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Initially, neural network based evaluation functions generally consisted of one neural network for the entire evaluation function, with input features selected from the board and whose output is an integer, normalized to the centipawn scale so that a value of 100 is roughly equivalent to a material advantage of a pawn.
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This consists in using the last computed approximate values of the root for approximating the function by a polynomial of low degree, which takes the same values at these approximate roots. Then the root of the polynomial is computed and used as a new approximate value of the root of the function, and the process is iterated.
Horner's method evaluates a polynomial using repeated bracketing: + + + + + = + (+ (+ (+ + (+)))). This method reduces the number of multiplications and additions to just Horner's method is so common that a computer instruction "multiply–accumulate operation" has been added to many computer processors, which allow doing the addition and multiplication operations in one combined step.
Regardless if ΔP is infinitesimal or finite, there is (at least—in the case of the derivative—theoretically) a point range, where the boundaries are P ± (0.5) ΔP (depending on the orientation—ΔF(P), δF(P) or ∇F(P)):
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The solution with the function value can be found after 325 function evaluations. Using the Nelder–Mead method from starting point x 0 = ( − 1 , 1 ) {\displaystyle x_{0}=(-1,1)} with a regular initial simplex a minimum is found with function value 1.36 ⋅ 10 − 10 {\displaystyle 1.36\cdot 10^{-10}} after 185 function evaluations.
A specific element x of X is a value of the variable, and the corresponding element of Y is the value of the function at x, or the image of x under the function. The image of a function, sometimes called its range, is the set of the images of all elements in the domain. [6] [7] [8] [9]