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The elementary functions are constructed by composing arithmetic operations, the exponential function (), the natural logarithm (), trigonometric functions (,), and their inverses. The complexity of an elementary function is equivalent to that of its inverse, since all elementary functions are analytic and hence invertible by means of Newton's ...
The "decimal" data type of the C# and Python programming languages, and the decimal formats of the IEEE 754-2008 standard, are designed to avoid the problems of binary floating-point representations when applied to human-entered exact decimal values, and make the arithmetic always behave as expected when numbers are printed in decimal.
C# has a built-in data type decimal consisting of 128 bits resulting in 28–29 significant digits. It has an approximate range of ±1.0 × 10 −28 to ±7.9228 × 10 28. [1] Starting with Python 2.4, Python's standard library includes a Decimal class in the module decimal. [2] Ruby's standard library includes a BigDecimal class in the module ...
For multiplication, the most straightforward algorithms used for multiplying numbers by hand (as taught in primary school) require (N 2) operations, but multiplication algorithms that achieve O(N log(N) log(log(N))) complexity have been devised, such as the Schönhage–Strassen algorithm, based on fast Fourier transforms, and there are also ...
This assumption implies that the product of two representatives mod N is less than RN, the exact hypothesis necessary for REDC to generate correct output. In particular, the product of aR mod N and bR mod N is REDC((aR mod N)(bR mod N)). The combined operation of multiplication and REDC is often called Montgomery multiplication.
Scalar multiplication of a vector by a factor of 3 stretches the vector out. The scalar multiplications −a and 2a of a vector a. In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra [1] [2] [3] (or more generally, a module in abstract algebra [4] [5]).
mXparser is an open-source mathematical expressions parser/evaluator providing abilities to calculate various expressions at a run time. [1] Expressions definitions are given as plain text, then verified in terms of grammar / syntax, finally calculated.
where f is the function for multiplying, P is the coordinate to multiply, d is the number of times to add the coordinate to itself. Example: 100P can be written as 2(2[P + 2(2[2(P + 2P)])]) and thus requires six point double operations and two point addition operations. 100P would be equal to f(P, 100).