libm: fix double rounding description

[www] / libm / index.html

diff --git a/libm/index.html b/libm/index.html
index abdd7f8..fa908a3 100644 (file)
--- a/libm/index.html
+++ b/libm/index.html
@@ -146,7 +146,7 @@ m68k and m88k cpus use different ld80 than the intel ones)
 
 
 <ul>
-<li>Double rounding (x87 issue):
+<li>Double rounding:
 <p>
 If a value rounded twice the result can be different
 than rounding just once.
@@ -162,10 +162,12 @@ results in extended precision, this only affects x87 instructions, not see2 etc)
 (afaik freebsd and openbsd use double precision by default)
 <p>
 So x = a+b may give different results depending on
-the fpu setting.
+the x87 fpu precision setting.
 (only happens in round to nearest rounding mode,
 but that's the most common one)
 <p>
+(double rounding can happen with float vs double as well)
+<p>
 C99 annex F prohibits double rounding,
 but that's non-normative.
 
@@ -188,7 +190,11 @@ C does not require consistent evaluation
 precision: the compiler may store intermediate
 results and round them to double while keep
 other parts in higher precision.
-So (a+b)==(a+b) may be false when the two sides
+So
+<pre>
+(a+b)==(a+b)
+</pre>
+may be false when the two sides
 are kept in different precision.
 (This is not an x87 specific problem, it matters whenever there
 is a higher precision fp type than the currently used one.
@@ -202,8 +208,11 @@ C99 has a way to control this (see
 when a result is stored in a variable or a
 type cast is used, then it is guaranteed that
 the precision is appropriate to that type.
-So in (double)(a+b)==(double)(a+b) both
-sides are guaranteed to be in double precision
+So in
+<pre>
+(double)(a+b)==(double)(a+b)
+</pre>
+both sides are guaranteed to be in double precision
 when the comparision is done.
 <p>
 (This still does not solve the x87 double rounding
@@ -252,7 +261,7 @@ different precision than at runtime).
 C99 actually allows most of these optimizations
 but they can be turned off with STDC pragmas (see
 <a href="http://repo.or.cz/w/c-standard.git/blob_plain/HEAD:/n1256.html#6.10.6">6.10.6</a>).
-Unfortunately <a href="http://gcc.gnu.org/c99status.html">gcc does not support these pragmas</s>.
+Unfortunately <a href="http://gcc.gnu.org/c99status.html">gcc does not support these pragmas</a>.
 <p>
 FENV_ACCESS ON tells the compiler that the code wants
 to access the floating point environment (eg. set different rounding mode)
@@ -300,11 +309,18 @@ in the libm code: scalbn even depends on signed int overflow.
 
 <li>Complex arithmetics
 <p>
-With gcc x*I turns into (x+0*I)*(0+I) = 0*x + x*I,
-so if x=inf then the result is nan+inf*I instead of inf*I
+gcc turns
+<pre>
+x*I
+</pre>
+into
+<pre>
+(x+0*I)*(0+I) = 0*x + x*I
+</pre>
+So if x=inf then the result is nan+inf*I instead of inf*I
 (an fmul instruction is generated for 0*x)
 <p>
-(There were various complex constant folding issues as well).
+(There were various complex constant folding issues as well in gcc).
 <p>
 So a+b*I cannot be used to create complex numbers,
 instead some double[2] manipulation should be used to