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A power of two is a number of the form 2 n where n is an integer, that is, ... The range of a long variable in the Java and C# programming languages.
Yao's method collects in u first those x i that appear to the highest power ; in the next round those with power are collected in u as well etc. The variable y is multiplied h − 1 {\displaystyle h-1} times with the initial u , h − 2 {\displaystyle h-2} times with the next highest powers, and so on.
modulo power of two: undefined behavior C#: modulo power of 2 in unchecked context; System.OverflowException is raised in checked context [10] Java: modulo power of two (char is the only unsigned primitive type in Java) modulo power of two JavaScript: all numbers are double-precision floating-point except the new BigInt: MATLAB: Builtin ...
In The Art of Computer Programming, Vol. 2, Seminumerical Algorithms, page 463, Donald Knuth notes that contrary to some assertions, this method does not always give the minimum possible number of multiplications. The smallest counterexample is for a power of 15, when the binary method needs six multiplications.
The binary number system expresses any number as a sum of powers of 2, and denotes it as a sequence of 0 and 1, separated by a binary point, where 1 indicates a power of 2 that appears in the sum; the exponent is determined by the place of this 1: the nonnegative exponents are the rank of the 1 on the left of the point (starting from 0), and ...
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Squaring is the same as raising to the power 2, and is denoted by a superscript 2; for instance, the square of 3 may be written as 3 2, which is the number 9. In some cases when superscripts are not available, as for instance in programming languages or plain text files, the notations x ^2 ( caret ) or x **2 may be used in place of x 2 .
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