问题
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]]]
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
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?
And why the second form is so much expensive?
回答1:
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
来源：https://stackoverflow.com/questions/4939616/the-basic-input-form-for-floating-point-numbers