Module: ppl/ppl Branch: bounded_arithmetic Commit: 0663324f51238dc36d88909734de6e7063ba4b73 URL: http://www.cs.unipr.it/git/gitweb.cgi?p=ppl/ppl.git;a=commit;h=0663324f51238...
Author: Alberto Gioia alberto.gioia1@studenti.unipr.it Date: Sat Aug 27 17:12:43 2011 +0200
Adjusted documentation about bitwise operator.
---
doc/definitions.dox | 92 ++++++++++++++++++++++++-------------------------- 1 files changed, 44 insertions(+), 48 deletions(-)
diff --git a/doc/definitions.dox b/doc/definitions.dox index ba80d9f..e48e010 100644 --- a/doc/definitions.dox +++ b/doc/definitions.dox @@ -2821,9 +2821,10 @@ The interval bitwise operators are defined as follow: \begin{cases} \bigl[0, \min(b, b')\bigr], & \text{if $a \geq 0$ and $a' \geq 0$}; \ - \bigl[0, \max(b, b')\bigr], - & \text{if $a \geq 0$ and $b' < 0$, - or $b < 0$ and $a' \geq 0$}; \ + \bigl[0, b\bigr], + & \text{if $a \geq 0$ and $b' < 0$}; \ + \bigl[0, b'\bigr], + & \text{if $b < 0$ and $a' \geq 0$}; \ \bigl[a + a', \min(b,b')\bigr], & \text{if $b < 0$ and $b' < 0$}. \ \end{cases} @@ -2834,9 +2835,10 @@ The interval bitwise operators are defined as follow: \begin{cases} \bigl[\max(a,a'), b + b'\bigr], & \text{if $a \geq 0$ and $a' \geq 0$}; \ - \bigl[\min(a,a'),-1\bigr], - & \text{if $a \geq 0$ and $b' < 0$, - or $b < 0$ and $a' \geq 0$}; \ + \bigl[a',-1\bigr], + & \text{if $a \geq 0$ and $b' < 0$}; \ + \bigl[a,-1\bigr], + & \text{if $b < 0$ and $a' \geq 0$}; \ \bigl[\max(a,a'),-1\bigr], & \text{if $b < 0$ and $b' < 0$}. \ \end{cases} @@ -2845,25 +2847,31 @@ The interval bitwise operators are defined as follow: \f[ [a,b] \bxorii [a',b'] \approx \begin{cases} - \bigl[0,|b + b'|\bigr], - & \text{if $a \geq 0$ and $a' \geq 0$, - or $b < 0$ and $b' < 0$}; \ - \bigl[-(|a| + |a'|),-1\bigr], - & \text{if $a \geq 0$ and $b' < 0$, - or $b < 0$ and $a' \geq 0$}. \ + \bigl[0,b + b'\bigr], + & \text{if $a \geq 0$ and $a' \geq 0$}; \ + \bigl[(-a + a'),-1\bigr], + & \text{if $a \geq 0$ and $b' < 0$}; \ + \bigl[(a - a'),-1\bigr], + & \text{if $b < 0$ and $a' \geq 0$}; \ + \bigl[0, |b + b'|\bigr], + & \text{if $b < 0$ and $b' < 0$}. \ \end{cases} \f]
\f[ [a,b] \olessthan ^{#} [a',b'] \approx \begin{cases} - \bigl[0, \bigl(b \times 2^{b'}\bigr)\bigr], + \bigl[\bigl(a \times 2^{a'}\bigr), + \bigl(b \times 2^{b'}\bigr)\bigr], & \text{if $a \geq 0$ and $a' \geq 0$}; \ - \bigl[0, \bigl(b / 2^{\left|b'\right|}\bigr)\bigr], + \bigl[\bigl(a / 2^{\left|a'\right|}\bigr, + \bigl(b / 2^{\left|b'\right|}\bigr)\bigr], & \text{if $a \geq 0$ and $b' < 0$}; \ - \bigl[\bigl(a \times 2^{a'}\bigr), 0\bigr], + \bigl[\bigl(a \times 2^{a'}\bigr), + \bigl(b \times 2^{b'}\bigr)\bigr], & \text{if $b < 0$ and $a' \geq 0$}; \ - \bigl[\bigl(a / 2^{\left|a'\right|}\bigr), 0\bigr], + \bigl[\bigl(a / 2^{\left|a'\right|}\bigr), + \bigl(b / 2^{\left|b'\right|}\bigr)\bigr], & \text{if $b < 0$ and $b' < 0$}. \ \end{cases} \f] @@ -2871,13 +2879,17 @@ The interval bitwise operators are defined as follow: \f[ [a,b] \ogreaterthan ^{#} [a',b'] \approx \begin{cases} - \bigl[0, \bigl(b / 2^{b'}\bigr)\bigr], + \bigl[\bigl(a / 2^{a'}\bigr), + \bigl(b / 2^{b'}\bigr)\bigr], & \text{if $a \geq 0$ and $a' \geq 0$}; \ - \bigl[0, \bigl(b \times 2^{\left|b'\right|}\bigr)\bigr], + \bigl[\bigl(a \times 2^{\left|a'\right|}\bigr), + \bigl(b \times 2^{\left|b'\right|}\bigr)\bigr], & \text{if $a \geq 0$ and $b' < 0$}; \ - \bigl[\bigl(a / 2^{a'}\bigr), 0\bigr], + \bigl[\bigl(a / 2^{a'}\bigr), + \bigl(b / 2^{b'}\bigr)\bigr], & \text{if $b < 0$ and $a' \geq 0$}; \ - \bigl[\bigl(a \times 2^{\left|a'\right|}\bigr), 0\bigr], + \bigl[\bigl(a \times 2^{\left|a'\right|}\bigr), + \bigl(b \times 2^{\left|b'\right|}\bigr)\bigr], & \text{if $b < 0$ and $b' < 0$}. \ \end{cases}
@@ -3060,10 +3072,8 @@ both greater than or equal zero, then \f$k + \sum_{v \in V}k_{v}v\f$ is approximated by:
\f[ - 0 - \leq \left(k + \sum_{v \in \cV}k_{v}v \right) - \leq + \approx \left(i + \sum_{v \in \cV}i_{v}v \right) \amlf 2^{\left(i' + \sum_{v \in \cV}i'_{v}v \right)} @@ -3074,10 +3084,8 @@ have discordand signs, with the first greater than or equal zero, then \f$k + \sum_{v \in V}k_{v}v\f$ is approximated by:
\f[ - 0 - \leq \left(k + \sum_{v \in \cV}k_{v}v \right) - \leq + \approx \left(i + \sum_{v \in \cV}i_{v}v \right) \adivlf 2^{\left(\left|i'\right| + \sum_{v \in \cV}\left|i'_{v}\right|v \right)} @@ -3088,26 +3096,22 @@ have discordand signs, with the first less than zero, then \f$k + \sum_{v \in V}k_{v}v\f$ is approximated by:
\f[ + \left(k + \sum_{v \in \cV}k_{v}v \right) + \approx \left(i + \sum_{v \in \cV}i_{v}v \right) \amlf 2^{\left(i' + \sum_{v \in \cV}i'_{v}v \right)} - \leq - \left(k + \sum_{v \in \cV}k_{v}v \right) - \leq - 0 \f]
Finally, if \f$i + \sum_{v \in V}i_{v}v\f$ and \f$i' + \sum_{v \in V}i'_{v}v\f$ are both less than zero, then \f$k + \sum_{v \in V}k_{v}v\f$ is approximated by:
\f[ + \left(k + \sum_{v \in \cV}k_{v}v \right) + \approx \left(i + \sum_{v \in \cV}i_{v}v \right) \adivlf 2^{\left(\left|i'\right| + \sum_{v \in \cV}\left|i'_{v}\right|v \right)} - \leq - \left(k + \sum_{v \in \cV}k_{v}v \right) - \leq - 0 \f]
\subsubsection bitwise_right_shift Bitwise RIGHT SHIFT @@ -3125,10 +3129,8 @@ both greater than or equal zero then \f$k + \sum_{v \in V}k_{v}v\f$ is approximated by:
\f[ - 0 - \leq \left(k + \sum_{v \in \cV}k_{v}v \right) - \leq + \approx \left(i + \sum_{v \in \cV}i_{v}v \right) \adivlf 2^{\left(i' + \sum_{v \in \cV}i'_{v}v \right)} @@ -3139,10 +3141,8 @@ have discordand signs, with the first greater than or equal zero, then \f$k + \sum_{v \in V}k_{v}v\f$ is approximated by:
\f[ - 0 - \leq \left(k + \sum_{v \in \cV}k_{v}v \right) - \leq + \approx \left(i + \sum_{v \in \cV}i_{v}v \right) \amlf 2^{\left(\left|i'\right| + \sum_{v \in \cV}\left|i'_{v}\right|v \right)} @@ -3153,26 +3153,22 @@ have discordand signs, with the first less than zero, then \f$k + \sum_{v \in V}k_{v}v\f$ is approximated by:
\f[ + \left(k + \sum_{v \in \cV}k_{v}v \right) + \approx \left(i + \sum_{v \in \cV}i_{v}v \right) \adivlf 2^{\left(i' + \sum_{v \in \cV}i'_{v}v \right)} - \leq - \left(k + \sum_{v \in \cV}k_{v}v \right) - \leq - 0 \f]
Finally, if \f$i + \sum_{v \in V}i_{v}v\f$ and \f$i' + \sum_{v \in V}i'_{v}v\f$ are both less than zero, then \f$k + \sum_{v \in V}k_{v}v\f$ is approximated by:
\f[ + \left(k + \sum_{v \in \cV}k_{v}v \right) + \approx \left(i + \sum_{v \in \cV}i_{v}v \right) \amlf 2^{\left(\left|i'\right| + \sum_{v \in \cV}\left|i'_{v}\right|v \right)} - \leq - \left(k + \sum_{v \in \cV}k_{v}v \right) - \leq - 0 \f]
Where \f$\asifp, \adifp, \amifp,\f$ and \f$\adivifp\f$ are the corresponding