Ejemplo n.º 1
0
    def _eval_power(b, e):
        """
        b is Real but not equal to rationals, integers, 0.5, oo, -oo, nan
        e is symbolic object but not equal to 0, 1

        (-p) ** r -> exp(r * log(-p)) -> exp(r * (log(p) + I*Pi)) ->
                  -> p ** r * (sin(Pi*r) + cos(Pi*r) * I)
        """
        if isinstance(e, Number):
            if isinstance(e, Integer):
                prec = b._prec
                return Real._new(mlib.mpf_pow_int(b._mpf_, e.p, prec, rnd), prec)
            e, prec = e._as_mpf_op(b._prec)
            b = b._mpf_
            try:
                y = mpf_pow(b, e, prec, rnd)
                return Real._new(y, prec)
            except mlib.ComplexResult:
                re, im = mlibc.mpc_pow((b, mlib.fzero), (e, mlib.fzero), prec, rnd)
                return Real._new(re, prec) + Real._new(im, prec) * S.ImaginaryUnit
Ejemplo n.º 2
0
    def _eval_power(b, e):
        """
        b is Real but not equal to rationals, integers, 0.5, oo, -oo, nan
        e is symbolic object but not equal to 0, 1

        (-p) ** r -> exp(r * log(-p)) -> exp(r * (log(p) + I*Pi)) ->
                  -> p ** r * (sin(Pi*r) + cos(Pi*r) * I)
        """
        if isinstance(e, Number):
            if isinstance(e, Integer):
                prec = b._prec
                return Real._new(mlib.mpf_pow_int(b._mpf_, e.p, prec, rnd), prec)
            e, prec = e._as_mpf_op(b._prec)
            b = b._mpf_
            try:
                y = mpf_pow(b, e, prec, rnd)
                return Real._new(y, prec)
            except mlib.ComplexResult:
                re, im = mlibc.mpc_pow((b, mlib.fzero), (e, mlib.fzero), prec, rnd)
                return Real._new(re, prec) + Real._new(im, prec) * S.ImaginaryUnit
Ejemplo n.º 3
0
def evalf_pow(v, prec, options):

    target_prec = prec
    base, exp = v.args

    # We handle x**n separately. This has two purposes: 1) it is much
    # faster, because we avoid calling evalf on the exponent, and 2) it
    # allows better handling of real/imaginary parts that are exactly zero
    if exp.is_Integer:
        p = exp.p
        # Exact
        if not p:
            return fone, None, prec, None
        # Exponentiation by p magnifies relative error by |p|, so the
        # base must be evaluated with increased precision if p is large
        prec += int(math.log(abs(p),2))
        re, im, re_acc, im_acc = evalf(base, prec+5, options)
        # Real to integer power
        if re and not im:
            return mpf_pow_int(re, p, target_prec), None, target_prec, None
        # (x*I)**n = I**n * x**n
        if im and not re:
            z = fpowi(im, p, target_prec)
            case = p % 4
            if case == 0: return z, None, target_prec, None
            if case == 1: return None, z, None, target_prec
            if case == 2: return mpf_neg(z), None, target_prec, None
            if case == 3: return None, mpf_neg(z), None, target_prec
        # Zero raised to an integer power
        if not re:
            return None, None, None, None
        # General complex number to arbitrary integer power
        re, im = libmpc.mpc_pow_int((re, im), p, prec)
        # Assumes full accuracy in input
        return finalize_complex(re, im, target_prec)

    # Pure square root
    if exp is S.Half:
        xre, xim, xre_acc, yim_acc = evalf(base, prec+5, options)
        # General complex square root
        if xim:
            re, im = libmpc.mpc_sqrt((xre or fzero, xim), prec)
            return finalize_complex(re, im, prec)
        if not xre:
            return None, None, None, None
        # Square root of a negative real number
        if mpf_lt(xre, fzero):
            return None, mpf_sqrt(mpf_neg(xre), prec), None, prec
        # Positive square root
        return mpf_sqrt(xre, prec), None, prec, None

    # We first evaluate the exponent to find its magnitude
    # This determines the working precision that must be used
    prec += 10
    yre, yim, yre_acc, yim_acc = evalf(exp, prec, options)
    # Special cases: x**0
    if not (yre or yim):
        return fone, None, prec, None

    ysize = fastlog(yre)
    # Restart if too big
    # XXX: prec + ysize might exceed maxprec
    if ysize > 5:
        prec += ysize
        yre, yim, yre_acc, yim_acc = evalf(exp, prec, options)

    # Pure exponential function; no need to evalf the base
    if base is S.Exp1:
        if yim:
            re, im = libmpc.mpc_exp((yre or fzero, yim), prec)
            return finalize_complex(re, im, target_prec)
        return mpf_exp(yre, target_prec), None, target_prec, None

    xre, xim, xre_acc, yim_acc = evalf(base, prec+5, options)
    # 0**y
    if not (xre or xim):
        return None, None, None, None

    # (real ** complex) or (complex ** complex)
    if yim:
        re, im = libmpc.mpc_pow((xre or fzero, xim or fzero), (yre or fzero, yim),
            target_prec)
        return finalize_complex(re, im, target_prec)
    # complex ** real
    if xim:
        re, im = libmpc.mpc_pow_mpf((xre or fzero, xim), yre, target_prec)
        return finalize_complex(re, im, target_prec)
    # negative ** real
    elif mpf_lt(xre, fzero):
        re, im = libmpc.mpc_pow_mpf((xre, fzero), yre, target_prec)
        return finalize_complex(re, im, target_prec)
    # positive ** real
    else:
        return mpf_pow(xre, yre, target_prec), None, target_prec, None