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integer floating-point operations ceil: returns the nearest integer not less than the given value floor: returns the nearest integer not greater than the given value trunc: returns the nearest integer not greater in magnitude than the given value round lround llround: returns the nearest integer, rounding away from zero in halfway cases nearbyint
Names and symbols used for integer division include div, /, \, and %. Definitions vary regarding integer division when the dividend or the divisor is negative: rounding may be toward zero (so called T-division) or toward −∞ (F-division); rarer styles can occur – see modulo operation for the details.
The floor of x is also called the integral part, integer part, greatest integer, or entier of x, and was historically denoted [x] (among other notations). [2] However, the same term, integer part, is also used for truncation towards zero, which differs from the floor function for negative numbers. For n an integer, ⌊n⌋ = ⌈n⌉ = n.
For to be an integer, we need to round / somehow. Rounding to the nearest integer will give the best approximation but can result in m / 2 k {\displaystyle m/2^{k}} being larger than 1 / n {\displaystyle 1/n} , which can cause underflows.
One method, more obscure than most, is to alternate direction when rounding a number with 0.5 fractional part. All others are rounded to the closest integer. Whenever the fractional part is 0.5, alternate rounding up or down: for the first occurrence of a 0.5 fractional part, round up, for the second occurrence, round down, and so on.
Long division is the standard algorithm used for pen-and-paper division of multi-digit numbers expressed in decimal notation. It shifts gradually from the left to the right end of the dividend, subtracting the largest possible multiple of the divisor (at the digit level) at each stage; the multiples then become the digits of the quotient, and the final difference is then the remainder.
This is different from the way rounding is usually done in signed integer division (which rounds towards 0). This discrepancy has led to bugs in a number of compilers. [8] For example, in the x86 instruction set, the SAR instruction (arithmetic right shift) divides a signed number by a power of two, rounding towards negative infinity. [9]
Rounding up and down to a multiple of a known power of 2, the next power of 2 and for detecting whether an operation crossed a power-of-2 boundary; Checking bounds; Counting total, leading and trailing zeros; Searching for bit strings; Permutations of bits and bytes in a word; Software algorithms for multiplication; Integer division