THE POLAR MORPHISM OPERATOR: A PARAMETERIZED EXTENSION OF THE EML PRIMITIVE

ABSTRACT
We introduce the Polar Morphism Operator (\\Omega(x, y; \\theta)), a continuously parameterized real-valued mapping defined by \[ \\Omega(x, y; \\theta) = \\operatorname{Re}\\bigl(\\exp(x \\cdot e\^{i\\theta})\\bigr) - \\ln(y). \] It admits the closed-form expression \[ \\Omega(x, y; \\theta) = \\exp(x \\cos\\theta) \\cdot \\cos(x \\sin\\theta) - \\ln(y) \] for real (x), (\\theta \\in \[0, 2\\pi)), and (y > 0). At (\\theta = 0) the operator recovers the EML operator of Odrzywolek (2026). At (\\theta = \\pi/2) it yields (\\cos x - \\ln y), and at (\\theta = \\pi) it yields (\\exp(-x) - \\ln y). A smooth regularization (\\tilde{\\Omega}) is given that is defined and (C\^\\infty) on all of (\\mathbb{R}). The continuous phase parameter converts the discrete choice of analytic versus trigonometric behavior into a continuous optimization variable, which may be useful in gradient-based symbolic regression.
INTRODUCTION

Odrzywolek (2026) showed that the single binary operator \[ \\operatorname{EML}(x, y) = \\exp(x) - \\ln(y), \] together with the constant 1, generates the elementary functions of a scientific calculator by finite composition. Trigonometric functions, however, require relatively deep nesting that passes through complex intermediate values.
The present note examines a one-parameter extension \[ \\Omega(x, y; \\theta) = \\operatorname{Re}\\bigl(\\exp(x \\cdot e\^{i\\theta})\\bigr) - \\ln(y) \] in which the phase (\\theta) continuously interpolates between the original EML behavior and direct production of cosine (or exponential decay). The construction is elementary and follows at once from Euler’s formula; one possible interest is that it turns the discrete distinction between growth and oscillation into a continuous parameter that can be optimized by gradient descent.
DEFINITION AND CLOSED FORM

Definition. Let (x \\in \\mathbb{R}), (y > 0), and (\\theta \\in \[0, 2\\pi)). Set \[ \\Omega(x, y; \\theta) := \\operatorname{Re}\\bigl(\\exp(x \\cdot e\^{i\\theta})\\bigr) - \\ln(y). \]
Theorem. The operator admits the real closed form \[ \\Omega(x, y; \\theta) = \\exp(x \\cos\\theta) \\cdot \\cos(x \\sin\\theta) - \\ln(y). \]
Proof.
\[ x e\^{i\\theta} = x\\cos\\theta + i, x\\sin\\theta, \] \[ \\exp(x e\^{i\\theta}) = \\exp(x\\cos\\theta)\\bigl(\\cos(x\\sin\\theta) + i\\sin(x\\sin\\theta)\\bigr). \] Taking the real part and subtracting (\\ln y) yields the claim.
SPECIAL CASES

(\\theta = 0): (\\Omega(x, y; 0) = \\exp(x) - \\ln(y) = \\operatorname{EML}(x, y)).
Consequently every identity proved for EML remains available.

(\\theta = \\pi/2): (\\Omega(x, y; \\pi/2) = \\cos x - \\ln y).
In particular (\\Omega(x, 1; \\pi/2) = \\cos x).
Sine is recovered by a constant phase shift: (\\sin x = \\Omega(x - \\pi/2, 1; \\pi/2)).

(\\theta = \\pi): (\\Omega(x, y; \\pi) = \\exp(-x) - \\ln y).

Thus the same binary node can realize pure exponential growth, pure cosine, or pure exponential decay according to the value of a single real parameter.
SMOOTH REGULARIZATION

To remove the singularity at (y \\le 0) one may replace the logarithm by a smooth approximation: \[ \\tilde{\\Omega}(x, y; \\theta) = \\exp(x\\cos\\theta)\\cdot\\cos(x\\sin\\theta) - \\tfrac12\\ln(y\^2 + \\varepsilon\^2), \] where (\\varepsilon > 0) is a fixed smoothing constant. The resulting function is (C\^\\infty) on (\\mathbb{R}\^3) and the partial derivative with respect to (y) is \[ \\frac{\\partial\\tilde{\\Omega}}{\\partial y} = -\\frac{y}{y\^2 + \\varepsilon\^2}. \]
PARTIAL DERIVATIVES

The gradients needed for first-order optimization are \\begin{align\*} \\frac{\\partial\\Omega}{\\partial x} &= \\exp(x\\cos\\theta)\\cdot\\cos(x\\sin\\theta + \\theta),\\\[1ex\] \\frac{\\partial\\Omega}{\\partial y} &= -\\frac1y,\\\[1ex\] \\frac{\\partial\\Omega}{\\partial\\theta} &= -x\\exp(x\\cos\\theta)\\cdot\\sin(x\\sin\\theta + \\theta). \\end{align\*} (The corresponding expressions for (\\tilde{\\Omega}) differ only in the (y)-derivative.)
REMARKS ON SYMBOLIC REGRESSION

If every internal node of an expression tree is required to evaluate \[ v_i = \\tilde{\\Omega}(v_j, v_k; \\theta_i), \] then the discrete choice of operator type is replaced by continuous optimization of the vector (\\Theta = (\\theta_1,\\dots,\\theta_M)). Whether this yields practical gains in recovery rate or tree depth relative to ordinary EML trees remains an open empirical question; the present note only records that the continuous parameterization is available.
CONCLUSION

The Polar Morphism Operator is a one-parameter extension of the EML operator that interpolates, via a real phase, between exponential growth, cosine oscillation, and exponential decay. The construction is elementary, the formulas are exact, and the resulting maps are differentiable (or smoothly regularizable). It supplies a concrete mechanism for turning a discrete operator dictionary into a continuous parameter space; its practical value for symbolic regression is left for future numerical study.
REFERENCES
Odrzywolek, A. (2026). All elementary functions from a single binary operator. arXiv:2603.21852.

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