Improve doc for floating point ‘=’ vs ‘eql’

* doc/lispref/numbers.texi (Float Basics, Comparison of Numbers):
Improve documentation of ‘=’ vs ‘eq’, ‘eql’ and ‘equal’
when NaNs and signed zeros are involved.

1 files changed, 21 insertions, 10 deletions

diff --git a/doc/lispref/numbers.texi b/doc/lispref/numbers.texi
index 6c51b849d35..70bb1030411 100644
--- a/doc/lispref/numbers.texi
+++ b/doc/lispref/numbers.texi
@@ -223,7 +223,7 @@ least one digit after any decimal point in a floating-point number;
 @samp{1500.} is an integer, not a floating-point number.
 
   Emacs Lisp treats @code{-0.0} as numerically equal to ordinary zero
-with respect to @code{equal} and @code{=}.  This follows the
+with respect to numeric comparisons like @code{=}.  This follows the
 @acronym{IEEE} floating-point standard, which says @code{-0.0} and
 @code{0.0} are numerically equal even though other operations can
 distinguish them.
@@ -232,19 +232,26 @@ distinguish them.
 @cindex negative infinity
 @cindex infinity
 @cindex NaN
-@findex eql
-@findex sxhash-eql
   The @acronym{IEEE} floating-point standard supports positive
 infinity and negative infinity as floating-point values.  It also
 provides for a class of values called NaN, or ``not a number'';
 numerical functions return such values in cases where there is no
 correct answer.  For example, @code{(/ 0.0 0.0)} returns a NaN@.
 A NaN is never numerically equal to any value, not even to itself.
-NaNs carry a sign and a significand, and non-numeric functions like
-@code{eql} and @code{sxhash-eql} treat two NaNs as equal when their
+NaNs carry a sign and a significand, and non-numeric functions treat
+two NaNs as equal when their
 signs and significands agree.  Significands of NaNs are
 machine-dependent and are not directly visible to Emacs Lisp.
 
+  When NaNs and signed zeros are involved, non-numeric functions like
+@code{eql}, @code{equal}, @code{sxhash-eql}, @code{sxhash-equal} and
+@code{gethash} determine whether values are indistinguishable, not
+whether they are numerically equal.  For example, when @var{x} and
+@var{y} are the same NaN, @code{(equal x y)} returns @code{t} whereas
+@code{(= x y)} uses numeric comparison and returns @code{nil};
+conversely, @code{(equal 0.0 -0.0)} returns @code{nil} whereas
+@code{(= 0.0 -0.0)} returns @code{t}.
+
 Here are read syntaxes for these special floating-point values:
 
 @table @asis
@@ -359,11 +366,15 @@ if so, @code{nil} otherwise.  The argument must be a number.
 @cindex comparing numbers
 
   To test numbers for numerical equality, you should normally use
-@code{=}, not @code{eq}.  There can be many distinct floating-point
-objects with the same numeric value.  If you use @code{eq} to
-compare them, then you test whether two values are the same
-@emph{object}.  By contrast, @code{=} compares only the numeric values
-of the objects.
+@code{=} instead of non-numeric comparison predicates like @code{eq},
+@code{eql} and @code{equal}.  Distinct floating-point objects can be
+numerically equal.  If you use @code{eq} to compare them, you test
+whether they are the same @emph{object}; if you use @code{eql} or
+@code{equal}, you test whether their values are
+@emph{indistinguishable}.  In contrast, @code{=} uses numeric
+comparison, and sometimes returns @code{t} when a non-numeric
+comparison would return @code{nil} and vice versa.  @xref{Float
+Basics}.
 
   In Emacs Lisp, each integer is a unique Lisp object.
 Therefore, @code{eq} is equivalent to @code{=} where integers are