Worked example

50 readings of an exponential growth, 1.5·e^0.4x plus noise, with one reading shifted by +1.0 — the kind of typo a spreadsheet hides. This is the unedited result.

Run it with your own data

The data as pasted (2 columns, no header)

0.000000 1.460547
0.102041 1.532290
0.204082 1.635777
0.306122 1.688375
0.408163 1.757118
0.510204 1.836631
0.612245 1.923800
0.714286 1.997967
0.816327 2.102720
0.918367 2.139591
1.020408 2.264726
1.122449 2.333984
1.224490 2.421820
1.326531 2.526846
1.428571 2.637451
1.530612 2.784890
1.632653 2.883479
1.734694 3.010339
1.836735 3.122851
1.938776 3.299406
2.040816 4.351414
2.142857 3.516691
2.244898 3.666332
2.346939 3.839499
2.448980 4.003699
2.551020 4.124989
2.653061 4.364569
2.755102 4.517771
2.857143 4.711867
2.959184 4.904914
3.061224 5.107934
3.163265 5.312488
3.265306 5.541624
3.367347 5.827899
3.469388 6.000055
3.571429 6.263820
3.673469 6.527235
3.775510 6.804391
3.877551 7.074070
3.979592 7.359835
4.081633 7.692664
4.183673 7.985576
4.285714 8.372111
4.387755 8.667919
4.489796 9.033332
4.591837 9.428727
4.693878 9.797121
4.795918 10.216069
4.897959 10.637018
5.000000 11.078941

Equation found

It predicts points it was not fitted to with an error of 0.0202 (cross-validated) using 2 parameters; the best polynomial needs 7 terms for 0.0193.

Why this one: equal held-out error within 5 %, fewer parameters (2 vs 7).

1 measurement did not fit and was left out of the final fit — see the rows to check.

The equation

\hat{y} = 1.4991 \cdot e^{0.40025 \cdot x}

Held-out error (cross-validated RMSE)0.02016 ± 0.00379
R²0.99995
Parameters2
Rows used49

1.499067485802664 * exp(0.40025242348258466 * x)

Every digit the engine computed. ln is the natural logarithm.

1.499067485802664 * np.exp(0.40025242348258466 * x)

Uses NumPy (import numpy as np); works on single values and on arrays.

=1.499067485802664 * EXP(0.40025242348258466 * x)

Replace each name with a cell reference, e.g. x → A2.

1.499067485802664 * exp(0.40025242348258466 * x)

Needs #include <math.h>; variables and result are double.

Also safe in 32-bit floating point (single-precision firmware, many PLCs): on your data the error stays below 0.0001 % of the measured range.

Parameters in normalised units, with standard errors

The engine fits the model in z = (x − mean) / std for each input and predicts y / std_y; the formula above is the same model rewritten in your units.

Parameter	Value	Standard error
p1	1.4759	0.00120907
p7	0.59478	0.000699444

Normalised form: (1.4759 * exp((0.59478 * z_x)))

Fit

Rows to check

Line numbers refer to the text you pasted or uploaded. t is the externally studentised residual from the screen; a row is flagged when |t| exceeds 3.54 (Bonferroni-corrected, 5 % family-wise).

Line	x	y	Model	t	Verdict
21	2.04082	4.35141	3.3929	49.5	Left out of the final fit

Alternatives

Ranked by cross-validated error with a parsimony rule: equations within one standard error of the best count as equally good and the simpler one ranks first. Several near-ties usually mean the data cannot tell those forms apart — choose by domain knowledge.

#	Equation (4 digits)	Parameters	Held-out error	R²	AICc
1	`1.499 * exp(0.4003 * x)`	2	0.02016 ± 0.00379	0.99995	-485.1
2	`2.773 * (267.7 / (x + 266.1))^(-107.8)`	3	0.01949 ± 0.00369	0.99996	-487.8
3	`0.06489 * (x + 22.95) * exp(0.362 * x)`	3	0.0195 ± 0.00369	0.99996	-487.8
4	`0.6232 * (ln(x + 16.2) - 0.3961) * exp(0.3801 * x)`	3	0.01951 ± 0.0037	0.99996	-487.9
5	`1.066 * exp(0.3925 * x) * atan(0.6729 * x + 5.632)`	3	0.01953 ± 0.00371	0.99996	-487.9
6	`1.493 * exp(0.4027 * x) * exp(-0.0003418 * exp(0.6729 * x))`	3	0.01949 ± 0.00362	0.99996	-487.2
7	`-5.883 * exp(0.6729 * x) * atan(-0.2581 * exp(-0.2752 * x))`	3	0.01951 ± 0.00373	0.99996	-488

Polynomial screen

#	Variables	Degree	Terms	Leave-one-out error	R²
1	x	6	7	0.01934	0.99996
2	x	5	6	0.01936	0.99996
3	x	7	8	0.01944	0.99996
4	x	4	5	0.01993	0.99996
5	x	3	4	0.02346	0.99994

Search details

Structures screened	167,542
Search strategy	exhaustive, simplest first
Operators	+ * / ^ exp log sin tan asin atan
Time budget used up	yes — a longer search may find more
Polynomial candidates	7
Computation time	94 s

Notes from the search

Screening stopped at the budget (167,542 structures, 85.79 % of the space, simplest structures first); a longer search time or fewer operators gives a fuller search.