The formulas for calculating the intraocular lens (IOL) power were originated in the 1980s with the birth of the first regression formulas (SRK and SRK II). Considered as obsolete since the 90's,  they left as inheritance the concept of the so called A-Constant, which is nothing more than the resulting coefficient after performing a multiple linear regression (P = A - 2.5L - 0.9K) being P the power of the intraocular lens that predicts the emetropia, K the power of the cornea and L the axial length .
A regression formula like the previous one could be improved considering that the eye is an optical system and the theoretical bases of this kind of optical system were already developed by Gauss in 1840, so Fyodorov incorporated the theory of paraxial optics to the calculation of the intraocular lens[3, 4]. From this moment we can say that the biggest evolution in the decrease of the prediction error in the calculation of the IOL power has come from the technological evolution more than from the theoretical development of any type of formula.
Formulas began to be proposed, and continue to be proposed, with the same theoretical basis (the vergence equation proposed by Fyodorov), and are known within the field of ophthalmology as "empirical" or "3rd / 4th generation" although it is correct to call them "Vergence Formulas". In my personal opinion, all the advances that continue using this equation should be disclosed as new approximations for the estimation of the ELPo rather than new formulas, being the ELPo the effective position of the IOL if it had a very reduced thickness so that its first principal plane matches the center of the lens.
An alternative to the vergence equation is the thick lens equation that takes into account the position of the principal planes of the cornea and intraocular lens,  Barrett described the application of the thick lens Gaussian equation to IOL calculation. Although it is true that this requires knowing the design parameters of the implanted IOL to know where the principal planes are located and although some current formulas are advertised as thick lens equations, this does not make sense when these parameters are not known (A-constant is necessary in the calculation).
The vergence equations still have a problem, we do not know which is the ELPo but we know that depends on the position of the IOL and its design. A migration from an obsolete concept but still used by manufacturers (A-Constant) to a theoretically more correct concept (ELPo) is required. We have reached a moment of technological development in which we can decompose the ELPo into fixed and variable parameters, for example, we can measure the distance between the cornea and the pupil plane (variable) leaving as static (constant) the distance between the pupil plane and the IOL plane (thin). Then new constants arise according to the empirical formulas that Retzlaff called "offset" in the development of the SRK/T,  Holladay "Surgeon Factor" , Barrett "Lens Factor" , etc. and that come to represent a decomposition of the ELPo into measurable variable parameters (corneal/iris distance) and that we need to predict (distance between the iris plane and the plane of the thin IOL).
The mathematical advantage or clinical disadvantage of these constants is that consist of a single parameter so the optimization can be easily achieved by simply changing the constant value in small steps (i.e. 0.001 in the case of the A-constant until the prediction error of the formula matches the patient's postoperative residual. This process requires time if the formula is not known, although if it is known, an algorithm can be implemented to perform the calculation in an automated way. In the case of not having the formula, the process would consist of:
Unlike the previous constants, Haigis' formula  uses a multiple regression to predict the ELPo, this means obtaining three coefficients a0,a1,a2. From the statistical point of view the calculation is the same as in the SRK formula to determine the A constant but instead of predicting the power of the IOL that reduces the prediction error to zero, what is predicted is the ELPo that reduces the prediction error to zero. To adjust these three constants we will need to perform a multiple regression between the ELPo (dependent) vs ACD (a1) and AXL (a2) resulting in a0 as the free coefficient of the adjustment.
In our online course "Analysis of Results in Refractive and Cataract Surgery" we have programmed the Constant Optimization for future incorporation to the course materials in order to teach you in a practical way all the process and the theoretical considerations in a more detailed way.
Holladay JT (1997) Standardizing constants for ultrasonic biometry, keratometry, and intraocular lens power calculations. J Cataract Refract Surg 23:1356–1370.
Sanders DR, Kraff MC (1980) Improvement of intraocular lens power calculation using empirical data. Am Intra-Ocular Implant Soc J 6:263–267.
Fyodorov SN, Kolinko A (1967) Estimation of optical power of the intraocular lens. Vestn Oftalmol 80:27–31
Fyodorov SN, Galin M, Linksz A (1975) Calculation of the optical power of intraocular lenses. Ivestigative Ophthalmol 14:625–628
Yoo YS, Whang WJ, Kim HS, et al (2020) New IOL formula using anterior segment three-dimensional optical coherence tomography. PLoS One 15:e0236137.
Koch DD, Hill W, Abulafia A, Wang L (2017) Pursuing perfection in intraocular lens calculations: I. Logical approach for classifying IOL calculation formulas. J Cataract Refract Surg 43:717–718.
Fernández J, Manuel Rodríguez-Vallejo JM, Tauste A, Piñero DP (2019) New approach for the calculation of the intraocular lens power based on the fictitious corneal refractive index estimation. J Ophthalmol November:8–13
Barrett GD (1987) Intraocular lens calculation formulas for new intraocular lens implants. J Cataract Refract Surg 13:389–396.
Retzlaff JA, Sanders DR, Kraff MC (1990) Development of the SRK/T intraocular lens implant power calculation formula. J Cataract Refract Surg 16:333–340.
Holladay JT, Musgrove KH, Prager TC, et al (1988) A three-part system for refining intraocular lens power calculations. J Cataract Refract Surg 14:17–24.
Barrett GD (1993) An improved universal theoretical formula for intraocular lens power prediction. J Cataract Refract Surg 19:713–720.
Schröder S, Leydolt C, Menapace R, et al (2016) Determination of Personalized IOL-Constants for the Haigis Formula under Consideration of Measurement Precision. PLoS One 11:e0158988.
We have prepared a complete course that will help you to understand the standards for reporting the clinical results of your practice. This is a practical course, besides the theoretical knowledge, we are going to supply you with the "Refractive Analysis Toolbox" for Matlab that will help you to do all the analyses by yourself. Don't be afraid about Matlab because we have reduced the complexity of the calculations to specific functions that give you the results in a easy way.