Why float has error

float may have errors due to limitations of binary representation, IEEE 754 standards, rounding errors, computer memory limitations, algorithm approximation and hardware limitations, etc. Detailed introduction: 1. Limitations of binary representation. Computers use binary internally to represent floating point numbers. However, many real numbers cannot be accurately represented by limited binary representation; 2. IEEE 754 standard. In order to solve the problem of floating point number representation, IEEE has formulated a A set of standards, namely IEEE 754, for the representation and operation of floating point numbers; 3. Rounding errors, etc.

The operating system for this tutorial: Windows 10 system, DELL G3 computer.

In computer science, it is a well-known fact that there are errors in floating-point number operations, and this error is mainly caused by the way floating-point numbers are represented internally by the computer. Regarding the question of "why float has errors", we can have an in-depth discussion from the following aspects:

1. Limitations of binary representation: Computers use binary internally to represent floating point numbers, but many real numbers cannot be represented by finite numbers. The binary representation represents exactly that. For example, our commonly used decimal fraction 1/10 has an infinite loop of decimal parts, but it can only be represented approximately in binary, resulting in inherent loss of accuracy.

2. IEEE 754 standard: In order to solve the problem of floating-point number representation, IEEE (Institute of Electrical and Electronics Engineers) developed a set of standards, namely IEEE 754, for the representation and operation of floating-point numbers. This standard defines the format, operation rules and precision of floating point numbers. However, even following the IEEE 754 standard, there are still errors in floating point operations due to the limitations of binary representation.

3. Rounding error: When the computer performs floating point calculations, the results will be rounded to fit in the limited storage space. The rounding process can cause errors to accumulate, especially in complex mathematical operations, where this error will become more obvious.

4. Computer memory limitations: Due to limited computer memory, the results of floating point operations need to be stored in memory. Since memory space is limited, the precision of floating point numbers is also limited. Even when following the IEEE 754 standard, some larger floating point numbers may still have errors due to memory limitations.

5. Approximation of the algorithm: In the process of implementing floating-point number operations, the algorithm itself may also introduce errors. For example, some numerically stable algorithms may approximate results to prevent numerical overflow or underflow during calculations. This approximation may lead to errors.

6. Hardware limitations: The hardware architecture of the computer also affects the accuracy of floating point numbers. For example, the number of bits of the processor, the number and size of registers, etc. will all affect the accuracy of floating point operations.

In order to reduce errors in floating point number operations, we can take the following methods:

1. Choose appropriate algorithms and data types: Choose appropriate algorithms and data types for specific problems. Can effectively reduce errors. For example, for computing tasks that require high precision, you can choose to use fixed-point arithmetic or other numerically stable algorithms.

2. Numerical stabilization: When performing floating-point number operations, you need to pay attention to the issue of numerical stabilization. For example, scaling techniques can be used to avoid numerical overflow or underflow problems.

3. Code optimization: Code optimization for specific hardware platforms can also improve the calculation accuracy of floating point numbers. For example, optimizing the implementation of algorithms, adjusting the storage method of data, etc.

4. Use an appropriate rounding strategy: Choosing an appropriate rounding strategy during floating-point operations can reduce the impact of rounding errors. For example, you can use different rounding methods such as rounding, rounding up, or rounding down to reduce errors.

5. Multiple iterations and stepwise approximation method: For some problems that require high-precision results, multiple iterations and stepwise approximation methods can be used to gradually approach the real results. This method can gradually reduce the error through multiple iterations until the required accuracy is achieved.

In short, although there are errors in floating-point operations, we can do so by choosing appropriate algorithms and data types, numerical stabilization, code optimization, using appropriate rounding strategies, and multiple iterations and stepwise approximation methods. To reduce errors and improve the accuracy of floating point operations.

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