frog.py

import numpy as np
import matplotlib.pyplot as plt


randomGenerator = np.random.default_rng()  # Require numpy v1.17+


ite = 1000  # Number of loops for the average
maxDistance = 1000


totalDist = np.arange(1, maxDistance)

randomLeap = np.empty_like(totalDist)
totalLeaps = np.zeros_like(totalDist)

for i in range(ite):

    print(f" - Iteration {i+1}/{ite}", end="\r")

    distLeft = totalDist.copy()

    leaps = np.zeros_like(totalDist)

    stillLeaping = np.ones_like(totalDist, dtype=bool)

    while np.any(stillLeaping):

        randomLeap[stillLeaping] = randomGenerator.integers(
            1, distLeft[stillLeaping], endpoint=True)

        np.subtract(distLeft, randomLeap, where=stillLeaping, out=distLeft)

        leaps[stillLeaping] += 1

        stillLeaping[distLeft == 0] = False  # Arrived at the end

    np.add(totalLeaps, leaps, out=totalLeaps)

print()

averageLeaps = totalLeaps/ite

A, B = np.polyfit(np.log(totalDist), averageLeaps, 1)

print(f"Average number of leaps ≈ {A:.5}*ln(Total distance)+{B:.5}")

plt.plot(totalDist, A*np.log(totalDist)+B, linewidth=2, label=f"y≈{A:.5}*ln(x)+{B:.5}")

plt.scatter(totalDist, averageLeaps, s=2, marker="*", color="green")

plt.grid(True)

# plt.xscale("symlog")

plt.xlabel("Total distance")
plt.ylabel("Leaps")

plt.legend()

plt.title("Average number of leaps")

plt.show()