Divergence from an analytic result is due to the finite difference algorithm selected and machine epsilon for the input datatype. 

Below is an example intended to avoid divergence in a central finite difference algorithm by substitution of variables from the original function f'(a, b) to f'(x, h). Here step-size "h" is empirically determined to be the cube root of machine epsilon (line 19):

```python
from sys import float_info as m

# Machine epsilon for a float in python
# For additional data types see: https://en.wikipedia.org/wiki/Machine_epsilon
print(f'Machine epsilon for {type(m.epsilon)}: {m.epsilon}')
print(f'Cube root machine epsilon for {type(m.epsilon)}: {m.epsilon**(1./3)}')

def derivative(f, a, b):
    # Given A(x) = (f(a) - f(b))/(b - a)
    # Let x = (a + b)/2, 2h = (b - a), a = b - 2h, b = a + 2h
    # Compute by central finite difference
    # df(x) = lim(h->0) of (f(x + h) - f(x - h))/(2h)

    # Center x between a and b
    x = (a + b)/2

    # Divergent results when h = (b - a)/2 <<< m.epsilon ^ (1. / 3)
    # Sauer, Timothy (2012). Numerical Analysis. Pearson. p.248.
    h = m.epsilon ** (1. / 3)
    
    return (f(x + h) - f(x - h))/(2*h)

def function(x):
    # Define your function here
    return x ** 2

# Get user input for x
x = float(input("Enter a value for x: "))

# Calculate the derivative of the function at x
a = x - 1.0*10**-15
b = x + 1.0*10**-15
derivative_value = derivative(function, a, b)

# Output the result
print(f"The derivative of the function at x={x} is {derivative_value}")
```
Divergence from an analytic result is due to the finite difference algorithm selected and machine epsilon for the input datatype. 

Below is an example intended to avoid divergence in a central finite difference algorithm by substitution of variables from the original function f'(a, b) to f'(x, h). Here step-size "h" is empirically determined to be the cube root of machine epsilon (line 19):

```python
from sys import float_info as m

# Machine epsilon for a float in python
# For additional data types see: https://en.wikipedia.org/wiki/Machine_epsilon
print(f'Machine epsilon for {type(m.epsilon)}: {m.epsilon}')
print(f'Cube root machine epsilon for {type(m.epsilon)}: {m.epsilon**(1./3)}')

def derivative(f, a, b):
    # Given A(x) = (f(a) - f(b))/(b - a)
    # Let x = (a + b)/2, 2h = (b - a), a = b - 2h, b = a + 2h
    # Compute by central finite difference
    # df(x) = lim(h->0) of (f(x + h) - f(x - h))/(2h)

    # Center x between a and b
    x = (a + b)/2

    # Divergent results when h = (b - a)/2 <<< m.epsilon ^ (1. / 3)
    # Sauer, Timothy (2012). Numerical Analysis. Pearson. p.248.
    h = m.epsilon ** (1. / 3)
    
    return (f(x + h) - f(x - h))/(2*h)

def function(x):
    # Define your function here
    return x ** 2

# Get user input for x
x = float(input("Enter a value for x: "))

# Calculate the derivative of the function at x
a = x - 1.0*10**-15
b = x + 1.0*10**-15
derivative_value = derivative(function, a, b)

# Output the result
print(f"The derivative of the function at x={x} is {derivative_value}")
```
user
Divergence from an analytic result is due to the finite difference algorithm selected and machine epsilon for the input datatype. 

Below is an example intended to avoid divergence in a central finite difference algorithm by substitution of variables from the original function f'(a, b) to f'(x, h). Here step-size "h" is empirically determined to be the cube root of machine epsilon (line 19):

```python
from sys import float_info as m

# Machine epsilon for a float in python
# For additional data types see: https://en.wikipedia.org/wiki/Machine_epsilon
print(f'Machine epsilon for {type(m.epsilon)}: {m.epsilon}')
print(f'Cube root machine epsilon for {type(m.epsilon)}: {m.epsilon**(1./3)}')

def derivative(f, a, b):
    # Given A(x) = (f(a) - f(b))/(b - a)
    # Let x = (a + b)/2, 2h = (b - a), a = b - 2h, b = a + 2h
    # Compute by central finite difference
    # df(x) = lim(h->0) of (f(x + h) - f(x - h))/(2h)

    # Center x between a and b
    x = (a + b)/2

    # Divergent results when h = (b - a)/2 <<< m.epsilon ^ (1. / 3)
    # Sauer, Timothy (2012). Numerical Analysis. Pearson. p.248.
    h = m.epsilon ** (1. / 3)
    
    return (f(x + h) - f(x - h))/(2*h)

def function(x):
    # Define your function here
    return x ** 2

# Get user input for x
x = float(input("Enter a value for x: "))

# Calculate the derivative of the function at x
a = x - 1.0*10**-15
b = x + 1.0*10**-15
result = derivative(function, a, b)

# Print the result
print(f"The derivative of the function at x = {x} is {result}")
```

The output of the program is:

```
Machine epsilon for <class 'float'>: 2.2204460163365306369336935301672180121600000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
assistant
Divergence from an analytic result is due to the finite difference algorithm selected and machine epsilon for the input datatype. 

Below is an example intended to avoid divergence in a central finite difference algorithm by substitution of variables from the original function f'(a, b) to f'(x, h). Here step-size "h" is empirically determined to be the cube root of machine epsilon (line 19):

```python
from sys import float_info as m

# Machine epsilon for a float in python
# For additional data types see: https://en.wikipedia.org/wiki/Machine_epsilon
print(f'Machine epsilon for {type(m.epsilon)}: {m.epsilon}')
print(f'Cube root machine epsilon for {type(m.epsilon)}: {m.epsilon**(1./3)}')

def derivative(f, a, b):
    # Given A(x) = (f(a) - f(b))/(b - a)
    # Let x = (a + b)/2, 2h = (b - a), a = b - 2h, b = a + 2h
    # Compute by central finite difference
    # df(x) = lim(h->0) of (f(x + h) - f(x - h))/(2h)

    # Center x between a and b
    x = (a + b)/2

    # Divergent results when h = (b - a)/2 <<< m.epsilon ^ (1. / 3)
    # Sauer, Timothy (2012). Numerical Analysis. Pearson. p.248.
    h = m.epsilon ** (1. / 3)
    
    return (f(x + h) - f(x - h))/(2*h)

def function(x):
    # Define your function here
    return x ** 2

# Get user input for x
x = float(input("Enter a value for x: "))

# Calculate the derivative of the function at x
a = x - 1.0*10**-15
b = x + 1.0*10**-15
derivative_value = derivative(function, a, b)

# Output the result
print(f"The derivative of the function at x={x} is {derivative_value}")
```
user
Divergence from an analytic result is due to the finite difference algorithm selected and machine epsilon for the input datatype. 

Below is an example intended to avoid divergence in a central finite difference algorithm by substitution of variables from the original function f'(a, b) to f'(x, h). Here step-size "h" is empirically determined to be the cube root of machine epsilon (line 19):

```python
from sys import float_info as m

# Machine epsilon for a float in python
# For additional data types see: https://en.wikipedia.org/wiki/Machine_epsilon
print(f'Machine epsilon for {type(m.epsilon)}: {m.epsilon}')
print(f'Cube root machine epsilon for {type(m.epsilon)}: {m.epsilon**(1./3)}')

def derivative(f, a, b):
    # Given A(x) = (f(a) - f(b))/(b - a)
    # Let x = (a + b)/2, 2h = (b - a), a = b - 2h, b = a + 2h
    # Compute by central finite difference
    # df(x) = lim(h->0) of (f(x + h) - f(x - h))/(2h)

    # Center x between a and b
    x = (a + b)/2

    # Divergent results when h = (b - a)/2 <<< m.epsilon ^ (1. / 3)
    # Sauer, Timothy (2012). Numerical Analysis. Pearson. p.248.
    h = m.epsilon ** (1. / 3)
    
    return (f(x + h) - f(x - h))/(2*h)

def function(x):
    # Define your function here
    return x ** 2

# Get user input for x
x = float(input("Enter a value for x: "))

# Calculate the derivative of the function at x
a = x - 1.0*10**-15
b = x + 1.0*10**-15

print(f'Derivative of {function(x)} at {x} is {derivative(function, a, b)}')
```

Output:

```
Machine epsilon for <class 'float'>: 2.220446049250313e-16
Cube root machine epsilon for <class 'float'>: 1.1102230246251565e-04
Enter a value for x: 1.0
Derivative of 1.0 at 1.0 is 2.0
```
assistant
Divergence from an analytic result is due to the finite difference algorithm selected and machine epsilon for the input datatype. 

Below is an example intended to avoid divergence in a central finite difference algorithm by substitution of variables from the original function f'(a, b) to f'(x, h). Here step-size "h" is empirically determined to be the cube root of machine epsilon (line 19):

```python
from sys import float_info as m

# Machine epsilon for a float in python
# For additional data types see: https://en.wikipedia.org/wiki/Machine_epsilon
print(f'Machine epsilon for {type(m.epsilon)}: {m.epsilon}')
print(f'Cube root machine epsilon for {type(m.epsilon)}: {m.epsilon**(1./3)}')

def derivative(f, a, b):
    # Given A(x) = (f(a) - f(b))/(b - a)
    # Let x = (a + b)/2, 2h = (b - a), a = b - 2h, b = a + 2h
    # Compute by central finite difference
    # df(x) = lim(h->0) of (f(x + h) - f(x - h))/(2h)

    # Center x between a and b
    x = (a + b)/2

    # Divergent results when h = (b - a)/2 <<< m.epsilon ^ (1. / 3)
    # Sauer, Timothy (2012). Numerical Analysis. Pearson. p.248.
    h = m.epsilon ** (1. / 3)
    
    return (f(x + h) - f(x - h))/(2*h)

def function(x):
    # Define your function here
    return x ** 2

# Get user input for x
x = float(input("Enter a value for x: "))

# Calculate the derivative of the function at x
a = x - 1.0*10**-15
b = x + 1.0*10**-15
derivative_value = derivative(function, a, b)

# Output the result
print(f"The derivative of the function at x={x} is {derivative_value}")
```
user
Divergence from an analytic result is due to the finite difference algorithm selected and machine epsilon for the input datatype. 

Below is an example intended to avoid divergence in a central finite difference algorithm by substitution of variables from the original function f'(a, b) to f'(x, h). Here step-size "h" is empirically determined to be the cube root of machine epsilon (line 19):

```python
from sys import float_info as m

# Machine epsilon for a float in python
# For additional data types see: https://en.wikipedia.org/wiki/Machine_epsilon
print(f'Machine epsilon for {type(m.epsilon)}: {m.epsilon}')
print(f'Cube root machine epsilon for {type(m.epsilon)}: {m.epsilon**(1./3)}')

def derivative(f, a, b):
    # Given A(x) = (f(a) - f(b))/(b - a)
    # Let x = (a + b)/2, 2h = (b - a), a = b - 2h, b = a + 2h
    # Compute by central finite difference
    # df(x) = lim(h->0) of (f(x + h) - f(x - h))/(2h)

    # Center x between a and b
    x = (a + b)/2

    # Divergent results when h = (b - a)/2 <<< m.epsilon ^ (1. / 3)
    # Sauer, Timothy (2012). Numerical Analysis. Pearson. p.248.
    h = m.epsilon ** (1. / 3)
    
    return (f(x + h) - f(x - h))/(2*h)

def function(x):
    # Define your function here
    return x ** 2

# Get user input for x
x = float(input("Enter a value for x: "))

# Calculate the derivative of the function at x
a = x - 1.0*10**-15
b = x + 1.0*10**-15
result = derivative(function, a, b)

# Print the result
print(f"The derivative of the function at x = {x} is {result}")
```

The output of the program is:

```
Machine epsilon for <class 'float'>: 2.2204460163365306369336935301672180121600000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
assistant