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libelefun.py

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+"""
+This module implements computation of elementary transcendental
+functions (powers, logarithms, trigonometric and hyperbolic
+functions, inverse trigonometric and hyperbolic) for real
+floating-point numbers.
+
+For complex and interval implementations of the same functions,
+see libmpc and libmpi.
+
+"""
+
+import math
+from bisect import bisect
+
+from .backend import xrange
+from .backend import MPZ, MPZ_ZERO, MPZ_ONE, MPZ_TWO, MPZ_FIVE, BACKEND
+
+from .libmpf import (
+    round_floor, round_ceiling, round_down, round_up,
+    round_nearest, round_fast,
+    ComplexResult,
+    bitcount, bctable, lshift, rshift, giant_steps, sqrt_fixed,
+    from_int, to_int, from_man_exp, to_fixed, to_float, from_float,
+    from_rational, normalize,
+    fzero, fone, fnone, fhalf, finf, fninf, fnan,
+    mpf_cmp, mpf_sign, mpf_abs,
+    mpf_pos, mpf_neg, mpf_add, mpf_sub, mpf_mul, mpf_div, mpf_shift,
+    mpf_rdiv_int, mpf_pow_int, mpf_sqrt,
+    reciprocal_rnd, negative_rnd, mpf_perturb,
+    isqrt_fast
+)
+
+from .libintmath import ifib
+
+
+#-------------------------------------------------------------------------------
+# Tuning parameters
+#-------------------------------------------------------------------------------
+
+# Cutoff for computing exp from cosh+sinh. This reduces the
+# number of terms by half, but also requires a square root which
+# is expensive with the pure-Python square root code.
+if BACKEND == 'python':
+    EXP_COSH_CUTOFF = 600
+else:
+    EXP_COSH_CUTOFF = 400
+# Cutoff for using more than 2 series
+EXP_SERIES_U_CUTOFF = 1500
+
+# Also basically determined by sqrt
+if BACKEND == 'python':
+    COS_SIN_CACHE_PREC = 400
+else:
+    COS_SIN_CACHE_PREC = 200
+COS_SIN_CACHE_STEP = 8
+cos_sin_cache = {}
+
+# Number of integer logarithms to cache (for zeta sums)
+MAX_LOG_INT_CACHE = 2000
+log_int_cache = {}
+
+LOG_TAYLOR_PREC = 2500  # Use Taylor series with caching up to this prec
+LOG_TAYLOR_SHIFT = 9    # Cache log values in steps of size 2^-N
+log_taylor_cache = {}
+# prec/size ratio of x for fastest convergence in AGM formula
+LOG_AGM_MAG_PREC_RATIO = 20
+
+ATAN_TAYLOR_PREC = 3000  # Same as for log
+ATAN_TAYLOR_SHIFT = 7   # steps of size 2^-N
+atan_taylor_cache = {}
+
+
+# ~= next power of two + 20
+cache_prec_steps = [22,22]
+for k in xrange(1, bitcount(LOG_TAYLOR_PREC)+1):
+    cache_prec_steps += [min(2**k,LOG_TAYLOR_PREC)+20] * 2**(k-1)
+
+
+#----------------------------------------------------------------------------#
+#                                                                            #
+#                   Elementary mathematical constants                        #
+#                                                                            #
+#----------------------------------------------------------------------------#
+
+def constant_memo(f):
+    """
+    Decorator for caching computed values of mathematical
+    constants. This decorator should be applied to a
+    function taking a single argument prec as input and
+    returning a fixed-point value with the given precision.
+    """
+    f.memo_prec = -1
+    f.memo_val = None
+    def g(prec, **kwargs):
+        memo_prec = f.memo_prec
+        if prec <= memo_prec:
+            return f.memo_val >> (memo_prec-prec)
+        newprec = int(prec*1.05+10)
+        f.memo_val = f(newprec, **kwargs)
+        f.memo_prec = newprec
+        return f.memo_val >> (newprec-prec)
+    g.__name__ = f.__name__
+    g.__doc__ = f.__doc__
+    return g
+
+def def_mpf_constant(fixed):
+    """
+    Create a function that computes the mpf value for a mathematical
+    constant, given a function that computes the fixed-point value.
+
+    Assumptions: the constant is positive and has magnitude ~= 1;
+    the fixed-point function rounds to floor.
+    """
+    def f(prec, rnd=round_fast):
+        wp = prec + 20
+        v = fixed(wp)
+        if rnd in (round_up, round_ceiling):
+            v += 1
+        return normalize(0, v, -wp, bitcount(v), prec, rnd)
+    f.__doc__ = fixed.__doc__
+    return f
+
+def bsp_acot(q, a, b, hyperbolic):
+    if b - a == 1:
+        a1 = MPZ(2*a + 3)
+        if hyperbolic or a&1:
+            return MPZ_ONE, a1 * q**2, a1
+        else:
+            return -MPZ_ONE, a1 * q**2, a1
+    m = (a+b)//2
+    p1, q1, r1 = bsp_acot(q, a, m, hyperbolic)
+    p2, q2, r2 = bsp_acot(q, m, b, hyperbolic)
+    return q2*p1 + r1*p2, q1*q2, r1*r2
+
+# the acoth(x) series converges like the geometric series for x^2
+# N = ceil(p*log(2)/(2*log(x)))
+def acot_fixed(a, prec, hyperbolic):
+    """
+    Compute acot(a) or acoth(a) for an integer a with binary splitting; see
+    http://numbers.computation.free.fr/Constants/Algorithms/splitting.html
+    """
+    N = int(0.35 * prec/math.log(a) + 20)
+    p, q, r = bsp_acot(a, 0,N, hyperbolic)
+    return ((p+q)<<prec)//(q*a)
+
+def machin(coefs, prec, hyperbolic=False):
+    """
+    Evaluate a Machin-like formula, i.e., a linear combination of
+    acot(n) or acoth(n) for specific integer values of n, using fixed-
+    point arithmetic. The input should be a list [(c, n), ...], giving
+    c*acot[h](n) + ...
+    """
+    extraprec = 10
+    s = MPZ_ZERO
+    for a, b in coefs:
+        s += MPZ(a) * acot_fixed(MPZ(b), prec+extraprec, hyperbolic)
+    return (s >> extraprec)
+
+# Logarithms of integers are needed for various computations involving
+# logarithms, powers, radix conversion, etc
+
+@constant_memo
+def ln2_fixed(prec):
+    """
+    Computes ln(2). This is done with a hyperbolic Machin-type formula,
+    with binary splitting at high precision.
+    """
+    return machin([(18, 26), (-2, 4801), (8, 8749)], prec, True)
+
+@constant_memo
+def ln10_fixed(prec):
+    """
+    Computes ln(10). This is done with a hyperbolic Machin-type formula.
+    """
+    return machin([(46, 31), (34, 49), (20, 161)], prec, True)
+
+
+r"""
+For computation of pi, we use the Chudnovsky series:
+
+             oo
+             ___        k
+      1     \       (-1)  (6 k)! (A + B k)
+    ----- =  )     -----------------------
+    12 pi   /___               3  3k+3/2
+                    (3 k)! (k!)  C
+            k = 0
+
+where A, B, and C are certain integer constants. This series adds roughly
+14 digits per term. Note that C^(3/2) can be extracted so that the
+series contains only rational terms. This makes binary splitting very
+efficient.
+
+The recurrence formulas for the binary splitting were taken from
+ftp://ftp.gmplib.org/pub/src/gmp-chudnovsky.c
+
+Previously, Machin's formula was used at low precision and the AGM iteration
+was used at high precision. However, the Chudnovsky series is essentially as
+fast as the Machin formula at low precision and in practice about 3x faster
+than the AGM at high precision (despite theoretically having a worse
+asymptotic complexity), so there is no reason not to use it in all cases.
+
+"""
+
+# Constants in Chudnovsky's series
+CHUD_A = MPZ(13591409)
+CHUD_B = MPZ(545140134)
+CHUD_C = MPZ(640320)
+CHUD_D = MPZ(12)
+
+def bs_chudnovsky(a, b, level, verbose):
+    """
+    Computes the sum from a to b of the series in the Chudnovsky
+    formula. Returns g, p, q where p/q is the sum as an exact
+    fraction and g is a temporary value used to save work
+    for recursive calls.
+    """
+    if b-a == 1:
+        g = MPZ((6*b-5)*(2*b-1)*(6*b-1))
+        p = b**3 * CHUD_C**3 // 24
+        q = (-1)**b * g * (CHUD_A+CHUD_B*b)
+    else:
+        if verbose and level < 4:
+            print("  binary splitting", a, b)
+        mid = (a+b)//2
+        g1, p1, q1 = bs_chudnovsky(a, mid, level+1, verbose)
+        g2, p2, q2 = bs_chudnovsky(mid, b, level+1, verbose)
+        p = p1*p2
+        g = g1*g2
+        q = q1*p2 + q2*g1
+    return g, p, q
+
+@constant_memo
+def pi_fixed(prec, verbose=False, verbose_base=None):
+    """
+    Compute floor(pi * 2**prec) as a big integer.
+
+    This is done using Chudnovsky's series (see comments in
+    libelefun.py for details).
+    """
+    # The Chudnovsky series gives 14.18 digits per term
+    N = int(prec/3.3219280948/14.181647462 + 2)
+    if verbose:
+        print("binary splitting with N =", N)
+    g, p, q = bs_chudnovsky(0, N, 0, verbose)
+    sqrtC = isqrt_fast(CHUD_C<<(2*prec))
+    v = p*CHUD_C*sqrtC//((q+CHUD_A*p)*CHUD_D)
+    return v
+
+def degree_fixed(prec):
+    return pi_fixed(prec)//180
+
+def bspe(a, b):
+    """
+    Sum series for exp(1)-1 between a, b, returning the result
+    as an exact fraction (p, q).
+    """
+    if b-a == 1:
+        return MPZ_ONE, MPZ(b)
+    m = (a+b)//2
+    p1, q1 = bspe(a, m)
+    p2, q2 = bspe(m, b)
+    return p1*q2+p2, q1*q2
+
+@constant_memo
+def e_fixed(prec):
+    """
+    Computes exp(1). This is done using the ordinary Taylor series for
+    exp, with binary splitting. For a description of the algorithm,
+    see:
+
+        http://numbers.computation.free.fr/Constants/
+            Algorithms/splitting.html
+    """
+    # Slight overestimate of N needed for 1/N! < 2**(-prec)
+    # This could be tightened for large N.
+    N = int(1.1*prec/math.log(prec) + 20)
+    p, q = bspe(0,N)
+    return ((p+q)<<prec)//q
+
+@constant_memo
+def phi_fixed(prec):
+    """
+    Computes the golden ratio, (1+sqrt(5))/2
+    """
+    prec += 10
+    a = isqrt_fast(MPZ_FIVE<<(2*prec)) + (MPZ_ONE << prec)
+    return a >> 11
+
+mpf_phi    = def_mpf_constant(phi_fixed)
+mpf_pi     = def_mpf_constant(pi_fixed)
+mpf_e      = def_mpf_constant(e_fixed)
+mpf_degree = def_mpf_constant(degree_fixed)
+mpf_ln2    = def_mpf_constant(ln2_fixed)
+mpf_ln10   = def_mpf_constant(ln10_fixed)
+
+
+@constant_memo
+def ln_sqrt2pi_fixed(prec):
+    wp = prec + 10
+    # ln(sqrt(2*pi)) = ln(2*pi)/2
+    return to_fixed(mpf_log(mpf_shift(mpf_pi(wp), 1), wp), prec-1)
+
+@constant_memo
+def sqrtpi_fixed(prec):
+    return sqrt_fixed(pi_fixed(prec), prec)
+
+mpf_sqrtpi   = def_mpf_constant(sqrtpi_fixed)
+mpf_ln_sqrt2pi   = def_mpf_constant(ln_sqrt2pi_fixed)
+
+
+#----------------------------------------------------------------------------#
+#                                                                            #
+#                                    Powers                                  #
+#                                                                            #
+#----------------------------------------------------------------------------#
+
+def mpf_pow(s, t, prec, rnd=round_fast):
+    """
+    Compute s**t. Raises ComplexResult if s is negative and t is
+    fractional.
+    """
+    ssign, sman, sexp, sbc = s
+    tsign, tman, texp, tbc = t
+    if ssign and texp < 0:
+        raise ComplexResult("negative number raised to a fractional power")
+    if texp >= 0:
+        return mpf_pow_int(s, (-1)**tsign * (tman<<texp), prec, rnd)
+    # s**(n/2) = sqrt(s)**n
+    if texp == -1:
+        if tman == 1:
+            if tsign:
+                return mpf_div(fone, mpf_sqrt(s, prec+10,
+                    reciprocal_rnd[rnd]), prec, rnd)
+            return mpf_sqrt(s, prec, rnd)
+        else:
+            if tsign:
+                return mpf_pow_int(mpf_sqrt(s, prec+10,
+                    reciprocal_rnd[rnd]), -tman, prec, rnd)
+            return mpf_pow_int(mpf_sqrt(s, prec+10, rnd), tman, prec, rnd)
+    # General formula: s**t = exp(t*log(s))
+    # TODO: handle rnd direction of the logarithm carefully
+    c = mpf_log(s, prec+10, rnd)
+    return mpf_exp(mpf_mul(t, c), prec, rnd)
+
+def int_pow_fixed(y, n, prec):
+    """n-th power of a fixed point number with precision prec
+
+       Returns the power in the form man, exp,
+       man * 2**exp ~= y**n
+    """
+    if n == 2:
+        return (y*y), 0
+    bc = bitcount(y)
+    exp = 0
+    workprec = 2 * (prec + 4*bitcount(n) + 4)
+    _, pm, pe, pbc = fone
+    while 1:
+        if n & 1:
+            pm = pm*y
+            pe = pe+exp
+            pbc += bc - 2
+            pbc = pbc + bctable[int(pm >> pbc)]
+            if pbc > workprec:
+                pm = pm >> (pbc-workprec)
+                pe += pbc - workprec
+                pbc = workprec
+            n -= 1
+            if not n:
+                break
+        y = y*y
+        exp = exp+exp
+        bc = bc + bc - 2
+        bc = bc + bctable[int(y >> bc)]
+        if bc > workprec:
+            y = y >> (bc-workprec)
+            exp += bc - workprec
+            bc = workprec
+        n = n // 2
+    return pm, pe
+
+# froot(s, n, prec, rnd) computes the real n-th root of a
+# positive mpf tuple s.
+# To compute the root we start from a 50-bit estimate for r
+# generated with ordinary floating-point arithmetic, and then refine
+# the value to full accuracy using the iteration
+
+#            1  /                     y       \
+#   r     = --- | (n-1)  * r   +  ----------  |
+#    n+1     n  \           n     r_n**(n-1)  /
+
+# which is simply Newton's method applied to the equation r**n = y.
+# With giant_steps(start, prec+extra) = [p0,...,pm, prec+extra]
+# and y = man * 2**-shift  one has
+# (man * 2**exp)**(1/n) =
+# y**(1/n) * 2**(start-prec/n) * 2**(p0-start) * ... * 2**(prec+extra-pm) *
+# 2**((exp+shift-(n-1)*prec)/n -extra))
+# The last factor is accounted for in the last line of froot.
+
+def nthroot_fixed(y, n, prec, exp1):
+    start = 50
+    try:
+        y1 = rshift(y, prec - n*start)
+        r = MPZ(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 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_cbrt(s, prec, rnd=round_fast):
+    """cubic root of a positive number"""
+    return mpf_nthroot(s, 3, prec, rnd)
+
+#----------------------------------------------------------------------------#
+#                                                                            #
+#                                Logarithms                                  #
+#                                                                            #
+#----------------------------------------------------------------------------#
+
+
+def log_int_fixed(n, prec, ln2=None):
+    """
+    Fast computation of log(n), caching the value for small n,
+    intended for zeta sums.
+    """
+    if n in log_int_cache:
+        value, vprec = log_int_cache[n]
+        if vprec >= prec:
+            return value >> (vprec - prec)
+    wp = prec + 10
+    if wp <= LOG_TAYLOR_SHIFT:
+        if ln2 is None:
+            ln2 = ln2_fixed(wp)
+        r = bitcount(n)
+        x = n << (wp-r)
+        v = log_taylor_cached(x, wp) + r*ln2
+    else:
+        v = to_fixed(mpf_log(from_int(n), wp+5), wp)
+    if n < MAX_LOG_INT_CACHE:
+        log_int_cache[n] = (v, wp)
+    return v >> (wp-prec)
+
+def agm_fixed(a, b, prec):
+    """
+    Fixed-point computation of agm(a,b), assuming
+    a, b both close to unit magnitude.
+    """
+    i = 0
+    while 1:
+        anew = (a+b)>>1
+        if i > 4 and abs(a-anew) < 8:
+            return a
+        b = isqrt_fast(a*b)
+        a = anew
+        i += 1
+    return a
+
+def log_agm(x, prec):
+    """
+    Fixed-point computation of -log(x) = log(1/x), suitable
+    for large precision. It is required that 0 < x < 1. The
+    algorithm used is the Sasaki-Kanada formula
+
+        -log(x) = pi/agm(theta2(x)^2,theta3(x)^2). [1]
+
+    For faster convergence in the theta functions, x should
+    be chosen closer to 0.
+
+    Guard bits must be added by the caller.
+
+    HYPOTHESIS: if x = 2^(-n), n bits need to be added to
+    account for the truncation to a fixed-point number,
+    and this is the only significant cancellation error.
+
+    The number of bits lost to roundoff is small and can be
+    considered constant.
+
+    [1] Richard P. Brent, "Fast Algorithms for High-Precision
+        Computation of Elementary Functions (extended abstract)",
+        http://wwwmaths.anu.edu.au/~brent/pd/RNC7-Brent.pdf
+
+    """
+    x2 = (x*x) >> prec
+    # Compute jtheta2(x)**2
+    s = a = b = x2
+    while a:
+        b = (b*x2) >> prec
+        a = (a*b) >> prec
+        s += a
+    s += (MPZ_ONE<<prec)
+    s = (s*s)>>(prec-2)
+    s = (s*isqrt_fast(x<<prec))>>prec
+    # Compute jtheta3(x)**2
+    t = a = b = x
+    while a:
+        b = (b*x2) >> prec
+        a = (a*b) >> prec
+        t += a
+    t = (MPZ_ONE<<prec) + (t<<1)
+    t = (t*t)>>prec
+    # Final formula
+    p = agm_fixed(s, t, prec)
+    return (pi_fixed(prec) << prec) // p
+
+def log_taylor(x, prec, r=0):
+    """
+    Fixed-point calculation of log(x). It is assumed that x is close
+    enough to 1 for the Taylor series to converge quickly. Convergence
+    can be improved by specifying r > 0 to compute
+    log(x^(1/2^r))*2^r, at the cost of performing r square roots.
+
+    The caller must provide sufficient guard bits.
+    """
+    for i in xrange(r):
+        x = isqrt_fast(x<<prec)
+    one = MPZ_ONE << prec
+    v = ((x-one)<<prec)//(x+one)
+    sign = v < 0
+    if sign:
+        v = -v
+    v2 = (v*v) >> prec
+    v4 = (v2*v2) >> prec
+    s0 = v
+    s1 = v//3
+    v = (v*v4) >> prec
+    k = 5
+    while v:
+        s0 += v // k
+        k += 2
+        s1 += v // k
+        v = (v*v4) >> prec
+        k += 2
+    s1 = (s1*v2) >> prec
+    s = (s0+s1) << (1+r)
+    if sign:
+        return -s
+    return s
+
+def log_taylor_cached(x, prec):
+    """
+    Fixed-point computation of log(x), assuming x in (0.5, 2)
+    and prec <= LOG_TAYLOR_PREC.
+    """
+    n = x >> (prec-LOG_TAYLOR_SHIFT)
+    cached_prec = cache_prec_steps[prec]
+    dprec = cached_prec - prec
+    if (n, cached_prec) in log_taylor_cache:
+        a, log_a = log_taylor_cache[n, cached_prec]
+    else:
+        a = n << (cached_prec - LOG_TAYLOR_SHIFT)
+        log_a = log_taylor(a, cached_prec, 8)
+        log_taylor_cache[n, cached_prec] = (a, log_a)
+    a >>= dprec
+    log_a >>= dprec
+    u = ((x - a) << prec) // a
+    v = (u << prec) // ((MPZ_TWO << prec) + u)
+    v2 = (v*v) >> prec
+    v4 = (v2*v2) >> prec
+    s0 = v
+    s1 = v//3
+    v = (v*v4) >> prec
+    k = 5
+    while v:
+        s0 += v//k
+        k += 2
+        s1 += v//k
+        v = (v*v4) >> prec
+        k += 2
+    s1 = (s1*v2) >> prec
+    s = (s0+s1) << 1
+    return log_a + 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
+    #------------------------------------------------------------------
+    # 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 = (MPZ_ONE<<bc) - man
+        else:
+            tman = man - (MPZ_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_log_hypot(a, b, prec, rnd):
+    """
+    Computes log(sqrt(a^2+b^2)) accurately.
+    """
+    # If either a or b is inf/nan/0, assume it to be a
+    if not b[1]:
+        a, b = b, a
+    # a is inf/nan/0
+    if not a[1]:
+        # both are inf/nan/0
+        if not b[1]:
+            if a == b == fzero:
+                return fninf
+            if fnan in (a, b):
+                return fnan
+            # at least one term is (+/- inf)^2
+            return finf
+        # only a is inf/nan/0
+        if a == fzero:
+            # log(sqrt(0+b^2)) = log(|b|)
+            return mpf_log(mpf_abs(b), prec, rnd)
+        if a == fnan:
+            return fnan
+        return finf
+    # Exact
+    a2 = mpf_mul(a,a)
+    b2 = mpf_mul(b,b)
+    extra = 20
+    # Not exact
+    h2 = mpf_add(a2, b2, prec+extra)
+    cancelled = mpf_add(h2, fnone, 10)
+    mag_cancelled = cancelled[2]+cancelled[3]
+    # Just redo the sum exactly if necessary (could be smarter
+    # and avoid memory allocation when a or b is precisely 1
+    # and the other is tiny...)
+    if cancelled == fzero or mag_cancelled < -extra//2:
+        h2 = mpf_add(a2, b2, prec+extra-min(a2[2],b2[2]))
+    return mpf_shift(mpf_log(h2, prec, rnd), -1)
+
+
+#----------------------------------------------------------------------
+# Inverse tangent
+#
+
+def atan_newton(x, prec):
+    if prec >= 100:
+        r = math.atan(int((x>>(prec-53)))/2.0**53)
+    else:
+        r = math.atan(int(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_taylor_get_cached(n, prec):
+    # Taylor series with caching wins up to huge precisions
+    # To avoid unnecessary precomputation at low precision, we
+    # do it in steps
+    # Round to next power of 2
+    prec2 = (1<<(bitcount(prec-1))) + 20
+    dprec = prec2 - prec
+    if (n, prec2) in atan_taylor_cache:
+        a, atan_a = atan_taylor_cache[n, prec2]
+    else:
+        a = n << (prec2 - ATAN_TAYLOR_SHIFT)
+        atan_a = atan_newton(a, prec2)
+        atan_taylor_cache[n, prec2] = (a, atan_a)
+    return (a >> dprec), (atan_a >> dprec)
+
+def atan_taylor(x, prec):
+    n = (x >> (prec-ATAN_TAYLOR_SHIFT))
+    a, atan_a = atan_taylor_get_cached(n, prec)
+    d = x - a
+    s0 = v = (d << prec) // ((a**2 >> prec) + (a*d >> prec) + (MPZ_ONE << prec))
+    v2 = (v**2 >> prec)
+    v4 = (v2 * v2) >> prec
+    s1 = v//3
+    v = (v * v4) >> prec
+    k = 5
+    while v:
+        s0 += v // k
+        k += 2
+        s1 += v // k
+        v = (v * v4) >> prec
+        k += 2
+    s1 = (s1 * v2) >> prec
+    s = s0 - s1
+    return atan_a + s
+
+def atan_inf(sign, prec, rnd):
+    if not sign:
+        return mpf_shift(mpf_pi(prec, rnd), -1)
+    return mpf_neg(mpf_shift(mpf_pi(prec, negative_rnd[rnd]), -1))
+
+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)
+
+# TODO: cleanup the special cases
+def mpf_atan2(y, x, prec, rnd=round_fast):
+    xsign, xman, xexp, xbc = x
+    ysign, yman, yexp, ybc = y
+    if not yman:
+        if y == fzero and x != fnan:
+            if mpf_sign(x) >= 0:
+                return fzero
+            return mpf_pi(prec, rnd)
+        if y in (finf, fninf):
+            if x in (finf, fninf):
+                return fnan
+            # pi/2
+            if y == finf:
+                return mpf_shift(mpf_pi(prec, rnd), -1)
+            # -pi/2
+            return mpf_neg(mpf_shift(mpf_pi(prec, negative_rnd[rnd]), -1))
+        return fnan
+    if ysign:
+        return mpf_neg(mpf_atan2(mpf_neg(y), x, prec, negative_rnd[rnd]))
+    if not xman:
+        if x == fnan:
+            return fnan
+        if x == finf:
+            return fzero
+        if x == fninf:
+            return mpf_pi(prec, rnd)
+        if y == fzero:
+            return fzero
+        return mpf_shift(mpf_pi(prec, rnd), -1)
+    tquo = mpf_atan(mpf_div(y, x, prec+4), prec+4)
+    if xsign:
+        return mpf_add(mpf_pi(prec+4), tquo, prec, rnd)
+    else:
+        return mpf_pos(tquo, 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")
+    # 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)
+
+def mpf_acos(x, prec, rnd=round_fast):
+    # acos(x) = 2*atan(sqrt(1-x**2)/(1+x))
+    sign, man, exp, bc = x
+    if bc + exp > 0:
+        if x not in (fone, fnone):
+            raise ComplexResult("acos(x) is real only for -1 <= x <= 1")
+        if x == fnone:
+            return mpf_pi(prec, rnd)
+    wp = prec + 15
+    a = mpf_mul(x, x)
+    b = mpf_sqrt(mpf_sub(fone, a, wp), wp)
+    c = mpf_div(b, mpf_add(fone, x, wp), wp)
+    return mpf_shift(mpf_atan(c, prec, rnd), 1)
+
+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_acosh(x, prec, rnd=round_fast):
+    # acosh(x) = log(x+sqrt(x**2-1))
+    wp = prec + 15
+    if mpf_cmp(x, fone) == -1:
+        raise ComplexResult("acosh(x) is real only for x >= 1")
+    q = mpf_sqrt(mpf_add(mpf_mul(x,x), fnone, wp), wp)
+    return mpf_log(mpf_add(x, q, wp), prec, rnd)
+
+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_fibonacci(x, prec, rnd=round_fast):
+    sign, man, exp, bc = x
+    if not man:
+        if x == fninf:
+            return fnan
+        return x
+    # F(2^n) ~= 2^(2^n)
+    size = abs(exp+bc)
+    if exp >= 0:
+        # Exact
+        if size < 10 or size <= bitcount(prec):
+            return from_int(ifib(to_int(x)), prec, rnd)
+    # Use the modified Binet formula
+    wp = prec + size + 20
+    a = mpf_phi(wp)
+    b = mpf_add(mpf_shift(a, 1), fnone, wp)
+    u = mpf_pow(a, x, wp)
+    v = mpf_cos_pi(x, wp)
+    v = mpf_div(v, u, wp)
+    u = mpf_sub(u, v, wp)
+    u = mpf_div(u, b, prec, rnd)
+    return u
+
+
+#-------------------------------------------------------------------------------
+# Exponential-type functions
+#-------------------------------------------------------------------------------
+
+def exponential_series(x, prec, type=0):
+    """
+    Taylor series for cosh/sinh or cos/sin.
+
+    type = 0 -- returns exp(x)  (slightly faster than cosh+sinh)
+    type = 1 -- returns (cosh(x), sinh(x))
+    type = 2 -- returns (cos(x), sin(x))
+    """
+    if x < 0:
+        x = -x
+        sign = 1
+    else:
+        sign = 0
+    r = int(0.5*prec**0.5)
+    xmag = bitcount(x) - prec
+    r = max(0, xmag + r)
+    extra = 10 + 2*max(r,-xmag)
+    wp = prec + extra
+    x <<= (extra - r)
+    one = MPZ_ONE << wp
+    alt = (type == 2)
+    if prec < EXP_SERIES_U_CUTOFF:
+        x2 = a = (x*x) >> wp
+        x4 = (x2*x2) >> wp
+        s0 = s1 = MPZ_ZERO
+        k = 2
+        while a:
+            a //= (k-1)*k; s0 += a; k += 2
+            a //= (k-1)*k; s1 += a; k += 2
+            a = (a*x4) >> wp
+        s1 = (x2*s1) >> wp
+        if alt:
+            c = s1 - s0 + one
+        else:
+            c = s1 + s0 + one
+    else:
+        u = int(0.3*prec**0.35)
+        x2 = a = (x*x) >> wp
+        xpowers = [one, x2]
+        for i in xrange(1, u):
+            xpowers.append((xpowers[-1]*x2)>>wp)
+        sums = [MPZ_ZERO] * u
+        k = 2
+        while a:
+            for i in xrange(u):
+                a //= (k-1)*k
+                if alt and k & 2: sums[i] -= a
+                else:             sums[i] += a
+                k += 2
+            a = (a*xpowers[-1]) >> wp
+        for i in xrange(1, u):
+            sums[i] = (sums[i]*xpowers[i]) >> wp
+        c = sum(sums) + one
+    if type == 0:
+        s = isqrt_fast(c*c - (one<<wp))
+        if sign:
+            v = c - s
+        else:
+            v = c + s
+        for i in xrange(r):
+            v = (v*v) >> wp
+        return v >> extra
+    else:
+        # Repeatedly apply the double-angle formula
+        # cosh(2*x) = 2*cosh(x)^2 - 1
+        # cos(2*x) = 2*cos(x)^2 - 1
+        pshift = wp-1
+        for i in xrange(r):
+            c = ((c*c) >> pshift) - one
+        # With the abs, this is the same for sinh and sin
+        s = isqrt_fast(abs((one<<wp) - c*c))
+        if sign:
+            s = -s
+        return (c>>extra), (s>>extra)
+
+def exp_basecase(x, prec):
+    """
+    Compute exp(x) as a fixed-point number. Works for any x,
+    but for speed should have |x| < 1. For an arbitrary number,
+    use exp(x) = exp(x-m*log(2)) * 2^m where m = floor(x/log(2)).
+    """
+    if prec > EXP_COSH_CUTOFF:
+        return exponential_series(x, prec, 0)
+    r = int(prec**0.5)
+    prec += r
+    s0 = s1 = (MPZ_ONE << prec)
+    k = 2
+    a = x2 = (x*x) >> prec
+    while a:
+        a //= k; s0 += a; k += 1
+        a //= k; s1 += a; k += 1
+        a = (a*x2) >> prec
+    s1 = (s1*x) >> prec
+    s = s0 + s1
+    u = r
+    while r:
+        s = (s*s) >> prec
+        r -= 1
+    return s >> u
+
+def exp_expneg_basecase(x, prec):
+    """
+    Computation of exp(x), exp(-x)
+    """
+    if prec > EXP_COSH_CUTOFF:
+        cosh, sinh = exponential_series(x, prec, 1)
+        return cosh+sinh, cosh-sinh
+    a = exp_basecase(x, prec)
+    b = (MPZ_ONE << (prec+prec)) // a
+    return a, b
+
+def cos_sin_basecase(x, prec):
+    """
+    Compute cos(x), sin(x) as fixed-point numbers, assuming x
+    in [0, pi/2). For an arbitrary number, use x' = x - m*(pi/2)
+    where m = floor(x/(pi/2)) along with quarter-period symmetries.
+    """
+    if prec > COS_SIN_CACHE_PREC:
+        return exponential_series(x, prec, 2)
+    precs = prec - COS_SIN_CACHE_STEP
+    t = x >> precs
+    n = int(t)
+    if n not in cos_sin_cache:
+        w = t<<(10+COS_SIN_CACHE_PREC-COS_SIN_CACHE_STEP)
+        cos_t, sin_t = exponential_series(w, 10+COS_SIN_CACHE_PREC, 2)
+        cos_sin_cache[n] = (cos_t>>10), (sin_t>>10)
+    cos_t, sin_t = cos_sin_cache[n]
+    offset = COS_SIN_CACHE_PREC - prec
+    cos_t >>= offset
+    sin_t >>= offset
+    x -= t << precs
+    cos = MPZ_ONE << prec
+    sin = x
+    k = 2
+    a = -((x*x) >> prec)
+    while a:
+        a //= k; cos += a; k += 1; a = (a*x) >> prec
+        a //= k; sin += a; k += 1; a = -((a*x) >> prec)
+    return ((cos*cos_t-sin*sin_t) >> prec), ((sin*cos_t+cos*sin_t) >> prec)
+
+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_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 mod_pi2(man, exp, mag, wp):
+    # Reduce to standard interval
+    if mag > 0:
+        i = 0
+        while 1:
+            cancellation_prec = 20 << i
+            wpmod = wp + mag + cancellation_prec
+            pi2 = pi_fixed(wpmod-1)
+            pi4 = pi2 >> 1
+            offset = wpmod + exp
+            if offset >= 0:
+                t = man << offset
+            else:
+                t = man >> (-offset)
+            n, y = divmod(t, pi2)
+            if y > pi4:
+                small = pi2 - y
+            else:
+                small = y
+            if small >> (wp+mag-10):
+                n = int(n)
+                t = y >> mag
+                wp = wpmod - mag
+                break
+            i += 1
+    else:
+        wp += (-mag)
+        offset = exp + wp
+        if offset >= 0:
+            t = man << offset
+        else:
+            t = man >> (-offset)
+        n = 0
+    return t, n, wp
+
+
+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_cos(x, prec, rnd=round_fast): return mpf_cos_sin(x, prec, rnd, 1)
+def mpf_sin(x, prec, rnd=round_fast): return mpf_cos_sin(x, prec, rnd, 2)
+def mpf_tan(x, prec, rnd=round_fast): return mpf_cos_sin(x, prec, rnd, 3)
+def mpf_cos_sin_pi(x, prec, rnd=round_fast): return mpf_cos_sin(x, prec, rnd, 0, 1)
+def mpf_cos_pi(x, prec, rnd=round_fast): return mpf_cos_sin(x, prec, rnd, 1, 1)
+def mpf_sin_pi(x, prec, rnd=round_fast): return mpf_cos_sin(x, prec, rnd, 2, 1)
+def mpf_cosh(x, prec, rnd=round_fast): return mpf_cosh_sinh(x, prec, rnd)[0]
+def mpf_sinh(x, prec, rnd=round_fast): return mpf_cosh_sinh(x, prec, rnd)[1]
+def mpf_tanh(x, prec, rnd=round_fast): return mpf_cosh_sinh(x, prec, rnd, tanh=1)
+
+
+# Low-overhead fixed-point versions
+
+def cos_sin_fixed(x, prec, pi2=None):
+    if pi2 is None:
+        pi2 = pi_fixed(prec-1)
+    n, t = divmod(x, pi2)
+    n = int(n)
+    c, s = cos_sin_basecase(t, prec)
+    m = n & 3
+    if m == 0: return c, s
+    if m == 1: return -s, c
+    if m == 2: return -c, -s
+    if m == 3: return s, -c
+
+def exp_fixed(x, prec, ln2=None):
+    if ln2 is None:
+        ln2 = ln2_fixed(prec)
+    n, t = divmod(x, ln2)
+    n = int(n)
+    v = exp_basecase(t, prec)
+    if n >= 0:
+        return v << n
+    else:
+        return v >> (-n)
+
+
+if BACKEND == 'sage':
+    try:
+        import sage.libs.mpmath.ext_libmp as _lbmp
+        mpf_sqrt = _lbmp.mpf_sqrt
+        mpf_exp = _lbmp.mpf_exp
+        mpf_log = _lbmp.mpf_log
+        mpf_cos = _lbmp.mpf_cos
+        mpf_sin = _lbmp.mpf_sin
+        mpf_pow = _lbmp.mpf_pow
+        exp_fixed = _lbmp.exp_fixed
+        cos_sin_fixed = _lbmp.cos_sin_fixed
+        log_int_fixed = _lbmp.log_int_fixed
+    except (ImportError, AttributeError):
+        print("Warning: Sage imports in libelefun failed")