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Following my earlier post on the terrible performances of ChatGPT for a grasshopping riddle, Bob Carpenter sent me his interaction with GPT4, using the same entry. Here is the python code outcome he obtained. The code is running  and the answer is correct.

BC: Thanks. Can you write a python program to simulate the final position of the grasshopper to within machine tolerance using single-precision floating point arithmetic?

GPT4: Sure, here’s a simple Python program that simulates the grasshopper’s journey using single-precision floating point arithmetic, specifically the float32 data type from the numpy library. It continues until the jump size is less than the machine epsilon for float32, which is the smallest representable positive number such that 1.0 + eps != 1.0.

import numpy as np
import random

# Initialize position and jump size
position = np.float32(0.0)
jump = np.float32(0.5)

# Get machine epsilon for float32
eps = np.finfo(np.float32).eps

# Perform jumps until jump size is less than machine epsilon
while jump > eps:
# Choose direction randomly
direction = random.choice([-1, 1])
# Update position
position += direction * jump
# Halve the jump size
jump /= 2

Remember that this program only simulates a single instance of the grasshopper’s journey. If you ran it multiple times, you would get different results each time due to the randomness in the choice of direction. However, on average, you should expect the final positions to be around 0, as per our mathematical analysis earlier. The absolute distance from the origin, however, will vary and, on average, should be close to 1/2.
print(“Final position:”, position)

BC: can you write a program to calculate the expected length of the final position from the origin?

import numpy as np
import random

# Define the number of simulations
num_simulations = 100000

# Initialize an array to store the final positions
final_positions = np.zeros(num_simulations, dtype=np.float32)

# Perform each simulation
for i in range(num_simulations):
# Initialize position and jump size
position = np.float32(0.0)
jump = np.float32(0.5)

# Get machine epsilon for float32
eps = np.finfo(np.float32).eps

# Perform jumps until jump size is less than machine epsilon
while jump > eps:
# Choose direction randomly
direction = random.choice([-1, 1])
# Update position
position += direction * jump
# Halve the jump size
jump /= 2

# Store the final position
final_positions[i] = position

# Calculate the expected length of the final position from the origin
expected_length = np.mean(np.abs(final_positions))

print(“Expected length of the final position from the origin:”, expected_length)