| ## Safe Numerics

## Introduction

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

- be demonstrably and verifiably correct.

- detect every user error such as input, assignment, etc.

- be efficient as possible subject to the constraints above.

### Problem

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 `int`, `unsigned int` 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.

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

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, [INT32-C] seems to recommend the following approach:

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

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.

    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;
    }

This example addresses only the problem of undefined/erroneous behavior related to overflow of the addition operation as applied to the type `int`. Similar problems occur with other built-in integer types such as `unsigned`, `long`, 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.

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.

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

### Solution

This library implements special versions of `int`, `unsigned`, etc. which behave exactly like the original ones except 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:

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

| Note
Library code in this document resides in the namespace `boost::numeric`. This namespace has generally been eliminated from text, code and examples in order to improve readability of the text.

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.

- the expression will yield the correct mathematical result

- the expression will emit a compilation error.

- the expression will invoke a runtime exception.

In other words, the library absolutely guarantees that no integer arithmetic expression will yield incorrect results.

### How It Works

The library implements special versions of `int`, `unsigned`, etc. named `safe<int>`, `safe<unsigned int>` etc. These behave exactly like the underlying types except 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 `signed` to `unsigned` 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.

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.

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.

### Additional Features

Operation of safe types is determined by template parameters which specify a pair of policy classes 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:

- Select or define an exception policy class to specify handling of exceptions.

  - Throw exception on runtime, trap at compile time if possible.

  - Trap at compile time all operations which could possibly fail at runtime.

  - Specify custom functions which should be called in case errors are detected at runtime.

- Select or define a promotion policy class to alter the C/C++ type promotion rules. This can be used to

  - 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.

  - 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.

  - 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.

- 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.

### Requirements

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

The following Boost Libraries must be installed in order to use this library

- mp11

- integer

- config

- tribool

- enable_if

The Safe Numerics library is delivered with an exhaustive suite of test programs.

### Scope

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.