tensorflow/

expr.rs

//! This module builds computation graphs.
//!
//! This module is unfinished.
#![cfg(feature = "tensorflow_unstable")]

use super::Graph;
use super::Operation;
use super::Shape;
use super::Status;
use super::Tensor;
use super::TensorType;
use std::cmp::Eq;
use std::collections::HashMap;
use std::convert::From;
use std::fmt::Debug;
use std::fmt::Display;
use std::fmt::Error;
use std::fmt::Formatter;
use std::hash::Hash;
use std::hash::Hasher;
use std::marker::PhantomData;
use std::ops;
use std::rc::Rc;

/// Denotes operator precedence.
/// Used for displaying expressions as strings.
#[derive(Ord, PartialOrd, Eq, PartialEq, Debug, Copy, Clone)]
pub enum OpLevel {
    /// Assignment.
    Assign,

    /// Addition and subtraction.
    Add,

    /// Multiplication, division, and remainder.
    Mul,

    /// Unary operators like negation.
    Unary,

    /// Variables and constants.
    Atom,
}

////////////////////////

/// A operation in an expression tree, which is a thin wrapper around an ExprImpl.
///
/// This is separate from ExprImpl because we want expressions to be wrapped in an Rc,
/// and we can't directly implement std::ops::Add, etc., for Rc<E: ExprImpl<Tgt;>.
#[derive(Debug, Clone)]
pub struct Expr<T: TensorType> {
    expr: Rc<dyn ExprImpl<T>>,
}

impl<T: TensorType> Expr<T> {
    /// Wraps an ExprImpl.
    pub fn new<I>(expr: I) -> Expr<T>
    where
        I: ExprImpl<T> + 'static,
    {
        Expr {
            expr: Rc::new(expr),
        }
    }
}

impl<T: TensorType> ops::Deref for Expr<T> {
    type Target = dyn ExprImpl<T>;

    fn deref(&self) -> &Self::Target {
        self.expr.deref()
    }
}

impl<T: TensorType> Display for Expr<T> {
    fn fmt(&self, f: &mut Formatter) -> Result<(), Error> {
        Display::fmt(&self.expr, f)
    }
}

impl<T: TensorType> From<T> for Expr<T> {
    fn from(value: T) -> Self {
        Expr::new(value)
    }
}

////////////////////////

/// Enum of an expr's possible shape states
#[derive(Debug)]
pub enum ShapeHint<'a> {
    /// Unknown shape
    Unknown,

    /// Well defined shape that exactly matches contained value
    Exactly(&'a [u64]),
}

////////////////////////

/// Trait implemented by all expression types.
/// Most users will want to store an Expr instead.
pub trait ExprImpl<T: TensorType>: Display + Debug {
    /// Returns the precedence level for this operator.
    fn op_level(&self) -> OpLevel;

    /// Returns the child expressions.
    ///
    /// For example, the child expressions of `x + y` would be `x` and `y`.
    fn children(&self) -> Vec<Box<dyn AnyExpr>>; // TODO: return an iterator

    /// Creates an operation for the expression.
    ///
    /// The implementation must use the operations in the `children` parameter
    /// rather than creating child operations itself.
    fn create_operation(
        &self,
        graph: &mut Graph,
        children: &[Operation],
        id_gen: &mut dyn FnMut() -> String,
    ) -> Result<Operation, Status>;

    /// Returns the derivative of the expression with respect to the given variable.
    fn derivative_by_variable(&self, var: &str) -> Result<Expr<T>, Status>;

    /// Returns a hint about the expression's shape.
    fn shape_hint(&self) -> ShapeHint {
        ShapeHint::Unknown
    }
}

impl<T: TensorType> ExprImpl<T> for T {
    fn op_level(&self) -> OpLevel {
        OpLevel::Atom
    }

    fn children(&self) -> Vec<Box<dyn AnyExpr>> {
        vec![]
    }

    fn create_operation(
        &self,
        graph: &mut Graph,
        _children: &[Operation],
        id_gen: &mut dyn FnMut() -> String,
    ) -> Result<Operation, Status> {
        let mut nd = graph.new_operation("Const", &id_gen())?;
        nd.set_attr_type("dtype", T::data_type())?;
        let mut value = Tensor::new(&[1]);
        value[0] = self.clone();
        nd.set_attr_tensor("value", value)?;
        nd.finish()
    }

    fn derivative_by_variable(&self, _var: &str) -> Result<Expr<T>, Status> {
        Ok(Expr::from(T::zero()))
    }
}

////////////////////////

macro_rules! impl_bin_op {
  ($name:ident, $fn_name:ident, $op:expr, $op_level:ident, $assoc:expr,
      $tf_op:expr, $doc:expr, $($ximpl:tt)*) => {
    #[doc = $doc]
    #[derive(Debug)]
    pub struct $name<T: TensorType> {
      left: Expr<T>,
      right: Expr<T>,
    }

    impl<T: TensorType> ops::$name for Expr<T> {
      type Output = Expr<T>;

      fn $fn_name(self, rhs: Expr<T>) -> Expr<T> {
        Expr::new($name {
            left: self,
            right: rhs,
        })
      }
    }

    impl<T: TensorType> ops::$name<T> for Expr<T> {
      type Output = Expr<T>;

      fn $fn_name(self, rhs: T) -> Expr<T> {
        Expr::new($name {
            left: self,
            right: Expr::from(rhs),
        })
      }
    }

    impl<T: TensorType> Display for $name<T> {
      fn fmt(&self, f: &mut Formatter) -> Result<(), Error> {
        if self.left.op_level() < OpLevel::$op_level {
          write!(f, "({})", self.left)?;
        } else {
          write!(f, "{}", self.left)?;
        }
        write!(f, concat!(" ", $op, " "))?;
        let paren = if $assoc {
          self.right.op_level() < OpLevel::$op_level
        } else {
          self.right.op_level() <= OpLevel::$op_level
        };
        if paren {
          write!(f, "({})", self.right)
        } else {
          write!(f, "{}", self.right)
        }
      }
    }

    impl<T: TensorType> ExprImpl<T> for $name<T> {
      fn op_level(&self) -> OpLevel {
        OpLevel::$op_level
      }

      fn children(&self) -> Vec<Box<dyn AnyExpr>> {
        vec![Box::new(self.left.clone()), Box::new(self.right.clone())]
      }

      fn create_operation(&self, graph: &mut Graph, children: &[Operation],
          id_gen: &mut dyn FnMut() -> String) -> Result<Operation, Status> {
        let mut nd = graph.new_operation($tf_op, &id_gen())?;
        nd.add_input(children[0].clone());
        nd.add_input(children[1].clone());
        nd.finish()
      }

      $($ximpl)*
    }
  }
}

impl_bin_op!(
    Add,
    add,
    "+",
    Add,
    true,
    "Add",
    "Expression resulting from adding two subexpressions.",
    fn derivative_by_variable(&self, var: &str) -> Result<Expr<T>, Status> {
        Ok(self.left.derivative_by_variable(var)? + self.right.derivative_by_variable(var)?)
    }
);
impl_bin_op!(
    Sub,
    sub,
    "-",
    Add,
    false,
    "Sub",
    "Expression resulting from subtracting two subexpressions.",
    fn derivative_by_variable(&self, var: &str) -> Result<Expr<T>, Status> {
        Ok(self.left.derivative_by_variable(var)? - self.right.derivative_by_variable(var)?)
    }
);
impl_bin_op!(
    Mul,
    mul,
    "*",
    Mul,
    true,
    "Mul",
    "Expression resulting from multiplying two subexpressions.",
    fn derivative_by_variable(&self, var: &str) -> Result<Expr<T>, Status> {
        Ok(self.left.derivative_by_variable(var)? * self.right.clone()
            + self.left.clone() * self.right.derivative_by_variable(var)?)
    }
);
impl_bin_op!(
    Div,
    div,
    "/",
    Mul,
    false,
    "Div",
    "Expression resulting from dividing two subexpressions.",
    fn derivative_by_variable(&self, var: &str) -> Result<Expr<T>, Status> {
        let num = self.left.derivative_by_variable(var)? * self.right.clone()
            - self.left.clone() * self.right.derivative_by_variable(var)?;
        let denom = self.right.clone() * self.right.clone();
        Ok(num / denom)
    }
);
impl_bin_op!(
    Rem,
    rem,
    "%",
    Mul,
    false,
    "Mod",
    "Expression resulting from taking a modulus.",
    fn derivative_by_variable(&self, var: &str) -> Result<Expr<T>, Status> {
        Ok(self.left.derivative_by_variable(var)?
            - TruncateDiv::new_expr(self.left.clone(), self.right.clone())
                * self.right.derivative_by_variable(var)?)
    }
);

////////////////////////

/// Expression that assigns a value to a variable.
#[derive(Debug)]
pub struct TruncateDiv<T: TensorType> {
    left: Expr<T>,
    right: Expr<T>,
}

impl<T: TensorType> TruncateDiv<T> {
    fn new(left: Expr<T>, right: Expr<T>) -> Self {
        TruncateDiv { left, right }
    }

    /// Creates an expression that divides `left` by `right` and rounds toward zero.
    pub fn new_expr(left: Expr<T>, right: Expr<T>) -> Expr<T> {
        Expr::new(TruncateDiv::new(left, right))
    }
}

impl<T: TensorType> Display for TruncateDiv<T> {
    fn fmt(&self, f: &mut Formatter) -> Result<(), Error> {
        write!(f, "{} // {}", self.left, self.right)
    }
}

impl<T: TensorType> ExprImpl<T> for TruncateDiv<T> {
    fn op_level(&self) -> OpLevel {
        OpLevel::Mul
    }

    fn children(&self) -> Vec<Box<dyn AnyExpr>> {
        vec![Box::new(self.left.clone()), Box::new(self.right.clone())]
    }

    fn create_operation(
        &self,
        graph: &mut Graph,
        children: &[Operation],
        id_gen: &mut dyn FnMut() -> String,
    ) -> Result<Operation, Status> {
        let mut nd = graph.new_operation("TruncateDiv", &id_gen())?;
        nd.add_input(children[0].clone());
        nd.add_input(children[1].clone());
        nd.finish()
    }

    fn derivative_by_variable(&self, var: &str) -> Result<Expr<T>, Status> {
        // Mod(x, y) = x - TruncateDiv(x, y) * y
        // TruncateDiv(x, y) = (x - Mod(x, y)) / y
        // d/dt TruncateDiv(x, y) = (y * d/dt (x - Mod(x, y)) - (x - Mod(x, y)) dy/dt) / (y * y)
        let diff = self.left.clone() - self.left.clone() % self.right.clone();
        let term1 = self.right.clone() * diff.derivative_by_variable(var)?;
        let term2 = diff * self.right.derivative_by_variable(var)?;
        Ok((term1 - term2) / (self.right.clone() * self.right.clone()))
    }
}

////////////////////////

/// Expression resulting from negation of an expression.
#[derive(Debug)]
pub struct Neg<T: TensorType> {
    expr: Expr<T>,
}

impl<T: TensorType> ops::Neg for Expr<T> {
    type Output = Expr<T>;

    fn neg(self) -> Expr<T> {
        Expr::new(Neg { expr: self })
    }
}

impl<T: TensorType> Display for Neg<T> {
    fn fmt(&self, f: &mut Formatter) -> Result<(), Error> {
        write!(f, "-")?;
        if self.expr.op_level() <= OpLevel::Unary {
            write!(f, "({})", self.expr)
        } else {
            write!(f, "{}", self.expr)
        }
    }
}

impl<T: TensorType> ExprImpl<T> for Neg<T> {
    fn op_level(&self) -> OpLevel {
        OpLevel::Unary
    }

    fn children(&self) -> Vec<Box<dyn AnyExpr>> {
        vec![Box::new(self.expr.clone())]
    }

    fn create_operation(
        &self,
        graph: &mut Graph,
        children: &[Operation],
        id_gen: &mut dyn FnMut() -> String,
    ) -> Result<Operation, Status> {
        let mut nd = graph.new_operation("Neg", &id_gen())?;
        nd.add_input(children[0].clone());
        nd.finish()
    }

    fn derivative_by_variable(&self, var: &str) -> Result<Expr<T>, Status> {
        Ok(-self.expr.derivative_by_variable(var)?)
    }
}

////////////////////////

/// Expression for a variable.
#[derive(Debug)]
pub struct Variable<T: TensorType> {
    shape: Vec<u64>,
    name: String,
    phantom: PhantomData<T>,
}

impl<T: TensorType> Variable<T> {
    fn new(shape: &[u64], name: &str) -> Self {
        Variable {
            shape: Vec::from(shape),
            name: name.to_string(),
            phantom: PhantomData,
        }
    }

    /// Creates an `Expr` for a variable.
    pub fn new_expr(shape: &[u64], name: &str) -> Expr<T> {
        Expr::new(Variable::new(shape, name))
    }
}

impl<T: TensorType> Display for Variable<T> {
    fn fmt(&self, f: &mut Formatter) -> Result<(), Error> {
        write!(f, "{}", self.name)
    }
}

impl<T: TensorType> ExprImpl<T> for Variable<T> {
    fn op_level(&self) -> OpLevel {
        OpLevel::Atom
    }

    fn children(&self) -> Vec<Box<dyn AnyExpr>> {
        vec![]
    }

    fn create_operation(
        &self,
        graph: &mut Graph,
        _children: &[Operation],
        _id_gen: &mut dyn FnMut() -> String,
    ) -> Result<Operation, Status> {
        let mut nd = graph.new_operation("Variable", &self.name)?;
        let shape = self
            .shape
            .iter()
            .map(|dim_size| Some(*dim_size as i64))
            .collect();

        nd.set_attr_type("dtype", T::data_type()).unwrap();
        nd.set_attr_shape("shape", &Shape(Some(shape))).unwrap();
        nd.finish()
    }

    fn derivative_by_variable(&self, var: &str) -> Result<Expr<T>, Status> {
        Ok(if var == self.name {
            Expr::from(T::one())
        } else {
            Expr::from(T::zero())
        })
    }

    fn shape_hint(&self) -> ShapeHint {
        ShapeHint::Exactly(&self.shape)
    }
}

////////////////////////

/// Expression for a placeholder.
#[derive(Debug)]
pub struct Placeholder<T: TensorType> {
    shape: Vec<u64>,
    name: String,
    phantom: PhantomData<T>,
}

impl<T: TensorType> Placeholder<T> {
    fn new(shape: &[u64], name: &str) -> Self {
        Placeholder {
            shape: Vec::from(shape),
            name: name.to_string(),
            phantom: PhantomData,
        }
    }

    /// Creates an `Expr` for a placeholder.
    pub fn new_expr(shape: &[u64], name: &str) -> Expr<T> {
        Expr::new(Placeholder::new(shape, name))
    }
}

impl<T: TensorType> Display for Placeholder<T> {
    fn fmt(&self, f: &mut Formatter) -> Result<(), Error> {
        write!(f, "{}", self.name)
    }
}

impl<T: TensorType> ExprImpl<T> for Placeholder<T> {
    fn op_level(&self) -> OpLevel {
        OpLevel::Atom
    }

    fn children(&self) -> Vec<Box<dyn AnyExpr>> {
        vec![]
    }

    fn create_operation(
        &self,
        graph: &mut Graph,
        _children: &[Operation],
        _id_gen: &mut dyn FnMut() -> String,
    ) -> Result<Operation, Status> {
        let mut nd = graph.new_operation("Placeholder", &self.name)?;
        let shape = self
            .shape
            .iter()
            .map(|dim_size| Some(*dim_size as i64))
            .collect();

        nd.set_attr_type("dtype", T::data_type()).unwrap();
        nd.set_attr_shape("shape", &Shape(Some(shape))).unwrap();
        nd.finish()
    }

    fn derivative_by_variable(&self, _var: &str) -> Result<Expr<T>, Status> {
        Ok(Expr::from(T::zero()))
    }

    fn shape_hint(&self) -> ShapeHint {
        ShapeHint::Exactly(&self.shape)
    }
}

////////////////////////

/// Expression for a constant.
#[derive(Debug)]
pub struct Constant<T: TensorType> {
    tensor: Tensor<T>,
}

impl<T: TensorType> Constant<T> {
    /// Creates a constant with the given value.
    pub fn new(tensor: Tensor<T>) -> Self {
        Constant { tensor }
    }

    /// Creates a constant with the given value.
    pub fn new_expr(tensor: Tensor<T>) -> Expr<T> {
        Expr::new(Constant { tensor })
    }
}

impl<T: TensorType> Display for Constant<T> {
    fn fmt(&self, f: &mut Formatter) -> Result<(), Error> {
        write!(f, "{}", self.tensor)
    }
}

impl<T: TensorType> ExprImpl<T> for Constant<T> {
    fn op_level(&self) -> OpLevel {
        OpLevel::Atom
    }

    fn children(&self) -> Vec<Box<dyn AnyExpr>> {
        vec![]
    }

    fn create_operation(
        &self,
        graph: &mut Graph,
        _children: &[Operation],
        id_gen: &mut dyn FnMut() -> String,
    ) -> Result<Operation, Status> {
        let mut nd = graph.new_operation("Const", &id_gen())?;

        nd.set_attr_type("dtype", T::data_type())?;
        nd.set_attr_tensor("value", self.tensor.clone())?;
        nd.finish()
    }

    fn derivative_by_variable(&self, _var: &str) -> Result<Expr<T>, Status> {
        Ok(Expr::from(T::zero()))
    }
}

////////////////////////

/// Expression that assigns a value to a variable.
#[derive(Debug)]
pub struct Assign<T: TensorType> {
    variable: Expr<T>,
    value: Expr<T>,
}

impl<T: TensorType> Assign<T> {
    fn new(variable: Expr<T>, value: Expr<T>) -> Self {
        Assign { variable, value }
    }

    /// Creates an expression that assigns `value` to `variable`.
    pub fn new_expr(variable: Expr<T>, value: Expr<T>) -> Expr<T> {
        Expr::new(Assign::new(variable, value))
    }

    /// Creates an expression that takes values from `iterable` to fill `variable`.
    pub fn to(variable: Expr<T>, iterable: impl Iterator<Item = T>) -> crate::Result<Expr<T>> {
        let constant = if let ShapeHint::Exactly(shape) = variable.expr.shape_hint() {
            let values: Vec<_> = iterable
                .take(shape.iter().product::<u64>() as usize)
                .collect();

            Constant::new_expr(Tensor::new(shape).with_values(&values)?)
        } else {
            return Err(invalid_arg!(
                "Cannot assign to expression {} with unknown size!",
                variable
            ));
        };

        Ok(Assign::new_expr(variable, constant))
    }
}

impl<T: TensorType> Display for Assign<T> {
    fn fmt(&self, f: &mut Formatter) -> Result<(), Error> {
        write!(f, "{} = {}", self.variable, self.value)
    }
}

impl<T: TensorType> ExprImpl<T> for Assign<T> {
    fn op_level(&self) -> OpLevel {
        OpLevel::Assign
    }

    fn children(&self) -> Vec<Box<dyn AnyExpr>> {
        vec![
            Box::new(self.variable.clone()),
            Box::new(self.value.clone()),
        ]
    }

    fn create_operation(
        &self,
        graph: &mut Graph,
        children: &[Operation],
        id_gen: &mut dyn FnMut() -> String,
    ) -> Result<Operation, Status> {
        let mut nd = graph.new_operation("Assign", &id_gen())?;
        nd.add_input(children[0].clone());
        nd.add_input(children[1].clone());
        nd.finish()
    }

    fn derivative_by_variable(&self, _var: &str) -> Result<Expr<T>, Status> {
        Err(invalid_arg!("Cannot take the derivative of an assignment"))
    }
}

////////////////////////

// TODO: See if we can make this private.
/// An `AnyExpr` is just an `Expr<T>` for some unknown `T`.
/// Clients *should not* implement this.
pub trait AnyExpr: Debug {
    /// Returns a pointer usable as a map key which identifies this expression.
    fn key(&self) -> *const ();

    /// Returns the child expressions.
    ///
    /// For example, the child expressions of `x + y` would be `x` and `y`.
    fn children(&self) -> Vec<Box<dyn AnyExpr>>; // TODO: return an iterator

    /// Creates an operation for the expression.
    ///
    /// The implementation must use the operations in the `children` parameter
    /// rather than creating child operations itself.
    fn create_operation(
        &self,
        graph: &mut Graph,
        children: &[Operation],
        id_gen: &mut dyn FnMut() -> String,
    ) -> Result<Operation, Status>;

    /// Returns a boxed clone.
    ///
    /// This is used rather than the `Clone` trait because that would prevent
    /// `AnyExpr` values from being used as trait objects.
    fn clone_box(&self) -> Box<dyn AnyExpr>;
}

impl<T: TensorType> AnyExpr for Expr<T> {
    #[allow(trivial_casts)]
    fn key(&self) -> *const () {
        self.expr.as_ref() as *const dyn ExprImpl<T> as *const ()
    }

    fn children(&self) -> Vec<Box<dyn AnyExpr>> {
        self.expr.children()
    }

    fn create_operation(
        &self,
        graph: &mut Graph,
        children: &[Operation],
        id_gen: &mut dyn FnMut() -> String,
    ) -> Result<Operation, Status> {
        self.expr.create_operation(graph, children, id_gen)
    }

    fn clone_box(&self) -> Box<dyn AnyExpr> {
        Box::new(self.clone())
    }
}

#[derive(Debug)]
struct Key(Box<dyn AnyExpr>);

impl PartialEq for Key {
    fn eq(&self, other: &Key) -> bool {
        self.0.key() == other.0.key()
    }
}

impl Eq for Key {}

impl Hash for Key {
    fn hash<H>(&self, state: &mut H)
    where
        H: Hasher,
    {
        state.write_isize(self.0.key() as isize)
    }
}

/// A `Compiler` compiles `Expr`s to `Operation`s.
#[derive(Debug)]
pub struct Compiler<'l> {
    graph: &'l mut Graph,
    operations: HashMap<Key, Operation>,
    next_id: i32,
}

impl<'l> Compiler<'l> {
    /// Creates a compiler for the given graph.
    pub fn new(graph: &'l mut Graph) -> Self {
        Compiler {
            graph,
            operations: HashMap::new(),
            next_id: 0,
        }
    }

    /// Compiles the expression.
    pub fn compile<T: TensorType>(&mut self, expr: Expr<T>) -> Result<Operation, Status> {
        self.compile_any(Box::new(expr))
    }

    /// Compiles the expression.
    pub fn compile_any(&mut self, expr: Box<dyn AnyExpr>) -> Result<Operation, Status> {
        let mut child_operations = vec![];
        for child in expr.children() {
            let key = Key(child.clone_box());
            // The result is mapped separately from the match statement below to avoid
            // reference lifetime isues.
            let value = self.operations.get(&key).cloned();
            child_operations.push(match value {
                Some(v) => v,
                None => self.compile_any(child)?,
            });
        }
        let mut next_id = self.next_id;
        let result = expr.create_operation(self.graph, &child_operations, &mut || {
            let id = format!("operation_{}", next_id);
            next_id += 1;
            id
        });
        self.next_id = next_id;
        let operation = result?;
        self.operations.insert(Key(expr), operation.clone());
        Ok(operation)
    }
}

////////////////////////

#[cfg(test)]
mod tests {
    use super::super::Graph;
    use super::*;

    #[test]
    fn test_display() {
        assert_eq!("1 + 2 + 3", format!("{}", (Expr::from(1) + 2) + 3));
        assert_eq!(
            "1 + 2 + 3",
            format!("{}", Expr::from(1) + (Expr::from(2) + 3))
        );
        assert_eq!("1 + 2 - 3", format!("{}", (Expr::from(1) + 2) - 3));
        assert_eq!(
            "1 - (2 + 3)",
            format!("{}", Expr::from(1) - (Expr::from(2) + 3))
        );

        assert_eq!("(1 + 2) * 3", format!("{}", (Expr::from(1) + 2) * 3));
        assert_eq!(
            "1 * (2 + 3)",
            format!("{}", Expr::from(1) * (Expr::from(2) + 3))
        );
        assert_eq!("1 * 2 * 3", format!("{}", (Expr::from(1) * 2) * 3));
        assert_eq!(
            "1 * 2 * 3",
            format!("{}", Expr::from(1) * (Expr::from(2) * 3))
        );

        assert_eq!("(1 + 2) / 3", format!("{}", (Expr::from(1) + 2) / 3));
        assert_eq!(
            "1 / (2 + 3)",
            format!("{}", Expr::from(1) / (Expr::from(2) + 3))
        );
        assert_eq!("1 * 2 / 3", format!("{}", (Expr::from(1) * 2) / 3));
        assert_eq!(
            "1 / (2 * 3)",
            format!("{}", Expr::from(1) / (Expr::from(2) * 3))
        );

        assert_eq!("(1 + 2) % 3", format!("{}", (Expr::from(1) + 2) % 3));
        assert_eq!(
            "1 % (2 + 3)",
            format!("{}", Expr::from(1) % (Expr::from(2) + 3))
        );
        assert_eq!("1 * 2 % 3", format!("{}", (Expr::from(1) * 2) % 3));
        assert_eq!(
            "1 % (2 * 3)",
            format!("{}", Expr::from(1) % (Expr::from(2) * 3))
        );

        assert_eq!("-1", format!("{}", -Expr::from(1)));
        assert_eq!("-(-1)", format!("{}", -(-Expr::from(1))));
        assert_eq!("-(1 + 2)", format!("{}", -(Expr::from(1) + 2)));

        assert_eq!("x", format!("{}", <Variable<f32>>::new(&vec![2, 3], "x")));

        assert_eq!(
            "x",
            format!("{}", <Placeholder<f32>>::new(&vec![2, 3], "x"))
        );

        assert_eq!(
            "x = 1 + 2",
            format!(
                "{}",
                Assign::new(
                    <Placeholder<f32>>::new_expr(&vec![2, 3], "x"),
                    Expr::from(1.0f32) + 2.0f32
                )
            )
        );
    }

    #[test]
    fn test_compile() {
        let mut g = Graph::new();

        let x = <Placeholder<f32>>::new_expr(&vec![2, 3], "x");
        let w = <Variable<f32>>::new_expr(&vec![2, 3], "w");

        let mut compiler = Compiler::new(&mut g);

        compiler
            .compile(x * w.clone() / w.clone() % w.clone() + w.clone() - w.clone())
            .unwrap();

        compiler
            .compile(Assign::to(w, ::std::iter::repeat(1.)).unwrap())
            .unwrap();
    }

    #[test]
    fn test_derivative_by_variable() {
        let x = <Variable<f32>>::new_expr(&[], "x");
        let y = <Variable<f32>>::new_expr(&[], "y");
        for &(ref expected, ref expression) in [
            ("0", Expr::from(1.0f32)),
            ("1", x.clone()),
            ("0", y.clone()),
            ("1 + 0", x.clone() + y.clone()),
            ("1 - 0", x.clone() - y.clone()),
            ("1 * x + x * 1", x.clone() * x.clone()),
            ("1 * y + x * 0", x.clone() * y.clone()),
            (
                "(1 * x + x * 1) * x + x * x * 1",
                x.clone() * x.clone() * x.clone(),
            ),
            ("(1 * y - x * 0) / (y * y)", x.clone() / y.clone()),
            ("1 - x // y * 0", x.clone() % y.clone()),
            ("0 - y // x * 1", y.clone() % x.clone()),
            (
                "(y * (1 - (1 - x // y * 0)) - (x - x % y) * 0) / (y * y)",
                TruncateDiv::new_expr(x.clone(), y.clone()),
            ),
        ]
        .iter()
        {
            assert_eq!(
                *expected,
                format!("{}", expression.derivative_by_variable("x").unwrap())
            );
        }
    }
}