Python functions.cosh函数代码示例

def test_C99CodePrinter__precision():
    n = symbols('n', integer=True)
    f32_printer = C99CodePrinter(dict(type_aliases={real: float32}))
    f64_printer = C99CodePrinter(dict(type_aliases={real: float64}))
    f80_printer = C99CodePrinter(dict(type_aliases={real: float80}))
    assert f32_printer.doprint(sin(x+2.1)) == 'sinf(x + 2.1F)'
    assert f64_printer.doprint(sin(x+2.1)) == 'sin(x + 2.1000000000000001)'
    assert f80_printer.doprint(sin(x+Float('2.0'))) == 'sinl(x + 2.0L)'

    for printer, suffix in zip([f32_printer, f64_printer, f80_printer], ['f', '', 'l']):
        def check(expr, ref):
            assert printer.doprint(expr) == ref.format(s=suffix, S=suffix.upper())
        check(Abs(n), 'abs(n)')
        check(Abs(x + 2.0), 'fabs{s}(x + 2.0{S})')
        check(sin(x + 4.0)**cos(x - 2.0), 'pow{s}(sin{s}(x + 4.0{S}), cos{s}(x - 2.0{S}))')
        check(exp(x*8.0), 'exp{s}(8.0{S}*x)')
        check(exp2(x), 'exp2{s}(x)')
        check(expm1(x*4.0), 'expm1{s}(4.0{S}*x)')
        check(Mod(n, 2), '((n) % (2))')
        check(Mod(2*n + 3, 3*n + 5), '((2*n + 3) % (3*n + 5))')
        check(Mod(x + 2.0, 3.0), 'fmod{s}(1.0{S}*x + 2.0{S}, 3.0{S})')
        check(Mod(x, 2.0*x + 3.0), 'fmod{s}(1.0{S}*x, 2.0{S}*x + 3.0{S})')
        check(log(x/2), 'log{s}((1.0{S}/2.0{S})*x)')
        check(log10(3*x/2), 'log10{s}((3.0{S}/2.0{S})*x)')
        check(log2(x*8.0), 'log2{s}(8.0{S}*x)')
        check(log1p(x), 'log1p{s}(x)')
        check(2**x, 'pow{s}(2, x)')
        check(2.0**x, 'pow{s}(2.0{S}, x)')
        check(x**3, 'pow{s}(x, 3)')
        check(x**4.0, 'pow{s}(x, 4.0{S})')
        check(sqrt(3+x), 'sqrt{s}(x + 3)')
        check(Cbrt(x-2.0), 'cbrt{s}(x - 2.0{S})')
        check(hypot(x, y), 'hypot{s}(x, y)')
        check(sin(3.*x + 2.), 'sin{s}(3.0{S}*x + 2.0{S})')
        check(cos(3.*x - 1.), 'cos{s}(3.0{S}*x - 1.0{S})')
        check(tan(4.*y + 2.), 'tan{s}(4.0{S}*y + 2.0{S})')
        check(asin(3.*x + 2.), 'asin{s}(3.0{S}*x + 2.0{S})')
        check(acos(3.*x + 2.), 'acos{s}(3.0{S}*x + 2.0{S})')
        check(atan(3.*x + 2.), 'atan{s}(3.0{S}*x + 2.0{S})')
        check(atan2(3.*x, 2.*y), 'atan2{s}(3.0{S}*x, 2.0{S}*y)')

        check(sinh(3.*x + 2.), 'sinh{s}(3.0{S}*x + 2.0{S})')
        check(cosh(3.*x - 1.), 'cosh{s}(3.0{S}*x - 1.0{S})')
        check(tanh(4.0*y + 2.), 'tanh{s}(4.0{S}*y + 2.0{S})')
        check(asinh(3.*x + 2.), 'asinh{s}(3.0{S}*x + 2.0{S})')
        check(acosh(3.*x + 2.), 'acosh{s}(3.0{S}*x + 2.0{S})')
        check(atanh(3.*x + 2.), 'atanh{s}(3.0{S}*x + 2.0{S})')
        check(erf(42.*x), 'erf{s}(42.0{S}*x)')
        check(erfc(42.*x), 'erfc{s}(42.0{S}*x)')
        check(gamma(x), 'tgamma{s}(x)')
        check(loggamma(x), 'lgamma{s}(x)')

        check(ceiling(x + 2.), "ceil{s}(x + 2.0{S})")
        check(floor(x + 2.), "floor{s}(x + 2.0{S})")
        check(fma(x, y, -z), 'fma{s}(x, y, -z)')
        check(Max(x, 8.0, x**4.0), 'fmax{s}(8.0{S}, fmax{s}(x, pow{s}(x, 4.0{S})))')
        check(Min(x, 2.0), 'fmin{s}(2.0{S}, x)')

    def f(rv):
        if not rv.is_Mul:
            return rv
        commutative_part, noncommutative_part = rv.args_cnc()
        # Since as_powers_dict loses order information,
        # if there is more than one noncommutative factor,
        # it should only be used to simplify the commutative part.
        if (len(noncommutative_part) > 1):
            return f(Mul(*commutative_part))*Mul(*noncommutative_part)
        rvd = rv.as_powers_dict()
        newd = rvd.copy()

        def signlog(expr, sign=1):
            if expr is S.Exp1:
                return sign, 1
            elif isinstance(expr, exp):
                return sign, expr.args[0]
            elif sign == 1:
                return signlog(-expr, sign=-1)
            else:
                return None, None

        ee = rvd[S.Exp1]
        for k in rvd:
            if k.is_Add and len(k.args) == 2:
                # k == c*(1 + sign*E**x)
                c = k.args[0]
                sign, x = signlog(k.args[1]/c)
                if not x:
                    continue
                m = rvd[k]
                newd[k] -= m
                if ee == -x*m/2:
                    # sinh and cosh
                    newd[S.Exp1] -= ee
                    ee = 0
                    if sign == 1:
                        newd[2*c*cosh(x/2)] += m
                    else:
                        newd[-2*c*sinh(x/2)] += m
                elif newd[1 - sign*S.Exp1**x] == -m:
                    # tanh
                    del newd[1 - sign*S.Exp1**x]
                    if sign == 1:
                        newd[-c/tanh(x/2)] += m
                    else:
                        newd[-c*tanh(x/2)] += m
                else:
                    newd[1 + sign*S.Exp1**x] += m
                    newd[c] += m

        return Mul(*[k**newd[k] for k in newd])

    def f(rv):
        if not rv.is_Mul:
            return rv
        rvd = rv.as_powers_dict()
        newd = rvd.copy()

        def signlog(expr, sign=1):
            if expr is S.Exp1:
                return sign, 1
            elif isinstance(expr, exp):
                return sign, expr.args[0]
            elif sign == 1:
                return signlog(-expr, sign=-1)
            else:
                return None, None

        ee = rvd[S.Exp1]
        for k in rvd:
            if k.is_Add and len(k.args) == 2:
                # k == c*(1 + sign*E**x)
                c = k.args[0]
                sign, x = signlog(k.args[1]/c)
                if not x:
                    continue
                m = rvd[k]
                newd[k] -= m
                if ee == -x*m/2:
                    # sinh and cosh
                    newd[S.Exp1] -= ee
                    ee = 0
                    if sign == 1:
                        newd[2*c*cosh(x/2)] += m
                    else:
                        newd[-2*c*sinh(x/2)] += m
                elif newd[1 - sign*S.Exp1**x] == -m:
                    # tanh
                    del newd[1 - sign*S.Exp1**x]
                    if sign == 1:
                        newd[-c/tanh(x/2)] += m
                    else:
                        newd[-c*tanh(x/2)] += m
                else:
                    newd[1 + sign*S.Exp1**x] += m
                    newd[c] += m

        return Mul(*[k**newd[k] for k in newd])

    def build_ideal(x, terms):
        """
        Build generators for our ideal. Terms is an iterable with elements of
        the form (fn, coeff), indicating that we have a generator fn(coeff*x).

        If any of the terms is trigonometric, sin(x) and cos(x) are guaranteed
        to appear in terms. Similarly for hyperbolic functions. For tan(n*x),
        sin(n*x) and cos(n*x) are guaranteed.
        """
        I = []
        y = Dummy('y')
        for fn, coeff in terms:
            for c, s, t, rel in (
                    [cos, sin, tan, cos(x)**2 + sin(x)**2 - 1],
                    [cosh, sinh, tanh, cosh(x)**2 - sinh(x)**2 - 1]):
                if coeff == 1 and fn in [c, s]:
                    I.append(rel)
                elif fn == t:
                    I.append(t(coeff*x)*c(coeff*x) - s(coeff*x))
                elif fn in [c, s]:
                    cn = fn(coeff*y).expand(trig=True).subs(y, x)
                    I.append(fn(coeff*x) - cn)
        return list(set(I))

def _trigpats():
    global _trigpat
    a, b, c = symbols('a b c', cls=Wild)
    d = Wild('d', commutative=False)

    # for the simplifications like sinh/cosh -> tanh:
    # DO NOT REORDER THE FIRST 14 since these are assumed to be in this
    # order in _match_div_rewrite.
    matchers_division = (
        (a*sin(b)**c/cos(b)**c, a*tan(b)**c, sin(b), cos(b)),
        (a*tan(b)**c*cos(b)**c, a*sin(b)**c, sin(b), cos(b)),
        (a*cot(b)**c*sin(b)**c, a*cos(b)**c, sin(b), cos(b)),
        (a*tan(b)**c/sin(b)**c, a/cos(b)**c, sin(b), cos(b)),
        (a*cot(b)**c/cos(b)**c, a/sin(b)**c, sin(b), cos(b)),
        (a*cot(b)**c*tan(b)**c, a, sin(b), cos(b)),
        (a*(cos(b) + 1)**c*(cos(b) - 1)**c,
            a*(-sin(b)**2)**c, cos(b) + 1, cos(b) - 1),
        (a*(sin(b) + 1)**c*(sin(b) - 1)**c,
            a*(-cos(b)**2)**c, sin(b) + 1, sin(b) - 1),

        (a*sinh(b)**c/cosh(b)**c, a*tanh(b)**c, S.One, S.One),
        (a*tanh(b)**c*cosh(b)**c, a*sinh(b)**c, S.One, S.One),
        (a*coth(b)**c*sinh(b)**c, a*cosh(b)**c, S.One, S.One),
        (a*tanh(b)**c/sinh(b)**c, a/cosh(b)**c, S.One, S.One),
        (a*coth(b)**c/cosh(b)**c, a/sinh(b)**c, S.One, S.One),
        (a*coth(b)**c*tanh(b)**c, a, S.One, S.One),

        (c*(tanh(a) + tanh(b))/(1 + tanh(a)*tanh(b)),
            tanh(a + b)*c, S.One, S.One),
    )

    matchers_add = (
        (c*sin(a)*cos(b) + c*cos(a)*sin(b) + d, sin(a + b)*c + d),
        (c*cos(a)*cos(b) - c*sin(a)*sin(b) + d, cos(a + b)*c + d),
        (c*sin(a)*cos(b) - c*cos(a)*sin(b) + d, sin(a - b)*c + d),
        (c*cos(a)*cos(b) + c*sin(a)*sin(b) + d, cos(a - b)*c + d),
        (c*sinh(a)*cosh(b) + c*sinh(b)*cosh(a) + d, sinh(a + b)*c + d),
        (c*cosh(a)*cosh(b) + c*sinh(a)*sinh(b) + d, cosh(a + b)*c + d),
    )

    # for cos(x)**2 + sin(x)**2 -> 1
    matchers_identity = (
        (a*sin(b)**2, a - a*cos(b)**2),
        (a*tan(b)**2, a*(1/cos(b))**2 - a),
        (a*cot(b)**2, a*(1/sin(b))**2 - a),
        (a*sin(b + c), a*(sin(b)*cos(c) + sin(c)*cos(b))),
        (a*cos(b + c), a*(cos(b)*cos(c) - sin(b)*sin(c))),
        (a*tan(b + c), a*((tan(b) + tan(c))/(1 - tan(b)*tan(c)))),

        (a*sinh(b)**2, a*cosh(b)**2 - a),
        (a*tanh(b)**2, a - a*(1/cosh(b))**2),
        (a*coth(b)**2, a + a*(1/sinh(b))**2),
        (a*sinh(b + c), a*(sinh(b)*cosh(c) + sinh(c)*cosh(b))),
        (a*cosh(b + c), a*(cosh(b)*cosh(c) + sinh(b)*sinh(c))),
        (a*tanh(b + c), a*((tanh(b) + tanh(c))/(1 + tanh(b)*tanh(c)))),

    )

    # Reduce any lingering artifacts, such as sin(x)**2 changing
    # to 1-cos(x)**2 when sin(x)**2 was "simpler"
    artifacts = (
        (a - a*cos(b)**2 + c, a*sin(b)**2 + c, cos),
        (a - a*(1/cos(b))**2 + c, -a*tan(b)**2 + c, cos),
        (a - a*(1/sin(b))**2 + c, -a*cot(b)**2 + c, sin),

        (a - a*cosh(b)**2 + c, -a*sinh(b)**2 + c, cosh),
        (a - a*(1/cosh(b))**2 + c, a*tanh(b)**2 + c, cosh),
        (a + a*(1/sinh(b))**2 + c, a*coth(b)**2 + c, sinh),

        # same as above but with noncommutative prefactor
        (a*d - a*d*cos(b)**2 + c, a*d*sin(b)**2 + c, cos),
        (a*d - a*d*(1/cos(b))**2 + c, -a*d*tan(b)**2 + c, cos),
        (a*d - a*d*(1/sin(b))**2 + c, -a*d*cot(b)**2 + c, sin),

        (a*d - a*d*cosh(b)**2 + c, -a*d*sinh(b)**2 + c, cosh),
        (a*d - a*d*(1/cosh(b))**2 + c, a*d*tanh(b)**2 + c, cosh),
        (a*d + a*d*(1/sinh(b))**2 + c, a*d*coth(b)**2 + c, sinh),
    )

    _trigpat = (a, b, c, d, matchers_division, matchers_add,
        matchers_identity, artifacts)
    return _trigpat

 def _expr_big_minus(cls, a, z, n):
     return cosh(2*a*asinh(sqrt(z)) + 2*a*pi*I*n)

 def _expr_big(cls, a, z, n):
     return cosh(2*a*acosh(sqrt(z)) + a*pi*I*(2*n - 1))

 def _expr_small_minus(cls, a, z):
     return cosh(2*a*asinh(sqrt(z)))

def exptrigsimp(expr, simplify=True):
    """
    Simplifies exponential / trigonometric / hyperbolic functions.
    When ``simplify`` is True (default) the expression obtained after the
    simplification step will be then be passed through simplify to
    precondition it so the final transformations will be applied.

    Examples
    ========

    >>> from sympy import exptrigsimp, exp, cosh, sinh
    >>> from sympy.abc import z

    >>> exptrigsimp(exp(z) + exp(-z))
    2*cosh(z)
    >>> exptrigsimp(cosh(z) - sinh(z))
    exp(-z)
    """
    from sympy.simplify.fu import hyper_as_trig, TR2i
    from sympy.simplify.simplify import bottom_up

    def exp_trig(e):
        # select the better of e, and e rewritten in terms of exp or trig
        # functions
        choices = [e]
        if e.has(*_trigs):
            choices.append(e.rewrite(exp))
        choices.append(e.rewrite(cos))
        return min(*choices, key=count_ops)
    newexpr = bottom_up(expr, exp_trig)

    if simplify:
        newexpr = newexpr.simplify()

    # conversion from exp to hyperbolic
    ex = newexpr.atoms(exp, S.Exp1)
    ex = [ei for ei in ex if 1/ei not in ex]
    ## sinh and cosh
    for ei in ex:
        e2 = ei**-2
        if e2 in ex:
            a = e2.args[0]/2 if not e2 is S.Exp1 else S.Half
            newexpr = newexpr.subs((e2 + 1)*ei, 2*cosh(a))
            newexpr = newexpr.subs((e2 - 1)*ei, 2*sinh(a))
    ## exp ratios to tan and tanh
    for ei in ex:
        n, d = ei - 1, ei + 1
        et = n/d
        etinv = d/n  # not 1/et or else recursion errors arise
        a = ei.args[0] if ei.func is exp else S.One
        if a.is_Mul or a is S.ImaginaryUnit:
            c = a.as_coefficient(I)
            if c:
                t = S.ImaginaryUnit*tan(c/2)
                newexpr = newexpr.subs(etinv, 1/t)
                newexpr = newexpr.subs(et, t)
                continue
        t = tanh(a/2)
        newexpr = newexpr.subs(etinv, 1/t)
        newexpr = newexpr.subs(et, t)

    # sin/cos and sinh/cosh ratios to tan and tanh, respectively
    if newexpr.has(HyperbolicFunction):
        e, f = hyper_as_trig(newexpr)
        newexpr = f(TR2i(e))
    if newexpr.has(TrigonometricFunction):
        newexpr = TR2i(newexpr)

    # can we ever generate an I where there was none previously?
    if not (newexpr.has(I) and not expr.has(I)):
        expr = newexpr
    return expr

def test_jscode_functions():
    assert jscode(sin(x) ** cos(x)) == "Math.pow(Math.sin(x), Math.cos(x))"
    assert jscode(sinh(x) * cosh(x)) == "Math.sinh(x)*Math.cosh(x)"
    assert jscode(Max(x, y) + Min(x, y)) == "Math.max(x, y) + Math.min(x, y)"
    assert jscode(tanh(x)*acosh(y)) == "Math.tanh(x)*Math.acosh(y)"
    assert jscode(asin(x)-acos(y)) == "-Math.acos(y) + Math.asin(x)"