not-yet-known not-yet-known not-yet-known unknown We present a novel, closed-form approximation to the standard normal cumulative distribution function (CDF), denoted the Prime Density function. Departing from traditional methods based on the error function or rational polynomial approximants[?, ?, ?], the Prime Density function is constructed as a logistic sigmoid with a cubic argument. Its structure enables a balance of symbolic tractability, numerical precision, and computational efficiency. The final optimized form, Φ ( x ) ≈ 1 1 + exp ( − ( 1 . 5 9 7 0 8 x + 0 . 0 7 0 9 5 x 3 ) ) , (1) was derived via a hybrid global-local optimization approach combining Differential Evolution and the Nelder–Mead simplex method, minimizing approximation error over the real line. The function is continuously differentiable, strictly increasing, and analytically invertible, making it suitable for real-time systems, symbolic manipulation, and hardware-constrained applications. Despite its simplicity, the approximation achieves a maximum absolute error below 1 . 7 × 1 0 − 4 , outperforming classical logistic fits and rivaling more complex rational approximations. Comparative analysis against standard benchmarks confirms the Prime Density function’s robustness in domains requiring high-throughput normal CDF evaluation, including probabilistic machine learning, biomedical statistics, and financial modeling. This work positions Prime Density as a practical and theoretically grounded alternative to conventional numerical methods, bridging the gap between analytic simplicity and modern performance demands.

We introduce the Prime Density function, a novel logistic-cubic closed-form approximation to the standard normal cumulative distribution function (CDF). Existing approximations either lack analytic simplicity and invertibility or compromise accuracy, particularly in distribution tails. To overcome these limitations, the Prime Density function employs a logistic sigmoid function with a cubic polynomial argument. Parameters were rigorously optimized through a hybrid global-local procedure combining Differential Evolution and Nelder-Mead methods, minimizing maximum absolute error and root-mean-square deviation across the real line. Our optimized approximation achieves a maximum absolute error below 1.7 × 10 −4 , surpassing classical logistic and rivaling complex rational approximations. The function maintains analytical invertibility, differentiability, and symbolic simplicity, providing distinct computational advantages for real-time analytics, symbolic computation, and resource-limited hardware. Empirical evaluations across diverse datasets-including environmental pollutant indices (PM2.5 AQI), financial returns (S&P 500), and biomedical markers (glucose and triglycerides)-demonstrate the Prime Density function's superior empirical flexibility and precision compared to traditional approximations. The results position the Prime Density function as a practical, rigorously validated, and computationally efficient alternative, effectively bridging analytic simplicity and high-performance demands.

Objectives: Much less attention has been given to the right heart and pulmonary circulation compared to the left heart and systemic circulation in patients with pre-eclampsia (PEC). We used transthoracic echocardiography (TTE) to estimate pulmonary artery pressure and right ventricular function in women with PEC. Methods: A case-control study at a tertiary care academic center. Ten early PEC (<34 week gestation) and nine late PEC (≥34 weeks gestation) patients with eleven early and ten late gestational age-matched controls. Two dimensional TTE was performed on all patients. The estimated mean PA pressure (eMPAP) was calculated based on pulmonary artery acceleration time (PAAT). Pulmonary vascular resistance (ePVR) was estimated from eMPAP and right ventricular (RV) cardiac output. RV myocardial performance index (RV MPI), tricuspid annular plane systolic excursion (TAPSE), tissue tricuspid annular displacement (TTAD) and lateral tricuspid annular tissue peak systolic velocity (S’) were measured. Results Compared to early controls, in early PEC the eMPAP and ePVR were elevated, PAAT was reduced, RV MPI was increased, TTAD was reduced and TAPSE and TV S’ were unchanged. Compared to late controls, in late PEC, estimated MPAP and estimated PVR were elevated, PAAT was reduced and RVMPI was increased, while TAPSE, TTAD and TV S’ were unchanged. Conclusions: Early PEC is associated with increased eMPAP and ePVR. A subclinical decrement of RV function is noticed. TTE is a useful screening tool for early detection of PH and RV dysfunction in PEC.