Directional Quaternion Step Differentiation and a Bicomplex Double-Step Calculus for Cancellation-Free First and Second Derivatives

Accurate derivative information is central to sensitivity analysis and optimization, yet standard finite differences can lose many digits when the step size is small because of subtractive cancellation. Complex-step differentiation largely resolves this issue for first derivatives, but robust second derivatives and mixed partials remain delicate: several practical complex-step variants for f′′ still subtract nearly equal quantities, and quaternion-step rules are often presented as separate constructions. We develop a unified slice-based framework that extracts first and second derivatives from a single evaluation by projecting algebraic coefficients in commutative subalgebras of the complexified quaternions. First, we formulate a directional quaternion-steprule parameterized by an arbitrary unit pure quaternion u and provide an explicit projection operator that makes the underlying complex slice Cu≃C transparent; the resulting first-derivative formula is rotation invariant and recovers classical j-step and planar (j,k)-step rules as special cases. Second, we construct a bicomplex double-step calculus in the commuting imaginary units i and u and show that one evaluation at z+(i+u)h separates derivative information into distinct coefficients, with the iu-component equal to h2f′′(z)+O(h4), giving a subtraction-free O(h2) approximation of f′′. For bivariate analytic functions we additionally derive one-shot identities for fx, fy, and fxy from f(x+uh,y+ih) and supply practical extraction identities, step-size guidance for h2-scaled coefficients, and branch-consistency diagnostics for non-entire functions. The “cancellation-free” property here refers to avoiding the subtraction of nearly equal real quantities at the level of the differentiation formula; in floating-point arithmetic, coefficient extraction and the 1/h2 scaling for second-order quantities still interact with roundoff, and we quantify the resulting stable regimes numerically.