from sympy import *
init_printing()

def evaluate(exprs, x, x0):
    """
    Evaluate each expression in exprs at the point x = x0.

    >>> x, y = symbols('x y')
    >>> exprs = [x**2, cos(x), x*y]
    >>> evaluate(exprs, x, 1)
    [1, cos(1), y]
    >>> evaluate(exprs, y, 0)
    [x**2, cos(x), 0]
    """
    return [expr.subs(x, x0) for expr in exprs]

x, y = symbols('x y')

exprs = [x**2, cos(x), x*y]

evaluate(exprs, x, 1)

evaluate(exprs, y, 0)

def uparrow(x, n):
    """
    Computes x**(x**(...x)), with n copies of x.

    >>> x = symbols('x')
    >>> uparrow(x, 3)
    x**(x**x)
    >>> uparrow(x, 1)
    x
    >>> uparrow(x**x, 3)
    (x**x)**((x**x)**(x**x))
    """
    expr = x
    for i in range(n - 1):
        expr = x**expr
    return expr

x = symbols('x')

uparrow(x, 3)

uparrow(x, 1)

uparrow(x**x, 3)

def nest(expr, x, n):
    """
    Nests expr into itself (in the variable x) n times.

    >>> x, y = symbols('x y')
    >>> nest(x**x, x, 3)
    ((x**x)**(x**x))**((x**x)**(x**x))
    >>> nest(sin(x)*cos(y), x, 2)
    sin(sin(x)*cos(y))*cos(y)
    >>> nest(sin(x)*cos(y), y, 2)
    sin(x)*cos(sin(x)*cos(y))
    >>> nest(x**2, x, 1)
    x**2
    """
    origexpr = expr
    for i in range(n - 1):
        expr = expr.subs(x, origexpr)
    return expr

x, y = symbols('x y')

nest(x**x, x, 3)

nest(sin(x)*cos(y), x, 2)

nest(sin(x)*cos(y), y, 2)

nest(x**2, x, 1)

def trig_rewrite(expr, x):
    """
    Rewrite all trig functions t(x) in terms of sin(x) and cos(x)

    >>> x, y = symbols('x y')
    >>> trig_rewrite(tan(x), x)
    sin(x)/cos(x)
    >>> trig_rewrite(tan(x) + cos(x)*sec(x), x)
    sin(x)/cos(x) + 1
    >>> trig_rewrite(cot(x) + sin(x)*csc(x), x)
    1 + cos(x)/sin(x)
    >>> trig_rewrite(tan(x)*tan(y), x)
    sin(x)*tan(y)/cos(x)
    >>> trig_rewrite(tan(x)*tan(y), y)
    sin(y)*tan(x)/cos(y)
    """
    return expr.subs([(tan(x), sin(x)/cos(x)), (sec(x), 1/cos(x)), (csc(x), 1/sin(x)), (cot(x), cos(x)/sin(x))])

x, y = symbols('x y')

trig_rewrite(tan(x), x)

trig_rewrite(tan(x) + cos(x)*sec(x), x)

trig_rewrite(cot(x) + sin(x)*csc(x), x)

trig_rewrite(tan(x)*tan(y), x)

trig_rewrite(tan(x)*tan(y), y)

def series_reduce(expr, x, p):
    """
    Remove all powers of x in expr with power greater than p.

    You may assume that there are no powers of x greater than 10.

    Bonus: which functions are represented by the series expansions below (you
    can use expr.series(x, 0, 10) to check if you are right)?

    >>> x, y = symbols('x y')
    >>> series_reduce(1 - x**2/2 + x**4/24 - x**6/720 + x**8/40320, x, 5)
    x**4/24 - x**2/2 + 1
    >>> series_reduce(1 + x + x**2 + x**3 + x**4 + x**5 + x**6 + x**7 + x**8 + x**9 + x**10, x, 0)
    1
    >>> series_reduce(x*y + x**3*y**3/3 + 2*x**5*y**5/15 + 17*x**7*y**7/315 + 62*x**9*y**9/2835, x, 5)
    2*x**5*y**5/15 + x**3*y**3/3 + x*y
    """
    return expr.subs([(x**i, 0) for i in range(11) if  i > p])

x, y = symbols('x y')

series_reduce(1 - x**2/2 + x**4/24 - x**6/720 + x**8/40320, x, 5)

series_reduce(1 + x + x**2 + x**3 + x**4 + x**5 + x**6 + x**7 + x**8 + x**9 + x**10, x, 0)

series_reduce(x*y + x**3*y**3/3 + 2*x**5*y**5/15 + 17*x**7*y**7/315 + 62*x**9*y**9/2835, x, 5)

from IPython.core.display import Image 
Image(filename='../imgs/comic2-1040.png') 

'1.2345'.split('.', 2)[1]

str(1.2345).split('.', 2)[1].find('345')

str(1.2345).split('.', 2)[1].find('345') + 1

'abcd'.find('def')

def find_999999(expr, limit=100000):
    """
    Find the first place in the decimal expr where 999999 appears.

    Only checks up to limit digits. 

    Returns False when 999999 does not appear.

    >>> find_999999(pi)
    762
    >>> find_999999(E)
    False
    >>> find_999999(E, 1000000) # This one will take a few seconds to compute
    384340
    """
    found = str(expr.evalf(limit)).split('.', 2)[1].find('999999') + 1
    if found == 0:
        return False
    return found

find_999999(pi)

find_999999(E)

find_999999(E, 1000000)