The IEEE 754 standard is the globally recognized standard for representing floating-point numbers in computers. It governs both the formatting and precision of floating-point numbers and is the basis for most modern processors, both CPUs and GPUs.
The most important points of the IEEE 754 standard:
1. Floating point formats
2. Format accuracy
3. Rounding errors and accuracy
4. Special values
5. Sources of error in practice
6. Comparison with decimal numbers
7. Use on GPUs
8. Conclusion on IEEE 754 accuracy:
1.) Floating point formats
The standard defines two primary floating-point formats:
- Single Precision (32-bit) : This is the most common floating-point format in many applications, including graphics and scientific computing. It uses 32 bits:
- 1 bit for the sign (positive or negative)
- 8 bits for the exponent
- 23 bits for the mantissa (also called the significand)
- Double Precision (64-bit) : A higher-precision format used in many scientific and financial applications. It uses 64 bits:
- 1 bit for the sign
- 11 bits for the exponent
- 52 bits for the mantissa
2.) Accuracy of formats
Precision refers to how many significant decimal or binary digits can be represented by the floating-point format. - Single Precision (32-bit) : - The exponent has a range of -126 to +127 (in base 2), which means that single-precision floating-point numbers can be represented with an accuracy of about 7 decimal places . - The mantissa consists of 23 bits, which allows a precision of about 7 decimal places in the representation of the number. - Double Precision (64-bit) : - The exponent has a range of -1022 to +1023, which allows the representation of numbers in the range 10-308 to 10308. - The mantissa consists of 52 bits, which allows a more precise representation of about 15 decimal places .
3.) Rounding errors and accuracy
Because the IEEE 754 standard only represents floating-point numbers with a limited number of bits, rounding errors occur when numbers that don't fit the format exactly are represented. This leads to errors in the last decimal places of a number. This is particularly noticeable when calculating with very small or very large numbers.
For example:
- The number '1/3' can be approximately represented in double precision format, but the value is rounded to the nearest available bits by the standard. This leads to a small rounding error that can become greater during subsequent calculations.
4.) Special values
The IEEE 754 standard also defines some special values:
- NaN (Not a Number) : Used to represent invalid or undefined mathematical operations, such as `0/0` or `sqrt(-1)`.
- Infinity : Used to represent overflow values, such as division by zero.
- Subnormal numbers (denormals) : Numbers smaller than the smallest normalized value. These are represented using "subnormalization" to provide greater precision for very small values, albeit with slightly reduced accuracy.
5.) Sources of error in practice
- Adding and subtracting very different numbers can lead to rounding errors , as the smaller number may be "lost."
- Accumulation of errors : In long calculations or in algorithms that require many iterations (such as Monte Carlo simulations), small rounding errors caused by the many operations can lead to significant errors.
6.) Comparison with decimals
Floating-point numbers in IEEE 754 format use the binary system to represent numbers. This sometimes leads to inaccuracies in the representation of decimal numbers. For example:
- The number '0.1' cannot be represented exactly in binary, resulting in it appearing in floating-point representations like '0.10000000149011612'.
7.) Use on GPUs
- In GPU programming (e.g., with OpenGL or CUDA), the IEEE 754 standard is also widely used. However, on GPUs, computing performance is optimized for parallelism, which means that in some cases, precision can be slightly reduced to maximize computing power. This sometimes leads to small deviations in calculations compared to CPU-based calculations.
8.) Conclusion on IEEE 754 accuracy:
The precision of 32-bit (single-precision) floating-point numbers is sufficient for many general-purpose applications, but only offers approximately 7 decimal places of precision. When higher precision is required, such as in scientific computing or in areas requiring exact calculations, the 64-bit double-precision format is preferred, as it offers approximately 15 decimal places of precision .
Although the IEEE 754 standard has widespread application, developers must always consider the limitations of floating-point numbers and be aware of potential rounding errors , especially for very small or very large numbers.
| (Image-1) Tell me something about IEEE 754 floating point precision and decimal place!? |
 |
