expression.hpp

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// Copyright (c) Sleipnir contributors
 
#pragma once
 
#include <stdint.h>
 
#include <algorithm>
#include <array>
#include <cmath>
#include <memory>
#include <numbers>
#include <string_view>
#include <utility>
 
#include <gch/small_vector.hpp>
 
#include "sleipnir/autodiff/expression_type.hpp"
#include "sleipnir/util/intrusive_shared_ptr.hpp"
#include "sleipnir/util/pool.hpp"
 
namespace slp::detail {
 
// The global pool allocator uses a thread-local static pool resource, which
// isn't guaranteed to be initialized properly across DLL boundaries on Windows
#ifdef _WIN32
inline constexpr bool USE_POOL_ALLOCATOR = false;
#else
inline constexpr bool USE_POOL_ALLOCATOR = true;
#endif
 
template <typename Scalar>
struct Expression;
 
template <typename Scalar>
constexpr void inc_ref_count(Expression<Scalar>* expr);
template <typename Scalar>
void dec_ref_count(Expression<Scalar>* expr);
 
/// Typedef for intrusive shared pointer to Expression.
///
/// @tparam Scalar Scalar type.
template <typename Scalar>
using ExpressionPtr = IntrusiveSharedPtr<Expression<Scalar>>;
 
/// Creates an intrusive shared pointer to an expression from the global pool
/// allocator.
///
/// @tparam T The derived expression type.
/// @param args Constructor arguments for Expression.
template <typename T, typename... Args>
static ExpressionPtr<typename T::Scalar> make_expression_ptr(Args&&... args) {
  if constexpr (USE_POOL_ALLOCATOR) {
    return allocate_intrusive_shared<T>(global_pool_allocator<T>(),
                                        std::forward<Args>(args)...);
  } else {
    return make_intrusive_shared<T>(std::forward<Args>(args)...);
  }
}
 
template <typename Scalar, ExpressionType T>
struct BinaryMinusExpression;
 
template <typename Scalar, ExpressionType T>
struct BinaryPlusExpression;
 
template <typename Scalar>
struct ConstantExpression;
 
template <typename Scalar, ExpressionType T>
struct DivExpression;
 
template <typename Scalar, ExpressionType T>
struct MultExpression;
 
template <typename Scalar, ExpressionType T>
struct UnaryMinusExpression;
 
/// Creates an intrusive shared pointer to a constant expression.
///
/// @tparam Scalar Scalar type.
/// @param value The expression value.
template <typename Scalar>
ExpressionPtr<Scalar> constant_ptr(Scalar value);
 
/// An autodiff expression node.
///
/// @tparam Scalar Scalar type.
template <typename Scalar_>
struct Expression {
  /// Scalar type alias.
  using Scalar = Scalar_;
 
  /// The value of the expression node.
  Scalar val{0};
 
  /// The adjoint of the expression node, used during autodiff.
  Scalar adjoint{0};
 
  /// Counts incoming edges for this node.
  uint32_t incoming_edges = 0;
 
  /// This expression's column in a Jacobian, or -1 otherwise.
  int32_t col = -1;
 
  /// The adjoint of the expression node, used during gradient expression tree
  /// generation.
  ExpressionPtr<Scalar> adjoint_expr;
 
  /// Reference count for intrusive shared pointer.
  uint32_t ref_count = 0;
 
  /// Expression arguments.
  std::array<ExpressionPtr<Scalar>, 2> args{nullptr, nullptr};
 
  /// Constructs a constant expression with a value of zero.
  constexpr Expression() = default;
 
  /// Constructs a nullary expression (an operator with no arguments).
  ///
  /// @param value The expression value.
  explicit constexpr Expression(Scalar value) : val{value} {}
 
  /// Constructs an unary expression (an operator with one argument).
  ///
  /// @param lhs Unary operator's operand.
  explicit constexpr Expression(ExpressionPtr<Scalar> lhs)
      : args{std::move(lhs), nullptr} {}
 
  /// Constructs a binary expression (an operator with two arguments).
  ///
  /// @param lhs Binary operator's left operand.
  /// @param rhs Binary operator's right operand.
  constexpr Expression(ExpressionPtr<Scalar> lhs, ExpressionPtr<Scalar> rhs)
      : args{std::move(lhs), std::move(rhs)} {}
 
  virtual ~Expression() = default;
 
  /// Returns true if the expression is the given constant.
  ///
  /// @param constant The constant.
  /// @return True if the expression is the given constant.
  constexpr bool is_constant(Scalar constant) const {
    return type() == ExpressionType::CONSTANT && val == constant;
  }
 
  /// Expression-Expression multiplication operator.
  ///
  /// @param lhs Operator left-hand side.
  /// @param rhs Operator right-hand side.
  friend ExpressionPtr<Scalar> operator*(const ExpressionPtr<Scalar>& lhs,
                                         const ExpressionPtr<Scalar>& rhs) {
    using enum ExpressionType;
 
    // Prune expression
    if (lhs->is_constant(Scalar(0))) {
      // Return zero
      return lhs;
    } else if (rhs->is_constant(Scalar(0))) {
      // Return zero
      return rhs;
    } else if (lhs->is_constant(Scalar(1))) {
      return rhs;
    } else if (rhs->is_constant(Scalar(1))) {
      return lhs;
    }
 
    // Evaluate constant
    if (lhs->type() == CONSTANT && rhs->type() == CONSTANT) {
      return constant_ptr(lhs->val * rhs->val);
    }
 
    // Evaluate expression type
    if (lhs->type() == CONSTANT) {
      if (rhs->type() == LINEAR) {
        return make_expression_ptr<MultExpression<Scalar, LINEAR>>(lhs, rhs);
      } else if (rhs->type() == QUADRATIC) {
        return make_expression_ptr<MultExpression<Scalar, QUADRATIC>>(lhs, rhs);
      } else {
        return make_expression_ptr<MultExpression<Scalar, NONLINEAR>>(lhs, rhs);
      }
    } else if (rhs->type() == CONSTANT) {
      if (lhs->type() == LINEAR) {
        return make_expression_ptr<MultExpression<Scalar, LINEAR>>(lhs, rhs);
      } else if (lhs->type() == QUADRATIC) {
        return make_expression_ptr<MultExpression<Scalar, QUADRATIC>>(lhs, rhs);
      } else {
        return make_expression_ptr<MultExpression<Scalar, NONLINEAR>>(lhs, rhs);
      }
    } else if (lhs->type() == LINEAR && rhs->type() == LINEAR) {
      return make_expression_ptr<MultExpression<Scalar, QUADRATIC>>(lhs, rhs);
    } else {
      return make_expression_ptr<MultExpression<Scalar, NONLINEAR>>(lhs, rhs);
    }
  }
 
  /// Expression-Expression division operator.
  ///
  /// @param lhs Operator left-hand side.
  /// @param rhs Operator right-hand side.
  friend ExpressionPtr<Scalar> operator/(const ExpressionPtr<Scalar>& lhs,
                                         const ExpressionPtr<Scalar>& rhs) {
    using enum ExpressionType;
 
    // Prune expression
    if (lhs->is_constant(Scalar(0))) {
      // Return zero
      return lhs;
    } else if (rhs->is_constant(Scalar(1))) {
      return lhs;
    }
 
    // Evaluate constant
    if (lhs->type() == CONSTANT && rhs->type() == CONSTANT) {
      return constant_ptr(lhs->val / rhs->val);
    }
 
    // Evaluate expression type
    if (rhs->type() == CONSTANT) {
      if (lhs->type() == LINEAR) {
        return make_expression_ptr<DivExpression<Scalar, LINEAR>>(lhs, rhs);
      } else if (lhs->type() == QUADRATIC) {
        return make_expression_ptr<DivExpression<Scalar, QUADRATIC>>(lhs, rhs);
      } else {
        return make_expression_ptr<DivExpression<Scalar, NONLINEAR>>(lhs, rhs);
      }
    } else {
      return make_expression_ptr<DivExpression<Scalar, NONLINEAR>>(lhs, rhs);
    }
  }
 
  /// Expression-Expression addition operator.
  ///
  /// @param lhs Operator left-hand side.
  /// @param rhs Operator right-hand side.
  friend ExpressionPtr<Scalar> operator+(const ExpressionPtr<Scalar>& lhs,
                                         const ExpressionPtr<Scalar>& rhs) {
    using enum ExpressionType;
 
    // Prune expression
    if (lhs == nullptr || lhs->is_constant(Scalar(0))) {
      return rhs;
    } else if (rhs == nullptr || rhs->is_constant(Scalar(0))) {
      return lhs;
    }
 
    // Evaluate constant
    if (lhs->type() == CONSTANT && rhs->type() == CONSTANT) {
      return constant_ptr(lhs->val + rhs->val);
    }
 
    auto type = std::max(lhs->type(), rhs->type());
    if (type == LINEAR) {
      return make_expression_ptr<BinaryPlusExpression<Scalar, LINEAR>>(lhs,
                                                                       rhs);
    } else if (type == QUADRATIC) {
      return make_expression_ptr<BinaryPlusExpression<Scalar, QUADRATIC>>(lhs,
                                                                          rhs);
    } else {
      return make_expression_ptr<BinaryPlusExpression<Scalar, NONLINEAR>>(lhs,
                                                                          rhs);
    }
  }
 
  /// Expression-Expression compound addition operator.
  ///
  /// @param lhs Operator left-hand side.
  /// @param rhs Operator right-hand side.
  friend ExpressionPtr<Scalar> operator+=(ExpressionPtr<Scalar>& lhs,
                                          const ExpressionPtr<Scalar>& rhs) {
    return lhs = lhs + rhs;
  }
 
  /// Expression-Expression subtraction operator.
  ///
  /// @param lhs Operator left-hand side.
  /// @param rhs Operator right-hand side.
  friend ExpressionPtr<Scalar> operator-(const ExpressionPtr<Scalar>& lhs,
                                         const ExpressionPtr<Scalar>& rhs) {
    using enum ExpressionType;
 
    // Prune expression
    if (lhs->is_constant(Scalar(0))) {
      if (rhs->is_constant(Scalar(0))) {
        // Return zero
        return rhs;
      } else {
        return -rhs;
      }
    } else if (rhs->is_constant(Scalar(0))) {
      return lhs;
    }
 
    // Evaluate constant
    if (lhs->type() == CONSTANT && rhs->type() == CONSTANT) {
      return constant_ptr(lhs->val - rhs->val);
    }
 
    auto type = std::max(lhs->type(), rhs->type());
    if (type == LINEAR) {
      return make_expression_ptr<BinaryMinusExpression<Scalar, LINEAR>>(lhs,
                                                                        rhs);
    } else if (type == QUADRATIC) {
      return make_expression_ptr<BinaryMinusExpression<Scalar, QUADRATIC>>(lhs,
                                                                           rhs);
    } else {
      return make_expression_ptr<BinaryMinusExpression<Scalar, NONLINEAR>>(lhs,
                                                                           rhs);
    }
  }
 
  /// Unary minus operator.
  ///
  /// @param lhs Operand of unary minus.
  friend ExpressionPtr<Scalar> operator-(const ExpressionPtr<Scalar>& lhs) {
    using enum ExpressionType;
 
    // Prune expression
    if (lhs->is_constant(Scalar(0))) {
      // Return zero
      return lhs;
    }
 
    // Evaluate constant
    if (lhs->type() == CONSTANT) {
      return constant_ptr(-lhs->val);
    }
 
    if (lhs->type() == LINEAR) {
      return make_expression_ptr<UnaryMinusExpression<Scalar, LINEAR>>(lhs);
    } else if (lhs->type() == QUADRATIC) {
      return make_expression_ptr<UnaryMinusExpression<Scalar, QUADRATIC>>(lhs);
    } else {
      return make_expression_ptr<UnaryMinusExpression<Scalar, NONLINEAR>>(lhs);
    }
  }
 
  /// Unary plus operator.
  ///
  /// @param lhs Operand of unary plus.
  friend ExpressionPtr<Scalar> operator+(const ExpressionPtr<Scalar>& lhs) {
    return lhs;
  }
 
  /// Either nullary operator with no arguments, unary operator with one
  /// argument, or binary operator with two arguments. This operator is used to
  /// update the node's value.
  ///
  /// @param lhs Left argument to binary operator.
  /// @param rhs Right argument to binary operator.
  /// @return The node's value.
  virtual Scalar value([[maybe_unused]] Scalar lhs,
                       [[maybe_unused]] Scalar rhs) const = 0;
 
  /// Returns the type of this expression (constant, linear, quadratic, or
  /// nonlinear).
  ///
  /// @return The type of this expression.
  virtual ExpressionType type() const = 0;
 
  /// Returns the name of this expression.
  ///
  /// @return The name of this expression.
  virtual std::string_view name() const = 0;
 
  /// Returns ∂/∂l as a Scalar.
  ///
  /// @param lhs Left argument to binary operator.
  /// @param rhs Right argument to binary operator.
  /// @param parent_adjoint Adjoint of parent expression.
  /// @return ∂/∂l as a Scalar.
  virtual Scalar grad_l([[maybe_unused]] Scalar lhs,
                        [[maybe_unused]] Scalar rhs,
                        [[maybe_unused]] Scalar parent_adjoint) const {
    return Scalar(0);
  }
 
  /// Returns ∂/∂r as a Scalar.
  ///
  /// @param lhs Left argument to binary operator.
  /// @param rhs Right argument to binary operator.
  /// @param parent_adjoint Adjoint of parent expression.
  /// @return ∂/∂r as a Scalar.
  virtual Scalar grad_r([[maybe_unused]] Scalar lhs,
                        [[maybe_unused]] Scalar rhs,
                        [[maybe_unused]] Scalar parent_adjoint) const {
    return Scalar(0);
  }
 
  /// Returns ∂/∂l as an Expression.
  ///
  /// @param lhs Left argument to binary operator.
  /// @param rhs Right argument to binary operator.
  /// @param parent_adjoint Adjoint of parent expression.
  /// @return ∂/∂l as an Expression.
  virtual ExpressionPtr<Scalar> grad_expr_l(
      [[maybe_unused]] const ExpressionPtr<Scalar>& lhs,
      [[maybe_unused]] const ExpressionPtr<Scalar>& rhs,
      [[maybe_unused]] const ExpressionPtr<Scalar>& parent_adjoint) const {
    return constant_ptr(Scalar(0));
  }
 
  /// Returns ∂/∂r as an Expression.
  ///
  /// @param lhs Left argument to binary operator.
  /// @param rhs Right argument to binary operator.
  /// @param parent_adjoint Adjoint of parent expression.
  /// @return ∂/∂r as an Expression.
  virtual ExpressionPtr<Scalar> grad_expr_r(
      [[maybe_unused]] const ExpressionPtr<Scalar>& lhs,
      [[maybe_unused]] const ExpressionPtr<Scalar>& rhs,
      [[maybe_unused]] const ExpressionPtr<Scalar>& parent_adjoint) const {
    return constant_ptr(Scalar(0));
  }
};
 
template <typename Scalar>
ExpressionPtr<Scalar> constant_ptr(Scalar value) {
  return make_expression_ptr<ConstantExpression<Scalar>>(value);
}
 
template <typename Scalar>
ExpressionPtr<Scalar> cbrt(const ExpressionPtr<Scalar>& x);
template <typename Scalar>
ExpressionPtr<Scalar> exp(const ExpressionPtr<Scalar>& x);
template <typename Scalar>
ExpressionPtr<Scalar> sin(const ExpressionPtr<Scalar>& x);
template <typename Scalar>
ExpressionPtr<Scalar> sinh(const ExpressionPtr<Scalar>& x);
template <typename Scalar>
ExpressionPtr<Scalar> sqrt(const ExpressionPtr<Scalar>& x);
 
/// Derived expression type for binary minus operator.
///
/// @tparam Scalar Scalar type.
/// @tparam T Expression type.
template <typename Scalar, ExpressionType T>
struct BinaryMinusExpression final : Expression<Scalar> {
  /// Constructs a binary expression (an operator with two arguments).
  ///
  /// @param lhs Binary operator's left operand.
  /// @param rhs Binary operator's right operand.
  constexpr BinaryMinusExpression(ExpressionPtr<Scalar> lhs,
                                  ExpressionPtr<Scalar> rhs)
      : Expression<Scalar>{std::move(lhs), std::move(rhs)} {}
 
  Scalar value(Scalar lhs, Scalar rhs) const override { return lhs - rhs; }
 
  ExpressionType type() const override { return T; }
 
  std::string_view name() const override { return "binary minus"; }
 
  Scalar grad_l(Scalar, Scalar, Scalar parent_adjoint) const override {
    return parent_adjoint;
  }
 
  Scalar grad_r(Scalar, Scalar, Scalar parent_adjoint) const override {
    return -parent_adjoint;
  }
 
  ExpressionPtr<Scalar> grad_expr_l(
      const ExpressionPtr<Scalar>&, const ExpressionPtr<Scalar>&,
      const ExpressionPtr<Scalar>& parent_adjoint) const override {
    return parent_adjoint;
  }
 
  ExpressionPtr<Scalar> grad_expr_r(
      const ExpressionPtr<Scalar>&, const ExpressionPtr<Scalar>&,
      const ExpressionPtr<Scalar>& parent_adjoint) const override {
    return -parent_adjoint;
  }
};
 
/// Derived expression type for binary plus operator.
///
/// @tparam Scalar Scalar type.
/// @tparam T Expression type.
template <typename Scalar, ExpressionType T>
struct BinaryPlusExpression final : Expression<Scalar> {
  /// Constructs a binary expression (an operator with two arguments).
  ///
  /// @param lhs Binary operator's left operand.
  /// @param rhs Binary operator's right operand.
  constexpr BinaryPlusExpression(ExpressionPtr<Scalar> lhs,
                                 ExpressionPtr<Scalar> rhs)
      : Expression<Scalar>{std::move(lhs), std::move(rhs)} {}
 
  Scalar value(Scalar lhs, Scalar rhs) const override { return lhs + rhs; }
 
  ExpressionType type() const override { return T; }
 
  std::string_view name() const override { return "binary plus"; }
 
  Scalar grad_l(Scalar, Scalar, Scalar parent_adjoint) const override {
    return parent_adjoint;
  }
 
  Scalar grad_r(Scalar, Scalar, Scalar parent_adjoint) const override {
    return parent_adjoint;
  }
 
  ExpressionPtr<Scalar> grad_expr_l(
      const ExpressionPtr<Scalar>&, const ExpressionPtr<Scalar>&,
      const ExpressionPtr<Scalar>& parent_adjoint) const override {
    return parent_adjoint;
  }
 
  ExpressionPtr<Scalar> grad_expr_r(
      const ExpressionPtr<Scalar>&, const ExpressionPtr<Scalar>&,
      const ExpressionPtr<Scalar>& parent_adjoint) const override {
    return parent_adjoint;
  }
};
 
/// Derived expression type for cbrt().
///
/// @tparam Scalar Scalar type.
template <typename Scalar>
struct CbrtExpression final : Expression<Scalar> {
  /// Constructs an unary expression (an operator with one argument).
  ///
  /// @param lhs Unary operator's operand.
  explicit constexpr CbrtExpression(ExpressionPtr<Scalar> lhs)
      : Expression<Scalar>{std::move(lhs)} {}
 
  Scalar value(Scalar x, Scalar) const override {
    using std::cbrt;
    return cbrt(x);
  }
 
  ExpressionType type() const override { return ExpressionType::NONLINEAR; }
 
  std::string_view name() const override { return "cbrt"; }
 
  Scalar grad_l(Scalar x, Scalar, Scalar parent_adjoint) const override {
    using std::cbrt;
 
    Scalar c = cbrt(x);
    return parent_adjoint / (Scalar(3) * c * c);
  }
 
  ExpressionPtr<Scalar> grad_expr_l(
      const ExpressionPtr<Scalar>& x, const ExpressionPtr<Scalar>&,
      const ExpressionPtr<Scalar>& parent_adjoint) const override {
    auto c = cbrt(x);
    return parent_adjoint / (constant_ptr(Scalar(3)) * c * c);
  }
};
 
/// cbrt() for Expressions.
///
/// @tparam Scalar Scalar type.
/// @param x The argument.
template <typename Scalar>
ExpressionPtr<Scalar> cbrt(const ExpressionPtr<Scalar>& x) {
  using enum ExpressionType;
  using std::cbrt;
 
  // Evaluate constant
  if (x->type() == CONSTANT) {
    if (x->val == Scalar(0)) {
      // Return zero
      return x;
    } else if (x->val == Scalar(-1) || x->val == Scalar(1)) {
      return x;
    } else {
      return constant_ptr(cbrt(x->val));
    }
  }
 
  return make_expression_ptr<CbrtExpression<Scalar>>(x);
}
 
/// Derived expression type for constant.
///
/// @tparam Scalar Scalar type.
template <typename Scalar>
struct ConstantExpression final : Expression<Scalar> {
  /// Constructs a nullary expression (an operator with no arguments).
  ///
  /// @param value The expression value.
  explicit constexpr ConstantExpression(Scalar value)
      : Expression<Scalar>{value} {}
 
  Scalar value(Scalar, Scalar) const override { return this->val; }
 
  ExpressionType type() const override { return ExpressionType::CONSTANT; }
 
  std::string_view name() const override { return "constant"; }
};
 
/// Derived expression type for decision variable.
///
/// @tparam Scalar Scalar type.
template <typename Scalar>
struct DecisionVariableExpression final : Expression<Scalar> {
  /// Constructs a decision variable expression with a value of zero.
  constexpr DecisionVariableExpression() = default;
 
  /// Constructs a nullary expression (an operator with no arguments).
  ///
  /// @param value The expression value.
  explicit constexpr DecisionVariableExpression(Scalar value)
      : Expression<Scalar>{value} {}
 
  Scalar value(Scalar, Scalar) const override { return this->val; }
 
  ExpressionType type() const override { return ExpressionType::LINEAR; }
 
  std::string_view name() const override { return "decision variable"; }
};
 
/// Derived expression type for binary division operator.
///
/// @tparam Scalar Scalar type.
/// @tparam T Expression type.
template <typename Scalar, ExpressionType T>
struct DivExpression final : Expression<Scalar> {
  /// Constructs a binary expression (an operator with two arguments).
  ///
  /// @param lhs Binary operator's left operand.
  /// @param rhs Binary operator's right operand.
  constexpr DivExpression(ExpressionPtr<Scalar> lhs, ExpressionPtr<Scalar> rhs)
      : Expression<Scalar>{std::move(lhs), std::move(rhs)} {}
 
  Scalar value(Scalar lhs, Scalar rhs) const override { return lhs / rhs; }
 
  ExpressionType type() const override { return T; }
 
  std::string_view name() const override { return "division"; }
 
  Scalar grad_l(Scalar, Scalar rhs, Scalar parent_adjoint) const override {
    return parent_adjoint / rhs;
  };
 
  Scalar grad_r(Scalar lhs, Scalar rhs, Scalar parent_adjoint) const override {
    return parent_adjoint * -lhs / (rhs * rhs);
  }
 
  ExpressionPtr<Scalar> grad_expr_l(
      const ExpressionPtr<Scalar>&, const ExpressionPtr<Scalar>& rhs,
      const ExpressionPtr<Scalar>& parent_adjoint) const override {
    return parent_adjoint / rhs;
  }
 
  ExpressionPtr<Scalar> grad_expr_r(
      const ExpressionPtr<Scalar>& lhs, const ExpressionPtr<Scalar>& rhs,
      const ExpressionPtr<Scalar>& parent_adjoint) const override {
    return parent_adjoint * -lhs / (rhs * rhs);
  }
};
 
/// Derived expression type for binary multiplication operator.
///
/// @tparam Scalar Scalar type.
/// @tparam T Expression type.
template <typename Scalar, ExpressionType T>
struct MultExpression final : Expression<Scalar> {
  /// Constructs a binary expression (an operator with two arguments).
  ///
  /// @param lhs Binary operator's left operand.
  /// @param rhs Binary operator's right operand.
  constexpr MultExpression(ExpressionPtr<Scalar> lhs, ExpressionPtr<Scalar> rhs)
      : Expression<Scalar>{std::move(lhs), std::move(rhs)} {}
 
  Scalar value(Scalar lhs, Scalar rhs) const override { return lhs * rhs; }
 
  ExpressionType type() const override { return T; }
 
  std::string_view name() const override { return "multiplication"; }
 
  Scalar grad_l([[maybe_unused]] Scalar lhs, Scalar rhs,
                Scalar parent_adjoint) const override {
    return parent_adjoint * rhs;
  }
 
  Scalar grad_r(Scalar lhs, [[maybe_unused]] Scalar rhs,
                Scalar parent_adjoint) const override {
    return parent_adjoint * lhs;
  }
 
  ExpressionPtr<Scalar> grad_expr_l(
      [[maybe_unused]] const ExpressionPtr<Scalar>& lhs,
      const ExpressionPtr<Scalar>& rhs,
      const ExpressionPtr<Scalar>& parent_adjoint) const override {
    return parent_adjoint * rhs;
  }
 
  ExpressionPtr<Scalar> grad_expr_r(
      const ExpressionPtr<Scalar>& lhs,
      [[maybe_unused]] const ExpressionPtr<Scalar>& rhs,
      const ExpressionPtr<Scalar>& parent_adjoint) const override {
    return parent_adjoint * lhs;
  }
};
 
/// Derived expression type for unary minus operator.
///
/// @tparam Scalar Scalar type.
/// @tparam T Expression type.
template <typename Scalar, ExpressionType T>
struct UnaryMinusExpression final : Expression<Scalar> {
  /// Constructs an unary expression (an operator with one argument).
  ///
  /// @param lhs Unary operator's operand.
  explicit constexpr UnaryMinusExpression(ExpressionPtr<Scalar> lhs)
      : Expression<Scalar>{std::move(lhs)} {}
 
  Scalar value(Scalar lhs, Scalar) const override { return -lhs; }
 
  ExpressionType type() const override { return T; }
 
  std::string_view name() const override { return "unary minus"; }
 
  Scalar grad_l(Scalar, Scalar, Scalar parent_adjoint) const override {
    return -parent_adjoint;
  }
 
  ExpressionPtr<Scalar> grad_expr_l(
      const ExpressionPtr<Scalar>&, const ExpressionPtr<Scalar>&,
      const ExpressionPtr<Scalar>& parent_adjoint) const override {
    return -parent_adjoint;
  }
};
 
/// Refcount increment for intrusive shared pointer.
///
/// @tparam Scalar Scalar type.
/// @param expr The shared pointer's managed object.
template <typename Scalar>
constexpr void inc_ref_count(Expression<Scalar>* expr) {
  ++expr->ref_count;
}
 
/// Refcount decrement for intrusive shared pointer.
///
/// @tparam Scalar Scalar type.
/// @param expr The shared pointer's managed object.
template <typename Scalar>
void dec_ref_count(Expression<Scalar>* expr) {
  // If a deeply nested tree is being deallocated all at once, calling the
  // Expression destructor when expr's refcount reaches zero can cause a stack
  // overflow. Instead, we iterate over its children to decrement their
  // refcounts and deallocate them.
  gch::small_vector<Expression<Scalar>*> stack;
  stack.emplace_back(expr);
 
  while (!stack.empty()) {
    auto elem = stack.back();
    stack.pop_back();
 
    // Decrement the current node's refcount. If it reaches zero, deallocate the
    // node and enqueue its children so their refcounts are decremented too.
    if (--elem->ref_count == 0) {
      if (elem->adjoint_expr != nullptr) {
        stack.emplace_back(elem->adjoint_expr.get());
      }
      for (auto& arg : elem->args) {
        if (arg != nullptr) {
          stack.emplace_back(arg.get());
        }
      }
 
      // Not calling the destructor here is safe because it only decrements
      // refcounts, which was already done above.
      if constexpr (USE_POOL_ALLOCATOR) {
        auto alloc = global_pool_allocator<Expression<Scalar>>();
        std::allocator_traits<decltype(alloc)>::deallocate(
            alloc, elem, sizeof(Expression<Scalar>));
      }
    }
  }
}
 
/// Derived expression type for abs().
///
/// @tparam Scalar Scalar type.
template <typename Scalar>
struct AbsExpression final : Expression<Scalar> {
  /// Constructs an unary expression (an operator with one argument).
  ///
  /// @param lhs Unary operator's operand.
  explicit constexpr AbsExpression(ExpressionPtr<Scalar> lhs)
      : Expression<Scalar>{std::move(lhs)} {}
 
  Scalar value(Scalar x, Scalar) const override {
    using std::abs;
    return abs(x);
  }
 
  ExpressionType type() const override { return ExpressionType::NONLINEAR; }
 
  std::string_view name() const override { return "abs"; }
 
  Scalar grad_l(Scalar x, Scalar, Scalar parent_adjoint) const override {
    if (x < Scalar(0)) {
      return -parent_adjoint;
    } else if (x > Scalar(0)) {
      return parent_adjoint;
    } else {
      return Scalar(0);
    }
  }
 
  ExpressionPtr<Scalar> grad_expr_l(
      const ExpressionPtr<Scalar>& x, const ExpressionPtr<Scalar>&,
      const ExpressionPtr<Scalar>& parent_adjoint) const override {
    if (x->val < Scalar(0)) {
      return -parent_adjoint;
    } else if (x->val > Scalar(0)) {
      return parent_adjoint;
    } else {
      return constant_ptr(Scalar(0));
    }
  }
};
 
/// abs() for Expressions.
///
/// @tparam Scalar Scalar type.
/// @param x The argument.
template <typename Scalar>
ExpressionPtr<Scalar> abs(const ExpressionPtr<Scalar>& x) {
  using enum ExpressionType;
  using std::abs;
 
  // Prune expression
  if (x->is_constant(Scalar(0))) {
    // Return zero
    return x;
  }
 
  // Evaluate constant
  if (x->type() == CONSTANT) {
    return constant_ptr(abs(x->val));
  }
 
  return make_expression_ptr<AbsExpression<Scalar>>(x);
}
 
/// Derived expression type for acos().
///
/// @tparam Scalar Scalar type.
template <typename Scalar>
struct AcosExpression final : Expression<Scalar> {
  /// Constructs an unary expression (an operator with one argument).
  ///
  /// @param lhs Unary operator's operand.
  explicit constexpr AcosExpression(ExpressionPtr<Scalar> lhs)
      : Expression<Scalar>{std::move(lhs)} {}
 
  Scalar value(Scalar x, Scalar) const override {
    using std::acos;
    return acos(x);
  }
 
  ExpressionType type() const override { return ExpressionType::NONLINEAR; }
 
  std::string_view name() const override { return "acos"; }
 
  Scalar grad_l(Scalar x, Scalar, Scalar parent_adjoint) const override {
    using std::sqrt;
    return -parent_adjoint / sqrt(Scalar(1) - x * x);
  }
 
  ExpressionPtr<Scalar> grad_expr_l(
      const ExpressionPtr<Scalar>& x, const ExpressionPtr<Scalar>&,
      const ExpressionPtr<Scalar>& parent_adjoint) const override {
    return -parent_adjoint / sqrt(constant_ptr(Scalar(1)) - x * x);
  }
};
 
/// acos() for Expressions.
///
/// @tparam Scalar Scalar type.
/// @param x The argument.
template <typename Scalar>
ExpressionPtr<Scalar> acos(const ExpressionPtr<Scalar>& x) {
  using enum ExpressionType;
  using std::acos;
 
  // Prune expression
  if (x->is_constant(Scalar(0))) {
    return constant_ptr(Scalar(std::numbers::pi) / Scalar(2));
  }
 
  // Evaluate constant
  if (x->type() == CONSTANT) {
    return constant_ptr(acos(x->val));
  }
 
  return make_expression_ptr<AcosExpression<Scalar>>(x);
}
 
/// Derived expression type for asin().
///
/// @tparam Scalar Scalar type.
template <typename Scalar>
struct AsinExpression final : Expression<Scalar> {
  /// Constructs an unary expression (an operator with one argument).
  ///
  /// @param lhs Unary operator's operand.
  explicit constexpr AsinExpression(ExpressionPtr<Scalar> lhs)
      : Expression<Scalar>{std::move(lhs)} {}
 
  Scalar value(Scalar x, Scalar) const override {
    using std::asin;
    return asin(x);
  }
 
  ExpressionType type() const override { return ExpressionType::NONLINEAR; }
 
  std::string_view name() const override { return "asin"; }
 
  Scalar grad_l(Scalar x, Scalar, Scalar parent_adjoint) const override {
    using std::sqrt;
    return parent_adjoint / sqrt(Scalar(1) - x * x);
  }
 
  ExpressionPtr<Scalar> grad_expr_l(
      const ExpressionPtr<Scalar>& x, const ExpressionPtr<Scalar>&,
      const ExpressionPtr<Scalar>& parent_adjoint) const override {
    return parent_adjoint / sqrt(constant_ptr(Scalar(1)) - x * x);
  }
};
 
/// asin() for Expressions.
///
/// @tparam Scalar Scalar type.
/// @param x The argument.
template <typename Scalar>
ExpressionPtr<Scalar> asin(const ExpressionPtr<Scalar>& x) {
  using enum ExpressionType;
  using std::asin;
 
  // Prune expression
  if (x->is_constant(Scalar(0))) {
    // Return zero
    return x;
  }
 
  // Evaluate constant
  if (x->type() == CONSTANT) {
    return constant_ptr(asin(x->val));
  }
 
  return make_expression_ptr<AsinExpression<Scalar>>(x);
}
 
/// Derived expression type for atan().
///
/// @tparam Scalar Scalar type.
template <typename Scalar>
struct AtanExpression final : Expression<Scalar> {
  /// Constructs an unary expression (an operator with one argument).
  ///
  /// @param lhs Unary operator's operand.
  explicit constexpr AtanExpression(ExpressionPtr<Scalar> lhs)
      : Expression<Scalar>{std::move(lhs)} {}
 
  Scalar value(Scalar x, Scalar) const override {
    using std::atan;
    return atan(x);
  }
 
  ExpressionType type() const override { return ExpressionType::NONLINEAR; }
 
  std::string_view name() const override { return "atan"; }
 
  Scalar grad_l(Scalar x, Scalar, Scalar parent_adjoint) const override {
    return parent_adjoint / (Scalar(1) + x * x);
  }
 
  ExpressionPtr<Scalar> grad_expr_l(
      const ExpressionPtr<Scalar>& x, const ExpressionPtr<Scalar>&,
      const ExpressionPtr<Scalar>& parent_adjoint) const override {
    return parent_adjoint / (constant_ptr(Scalar(1)) + x * x);
  }
};
 
/// atan() for Expressions.
///
/// @tparam Scalar Scalar type.
/// @param x The argument.
template <typename Scalar>
ExpressionPtr<Scalar> atan(const ExpressionPtr<Scalar>& x) {
  using enum ExpressionType;
  using std::atan;
 
  // Prune expression
  if (x->is_constant(Scalar(0))) {
    // Return zero
    return x;
  }
 
  // Evaluate constant
  if (x->type() == CONSTANT) {
    return constant_ptr(atan(x->val));
  }
 
  return make_expression_ptr<AtanExpression<Scalar>>(x);
}
 
/// Derived expression type for atan2().
///
/// @tparam Scalar Scalar type.
template <typename Scalar>
struct Atan2Expression final : Expression<Scalar> {
  /// Constructs a binary expression (an operator with two arguments).
  ///
  /// @param lhs Binary operator's left operand.
  /// @param rhs Binary operator's right operand.
  constexpr Atan2Expression(ExpressionPtr<Scalar> lhs,
                            ExpressionPtr<Scalar> rhs)
      : Expression<Scalar>{std::move(lhs), std::move(rhs)} {}
 
  Scalar value(Scalar y, Scalar x) const override {
    using std::atan2;
    return atan2(y, x);
  }
 
  ExpressionType type() const override { return ExpressionType::NONLINEAR; }
 
  std::string_view name() const override { return "atan2"; }
 
  Scalar grad_l(Scalar y, Scalar x, Scalar parent_adjoint) const override {
    return parent_adjoint * x / (y * y + x * x);
  }
 
  Scalar grad_r(Scalar y, Scalar x, Scalar parent_adjoint) const override {
    return parent_adjoint * -y / (y * y + x * x);
  }
 
  ExpressionPtr<Scalar> grad_expr_l(
      const ExpressionPtr<Scalar>& y, const ExpressionPtr<Scalar>& x,
      const ExpressionPtr<Scalar>& parent_adjoint) const override {
    return parent_adjoint * x / (y * y + x * x);
  }
 
  ExpressionPtr<Scalar> grad_expr_r(
      const ExpressionPtr<Scalar>& y, const ExpressionPtr<Scalar>& x,
      const ExpressionPtr<Scalar>& parent_adjoint) const override {
    return parent_adjoint * -y / (y * y + x * x);
  }
};
 
/// atan2() for Expressions.
///
/// @tparam Scalar Scalar type.
/// @param y The y argument.
/// @param x The x argument.
template <typename Scalar>
ExpressionPtr<Scalar> atan2(const ExpressionPtr<Scalar>& y,
                            const ExpressionPtr<Scalar>& x) {
  using enum ExpressionType;
  using std::atan2;
 
  // Prune expression
  if (y->is_constant(Scalar(0))) {
    // Return zero
    return y;
  } else if (x->is_constant(Scalar(0))) {
    return constant_ptr(Scalar(std::numbers::pi) / Scalar(2));
  }
 
  // Evaluate constant
  if (y->type() == CONSTANT && x->type() == CONSTANT) {
    return constant_ptr(atan2(y->val, x->val));
  }
 
  return make_expression_ptr<Atan2Expression<Scalar>>(y, x);
}
 
/// Derived expression type for cos().
///
/// @tparam Scalar Scalar type.
template <typename Scalar>
struct CosExpression final : Expression<Scalar> {
  /// Constructs an unary expression (an operator with one argument).
  ///
  /// @param lhs Unary operator's operand.
  explicit constexpr CosExpression(ExpressionPtr<Scalar> lhs)
      : Expression<Scalar>{std::move(lhs)} {}
 
  Scalar value(Scalar x, Scalar) const override {
    using std::cos;
    return cos(x);
  }
 
  ExpressionType type() const override { return ExpressionType::NONLINEAR; }
 
  std::string_view name() const override { return "cos"; }
 
  Scalar grad_l(Scalar x, Scalar, Scalar parent_adjoint) const override {
    using std::sin;
    return parent_adjoint * -sin(x);
  }
 
  ExpressionPtr<Scalar> grad_expr_l(
      const ExpressionPtr<Scalar>& x, const ExpressionPtr<Scalar>&,
      const ExpressionPtr<Scalar>& parent_adjoint) const override {
    return parent_adjoint * -sin(x);
  }
};
 
/// cos() for Expressions.
///
/// @tparam Scalar Scalar type.
/// @param x The argument.
template <typename Scalar>
ExpressionPtr<Scalar> cos(const ExpressionPtr<Scalar>& x) {
  using enum ExpressionType;
  using std::cos;
 
  // Prune expression
  if (x->is_constant(Scalar(0))) {
    return constant_ptr(Scalar(1));
  }
 
  // Evaluate constant
  if (x->type() == CONSTANT) {
    return constant_ptr(cos(x->val));
  }
 
  return make_expression_ptr<CosExpression<Scalar>>(x);
}
 
/// Derived expression type for cosh().
///
/// @tparam Scalar Scalar type.
template <typename Scalar>
struct CoshExpression final : Expression<Scalar> {
  /// Constructs an unary expression (an operator with one argument).
  ///
  /// @param lhs Unary operator's operand.
  explicit constexpr CoshExpression(ExpressionPtr<Scalar> lhs)
      : Expression<Scalar>{std::move(lhs)} {}
 
  Scalar value(Scalar x, Scalar) const override {
    using std::cosh;
    return cosh(x);
  }
 
  ExpressionType type() const override { return ExpressionType::NONLINEAR; }
 
  std::string_view name() const override { return "cosh"; }
 
  Scalar grad_l(Scalar x, Scalar, Scalar parent_adjoint) const override {
    using std::sinh;
    return parent_adjoint * sinh(x);
  }
 
  ExpressionPtr<Scalar> grad_expr_l(
      const ExpressionPtr<Scalar>& x, const ExpressionPtr<Scalar>&,
      const ExpressionPtr<Scalar>& parent_adjoint) const override {
    return parent_adjoint * sinh(x);
  }
};
 
/// cosh() for Expressions.
///
/// @tparam Scalar Scalar type.
/// @param x The argument.
template <typename Scalar>
ExpressionPtr<Scalar> cosh(const ExpressionPtr<Scalar>& x) {
  using enum ExpressionType;
  using std::cosh;
 
  // Prune expression
  if (x->is_constant(Scalar(0))) {
    return constant_ptr(Scalar(1));
  }
 
  // Evaluate constant
  if (x->type() == CONSTANT) {
    return constant_ptr(cosh(x->val));
  }
 
  return make_expression_ptr<CoshExpression<Scalar>>(x);
}
 
/// Derived expression type for erf().
///
/// @tparam Scalar Scalar type.
template <typename Scalar>
struct ErfExpression final : Expression<Scalar> {
  /// Constructs an unary expression (an operator with one argument).
  ///
  /// @param lhs Unary operator's operand.
  explicit constexpr ErfExpression(ExpressionPtr<Scalar> lhs)
      : Expression<Scalar>{std::move(lhs)} {}
 
  Scalar value(Scalar x, Scalar) const override {
    using std::erf;
    return erf(x);
  }
 
  ExpressionType type() const override { return ExpressionType::NONLINEAR; }
 
  std::string_view name() const override { return "erf"; }
 
  Scalar grad_l(Scalar x, Scalar, Scalar parent_adjoint) const override {
    using std::exp;
    return parent_adjoint * Scalar(2.0 * std::numbers::inv_sqrtpi) *
           exp(-x * x);
  }
 
  ExpressionPtr<Scalar> grad_expr_l(
      const ExpressionPtr<Scalar>& x, const ExpressionPtr<Scalar>&,
      const ExpressionPtr<Scalar>& parent_adjoint) const override {
    return parent_adjoint *
           constant_ptr(Scalar(2.0 * std::numbers::inv_sqrtpi)) * exp(-x * x);
  }
};
 
/// erf() for Expressions.
///
/// @tparam Scalar Scalar type.
/// @param x The argument.
template <typename Scalar>
ExpressionPtr<Scalar> erf(const ExpressionPtr<Scalar>& x) {
  using enum ExpressionType;
  using std::erf;
 
  // Prune expression
  if (x->is_constant(Scalar(0))) {
    // Return zero
    return x;
  }
 
  // Evaluate constant
  if (x->type() == CONSTANT) {
    return constant_ptr(erf(x->val));
  }
 
  return make_expression_ptr<ErfExpression<Scalar>>(x);
}
 
/// Derived expression type for exp().
///
/// @tparam Scalar Scalar type.
template <typename Scalar>
struct ExpExpression final : Expression<Scalar> {
  /// Constructs an unary expression (an operator with one argument).
  ///
  /// @param lhs Unary operator's operand.
  explicit constexpr ExpExpression(ExpressionPtr<Scalar> lhs)
      : Expression<Scalar>{std::move(lhs)} {}
 
  Scalar value(Scalar x, Scalar) const override {
    using std::exp;
    return exp(x);
  }
 
  ExpressionType type() const override { return ExpressionType::NONLINEAR; }
 
  std::string_view name() const override { return "exp"; }
 
  Scalar grad_l(Scalar x, Scalar, Scalar parent_adjoint) const override {
    using std::exp;
    return parent_adjoint * exp(x);
  }
 
  ExpressionPtr<Scalar> grad_expr_l(
      const ExpressionPtr<Scalar>& x, const ExpressionPtr<Scalar>&,
      const ExpressionPtr<Scalar>& parent_adjoint) const override {
    return parent_adjoint * exp(x);
  }
};
 
/// exp() for Expressions.
///
/// @tparam Scalar Scalar type.
/// @param x The argument.
template <typename Scalar>
ExpressionPtr<Scalar> exp(const ExpressionPtr<Scalar>& x) {
  using enum ExpressionType;
  using std::exp;
 
  // Prune expression
  if (x->is_constant(Scalar(0))) {
    return constant_ptr(Scalar(1));
  }
 
  // Evaluate constant
  if (x->type() == CONSTANT) {
    return constant_ptr(exp(x->val));
  }
 
  return make_expression_ptr<ExpExpression<Scalar>>(x);
}
 
template <typename Scalar>
ExpressionPtr<Scalar> hypot(const ExpressionPtr<Scalar>& x,
                            const ExpressionPtr<Scalar>& y);
 
/// Derived expression type for hypot().
///
/// @tparam Scalar Scalar type.
template <typename Scalar>
struct HypotExpression final : Expression<Scalar> {
  /// Constructs a binary expression (an operator with two arguments).
  ///
  /// @param lhs Binary operator's left operand.
  /// @param rhs Binary operator's right operand.
  constexpr HypotExpression(ExpressionPtr<Scalar> lhs,
                            ExpressionPtr<Scalar> rhs)
      : Expression<Scalar>{std::move(lhs), std::move(rhs)} {}
 
  Scalar value(Scalar x, Scalar y) const override {
    using std::hypot;
    return hypot(x, y);
  }
 
  ExpressionType type() const override { return ExpressionType::NONLINEAR; }
 
  std::string_view name() const override { return "hypot"; }
 
  Scalar grad_l(Scalar x, Scalar y, Scalar parent_adjoint) const override {
    using std::hypot;
    return parent_adjoint * x / hypot(x, y);
  }
 
  Scalar grad_r(Scalar x, Scalar y, Scalar parent_adjoint) const override {
    using std::hypot;
    return parent_adjoint * y / hypot(x, y);
  }
 
  ExpressionPtr<Scalar> grad_expr_l(
      const ExpressionPtr<Scalar>& x, const ExpressionPtr<Scalar>& y,
      const ExpressionPtr<Scalar>& parent_adjoint) const override {
    return parent_adjoint * x / hypot(x, y);
  }
 
  ExpressionPtr<Scalar> grad_expr_r(
      const ExpressionPtr<Scalar>& x, const ExpressionPtr<Scalar>& y,
      const ExpressionPtr<Scalar>& parent_adjoint) const override {
    return parent_adjoint * y / hypot(x, y);
  }
};
 
/// hypot() for Expressions.
///
/// @tparam Scalar Scalar type.
/// @param x The x argument.
/// @param y The y argument.
template <typename Scalar>
ExpressionPtr<Scalar> hypot(const ExpressionPtr<Scalar>& x,
                            const ExpressionPtr<Scalar>& y) {
  using enum ExpressionType;
  using std::hypot;
 
  // Prune expression
  if (x->is_constant(Scalar(0))) {
    return y;
  } else if (y->is_constant(Scalar(0))) {
    return x;
  }
 
  // Evaluate constant
  if (x->type() == CONSTANT && y->type() == CONSTANT) {
    return constant_ptr(hypot(x->val, y->val));
  }
 
  return make_expression_ptr<HypotExpression<Scalar>>(x, y);
}
 
/// Derived expression type for log().
///
/// @tparam Scalar Scalar type.
template <typename Scalar>
struct LogExpression final : Expression<Scalar> {
  /// Constructs an unary expression (an operator with one argument).
  ///
  /// @param lhs Unary operator's operand.
  explicit constexpr LogExpression(ExpressionPtr<Scalar> lhs)
      : Expression<Scalar>{std::move(lhs)} {}
 
  Scalar value(Scalar x, Scalar) const override {
    using std::log;
    return log(x);
  }
 
  ExpressionType type() const override { return ExpressionType::NONLINEAR; }
 
  std::string_view name() const override { return "log"; }
 
  Scalar grad_l(Scalar x, Scalar, Scalar parent_adjoint) const override {
    return parent_adjoint / x;
  }
 
  ExpressionPtr<Scalar> grad_expr_l(
      const ExpressionPtr<Scalar>& x, const ExpressionPtr<Scalar>&,
      const ExpressionPtr<Scalar>& parent_adjoint) const override {
    return parent_adjoint / x;
  }
};
 
/// log() for Expressions.
///
/// @tparam Scalar Scalar type.
/// @param x The argument.
template <typename Scalar>
ExpressionPtr<Scalar> log(const ExpressionPtr<Scalar>& x) {
  using enum ExpressionType;
  using std::log;
 
  // Prune expression
  if (x->is_constant(Scalar(0))) {
    // Return zero
    return x;
  }
 
  // Evaluate constant
  if (x->type() == CONSTANT) {
    return constant_ptr(log(x->val));
  }
 
  return make_expression_ptr<LogExpression<Scalar>>(x);
}
 
/// Derived expression type for log10().
///
/// @tparam Scalar Scalar type.
template <typename Scalar>
struct Log10Expression final : Expression<Scalar> {
  /// Constructs an unary expression (an operator with one argument).
  ///
  /// @param lhs Unary operator's operand.
  explicit constexpr Log10Expression(ExpressionPtr<Scalar> lhs)
      : Expression<Scalar>{std::move(lhs)} {}
 
  Scalar value(Scalar x, Scalar) const override {
    using std::log10;
    return log10(x);
  }
 
  ExpressionType type() const override { return ExpressionType::NONLINEAR; }
 
  std::string_view name() const override { return "log10"; }
 
  Scalar grad_l(Scalar x, Scalar, Scalar parent_adjoint) const override {
    return parent_adjoint / (Scalar(std::numbers::ln10) * x);
  }
 
  ExpressionPtr<Scalar> grad_expr_l(
      const ExpressionPtr<Scalar>& x, const ExpressionPtr<Scalar>&,
      const ExpressionPtr<Scalar>& parent_adjoint) const override {
    return parent_adjoint / (constant_ptr(Scalar(std::numbers::ln10)) * x);
  }
};
 
/// log10() for Expressions.
///
/// @tparam Scalar Scalar type.
/// @param x The argument.
template <typename Scalar>
ExpressionPtr<Scalar> log10(const ExpressionPtr<Scalar>& x) {
  using enum ExpressionType;
  using std::log10;
 
  // Prune expression
  if (x->is_constant(Scalar(0))) {
    // Return zero
    return x;
  }
 
  // Evaluate constant
  if (x->type() == CONSTANT) {
    return constant_ptr(log10(x->val));
  }
 
  return make_expression_ptr<Log10Expression<Scalar>>(x);
}
 
/// Derived expression type for max().
///
/// Returns the greater of a and b. If the values are equivalent, returns a.
///
/// @tparam Scalar Scalar type.
template <typename Scalar>
struct MaxExpression final : Expression<Scalar> {
  /// Constructs a binary expression (an operator with two arguments).
  ///
  /// @param lhs Binary operator's left operand.
  /// @param rhs Binary operator's right operand.
  constexpr MaxExpression(ExpressionPtr<Scalar> lhs, ExpressionPtr<Scalar> rhs)
      : Expression<Scalar>{std::move(lhs), std::move(rhs)} {}
 
  Scalar value(Scalar a, Scalar b) const override {
    using std::max;
    return max(a, b);
  }
 
  ExpressionType type() const override { return ExpressionType::NONLINEAR; }
 
  std::string_view name() const override { return "max"; }
 
  Scalar grad_l(Scalar a, Scalar b, Scalar parent_adjoint) const override {
    if (a >= b) {
      return parent_adjoint;
    } else {
      return Scalar(0);
    }
  }
 
  Scalar grad_r(Scalar a, Scalar b, Scalar parent_adjoint) const override {
    if (b > a) {
      return parent_adjoint;
    } else {
      return Scalar(0);
    }
  }
 
  ExpressionPtr<Scalar> grad_expr_l(
      const ExpressionPtr<Scalar>& a, const ExpressionPtr<Scalar>& b,
      const ExpressionPtr<Scalar>& parent_adjoint) const override {
    if (a->val >= b->val) {
      return parent_adjoint;
    } else {
      return constant_ptr(Scalar(0));
    }
  }
 
  ExpressionPtr<Scalar> grad_expr_r(
      const ExpressionPtr<Scalar>& a, const ExpressionPtr<Scalar>& b,
      const ExpressionPtr<Scalar>& parent_adjoint) const override {
    if (b->val > a->val) {
      return parent_adjoint;
    } else {
      return constant_ptr(Scalar(0));
    }
  }
};
 
/// max() for Expressions.
///
/// @tparam Scalar Scalar type.
/// @param a The a argument.
/// @param b The b argument.
template <typename Scalar>
ExpressionPtr<Scalar> max(const ExpressionPtr<Scalar>& a,
                          const ExpressionPtr<Scalar>& b) {
  using enum ExpressionType;
  using std::max;
 
  // Evaluate constant
  if (a->type() == CONSTANT && b->type() == CONSTANT) {
    return constant_ptr(max(a->val, b->val));
  }
 
  return make_expression_ptr<MaxExpression<Scalar>>(a, b);
}
 
/// Derived expression type for min().
///
/// Returns the lesser of a and b. If the values are equivalent, returns a.
///
/// @tparam Scalar Scalar type.
template <typename Scalar>
struct MinExpression final : Expression<Scalar> {
  /// Constructs a binary expression (an operator with two arguments).
  ///
  /// @param lhs Binary operator's left operand.
  /// @param rhs Binary operator's right operand.
  constexpr MinExpression(ExpressionPtr<Scalar> lhs, ExpressionPtr<Scalar> rhs)
      : Expression<Scalar>{std::move(lhs), std::move(rhs)} {}
 
  Scalar value(Scalar a, Scalar b) const override {
    using std::min;
    return min(a, b);
  }
 
  ExpressionType type() const override { return ExpressionType::NONLINEAR; }
 
  std::string_view name() const override { return "min"; }
 
  Scalar grad_l(Scalar a, Scalar b, Scalar parent_adjoint) const override {
    if (a <= b) {
      return parent_adjoint;
    } else {
      return Scalar(0);
    }
  }
 
  Scalar grad_r([[maybe_unused]] Scalar a, [[maybe_unused]] Scalar b,
                Scalar parent_adjoint) const override {
    if (b < a) {
      return parent_adjoint;
    } else {
      return Scalar(0);
    }
  }
 
  ExpressionPtr<Scalar> grad_expr_l(
      const ExpressionPtr<Scalar>& a, const ExpressionPtr<Scalar>& b,
      const ExpressionPtr<Scalar>& parent_adjoint) const override {
    if (a->val <= b->val) {
      return parent_adjoint;
    } else {
      return constant_ptr(Scalar(0));
    }
  }
 
  ExpressionPtr<Scalar> grad_expr_r(
      const ExpressionPtr<Scalar>& a, const ExpressionPtr<Scalar>& b,
      const ExpressionPtr<Scalar>& parent_adjoint) const override {
    if (b->val < a->val) {
      return parent_adjoint;
    } else {
      return constant_ptr(Scalar(0));
    }
  }
};
 
/// min() for Expressions.
///
/// @tparam Scalar Scalar type.
/// @param a The a argument.
/// @param b The b argument.
template <typename Scalar>
ExpressionPtr<Scalar> min(const ExpressionPtr<Scalar>& a,
                          const ExpressionPtr<Scalar>& b) {
  using enum ExpressionType;
  using std::min;
 
  // Evaluate constant
  if (a->type() == CONSTANT && b->type() == CONSTANT) {
    return constant_ptr(min(a->val, b->val));
  }
 
  return make_expression_ptr<MinExpression<Scalar>>(a, b);
}
 
template <typename Scalar>
ExpressionPtr<Scalar> pow(const ExpressionPtr<Scalar>& base,
                          const ExpressionPtr<Scalar>& power);
 
/// Derived expression type for pow().
///
/// @tparam Scalar Scalar type.
/// @tparam T Expression type.
template <typename Scalar, ExpressionType T>
struct PowExpression final : Expression<Scalar> {
  /// Constructs a binary expression (an operator with two arguments).
  ///
  /// @param lhs Binary operator's left operand.
  /// @param rhs Binary operator's right operand.
  constexpr PowExpression(ExpressionPtr<Scalar> lhs, ExpressionPtr<Scalar> rhs)
      : Expression<Scalar>{std::move(lhs), std::move(rhs)} {}
 
  Scalar value(Scalar base, Scalar power) const override {
    using std::pow;
    return pow(base, power);
  }
 
  ExpressionType type() const override { return T; }
 
  std::string_view name() const override { return "pow"; }
 
  Scalar grad_l(Scalar base, Scalar power,
                Scalar parent_adjoint) const override {
    using std::pow;
    return parent_adjoint * pow(base, power - Scalar(1)) * power;
  }
 
  Scalar grad_r(Scalar base, Scalar power,
                Scalar parent_adjoint) const override {
    using std::log;
    using std::pow;
 
    // Since x log(x) -> 0 as x -> 0
    if (base == Scalar(0)) {
      return Scalar(0);
    } else {
      return parent_adjoint * pow(base, power) * log(base);
    }
  }
 
  ExpressionPtr<Scalar> grad_expr_l(
      const ExpressionPtr<Scalar>& base, const ExpressionPtr<Scalar>& power,
      const ExpressionPtr<Scalar>& parent_adjoint) const override {
    return parent_adjoint * pow(base, power - constant_ptr(Scalar(1))) * power;
  }
 
  ExpressionPtr<Scalar> grad_expr_r(
      const ExpressionPtr<Scalar>& base, const ExpressionPtr<Scalar>& power,
      const ExpressionPtr<Scalar>& parent_adjoint) const override {
    // Since x log(x) -> 0 as x -> 0
    if (base->val == Scalar(0)) {
      // Return zero
      return base;
    } else {
      return parent_adjoint * pow(base, power) * log(base);
    }
  }
};
 
/// pow() for Expressions.
///
/// @tparam Scalar Scalar type.
/// @param base The base.
/// @param power The power.
template <typename Scalar>
ExpressionPtr<Scalar> pow(const ExpressionPtr<Scalar>& base,
                          const ExpressionPtr<Scalar>& power) {
  using enum ExpressionType;
  using std::pow;
 
  // Prune expression
  if (base->is_constant(Scalar(0))) {
    // Return zero
    return base;
  } else if (base->is_constant(Scalar(1))) {
    // Return one
    return base;
  }
  if (power->is_constant(Scalar(0))) {
    return constant_ptr(Scalar(1));
  } else if (power->is_constant(Scalar(1))) {
    return base;
  }
 
  // Evaluate constant
  if (base->type() == CONSTANT && power->type() == CONSTANT) {
    return constant_ptr(pow(base->val, power->val));
  }
 
  if (power->is_constant(Scalar(2))) {
    if (base->type() == LINEAR) {
      return make_expression_ptr<MultExpression<Scalar, QUADRATIC>>(base, base);
    } else {
      return make_expression_ptr<MultExpression<Scalar, NONLINEAR>>(base, base);
    }
  }
 
  return make_expression_ptr<PowExpression<Scalar, NONLINEAR>>(base, power);
}
 
/// Derived expression type for sign().
///
/// @tparam Scalar Scalar type.
template <typename Scalar>
struct SignExpression final : Expression<Scalar> {
  /// Constructs an unary expression (an operator with one argument).
  ///
  /// @param lhs Unary operator's operand.
  explicit constexpr SignExpression(ExpressionPtr<Scalar> lhs)
      : Expression<Scalar>{std::move(lhs)} {}
 
  Scalar value(Scalar x, Scalar) const override {
    if (x < Scalar(0)) {
      return Scalar(-1);
    } else if (x == Scalar(0)) {
      return Scalar(0);
    } else {
      return Scalar(1);
    }
  }
 
  ExpressionType type() const override { return ExpressionType::NONLINEAR; }
 
  std::string_view name() const override { return "sign"; }
};
 
/// sign() for Expressions.
///
/// @tparam Scalar Scalar type.
/// @param x The argument.
template <typename Scalar>
ExpressionPtr<Scalar> sign(const ExpressionPtr<Scalar>& x) {
  using enum ExpressionType;
 
  // Evaluate constant
  if (x->type() == CONSTANT) {
    if (x->val < Scalar(0)) {
      return constant_ptr(Scalar(-1));
    } else if (x->val == Scalar(0)) {
      // Return zero
      return x;
    } else {
      return constant_ptr(Scalar(1));
    }
  }
 
  return make_expression_ptr<SignExpression<Scalar>>(x);
}
 
/// Derived expression type for sin().
///
/// @tparam Scalar Scalar type.
template <typename Scalar>
struct SinExpression final : Expression<Scalar> {
  /// Constructs an unary expression (an operator with one argument).
  ///
  /// @param lhs Unary operator's operand.
  explicit constexpr SinExpression(ExpressionPtr<Scalar> lhs)
      : Expression<Scalar>{std::move(lhs)} {}
 
  Scalar value(Scalar x, Scalar) const override {
    using std::sin;
    return sin(x);
  }
 
  ExpressionType type() const override { return ExpressionType::NONLINEAR; }
 
  std::string_view name() const override { return "sin"; }
 
  Scalar grad_l(Scalar x, Scalar, Scalar parent_adjoint) const override {
    using std::cos;
    return parent_adjoint * cos(x);
  }
 
  ExpressionPtr<Scalar> grad_expr_l(
      const ExpressionPtr<Scalar>& x, const ExpressionPtr<Scalar>&,
      const ExpressionPtr<Scalar>& parent_adjoint) const override {
    return parent_adjoint * cos(x);
  }
};
 
/// sin() for Expressions.
///
/// @tparam Scalar Scalar type.
/// @param x The argument.
template <typename Scalar>
ExpressionPtr<Scalar> sin(const ExpressionPtr<Scalar>& x) {
  using enum ExpressionType;
  using std::sin;
 
  // Prune expression
  if (x->is_constant(Scalar(0))) {
    // Return zero
    return x;
  }
 
  // Evaluate constant
  if (x->type() == CONSTANT) {
    return constant_ptr(sin(x->val));
  }
 
  return make_expression_ptr<SinExpression<Scalar>>(x);
}
 
/// Derived expression type for sinh().
///
/// @tparam Scalar Scalar type.
template <typename Scalar>
struct SinhExpression final : Expression<Scalar> {
  /// Constructs an unary expression (an operator with one argument).
  ///
  /// @param lhs Unary operator's operand.
  explicit constexpr SinhExpression(ExpressionPtr<Scalar> lhs)
      : Expression<Scalar>{std::move(lhs)} {}
 
  Scalar value(Scalar x, Scalar) const override {
    using std::sinh;
    return sinh(x);
  }
 
  ExpressionType type() const override { return ExpressionType::NONLINEAR; }
 
  std::string_view name() const override { return "sinh"; }
 
  Scalar grad_l(Scalar x, Scalar, Scalar parent_adjoint) const override {
    using std::cosh;
    return parent_adjoint * cosh(x);
  }
 
  ExpressionPtr<Scalar> grad_expr_l(
      const ExpressionPtr<Scalar>& x, const ExpressionPtr<Scalar>&,
      const ExpressionPtr<Scalar>& parent_adjoint) const override {
    return parent_adjoint * cosh(x);
  }
};
 
/// sinh() for Expressions.
///
/// @tparam Scalar Scalar type.
/// @param x The argument.
template <typename Scalar>
ExpressionPtr<Scalar> sinh(const ExpressionPtr<Scalar>& x) {
  using enum ExpressionType;
  using std::sinh;
 
  // Prune expression
  if (x->is_constant(Scalar(0))) {
    // Return zero
    return x;
  }
 
  // Evaluate constant
  if (x->type() == CONSTANT) {
    return constant_ptr(sinh(x->val));
  }
 
  return make_expression_ptr<SinhExpression<Scalar>>(x);
}
 
/// Derived expression type for sqrt().
///
/// @tparam Scalar Scalar type.
template <typename Scalar>
struct SqrtExpression final : Expression<Scalar> {
  /// Constructs an unary expression (an operator with one argument).
  ///
  /// @param lhs Unary operator's operand.
  explicit constexpr SqrtExpression(ExpressionPtr<Scalar> lhs)
      : Expression<Scalar>{std::move(lhs)} {}
 
  Scalar value(Scalar x, Scalar) const override {
    using std::sqrt;
    return sqrt(x);
  }
 
  ExpressionType type() const override { return ExpressionType::NONLINEAR; }
 
  std::string_view name() const override { return "sqrt"; }
 
  Scalar grad_l(Scalar x, Scalar, Scalar parent_adjoint) const override {
    using std::sqrt;
    return parent_adjoint / (Scalar(2) * sqrt(x));
  }
 
  ExpressionPtr<Scalar> grad_expr_l(
      const ExpressionPtr<Scalar>& x, const ExpressionPtr<Scalar>&,
      const ExpressionPtr<Scalar>& parent_adjoint) const override {
    return parent_adjoint / (constant_ptr(Scalar(2)) * sqrt(x));
  }
};
 
/// sqrt() for Expressions.
///
/// @tparam Scalar Scalar type.
/// @param x The argument.
template <typename Scalar>
ExpressionPtr<Scalar> sqrt(const ExpressionPtr<Scalar>& x) {
  using enum ExpressionType;
  using std::sqrt;
 
  // Evaluate constant
  if (x->type() == CONSTANT) {
    if (x->val == Scalar(0)) {
      // Return zero
      return x;
    } else if (x->val == Scalar(1)) {
      return x;
    } else {
      return constant_ptr(sqrt(x->val));
    }
  }
 
  return make_expression_ptr<SqrtExpression<Scalar>>(x);
}
 
/// Derived expression type for tan().
///
/// @tparam Scalar Scalar type.
template <typename Scalar>
struct TanExpression final : Expression<Scalar> {
  /// Constructs an unary expression (an operator with one argument).
  ///
  /// @param lhs Unary operator's operand.
  explicit constexpr TanExpression(ExpressionPtr<Scalar> lhs)
      : Expression<Scalar>{std::move(lhs)} {}
 
  Scalar value(Scalar x, Scalar) const override {
    using std::tan;
    return tan(x);
  }
 
  ExpressionType type() const override { return ExpressionType::NONLINEAR; }
 
  std::string_view name() const override { return "tan"; }
 
  Scalar grad_l(Scalar x, Scalar, Scalar parent_adjoint) const override {
    using std::cos;
 
    auto c = cos(x);
    return parent_adjoint / (c * c);
  }
 
  ExpressionPtr<Scalar> grad_expr_l(
      const ExpressionPtr<Scalar>& x, const ExpressionPtr<Scalar>&,
      const ExpressionPtr<Scalar>& parent_adjoint) const override {
    auto c = cos(x);
    return parent_adjoint / (c * c);
  }
};
 
/// tan() for Expressions.
///
/// @tparam Scalar Scalar type.
/// @param x The argument.
template <typename Scalar>
ExpressionPtr<Scalar> tan(const ExpressionPtr<Scalar>& x) {
  using enum ExpressionType;
  using std::tan;
 
  // Prune expression
  if (x->is_constant(Scalar(0))) {
    // Return zero
    return x;
  }
 
  // Evaluate constant
  if (x->type() == CONSTANT) {
    return constant_ptr(tan(x->val));
  }
 
  return make_expression_ptr<TanExpression<Scalar>>(x);
}
 
/// Derived expression type for tanh().
///
/// @tparam Scalar Scalar type.
template <typename Scalar>
struct TanhExpression final : Expression<Scalar> {
  /// Constructs an unary expression (an operator with one argument).
  ///
  /// @param lhs Unary operator's operand.
  explicit constexpr TanhExpression(ExpressionPtr<Scalar> lhs)
      : Expression<Scalar>{std::move(lhs)} {}
 
  Scalar value(Scalar x, Scalar) const override {
    using std::tanh;
    return tanh(x);
  }
 
  ExpressionType type() const override { return ExpressionType::NONLINEAR; }
 
  std::string_view name() const override { return "tanh"; }
 
  Scalar grad_l(Scalar x, Scalar, Scalar parent_adjoint) const override {
    using std::cosh;
 
    auto c = cosh(x);
    return parent_adjoint / (c * c);
  }
 
  ExpressionPtr<Scalar> grad_expr_l(
      const ExpressionPtr<Scalar>& x, const ExpressionPtr<Scalar>&,
      const ExpressionPtr<Scalar>& parent_adjoint) const override {
    auto c = cosh(x);
    return parent_adjoint / (c * c);
  }
};
 
/// tanh() for Expressions.
///
/// @tparam Scalar Scalar type.
/// @param x The argument.
template <typename Scalar>
ExpressionPtr<Scalar> tanh(const ExpressionPtr<Scalar>& x) {
  using enum ExpressionType;
  using std::tanh;
 
  // Prune expression
  if (x->is_constant(Scalar(0))) {
    // Return zero
    return x;
  }
 
  // Evaluate constant
  if (x->type() == CONSTANT) {
    return constant_ptr(tanh(x->val));
  }
 
  return make_expression_ptr<TanhExpression<Scalar>>(x);
}
 
}  // namespace slp::detail