浮点数的基本输入形式? [英] The basic input form for floating-point numbers?

问题描述

我刚刚发现了两种浮点数输入形式之间的根本区别:

I just have discovered the fundamental difference between two input forms for floating-point numbers:

In[8]:= 1.5*^-334355//Hold//FullForm
1.5*10^-334355//Hold//FullForm
Out[8]//FullForm= Hold[1.5000000000000000000000000000000001`15.954589770191005*^-334355]
Out[9]//FullForm= Hold[Times[1.5`,Power[10,-334355]]]

这两种形式在内存和时间消耗上有很大的不同:

These two forms differ very much in memory and time consumption:

In[7]:= start = MaxMemoryUsed[];
1.5*^-33432242 // Timing
start = MaxMemoryUsed[] - start
1.5*10^-33432242 // Timing
MaxMemoryUsed[] - start

Out[8]= {1.67401*10^-16, 1.500000000000000*10^-33432242}

Out[9]= 0

Out[10]= {7.741, 1.500000000000000*10^-33432242}

Out[11]= 34274192

但我无法找到记录 *^ 表单的位置.它是浮点数的真正基本输入形式吗?其他基数的数字如何?

But I cannot find out where the form *^ is documented. Is it a real basic input form for floating-point numbers? How is about numbers in other bases?

为什么第二种形式这么贵?

And why the second form is so much expensive?

推荐答案

关于时间和内存消耗——这些是评估的结果，与不同的形式无关.当 10 显式存在时，您对 10 的幂使用整数算法，因此时间/内存效率低下.当我们从一开始就使用机器精度时，效果就消失了:

Regarding the time and memory consumption - these are the consequences of evaluation, have nothing to do with different forms. You use integer arithmetic for the power of 10 when 10 is present explicitly, thus the time/memory inefficiency. When we use machine precision from the start, the effect disappears:

In[1]:= MaxMemoryUsed[]
1.5*^-33432242 // Timing
MaxMemoryUsed[]
1.5*10.^-33432242 // Timing
MaxMemoryUsed[]

Out[1]= 17417696

Out[2]= {0., 1.500000000000000*10^-33432242}

Out[3]= 17417696

Out[4]= {0., 1.500000000043239*10^-33432242}

Out[5]= 17417696

HTH

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