Merge pull request #1237 from WarrenWeckesser/doc-cauchy-cdf

DOC: Update description of the Cauchy cdf and its complement.

doc/distributions/cauchy.qbk

@@ -105,28 +105,23 @@ In the following table __x0 is the location parameter of the distribution,
 
 [table
 [[Function][Implementation Notes]]
-[[pdf][Using the relation: ['pdf = 1 / ([pi] * [gamma] * (1 + ((x - __x0) / [gamma])[super 2]) ]]]
+[[pdf][Using the relation: ['pdf = 1 / ([pi] * [gamma] * (1 + ((x - __x0) / [gamma])[super 2])) ]]]
 [[cdf and its complement][
 The cdf is normally given by:
 
 [expression p = 0.5 + atan(x)/[pi]]
 
 But that suffers from cancellation error as x -> -[infin].
-So recall that for `x < 0`:
 
-[expression atan(x) = -[pi]/2 - atan(1/x)]
+Instead, the mathematically equivalent expression based on the function atan2
+is used:
 
-Substituting into the above we get:
+[expression p = atan2(1, -x)/[pi]]
 
-[expression p = -atan(1/x) / [pi]  ; x < 0]
+By symmetry, the complement is
 
-So the procedure is to calculate the cdf for -fabs(x)
-using the above formula.  Note that to factor in the location and scale
-parameters you must substitute (x - __x0) / [gamma] for x in the above.
+[expression q = atan2(1, x)/[pi]]
 
-This procedure yields the smaller of /p/ and /q/, so the result
-may need subtracting from 1 depending on whether we want the complement
-or not, and whether /x/ is less than __x0 or not.
 ]]
 [[quantile][The same procedure is used irrespective of whether we're starting
             from the probability or its complement.  First the argument /p/ is