doc/algorithms.tex: Analyse cases where Re(x^y)>0 and Im(x^y) = 0.

tests/pow.dat: add tests with Re(x^y)>0 and Im(x^y) = 0, fix tests with undefined sign of the zero part.
src/pow.c: fix sign of zero when Re(x^y)>0 and Im(x^y) = 0.


git-svn-id: svn://scm.gforge.inria.fr/svn/mpc/trunk@624 211d60ee-9f03-0410-a15a-8952a2c7a4e4

3 files changed, 327 insertions, 10 deletions

diff --git a/doc/algorithms.tex b/doc/algorithms.tex
index a1d09e2..14b29ef 100644
--- a/doc/algorithms.tex
+++ b/doc/algorithms.tex
@@ -1188,7 +1188,7 @@ is the limit of $1 + \epsilon i$ for $\epsilon > 0$ tending to zero.
 Similarly, $y = -\delta + \gamma i$ with $\delta, \gamma > 0$.
 Now $\log x \approx \epsilon^2/2 + \epsilon i$, thus
 $y \log x \approx (-\epsilon^2 \delta/2 - \epsilon \gamma)
-+ i (\epsilon^2 \gamma/2 - epsilon \delta)$.
++ i (\epsilon^2 \gamma/2 - \epsilon \delta)$.
 Thus if we neglect terms of order $3$ or more, 
 whatever the relative growth of $\epsilon, \delta, \gamma$, the real and
 imaginary parts of $y \log x$ are negative, thus we decide $x^y$ is rounded
@@ -1204,6 +1204,190 @@ thus we also round $x^y$ to $1 - 0 i$.
 % x = 1 - epsilon i, y = -delta-gamma*i
 % y log(x) = -epsilon gamma + i epsilon delta + O(epsilon^2)
 
+
+
+
+The sign of zero parts are chosen so that they are consistent with the formula
+$x^y = \exp(y\log x)$.
+Let $\phi \in [-\pi, +\pi]$ the argument of $x = |x| e^{i\phi}$, $y_1$
+(resp. $y_2$) the real (resp. imaginary) part of $y = y_1 + y_2 i$.
+Then 
+\[
+x^y=\exp(A(x,y)) (\cos B(x,y)+\sin B(x,y) i)
+\] where
+\begin {eqnarray*}
+A(x,y) & = & y_1\log|x|-y_2\phi,\\
+B(x,y) & = & y_2\log|x|+y_1\phi.
+\end {eqnarray*}
+As $|x^y| = \exp(A(x,y))$ is positive, the value of $B(x,y)$ determines the
+sign of each part of $x^y$.
+Special study is needed around $B(x, y)$ values of the form $k \pi/2$ to
+determine the sign of the zero part: let $Q_{x_0}$ (resp. $Q_{y_0}$) the
+quadrant where $x_0$ (resp. $y_0$) lies, when $B(x_0,y_0)$ lies around zero,
+if $B(x,y)$ stays non negative for $x$ and $y$ in an open domain of $Q_{x_0}
+\times Q_{y_0}$ containing $(x_0, y_0)$ then $\sin B(x_0, y_0) = \sin(+0) =
++0$, if it stays non positive then $\sin B(x_0, y_0) = \sin (-0) = -0$.
+If $B(x,y)$ tends to $+\pi$ (resp. $-\pi$) staying lesser (resp. greater) than
+it when $(x,y)$ tends to $(x_0, y_0)$, then $\sin B(x_0,y_0) = +0$.
+Conversely, if $B(x,y)$ tend to $+\pi$ (resp. $-\pi$) remaining greater (resp.
+lesser) than it, then $\sin B(x_0,y_0) = -0$.
+The same applies to the real part when $B(x,y)$ lies around $\pm \pi/2$.
+
+First, let us solve $B(x,y) = 0$.
+\begin {enumerate}
+\item
+If
+\[
+y_2 \log |x| = y_1 \phi = +0,
+\]
+then
+\[
+x^y = +\exp(A(x,y)) +0i,
+\]
+the solutions are summarized in the following table:
+\[
+\begin {array}{l c r c l l l l}
+(x_1 -0i)^{y_1 -0i} & = & x_1^{y_1} +0i 
+& \text {with} & 0 < x_1 < 1 & & y_1 <0 \\
+
+(x_1 -0i)^{-0 -0i}  & = & +1 +0i
+& \text {with} & |x_1| < 1 \\
+
+x^{-0 -0i} & = & +1 +0i
+& \text {with} & |x| < 1 & x_2 < 0 \\
+
+x^{+0 -0i} & = & +1 +0i
+& \text {with} & |x| < 1 & x_2 > 0 \\
+
+(x_1 +0i)^{+0 -0i} & = & +1 +0i
+& \text {with} & |x_1| < 1\\
+
+(x_1 +0i)^{y_1 -0i} & = & x_1^{y_1} +0i
+& \text {with} & 0 < x_1 < 1 & & y_1 > 0 \\
+
+\hline
+
+(x_1 -0i)^{y_1 +0i} & = & x_1^{y_1} +0i
+& \text {with} & x_1 \geq 1 & & y_1 <0 \\
+
+(x_1 -0i)^{-0 +0i} & = & +1 +0i
+& \text {with} & |x_1| \geq 1 \\
+
+x^{-0 +0i} & = & +1 +0i
+& \text {with} & |x| \geq 1 & x_2 < 0 \\
+
+x^{+0 +0i} & = & +1 +0i
+& \text {with} & |x| \geq 1 & x_2 > 0 \\
+
+(x_1 +0i)^{+0 +0i} & = & +1 +0i
+& \text {with} & |x_1| \geq 1\\
+
+(x_1 +0i)^{y_1 +0i} & = & x_1^{y_1} +0i
+& \text {with} & x_1 \geq 1 & & y_1 > 0 \\
+
+\hline
+
+(+1 -0i)^{y_1 +y_2i} & = & +1 +0i
+& \text {with} & & & y_1 <0 & y_2 > 0\\
+
+(\pm 1 -0i)^{-0 +y_2i} & = & \exp (y_2 \phi) +0i
+& \text {with} & & & & y_2 > 0\\
+
+x^{-0 +y_2i} & = & \exp (y_2 \phi) +0i
+& \text {with} & |x| = 1 & x_2 < 0 & & y_2 > 0 \\
+
+x^{+0 +y_2i} & = & \exp (y_2 \phi) +0i
+& \text {with} & |x| = 1 & x_2 > 0 & & y_2 > 0\\
+
+(\pm1 +0i)^{+0 +y_2i} & = & \exp (y_2 \phi) +0i
+& \text {with} & & & & y_2 > 0\\
+
+(+1 +0i)^{y_1 +y_2i} & = & +1 +0i
+& \text {with} & & & y_1 > 0 & y_2 > 0
+\end {array}
+\]
+
+\item
+If
+\[
+y_2 \log |x| = y_1 \phi = -0,
+\]
+then
+\[
+x^y = +\exp(A(x,y)) -0i,
+\]
+the solutions are summarized in the following table:
+\[
+\begin {array}{l c r c l l l l}
+(+1 +0i)^{y_1 +y_2i} & = & +1 -0i
+& \text {with} & & & y_1 < 0 & y_2 < 0\\
+
+(\pm 1 +0i)^{-0 +y_2i} & = & \exp(y_2 \phi) -0i
+& \text {with} & & & & y_2 <0 \\
+
+x^{-0 +y_2i} & = & \exp(y_2 \phi) -0i
+& \text {with} & |x| = 1 & x_2 > 0 & & y_2 <0 \\
+
+x^{+0 +y_2i} & = & \exp(y_2 \phi) -0i
+& \text {with} & |x| = 1 & x_2 < 0 & & y_2 <0 \\
+
+(\pm 1 -0i)^{+0 +y_2i} & = & \exp(y_2 \phi) -0i
+& \text {with} & & & & y_2 < 0\\
+
+(+1 -0i)^{y_1 +y_2i} & = & +1 -0i
+& \text {with} & & & y_1 > 0 & y_2 < 0\\
+
+\hline
+
+(x_1 +0i)^{y_1 -0i} & = & x^{y_1} -0i
+& \text {with} & x_1 \geq 1 & & y_1 <0 \\
+
+(x_1 +0i)^{-0 -0i} & = & +1 -0i
+& \text {with} & |x_1| \geq 1 \\
+
+x^{-0 -0i} & = & +1 -0i
+& \text {with} & |x| \geq 1 & x_2 > 0 \\
+
+x^{+0 -0i} & = & +1 -0i
+& \text {with} & |x| \geq 1 & x_2 < 0 \\
+
+(x_1 -0i)^{+0 -0i} & = & +1 -0i
+& \text {with} & |x_1| \geq 1\\
+
+(x_1 -0i)^{y_1 -0i} & = & x_1^{y_1} -0i
+& \text {with} & x_1 \geq 1 & & y_1 > 0 \\
+
+\hline
+
+(x_1 +0i)^{y_1 +0i} & = & x_1^{y_1} -0i
+& \text {with} & 0 < x_1 < 1 & & y_1 <0 \\
+
+(x_1 +0i)^{-0 +0i} & = & +1 -0i
+& \text {with} & |x_1| < 1 \\
+
+x^{-0 +0i} & = & +1 -0i
+& \text {with} & |x| < 1 & x_2 > 0 \\
+
+x^{+0 +0i} & = & +1 -0i
+& \text {with} & |x| < 1 & x_2 < 0 \\
+
+(x_1 -0i)^{+0 +0i} & = & +1 -0i
+& \text {with} & |x_1| < 1 \\
+
+(x_1 -0i)^{y_1 +0i} & = & x_1^{y_1} -0i
+& \text {with} & 0 < x_1 < 1 & & y_1 > 0 \\
+\end {array}
+\]
+
+\item
+Is it possible that 
+\[
+\frac{\log |x|}{\phi} = -\frac{y_1}{y_2} \neq 0?
+\]
+In other words, can $\frac{\log |x|}{\arg (x)}$ be a rational number when $x$
+is a dyadic complex?
+\end {enumerate}
+
 \bibliographystyle{acm}
 \bibliography{algorithms}
 
diff --git a/src/pow.c b/src/pow.c
index 81c50df..70a7b9c 100644
--- a/src/pow.c
+++ b/src/pow.c
@@ -463,7 +463,10 @@ mpc_pow (mpc_ptr z, mpc_srcptr x, mpc_srcptr y, mpc_rnd_t rnd)
       /* Special case 1^y = 1 */
       if (mpfr_cmp_ui (MPC_RE(x), 1) == 0)
         {
-          ret = mpc_set_ui (z, 1, rnd);
+          ret = mpc_set_ui (z, +1, rnd);
+          if (mpfr_signbit (MPC_IM (y))
+              &&  mpfr_signbit (MPC_RE (y)) != mpfr_signbit (MPC_IM (x)))
+            mpc_conj (z, z, MPC_RNDNN);
           goto end;
         }
 
@@ -635,8 +638,12 @@ mpc_pow (mpc_ptr z, mpc_srcptr x, mpc_srcptr y, mpc_rnd_t rnd)
 
   if (z_real)
     {
+      int s = mpfr_signbit (MPC_IM (y))
+        && mpfr_signbit (MPC_RE (y)) != mpfr_signbit (MPC_IM (x));
       ret = mpfr_set (MPC_RE(z), MPC_RE(u), MPC_RND_RE(rnd));
       ret = MPC_INEX(ret, mpfr_set_ui (MPC_IM(z), 0, MPC_RND_IM(rnd)));
+      if (s)
+        mpc_conj (z, z, MPC_RNDNN);
     }
   else if (z_imag)
     {
diff --git a/tests/pow.dat b/tests/pow.dat
index 7baaad0..281853a 100644
--- a/tests/pow.dat
+++ b/tests/pow.dat
@@ -169,6 +169,128 @@
 # (1-0*i)^(-0-0*i) -> 1+0*i
 0 0 53   +1 53   +0    53   +1 53   -0  53   -0 53   -0 N N
 
+# (x-0i)^(y-0i) = +a +0i when 0 < x < 1 and y <= -0
+0 0 53     4 53 +0      53 +0.5 53   -0    53 -2 53 -0 N N
+0 0 53     1 53 +0      53 +0.5 53   -0    53 -0 53 -0 N N
+# x^(-0-0i) = +1 +0i when |x| < 1 and Im(x) < 0
+0 0 53     1 53 +0      53 -0.5 53 -0.5    53 -0 53 -0 N N
+0 0 53     1 53 +0      53   -0 53 -0.5    53 -0 53 -0 N N
+0 0 53     1 53 +0      53   +0 53 -0.5    53 -0 53 -0 N N
+0 0 53     1 53 +0      53 +0.5 53 -0.5    53 -0 53 -0 N N
+# x^(+0-0i) = +1 +0i when |x| < 1 and Im(x) > 0
+0 0 53     1 53 +0      53 +0.5 53 +0.5    53 +0 53 -0 N N
+0 0 53     1 53 +0      53   +0 53 +0.5    53 +0 53 -0 N N
+0 0 53     1 53 +0      53   -0 53 +0.5    53 +0 53 -0 N N
+0 0 53     1 53 +0      53 -0.5 53 +0.5    53 +0 53 -0 N N
+# (x+0i)^(y-0i) = +a +0i when 0 < x < 1 and y >= +0
+0 0 53     1 53 +0      53 +0.5 53   +0    53 +0 53 -0 N N
+0 0 53  0.25 53 +0      53 +0.5 53   +0    53 +2 53 -0 N N
+
+# (x-0i)^(y+0i) = +a +0i when x >= 1 and y <= -0
+0 0 53   0.5 53 +0      53   +2 53   -0    53 -1 53 +0 N N
+0 0 53     1 53 +0      53   +3 53   -0    53 -0 53 +0 N N
+0 0 53     1 53 +0      53   +1 53   -0    53 -1 53 +0 N N
+0 0 53     1 53 +0      53   +1 53   -0    53 -0 53 +0 N N
+# x^(-0+0i) = +1 +0i when |x| >= 1 and Im(x) < 0
+0 0 53     1 53 +0      53 +0.5 53 -0.5    53 -0 53 +0 N N
+0 0 53     1 53 +0      53   +0 53 -0.5    53 -0 53 +0 N N
+0 0 53     1 53 +0      53   -0 53 -0.5    53 -0 53 +0 N N
+0 0 53     1 53 +0      53 -0.5 53 -0.5    53 -0 53 +0 N N
+# x^(+0+0i) = +1 +0i when |x| >= 1 and Im(x) > 0
+0 0 53     1 53 +0      53 +0.5 53 +0.5    53 +0 53 +0 N N
+0 0 53     1 53 +0      53   +0 53   -1    53 +0 53 +0 N N
+0 0 53     1 53 +0      53   -0 53   +1    53 +0 53 +0 N N
+0 0 53     1 53 +0      53 -0.5 53 +0.5    53 +0 53 +0 N N
+# (x+0i)^(y+0i) = +a +0i when x >= 1 and y >= +0
+0 0 53     1 53   +0    53   +2 53   +0    53 +0 53 +0 N N
+0 0 53     4 53   +0    53   +2 53   +0    53 +2 53 +0 N N
+0 0 53     1 53   +0    53   +1 53   +0    53 +0 53 +0 N N
+0 0 53     1 53   +0    53   +1 53   +0    53 +2 53 +0 N N
+
+# (+1-0i)^(y1+y2 i) = +1 +0i when y1 <= -0 and y2 > 0
+0 0 53                     1 53 +0      53 +1 53 -0    53 -2 53 +1 N N
+0 0 53                     1 53 +0      53 +1 53 -0    53 -1 53 +2 N N
+# (+/-1-0i)^(+0+y i) = +a +0i when y > 0
+0 0 53                     1 53 +0      53 +1 53 -0    53 -0 53 +1 N N
+- 0 53 +0x10BBEEE9177E19p-43 53 +0      53 -1 53 -0    53 -0 53 +2 N N
+# x^(-0-y i) = +a +0i when |x| = 1, Im(x) < 0 and y > 0
++ 0 53 +0x1724046EB0933Ap-48 53 +0      53 -1 53 -0    53 -0 53 +1 N N
++ 0 53 +0x1724046EB0933Ap-48 53 +0      53 -0 53 -1    53 -0 53 +2 N N
+- 0 53 +0x1BD4567B975381p-46 53 +0      53 +0 53 -1    53 -0 53 +3 N N
+0 0 53                     1 53 +0      53 +1 53 -0    53 -0 53 +4 N N
+# x^(+0+y i) = +a +0i when |x| = 1, Im(x) > 0 and y > 0
+0 0 53                     1 53 +0      53 +1 53 +0    53 +0 53 +1 N N
++ 0 53 +0x1620227B598EF9p-57 53 +0      53 +0 53 +1    53 +0 53 +2 N N
+- 0 53 +0x1265D4E92B6B9Bp-59 53 +0      53 -0 53 +1    53 +0 53 +3 N N
++ 0 53 +0x1D4102BC3F7D4Cp-71 53 +0      53 -1 53 +0    53 +0 53 +4 N N
+# (+/-1+0i)^(0+yi) = +a +0i when y > 0
++ 0 53 +0x1E989F5D6DFF5Cp-62 53 +0      53 -1 53 +0    53 +0 53 +2 N N
+0 0 53                     1 53 +0      53 +1 53 +0    53 +0 53 +2 N N
+# (+1 +0i)^(y1+y2 i) = +1 +0i when y1 >= +0 and y2 >0
+0 0 53                     1 53 +0      53 +1 53 +0    53 +2 53 +2 N N
+0 0 53                     1 53 +0      53 +1 53 +0    53 +0 53 +2 N N
+
+# (+1 +0i)^y = +1 -0i when Re(y) <= -0 and Im(y) < 0
+0 0 53                     1 53 -0      53 +1 53 +0    53 -1 53 -1 N N
+# (+/-1 +0i)^(-0 +yi) = +a -0i when y < 0
+0 0 53                     1 53 -0      53 +1 53 +0    53 -0 53 -1 N N
++ 0 53 +0x1724046EB0933Ap-48 53 -0      53 -1 53 +0    53 -0 53 -1 N N
+# x^(-0+y i) = +a -0i when |x| = 1, Im(x) >= +0 and y < 0
++ 0 53 +0x1724046EB0933Ap-48 53 -0      53 +0 53 +1    53 -0 53 -2 N N
+- 0 53 +0x1BD4567B975381p-46 53 -0      53 -0 53 +1    53 -0 53 -3 N N
+# x^(+0+y i) = +a -0i when |x| = 1, Im(x) <= -0 and y < 0
+- 0 53 +0x1A9BCC46F767DFp-55 53 -0      53 +0 53 -1    53 +0 53 -1 N N
++ 0 53 +0x1620227B598EF9p-57 53 -0      53 -0 53 -1    53 +0 53 -2 N N
+# (+/-1 -0i)^(+0+y i) = +a -0i when y < 0
+0 0 53                     1 53 -0      53 +1 53 -0    53 +0 53 -1 N N
++ 0 53 +0x1620227B598EF9p-57 53 -0      53 -1 53 -0    53 +0 53 -1 N N
+# (+1 -0i)^y = +1 -0i when Re(y) > 0 and Im(y) < 0
+0 0 53                     1 53 -0      53 +1 53 -0    53 +2 53 -3 N N
+
+# (x +0i)^(y -0i) = +a -0i when x >= 1 and y < 0
+0 0 53   0.5 53 -0      53   +2 53   +0    53 -1 53 -0 N N
+0 0 53     1 53 -0      53   +1 53   +0    53 -1 53 -0 N N
+# x^(-0 -0i) = +1 -0i when |x| >= 1 and Im (x) >= +0
+0 0 53     1 53 -0      53 +1.5 53 +0      53 -0 53 -0 N N
+0 0 53     1 53 -0      53   +1 53 +0      53 -0 53 -0 N N
+0 0 53     1 53 -0      53   -1 53 +0      53 -0 53 -0 N N
+0 0 53     1 53 -0      53 -1.5 53 +0      53 -0 53 -0 N N
+0 0 53     1 53 -0      53 +1.5 53 +4      53 -0 53 -0 N N
+0 0 53     1 53 -0      53   +1 53 +4      53 -0 53 -0 N N
+0 0 53     1 53 -0      53   -1 53 +4      53 -0 53 -0 N N
+0 0 53     1 53 -0      53 -1.5 53 +4      53 -0 53 -0 N N
+# x^(+0 -0i) = +1 -0i when |x| >= 1 and Im (x) <= -0
+0 0 53     1 53 -0      53 +1.5 53 -0      53 +0 53 -0 N N
+0 0 53     1 53 -0      53   +1 53 -0      53 +0 53 -0 N N
+0 0 53     1 53 -0      53   -1 53 -0      53 +0 53 -0 N N
+0 0 53     1 53 -0      53 -1.5 53 -0      53 +0 53 -0 N N
+0 0 53     1 53 -0      53 +1.5 53 -4      53 +0 53 -0 N N
+0 0 53     1 53 -0      53   +1 53 -4      53 +0 53 -0 N N
+0 0 53     1 53 -0      53   -1 53 -4      53 +0 53 -0 N N
+0 0 53     1 53 -0      53 -1.5 53 -4      53 +0 53 -0 N N
+# (x -0i)^(y -0i) = x^y -0i when x >= 1 and y > 0
+0 0 53     9 53 -0      53   +3 53   -0    53 +2 53 -0 N N
+0 0 53     1 53 -0      53   +1 53   -0    53 +2 53 -0 N N
+
+# (x +0i)^(y +0i) = x^y -0i when 0 < x < 1 and y < 0
+0 0 53     2 53 -0      53 +0.5 53   +0    53 -1 53 +0 N N
+# x^(-0+0i) = +1 -0i when |x| < 1 and Im(x) >= +0
+0 0 53     1 53 -0      53 -0.5 53   +0    53 -0 53 +0 N N
+0 0 53     1 53 -0      53 -0.1 53 +0.3    53 -0 53 +0 N N
+0 0 53     1 53 -0      53 -0.0 53 +0.3    53 -0 53 +0 N N
+0 0 53     1 53 -0      53 +0.0 53 +0.3    53 -0 53 +0 N N
+0 0 53     1 53 -0      53 +0.1 53 +0.3    53 -0 53 +0 N N
+0 0 53     1 53 -0      53 +0.5 53   +0    53 -0 53 +0 N N
+# x^(+0+0i) = +1 -0i when |x| < 1 and Im(x) <= -0
+0 0 53     1 53 -0      53 -0.5 53   -0    53 +0 53 +0 N N
+0 0 53     1 53 -0      53 -0.1 53 -0.3    53 +0 53 +0 N N
+0 0 53     1 53 -0      53 -0.0 53 -0.3    53 +0 53 +0 N N
+0 0 53     1 53 -0      53 +0.0 53 -0.3    53 +0 53 +0 N N
+0 0 53     1 53 -0      53 +0.1 53 -0.3    53 +0 53 +0 N N
+0 0 53     1 53 -0      53 +0.5 53   -0    53 +0 53 +0 N N
+# (x-0i)^(y+0i) = x^y -0i when 0 < x < 1 and y > 0
+0 0 53 +0.25 53 -0      53 +0.5 53   -0    53 +2 53 +0 N N
+
 # exact cases
 # (-4)^(1/4) = 1+i
 0 0 2 1 2 1 2 -4 2 0 2 0x1p-2 2 0 N N
@@ -195,14 +317,18 @@
 # x = -2 - epsilon*i, y = -3 - delta*i
 # log(x) = log(2) - [Pi-epsilon/2]*i + O(epsilon^2)
 # y*log(x) ~ -3*log(2) + [3*Pi-3*epsilon/2-delta*log(2)]*i
-0 0    2 -0x1p-3 2 +0    2 -2 2 -0    2 -3 2 -0    N N
-0 0 2 +0 2 -2    2 +0 2 0x1p-1    2 -1 2 -0    N N
-0 - 2 +0 3 -5    2 +0 53 0xCCCCCCCCCCCCDp-54    2 -1 2 -0    N N
-- 0 5 -25 2 +0   2 +0 53 0xCCCCCCCCCCCCDp-54    2 -2 2 -0    N N
-- - 53 -0x85649E3220691p-63 53 -0x14A25D455A9D0Dp-60 3 5 2 3 2 -3 2 +0 N N
-+ 0 53 0xABCC77118461Dp-74 2 +0 3 5 3 5 2 -8 2 0 N N
-+ 0 53 -0x127DB86014739Dp-93 2 +0 2 -1 2 -0 2 1 4 -9 N N
-+ + 24 0xC1F98Dp-21 24 0x12FF89p-2 24 -7 24 +0 24 0xCFFFF3p-21 24 +0 N N
+0 0  2              -0x1p-3  2                    +0      2 -2  2                  -0   2           -3  2 -0 N N
+0 0  2                   +0  2                    -2      2 +0  2              0x1p-1   2           -1  2 -0 N N
+
+0 -  2                   +0  3                    -5      2 +0 53 0xCCCCCCCCCCCCDp-54   2           -1  2 -0 N N
+# undefined zero sign in result
+- 0  5                  -25  2                     0      2 +0 53 0xCCCCCCCCCCCCDp-54   2           -2  2 -0 N N
+
+- - 53 -0x85649E3220691p-63 53 -0x14A25D455A9D0Dp-60      3  5  2                   3   2           -3  2 +0 N N
++ 0 53  0xABCC77118461Dp-74  2                    +0      3  5  3                   5   2           -8  2 +0 N N
+
++ 0 53 -0x127DB86014739Dp-93 2                    -0      2 -1  2                  -0   2            1  4 -9 N N
++ + 24  0xC1F98Dp-21        24           0x12FF89p-2     24 -7 24                  +0  24 0xCFFFF3p-21 24 +0 N N
 # underflow case
 - - 24 +0 24 +0 24 2 24 0x44CCCDp-20 24 -0x7FFFF200 24 -0x7FFFF200 N N
 - 0 53 0x14D55AFA6E0BB0p210433620 53 0 53 +0 53 0x44CCCCFFFFFFFp-48 53 0x5F5E100 53 +0 N N