"Power." Wolfram Language & System Documentation Center. Wolfram Research (1988), Power, Wolfram Language function, (updated 2021). Ĭite this as: Wolfram Research (1988), Power, Wolfram Language function, (updated 2021). The special case CubeRoot corresponds to Surd. To obtain a real-valued n root, Surd can be used. Because of this branch cut, Power returns a complex root by default instead of the real one for negative real x and odd positive n. Power has a branch cut discontinuity for y running from to 0 in the complex x plane for noninteger y.Exponentiation using the base of the natural logarithm E can be input as Exp but is represented using Power. The function Sqrt is represented using Power.PowerExpand can be used to do formal expansion and associated simplification, and ExpToTrig can be used to get trigonometric forms of Power expressions. Many expressions involving Power, Exp, Log, and related functions are automatically simplified or else may be simplified using Simplify or FullSimplify. The rules for combining quantities containing powers are called the exponent laws, and the process of raising a base to a given power is known as exponentiation. The operation of taking an expression to the second power is known as “squaring ” and the operation of taking an expression to the third power is known as “cubing ”.The inverse of a power function is given by Log, so solving the equation for gives a principal solution of. A number to the first power is equal to itself ( ), and 1 to any complex power is equal to 1 ( ). The expression Power is commonly represented using the shorthand syntax x^ y or written in 2D typeset form as x y. ![]() Power is a mathematical function that raises an expression to a given power.The recursion has to bottom out so we provide some base size below which we use the numeric approximation method noted already. We use integer multiplication, squaring, and the integer Quotient function. Last step is to iteratively repair that estimate. Recursively compute the integer sqrt of b, multiply by 2^n, and use the usual Taylor approximation to get a correction that estimates sqrt(a). The idea is to split the number into an upper and lower part, using a power of 4 for the split size so that we can shift back by a factor of 2. My guess is it is similar to Zmmermann's, but he may well have had some extra efficiencies. floorSqrt :=įor variety, here is a top-level implementation of an integer-based method. For purposes of Floor extraction, it suffices to use as precision half the digit size. A direct approach, as already noted, would do the square root numerically. Sqrt is using exact methods in an effort to pull out "small" squares. Is there any way to speed up the calculation of integer square roots, preferably approaching GMP's raw performance? Instead I suspect that its Floor] function falls back on a generic algorithm that must calculate its argument to sufficient precision using arbitrary-precision floating-point approximation. The frustrating thing is that Mathematica already depends on the GMP library for its arbitrary-precision integers, but I cannot seem to make Mathematica use the GMP implementation of the integer square root. On the same machine, this is approximately 160 times faster: $ python3 benchmark.py Here's an example python program doing just that: from gmpy2 import mpz, isqrt ![]() However, the same machine can calculate the integer square root much faster using the GMP library. On my Core i7 linux machine, calculating the integer (floor) square root using the straightforward method takes 2.68 seconds: In:= n = 10^1000000 - 3^2095903 ![]() ![]() I'm looking for the highest-performance method of calculating integer square roots in Mathematica of very big arbitrary-precision numbers.Īs an example testcase, I use: n = 10^1000000 - 3 ^ 2095903
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