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If the number of bits used is fixed, the range of integers that can be represented would be fixed and can potentially overflow. In python, integers have arbitrary precision and therefore we can represent an arbitrarily large range of integers only limited by memory available. In python 2, there are actually two integers types: int and long , where int is the C-style fixed-precision integer and long is the arbitrary-precision integer.
Can Integer Operations Overflow in Python?
Operations are automatically promoted to long if int is not sufficient, so there's no risk of overflowing. In python 3, int is the only integer type and it is arbitrary-precision. To see a bit under the hood, let's examine how much the storage size changes for different integers in python. Certainly not the most compact representation, as a raw bit array i. However we get the benefits of arbitrary precision and many others in python.
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Though the vast majority of computers can perform multiple-precision arithmetic on operands in memory, allowing numbers to be arbitrarily long and overflow to be avoided, the register width limits the sizes of numbers that can be operated on e. Typical binary register widths for unsigned integers include:.
When an arithmetic operation produces a result larger than the maximum above for an N-bit integer, an overflow reduces the result to modulo N-th power of 2, retaining only the least significant bits of the result and effectively causing a wrap around. Such wraparound may cause security problems—if an overflowed value is used as the number of bytes to allocate for a buffer, the buffer will be allocated unexpectedly small, potentially leading to a buffer overflow which, depending on the usage of the buffer, might in turn cause arbitrary code execution.
If the variable has a signed integer type, a program may make the assumption that a variable always contains a positive value. A solution for this particular problem is to use unsigned integer types for values that a program expects and assumes will never be negative.
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The carry flag is set when the result of an addition or subtraction, considering the operands and result as unsigned numbers, does not fit in the given number of bits. This indicates an overflow with a carry or borrow from the most significant bit. An immediately following add with carry or subtract with borrow operation would use the contents of this flag to modify a register or a memory location that contains the higher part of a multi-word value.
The overflow flag is set when the result of an operation on signed numbers does not have the sign that one would predict from the signs of the operands, e. This indicates that an overflow has occurred and the signed result represented in two's complement form would not fit in the given number of bits. For an unsigned type, when the ideal result of an operation is outside the types representable range and the returned result is obtained by wrapping, then this event is commonly defined as an overflow.
In contrast, the C11 standard defines that this event is not an overflow and states "a computation involving unsigned operands can never overflow. When the ideal result of an integer operation is outside the types representable range and the returned result is obtained by clamping, then this event is commonly defined as a saturation. Usage varies as to whether a saturation is or is not an overflow. To eliminate ambiguity, the terms wrapping overflow  and saturating overflow  can be used.
The term underflow is most commonly used for floating-point math and not for integer math .
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But, many references can be found to integer underflow. When the term integer underflow is used, the definition of overflow may include all types of overflows or it may only include cases where the ideal result was closer to positive infinity than the output type's representable value closest to positive infinity. When the ideal result of an operation is not an exact integer, the meaning of overflow can be ambiguous in edge cases.
Consider the case where the ideal result has value If overflow is defined as the ideal value being outside the representable range of the output type, then this case would be classified as an overflow. For operations that have well defined rounding behavior, overflow classification may need to be postponed until after rounding is applied. The C11 standard  defines that conversions from floating point to integer must round toward zero.
If C is used to convert the floating point value Since the rounded integer is in the outputs range, the C standard would not classify this conversion as an overflow.
From Overwhelmed to Overflowing – Part 2
Programming languages implement various mitigation methods against an accidental overflow: Ada , Seed7 and certain variants of functional languages , trigger an exception condition on overflow, while Python since 2. Run-time overflow detection implementation UBSan is also available for C compilers.
In languages with native support for Arbitrary-precision arithmetic and type safety such as Python or Common Lisp , numbers are promoted to a larger size automatically when overflows occur, or exceptions thrown conditions signaled when a range constraint exists. Using such languages may thus be helpful to mitigate this issue.
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However, in some such languages, situations are still possible where an integer overflow can occur. An example is explicit optimization of a code path which is considered a bottleneck by the profiler.
In the case of Common Lisp, this is possible by using an explicit declaration to type-annotate a variable to a machine-size word fixnum  and lower the type safety level to zero  for a particular code block. In Java 8, there are overloaded methods, for example like Math. An example of this is a grayscale image where 0 represents black, 1 represents white, and values in-between represent varying shades of gray.
One operation that one may want to support is brightening the image by multiplying every pixel by a constant. Saturated arithmetic allows one to just blindly multiply every pixel by that constant without worrying about overflow by just sticking to a reasonable outcome that all these pixels larger than 1 i. Unanticipated arithmetic overflow is a fairly common cause of program errors.
Such overflow bugs may be hard to discover and diagnose because they may manifest themselves only for very large input data sets, which are less likely to be used in validation tests. Taking the arithmetic mean of two numbers by adding them and dividing by two, as done in many search algorithms , causes error if the sum although not the resulting mean is too large to be represented and hence overflows.
An unhandled arithmetic overflow in the engine steering software was the primary cause of the crash of the maiden flight of the Ariane 5 rocket. Frustratingly, the part of the software in which the overflow error occurred was not even required to be running for the Ariane 5 at the time that it caused the rocket to fail—it was a launch-regime process for a smaller predecessor of the Ariane 5 that had remained in the software when it was adapted for the new rocket.