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Round-to-nearest: () is set to the nearest floating-point number to . When there is a tie, the floating-point number whose last stored digit is even (also, the last digit, in binary form, is equal to 0) is used.
Since 2 10 = 1024, the complete range of the positive normal floating-point numbers in this format is from 2 −1022 ≈ 2 × 10 −308 to approximately 2 1024 ≈ 2 × 10 308. The number of normal floating-point numbers in a system (B, P, L, U) where B is the base of the system, P is the precision of the significand (in base B),
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 ...
The IEEE 754 specification—followed by all modern floating-point hardware—requires that the result of an elementary arithmetic operation (addition, subtraction, multiplication, division, and square root since 1985, and FMA since 2008) be correctly rounded, which implies that in rounding to nearest, the rounded result is within 0.5 ulp of ...
In floating-point arithmetic, rounding aims to turn a given value x into a value y with a specified number of significant digits. In other words, y should be a multiple of a number m that depends on the magnitude of x. The number m is a power of the base (usually 2 or 10) of the floating-point representation.
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 ...
C99 adds several functions and types for fine-grained control of floating-point environment. [3] These functions can be used to control a variety of settings that affect floating-point computations, for example, the rounding mode, on what conditions exceptions occur, when numbers are flushed to zero, etc.
Trailing zeros shown where they are significant for the six-digit floating-point number. y = 3.14159 - 0.00000 y = input[i] - c t = 10000.0 + 3.14159 t = sum + y = 10003.14159 Normalization done, next round off to six digits.