arctanF

arctanf refers to the arctangent function applied to a real-valued input, typically in the context of single-precision floating-point arithmetic in programming languages. In many libraries the function is named atanf and takes a float, returning a float. It computes the principal value of the inverse tangent, with range -π/2 to π/2 (open at the ends) and domain all real numbers. As an odd function, arctan(-x) = -arctan(x). The derivative is d/dx arctan x = 1/(1+x^2); by the chain rule, d/dx arctan f(x) = f'(x)/(1+f(x)^2). The function approaches ±π/2 as x → ±∞.

Numerical evaluation relies on range reduction and polynomial or rational approximations to balance accuracy and performance. A common expansion around zero is the Maclaurin series arctan x = x - x^3/3 + x^5/5 - x^7/7 + ... which converges for |x| ≤ 1, with the endpoint values at x = ±1 equal to ±π/4. In practice, implementations combine several techniques to maintain precision across the real line and for inputs with limited precision.

Usage and notation: arctanf is used in graphics, physics, and geometry computations where single-precision speed is important. It is essential to maintain consistency about units: arctan returns values in radians by default in most languages, not degrees. When applying arctanf to a function, the chain rule applies as for arctan, using the function’s derivative. Different languages may expose the same mathematical operation under names such as atan, atanf, or arctan_f.