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For big-integers of "50 to 1,000,000 digits", Burnikel-Ziegler Karatsuba division and Karatsuba multiplication are recommended by the algorithm's creator. [2] An example algorithm for 64-bit unsigned integers is below. The algorithm: Normalizes the input inside u64_isqrt. Calls u64_normalized_isqrt_rem, which requires a normalized input.
It is also used in the WAP environment, where it is called variable length unsigned integer or uintvar. The DWARF debugging format [ 4 ] defines a variant called LEB128 (or ULEB128 for unsigned numbers), where the least significant group of 7 bits is encoded in the first byte, and the most significant bits are in the last byte (so effectively ...
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.
In particular, multiplying or adding two integers may result in a value that is unexpectedly small, and subtracting from a small integer may cause a wrap to a large positive value (for example, 8-bit integer addition 255 + 2 results in 1, which is 257 mod 2 8, and similarly subtraction 0 − 1 results in 255, a two's complement representation ...
For unsigned integers, the bitwise complement of a number is the "mirror reflection" of the number across the half-way point of the unsigned integer's range. For example, for 8-bit unsigned integers, NOT x = 255 - x, which can be visualized on a graph as a downward line that effectively "flips" an increasing range from 0 to 255, to a decreasing ...
Booth's algorithm can be implemented by repeatedly adding (with ordinary unsigned binary addition) one of two predetermined values A and S to a product P, then performing a rightward arithmetic shift on P. Let m and r be the multiplicand and multiplier, respectively; and let x and y represent the number of bits in m and r.
For example, suppose we want to multiply two unsigned 8-bit integers together: a[7:0] and b[7:0]. We can produce eight partial products by performing eight 1-bit multiplications, one for each bit in multiplicand a :
Adding an integer to a sequence number is simple unsigned integer addition, followed by unsigned modulo operation to bring the result back into range (usually implicit in the unsigned addition, on most architectures): s' = (s + n) modulo 2 SERIAL_BITS. Addition of a value below 0 or above 2 SERIAL_BITS−1 − 1 is undefined. Basically, adding ...