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A floating-point number is a rational number, because it can be represented as one integer divided by another; for example 1.45 × 10 3 is (145/100)×1000 or 145,000 /100. The base determines the fractions that can be represented; for instance, 1/5 cannot be represented exactly as a floating-point number using a binary base, but 1/5 can be ...
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 ...
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.
A bit is a binary digit that represents one of two states. The concept of a bit can be understood as a value of either 1 or 0, on or off, yes or no, true or false, or encoded by a switch or toggle of some kind. While a single bit, on its own, is able to represent only two values, a string of bits may be used to represent larger values. For ...
The number 0.15625 represented as a single-precision IEEE 754-1985 floating-point number. See text for explanation. The three fields in a 64bit IEEE 754 float. Floating-point numbers in IEEE 754 format consist of three fields: a sign bit, a biased exponent, and a fraction. The following example illustrates the meaning of each.
Subnormal numbers ensure that for finite floating-point numbers x and y, x − y = 0 if and only if x = y, as expected, but which did not hold under earlier floating-point representations. [ 43 ] On the design rationale of the x87 80-bit format , Kahan notes: "This Extended format is designed to be used, with negligible loss of speed, for all ...
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).