# Eliminating Runtime Penalty

Up until now, we've mostly focused on detecting when incorrect results are produced and handling these occurrences either by throwing an exception or invoking some designated function. We've achieved our goal of detecting and handling arithmetically incorrect behavior - but at cost of checking many arithmetic operations at runtime. It is a fact that many C++ programmers will find this trade-off unacceptable. So the question arises as to how we might minimize or eliminate this runtime penalty.

The first step is to determine what parts of a program might invoke exceptions. The following program is similar to previous examples but uses a special exception policy: `loose_trap_policy`.

Now, any expression which might fail at runtime is flagged with a compile time error. There is no longer any need for `try/catch` blocks. Since this program does not compile, the library absolutely guarantees that no arithmetic expression will yield incorrect results. Furthermore, it is absolutely guaranteed that no exception will ever be thrown. This is our original goal.

Now all we need to do is make the program compile. There are a couple of ways to achieve this.

## Using safe_range and safe_literal

When trying to avoid arithmetic errors of the above type, programmers will select data types which are wide enough to hold values large enough to be certain that results won't overflow, but are not so large as to make the program needlessly inefficient. In the example below, we presume we know that the values we want to work with fall in the range [-24,82]. So we "know" the program will always result in a correct result. But since we trust no one, and since the program could change and the expressions be replaced with other ones, we'll still use the `loose_trap_policy` exception policy to verify at compile time that what we "know" to be true is in fact true.

- ``safe_signed_range`` defines a type which is limited to the indicated range. Out of range assignments will be detected at compile time if possible (as in this case) or at run time if necessary.

- A safe range could be defined with the same minimum and maximum value effectively restricting the type to holding one specific value. This is what `safe_signed_literal` does.

- Defining constants with `safe_signed_literal` enables the library to correctly anticipate the correct range of the results of arithmetic expressions at compile time.

- The usage of ``loose_trap_policy`` will mean that any assignment to z which could be outside its legal range will result in a compile time error.

- All safe integer operations are implemented as constant expressions. The usage of `constexpr` will guarantee that `z` will be available at compile time for any subsequent use.

- So if this program compiles, it's guaranteed to return a valid result.

The output uses a custom output manipulator, `safe_format`, for safe types to display the underlying type and its range as well as current value. This program produces the following run time output.

    example 83:
    x = <signed char>[10,10] = 10
    y = <signed char>[67,67] = 67
    x + y = <int>[77,77] = 77
    z = <signed char>[-24,82] = 77

Take note of the various variable types:

- `x` and `y` are safe types with fixed ranges which encompass one single value. They can hold only that value which they have been assigned at compile time.

- `The sum x + y can also be determined at compile time.`

- The type of z is defined so that It can hold only values in the closed range -24,82. We can assign the sum of x + y because it is in the range that `z` is guaranteed to hold. If the sum could not be be guaranteed to fall in the range of `z`, we would get a compile time error due to the fact we are using the `loose_trap_policy` exception policy.

All this information regarding the range and values of variables has been determined at compile time. There is no runtime overhead. The usage of safe types does not alter the calculations or results in anyway. So `safe_t` and `const_safe_t` could be redefined to `int` and `const int` respectively and the program would operate identically - although it might We could compile the program for another machine - as is common when building embedded systems and know (assuming the target machine architecture was the same as our native one) that no erroneous results would ever be produced.

## Using Automatic Type Promotion

The C++ standard describes how binary operations on different integer types are handled. Here is a simplified version of the rules:

- promote any operand smaller than `int` to an `int` or `unsigned int`.

- if the size of the signed operand is larger than the size of the signed operand, the type of the result will be signed. Otherwise, the type of the result will be unsigned.

- Convert the type each operand to the type of the result, expanding the size as necessary.

- Perform the operation the two resultant operands.

So the type of the result of some binary operation may be different than the types of either or both of the original operands.

If the values are large, the result can exceed the size that the resulting integer type can hold. This is what we call "overflow". The C/C++ standard characterizes this as undefined behavior and leaves to compiler implementors the decision as to how such a situation will be handled. Usually, this means just truncating the result to fit into the result type - which sometimes will make the result arithmetically incorrect. However, depending on the compiler and compile time switch settings, such cases may result in some sort of run time exception or silently producing some arbitrary result.

The complete signature for a safe integer type is:

    template <
        class T,                  // underlying integer type
        class P = native,         // type promotion policy class
        class E = default_exception_policy // error handling policy class
    >
    safe;

The promotion rules for arithmetic operations are implemented in the default `native` type promotion policy are consistent with those of standard C++

Up until now, we've focused on detecting when an arithmetic error occurs and invoking an exception or other kind of error handler.

But now we look at another option. Using the `automatic` type promotion policy, we can change the rules of C++ arithmetic for safe types to something like the following:

- for any C++ numeric type, we know from `std::numeric_limits` what the maximum and minimum values that a variable can be - this defines a closed interval.

- For any binary operation on these types, we can calculate the interval of the result at compile time.

- From this interval we can select a new type which can be guaranteed to hold the result and use this for the calculation. This is more or less equivalent to the following code:

      int x, y;
      int z = x + y               // could overflow
      // so replace with the following:
      int x, y;
      long z = (long)x + (long)y; // can never overflow

  One could do this by editing his code manually as above, but such a task would be tedious, error prone, non-portable and leave the resulting code hard to read and verify. Using the `automatic` type promotion policy will achieve the equivalent result without these problems.

When using the `automatic` type promotion policy, with a given a binary operation, we silently promote the types of the operands to a wider result type so the result cannot overflow. This is a fundamental departure from the C++ Standard behavior.

If the interval of the result cannot be guaranteed to fit in the largest type that the machine can handle (usually 64 bits these days), the largest available integer type with the correct result sign is used. So even with our "automatic" type promotion scheme, it's still possible to overflow. So while our `automatic` type promotion policy might eliminate exceptions in our example above, it wouldn't be guaranteed to eliminate them for all programs.

Using the `loose_trap_policy` exception policy will produce a compile time error anytime it's possible for an error to occur.

This small example illustrates how to use automatic type promotion to eliminate all runtime penalty.

- the `automatic` type promotion policy has rendered the result of the sum of two `integers` as a `safe<long`> type.

- our program compiles without error - even when using the `loose_trap_policy` exception policy. This is because since a `long` can always hold the result of the sum of two integers.

- We do not need to use the `try/catch` idiom to handle arithmetic errors - we will have no exceptions.

- We only needed to change two lines of code to achieve our goal of guaranteed program correctness with no runtime penalty.

The above program produces the following output:

    example 82:
    x = <int>[-2147483648,2147483647] = 2147483647
    y = <int>[-2147483648,2147483647] = 2
    x + y = <long>[-4294967296,4294967294] = 2147483649

Note that if any time in the future we were to change safe<int> to safe<long long> the program could now overflow. But since we're using `loose_trap_policy` the modified program would fail to compile. At this point we'd have to alter our yet program again to eliminate run time penalty or set aside our goal of zero run time overhead and change the exception policy to `default_exception_policy` .

Note that once we use automatic type promotion, our programming language isn't C/C++ anymore. So don't be tempted to so something like the following:

    // DON'T DO THIS !
    #if defined(NDEBUG)
    using safe_t = boost::numeric::safe<
        int,
        boost::numeric::automatic, // note use of "automatic" policy!!!
        boost::numeric::loose_trap_policy
    >;
    #else
    using safe_t = boost::numeric::safe<int>;
    #endif

## Mixing Approaches

For purposes of exposition, we've divided the discussion of how to eliminate runtime penalties by the different approaches available. A realistic program could likely include all techniques mentioned above. Consider the following:

- As before, we define a type `safe_t` to reflect our view of legal values for this program. This uses the `automatic` type promotion policy as well as the `loose_trap_policy` exception policy to enforce elimination of runtime penalties.

- The function `f` accepts only arguments of type `safe_t` so there is no need to check the input values. This performs the functionality of programming by contract with no runtime cost.

- In addition, we define `input_safe_t` to be used when reading variables from the program console. Clearly, these can only be checked at runtime so they use the throw_exception policy. When variables are read from the console they are checked for legal values. We need no ad hoc code to do this, as these types are guaranteed to contain legal values and will throw an exception when this guarantee is violated. In other words, we automatically get checking of input variables with no additional programming.

- On calling of the function `f`, arguments of type `input_safe_t` are converted to values of type `safe_t` . In this particular example, it can be determined at compile time that construction of an instance of a `safe_t` from an `input_safe_t` can never fail. Hence, no `try/catch` block is necessary. The usage of the `loose_trap_policy` policy for `safe_t` types guarantees this to be true at compile time.

Here is the output from the program when values 12 and 32 are input from the console:

    example 84:
    type in values in format x y:33 45
    x<signed char>[-24,82] = 33
    y<signed char>[-24,82] = 45
    z = <short>[-48,164] = 78
    (x + y) = <short>[-48,164] = 78
    (x - y) = <signed char>[-106,106] = -12
    <short>[-48,164] = 78