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Augmented version above showing both signs of representable values. Over the years, a variety of floating-point representations have been used in computers. In 1985, the IEEE 754 Standard for Floating-Point Arithmetic was established, and since the 1990s, the most commonly encountered representations are those defined by the IEEE.
Like the binary floating-point formats, the number is divided into a sign, an exponent, and a significand. Unlike binary floating-point, numbers are not necessarily normalized; values with few significant digits have multiple possible representations: 1×10 2 =0.1×10 3 =0.01×10 4, etc. When the significand is zero, the exponent can be any ...
Because the significand is not normalized, most values with less than 34 significant digits have multiple possible representations; 1 × 10 2 = 0.1 × 10 3 = 0.01 × 10 4, etc. This set of representations for a same value is called a cohort. Zero has 12288 possible representations (24576 if both signed zeros are included, in two different cohorts).
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 ...
decimal32 supports 'normal' values, which can have 7 digit precision from ±1.000 000 × 10 ^ −95 up to ±9.999 999 × 10 ^ +96, plus 'subnormal' values with ramp-down relative precision down to ±1. × 10 ^ −101 (one digit), signed zeros, signed infinities and NaN (Not a Number). The encoding is somewhat complex, see below.
In base ten, a sixteen-bit integer is certainly adequate as it allows up to 32767. However, this example cheats, in that the value of n is not itself limited to a single digit. This has the consequence that the method will fail for n > 3200 or so. In a more general implementation, n would also use a multi-digit
In a normal floating-point value, there are no leading zeros in the significand (also commonly called mantissa); rather, leading zeros are removed by adjusting the exponent (for example, the number 0.0123 would be written as 1.23 × 10 −2). Conversely, a denormalized floating-point value has a significand with a leading digit of zero.
Here the 'IEEE 754 double value' resulting of the 15 bit figure is 3.330560653658221E-15, which is rounded by Excel for the 'user interface' to 15 digits 3.33056065365822E-15, and then displayed with 30 decimals digits gets one 'fake zero' added, thus the 'binary' and 'decimal' values in the sample are identical only in display, the values ...