expanded versions VFx_ratio, VDx_ratio, VEx_ratio:
xi = (A*X_i + B),
Y_i = P/Q as above with xi instead of X_i

A polynomial P of degree degP is divided by a polynomial Q of degree degQ. The coefficients p_k and q_k are taken from the vector Coeff. Coeff must contain the constant term p₀ as element no. 0, the linear coefficient p₁ as element no. 1, and so one until p_degP as element no. degP immediately followed by q₀ as element no. (degP+1), q₁ as element no. (degP+2) etc., until q_degQ as element no. (degP+degQ+1). This means that Coeff contains a total of (degP+degQ+2) elements.

"unprotected" versions (prefix VFu_,   VFux_, etc.):
These functions do not perform any error handling, which makes them much faster (up to 50%) than the standard versions. They do, however, indicate in their return value, if an error occurred.

Even in the case that the individual polynomials P and Q might overflow, their ratio still might yield a perfectly legal result. Therefore, OptiVec offers additional versions which internally employ higher accuracy (and range): VF_ratio_d and VF_ratio_e work in double or extended precision, resp., before converting the result back to single precision. Likewise, VD_ratio_e works in extended precision (even if the compiler does not support extended precision!), before converting the result back to double. These versions exist only on the CPU, not in CUDA.

When rational models are used for the approximation of mathematical functions, sometimes one of the polynomials only consists of odd terms, while the other polynomial has only even terms. Examples are the rational representation of the tangent function (P has only odd terms, Q has only even terms) or of the cotangent function (P has only even terms, Q has only odd terms). These cases are covered by the following OptiVec functions:

VF_ratioOddEven: P has only odd terms, Q has only even terms:
P_i = p₁ * X_i + p₁ * X_i³ + ... + p_degP * X_i^degP, with degP = 2n+1
Q_i = q₀ + q₂ * X_i² + ... + q_degQ * X_i^degQ, with degQ = 2n
Coeff here contains (degP+1)/2 + (degQ/2)+1 coefficients.

VF_ratioEvenOdd: P has only even, Q has only odd terms:
P_i = p₀ + p₂ * X_i² + ... + p_degP * X_i^degP, with degP = 2n
Q_i = q₁ * X_i + q₁ * X_i³ + ... + q_degQ * X_i^degQ, with degQ = 2n+1
Coeff here contains (degP/2)+1 + (degQ+1)/2 coefficients.

While mathematically legal, round-off error makes polynomials and, by extension, rations of high degrees practically useless. In the present implementation, VF_ratio, VF_ratio_d, and VD_ratio allow degP+degQ ± 32. Above that, an error message "Invalid Parameter(s)" is emitted and program execution halted. Only the VE_ versions along with VF_ratio_e and VD_ratio_e allow to evaluate ratios of higher degrees.