Add example plot to examples of docs

docs/examples.md

@@ -0,0 +1,109 @@
+# Examples
+
+### Preamble
+
+```python
+import numpy as np
+from pysr import *
+```
+
+We'll also set up some default options that will
+make these simple searches go faster (but are less optimal
+for more complex searches).
+
+```python
+kwargs = dict(populations=5, niterations=5, annealing=True)
+```
+
+1. Simple search
+
+Here's a simple example where we turn off multiprocessing,
+and find the expression `2 cos(x3) + x0^2 - 2`.
+
+```python
+X = 2 * np.random.randn(100, 5)
+y = 2 * np.cos(X[:, 3]) + X[:, 0] ** 2 - 2
+expressions = pysr(X, y, binary_operators=["+", "-", "*", "/"], **kwargs)
+print(best(expressions))
+```
+
+2. Custom operator
+
+Here, we define a custom operator and use it to find an expression:
+
+```python
+X = 2 * np.random.randn(100, 5)
+y = 1 / X[:, 0]
+expressions = pysr(
+    X,
+    y,
+    binary_operators=["plus", "mult"],
+    unary_operators=["inv(x) = 1/x"],
+    **kwargs
+)
+print(best(expressions))
+```
+
+3. Multiple outputs
+
+Here, we do the same thing, but with multiple expressions at once,
+each requiring a different feature.
+```python
+X = 2 * np.random.randn(100, 5)
+y = 1 / X[:, [0, 1, 2]]
+expressions = pysr(
+    X,
+    y,
+    binary_operators=["plus", "mult"],
+    unary_operators=["inv(x) = 1/x"],
+    **kwargs
+)
+```
+
+4. Plotting an expression
+
+Here, let's use the same equations, but get a format we can actually
+use and test. We can add this option after a search via the `get_hof`
+function:
+
+```python
+expressions = get_hof(extra_sympy_mappings={"inv": lambda x: 1/x})
+```
+If you look at the lists of expressions before and after, you will
+see that the sympy format now has replaced `inv` with `1/`.
+
+For now, let's consider the expressions for output 0:
+```python
+expressions = expressions[0]
+```
+This is a pandas table, which we can filter:
+```python
+best_expression = expressions.iloc[expressions.MSE.argmin()]
+```
+We can see the LaTeX version of this with:
+```python
+import sympy
+sympy.latex(best_expression.sympy_format)
+```
+
+We can access the numpy version with:
+```python
+f = best_expression.lambda_format
+print(f)
+```
+
+Which shows a PySR object on numpy code:
+```
+>> PySRFunction(X=>1/x0)
+```
+
+Let's plot this against the truth:
+```python
+from matplotlib import pyplot as plt
+plt.scatter(y[:, 0], f(X))
+plt.xlabel('Truth')
+plt.ylabel('Prediction')
+plt.show()
+```
+Which gives us:
+![](./images/example_plot.png)