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Floating-point arithmetic

In computing, most numerical calculations are done with floating-point arithmetic, which since the 1980s has been standardized by the IEEE 754 specification. In IEEE floating-point arithmetic, numbers are represented using a sign (positive or negative), a fixed-precision significand and an integer exponent. Numbers whose exponent is too large to represent instead "overflow" to positive or negative infinity (+∞ or −∞), while numbers whose exponent is too small to represent instead " underflow " to positive or negative zero (+0 or −0). A NaN (not a number) value represents undefined results.

In IEEE arithmetic, division of 0/0 or ∞/∞ results in NaN, but otherwise division always produces a well-defined result. Dividing any non-zero number by positive zero (+0) results in an infinity of the same sign as the dividend. Dividing any non-zero number by negative zero (−0) results in an infinity of the opposite sign as the dividend. This definition preserves the sign of the result in case of arithmetic underflow.

For example, using single-precision IEEE arithmetic, if x = −2, then x /2 underflows to −0, and dividing 1 by this result produces 1/(x /2) = −∞. The exact result −2 is too large to represent as a single-precision number, so an infinity of the same sign is used instead to indicate overflow.