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numerics - Is there a difference between Divide[a,b] and a/b?


In this comment it was asserted that Divide[a,b] and a/b are different, though the documentation indicates that they are the same. In particular, it was asserted that a/b is evaluated as a * 1/b, whereas Divide[a,b] performs the division directly. It was further asserted that this could result in an observable difference in behavior for machine-precision numbers.


The question is: is there a difference? If so, what is it exactly? How does it manifest itself? Is there an example where Divide gives a different result than /?



I tried a few edge cases with $MaxMachineNumber and $MinMachineNumber, but didn't find any differences in behavior between the two.



Answer



Oleksandr is correct about the way evaluation works. a/b seems to be interpreted (parsed) directly as Times[a, Power[b,-1]], or more readably: $a\times b^{-1}$. Divide[a,b] is interpreted as is. Evaluation then proceeds from these forms, and the arithmetic is carried out differently for the two cases: either $a\times (1/b)$ or $a/b$.


Here are some examples that illustrate the evaluation sequence:


In[95]:= 
On[]
Divide[4,8]
Off[]

During evaluation of In[95]:= On::trace: On[] --> Null. >>

During evaluation of In[95]:= Divide::trace: 4/8 --> 1/2. >>

Out[96]= 1/2

In[98]:= 
On[]
4/8
Off[]

During evaluation of In[98]:= On::trace: On[] --> Null. >>

During evaluation of In[98]:= Power::trace: 1/8 --> 1/8. >>
During evaluation of In[98]:= Times::trace: 4/8 --> 4/8. >>
During evaluation of In[98]:= Times::trace: 4/8 --> 1/2. >>

Out[99]= 1/2

This can indeed theoretically lead to different machine precision results. Let's find out if it really does! We are going to compare the complete binary representation of the results, and we won't use == or === (which both have some tolerance).


Table[{k, RealDigits[k/137., 2] === RealDigits[Divide[k, 137.], 2]}, {k, 1, 20}]

(* ==> {{1, True}, {2, True}, {3, True}, {4, True}, {5, True}, {6, True}, {7, True}, 

        {8, True}, {9, True}, {10, True}, {11, True}, {12, True}, {13, True}, 
        {14, True}, {15, False}, {16, True}, {17, True}, {18, True}, {19, True}, {20, True}} 
 *)

So Divide[15, 137.] and 15/137. really do lead to different results. Conclusion: yes, there is an observable difference.


Again, keep in mind that even === has some tolerance (Internal`$SameQTolerance) when comparing machine precision numbers (though less than ==, Internal`$EqualTolerance) and 15/137. === Divide[15, 137.] returns True. The difference is there though as evidenced by the full 53-bit binary representation.


So will you ever see the effects of this in practice? Theoretically the error may accumulate, and there are some functions which do not honour these tolerances and perform strict comparisons (try e.g. Union[{15/137., Divide[15, 137.]}], which returns a list of length 2).


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