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<section id="safe_numerics.introduction">
  <title>Introduction</title>

  <?dbhtml stop-chunking?>

  <para>This library is intended as a drop-in replacement for all built-in
  integer types in any program which must:</para>

  <itemizedlist>
    <listitem>
      <para>be demonstrably and verifiably correct.</para>
    </listitem>

    <listitem>
      <para>detect every user error such as input, assignment, etc.</para>
    </listitem>

    <listitem>
      <para>be efficient as possible subject to the constraints above.</para>
    </listitem>
  </itemizedlist>

  <section id="safe_numerics.introduction.problem">
    <title>Problem</title>

    <para>Arithmetic operations in C/C++ are NOT guaranteed to yield a correct
    mathematical result. This feature is inherited from the early days of C.
    The behavior of <code>int</code>, <code>unsigned int</code> and others
    were designed to map closely to the underlying hardware. Computer hardware
    implements these types as a fixed number of bits. When the result of
    arithmetic operations exceeds this number of bits, the result will not be
    arithmetically correct. The following example illustrates just one example
    where this causes problems.</para>

    <programlisting>int f(int x, int y){
    // this returns an invalid result for some legal values of x and y !
    return x + y;
}
</programlisting>

    <para>It is incumbent upon the C/C++ programmer to guarantee that this
    behavior does not result in incorrect or unexpected operation of the
    program. There are no language facilities which implement such a
    guarantee. A programmer needs to examine each expression individually to
    know that his program will not return an invalid result. There are a
    number of ways to do this. In the above instance,
    <citation>INT32-C</citation> seems to recommend the following
    approach:</para>

    <programlisting>int f(int x, int y){
  if (((y &gt; 0) &amp;&amp; (x &gt; (INT_MAX - y))) 
  || ((y &lt; 0) &amp;&amp; (x &lt; (INT_MIN - y)))) {
    /* Handle error */
  }
  return x + y;
}
</programlisting>

    <para>This will indeed trap the error. However, it would be tedious and
    laborious for a programmer to alter his code in this manner. Altering code
    in this way for all arithmetic operations would likely render the code
    unreadable and add another source of potential programming errors. This
    approach is clearly not functional when the expression is even a little
    more complex as is shown in the following example.</para>

    <programlisting>int f(int x, int y, int z){
    // this returns an invalid result for some legal values of x and y !
    return x + y * z;
}
</programlisting>

    <para>This example addresses only the problem of undefined/erroneous
    behavior related to overflow of the addition operation as applied to the
    type <code>int</code>. Similar problems occur with other built-in integer
    types such as <code>unsigned</code>, <code>long</code>, etc. And it also
    applies to other operations such as subtraction, multiplication etc. .
    C/C++ often automatically and silently converts some integer types to
    others in the course of implementing binary operations. Sometimes such
    conversions can silently change arithmetic values which inject errors. The
    C/C++ standards designate some behavior such as right shifting a negative
    number as "implementation defined behavior". These days machines usually
    do what the programmer expects - but such behavior is not guaranteed.
    Relying on such behavior will create a program which cannot be guaranteed
    to be portable. And then there is undefined behavior. In this case,
    compiler writer is under no obligation to do anything in particular.
    Sometimes this will unexpectedly break the program. At the very least, the
    program is rendered non-portable. Finally there is the case of behavior
    that is arithmetically wrong to begin with - for example divide by zero.
    Some runtime environments will just terminate the program, others may
    throw some sort of exception. In any case, the execution has failed in a
    manner from which there is no recovery.</para>

    <para>All of the above conditions are obstacles to creation of a program
    which will never fail. The Safe Numerics Library addresses all of these
    conditions, at least as far as integer operations are concerned.</para>

    <para>Since the problems and their solution are similar, we'll confine the
    current discussion to just the one example shown above.</para>
  </section>

  <section id="safe_numerics.introduction.solution">
    <title>Solution</title>

    <para>This library implements special versions of <code>int</code>,
    <code>unsigned</code>, etc. which behave exactly like the original ones
    <emphasis role="bold">except</emphasis> that the results of these
    operations are guaranteed to be either to be arithmetically correct or
    invoke an error. Using this library, the above example would be rendered
    as:</para>

    <programlisting>#include &lt;boost/safe_numerics/safe_integer.hpp&gt;
using namespace boost::numeric;
safe&lt;int&gt; f(safe&lt;int&gt; x, safe&lt;int&gt; y){
  return x + y; // throw exception if correct result cannot be returned
}
</programlisting>

    <para><note>
        <para>Library code in this document resides in the namespace
        <code>boost::numeric</code>. This namespace has generally been
        eliminated from text, code and examples in order to improve
        readability of the text.</para>
      </note>The addition expression is checked at runtime or (if possible) at
    compile time to trap any possible errors resulting in incorrect arithmetic
    behavior. Arithmetic expressions will not produce an erroneous result.
    Instead, one and only one of the following is guaranteed to occur.</para>

    <para><itemizedlist>
        <listitem>
          <para>the expression will yield the correct mathematical
          result</para>
        </listitem>

        <listitem>
          <para>the expression will emit a compilation error.</para>
        </listitem>

        <listitem>
          <para>the expression will invoke a runtime exception.</para>
        </listitem>
      </itemizedlist></para>

    <para>In other words, the <emphasis role="bold">library absolutely
    guarantees that no integer arithmetic expression will yield incorrect
    results</emphasis>.</para>
  </section>

  <section id="safe_numerics.introduction.implementation">
    <title>How It Works</title>

    <para>The library implements special versions of <code>int</code>,
    <code>unsigned</code>, etc. named <code>safe&lt;int&gt;</code>,
    <code>safe&lt;unsigned int&gt;</code> etc. These behave exactly like the
    underlying types <emphasis role="bold">except</emphasis> that expressions
    using these types fulfill the above guarantee. These "safe" types are
    meant to be "drop-in" replacements for the built-in types of the same
    name. So things which are legal - such as assignment of a
    <code>signed</code> to <code>unsigned</code> value - are not trapped at
    compile time as they are legal C/C++ code. Instead, they are checked at
    runtime to trap the case where this (legal) operation would lead to an
    arithmetically incorrect result.</para>

    <para>Note that the library addresses arithmetical errors generated by
    straightforward C/C++ expressions. Some of these arithmetic errors are
    defined as conforming to the C/C++ standards while others are not. So
    characterizing this library as only addressing undefined behavior of C/C++
    numeric expressions would be misleading.</para>

    <para>Facilities particular to C++14 are employed to minimize any runtime
    overhead. In many cases there is no runtime overhead at all. In other
    cases, a program using the library can be slightly altered to achieve the
    above guarantee without any runtime overhead.</para>
  </section>

  <section id="safe_numerics.introduction.additional_features">
    <title>Additional Features</title>

    <para>Operation of safe types is determined by template parameters which
    specify a pair of <link linkend="safe_numerics.promotion_policies">policy
    classes</link> which specify the behavior for type promotion and error
    handling. In addition to the usage serving as a drop-in replacement for
    standard integer types, users of the library can:</para>

    <para><itemizedlist>
        <listitem>
          <para>Select or define an exception policy class to specify handling
          of exceptions.<itemizedlist>
              <listitem>
                <para>Throw exception on runtime, trap at compile time if
                possible.</para>
              </listitem>

              <listitem>
                <para>Trap at compile time all operations which could possibly
                fail at runtime.</para>
              </listitem>

              <listitem>
                <para>Specify custom functions which should be called in case
                errors are detected at runtime.</para>
              </listitem>
            </itemizedlist></para>
        </listitem>

        <listitem>
          <para>Select or define a promotion policy class to alter the C/C++
          type promotion rules. This can be used to <itemizedlist>
              <listitem>
                <para>Use C/C++ native type promotion rules so that, except
                for throwing/trapping of exceptions on operations resulting in
                incorrect arithmetic behavior, programs will operate
                identically when using/not using safe types. This might be
                used if safe types are only enabled during debug and
                testing.</para>
              </listitem>

              <listitem>
                <para>Replace C/C++ native promotion rules with ones which are
                arithmetically equivalent but minimize the need for runtime
                checking of arithmetic results. Such a policy will effectively
                change the semantics of a C++ program. It's not really C++ any
                more. The program cannot be expected to function the same as
                when normal integer types are used.</para>
              </listitem>

              <listitem>
                <para>Replace C/C++ native promotion rules with ones which
                emulate other machine architectures. This is designed to
                permit the testing of C/C++ code destined to be run on another
                machine on one's development platform. Such a situation often
                occurs while developing code for embedded systems.</para>
              </listitem>
            </itemizedlist></para>
        </listitem>

        <listitem>
          <para>Enforce other program requirements using bounded integer
          types. The library includes the types for ranges and literals.
          Operations which violate these requirements will be trapped at
          either compile time or runtime and not silently return invalid
          values. These types can be used to improve program correctness and
          performance.</para>
        </listitem>
      </itemizedlist></para>
  </section>

  <section id="safe_numerics.introduction.requirements">
    <title>Requirements</title>

    <para>This library is composed entirely of C++ Headers. It requires a
    compiler compatible with the C++14 standard.</para>

    <para>The following Boost Libraries must be installed in order to use this
    library</para>

    <para><itemizedlist>
        <listitem>
          <para>mp11</para>
        </listitem>

        <listitem>
          <para>integer</para>
        </listitem>

        <listitem>
          <para>config</para>
        </listitem>

        <listitem>
          <para>tribool</para>
        </listitem>

        <listitem>
          <para>enable_if</para>
        </listitem>
      </itemizedlist>The Safe Numerics library is delivered with an exhaustive
    suite of test programs.</para>
  </section>

  <section id="safe_numerics.introduction.scope">
    <title>Scope</title>

    <para>This library currently applies only to built-in integer types.
    Analogous issues arise for floating point types but they are not currently
    addressed by this version of the library. User or library defined types
    such as arbitrary precision integers can also have this problem. Extension
    of this library to these other types is not currently under development
    but may be addressed in the future. This is one reason why the library
    name is "safe numeric" rather than "safe integer" library.</para>
  </section>
</section>
