symjit 2.16.0

use std::collections::{HashMap, HashSet};

use anyhow::{anyhow, Result};
use num_complex::Complex;

use crate::applet::{recast_as_f64, recast_as_f64_mut};
use crate::code::VirtualTable;
use crate::composer::Composer;
use crate::config::{Config, SLICE_CAP};
use crate::defuns::Defuns;
use crate::expr::Expr;
use crate::instruction::{BuiltinSymbol, Instruction, Slot, SymbolicaModel};
use crate::model::{CellModel, Equation, Program, Variable};
use crate::parser::Parser;
use crate::symbol::Loc;
use crate::types::Element;
use crate::utils::Compiled;
use crate::Application;

// #[derive(Debug)]
pub struct Compiler {
    config: Config,
}

#[cfg(not(target_arch = "x86_64"))]
#[allow(non_camel_case_types)]
type __m256d = [f64; 4];

/// The central hub of the Rust interface. It compiles a list of
/// variables and expressions into a callable object (of type `Application`).
///
/// # Workflow
///
/// 1. Create terminals (variables and constants) and compose expressions using `Expr` methods:
///    * Constructors: `var`, `from`, `unary`, `binary`, ...
///    * Standard algebraic operations: `add`, `mul`, ...
///    * Standard operators `+`, `-`, `*`, `/`, `%`, `&`, `|`, `^`, `!`.
///    * Unary functions such as `sin`, `exp`, and other standard mathematical functions.
///    * Binary functions such as `pow`, `min`, ...
///    * IfElse operation `ifelse(cond, true_val, false_val)`.
///    * Heavide function: `heaviside(x)`, which returns 1 if `x >= 0`; otherwise 0.
///    * Comparison methods `eq`, `ne`, `lt`, `le`, `gt`, and `ge`.
///    * Looping constructs `sum` and `prod`.
/// 2. Create a new `Compiler` object (say, `comp`) using one of its constructors: `new()`
///    or `with_compile_type(ty: CompilerType)`.
/// 3. Fine-tune the optimization passes using `opt_level`, `simd`, `fastmath`,
///    and `cse` methods (optional).
/// 4. Define user-defined functions by called `comp.def_unary` and `comp.def_binary`
///    (optional).
/// 5. Compile by calling `comp.compile` or `comp.compile_params`. The result is of
///    type `Application` (say, `app`).
/// 6. Execute the compiled code using one of the `app`'s `call` functions:
///    * `call(&[f64])`: scalar call.
///    * `call_params(&[f64], &[f64])`: scalar call with parameters.
///    * `call_simd(&[__m256d])`: simd call.
///    * `call_simd_params(&[__m256d], &[f64])`: simd call with parameters.
/// 7. Optionally, generate a standalone fast function to execute.
///
///
/// # Examples
///
/// ```rust
/// use anyhow::Result;
/// use symjit::{Compiler, Expr};
///
/// pub fn main() -> Result<()> {
///     let x = Expr::var("x");
///     let y = Expr::var("y");
///     let u = &x + &y;
///     let v = &x * &y;
///
///     let mut config = Config::default();
///     config.set_opt_level(2);
///     let mut comp = Compiler::with_config(config);
///     let mut app = comp.compile(&[x, y], &[u, v])?;
///     let res = app.call(&[3.0, 5.0]);
///     println!("{:?}", &res);
///
///     Ok(())
/// }
/// ```
impl Compiler {
    /// Creates a new `Compiler` object with default settings.
    pub fn new() -> Compiler {
        Compiler {
            config: Config::default(),
        }
    }

    pub fn with_config(config: Config) -> Compiler {
        Compiler { config }
    }

    /// Compiles a model.
    ///
    /// `states` is a list of variables, created by `Expr::var`.
    /// `obs` is a list of expressions.
    pub fn compile(&mut self, states: &[Expr], obs: &[Expr]) -> Result<Application> {
        self.compile_params(states, obs, &[])
    }

    /// Compiles a model with parameters.
    ///
    /// `states` is a list of variables, created by `Expr::var`.
    /// `obs` is a list of expressions.
    /// `params` is a list of parameters, created by `Expr::var`.
    ///
    /// Note: for scalar functions, the difference between states and params
    ///     is mostly by convenion. However, they are different in SIMD cases,
    ///     as params are always f64.
    pub fn compile_params(
        &mut self,
        states: &[Expr],
        obs: &[Expr],
        params: &[Expr],
    ) -> Result<Application> {
        let mut vars: Vec<Variable> = Vec::new();

        for state in states.iter() {
            let v = state.to_variable()?;
            vars.push(v);
        }

        let mut ps: Vec<Variable> = Vec::new();

        for p in params.iter() {
            let v = p.to_variable()?;
            ps.push(v);
        }

        let mut eqs: Vec<Equation> = Vec::new();

        for (i, expr) in obs.iter().enumerate() {
            let name = format!("${}", i);
            let lhs = Expr::var(&name);
            eqs.push(Expr::equation(&lhs, expr));
        }

        let ml = CellModel {
            iv: Expr::var("$_").to_variable()?,
            params: ps,
            states: vars,
            algs: Vec::new(),
            odes: Vec::new(),
            obs: eqs,
        };

        let prog = Program::new(&ml, self.config.clone())?;
        // let df = Defuns::new();
        let app = Application::new(prog, HashSet::new())?;
        // app.prepare_simd();

        // #[cfg(target_arch = "aarch64")]
        // if let Ok(app) = &app {
        //     // this is a hack to give enough delay to prevent a bus error
        //     app.dump("dump.bin", "scalar");
        //     std::fs::remove_file("dump.bin")?;
        // };

        Ok(app)
    }
}

pub enum FastFunc<'a> {
    F1(extern "C" fn(f64) -> f64, &'a Application),
    F2(extern "C" fn(f64, f64) -> f64, &'a Application),
    F3(extern "C" fn(f64, f64, f64) -> f64, &'a Application),
    F4(extern "C" fn(f64, f64, f64, f64) -> f64, &'a Application),
    F5(
        extern "C" fn(f64, f64, f64, f64, f64) -> f64,
        &'a Application,
    ),
    F6(
        extern "C" fn(f64, f64, f64, f64, f64, f64) -> f64,
        &'a Application,
    ),
    F7(
        extern "C" fn(f64, f64, f64, f64, f64, f64, f64) -> f64,
        &'a Application,
    ),
    F8(
        extern "C" fn(f64, f64, f64, f64, f64, f64, f64, f64) -> f64,
        &'a Application,
    ),
}

impl Application {
    /// Calls the compiled function.
    ///
    /// `args` is a slice of f64 values, corresponding to the states.
    ///
    /// The output is a `Vec<f64>`, corresponding to the observables (the expressions passed
    /// to `compile`).
    pub fn call(&mut self, args: &[f64]) -> Vec<f64> {
        if let Some(f) = &mut self.compiled {
            {
                let mem = f.mem_mut();
                let states = &mut mem[self.first_state..self.first_state + self.count_states];
                states.copy_from_slice(args);
            }

            f.exec(&self.params[..]);

            let obs = {
                let mem = f.mem();
                &mem[self.first_obs..self.first_obs + self.count_obs]
            };

            obs.to_vec()
        } else {
            Vec::new()
        }
    }

    /// Sets the params and calls the compiled function.
    ///
    /// `args` is a slice of f64 values, corresponding to the states.
    /// `params` is a slice of f64 values, corresponding to the params.
    ///
    /// The output is a `Vec<f64>`, corresponding to the observables (the expressions passed
    /// to `compile`).
    pub fn call_params(&mut self, args: &[f64], params: &[f64]) -> Vec<f64> {
        if let Some(f) = &mut self.compiled {
            {
                let mem = f.mem_mut();
                let states = &mut mem[self.first_state..self.first_state + self.count_states];
                states.copy_from_slice(args);
            }

            f.exec(params);

            let obs = {
                let mem = f.mem();
                &mem[self.first_obs..self.first_obs + self.count_obs]
            };

            obs.to_vec()
        } else {
            Vec::new()
        }
    }

    pub fn interpret<T>(&mut self, args: &[T], outs: &mut [T])
    where
        T: Element,
    {
        let args = recast_as_f64(args);
        let outs = recast_as_f64_mut(outs);

        let mut regs = [0.0; 32];
        self.bytecode
            .mir
            .exec_instruction(outs, &mut self.bytecode.stack, &mut regs, args);
    }

    pub fn interpret_matrix(&mut self, args: &[f64], outs: &mut [f64], n: usize) {
        let count_params = self.count_params;
        let count_obs = self.count_obs;

        for i in 0..n {
            self.interpret(
                &args[i * count_params..(i + 1) * count_params],
                &mut outs[i * count_obs..(i + 1) * count_obs],
            );
        }
    }

    /// Generic evaluate function for compiled Symbolica expressions
    pub fn evaluate<T>(&self, args: &[T], outs: &mut [T])
    where
        T: Element,
    {
        self.as_applet().evaluate(args, outs);
    }

    /// Generic evaluate_single function for compiled Symbolica expressions
    #[inline(always)]
    pub fn evaluate_single<T>(&self, args: &[T]) -> T
    where
        T: Element + Copy,
    {
        self.as_applet().evaluate_single(args)
    }

    /// Generic evaluate function for compiled Symbolica expressions
    /// The main entry point to compute matrices.
    /// The actual dispatched method depends on the configuration and the
    /// type of the arguments.
    pub fn evaluate_matrix<T>(&self, args: &[T], outs: &mut [T], n: usize)
    where
        T: Element,
    {
        self.as_applet().evaluate_matrix(args, outs, n);
    }

    /// Returns a fast function.
    ///
    /// `Application` call functions need to copy the input argument slice into
    /// the function memory area and then copy the output to a `Vec`. This process
    /// is acceptable for large and complex functions but incurs a penalty for
    /// small functions. Therefore, for a certain subset of applications, Symjit
    /// can compile a fast funcction and return a function pointer. Examples:
    ///
    /// ```rust
    /// fn test_fast() -> Result<()> {
    ///     let x = Expr::var("x");
    ///     let y = Expr::var("y");
    ///     let z = Expr::var("z");
    ///     let u = &x * &(&y - &z).pow(&Expr::from(2));
    ///
    ///     let mut comp = Compiler::new();
    ///     let mut app = comp.compile(&[x, y, z], &[u])?;
    ///     let f = app.fast_func()?;
    ///
    ///     if let FastFunc::F3(f, _) = f {
    ///         let res = f(3.0, 5.0, 9.0);
    ///         println!("fast\t{:?}", &res);
    ///     }
    ///
    ///     Ok(())
    /// }
    /// ```
    ///
    /// The conditions for a fast function are:
    ///
    /// * A fast function can have 1 to 8 arguments.
    /// * No SIMD and no parameters.
    /// * It returns only a single value.
    ///
    /// If these conditions are met, you can generate a fast functin by calling
    /// `app.fast_func()`, with a return type of `Result<FastFunc>`. `FastFunc` is an
    /// enum with eight variants `F1, `F2`, ..., `F8`, corresponding to
    /// functions with 1 to 8 arguments.
    ///
    pub fn fast_func(&mut self) -> Result<FastFunc<'_>> {
        let f = self.get_fast();

        if let Some(f) = f {
            match self.count_states {
                1 => {
                    let g: extern "C" fn(f64) -> f64 = unsafe { std::mem::transmute(f) };
                    Ok(FastFunc::F1(g, self))
                }
                2 => {
                    let g: extern "C" fn(f64, f64) -> f64 = unsafe { std::mem::transmute(f) };
                    Ok(FastFunc::F2(g, self))
                }
                3 => {
                    let g: extern "C" fn(f64, f64, f64) -> f64 = unsafe { std::mem::transmute(f) };
                    Ok(FastFunc::F3(g, self))
                }
                4 => {
                    let g: extern "C" fn(f64, f64, f64, f64) -> f64 =
                        unsafe { std::mem::transmute(f) };
                    Ok(FastFunc::F4(g, self))
                }
                5 => {
                    let g: extern "C" fn(f64, f64, f64, f64, f64) -> f64 =
                        unsafe { std::mem::transmute(f) };
                    Ok(FastFunc::F5(g, self))
                }
                6 => {
                    let g: extern "C" fn(f64, f64, f64, f64, f64, f64) -> f64 =
                        unsafe { std::mem::transmute(f) };
                    Ok(FastFunc::F6(g, self))
                }
                7 => {
                    let g: extern "C" fn(f64, f64, f64, f64, f64, f64, f64) -> f64 =
                        unsafe { std::mem::transmute(f) };
                    Ok(FastFunc::F7(g, self))
                }
                8 => {
                    let g: extern "C" fn(f64, f64, f64, f64, f64, f64, f64, f64) -> f64 =
                        unsafe { std::mem::transmute(f) };
                    Ok(FastFunc::F8(g, self))
                }
                _ => Err(anyhow!("not a fast function")),
            }
        } else {
            Err(anyhow!("not a fast function"))
        }
    }
}

/************************* Symbolica *****************************/

/// Translates Symbolica IR (generated by export_instructions) into a Symjit Model
#[derive(Debug, Clone)]
pub struct Translator {
    config: Config,
    ssa: Vec<Instruction>,
    consts: Vec<Complex<f64>>, // constants
    count_params: usize,
    count_statics: usize,
    eqs: Vec<Equation>,            // Symjit Equations (output)
    temps: HashMap<usize, Slot>,   // Temp idx => Static idx
    counts: HashMap<usize, usize>, // Static idx => number of usage on the RHS
    cache: HashMap<usize, Expr>,   // cache of Static variables (Static idx => Expr)
    outs: HashMap<usize, Expr>,    // cache of Outs (Out idx => Expr)
    reals: HashSet<Loc>,
    num_params: usize,
    has_jump: bool,
    last_label: usize,
    depth: usize,
    conds: Vec<Slot>,
}

impl Composer for Translator {
    fn append_constant(&mut self, z: Complex<f64>) -> Result<usize> {
        self.consts.push(z);
        Ok(self.consts.len() - 1)
    }

    fn append_add(&mut self, lhs: &Slot, args: &[Slot], num_reals: usize) -> Result<()> {
        let args = self.consume_list(args)?;
        let lhs = self.produce(lhs)?;
        self.ssa.push(Instruction::Add(lhs, args, num_reals));
        Ok(())
    }

    fn append_mul(&mut self, lhs: &Slot, args: &[Slot], num_reals: usize) -> Result<()> {
        let args = self.consume_list(args)?;
        let lhs = self.produce(lhs)?;
        self.ssa.push(Instruction::Mul(lhs, args, num_reals));
        Ok(())
    }

    fn append_pow(&mut self, lhs: &Slot, arg: &Slot, p: i64, is_real: bool) -> Result<()> {
        let arg = self.consume(arg)?;
        let lhs = self.produce(lhs)?;
        self.ssa.push(Instruction::Pow(lhs, arg, p, is_real));
        Ok(())
    }

    fn append_powf(&mut self, lhs: &Slot, arg: &Slot, p: &Slot, is_real: bool) -> Result<()> {
        let arg = self.consume(arg)?;
        let p = self.consume(p)?;
        let lhs = self.produce(lhs)?;
        self.ssa.push(Instruction::Powf(lhs, arg, p, is_real));
        Ok(())
    }

    fn append_assign(&mut self, lhs: &Slot, rhs: &Slot) -> Result<()> {
        let rhs = self.consume(rhs)?;
        let lhs = self.produce(lhs)?;
        self.ssa.push(Instruction::Assign(lhs, rhs));
        Ok(())
    }

    fn append_label(&mut self, id: usize) -> Result<()> {
        self.ssa.push(Instruction::Label(id));
        Ok(())
    }

    fn append_if_else(&mut self, cond: &Slot, id: usize) -> Result<()> {
        self.has_jump = true;
        let cond = self.consume(cond)?;
        self.ssa.push(Instruction::IfElse(cond, id));
        Ok(())
    }

    fn append_goto(&mut self, id: usize) -> Result<()> {
        self.last_label = self.last_label.max(id);
        self.ssa.push(Instruction::Goto(id));
        Ok(())
    }

    fn append_external_fun(&mut self, lhs: &Slot, op: &str, args: &[Slot]) -> Result<()> {
        let args = self.consume_list(args)?;
        let lhs = self.produce(lhs)?;
        self.ssa
            .push(Instruction::ExternalFun(lhs, op.to_string(), args));
        Ok(())
    }

    fn append_fun(
        &mut self,
        lhs: &Slot,
        fun: &BuiltinSymbol,
        arg: &Slot,
        is_real: bool,
    ) -> Result<()> {
        let arg = self.consume(arg)?;
        let lhs = self.produce(lhs)?;
        self.ssa.push(Instruction::Fun(lhs, *fun, arg, is_real));
        Ok(())
    }

    fn append_join(
        &mut self,
        lhs: &Slot,
        cond: &Slot,
        true_val: &Slot,
        false_val: &Slot,
    ) -> Result<()> {
        let cond = self.consume(cond)?;
        let true_val = self.consume(true_val)?;
        let false_val = self.consume(false_val)?;
        let lhs = self.produce(lhs)?;
        self.ssa
            .push(Instruction::Join(lhs, cond, true_val, false_val));
        Ok(())
    }

    fn set_num_params(&mut self, num_params: usize) {
        self.num_params = num_params
    }

    fn compile(&mut self) -> Result<Application> {
        let (ml, reals) = self.translate()?;
        let prog = Program::new(&ml, self.config.clone())?;
        let mut app = Application::new(prog, reals)?;
        app.prepare_simd();
        Ok(app)
    }
}

impl Translator {
    pub fn new(config: Config) -> Translator {
        Translator {
            config,
            ssa: Vec::new(),
            consts: Vec::new(),
            count_params: 0,
            count_statics: 0,
            eqs: Vec::new(),
            temps: HashMap::new(),
            counts: HashMap::new(),
            cache: HashMap::new(),
            outs: HashMap::new(),
            reals: HashSet::new(),
            num_params: 0,
            has_jump: false,
            last_label: 0,
            depth: 0,
            conds: Vec::new(),
        }
    }

    pub fn parse_model(&mut self, model: &SymbolicaModel) -> Result<()> {
        for c in model.2.iter() {
            let val = Complex::new(c.value().re, c.value().im);
            self.consts.push(val);
        }

        self.convert(model)?;
        Ok(())
    }

    /// The first pass by converting Symbolica IR into
    /// Static-Single-Assingment (SSA) Form
    fn convert(&mut self, model: &SymbolicaModel) -> Result<()> {
        for line in model.0.iter() {
            match line {
                Instruction::Add(lhs, args, num_reals) => self.append_add(lhs, args, *num_reals)?,
                Instruction::Mul(lhs, args, num_reals) => self.append_mul(lhs, args, *num_reals)?,
                Instruction::Pow(lhs, arg, p, is_real) => {
                    self.append_pow(lhs, arg, *p, *is_real)?
                }
                Instruction::Powf(lhs, arg, p, is_real) => {
                    self.append_powf(lhs, arg, p, *is_real)?
                }
                Instruction::Assign(lhs, rhs) => self.append_assign(lhs, rhs)?,
                Instruction::Fun(lhs, fun, arg, is_real) => {
                    self.append_fun(lhs, fun, arg, *is_real)?
                }
                Instruction::Join(lhs, cond, true_val, false_val) => {
                    self.depth -= 1;
                    self.append_join(lhs, cond, true_val, false_val)?
                }
                Instruction::Label(id) => self.append_label(*id)?,
                Instruction::IfElse(cond, id) => {
                    self.append_if_else(cond, *id)?;
                    self.depth += 1;
                }
                Instruction::Goto(id) => self.append_goto(*id)?,
                Instruction::ExternalFun(lhs, op, args) => {
                    self.append_external_fun(lhs, op, args)?
                }
            }
        }

        Ok(())
    }

    fn create_static(&mut self) -> Result<Slot> {
        let s = Slot::Static(self.count_statics);
        self.counts.insert(self.count_statics, 0);
        self.count_statics += 1;
        Ok(s)
    }

    /// Produces a new Static variable if needed.
    /// slot should be an LHS.
    fn produce(&mut self, slot: &Slot) -> Result<Slot> {
        match slot {
            Slot::Temp(idx) => {
                if self.depth > 0 {
                    if let Some(Slot::Static(s)) = self.temps.get(idx) {
                        *self.counts.get_mut(s).unwrap() += 1;
                        return Ok(Slot::Static(*s));
                    }
                }

                let s = self.create_static()?;
                self.temps.insert(*idx, s);
                Ok(s)
            }
            Slot::Out(idx) => Ok(Slot::Out(*idx)),
            _ => Err(anyhow!("unacceptable lhs.")),
        }
    }

    /// Consumes a slot.
    /// slot should be an RHS.
    fn consume(&mut self, slot: &Slot) -> Result<Slot> {
        match slot {
            Slot::Temp(idx) => {
                if let Some(Slot::Static(s)) = self.temps.get(idx) {
                    *self.counts.get_mut(s).unwrap() += 1;
                    Ok(Slot::Static(*s))
                } else {
                    Err(anyhow!("Not a static reg."))
                }
            }
            Slot::Out(idx) => Ok(Slot::Out(*idx)),
            Slot::Param(idx) => Ok(Slot::Param(*idx)),
            Slot::Const(idx) => Ok(Slot::Const(*idx)),
            Slot::Static(_) | Slot::Arg(_) => Err(anyhow!("Undefined Static/Arg.")),
        }
    }

    fn consume_list(&mut self, slots: &[Slot]) -> Result<Vec<Slot>> {
        slots.iter().map(|s| self.consume(s)).collect()
    }

    /// The second pass. It translates the SSA-form into a Symjit model.
    pub fn translate(&mut self) -> Result<(CellModel, HashSet<Loc>)> {
        let ssa = std::mem::take(&mut self.ssa);

        for line in ssa.iter() {
            match line {
                Instruction::Add(lhs, args, n) => self.translate_nary("plus", lhs, args, *n)?,
                Instruction::Mul(lhs, args, n) => self.translate_nary("times", lhs, args, *n)?,
                Instruction::Pow(lhs, arg, p, is_real) => {
                    let p = Expr::from(*p as f64);
                    self.translate_pow(lhs, arg, &p, *is_real)?
                }
                Instruction::Powf(lhs, arg, p, is_real) => {
                    let p = self.expr(p, false);
                    self.translate_pow(lhs, arg, &p, *is_real)?
                }
                Instruction::Assign(lhs, rhs) => self.translate_assign(lhs, rhs)?,
                Instruction::Fun(lhs, fun, arg, is_real) => {
                    self.translate_fun(lhs, fun, arg, *is_real)?
                }
                Instruction::Join(lhs, cond, true_val, false_val) => {
                    self.translate_join(lhs, cond, true_val, false_val)?
                }
                Instruction::Label(id) => self.translate_label(*id)?,
                Instruction::IfElse(cond, id) => self.translate_ifelse(cond, *id)?,
                Instruction::Goto(id) => self.translate_goto(*id)?,
                Instruction::ExternalFun(lhs, op, args) => {
                    self.translate_external_fun(lhs, op, args)?
                }
            }
        }

        // Important! Outs are cached and should be written to final outputs.
        for k in 0..self.outs.len() {
            let out = Expr::var(&format!("Out{}", k));

            if let Some(eq) = self.outs.get(&k) {
                self.eqs.push(Expr::equation(&out, eq));
            }
        }

        let mut params: Vec<Variable> = (0..=self.count_params.max(self.num_params.max(1) - 1))
            .map(|idx| self.expr(&Slot::Param(idx), false).to_variable().unwrap())
            .collect();

        let mut states: Vec<Variable> = Vec::new();

        if !self.config.symbolica() {
            (params, states) = (states, params)
        }

        Ok((
            CellModel {
                iv: Expr::var("$_").to_variable().unwrap(),
                params,
                states,
                algs: Vec::new(),
                odes: Vec::new(),
                obs: self.eqs.clone(),
            },
            self.reals.clone(),
        ))
    }

    // The counterpart of consume for the second-pass
    fn expr(&mut self, slot: &Slot, is_real: bool) -> Expr {
        match slot {
            Slot::Param(idx) => {
                if is_real {
                    self.reals.insert(Loc::Param(*idx as u32));
                }
                self.count_params = self.count_params.max(*idx);
                Expr::var(&format!("Param{}", idx))
            }
            Slot::Out(idx) => {
                if let Some(e) = self.outs.get(idx) {
                    e.clone()
                } else {
                    Expr::var(&format!("Out{}", idx))
                }
            }
            Slot::Temp(idx) => Expr::var(&format!("__Temp{}", idx)),
            Slot::Const(idx) => {
                let val = self.consts[*idx];
                if val.im != 0.0 {
                    Expr::binary("complex", &Expr::from(val.re), &Expr::from(val.im))
                } else {
                    Expr::from(self.consts[*idx].re)
                }
            }
            Slot::Static(idx) => self
                .cache
                .remove(idx)
                .unwrap_or(Expr::var(&format!("__Static{}", idx))),
            Slot::Arg(idx) => Expr::var(&format!("__Arg{}", idx)),
        }
    }

    // The counterpart of produce for the second-pass
    fn assign(&mut self, lhs: &Slot, rhs: Expr) -> Result<()> {
        if !self.has_jump {
            if let Slot::Static(idx) = lhs {
                // Important! If a static variable is used only once, it
                // is pushed into the cache to be incorporated into the
                // destination expression tree.
                if self.counts.get(idx).is_some_and(|c| *c == 1) {
                    self.cache.insert(*idx, rhs);
                    return Ok(());
                }
            }

            if let Slot::Out(idx) = lhs {
                self.outs.insert(*idx, rhs.clone());
                return Ok(());
            }
        }

        let lhs = self.expr(lhs, false);
        self.eqs.push(Expr::equation(&lhs, &rhs));
        Ok(())
    }

    fn translate_nary(&mut self, op: &str, lhs: &Slot, args: &[Slot], n: usize) -> Result<()> {
        let args: Vec<Expr> = args
            .iter()
            .enumerate()
            .map(|(i, x)| self.expr(x, i < n))
            .collect();
        let p: Vec<&Expr> = args.iter().collect();

        if n == 0 || n >= p.len() {
            self.assign(lhs, Expr::nary(op, &p))
        } else {
            let l = Expr::nary(op, &p[..n]);
            let r = Expr::nary(op, &p[n..]);
            self.assign(lhs, Expr::nary(op, &[&l, &r]))
        }
    }

    fn translate_pow(&mut self, lhs: &Slot, arg: &Slot, power: &Expr, is_real: bool) -> Result<()> {
        let arg = self.expr(arg, is_real);
        self.assign(lhs, Expr::binary("power", &arg, power))
    }

    fn translate_assign(&mut self, lhs: &Slot, rhs: &Slot) -> Result<()> {
        let rhs = self.expr(rhs, false);
        self.assign(lhs, rhs)
    }

    fn translate_fun(
        &mut self,
        lhs: &Slot,
        fun: &BuiltinSymbol,
        arg: &Slot,
        is_real: bool,
    ) -> Result<()> {
        let arg = self.expr(arg, is_real);

        let op = match fun.0 {
            2 => "exp",
            3 => "ln",
            4 => "sin",
            5 => "cos",
            6 => {
                if is_real {
                    "real_root"
                } else {
                    "root"
                }
            }
            7 => "conjugate",
            _ => return Err(anyhow!("function is not defined.")),
        };

        self.assign(lhs, Expr::unary(op, &arg))
    }

    fn translate_external_fun(&mut self, lhs: &Slot, op: &str, args: &[Slot]) -> Result<()> {
        let n = args.len();
        assert!(n <= SLICE_CAP);
        let args: Vec<Expr> = args.iter().map(|a| self.expr(a, false)).collect();

        if VirtualTable::from_str(op).is_ok() {
            if n == 1 {
                self.assign(lhs, Expr::unary(op, &args[0]))?;
            } else if n == 2 {
                self.assign(lhs, Expr::binary(op, &args[0], &args[1]))?;
            } else {
                return Err(anyhow!("wrong number of arguments to {:?}", op));
            }
        } else if self.config.is_intrinsic_unary(op) && n == 1 {
            self.assign(lhs, Expr::unary(op, &args[0]))?;
        } else if self.config.is_intrinsic_binary(op) && n == 2 {
            self.assign(lhs, Expr::binary(op, &args[0], &args[1]))?;
        } else {
            let temps: Vec<Slot> = (0..n).map(|_| self.create_static().unwrap()).collect();
            let slice: Vec<Slot> = (0..n).map(Slot::Arg).collect();

            for i in 0..n {
                self.assign(&temps[i], args[i].clone())?;
            }

            for i in 0..n {
                if let Slot::Static(idx) = temps[i] {
                    self.assign(&slice[i], Expr::var(&format!("__Static{}", idx)))?;
                }
            }

            let op = format!("${}", op);
            self.assign(
                lhs,
                Expr::binary(&op, &Expr::from(0), &Expr::from(n as i32)),
            )?;
        }

        Ok(())
    }

    fn translate_label(&mut self, id: usize) -> Result<()> {
        self.eqs.push(Expr::special(&Expr::Label { id }));
        Ok(())
    }

    fn translate_ifelse(&mut self, cond: &Slot, id: usize) -> Result<()> {
        // let cond = Expr::binary(
        //     "lt",
        //     &Expr::unary("abs", &self.expr(cond, false)),
        //     &Expr::from(f64::EPSILON),
        // );

        self.conds.push(*cond);
        let if_clause = Expr::binary("eq", &self.expr(cond, false), &Expr::from(0.0));
        self.eqs.push(Expr::special(&Expr::BranchIf {
            cond: Box::new(if_clause),
            id,
            is_else: false,
        }));
        Ok(())
    }

    fn translate_goto(&mut self, id: usize) -> Result<()> {
        if self.config.simd_branch() {
            // TODO: the commented out area should be uncommented, except it causes
            // a bug in some programs. The effects are not local and likely related
            // to instruction movements.

            let cond = self.conds.pop().unwrap();
            self.conds.push(cond);
            let if_clause = Expr::binary("eq", &self.expr(&cond, false), &Expr::from(0.0));
            self.eqs.push(Expr::special(&Expr::BranchIf {
                cond: Box::new(if_clause),
                id,
                is_else: true,
            }));
        } else {
            self.eqs.push(Expr::special(&Expr::Branch { id }));
        }

        Ok(())
    }

    fn translate_join(
        &mut self,
        lhs: &Slot,
        _cond: &Slot,
        true_val: &Slot,
        false_val: &Slot,
    ) -> Result<()> {
        // Join is essentially a Φ-function.
        let t = self.expr(true_val, false);
        let f = self.expr(false_val, false);
        let cond = self.conds.pop().unwrap();
        let mask = Expr::binary("eq", &self.expr(&cond, false), &Expr::from(0.0));
        self.assign(lhs, mask.ifelse(&f, &t))?;
        Ok(())
    }
}

impl Compiler {
    /// Compiles a Symbolica model.
    ///
    /// `json` is the JSON-encoded output of Symbolica `export_instructions`.
    ///
    /// Example:
    ///
    /// ```rust
    /// let params = vec![parse!("x"), parse!("y")];
    /// let eval = parse!("x + y^2")
    ///     .evaluator(&FunctionMap::new(), &params, OptimizationSettings::default())?
    ///
    /// let json = serde_json::to_string(&eval.export_instructions())?;
    /// let mut comp = Compiler::new();
    /// let mut app = comp.translate(&json)?;
    /// assert!(app.evaluate_single(&[2.0, 3.0]) == 11.0);
    /// ```
    pub fn translate(&mut self, json: String, num_params: usize) -> Result<Application> {
        let mut translator = Translator::new(self.config.clone());

        let model: SymbolicaModel = if json.starts_with("[[{") {
            serde_json::from_str(json.as_str())?
        } else {
            Parser::new(json).parse()?
        };

        translator.parse_model(&model)?;
        translator.set_num_params(num_params);
        let (ml, reals) = translator.translate()?;

        let prog = Program::new(&ml, translator.config)?;
        let mut app = Application::new(prog, reals)?;

        app.prepare_simd();
        Ok(app)
    }
}