[Tutor] nice()

[Smith]

I've been thinking about a function that was recently proposed at python-dev named 'areclose'. It is a function that is meant to tell whether two (or possible more) numbers are close to each other. It is a function similar to one that exists in Numeric. One such implementation is

def areclose(x,y,abs_tol=1e-8,rel_tol=1e-5):
    diff = abs(x-y)
    return diff <= ans_tol or diff <= rel_tol*max(abs(x),abs(y))

(This is the form given by Scott Daniels on python-dev.)

Anyway, one of the rationales for including such a function was: 

  When teaching some programming to total newbies, a common frustration
  is how to explain why a==b is False when a and b are floats computed
  by different routes which ``should'' give the same results (if
  arithmetic had infinite precision).  Decimals can help, but another
  approach I've found useful is embodied in Numeric.allclose(a,b) --
  which returns True if all items of the arrays are ``close'' (equal to
  within certain absolute and relative tolerances)
The problem with the above function, however, is that it *itself* has a comparison between floats and it will give undesired result for something like the following test:

###
>>> print areclose(2, 2.1, .1, 0) #see if 2 and 2.1 are within 0.1 of each other
False
>>>
###

Here is an alternative that might be a nice companion to the repr() and round() functions: nice(). It is a combination of Tim Peter's delightful 'case closed' presentation in the thread, "Rounding to n significant digits?" [1] and the hidden magic of "prints" simplification of floating point numbers when being asked to show them. 

It's default behavior is to return a number in the form that the number would have when being printed. An optional argument, however, allows the user to specify the number of digits to round the number to as counted from the most significant digit. (An alternative name, then, could be 'lround' but I think there is less baggage for the new user to think about if the name is something like nice()--a function that makes the floating point numbers "play nice." And I also think the name...sounds nice.) 

Here it is in action:

###
>>> 3*1.1==3.3
False
>>> nice(3*1.1)==nice(3.3)
True
>>> x=3.21/0.65; print x
4.93846153846
>>> print nice(x,2)
4.9
>>> x=x*1e5; print nice(x,2)
490000.0
###

Here's the function: 
###
def nice(x,leadingDigits=0):
 """Return x either as 'print' would show it (the default) or rounded to the
 specified digit as counted from the leftmost non-zero digit of the number,

 e.g. nice(0.00326,2) --> 0.0033"""
 assert leadingDigits>=0
 if leadingDigits==0:
  return float(str(x)) #just give it back like 'print' would give it
 leadingDigits=int(leadingDigits)
 return float('%.*e' % (leadingDigits,x)) #give it back as rounded by the %e format
###

Might something like this be useful? For new users, no arguments are needed other than x and floating points suddenly seem to behave in tests made using nice() values. It's also useful for those computing who want to show a physically meaningful value that has been rounded to the appropriate digit as counted from the most significant digit rather than from the decimal point. 

Some time back I had worked on the significant digit problem and had several math calls to figure out what the exponent was. The beauty of Tim's solution is that you just use built in string formatting to do the work. Nice.

/c

[1] http://mail.python.org/pipermail/tutor/2004-July/030324.html
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