MIT 6.0001作业1C代码排查：两版本逻辑一致却结果迥异

问题背景 #

需要计算36个月内存够房子首付的最优储蓄率，允许结果与目标首付误差在100美元以内。假设条件：

• 半年加薪率（semi_annual_raise）为0.07（7%）
• 投资年回报率（annual return）为0.04（4%）
• 首付为房价的0.25（25%）
• 目标房价为100万美元

两段代码及运行结果 #

第一段代码（运行错误） #

annual_salary=float(input("Enter your annual salary:"))
semi_annual_raise=0.07
total_cost=1000000
portion_down_payment = 0.25
current_saving=0
r=0.04
low=0
high=10000
guess=(low+high)/2
numGuesses=0
while abs(portion_down_payment*total_cost-current_saving)>=100:
    current_saving=0
    portion_saved=guess/10000
    for i in range (36):
        current_saving+=current_saving*r/12+annual_salary/12*portion_saved
        if i%6==0 and i>0:
            annual_salary+=annual_salary*semi_annual_raise
    
    if portion_down_payment*total_cost-current_saving>0:
        low=guess
    else:
        high=guess
    guess=(low+high)/2
    numGuesses+=1   
    if numGuesses > 1000:
        break

if numGuesses>1000:
    print("It is not possible to pay the down payment in three years.")
else:
    rate=guess/10000
    print("steps in bisection search is",numGuesses,"and the best saving rate is",rate)

输入年薪150000时，结果：

It is not possible to pay the down payment in three years.

第二段代码（运行正确） #

annual_salary=float(input("Enter your annual salary:"))
semi_annual_raise=0.07
total_cost=1000000
portion_down_payment = 0.25
r=0.04
low=0
high=10000
guess=(low+high)/2
numGuesses=0

def current_saving(annual_salary,guess,months):
    portion_saved=guess/10000
    saving=0
    r=0.04
    semi_annual_raise=0.07
    for i in range (months):
        saving+=saving*r/12+annual_salary/12*portion_saved
        if i%6==0 and i>0:
            annual_salary+=annual_salary*semi_annual_raise
    return(saving)
total=0
while abs(portion_down_payment*total_cost-total)>=100:
    total=current_saving(annual_salary,guess,36)
    if portion_down_payment*total_cost-total>0:
        low=guess
    else:
        high=guess
    guess=(low+high)/2
    numGuesses+=1   
    if numGuesses > 1000:
        break

if numGuesses>1000:
    print("It is not possible to pay the down payment in three years.")
else:
    rate=guess/10000
    print("steps in bisection search is",numGuesses,"and the best saving rate is",rate)

输入相同年薪时，结果：

steps in bisection search is 7 and the best saving rate is 0.44140625

错误定位 #

两段代码的核心差异在于年薪变量的作用域：

• 第一段代码中，annual_salary是全局变量，每次执行while循环内的for循环时，都会对它进行加薪修改。这意味着后续的二分迭代使用的是已经涨过多次工资的数值，而非用户初始输入的年薪。随着迭代次数增加，计算出的储蓄额会远高于实际基于初始工资的数值，导致二分搜索逻辑混乱，最终错误判断为无法凑够首付。
• 第二段代码中，函数current_saving接收的是初始的annual_salary作为参数，在函数内部修改的是局部变量，不会影响全局的初始年薪。每次二分迭代时，都是基于用户输入的原始工资重新计算储蓄额，逻辑正确。

修正后的第一段代码 #

annual_salary=float(input("Enter your annual salary:"))
initial_salary = annual_salary  # 保存初始年薪
semi_annual_raise=0.07
total_cost=1000000
portion_down_payment = 0.25
current_saving=0
r=0.04
low=0
high=10000
guess=(low+high)/2
numGuesses=0
while abs(portion_down_payment*total_cost-current_saving)>=100:
    current_saving=0
    portion_saved=guess/10000
    current_salary = initial_salary  # 每次迭代都使用初始工资
    for i in range (36):
        current_saving+=current_saving*r/12+current_salary/12*portion_saved
        if i%6==0 and i>0:
            current_salary+=current_salary*semi_annual_raise
    
    if portion_down_payment*total_cost-current_saving>0:
        low=guess
    else:
        high=guess
    guess=(low+high)/2
    numGuesses+=1   
    if numGuesses > 1000:
        break

if numGuesses>1000:
    print("It is not possible to pay the down payment in three years.")
else:
    rate=guess/10000
    print("steps in bisection search is",numGuesses,"and the best saving rate is",rate)

内容的提问来源于stack exchange，提问作者X Y