##def sqrt(x) :
##    guess = 1.0
##    while guess * guess != x:
##        guess = (guess + x/guess)/2
##    return guess



##def sqrt(x) :
##    guess = 1.0
##    n = 0
##    while abs(guess * guess - x) > 0.000000001 :
##        guess = (guess + x/guess)/2
##        n = n + 1
##        print(str(n) + "th iteration:", guess)
##    return guess


# math包专门用于浮点数

##from math import fabs
##def sqrt(x) :
##    guess = 1.0
##    while fabs(guess * guess - x) >= 0.0001 :
##        guess = (guess + x/guess)/2
##    return guess


### 两段说明浮点数计算误差的程序

##### 加 0.1 重复 10 次，能等于 1.0 吗？
##x = 0.0
##for n in range(10):
##    x = x + 0.1
##print("x == 1.0 is", x == 1.0)
##print("x is", x)


###### 希望循环体执行 100 次。实际做多少次？
##s = 0.0
##n = 0
##while s < 10.0 :
##    s = s + 0.1
##    n = n + 1
##print("Iterating", n , "times.","s =", s)


##ln2 = 0.0
##for n in range(1, 21):
##    ln2 += (-1)**(n-1) * 1/n 
##
##
##ln2 = 0.0
##n = 1
##while n <= 20:
##    ln2 += (-1)**(n-1) * 1/n 
##    n += 1



##### 计算 sin(x) 的近似值

###### 1 #######

##def term(x, n):
##    t = (-1)**n * x ** (2 * n + 1)
##    for i in range(2, 2 * n + 2):
##        t /= i
##    return t
##
##def mysin(x):
##    sn = 0.0
##    n = 0
##    while True:
##        t = term(x, n)
##        if abs(t) < 1e-6:
##            return sn
##        sn += t
##        n += 1

###### 2 ########

##def mysin(x):
##    sn = x
##    t = x
##    n = 0
##    while True:
##        n += 1
##        t *= - x * x / (2*n) / (2*n + 1)
##        if abs(t) < 1e-6:
##            return sn
##        sn += t
    

##### 简单的确定次数的输入循环 ###########

##print("Calculating average of 10 numbers")
##ss = 0.0
##for i in range(10):
##    x = float(input("Next number: "))
##    ss += x
##print("The average of these 10 numbers is", ss/10)


##### 输入控制的循环 ###########

##print("Calculating average.")
##ss = 0.0
##n = 0
##while True:
##    snum = input("Next number ('None' to stop): ")
##    if snum == "None":
##        break
##    ss += float(snum)
##    n += 1
##print("The average of these " + str(n) + " numbers is", ss/n)


##name = ""
##while name != "no more":
##    name = input("Hi, friend! What is your name? ")
##    print("Hello, " + name + "!\n")


##name = ""
##while True :
##    name = input("Hi, friend! What is your name? ")
##    if name == "no more":
##        break
##    print("Hello, " + name + "!\n")


##name = ""
##while True :
##    name = input("Hi, friend! What is your name? ")
##    if name == "no more":
##        break
##    s = "Hello, " + name + "!\n"
##    print(s)


######### 程序终止性的例子 ##########

##def collatz(n):
##    num = n
##    count = 0
##    while n != 1:
##        if n % 2 == 0:  # n is even
##            n = n//2
##        else:           # n is odd
##            n = n * 3 + 1
##        count += 1
##    print("For integer " + str(num) + " iterating " +
##          str(count) + " times.")
##
##for i in range(20, 30):
##    collatz(i)


###### 用 assert 检查输入的程序

##from math import fabs
##def sqrt(x) :
##    assert(x >= 0)
##    guess = 1.0
##    while fabs(guess * guess - x) >= 0.0001 :
##        guess = (guess + x/guess)/2
##    return guess


### Here we take n! == 1 for all n <= 0
##def fact(n) :
##    assert(type(n) == int and n >= 0)
##    prod = 1
##    while n > 1:
##        prod *= n
##        n -= 1
##    return prod
##
##for n in range(1, 25, 3):
##    print("Factorial of", n, "is", fact(n))