[Python] SciPy fails to stably determine parameters given large data scale differences

We are using scipy.optimize.curve_fit to fit parameters X_eff and Y_eff for an Automated Market Maker (AMM) model: y_out = (Y_eff * x_in) / (X_eff + x_in)

We're observing extreme sensitivity in the fitted X_eff and Y_eff values when our input data points (x_in, y_out) have significantly different scales, or when there are minor variations in the larger-scale data points.

Here is the code that demonstrates that with static hardcoded values:

import numpy as np
from scipy.optimize import curve_fit

def swap_model(x_in, X_eff, Y_eff):
denominator = X_eff + x_in
if np.isscalar(denominator):
return (Y_eff * x_in) / (denominator if not np.isclose(denominator, 0) else 1e-9)
else:
denominator_copy = np.copy(denominator)
denominator_copy[np.isclose(denominator_copy, 0)] = 1e-9
return (Y_eff * x_in) / denominator_copy

bounds = ([1e-6, 1e-6], [np.inf, np.inf])

x_test_value = 6507.3125

print("--- Case 1: Poor Fit ---")
x_data_bad = np.array([1, 2, 39000])
y_data_bad = np.array([8423.724335, 16846.531751, 319765062.090722])
y_actual_for_bad_case_test = 54311422.15982

try:
popt_bad, _ = curve_fit(swap_model, x_data_bad, y_data_bad, bounds=bounds, maxfev=5000)
X_eff_bad, Y_eff_bad = popt_bad
print(f"Fitted (Bad): X_eff = {X_eff_bad:.4f}, Y_eff = {Y_eff_bad:.4f}")
y_pred_bad = swap_model(x_test_value, X_eff_bad, Y_eff_bad)
error_bad = abs(y_actual_for_bad_case_test - y_pred_bad) / y_actual_for_bad_case_test * 100
print(f"Predicted for {x_test_value:.4f}: {y_pred_bad:.4f}, Actual: {y_actual_for_bad_case_test:.4f}, Error: {error_bad:.2f}%")
except RuntimeError as e: print(f"RuntimeError: {e}")
except Exception as e: print(f"Other exception in bad fit: {e}")


print("\n--- Case 2: Better Fit ---")
x_data_good = np.array([1, 2, 32000])
y_data_good = np.array([8430.169107, 16859.464909, 264727159.854379])
y_actual_for_good_case_test = 54549399.392518

try:
popt_good, _ = curve_fit(swap_model, x_data_good, y_data_good, bounds=bounds, maxfev=5000)
X_eff_good, Y_eff_good = popt_good
print(f"Fitted (Good): X_eff = {X_eff_good:,.4f}, Y_eff = {Y_eff_good:,.4f}")
y_pred_good = swap_model(x_test_value, X_eff_good, Y_eff_good)
error_good = abs(y_actual_for_good_case_test - y_pred_good) / y_actual_for_good_case_test * 100
print(f"Predicted for {x_test_value:.4f}: {y_pred_good:.4f}, Actual: {y_actual_for_good_case_test:.4f}, Error: {error_good:.2f}%")
except RuntimeError as e: print(f"RuntimeError: {e}")
except Exception as e: print(f"Other exception in good fit: {e}")


The same code but with comments in pastebin: https://pastebin.com/V2BZPbJB

The graph here the one that works:

and here the one that does not:



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