# 2. Symbolic manipulation#

The symbolic math framework ought to feel familiar to users experienced with other symbolic tools, such as sympy. To begin with, we will declare some input variables and combine them into a larger expression:

```
from wrenfold import sym
# Create two symbolic variables and construct an expression.
x, y = sym.symbols('x, y')
g = sym.cos(x * y)
print(g)
```

```
cos(x*y)
```

```
# Visualize the expression tree:
print(g.expression_tree_str())
```

```
Function (cos):
└─ Multiplication:
├─ Variable (x, unknown)
└─ Variable (y, unknown)
```

wrenfold represents mathematical operations as an expression tree. As operations are composed, the tree grows in depth:

```
# Use `g` as part of a larger expression:
f = g + x**3 * y
f
```

```
x**3*y + cos(x*y)
```

```
# Visualize the expression tree:
print(f.expression_tree_str())
```

```
Addition:
├─ Multiplication:
│ ├─ Variable (y, unknown)
│ └─ Power:
│ ├─ Variable (x, unknown)
│ └─ Integer (3)
└─ Function (cos):
└─ Multiplication:
├─ Variable (x, unknown)
└─ Variable (y, unknown)
```

Symbolic expressions are **immutable**. They can be combined to form new expressions, or we can
subject them to analytical operations that create new expressions. For instance, we can easily
obtain the derivatives of `f`

:

```
# Compute the derivative of `f` wrt `x`:
df = f.diff(x)
df
```

```
3*x**2*y - y*sin(x*y)
```

```
# Compute the second derivative of `f` wrt `y`:
f.diff(y, 2)
```

```
-x**2*cos(x*y)
```

Or collect powers:

```
# Collect powers of `y` in our derivative expression, `df`:
df.collect(y)
```

```
y*(3*x**2 - sin(x*y))
```

Or substitute numerical constants and evaluate into a floating point value:

```
val = df.subs(x, sym.E).subs(y, sym.integer(1) / 3)
print(val)
```

```
E**2 - sin(E/3)/3
```

```
val.eval()
```

```
7.126689299943595
```

As expressions are composed, they are automatically converted to canonical form:

```
# Constants are folded and coefficients are combined in additions:
-1 + x + x + 5 # result: 4 + 2*x
# Constants are distributed into additions:
(7 * x + y ** 2) * sym.rational(3, 7) # result: 3*x + 3*y**2/7
# Common terms in multiplications are converted to powers:
(x * y * x * x) / y # result: x**3
# Some power expressions simplify automatically:
(1 / x) ** 2 # result: x**(-2)
sym.sqrt(x) ** 2 # result: x
```

```
z = sym.symbols('z', nonnegative=True)
((3 * z) ** 4) ** (sym.one / 4)
```

```
3*z
```

While wrenfold is not intended to be a full computer algebra system, it does support a variety of common functions and operations. Symbolic expressions can be converted to and from SymPy in order to perform more advanced manipulations.