a+bi is not a negative real number."""
+    a, b = z
+    if b == fzero:
+        if a == fzero:
+            return (a, b)
+        # When a+bi is a negative real number, we get a real sqrt times i
+        if a[0]:
+            im = mpf_sqrt(mpf_neg(a), prec, rnd)
+            return (fzero, im)
+        else:
+            re = mpf_sqrt(a, prec, rnd)
+            return (re, fzero)
+    wp = prec+20
+    if not a[0]:                               # case a positive
+        t  = mpf_add(mpc_abs((a, b), wp), a, wp)  # t = abs(a+bi) + a
+        u = mpf_shift(t, -1)                      # u = t/2
+        re = mpf_sqrt(u, prec, rnd)               # re = sqrt(u)
+        v = mpf_shift(t, 1)                       # v = 2*t
+        w  = mpf_sqrt(v, wp)                      # w = sqrt(v)
+        im = mpf_div(b, w, prec, rnd)             # im = b / w
+    else:                                      # case a negative
+        t = mpf_sub(mpc_abs((a, b), wp), a, wp)   # t = abs(a+bi) - a
+        u = mpf_shift(t, -1)                      # u = t/2
+        im = mpf_sqrt(u, prec, rnd)               # im = sqrt(u)
+        v = mpf_shift(t, 1)                       # v = 2*t
+        w  = mpf_sqrt(v, wp)                      # w = sqrt(v)
+        re = mpf_div(b, w, prec, rnd)             # re = b/w
+        if b[0]:
+            re = mpf_neg(re)
+            im = mpf_neg(im)
+    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
+        re = MPZ(int(re))
+        im = MPZ(int(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(z, n, prec, rnd=round_fast):
+    """
+    Complex n-th root.
+
+    Use Newton method as in the real case when it is faster,
+    otherwise use z**(1/n)
+    """
+    a, b = z
+    if a[0] == 0 and b == fzero:
+        re = mpf_nthroot(a, n, prec, rnd)
+        return (re, fzero)
+    if n < 2:
+        if n == 0:
+            return mpc_one
+        if n == 1:
+            return mpc_pos((a, b), prec, rnd)
+        if n == -1:
+            return mpc_div(mpc_one, (a, b), prec, rnd)
+        inverse = mpc_nthroot((a, b), -n, prec+5, reciprocal_rnd[rnd])
+        return mpc_div(mpc_one, inverse, prec, rnd)
+    if n <= 20:
+        prec2 = int(1.2 * (prec + 10))
+        asign, aman, aexp, abc = a
+        bsign, bman, bexp, bbc = b
+        pf = mpc_abs((a,b), prec)
+        if pf[-2] + pf[-1] > -10  and pf[-2] + pf[-1] < prec:
+            af = to_fixed(a, prec2)
+            bf = to_fixed(b, prec2)
+            re, im = mpc_nthroot_fixed(af, bf, n, prec2)
+            extra = 10
+            re = from_man_exp(re, -prec2-extra, prec2, rnd)
+            im = from_man_exp(im, -prec2-extra, prec2, rnd)
+            return re, im
+    fn = from_int(n)
+    prec2 = prec+10 + 10
+    nth = mpf_rdiv_int(1, fn, prec2)
+    re, im = mpc_pow((a, b), (nth, fzero), prec2, rnd)
+    re = normalize(re[0], re[1], re[2], re[3], prec, rnd)
+    im = normalize(im[0], im[1], im[2], im[3], prec, rnd)
+    return re, im
+
+def mpc_cbrt(z, prec, rnd=round_fast):
+    """
+    Complex cubic root.
+    """
+    return mpc_nthroot(z, 3, prec, rnd)
+
+def mpc_exp(z, prec, rnd=round_fast):
+    """
+    Complex exponential function.
+
+    We use the direct formula exp(a+bi) = exp(a) * (cos(b) + sin(b)*i)
+    for the computation. This formula is very nice because it is
+    pefectly stable; since we just do real multiplications, the only
+    numerical errors that can creep in are single-ulp rounding errors.
+
+    The formula is efficient since mpmath's real exp is quite fast and
+    since we can compute cos and sin simultaneously.
+
+    It is no problem if a and b are large; if the implementations of
+    exp/cos/sin are accurate and efficient for all real numbers, then
+    so is this function for all complex numbers.
+    """
+    a, b = z
+    if a == fzero:
+        return mpf_cos_sin(b, prec, rnd)
+    if b == fzero:
+        return mpf_exp(a, prec, rnd), fzero
+    mag = mpf_exp(a, prec+4, rnd)
+    c, s = mpf_cos_sin(b, prec+4, rnd)
+    re = mpf_mul(mag, c, prec, rnd)
+    im = mpf_mul(mag, s, prec, rnd)
+    return re, im
+
+def mpc_log(z, prec, rnd=round_fast):
+    re = mpf_log_hypot(z[0], z[1], prec, rnd)
+    im = mpc_arg(z, prec, rnd)
+    return re, im
+
+def mpc_cos(z, prec, rnd=round_fast):
+    """Complex cosine. The formula used is cos(a+bi) = cos(a)*cosh(b) -
+    sin(a)*sinh(b)*i.
+
+    The same comments apply as for the complex exp: only real
+    multiplications are pewrormed, so no cancellation errors are
+    possible. The formula is also efficient since we can compute both
+    pairs (cos, sin) and (cosh, sinh) in single stwps."""
+    a, b = z
+    if b == fzero:
+        return mpf_cos(a, prec, rnd), fzero
+    if a == fzero:
+        return mpf_cosh(b, prec, rnd), fzero
+    wp = prec + 6
+    c, s = mpf_cos_sin(a, wp)
+    ch, sh = mpf_cosh_sinh(b, wp)
+    re = mpf_mul(c, ch, prec, rnd)
+    im = mpf_mul(s, sh, prec, rnd)
+    return re, mpf_neg(im)
+
+def mpc_sin(z, prec, rnd=round_fast):
+    """Complex sine. We have sin(a+bi) = sin(a)*cosh(b) +
+    cos(a)*sinh(b)*i. See the docstring for mpc_cos for additional
+    comments."""
+    a, b = z
+    if b == fzero:
+        return mpf_sin(a, prec, rnd), fzero
+    if a == fzero:
+        return fzero, mpf_sinh(b, prec, rnd)
+    wp = prec + 6
+    c, s = mpf_cos_sin(a, wp)
+    ch, sh = mpf_cosh_sinh(b, wp)
+    re = mpf_mul(s, ch, prec, rnd)
+    im = mpf_mul(c, sh, prec, rnd)
+    return re, im
+
+def mpc_tan(z, prec, rnd=round_fast):
+    """Complex tangent. Computed as tan(a+bi) = sin(2a)/M + sinh(2b)/M*i
+    where M = cos(2a) + cosh(2b)."""
+    a, b = z
+    asign, aman, aexp, abc = a
+    bsign, bman, bexp, bbc = b
+    if b == fzero: return mpf_tan(a, prec, rnd), fzero
+    if a == fzero: return fzero, mpf_tanh(b, prec, rnd)
+    wp = prec + 15
+    a = mpf_shift(a, 1)
+    b = mpf_shift(b, 1)
+    c, s = mpf_cos_sin(a, wp)
+    ch, sh = mpf_cosh_sinh(b, wp)
+    # TODO: handle cancellation when c ~=  -1 and ch ~= 1
+    mag = mpf_add(c, ch, wp)
+    re = mpf_div(s, mag, prec, rnd)
+    im = mpf_div(sh, mag, prec, rnd)
+    return re, im
+
+def mpc_cos_pi(z, prec, rnd=round_fast):
+    a, b = z
+    if b == fzero:
+        return mpf_cos_pi(a, prec, rnd), fzero
+    b = mpf_mul(b, mpf_pi(prec+5), prec+5)
+    if a == fzero:
+        return mpf_cosh(b, prec, rnd), fzero
+    wp = prec + 6
+    c, s = mpf_cos_sin_pi(a, wp)
+    ch, sh = mpf_cosh_sinh(b, wp)
+    re = mpf_mul(c, ch, prec, rnd)
+    im = mpf_mul(s, sh, prec, rnd)
+    return re, mpf_neg(im)
+
+def mpc_sin_pi(z, prec, rnd=round_fast):
+    a, b = z
+    if b == fzero:
+        return mpf_sin_pi(a, prec, rnd), fzero
+    b = mpf_mul(b, mpf_pi(prec+5), prec+5)
+    if a == fzero:
+        return fzero, mpf_sinh(b, prec, rnd)
+    wp = prec + 6
+    c, s = mpf_cos_sin_pi(a, wp)
+    ch, sh = mpf_cosh_sinh(b, wp)
+    re = mpf_mul(s, ch, prec, rnd)
+    im = mpf_mul(c, sh, prec, rnd)
+    return re, im
+
+def mpc_cos_sin(z, prec, rnd=round_fast):
+    a, b = z
+    if a == fzero:
+        ch, sh = mpf_cosh_sinh(b, prec, rnd)
+        return (ch, fzero), (fzero, sh)
+    if b == fzero:
+        c, s = mpf_cos_sin(a, prec, rnd)
+        return (c, fzero), (s, fzero)
+    wp = prec + 6
+    c, s = mpf_cos_sin(a, wp)
+    ch, sh = mpf_cosh_sinh(b, wp)
+    cre = mpf_mul(c, ch, prec, rnd)
+    cim = mpf_mul(s, sh, prec, rnd)
+    sre = mpf_mul(s, ch, prec, rnd)
+    sim = mpf_mul(c, sh, prec, rnd)
+    return (cre, mpf_neg(cim)), (sre, sim)
+
+def mpc_cos_sin_pi(z, prec, rnd=round_fast):
+    a, b = z
+    if b == fzero:
+        c, s = mpf_cos_sin_pi(a, prec, rnd)
+        return (c, fzero), (s, fzero)
+    b = mpf_mul(b, mpf_pi(prec+5), prec+5)
+    if a == fzero:
+        ch, sh = mpf_cosh_sinh(b, prec, rnd)
+        return (ch, fzero), (fzero, sh)
+    wp = prec + 6
+    c, s = mpf_cos_sin_pi(a, wp)
+    ch, sh = mpf_cosh_sinh(b, wp)
+    cre = mpf_mul(c, ch, prec, rnd)
+    cim = mpf_mul(s, sh, prec, rnd)
+    sre = mpf_mul(s, ch, prec, rnd)
+    sim = mpf_mul(c, sh, prec, rnd)
+    return (cre, mpf_neg(cim)), (sre, sim)
+
+def mpc_cosh(z, prec, rnd=round_fast):
+    """Complex hyperbolic cosine. Computed as cosh(z) = cos(z*i)."""
+    a, b = z
+    return mpc_cos((b, mpf_neg(a)), prec, rnd)
+
+def mpc_sinh(z, prec, rnd=round_fast):
+    """Complex hyperbolic sine. Computed as sinh(z) = -i*sin(z*i)."""
+    a, b = z
+    b, a = mpc_sin((b, a), prec, rnd)
+    return a, b
+
+def mpc_tanh(z, prec, rnd=round_fast):
+    """Complex hyperbolic tangent. Computed as tanh(z) = -i*tan(z*i)."""
+    a, b = z
+    b, a = mpc_tan((b, a), prec, rnd)
+    return a, b
+
+# TODO: avoid loss of accuracy
+def mpc_atan(z, prec, rnd=round_fast):
+    a, b = z
+    # atan(z) = (I/2)*(log(1-I*z) - log(1+I*z))
+    # x = 1-I*z = 1 + b - I*a
+    # y = 1+I*z = 1 - b + I*a
+    wp = prec + 15
+    x = mpf_add(fone, b, wp), mpf_neg(a)
+    y = mpf_sub(fone, b, wp), a
+    l1 = mpc_log(x, wp)
+    l2 = mpc_log(y, wp)
+    a, b = mpc_sub(l1, l2, prec, rnd)
+    # (I/2) * (a+b*I) = (-b/2 + a/2*I)
+    v = mpf_neg(mpf_shift(b,-1)), mpf_shift(a,-1)
+    # Subtraction at infinity gives correct real part but
+    # wrong imaginary part (should be zero)
+    if v[1] == fnan and mpc_is_inf(z):
+        v = (v[0], fzero)
+    return v
+
+beta_crossover = from_float(0.6417)
+alpha_crossover = from_float(1.5)
+
+def acos_asin(z, prec, rnd, n):
+    """ complex acos for n = 0, asin for n = 1
+    The algorithm is described in
+    T.E. Hull, T.F. Fairgrieve and P.T.P. Tang
+    'Implementing the Complex Arcsine and Arcosine Functions
+    using Exception Handling',
+    ACM Trans. on Math. Software Vol. 23 (1997), p299
+    The complex acos and asin can be defined as
+    acos(z) = acos(beta) - I*sign(a)* log(alpha + sqrt(alpha**2 -1))
+    asin(z) = asin(beta) + I*sign(a)* log(alpha + sqrt(alpha**2 -1))
+    where z = a + I*b
+    alpha = (1/2)*(r + s); beta = (1/2)*(r - s) = a/alpha
+    r = sqrt((a+1)**2 + y**2); s = sqrt((a-1)**2 + y**2)
+    These expressions are rewritten in different ways in different
+    regions, delimited by two crossovers alpha_crossover and beta_crossover,
+    and by abs(a) <= 1, in order to improve the numerical accuracy.
+    """
+    a, b = z
+    wp = prec + 10
+    # special cases with real argument
+    if b == fzero:
+        am = mpf_sub(fone, mpf_abs(a), wp)
+        # case abs(a) <= 1
+        if not am[0]:
+            if n == 0:
+                return mpf_acos(a, prec, rnd), fzero
+            else:
+                return mpf_asin(a, prec, rnd), fzero
+        # cases abs(a) > 1
+        else:
+            # case a < -1
+            if a[0]:
+                pi = mpf_pi(prec, rnd)
+                c = mpf_acosh(mpf_neg(a), prec, rnd)
+                if n == 0:
+                    return pi, mpf_neg(c)
+                else:
+                    return mpf_neg(mpf_shift(pi, -1)), c
+            # case a > 1
+            else:
+                c = mpf_acosh(a, prec, rnd)
+                if n == 0:
+                    return fzero, c
+                else:
+                    pi = mpf_pi(prec, rnd)
+                    return mpf_shift(pi, -1), mpf_neg(c)
+    asign = bsign = 0
+    if a[0]:
+        a = mpf_neg(a)
+        asign = 1
+    if b[0]:
+        b = mpf_neg(b)
+        bsign = 1
+    am = mpf_sub(fone, a, wp)
+    ap = mpf_add(fone, a, wp)
+    r = mpf_hypot(ap, b, wp)
+    s = mpf_hypot(am, b, wp)
+    alpha = mpf_shift(mpf_add(r, s, wp), -1)
+    beta = mpf_div(a, alpha, wp)
+    b2 = mpf_mul(b,b, wp)
+    # case beta <= beta_crossover
+    if not mpf_sub(beta_crossover, beta, wp)[0]:
+        if n == 0:
+            re = mpf_acos(beta, wp)
+        else:
+            re = mpf_asin(beta, wp)
+    else:
+        # to compute the real part in this region use the identity
+        # asin(beta) = atan(beta/sqrt(1-beta**2))
+        # beta/sqrt(1-beta**2) = (alpha + a) * (alpha - a)
+        # alpha + a is numerically accurate; alpha - a can have
+        # cancellations leading to numerical inaccuracies, so rewrite
+        # it in differente ways according to the region
+        Ax = mpf_add(alpha, a, wp)
+        # case a <= 1
+        if not am[0]:
+            # c = b*b/(r + (a+1)); d = (s + (1-a))
+            # alpha - a = (1/2)*(c + d)
+            # case n=0: re = atan(sqrt((1/2) * Ax * (c + d))/a)
+            # case n=1: re = atan(a/sqrt((1/2) * Ax * (c + d)))
+            c = mpf_div(b2, mpf_add(r, ap, wp), wp)
+            d = mpf_add(s, am, wp)
+            re = mpf_shift(mpf_mul(Ax, mpf_add(c, d, wp), wp), -1)
+            if n == 0:
+                re = mpf_atan(mpf_div(mpf_sqrt(re, wp), a, wp), wp)
+            else:
+                re = mpf_atan(mpf_div(a, mpf_sqrt(re, wp), wp), wp)
+        else:
+            # c = Ax/(r + (a+1)); d = Ax/(s - (1-a))
+            # alpha - a = (1/2)*(c + d)
+            # case n = 0: re = atan(b*sqrt(c + d)/2/a)
+            # case n = 1: re = atan(a/(b*sqrt(c + d)/2)
+            c = mpf_div(Ax, mpf_add(r, ap, wp), wp)
+            d = mpf_div(Ax, mpf_sub(s, am, wp), wp)
+            re = mpf_shift(mpf_add(c, d, wp), -1)
+            re = mpf_mul(b, mpf_sqrt(re, wp), wp)
+            if n == 0:
+                re = mpf_atan(mpf_div(re, a, wp), wp)
+            else:
+                re = mpf_atan(mpf_div(a, re, wp), wp)
+    # to compute alpha + sqrt(alpha**2 - 1), if alpha <= alpha_crossover
+    # replace it with 1 + Am1 + sqrt(Am1*(alpha+1)))
+    # where Am1 = alpha -1
+    # if alpha <= alpha_crossover:
+    if not mpf_sub(alpha_crossover, alpha, wp)[0]:
+        c1 = mpf_div(b2, mpf_add(r, ap, wp), wp)
+        # case a < 1
+        if mpf_neg(am)[0]:
+            # Am1 = (1/2) * (b*b/(r + (a+1)) + b*b/(s + (1-a))
+            c2 = mpf_add(s, am, wp)
+            c2 = mpf_div(b2, c2, wp)
+            Am1 = mpf_shift(mpf_add(c1, c2, wp), -1)
+        else:
+            # Am1 = (1/2) * (b*b/(r + (a+1)) + (s - (1-a)))
+            c2 = mpf_sub(s, am, wp)
+            Am1 = mpf_shift(mpf_add(c1, c2, wp), -1)
+        # im = log(1 + Am1 + sqrt(Am1*(alpha+1)))
+        im = mpf_mul(Am1, mpf_add(alpha, fone, wp), wp)
+        im = mpf_log(mpf_add(fone, mpf_add(Am1, mpf_sqrt(im, wp), wp), wp), wp)
+    else:
+        # im = log(alpha + sqrt(alpha*alpha - 1))
+        im = mpf_sqrt(mpf_sub(mpf_mul(alpha, alpha, wp), fone, wp), wp)
+        im = mpf_log(mpf_add(alpha, im, wp), wp)
+    if asign:
+        if n == 0:
+            re = mpf_sub(mpf_pi(wp), re, wp)
+        else:
+            re = mpf_neg(re)
+    if not bsign and n == 0:
+        im = mpf_neg(im)
+    if bsign and n == 1:
+        im = mpf_neg(im)
+    re = normalize(re[0], re[1], re[2], re[3], prec, rnd)
+    im = normalize(im[0], im[1], im[2], im[3], prec, rnd)
+    return re, im
+
+def mpc_acos(z, prec, rnd=round_fast):
+    return acos_asin(z, prec, rnd, 0)
+
+def mpc_asin(z, prec, rnd=round_fast):
+    return acos_asin(z, prec, rnd, 1)
+
+def mpc_asinh(z, prec, rnd=round_fast):
+    # asinh(z) = I * asin(-I z)
+    a, b = z
+    a, b =  mpc_asin((b, mpf_neg(a)), prec, rnd)
+    return mpf_neg(b), a
+
+def mpc_acosh(z, prec, rnd=round_fast):
+    # acosh(z) = -I * acos(z)   for Im(acos(z)) <= 0
+    #            +I * acos(z)   otherwise
+    a, b = mpc_acos(z, prec, rnd)
+    if b[0] or b == fzero:
+        return mpf_neg(b), a
+    else:
+        return b, mpf_neg(a)
+
+def mpc_atanh(z, prec, rnd=round_fast):
+    # atanh(z) = (log(1+z)-log(1-z))/2
+    wp = prec + 15
+    a = mpc_add(z, mpc_one, wp)
+    b = mpc_sub(mpc_one, z, wp)
+    a = mpc_log(a, wp)
+    b = mpc_log(b, wp)
+    v = mpc_shift(mpc_sub(a, b, wp), -1)
+    # Subtraction at infinity gives correct imaginary part but
+    # wrong real part (should be zero)
+    if v[0] == fnan and mpc_is_inf(z):
+        v = (fzero, v[1])
+    return v
+
+def mpc_fibonacci(z, prec, rnd=round_fast):
+    re, im = z
+    if im == fzero:
+        return (mpf_fibonacci(re, prec, rnd), fzero)
+    size = max(abs(re[2]+re[3]), abs(re[2]+re[3]))
+    wp = prec + size + 20
+    a = mpf_phi(wp)
+    b = mpf_add(mpf_shift(a, 1), fnone, wp)
+    u = mpc_pow((a, fzero), z, wp)
+    v = mpc_cos_pi(z, wp)
+    v = mpc_div(v, u, wp)
+    u = mpc_sub(u, v, wp)
+    u = mpc_div_mpf(u, b, prec, rnd)
+    return u
+
+def mpf_expj(x, prec, rnd='f'):
+    raise ComplexResult
+
+def mpc_expj(z, prec, rnd='f'):
+    re, im = z
+    if im == fzero:
+        return mpf_cos_sin(re, prec, rnd)
+    if re == fzero:
+        return mpf_exp(mpf_neg(im), prec, rnd), fzero
+    ey = mpf_exp(mpf_neg(im), prec+10)
+    c, s = mpf_cos_sin(re, prec+10)
+    re = mpf_mul(ey, c, prec, rnd)
+    im = mpf_mul(ey, s, prec, rnd)
+    return re, im
+
+def mpf_expjpi(x, prec, rnd='f'):
+    raise ComplexResult
+
+def mpc_expjpi(z, prec, rnd='f'):
+    re, im = z
+    if im == fzero:
+        return mpf_cos_sin_pi(re, prec, rnd)
+    sign, man, exp, bc = im
+    wp = prec+10
+    if man:
+        wp += max(0, exp+bc)
+    im = mpf_neg(mpf_mul(mpf_pi(wp), im, wp))
+    if re == fzero:
+        return mpf_exp(im, prec, rnd), fzero
+    ey = mpf_exp(im, prec+10)
+    c, s = mpf_cos_sin_pi(re, prec+10)
+    re = mpf_mul(ey, c, prec, rnd)
+    im = mpf_mul(ey, s, prec, rnd)
+    return re, im
+
+
+if BACKEND == 'sage':
+    try:
+        import sage.libs.mpmath.ext_libmp as _lbmp
+        mpc_exp = _lbmp.mpc_exp
+        mpc_sqrt = _lbmp.mpc_sqrt
+    except (ImportError, AttributeError):
+        print("Warning: Sage imports in libmpc failed")