cs.LGOct 14, 2026

Exhaustive Canonical Skeletons Generation: A Formal Dataset for Decomposable Symbolic Regression

Authors: Giorgio Morales, Anikó Ekárt

Organizations: Aston Centre for Artificial Intelligence Research and Application, Aston University, Birmingham, UK

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Abstract

Neural symbolic regression (SR) models require diverse datasets of mathematical expressions for robust training. However, existing datasets rely predominantly on stochastic sampling methods, which suffer from high structural redundancy and incomplete functional coverage. To address this, we introduce an exhaustive dataset of univariate canonical skeletons generated via a deterministic constrained context-free grammar. A canonical skeleton acts as a generic structural representation that subsumes more specific functional forms, eliminating redundant expressions and avoiding repeated computational efforts. Although our dataset consists of univariate skeletons, it is designed to support the decomposable SR paradigm to solve multivariate SR problems without assuming linear separability. Furthermore, we pair this dataset with a sampling strategy that generates clean numerical data without triggering numerical instabilities or relying on artificial protected operators. Canonicalizing two existing neural SR datasets reveals that our dataset is substantially more compact, eliminates structural repetitions, and achieves a uniform operator distribution free of simplicity bias. Finally, we establish a benchmark of 150 SR problems across three tiers, on which state-of-the-art SR methods are shown to struggle significantly.

Figures & tables

MetricExCaUSSeTGAPNeSymRes
Total Expressions71,0271,650,00010,000,000
Unique Skeletons71,027664128,514
Duplication Rate (%)0.0099.9698.71
Dictionary Size (Unique Ops)161211
Mean Unary Nesting Depth1.581.291.12
Max Unary Nesting Depth236
TABLE I : Quantitative comparison of SR datasets
Fig. 1 : SR datasets comparison
Fig. 2 : Extrapolation MSE on the (a) E, (b) M, and (c) H tiers
Fig. 3 : Extrapolation R2R^{2} on the (a) E, (b) M, and (c) H tiers
Appendix figures & tables22 assets

Supplementary material from the paper’s appendix.

Appendix

Fig. D.1 : A generation example using f(x)=1(x+4)3+e1x−4f(x)=\frac{1}{(x+4)^{3}}+e^{\frac{1}{x-4}} . (Top) Generated data on the entire domain [−10,10][-10,10] . (Bottom) Detailed view of how singularities are avoided.
Fig. D.2 : Various functions generated using the canonical skeleton e(x)=C+LC arctan⁡(P)+Ctanh⁡(L)\mathbf{e}(x)=C+\frac{L}{C\,\arctan(P)+C}\tanh(L)
IDSkeletonEquation
E1C∣L∣+LC|{L}|+L−1.31+5.79E-2∣4.54x+4.61∣+7.67E-2x-1.31+5.79\text{E-2}|{4.54x+4.61}|+7.67\text{E-2}x
E2CeLsinh⁡(P)+CCe^{L}\sinh(P)+C−0.30+2.17E-2e0.16xsinh⁡(−0.48x+0.11+3.11E-2x2)-0.30+2.17\text{E-2}e^{0.16x}\sinh(-0.48x+0.11+3.11\text{E-2}x^{2})
E3Cinv⁡(L)cosh⁡(P)+Casinh⁡(L)+CC\operatorname{inv}(L)\cosh(P)+C\operatorname{asinh}(L)+C−4.97E-2+0.13asinh⁡(x)+5.06E-2cosh⁡(−5.54E-3+3.05E-2x+7.70E-2x2)1.35x+1.63-4.97\text{E-2}+0.13\operatorname{asinh}(x)+\frac{5.06\text{E-2}\cosh(-5.54\text{E-3}+3.05\text{E-2}x+7.70\text{E-2}x^{2})}{1.35x+1.63}
E4Cinv⁡(Csin⁡(L)atan⁡(P)+L)+CC\operatorname{inv}(C\sin(L)\operatorname{atan}(P)+L)+C−1.260.85x+1.40sin⁡(2.91x)atan⁡(−3.21+0.32x3+0.64x−x2)+−4.89E-2\frac{-1.26}{0.85x+1.40\sin(2.91x)\operatorname{atan}(-3.21+0.32x^{3}+0.64x-x^{2})}+-4.89\text{E-2}
E5C+Llog⁡(P)C+L\log(P)−0.42+(0.13+3.54E-2x)log⁡(−2.06x+0.45+2.49x2)-0.42+(0.13+3.54\text{E-2}x)\log(-2.06x+0.45+2.49x^{2})
E6CPow⁡(C+Lasinh⁡(L))tanh⁡(P)+CC\operatorname{Pow}(C+L\operatorname{asinh}(L))\tanh(P)+C−0.14+−1.02E-13((−3.12+−4.11x)asinh⁡(−0.65+x))132tanh⁡(−8.37E-3x+1.85E-3+4.91E-3x2+7.59E-3x3)\begin{aligned} &-0.14+-1.02\text{E-13}((-3.12+-4.11x)\operatorname{asinh}(-0.65+x))^{\frac{13}{2}}\tanh(-8.37\text{E-3}x+1.85\text{E-3}\\ &\hskip 8.19447pt+4.91\text{E-3}x^{2}+7.59\text{E-3}x^{3})\end{aligned}
Appendix
TABLE E.1 : Generated Equations for the E tier
IDSkeletonEquation
M1Cinv⁡(C∣L∣+L)+Csin⁡(P)+CC\operatorname{inv}(C|{L}|+L)+C\sin(P)+C−2.51E-2+0.78sin⁡(0.60+0.98x+1.40x2)+0.78−4.23x+4.05+∣−1.90+3.63x∣-2.51\text{E-2}+0.78\sin(0.60+0.98x+1.40x^{2})+\frac{0.78}{-4.23x+4.05+|{-1.90+3.63x}|}
M2Cinv⁡(Clog⁡(P)+C)+C∣L∣+CC\operatorname{inv}(C\log(P)+C)+C|{L}|+C−1.66+0.313.09+4.66log⁡(−0.30x5+−4.93x2+0.59x+0.93x4+1.87+4.33x6−x3)+0.32∣1.92+x∣\begin{aligned} &-1.66+\frac{0.31}{3.09+4.66\log(-0.30x^{5}+-4.93x^{2}+0.59x+0.93x^{4}+1.87+4.33x^{6}-x^{3})}+0.32|{1.92+x}|\end{aligned}
M3Ccube⁡(Cinv⁡(L)asin⁡(P)+C)+Cinv⁡(L)+C\begin{aligned} &C\operatorname{cube}(C\operatorname{inv}(L)\operatorname{asin}(P)+C)+C\operatorname{inv}(L)+C\end{aligned}−0.42+1.26(−1.51asin⁡(−2.00E-3x2+0.20+4.10E-3x+9.58E-4x3)−2.70x+−2.81+0.66)3+3.09−2.95+3.17x\begin{aligned} &-0.42+1.26(\frac{-1.51\operatorname{asin}(-2.00\text{E-3}x^{2}+0.20+4.10\text{E-3}x+9.58\text{E-4}x^{3})}{-2.70x+-2.81}+0.66)^{3}+\frac{3.09}{-2.95+3.17x}\end{aligned}
M4CPow⁡(C+Lasinh⁡(L))+Ccosh⁡(P)+CC\operatorname{Pow}(C+L\operatorname{asinh}(L))+C\cosh(P)+C−0.45+1.54E-5(−4.16+(−3.34+0.58x)asinh⁡(−0.65+x))4+1.54E-5cosh⁡(4.61E-2x+6.66E-2+7.47E-2x2)\begin{aligned} &-0.45+1.54\text{E-5}(-4.16+(-3.34+0.58x)\operatorname{asinh}(-0.65+x))^{4}+1.54\text{E-5}\cosh(4.61\text{E-2}x+6.66\text{E-2}+7.47\text{E-2}x^{2})\end{aligned}
M5Cacosh⁡(L)acosh⁡(Cinv⁡(P)+L)+CC\operatorname{acosh}(L)\operatorname{acosh}(C\operatorname{inv}(P)+L)+C−2.23+0.43acosh⁡(−0.97x+10.75)acosh⁡(−0.21x+3.19−1−1.94x2+−4.34x5+2.77+4.17x4+x3+x)\begin{aligned} &-2.23+0.43\operatorname{acosh}{\left(-0.97x+10.75\right)}\operatorname{acosh}{\left(-0.21x+3.19-\frac{1}{-1.94x^{2}+-4.34x^{5}+2.77+4.17x^{4}+x^{3}+x}\right)}\end{aligned}
M6CCinv⁡(L)+Csinh⁡(C+Linv⁡(P))+CC\sqrt{C\operatorname{inv}(L)+C}\sinh(C+L\operatorname{inv}(P))+C1.3510.11+7.08E-4−2.28+−3.14xsinh⁡(−7.20E-2+0.39+2.98x−3.16x+3.17+4.67x2)+5.44E-2\begin{aligned} &1.35\sqrt{10.11+\frac{7.08\text{E-4}}{-2.28+-3.14x}}\sinh(-7.20\text{E-2}+\frac{0.39+2.98x}{-3.16x+3.17+4.67x^{2}})+5.44\text{E-2}\end{aligned}
Appendix
TABLE E.2 : Generated Equations for the M tier
IDSkeletonEquation
H1Cinv⁡(L)Pow⁡(Clog⁡(L)+Casin⁡(P)+C)+CC\operatorname{inv}{\left(L\right)}\operatorname{Pow}{\left(C\log{\left(L\right)}+C\operatorname{asin}{\left(P\right)}+C\right)}+C−0.53+8.41E-2(−1.16+−2.64asin⁡(−0.97+1.52E-2x2+4.42E-2x)+0.29log⁡(1.14x+11.49))4−1.02x+3.34-0.53+\frac{8.41\text{E-2}\left(-1.16+-2.64\operatorname{asin}{\left(-0.97+1.52\text{E-2}x^{2}+4.42\text{E-2}x\right)}+0.29\log{\left(1.14x+11.49\right)}\right)^{4}}{-1.02x+3.34}
H2CPow⁡(Ctanh⁡(L)+C)atan⁡(Cinv⁡(L)+C)+PC\operatorname{Pow}{\left(C\tanh{\left(L\right)}+C\right)}\operatorname{atan}{\left(C\operatorname{inv}{\left(L\right)}+C\right)}+P−0.53+−9.11E-6x5+1.19E-5x+1.39E-5x4+2.99E-6x6+2.99E-6x3+3.28E-6x2+6.97E-6(−4.81tanh⁡(0.80x)+1.39)6atan⁡(2.03+3.71−0.51x+−2.78)\begin{aligned} -0.53+&-9.11\text{E-6}x^{5}+1.19\text{E-5}x+1.39\text{E-5}x^{4}+2.99\text{E-6}x^{6}+2.99\text{E-6}x^{3}+\\ &3.28\text{E-6}x^{2}+6.97\text{E-6}\left(-4.81\tanh{\left(0.80x\right)}+1.39\right)^{6}\operatorname{atan}{(2.03+\frac{3.71}{-0.51x+-2.78})}\end{aligned}
H3Cinv⁡(L)Pow⁡(Csin⁡(L)+C)+CPow⁡(Catan⁡(P)+C)+CC\operatorname{inv}{\left(L\right)}\operatorname{Pow}{\left(C\sin{\left(L\right)}+C\right)}+C\operatorname{Pow}{\left(C\operatorname{atan}{\left(P\right)}+C\right)}+C−0.18atan⁡6(−3.57x+0.46+x2)+1.94+5.09sin⁡115(−1.63+2.17x)1.54+3.67x-0.18\operatorname{atan}^{6}{\left(-3.57x+0.46+x^{2}\right)}+1.94+\frac{5.09\sin^{\frac{11}{5}}{\left(-1.63+2.17x\right)}}{1.54+3.67x}
H4Cinv⁡(C+Llog⁡(L))+Catanh⁡(Cinv⁡(P)+C)+CC\operatorname{inv}{\left(C+L\log{\left(L\right)}\right)}+C\operatorname{atanh}{\left(C\operatorname{inv}{\left(P\right)}+C\right)}+C−2.99atanh⁡(0.57−2.19+−4.58x3+0.45x2+0.91x+x4+7.95E-2)+0.20+2.012.13+(−1.12x+2.93)log⁡(−1.31x+13.25)-2.99\operatorname{atanh}{\left(\frac{0.57}{-2.19+-4.58x^{3}+0.45x^{2}+0.91x+x^{4}}+7.95\text{E-2}\right)}+0.20+\frac{2.01}{2.13+\left(-1.12x+2.93\right)\log{\left(-1.31x+13.25\right)}}
H5CCinv⁡(P)atan⁡(L)+C+Cinv⁡(L)+CC\sqrt{C\operatorname{inv}{\left(P\right)}\operatorname{atan}{\left(L\right)}+C}+C\operatorname{inv}{\left(L\right)}+C−8.82+1.18x+2.74−3.81atan⁡(1.83x+4.25)−4.57x2+3.04x+4.18+10.37-8.82+\frac{1.18}{x}+2.74\sqrt{\frac{-3.81\operatorname{atan}{\left(1.83x+4.25\right)}}{-4.57x^{2}+3.04x+4.18}+10.37}
H6Cinv⁡(Catanh⁡(L)+C)+Ccosh⁡(Cinv⁡(P)+L)+CC\operatorname{inv}{\left(C\operatorname{atanh}{\left(L\right)}+C\right)}+C\cosh{\left(C\operatorname{inv}{\left(P\right)}+L\right)}+C−1.33E-4−1.92+3.47atanh⁡(−2.45E-3+0.10x)+−1.76E-2cosh⁡(0.60x+1.48−0.21+−1.17x+x2)+0.70\frac{-1.33\text{E-4}}{-1.92+3.47\operatorname{atanh}{\left(-2.45\text{E-3}+0.10x\right)}}+-1.76\text{E-2}\cosh{\left(0.60x+\frac{1.48}{-0.21+-1.17x+x^{2}}\right)}+0.70
Appendix
TABLE E.3 : Generated Equations for the H tier
Fig. E.1 : Plots for Benchmark Equations — E Tier (E1–E20)
Fig. E.2 : Plots for Benchmark Equations — E Tier (E21–E40)
Fig. E.3 : Plots for Benchmark Equations — E Tier (E41–E50)
Fig. E.4 : Plots for Benchmark Equations — M Tier (M1–M20)
Fig. E.5 : Plots for Benchmark Equations — M Tier (M21–M40)
Fig. E.6 : Plots for Benchmark Equations — M Tier (M41–M50)
Fig. E.7 : Plots for Benchmark Equations — H Tier (H1–H20)
Fig. E.8 : Plots for Benchmark Equations — H Tier (H21–H40)
Fig. E.9 : Plots for Benchmark Equations — H Tier (H41–H50)
MethodHyperparameterValues
AIFeynmanPolynomial fitting degree2, 3, 4, 5
Neural network training epochs500, 1000, 2000
BSRNumber of trees2, 4, 8
Number of iterations50, 100, 200, 500
Regularization parameter-0.1, -1, -2, -5
DSRNumber of samples100,000, 500,000, 1,000,000
Appendix
TABLE F.1 : Tuning hyperparameter grids
MethodHyperparameterEasyMediumHard
AIFeynmanPolynomial fitting degree323
Neural network training epochs200020001000
BSRNumber of trees222
Number of iterations50200100
Regularization parameter-0.1-0.1-2
DSRNumber of samples10000001000000100000
Appendix
TABLE F.2 : Selected optimal hyperparameter configurations
IDSetAIFeynmanBSRDSRE2EGPGOMEAGplearnITEANeSymReSOperonPySRTaylorGPTPSRUDSR
E1Int.1.72 ±\pm 00.846 ±\pm 00.089 ±\pm 03.4e-5 ±\pm 02.2e-7 ±\pm 1.9e-70.317 ±\pm 0.3741.6e-8 ±\pm 7.6e-90.831 ±\pm 9.8e-38.0e-4 ±\pm 9.4e-41.1e-12 ±\pm 2.8e-120.846 ±\pm 00.846 ±\pm 00.089 ±\pm 0
Ext.2.07 ±\pm 02.38 ±\pm 00.604 ±\pm 02.0e-5 ±\pm 07.3e-7 ±\pm 7.5e-71.25 ±\pm 0.8145.6e-7 ±\pm 1.1e-62.71 ±\pm 0.0243.0e-3 ±\pm 4.3e-30 ±\pm 1.5e-122.38 ±\pm 02.38 ±\pm 00.604 ±\pm 0
E2Int.1.36 ±\pm 00.223 ±\pm 00.115 ±\pm 03.3e-4 ±\pm 00.022 ±\pm 0.0670.164 ±\pm 0.0601.8e-10 ±\pm 7.5e-110.207 ±\pm 0.0166.8e-5 ±\pm 5.1e-53.5e-5 ±\pm 1.3e-50.092 ±\pm 00.223 ±\pm 00.115 ±\pm 0
Ext.5.88 ±\pm 08.00 ±\pm 05.93 ±\pm 00.087 ±\pm 012.94 ±\pm 36.217.96 ±\pm 0.8812.7e-5 ±\pm 1.5e-68.31 ±\pm 0.3180.824 ±\pm 2.310.090 ±\pm 0.0656.41 ±\pm 08.00 ±\pm 05.93 ±\pm 0
E3Int.1.09 ±\pm 00.216 ±\pm 00.020 ±\pm 03.3e-3 ±\pm 2.3e-49.9e-4 ±\pm 1.1e-30.164 ±\pm 0.0902.6e-5 ±\pm 5.7e-60.214 ±\pm 5.9e-42.3e-3 ±\pm 1.4e-30.022 ±\pm 0.0650.021 ±\pm 00.216 ±\pm 00.020 ±\pm 0
Ext.8.28 ±\pm 08.06 ±\pm 04.44 ±\pm 01.69 ±\pm 0.1711.1e4 ±\pm 1.8e48.06 ±\pm 00.020 ±\pm 7.7e-38.08 ±\pm 8.5e-34.6e6 ±\pm 1.2e73.3e4 ±\pm 9.1e44.42 ±\pm 08.06 ±\pm 04.44 ±\pm 0
Appendix
TABLE F.3 : Easy Tier — MSE Performance
IDSetAIFeynmanBSRDSRE2EGPGOMEAGplearnITEANeSymReSOperonPySRTaylorGPTPSRUDSR
M1Int.1.02 ±\pm 0177.3 ±\pm 325.10.381 ±\pm 00.181 ±\pm 0.0760.355 ±\pm 0.240—1.18 ±\pm 0.3291.07 ±\pm 1.4e-30.382 ±\pm 0.1310.268 ±\pm 0.087—1.07 ±\pm 00.381 ±\pm 0
Ext.0.351 ±\pm 0380.6 ±\pm 770.20.321 ±\pm 00.428 ±\pm 0.2650.317 ±\pm 0.0120.395 ±\pm 0.075360.8 ±\pm 1025.70.365 ±\pm 0.0420.370 ±\pm 0.0440.286 ±\pm 0.094—0.351 ±\pm 00.321 ±\pm 0
M2Int.9.2e-4 ±\pm 00.916 ±\pm 0.1190.148 ±\pm 08.8e-4 ±\pm 1.4e-47.7e-4 ±\pm 2.8e-40.281 ±\pm 0.3554.1e-4 ±\pm 2.2e-51.11 ±\pm 0.4048.2e-4 ±\pm 1.1e-42.7e-4 ±\pm 8.8e-5—0.161 ±\pm 00.148 ±\pm 0
Ext.3.7e-5 ±\pm 01.96 ±\pm 0.5720.588 ±\pm 06.7e-4 ±\pm 4.7e-42.5e-5 ±\pm 2.3e-51.03 ±\pm 1.151.3e-4 ±\pm 1.7e-42.72 ±\pm 0.1712.8e-5 ±\pm 2.1e-53.0e-5 ±\pm 1.1e-5—0.254 ±\pm 00.588 ±\pm 0
M3Int.1.15 ±\pm 02.1e4 ±\pm 4.7e40.101 ±\pm 00.024 ±\pm 0.0250.037 ±\pm 0.0350.079 ±\pm 7.2e-57.4e-5 ±\pm 1.9e-51.11 ±\pm 1.7e-30.094 ±\pm 0.0705.0e-4 ±\pm 2.1e-41.11 ±\pm 00.923 ±\pm 00.101 ±\pm 0
Ext.0.024 ±\pm 016.77 ±\pm 33.933.9e-3 ±\pm 03.4e-3 ±\pm 7.7e-37.6e-3 ±\pm 5.2e-32.9e-3 ±\pm 2.7e-44.9e-4 ±\pm 1.4e-40.013 ±\pm 6.4e-40.013 ±\pm 7.7e-35.3e-3 ±\pm 1.1e-30.013 ±\pm 01.8e-3 ±\pm 03.9e-3 ±\pm 0
Appendix
TABLE F.4 : Medium Tier — MSE Performance
IDSetAIFeynmanBSRDSRE2EGPGOMEAGplearnITEANeSymReSOperonPySRTaylorGPTPSRUDSR
H1Int.1.63 ±\pm 080.60 ±\pm 183.10.308 ±\pm 00.089 ±\pm 0.0240.024 ±\pm 0.0290.147 ±\pm 0.0600.064 ±\pm 1.5e-30.897 ±\pm 0.0160.065 ±\pm 0.0250.100 ±\pm 0.270—0.910 ±\pm 00.308 ±\pm 0
Ext.4.14 ±\pm 015.47 ±\pm 27.122.45 ±\pm 01.13 ±\pm 0.5760.924 ±\pm 0.0811.28 ±\pm 0.80824.42 ±\pm 19.542.13 ±\pm 0.1200.934 ±\pm 0.0320.981 ±\pm 0.284—1.81 ±\pm 02.45 ±\pm 0
H2Int.2.02 ±\pm 01.3e4 ±\pm 2.9e40.217 ±\pm 00.171 ±\pm 0.2150.014 ±\pm 0.0150.254 ±\pm 0.1815.1e-3 ±\pm 1.1e-40.448 ±\pm 2.9e-30.016 ±\pm 9.2e-30.056 ±\pm 0.1470.389 ±\pm 01.85 ±\pm 00.217 ±\pm 0
Ext.6.44 ±\pm 01.3e5 ±\pm 2.8e52.86 ±\pm 01.83 ±\pm 2.410.235 ±\pm 0.2465.31 ±\pm 1.4723.70 ±\pm 42.296.55 ±\pm 0.0401.06 ±\pm 2.680.841 ±\pm 1.575.86 ±\pm 018.42 ±\pm 02.86 ±\pm 0
H3Int.0.451 ±\pm 09.0e+12 ±\pm 2.2e+130.505 ±\pm 00.771 ±\pm 0.3040.104 ±\pm 0.0250.599 ±\pm 0.3330.045 ±\pm 5.6e-31.06 ±\pm 7.5e-40.206 ±\pm 0.0530.094 ±\pm 9.8e-3—1.06 ±\pm 00.505 ±\pm 0
Ext.0.015 ±\pm 04.2e+15 ±\pm 1.0e+160.644 ±\pm 00.456 ±\pm 0.0900.022 ±\pm 0.0200.298 ±\pm 0.11129.23 ±\pm 46.720.547 ±\pm 0.0160.056 ±\pm 0.0750.017 ±\pm 0.012—0.441 ±\pm 00.644 ±\pm 0
Appendix
TABLE F.5 : Hard Tier — MSE Performance
IDSetAIFeynmanBSRDSRE2EGPGOMEAGplearnITEANeSymReSOperonPySRTaylorGPTPSRUDSR
E1Int.-1.07 ±\pm 0-0.021 ±\pm 00.892 ±\pm 01.000 ±\pm 01.000 ±\pm 2.3e-70.617 ±\pm 0.4511.000 ±\pm 9.2e-9-2.4e-3 ±\pm 0.0120.999 ±\pm 1.1e-31.000 ±\pm 3.4e-12-0.021 ±\pm 0-0.021 ±\pm 00.892 ±\pm 0
Ext.-1.09 ±\pm 0-1.41 ±\pm 00.388 ±\pm 01.000 ±\pm 01.000 ±\pm 7.6e-7-0.263 ±\pm 0.8251.000 ±\pm 1.1e-6-1.74 ±\pm 0.0240.997 ±\pm 4.4e-31.000 ±\pm 1.5e-12-1.41 ±\pm 0-1.41 ±\pm 00.388 ±\pm 0
E2Int.-6.15 ±\pm 0-0.173 ±\pm 00.396 ±\pm 00.998 ±\pm 00.883 ±\pm 0.3520.134 ±\pm 0.3161.000 ±\pm 3.9e-10-0.088 ±\pm 0.0851.000 ±\pm 2.7e-41.000 ±\pm 6.6e-50.518 ±\pm 0-0.173 ±\pm 00.396 ±\pm 0
Ext.-0.100 ±\pm 0-0.496 ±\pm 0-0.109 ±\pm 00.984 ±\pm 0-1.42 ±\pm 6.78-0.490 ±\pm 0.1651.000 ±\pm 2.9e-7-0.556 ±\pm 0.0590.846 ±\pm 0.4330.983 ±\pm 0.012-0.200 ±\pm 0-0.496 ±\pm 0-0.109 ±\pm 0
E3Int.-4.09 ±\pm 0-6.2e-3 ±\pm 00.905 ±\pm 00.985 ±\pm 1.1e-30.995 ±\pm 5.3e-30.237 ±\pm 0.4211.000 ±\pm 2.7e-5-4.2e-4 ±\pm 2.7e-30.989 ±\pm 6.7e-30.897 ±\pm 0.3010.903 ±\pm 0-6.2e-3 ±\pm 00.905 ±\pm 0
Ext.-0.041 ±\pm 0-0.014 ±\pm 00.442 ±\pm 00.787 ±\pm 0.021-1442.9 ±\pm 2247.9-0.014 ±\pm 00.997 ±\pm 9.7e-4-0.016 ±\pm 1.1e-3-5.8e5 ±\pm 1.5e6-4093.2 ±\pm 1.1e40.444 ±\pm 0-0.014 ±\pm 00.442 ±\pm 0
Appendix
TABLE F.6 : Easy Tier — R2R^{2} Performance
IDSetAIFeynmanBSRDSRE2EGPGOMEAGplearnITEANeSymReSOperonPySRTaylorGPTPSRUDSR
M1Int.0.048 ±\pm 0-164.4 ±\pm 303.30.645 ±\pm 00.831 ±\pm 0.0710.669 ±\pm 0.224—-0.103 ±\pm 0.307-1.4e-3 ±\pm 1.3e-30.644 ±\pm 0.1230.750 ±\pm 0.081—-2.6e-4 ±\pm 00.645 ±\pm 0
Ext.-0.070 ±\pm 0-1159.8 ±\pm 2349.30.022 ±\pm 0-0.307 ±\pm 0.8090.035 ±\pm 0.037-0.206 ±\pm 0.230-1099.6 ±\pm 3128.8-0.114 ±\pm 0.128-0.128 ±\pm 0.1350.128 ±\pm 0.287—-0.070 ±\pm 00.022 ±\pm 0
M2Int.0.999 ±\pm 0-0.084 ±\pm 0.1410.824 ±\pm 00.999 ±\pm 1.7e-40.999 ±\pm 3.3e-40.667 ±\pm 0.4201.000 ±\pm 2.6e-5-0.312 ±\pm 0.4780.999 ±\pm 1.3e-41.000 ±\pm 1.0e-4—0.809 ±\pm 00.824 ±\pm 0
Ext.1.000 ±\pm 0-4.23 ±\pm 1.53-0.572 ±\pm 00.998 ±\pm 1.3e-31.000 ±\pm 6.2e-5-1.76 ±\pm 3.071.000 ±\pm 4.6e-4-6.28 ±\pm 0.4571.000 ±\pm 5.6e-51.000 ±\pm 3.0e-5—0.320 ±\pm 0-0.572 ±\pm 0
M3Int.-0.032 ±\pm 0-1.9e4 ±\pm 4.3e40.909 ±\pm 00.978 ±\pm 0.0230.967 ±\pm 0.0320.929 ±\pm 6.5e-51.000 ±\pm 1.7e-5-4.8e-4 ±\pm 1.5e-30.915 ±\pm 0.0631.000 ±\pm 1.9e-4-3.8e-5 ±\pm 00.168 ±\pm 00.909 ±\pm 0
Ext.-1.65 ±\pm 0-1833.9 ±\pm 3711.70.577 ±\pm 00.630 ±\pm 0.8410.172 ±\pm 0.5650.686 ±\pm 0.0290.946 ±\pm 0.015-0.424 ±\pm 0.070-0.395 ±\pm 0.8450.418 ±\pm 0.121-0.378 ±\pm 00.805 ±\pm 00.577 ±\pm 0
Appendix
TABLE F.7 : Medium Tier — R2R^{2} Performance
IDSetAIFeynmanBSRDSRE2EGPGOMEAGplearnITEANeSymReSOperonPySRTaylorGPTPSRUDSR
H1Int.-0.821 ±\pm 0-89.03 ±\pm 204.60.656 ±\pm 00.900 ±\pm 0.0270.973 ±\pm 0.0320.836 ±\pm 0.0670.929 ±\pm 1.7e-3-2.0e-3 ±\pm 0.0180.927 ±\pm 0.0280.889 ±\pm 0.302—-0.016 ±\pm 00.656 ±\pm 0
Ext.-5.45 ±\pm 0-23.13 ±\pm 42.29-2.82 ±\pm 0-0.762 ±\pm 0.897-0.442 ±\pm 0.126-1.00 ±\pm 1.26-37.08 ±\pm 30.47-2.32 ±\pm 0.188-0.456 ±\pm 0.049-0.530 ±\pm 0.443—-1.83 ±\pm 0-2.82 ±\pm 0
H2Int.-3.55 ±\pm 0-3.0e4 ±\pm 6.6e40.511 ±\pm 00.616 ±\pm 0.4830.969 ±\pm 0.0340.428 ±\pm 0.4070.988 ±\pm 2.5e-4-8.0e-3 ±\pm 6.5e-30.965 ±\pm 0.0210.873 ±\pm 0.3320.126 ±\pm 0-3.17 ±\pm 00.511 ±\pm 0
Ext.-4.77 ±\pm 0-1.1e5 ±\pm 2.5e5-1.56 ±\pm 0-0.638 ±\pm 2.160.789 ±\pm 0.221-3.76 ±\pm 1.32-20.24 ±\pm 37.90-4.87 ±\pm 0.0350.054 ±\pm 2.400.246 ±\pm 1.41-4.25 ±\pm 0-15.51 ±\pm 0-1.56 ±\pm 0
H3Int.0.574 ±\pm 0-8.5e+12 ±\pm 2.1e+130.522 ±\pm 00.270 ±\pm 0.2870.902 ±\pm 0.0240.433 ±\pm 0.3150.957 ±\pm 5.3e-3-9.3e-4 ±\pm 7.1e-40.805 ±\pm 0.0500.911 ±\pm 9.3e-3—-5.1e-3 ±\pm 00.522 ±\pm 0
Ext.-1.13 ±\pm 0-5.8e+17 ±\pm 1.4e+18-88.33 ±\pm 0-62.28 ±\pm 12.43-1.98 ±\pm 2.74-40.39 ±\pm 15.41-4054.1 ±\pm 6482.3-74.88 ±\pm 2.25-6.83 ±\pm 10.42-1.34 ±\pm 1.73—-60.24 ±\pm 0-88.33 ±\pm 0
Appendix
TABLE F.8 : Hard Tier — R2R^{2} Performance
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