예제 #1
0
def mpi_cos_sin(x, prec):
    a, b = x
    if a == b == fzero:
        return (fone, fone), (fzero, fzero)
    # Guaranteed to contain both -1 and 1
    if (finf in x) or (fninf in x):
        return (fnone, fone), (fnone, fone)
    wp = prec + 20
    ca, sa, na = cos_sin_quadrant(a, wp)
    cb, sb, nb = cos_sin_quadrant(b, wp)
    ca, cb = mpf_min_max([ca, cb])
    sa, sb = mpf_min_max([sa, sb])
    # Both functions are monotonic within one quadrant
    if na == nb:
        pass
    # Guaranteed to contain both -1 and 1
    elif nb - na >= 4:
        return (fnone, fone), (fnone, fone)
    else:
        # cos has maximum between a and b
        if na // 4 != nb // 4:
            cb = fone
        # cos has minimum
        if (na - 2) // 4 != (nb - 2) // 4:
            ca = fnone
        # sin has maximum
        if (na - 1) // 4 != (nb - 1) // 4:
            sb = fone
        # sin has minimum
        if (na - 3) // 4 != (nb - 3) // 4:
            sa = fnone
    # Perturb to force interval rounding
    more = from_man_exp((MPZ_ONE << wp) + (MPZ_ONE << 10), -wp)
    less = from_man_exp((MPZ_ONE << wp) - (MPZ_ONE << 10), -wp)

    def finalize(v, rounding):
        if bool(v[0]) == (rounding == round_floor):
            p = more
        else:
            p = less
        v = mpf_mul(v, p, prec, rounding)
        sign, man, exp, bc = v
        if exp + bc >= 1:
            if sign:
                return fnone
            return fone
        return v

    ca = finalize(ca, round_floor)
    cb = finalize(cb, round_ceiling)
    sa = finalize(sa, round_floor)
    sb = finalize(sb, round_ceiling)
    return (ca, cb), (sa, sb)
예제 #2
0
def mpi_cos_sin(x, prec):
    a, b = x
    if a == b == fzero:
        return (fone, fone), (fzero, fzero)
    # Guaranteed to contain both -1 and 1
    if (finf in x) or (fninf in x):
        return (fnone, fone), (fnone, fone)
    wp = prec + 20
    ca, sa, na = cos_sin_quadrant(a, wp)
    cb, sb, nb = cos_sin_quadrant(b, wp)
    ca, cb = mpf_min_max([ca, cb])
    sa, sb = mpf_min_max([sa, sb])
    # Both functions are monotonic within one quadrant
    if na == nb:
        pass
    # Guaranteed to contain both -1 and 1
    elif nb - na >= 4:
        return (fnone, fone), (fnone, fone)
    else:
        # cos has maximum between a and b
        if na//4 != nb//4:
            cb = fone
        # cos has minimum
        if (na-2)//4 != (nb-2)//4:
            ca = fnone
        # sin has maximum
        if (na-1)//4 != (nb-1)//4:
            sb = fone
        # sin has minimum
        if (na-3)//4 != (nb-3)//4:
            sa = fnone
    # Perturb to force interval rounding
    more = from_man_exp((MPZ_ONE<<wp) + (MPZ_ONE<<10), -wp)
    less = from_man_exp((MPZ_ONE<<wp) - (MPZ_ONE<<10), -wp)
    def finalize(v, rounding):
        if bool(v[0]) == (rounding == round_floor):
            p = more
        else:
            p = less
        v = mpf_mul(v, p, prec, rounding)
        sign, man, exp, bc = v
        if exp+bc >= 1:
            if sign:
                return fnone
            return fone
        return v
    ca = finalize(ca, round_floor)
    cb = finalize(cb, round_ceiling)
    sa = finalize(sa, round_floor)
    sb = finalize(sb, round_ceiling)
    return (ca,cb), (sa,sb)
예제 #3
0
파일: libhyper.py 프로젝트: Aang/sympy
def mpc_agm(a, b, prec, rnd=round_fast):
    """
    Complex AGM.

    TODO:
    * check that convergence works as intended
    * optimize
    * select a nonarbitrary branch
    """
    if mpc_is_infnan(a) or mpc_is_infnan(b):
        return fnan, fnan
    if mpc_zero in (a, b):
        return fzero, fzero
    if mpc_neg(a) == b:
        return fzero, fzero
    wp = prec+20
    eps = mpf_shift(fone, -wp+10)
    while 1:
        a1 = mpc_shift(mpc_add(a, b, wp), -1)
        b1 = mpc_sqrt(mpc_mul(a, b, wp), wp)
        a, b = a1, b1
        size = mpf_min_max([mpc_abs(a,10), mpc_abs(b,10)])[1]
        err = mpc_abs(mpc_sub(a, b, 10), 10)
        if size == fzero or mpf_lt(err, mpf_mul(eps, size)):
            return a
예제 #4
0
def mpc_agm(a, b, prec, rnd=round_fast):
    """
    Complex AGM.

    TODO:
    * check that convergence works as intended
    * optimize
    * select a nonarbitrary branch
    """
    if mpc_is_infnan(a) or mpc_is_infnan(b):
        return fnan, fnan
    if mpc_zero in (a, b):
        return fzero, fzero
    if mpc_neg(a) == b:
        return fzero, fzero
    wp = prec + 20
    eps = mpf_shift(fone, -wp + 10)
    while 1:
        a1 = mpc_shift(mpc_add(a, b, wp), -1)
        b1 = mpc_sqrt(mpc_mul(a, b, wp), wp)
        a, b = a1, b1
        size = mpf_min_max([mpc_abs(a, 10), mpc_abs(b, 10)])[1]
        err = mpc_abs(mpc_sub(a, b, 10), 10)
        if size == fzero or mpf_lt(err, mpf_mul(eps, size)):
            return a
예제 #5
0
def mpi_square(s, prec=0):
    sa, sb = s
    if mpf_ge(sa, fzero):
        a = mpf_mul(sa, sa, prec, round_floor)
        b = mpf_mul(sb, sb, prec, round_ceiling)
    elif mpf_le(sb, fzero):
        a = mpf_mul(sb, sb, prec, round_floor)
        b = mpf_mul(sa, sa, prec, round_ceiling)
    else:
        sa = mpf_neg(sa)
        sa, sb = mpf_min_max([sa, sb])
        a = fzero
        b = mpf_mul(sb, sb, prec, round_ceiling)
    return a, b
예제 #6
0
def mpi_square(s, prec=0):
    sa, sb = s
    if mpf_ge(sa, fzero):
        a = mpf_mul(sa, sa, prec, round_floor)
        b = mpf_mul(sb, sb, prec, round_ceiling)
    elif mpf_le(sb, fzero):
        a = mpf_mul(sb, sb, prec, round_floor)
        b = mpf_mul(sa, sa, prec, round_ceiling)
    else:
        sa = mpf_neg(sa)
        sa, sb = mpf_min_max([sa, sb])
        a = fzero
        b = mpf_mul(sb, sb, prec, round_ceiling)
    return a, b
예제 #7
0
def mpi_mul(s, t, prec=0):
    sa, sb = s
    ta, tb = t
    sas = mpf_sign(sa)
    sbs = mpf_sign(sb)
    tas = mpf_sign(ta)
    tbs = mpf_sign(tb)
    if sas == sbs == 0:
        # Should maybe be undefined
        if ta == fninf or tb == finf:
            return fninf, finf
        return fzero, fzero
    if tas == tbs == 0:
        # Should maybe be undefined
        if sa == fninf or sb == finf:
            return fninf, finf
        return fzero, fzero
    if sas >= 0:
        # positive * positive
        if tas >= 0:
            a = mpf_mul(sa, ta, prec, round_floor)
            b = mpf_mul(sb, tb, prec, round_ceiling)
            if a == fnan: a = fzero
            if b == fnan: b = finf
        # positive * negative
        elif tbs <= 0:
            a = mpf_mul(sb, ta, prec, round_floor)
            b = mpf_mul(sa, tb, prec, round_ceiling)
            if a == fnan: a = fninf
            if b == fnan: b = fzero
        # positive * both signs
        else:
            a = mpf_mul(sb, ta, prec, round_floor)
            b = mpf_mul(sb, tb, prec, round_ceiling)
            if a == fnan: a = fninf
            if b == fnan: b = finf
    elif sbs <= 0:
        # negative * positive
        if tas >= 0:
            a = mpf_mul(sa, tb, prec, round_floor)
            b = mpf_mul(sb, ta, prec, round_ceiling)
            if a == fnan: a = fninf
            if b == fnan: b = fzero
        # negative * negative
        elif tbs <= 0:
            a = mpf_mul(sb, tb, prec, round_floor)
            b = mpf_mul(sa, ta, prec, round_ceiling)
            if a == fnan: a = fzero
            if b == fnan: b = finf
        # negative * both signs
        else:
            a = mpf_mul(sa, tb, prec, round_floor)
            b = mpf_mul(sa, ta, prec, round_ceiling)
            if a == fnan: a = fninf
            if b == fnan: b = finf
    else:
        # General case: perform all cross-multiplications and compare
        # Since the multiplications can be done exactly, we need only
        # do 4 (instead of 8: two for each rounding mode)
        cases = [
            mpf_mul(sa, ta),
            mpf_mul(sa, tb),
            mpf_mul(sb, ta),
            mpf_mul(sb, tb)
        ]
        if fnan in cases:
            a, b = (fninf, finf)
        else:
            a, b = mpf_min_max(cases)
            a = mpf_pos(a, prec, round_floor)
            b = mpf_pos(b, prec, round_ceiling)
    return a, b
예제 #8
0
def mpi_mul(s, t, prec=0):
    sa, sb = s
    ta, tb = t
    sas = mpf_sign(sa)
    sbs = mpf_sign(sb)
    tas = mpf_sign(ta)
    tbs = mpf_sign(tb)
    if sas == sbs == 0:
        # Should maybe be undefined
        if ta == fninf or tb == finf:
            return fninf, finf
        return fzero, fzero
    if tas == tbs == 0:
        # Should maybe be undefined
        if sa == fninf or sb == finf:
            return fninf, finf
        return fzero, fzero
    if sas >= 0:
        # positive * positive
        if tas >= 0:
            a = mpf_mul(sa, ta, prec, round_floor)
            b = mpf_mul(sb, tb, prec, round_ceiling)
            if a == fnan: a = fzero
            if b == fnan: b = finf
        # positive * negative
        elif tbs <= 0:
            a = mpf_mul(sb, ta, prec, round_floor)
            b = mpf_mul(sa, tb, prec, round_ceiling)
            if a == fnan: a = fninf
            if b == fnan: b = fzero
        # positive * both signs
        else:
            a = mpf_mul(sb, ta, prec, round_floor)
            b = mpf_mul(sb, tb, prec, round_ceiling)
            if a == fnan: a = fninf
            if b == fnan: b = finf
    elif sbs <= 0:
        # negative * positive
        if tas >= 0:
            a = mpf_mul(sa, tb, prec, round_floor)
            b = mpf_mul(sb, ta, prec, round_ceiling)
            if a == fnan: a = fninf
            if b == fnan: b = fzero
        # negative * negative
        elif tbs <= 0:
            a = mpf_mul(sb, tb, prec, round_floor)
            b = mpf_mul(sa, ta, prec, round_ceiling)
            if a == fnan: a = fzero
            if b == fnan: b = finf
        # negative * both signs
        else:
            a = mpf_mul(sa, tb, prec, round_floor)
            b = mpf_mul(sa, ta, prec, round_ceiling)
            if a == fnan: a = fninf
            if b == fnan: b = finf
    else:
        # General case: perform all cross-multiplications and compare
        # Since the multiplications can be done exactly, we need only
        # do 4 (instead of 8: two for each rounding mode)
        cases = [mpf_mul(sa, ta), mpf_mul(sa, tb), mpf_mul(sb, ta), mpf_mul(sb, tb)]
        if fnan in cases:
            a, b = (fninf, finf)
        else:
            a, b = mpf_min_max(cases)
            a = mpf_pos(a, prec, round_floor)
            b = mpf_pos(b, prec, round_ceiling)
    return a, b