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The "decimal" data type of the C# and Python programming languages, and the decimal formats of the IEEE 754-2008 standard, are designed to avoid the problems of binary floating-point representations when applied to human-entered exact decimal values, and make the arithmetic always behave as expected when numbers are printed in decimal.
Huberto M. Sierra noted in his 1956 patent "Floating Decimal Point Arithmetic Control Means for Calculator": [1] Thus under some conditions, the major portion of the significant data digits may lie beyond the capacity of the registers.
This alternative definition is significantly more widespread: machine epsilon is the difference between 1 and the next larger floating point number.This definition is used in language constants in Ada, C, C++, Fortran, MATLAB, Mathematica, Octave, Pascal, Python and Rust etc., and defined in textbooks like «Numerical Recipes» by Press et al.
The standard defines five basic formats that are named for their numeric base and the number of bits used in their interchange encoding. There are three binary floating-point basic formats (encoded with 32, 64 or 128 bits) and two decimal floating-point basic formats (encoded with 64 or 128 bits).
Computers typically use binary arithmetic, but to make the example easier to read, it will be given in decimal. Suppose we are using six-digit decimal floating-point arithmetic, sum has attained the value 10000.0, and the next two values of input[i] are 3.14159 and 2.71828. The exact result is 10005.85987, which rounds to 10005.9.
A floating-point variable can represent a wider range of numbers than a fixed-point variable of the same bit width at the cost of precision. A signed 32-bit integer variable has a maximum value of 2 31 − 1 = 2,147,483,647, whereas an IEEE 754 32-bit base-2 floating-point variable has a maximum value of (2 − 2 −23) × 2 127 ≈ 3.4028235 ...
To approximate the greater range and precision of real numbers, we have to abandon signed integers and fixed-point numbers and go to a "floating-point" format. In the decimal system, we are familiar with floating-point numbers of the form (scientific notation): 1.1030402 × 10 5 = 1.1030402 × 100000 = 110304.02. or, more compactly: 1.1030402E5
Although the radix conversion from decimal floating-point to binary floating-point only incurs a small relative error, catastrophic cancellation may amplify it into a much larger one: double x = 1.000000000000001 ; // rounded to 1 + 5*2^{-52} double y = 1.000000000000002 ; // rounded to 1 + 9*2^{-52} double z = y - x ; // difference is exactly ...