### Demographic and biometric data comparison

This study included 100 eyes from 50 postoperative refractive surgery patients, with 25 individuals (50 eyes) having undergone FS-LASIK (10 males, 15 females) and 25 individuals (50 eyes) having undergone SMILE (12 males, 13 females). Table 1 presents the demographic data and various corneal biometric parameters of the patients. The sex distribution was compared using Pearson’s Chi-square test, revealing no statistically significant difference between the two groups (χ^{2} = 0.325, *p* = 0.569).

Preoperative age and parameters such as AL, CCT, and WTW were compared between the two groups using t-tests after verifying normal distribution (results: t = 0.6169, *p* = 0.5402; t = 1.934, *p* = 0.056; t = 1.919, *p* = 0.579, respectively). Mann–Whitney U test was employed to compare ACD and Km between the groups (results: z = -1.093, *p* = 0.274; z = -1.948, *p* = 0.051, respectively). And no significant differences were found, hence, the two groups were well-matched in terms of age, sex, and corneal parameters.

### FS-LASIK group formulas comparison

Table 2 presents a comparison of the mean, standard deviation, extreme values, and median of the AE within the FS-LASIK group as generated by the six formulas. In the FS-LASIK group, the mean AE values for six formulas were as follows (in descending order): BTK (0.28 ± 0.25 D), BTKNH (0.40 ± 0.35 D), Masket (0.52 ± 0.41 D), M-Masket (0.63 ± 0.55 D), Haigis-L (0.80 ± 0.52 D), and Shammas-PL (0.94 ± 0.56 D). Figure 1a presents a Violin plot depicting the AE of six formulas in the FS-LASIK groups, and one-way ANOVA analysis indicated no significant difference between BTK and BTKNH (*p* = 0.18), but there were significant differences when compared to Masket, M-Masket, Shammas-PL, and Haigis-L formulas (*p* = 0.007, *p* < 0.001, *p* < 0.001, and *p* < 0.001, respectively).

Figures 2a-f and 4a and Table 3 illustrate the distribution frequency of AE and PE within various ranges for the six formulas. BTK (41 cases, 82%) and BTKNH (37 cases, 74%) had the highest proportion of AE within 0.5D, with no significant difference between them (χ^{2} = 0.932, *p* = 0.334). Remarkably, all AE results for BTK were within 1.0 D, indicating excellent stability. Additionally, BTK (28 cases, 56%) and BTKNH (22 cases, 44%) had the highest proportion of PE within ± 0.25 D, closely resembling BUII results, with no significant difference between the two (χ^{2} = 1.44, *p* = 0.23).

### SMILE group formulas comparison

Table 4 presents a comparison of the key statistical measures for the AE within the SMILE group, as calculated using the six formulas. In the SMILE group, the mean AE values for the six formulas were as follows: BTK (0.36 ± 0.24 D), Masket (0.43 ± 0.33 D), BTKNH (0.52 ± 0.37 D), M-Masket (0.53 ± 0.39 D), Haigis-L (0.80 ± 0.56 D), and Shammas-PL (0.86 ± 0.65 D). Figure 1b illustrates the achieved AE of six formulas in the SMILE groups. The one-way ANOVA analysis demonstrated no significant difference between the BTK and Masket formulas, as well as between the BTK and BTKNH formulas (BTK-Masket: *p* = 0.411; BTK-BTKNH: *p* = 0.064). However, significant differences were found when comparing BTK with M-Masket, Shammas-PL, and Haigis-L formulas (*p* = 0.048, *p* < 0.001, and *p* < 0.001, respectively).

Similarly, Figs. 3a-f and 4b and Table 5 illustrate the distribution frequency of AE and PE within various ranges for the six formulas in the SMILE group. BTK (38 cases, 76%) had the highest proportion of AE within 0.5 D, followed by Masket (32 cases, 64%) and BTKNH (31 cases, 62%), with no significant difference between them (BTK-Masket: χ^{2} = 1.714, *p* = 0.19; BTK-BTKNH: χ^{2} = 2.291, *p* = 0.13). As in the FS-LASIK group, the BTK formula demonstrated excellent performance, with 100% of AE results within 1.0 D in the SMILE group. Moreover, the Masket formula had 96% of PE results within 1.0 D. Among the SMILE group, the formula with the highest proportion of PE within ± 0.25D was Masket (20 cases, 40%), and the chi-square test indicated a significant difference when compared to Haigis-L (Masket-Haigis-L: χ^{2} = 5.877, *p* = 0.015), but not when compared to other formulas (Masket-BTK: χ^{2} = 0.694, *p* = 0.405; Masket-BTKNH: χ^{2} = 1.604, *p* = 0.205; Masket-M-Masket: χ^{2} = 1.604, *p* = 0.205; Masket-Shammas-PL: χ^{2} = 2.941, *p* = 0.086).

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