diff --git a/include/triton/driver/module.h b/include/triton/driver/module.h
index d62142c7d..2e6a390e9 100755
--- a/include/triton/driver/module.h
+++ b/include/triton/driver/module.h
@@ -74,6 +74,7 @@ public:
   cu_module(driver::context* context, std::unique_ptr<llvm::Module> module);
   cu_module(driver::context* context, const std::string& source);
   std::unique_ptr<buffer> symbol(const char * name) const;
+  const std::string& source() const { return source_; }
 
 private:
   std::string source_;
diff --git a/include/triton/runtime/arg.h b/include/triton/runtime/arg.h
index 6e255f0e7..1a741077c 100644
--- a/include/triton/runtime/arg.h
+++ b/include/triton/runtime/arg.h
@@ -7,6 +7,9 @@
 #include <stdexcept>
 
 namespace triton{
+namespace ir{
+  class type;
+}
 
 namespace driver{
   class buffer;
@@ -26,6 +29,9 @@ enum arg_type {
   BUFFER_T
 };
 
+arg_type convert(ir::type *ty);
+
+
 inline size_t size_of(arg_type ty){
   switch(ty){
     case INT1_T: return 1;
diff --git a/include/triton/runtime/function.h b/include/triton/runtime/function.h
index 26253b7ee..240e4944f 100644
--- a/include/triton/runtime/function.h
+++ b/include/triton/runtime/function.h
@@ -8,6 +8,7 @@
 #include <string>
 #include <memory>
 #include <functional>
+#include <set>
 // codegen
 #include "triton/ir/context.h"
 #include "triton/codegen/target.h"
@@ -68,6 +69,11 @@ public:
     T D(const std::string& name) const {
       return convert<T>(defines.at(name));
     }
+    bool operator<(const options_t& other) const {
+      return std::make_pair(defines, num_warps) <
+             std::make_pair(other.defines, other.num_warps);
+    }
+    std::string to_str() const;
 
     std::map<std::string, std::string> defines;
     size_t num_warps;
@@ -79,41 +85,63 @@ public:
 private:
   class caller {
   public:
-    caller(ir::function *ir, std::shared_ptr<driver::module> program, const options_t& opt_);
-    void operator()(driver::stream *stream, const grid_t& grid, const std::vector<arg>& args) const;
+    // constructors
+    caller(driver::context* ctx, std::ifstream& ifs, const options_t& opt);
+    caller(ir::function *ir, std::shared_ptr<driver::module> program, const options_t& opt);
+    // serialization
+    void write(std::ofstream& ofs);
+    void read(driver::context* ctx, std::ifstream& ifs);
+    // accessors
     const options_t opt() const { return opt_; }
+    const driver::module* parent() const { return &*parent_; }
+    // entry points
+    void operator()(driver::stream *stream, const grid_t& grid, const std::vector<arg>& args) const;
 
   private:
     std::shared_ptr<driver::kernel> bin_;
     std::shared_ptr<driver::module> parent_;
     std::vector<arg_type> param_tys_;
     options_t opt_;
+    std::string name_;
   };
 
 private:
   typedef std::pair<driver::device*, std::vector<int64_t>> cache_key_t;
 
 private:
+  // cache
+  static std::string get_cache_prefix();
+  // make
   triton::lang::translation_unit *make_ast(const std::string &src);
   std::unique_ptr<ir::module> make_ir(Parser &parser);
   std::unique_ptr<driver::module> make_bin(ir::module &function, driver::context *context, const options_t &opt);
-  caller autotune(driver::stream *stream, const grid_fn_ty& grid, const std::vector<arg> &args);
+  caller *make(driver::stream *stream, options_t opt);
+  void precompile(driver::stream *stream, const options_space_t& tuning_space);
+  // autotune
+  function::cache_key_t get_key(driver::stream *stream, const std::vector<arg>& args);
+  caller* autotune(driver::stream *stream, const grid_fn_ty& grid, const std::vector<arg> &args);
 
 public:
   static std::string preheader();
 
 public:
-  function(const std::string& src, const options_space_t& opt = options_space_t());
+  function(const std::string& src, const options_space_t& opt, const std::string &cache_ref = "");
   void operator()(const std::vector<arg>& args, const grid_t& grid, driver::stream* stream);
   void operator()(const std::vector<arg>& args, const grid_fn_ty& grid, driver::stream *stream);
   void set_cst(const std::string& name, void* data, size_t n_bytes);
 
 private:
+  std::map<std::string, std::vector<char>> cst_;
+  // pre-compilation
   ir::context ctx_;
   std::string src_;
-  options_space_t opt_space_;
-  std::map<cache_key_t, caller> cache_;
-  std::map<std::string, std::vector<char>> cst_;
+  options_space_t opt_;
+  std::set<options_t> compiled_;
+  std::map<options_t, std::unique_ptr<caller>> callers_;
+  // caching
+  std::string cache_ref_;
+  std::string cache_path_;
+  std::map<cache_key_t, caller*> cache_;
 };
 
 }
diff --git a/include/triton/tools/sys/mkdir.hpp b/include/triton/tools/sys/mkdir.hpp
index e6c289535..5198a0098 100755
--- a/include/triton/tools/sys/mkdir.hpp
+++ b/include/triton/tools/sys/mkdir.hpp
@@ -61,6 +61,14 @@ namespace tools
         return (status==0 || errno==EEXIST)?0:-1;
     }
 
+    inline int mtime(std::string const & path)
+    {
+      struct stat st;
+      if(stat(path.c_str(), &st) != 0)
+        return 0;
+      return st.st_mtime;
+    }
+
 }
 
 }
diff --git a/lib/driver/module.cc b/lib/driver/module.cc
index 28940f563..51b722ecf 100755
--- a/lib/driver/module.cc
+++ b/lib/driver/module.cc
@@ -253,11 +253,11 @@ std::string cu_module::compile_llvm_module(std::unique_ptr<llvm::Module> module,
    return result;
 }
 
+
 cu_module::cu_module(driver::context * context, std::unique_ptr<llvm::Module> ll_module): cu_module(context, compile_llvm_module(std::move(ll_module), context->device())) { }
 
 cu_module::cu_module(driver::context * context, std::string const & source) : module(context, CUmodule(), true), source_(source){
   cu_context::context_switcher ctx(*context);
-//  std::cout << source << std::endl;
   // JIT compile source-code
   CUjit_option opt[] = {CU_JIT_ERROR_LOG_BUFFER_SIZE_BYTES, CU_JIT_ERROR_LOG_BUFFER};
   unsigned int errbufsize = 8096;
@@ -266,10 +266,10 @@ cu_module::cu_module(driver::context * context, std::string const & source) : mo
   try{
     dispatch::cuModuleLoadDataEx(&*cu_, source_.data(), 2, opt, optval);
   }catch(exception::cuda::base const &){
-//#ifdef TRITON_LOG_PTX_ERROR
+#ifdef TRITON_LOG_PTX_ERROR
     std::cerr << "Compilation Failed! Log: " << std::endl;
     std::cerr << errbuf << std::endl;
-//#endif
+#endif
     throw;
   }
 }
diff --git a/lib/runtime/function.cc b/lib/runtime/function.cc
index 25d355f2c..c5b109f6a 100644
--- a/lib/runtime/function.cc
+++ b/lib/runtime/function.cc
@@ -28,24 +28,28 @@
 #include "triton/ir/function.h"
 #include "triton/ir/print.h"
 #include "triton/tools/bench.hpp"
+#include "triton/tools/sha1.hpp"
+#include "triton/tools/sys/getenv.hpp"
+#include "triton/tools/sys/mkdir.hpp"
 #include "llvm/IR/Module.h"
 #include <mutex>
+#include <fstream>
 
 std::mutex mut;
 
 namespace triton{
 namespace runtime {
 
-// helpers
-void _parallel_loop_nest(std::vector<size_t> const & ranges,
-                       std::function<void(std::vector<size_t> const &)> const & f,
-                       size_t nthreads){
+/* --------------------- */
+/*    HELPERS            */
+/* --------------------- */
+
+void _loop_nest(std::vector<size_t> const & ranges,
+                       std::function<void(std::vector<size_t> const &)> const & f){
   size_t D = ranges.size();
   std::vector<size_t> values(D, 0);
-  // Start with innermost loop
   size_t i = D - 1;
   while(true){
-    //  Execute function
     f(values);
     while(values[i]++ == ranges[i] - 1){
       if(i == 0)
@@ -56,24 +60,31 @@ void _parallel_loop_nest(std::vector<size_t> const & ranges,
   }
 }
 
-template<class T>
-void _parallel_loop_nest(std::vector<std::vector<T>> const & iterates, std::function<void(std::vector<T>)> const & f, size_t nthreads){
-  //Ranges to iterate over
-  std::vector<size_t> ranges;
-  for(auto const & x: iterates)
-    ranges.push_back(x.size());
-  //Proxy function
-  auto proxy = [&](std::vector<size_t> const & idx){
-    std::vector<T> x(iterates.size());
-    for(size_t i = 0; i < x.size(); ++i)
-    x[i] = iterates[i][idx[i]];
-  f(x);
-  };
-  //Iterate
-  _parallel_loop_nest(ranges, proxy, nthreads);
+
+/* --------------------- */
+/*    OPTIONS            */
+/* --------------------- */
+
+std::string function::options_t::to_str() const{
+  std::string ret = "nw-" + std::to_string(num_warps);
+  for(const auto& x : defines){
+    ret += '-';
+    ret += x.first;
+    ret += '-';
+    ret += x.second;
+  }
+  // legalize
+  for(char& x: ret){
+    if(x == ' ' || x == '^' || x == ',' || x == ':')
+      x = '_';
+  }
+  return ret;
 }
 
-// caller
+
+/* --------------------- */
+/*     CALLER OBJECT     */
+/* --------------------- */
 
 arg_type convert(ir::type *ty) {
   if(ty->is_integer_ty(1))
@@ -97,8 +108,46 @@ arg_type convert(ir::type *ty) {
   throw std::runtime_error("unknown type");
 }
 
-function::caller::caller(ir::function *ir, std::shared_ptr<driver::module> parent, const options_t& opt)
-  : bin_(driver::kernel::create(&*parent, ir->get_name().c_str())), parent_(parent), opt_(opt) {
+void function::caller::write(std::ofstream &ofs) {
+  // write name
+  ofs << name_ << std::endl;
+  // write signature
+  for(size_t i = 0; i < param_tys_.size(); i++)
+    ofs << param_tys_[i] << " ";
+  ofs << std::endl;
+  // write module
+  std::string source = ((driver::cu_module*)(&*parent_))->source();
+  ofs << source;
+}
+
+void function::caller::read(driver::context* ctx, std::ifstream &ifs) {
+  // read name
+  std::getline(ifs, name_);
+  // read signature
+  std::string line;
+  std::getline(ifs, line);
+  std::istringstream current(line);
+  int param;
+  param_tys_.clear();
+  while(current >> param)
+    param_tys_.push_back((arg_type)param);
+  // read module
+  std::string src((std::istreambuf_iterator<char>(ifs)),
+                   std::istreambuf_iterator<char>());
+  parent_.reset(new driver::cu_module(ctx, src));
+  bin_.reset(driver::kernel::create(&*parent_, name_.c_str()));
+
+}
+
+function::caller::caller(driver::context* ctx, std::ifstream &ifs, const options_t& opt)
+  : opt_(opt) {
+  read(ctx, ifs);
+}
+
+function::caller::caller(ir::function *ir,
+                         std::shared_ptr<driver::module> parent, const options_t& opt)
+  : parent_(parent), opt_(opt), name_(ir->get_name()) {
+  bin_.reset(driver::kernel::create(&*parent, name_.c_str()));
   // extract signature
   ir::function_type* ty = ir->get_fn_type();
   for(size_t i = 0; i < ty->get_num_params(); i++)
@@ -109,6 +158,7 @@ function::caller::caller(ir::function *ir, std::shared_ptr<driver::module> paren
 void function::caller::operator ()(driver::stream *stream, const grid_t& _grid, const std::vector<arg>& args) const {
   if(args.size() != param_tys_.size())
     throw std::runtime_error("invalid number of arguments");
+  // set arguments
   for(size_t i = 0; i < args.size(); i++){
     arg arg_i = args.at(i);
     arg_type ty = arg_i.type();
@@ -119,99 +169,33 @@ void function::caller::operator ()(driver::stream *stream, const grid_t& _grid,
     else
       bin_->setArg(i, size_of(ty), arg_i.data());
   }
-  // sanity check
+  // set grid
   if(_grid.size() > 3)
     throw std::runtime_error("grid size must be no greater than 3");
   std::array<size_t, 3> grid;
   for(size_t i = 0; i < 3; i++)
     grid[i] = (i < _grid.size()) ? _grid[i] : 1;
+  // enqueue
   stream->enqueue(&*bin_, grid, {opt_.num_warps * 32, 1, 1});
 }
 
 
+/* --------------------- */
+/*    FUNCTION           */
+/* --------------------- */
+
+// create Triton-IR from AST
 std::unique_ptr<ir::module> function::make_ir(Parser& parser) {
-  // create Triton-IR from AST
   ir::module* module = new ir::module("", ctx_);
   Generator gen(&parser);
   gen.Gen(module);
   return std::unique_ptr<ir::module>(module);
 }
 
-
-function::caller function::autotune(driver::stream* stream, const grid_fn_ty& grid_fn,
-                                       const std::vector<arg>& args) {
-
-  // all tuning parameters are strings
-  std::vector<std::string> num_warps;
-  for(size_t i: opt_space_.num_warps)
-    num_warps.push_back(std::to_string(i));
-  std::vector<std::vector<std::string>> space;
-  space.push_back(num_warps);
-  for(const auto& i: opt_space_.defines)
-    space.push_back(i.second);
-
-  // exhaustive search
-  double best_ts = INFINITY;
-  std::unique_ptr<caller> ret;
-
-  auto benchmark = [&](std::vector<std::string> params) {
-    // extract options
-    options_t opt;
-    unsigned i = 0;
-    opt.num_warps = std::stoi(params[i++]);
-    for(auto it: opt_space_.defines){
-      opt.defines[it.first] = params[i++];
-    }
-    // pre-process
-    TokenSequence tokens;
-    Preprocessor cpp(&src_, true);
-    for(auto it: opt_space_.defines)
-      cpp.AddMacro(it.first, &opt.defines.at(it.first));
-    cpp.Process(tokens);
-
-    // parse
-    Parser parser(tokens);
-    parser.Parse();
-    // triton-ir code-gen
-    auto ir = make_ir(parser);
-    // binary code-gen
-    std::unique_ptr<driver::module> bin;
-    try{
-      bin = make_bin(*ir, stream->context(), opt);
-    }catch(const std::runtime_error& e){
-      return;
-    }
-    // kernel uses too much resources
-    if(!bin)
-      return;
-    // copy constants
-    std::unique_ptr<driver::buffer> buffer;
-    for(ir::alloc_const* alloc: ir->allocs()){
-      std::string name = alloc->get_name();
-      auto it = cst_.find(name);
-      if(it == cst_.end())
-        throw std::runtime_error("constant not set before execution");
-      buffer = bin->symbol(name.c_str());
-      stream->write(&*buffer, true, 0, it->second);
-    }
-    // benchmark
-    ir::function *tmp = ir->get_function_list()[0];
-    caller call(tmp, std::move(bin), opt);
-    double ts = tools::bench([&]() { call(stream, grid_fn(opt), args); }, stream, true);
-    // save best
-    if(ts < best_ts) {
-      best_ts = ts;
-      ret.reset(new caller(call));
-    }
-  };
-  _parallel_loop_nest<std::string>(space, benchmark, 1);
-  if(!ret)
-    throw std::runtime_error("could not find valid option in provided space");
-  return *ret;
-}
-
-
-std::unique_ptr<driver::module> function::make_bin(ir::module &module, driver::context *context, const options_t& opt) {
+// create Binary from Triton-IR
+std::unique_ptr<driver::module> function::make_bin(ir::module &module,
+                                                   driver::context *context,
+                                                   const options_t& opt) {
   std::unique_ptr<codegen::target> target = context->device()->make_target();
   // generate llvm code
   llvm::LLVMContext ctx;
@@ -236,8 +220,6 @@ std::unique_ptr<driver::module> function::make_bin(ir::module &module, driver::c
   dce.run(module);
   peephole.run(module);
   dce.run(module);
-//  ir::print(module, std::cout);
-//  exit(EXIT_FAILURE);
   align.run(module);
   cts.run(module);
   axes.run(module);
@@ -258,16 +240,135 @@ std::unique_ptr<driver::module> function::make_bin(ir::module &module, driver::c
     return std::unique_ptr<driver::module>();
   barriers.run(module);
   isel.visit(module, *llvm);
-  // return binary
   std::unique_ptr<driver::module> res(driver::module::create(context, std::move(llvm)));
-  // done
-//  exit(EXIT_FAILURE);
   return res;
 }
 
+
+// create Binary from options
+function::caller* function::make(driver::stream *stream, options_t opt) {
+  // cache path
+  std::string cache_path = cache_path_ + opt.to_str() + ".ptx";
+  int ref_mtime = tools::mtime(cache_ref_);
+  int ptx_mtime = tools::mtime(cache_path);
+  // if cached ptx is newer than reference library
+  if(!ref_mtime || !ptx_mtime || ref_mtime < ptx_mtime){
+    std::ifstream ifs(cache_path);
+    // file is empty -- invalid
+    if(ifs && ifs.peek() == std::ifstream::traits_type::eof())
+      return nullptr;
+    // load cached caller
+    if(ifs)
+      return new caller(stream->context(), ifs, opt);
+  }
+  // pre-process
+  TokenSequence tokens;
+  Preprocessor cpp(&src_, true);
+  for(auto it: opt.defines)
+    cpp.AddMacro(it.first, &it.second);
+  cpp.Process(tokens);
+  // src -> ast
+  Parser parser(tokens);
+  parser.Parse();
+  // ast -> triton-ir
+  auto ir = make_ir(parser);
+  // triton-ir -> binary
+  std::unique_ptr<driver::module> bin;
+  try{
+    bin = make_bin(*ir, stream->context(), opt);
+  }catch(const std::runtime_error&){
+    if(!cache_path_.empty())
+      std::ofstream ofs(cache_path);
+    return nullptr;
+  }
+  // create callable
+  ir::function *tmp = ir->get_function_list()[0];
+  caller* ret = new caller(tmp, std::move(bin), opt);
+  // serialize callable
+  if(!cache_path_.empty()){
+    std::ofstream ofs(cache_path);
+    ret->write(ofs);
+  }
+  return ret;
+}
+
+// precompile all kernels spanned by given options space
+void function::precompile(driver::stream* stream,
+                          const options_space_t& space) {
+  // all ranges
+  std::vector<size_t> ranges;
+  ranges.push_back(space.num_warps.size());
+  for(const auto& x: space.defines)
+    ranges.push_back(x.second.size());
+  // functor for source with given option
+  auto do_make = [&](std::vector<size_t> params) {
+    // compilation options
+    unsigned i = 0;
+    options_t opt;
+    opt.num_warps = space.num_warps[params[i++]];
+    for(auto D: space.defines)
+      opt.defines[D.first] = D.second[params[i++]];
+    // compile
+    caller* call = make(stream, opt);
+    if(!call)
+      return;
+    // copy constants
+    std::unique_ptr<driver::buffer> buffer;
+    for(const auto& cst: cst_){
+      buffer = call->parent()->symbol(cst.first.c_str());
+      stream->write(&*buffer, true, 0, cst.second);
+    }
+    callers_[opt].reset(call);
+  };
+  // multi-threaded compilation
+  _loop_nest(ranges, do_make);
+  if(callers_.empty())
+    throw std::runtime_error("could not find valid option in provided space");
+}
+
+// return auto-tuning key for given function arguments
+function::cache_key_t function::get_key(driver::stream *stream, const std::vector<arg>& args) {
+  cache_key_t ret;
+  ret.first = stream->context()->device();
+  for(size_t i = 0; i < args.size(); i++){
+    arg_type ty = args.at(i).type();
+    if(!is_int_type(ty))
+      continue;
+    long val = 0;
+    std::memcpy((void*)&val, args.at(i).data(), size_of(ty));
+    ret.second.push_back(val);
+  }
+  return ret;
+}
+// returns program with best compilation options for given parameter
+function::caller* function::autotune(driver::stream* stream, const grid_fn_ty& grid_fn,
+                                       const std::vector<arg>& args) {
+  // fast path -- no autotuning necessary
+  if(callers_.size() == 1)
+    return &*callers_.begin()->second;
+  // slow path -- autotuning necessary
+  double best_ts = INFINITY;
+  caller* ret = nullptr;
+  for(auto &x : callers_){
+    if(x.second == nullptr)
+      throw std::runtime_error("configuration not compiled");
+    caller* current = &*x.second;
+    double ts = tools::bench([&]() { (*current)(stream, grid_fn(x.first), args); },
+                                     stream, true);
+    ret = (ts < best_ts) ? current : ret;
+    best_ts = std::min(ts, best_ts);
+  }
+  return ret;
+}
+
+// set copy host buffer "data" into constant memory buffer "name"
+void function::set_cst(const std::string& name, void* data, size_t n_bytes) {
+   cst_[name] = std::vector<char>((char*)data, (char*)data + n_bytes);
+}
+
+
 std::string function::preheader() {
-return
-R"(
+  return  R"(
 #define bool _Bool
 #define true 1
 #define false 0
@@ -297,47 +398,65 @@ typedef long int64;
 )";
 }
 
-function::function(const std::string &src, const options_space_t& opt):  src_(src), opt_space_(opt) {
+std::string function::get_cache_prefix() {
+  //user-specified cache path
+  std::string result = tools::getenv("TRITON_CACHE_PATH");
+  if(!result.empty()){
+    if(tools::mkpath(result)==0)
+      return result;
+  }
+  //create in home
+  result = tools::getenv("HOME");
+  if(!result.empty())
+  {
+    result = result + "/.triton/cache/";
+    if(tools::mkpath(result)==0)
+      return result;
+  }
+  return "";
+}
+
+function::function(const std::string &src,
+                   const options_space_t& opt,
+                   const std::string &cache_ref):
+    src_(src), opt_(opt), cache_ref_(cache_ref) {
+  // hash source code
+  unsigned char hash[20];
+  sha1::calc((void*)src_.data(), src_.size(), hash);
+  // create cache path
+  char _hex[40];
+  sha1::toHexString(hash, _hex);
+  std::string hex(_hex, _hex + 40);
+  cache_path_ = get_cache_prefix() + hex + "/";
+  tools::mkpath(cache_path_);
+  // append pre-header to source
   src_ = preheader() + src_;
 }
 
-void function::operator()(const std::vector<arg>& args, const grid_fn_ty& grid_fn, driver::stream *stream) {
-  cache_key_t key;
-
-  /* figure out if the kernel should be re-tuned */
-  // re-tune if device is different
-  key.first = stream->context()->device();
-  // re-tune if any int argument is different
-  for(size_t i = 0; i < args.size(); i++){
-    arg_type ty = args.at(i).type();
-    if(is_int_type(ty)){
-      long val = 0;
-      std::memcpy((void*)&val, args.at(i).data(), size_of(ty));
-      key.second.push_back(val);
-    }
-  }
-
-  /* find existing configuration */
+void function::operator()(const std::vector<arg>& args,
+                          const grid_fn_ty& grid_fn,
+                          driver::stream *stream) {
+  // pre-compile kernels
+  if(callers_.empty())
+    precompile(stream, opt_);
+  // auto-tune if necessary
+  auto key = get_key(stream, args);
   auto it = cache_.find(key);
-  if(it != cache_.end()){
-    it->second(stream, grid_fn(it->second.opt()), args);
-    return;
-  }
-
-  /* re-tune and re-compile */
-  {
-    std::lock_guard<std::mutex> lock(mut);
-    cache_.insert({key, autotune(stream, grid_fn, args)});
+  if(it == cache_.end()){
+    auto best = autotune(stream, grid_fn, args);
+    it = cache_.insert({key, best}).first;
   }
+  // run
+  (*it->second)(stream, grid_fn(it->second->opt()), args);
 }
 
-void function::operator()(const std::vector<arg>& args, const grid_t& grid, driver::stream *stream) {
+void function::operator()(const std::vector<arg>& args,
+                          const grid_t& grid,
+                          driver::stream *stream) {
   return this->operator()(args, [&grid](const options_t&){ return grid; }, stream);
 }
 
-void function::set_cst(const std::string& name, void* data, size_t n_bytes) {
-   cst_[name] = std::vector<char>((char*)data, (char*)data + n_bytes);
-}
+
 
 }
 }
diff --git a/python/examples/einsum.py b/python/examples/einsum.py
index a971c6683..ce6d49210 100644
--- a/python/examples/einsum.py
+++ b/python/examples/einsum.py
@@ -17,14 +17,14 @@ MNK = [
         (2048, 2048, 2048),
         #(8192, 8192, 8192),
        
-        # (64, 64, 64000),
-        # (64, 64, 128000),
-        # (256, 256, 64000),
-        # (256, 256, 128000),
+        (64, 64, 64000),
+        (64, 64, 128000),
+        (256, 256, 64000),
+        (256, 256, 128000),
 
-        # (1536, 16, 1536),
-        # (1536, 32, 1536),
-        # (1536, 64, 1536),
+        (1536, 16, 1536),
+        (1536, 32, 1536),
+        (1536, 64, 1536),
         # (1536, 128, 1536),
         # (4096, 16, 4096),
         # (4096, 32, 4096),
@@ -33,9 +33,9 @@ MNK = [
     
         # (127008, 768, 576) 
       ]
-#for M, N, K in MNK:
-#    matmul = lambda a, b: torch.matmul(a, b)
-#    configs += [([M, K], [K, N], [M, N], matmul, 'mk,kn->mn', dict())]
+for M, N, K in MNK:
+    matmul = lambda a, b: torch.matmul(a, b)
+    configs += [([M, K], [K, N], [M, N], matmul, 'mk,kn->mn', dict())]
 #for M, N, K in MNK:
 #    matmul = lambda a, b: torch.matmul(a.t(), b)
 #    configs += [([M, K], [M, N], [K, N], None, 'mk,mn->kn', dict())]
@@ -175,15 +175,15 @@ for a_shape, b_shape, c_shape, torch_fn, expr, arrays in configs:
     a = torch.rand(*a_shape).type(dtype).cuda()
     b = torch.rand(*b_shape).type(dtype).cuda()
     # triton output
-    print(a.size(), b.size())
-    tc = triton.ops.einsum(expr, a, b, c_shape, arrays = arrays, bench = True)
+    tc = torch.empty(c_shape, device=a.device)
+    triton.ops.einsum(expr, a, b, tc, arrays = arrays, bench = True)
     # reference output
     if torch_fn:
         rc = torch_fn(a, b, **arrays)
     else:
         rc = torch.einsum(expr, a, b)
     # performance relative to equivalent matrix multiplication
-    ctx = triton.ctx_registry[tc]
+    ctx = triton.ops._einsum.registry[tc]
     B, M, N, K = ctx.matmul_B, ctx.matmul_M, ctx.matmul_N, ctx.matmul_K
     cmp_eqbmm = False
     if cmp_eqbmm:
diff --git a/python/src/bindings.cc b/python/src/bindings.cc
index 0d9d545bc..af281ff83 100644
--- a/python/src/bindings.cc
+++ b/python/src/bindings.cc
@@ -6,6 +6,7 @@
 #include <regex>
 #include <algorithm>
 #include "triton/runtime/function.h"
+#include "triton/runtime/arg.h"
 #include "triton/lang/code_gen.h"
 #include "triton/lang/parser.h"
 #include "triton/lang/cpp.h"
@@ -40,9 +41,10 @@ void delete_grid(size_t id) {
 /* Function map */
 
 void register_fn(size_t id,
-                   const std::string& src,
-                   const rt::function::options_space_t& opt) {
-  id_fn_map[id].reset(new rt::function(src, opt));
+                 const std::string& src,
+                 const rt::function::options_space_t& opt,
+                 const std::string &cache_ref) {
+  id_fn_map[id].reset(new rt::function(src, opt, cache_ref));
 }
 
 void delete_fn(size_t id) {
@@ -64,6 +66,7 @@ size_t make_op_id() {
   return id_fn_map.size();
 }
 
+
 /* TF scalar wrapper */
 size_t make_scalar_id() {
   size_t ret = i64scalar_map.size();
@@ -423,6 +426,37 @@ inline std::string to_torch_ty(ir::type *ty) {
   throw std::runtime_error("unknown type");
 }
 
+inline std::string to_torch_ty(rt::arg_type ty){
+  switch(ty){
+    case rt::INT1_T:   return "int64_t";
+    case rt::INT8_T:   return "int64_t";
+    case rt::INT16_T:  return "int64_t";
+    case rt::INT32_T:  return "int64_t";
+    case rt::INT64_T:  return "int64_t";
+    case rt::HALF_T:   return "double";
+    case rt::FLOAT_T:  return "double";
+    case rt::DOUBLE_T: return "double";
+    case rt::BUFFER_T: return "torch::Tensor";
+    default:           return "UNKNOWN";
+  }
+}
+
+inline std::string to_c_ty(rt::arg_type ty){
+  switch(ty){
+    case rt::INT1_T:   return "bool";
+    case rt::INT8_T:   return "int8_t";
+    case rt::INT16_T:  return "int16_t";
+    case rt::INT32_T:  return "int32_t";
+    case rt::INT64_T:  return "int64_t";
+    case rt::HALF_T:   return "half";
+    case rt::FLOAT_T:  return "float";
+    case rt::DOUBLE_T: return "double";
+    case rt::BUFFER_T: return "drv::cu_buffer";
+    default:           return "UNKNOWN";
+  }
+}
+
+
 inline std::string to_c_ty(ir::type *ty) {
   if(ty->is_integer_ty(1))
     return "bool";
@@ -448,33 +482,30 @@ inline std::string to_c_ty(ir::type *ty) {
 
 
 void gen_torch_signature(std::ostringstream& oss,
-                         ir::function* fn,
-                         const std::string& name) {
-  const auto& args = fn->args();
+                         const std::string& name,
+                         const std::vector<rt::arg_type>& args) {
   std::string ret_ty = "void";
   oss << ret_ty << " " << name << "(";
   oss << "int64_t id, ";
   oss << "int64_t bench, ";
   oss << "int64_t bench_id,  ";
   for(size_t i = 0; i < args.size(); i++) {
-    ir::argument* arg = args[i];
     if(i > 0)
       oss << ", ";
-    oss << to_torch_ty(arg->get_type()) << " " << arg->get_name();
+    oss << to_torch_ty(args[i]) << " " << "th_arg_" << i;
   }
   oss << ")";
 }
 
 void gen_torch_init_driver(std::ostringstream &oss,
-                           const std::vector<ir::argument*>&args) {
-  ir::argument* tensor = nullptr;
-  for(ir::argument* arg: args)
-    if(arg->get_type()->is_pointer_ty()){
-      tensor = arg;
+                           const std::vector<rt::arg_type>&args) {
+  // Find index of first buffer
+  size_t i;
+  for(i = 0; i < args.size(); i++)
+    if(args[i] == rt::BUFFER_T)
       break;
-    }
   oss <<  "  // Wrap CUDA handles" << std::endl;
-  oss <<  "  c10::DeviceIndex device = " << tensor->get_name() << ".storage().device().index();" << std::endl;
+  oss <<  "  c10::DeviceIndex device = th_arg_" << i << ".storage().device().index();" << std::endl;
   oss <<  "  // Get stream" << std::endl;
   oss <<  "  CUstream custream = (CUstream)at::cuda::getCurrentCUDAStream(device).stream();" << std::endl;
   oss <<  "  triton::driver::cu_stream stream(custream, false);" << std::endl;
@@ -482,28 +513,28 @@ void gen_torch_init_driver(std::ostringstream &oss,
 }
 
 void gen_torch_make_handles(std::ostream &os,
-                            const std::vector<ir::argument*>& args) {
+                            const std::vector<rt::arg_type>& args) {
   for(unsigned i = 0; i < args.size(); i++){
-    ir::argument *arg = args[i];
-    const std::string& name = arg->get_name();
-    ir::type* ty = arg->get_type();
-    if(!ty->is_pointer_ty())
-      os << "  " << to_c_ty(ty) << " arg_" << name << " = " << name << ";" << std::endl;
+    rt::arg_type arg = args[i];
+    const std::string th_name = "th_arg_" + std::to_string(i);
+    const std::string name = "arg_" + std::to_string(i);
+    if(arg != rt::BUFFER_T)
+      os << "  " << to_c_ty(arg) << " " << name << " = " << th_name << ";" << std::endl;
     else{
-      os << "  CHECK_INPUT(" << name << ");" << std::endl;
-      os << "  drv::cu_buffer arg_" + name + "(ctx, " + name + ".storage().size(), "
-            " (CUdeviceptr)((char*)" + name + ".storage().data() + " + name + ".storage_offset() * " + name + ".itemsize()), false);" << std::endl;
+      os << "  CHECK_INPUT(" << th_name << ");" << std::endl;
+      os << "  drv::cu_buffer " + name + "(ctx, " + th_name + ".storage().size(), "
+            " (CUdeviceptr)((char*)" + th_name + ".storage().data() + " + th_name + ".storage_offset() * " + th_name + ".itemsize()), false);" << std::endl;
     }
   }
 }
 
-void gen_torch_make_launch_function(std::ostream &os, const std::vector<ir::argument*>& args) {
+void gen_torch_make_launch_function(std::ostream &os, 
+                                    const std::vector<rt::arg_type>& args) {
   os << "  std::function<void()> run = [&](){\n  ";
   os << "    (*id_fn_map.at(id))({";
   for(unsigned i = 0; i < args.size() ; i++){
-    ir::argument *arg = args[i];
-    std::string name = "arg_" + arg->get_name();
-    if(arg->get_type()->is_pointer_ty())
+    std::string name = "arg_" + std::to_string(i);
+    if(args[i] == rt::BUFFER_T)
       name = "&" + name;
     if(i > 0)
       os << ", ";
@@ -531,15 +562,7 @@ void gen_torch_ret(std::ostream &os, const std::vector<std::string>& outputs) {
 }
 
 std::tuple<std::string,
-           std::string> make_torch_src(const std::string& src,
-                                       const runtime::function::options_space_t& opt) {
-  // triton-ir code-gen
-  ir::context ctx;
-  auto ir = std::shared_ptr<ir::module>(new ir::module("", ctx));
-  make_module(src, &*ir, opt);
-  // function
-  ir::function* fn = ir->get_function_list().front();
-  std::string name = fn->get_name();
+           std::string> make_torch_src(const std::string& name, std::vector<rt::arg_type> args) {
   // generate framework code
   std::ostringstream oss;
   oss << R"(
@@ -563,11 +586,11 @@ extern std::map<size_t, int64_t> i64scalar_map;
 
 )";
 
-  gen_torch_signature(oss, fn, name);
+  gen_torch_signature(oss, name, args);
   oss << " {" << std::endl;
-  gen_torch_init_driver(oss, fn->args());
-  gen_torch_make_handles(oss, fn->args());
-  gen_torch_make_launch_function(oss, fn->args());
+  gen_torch_init_driver(oss, args);
+  gen_torch_make_handles(oss, args);
+  gen_torch_make_launch_function(oss, args);
   //gen_torch_ret(oss);
   oss << "}" << std::endl;
 
@@ -578,6 +601,22 @@ extern std::map<size_t, int64_t> i64scalar_map;
   return {oss.str(), name};
 }
 
+/* Function signature */
+std::vector<rt::arg_type> get_fn_signature(const std::string& src,
+                                           const runtime::function::options_space_t& opt) {
+  // triton-ir code-gen
+  ir::context ctx;
+  auto ir = std::shared_ptr<ir::module>(new ir::module("", ctx));
+  make_module(src, &*ir, opt);
+  // function
+  ir::function* fn = ir->get_function_list().front();
+  // extract signature
+  std::vector<rt::arg_type> ret;
+  ir::function_type* ty = fn->get_fn_type();
+  for(size_t i = 0; i < ty->get_num_params(); i++)
+    ret.push_back(rt::convert(ty->get_param_ty(i)));
+  return ret;
+}
 
 typedef triton::runtime::function::options_t options_t;
 typedef triton::runtime::function::options_space_t options_space_t;
@@ -593,6 +632,17 @@ PYBIND11_MODULE(libtriton, m) {
           "Creates C++ source code for a custom PyTorch op ");
 
     // bindings for triton classes
+    pybind11::enum_<rt::arg_type>(m, "arg_type")
+        .value("int1", rt::INT1_T)
+        .value("int8", rt::INT8_T)
+        .value("int16", rt::INT16_T)
+        .value("int32", rt::INT32_T)
+        .value("int64", rt::INT64_T)
+        .value("half", rt::HALF_T)
+        .value("float", rt::FLOAT_T)
+        .value("double", rt::DOUBLE_T)
+        .value("buffer", rt::BUFFER_T);
+
     pybind11::class_<options_t>(m, "options")
         .def(pybind11::init<>())
         .def("d", &options_t::D<int>)
@@ -604,6 +654,7 @@ PYBIND11_MODULE(libtriton, m) {
         .def_readwrite("num_warps", &options_space_t::num_warps);
 
     // hooks into triton constructs since frameworks may not use pybind11
+    m.def("get_fn_signature", &get_fn_signature);
     m.def("register_grid", &register_grid);
     m.def("delete_grid", &delete_grid);
     m.def("register_fn", &register_fn);
diff --git a/python/triton/__init__.py b/python/triton/__init__.py
index cb4097e72..388484039 100644
--- a/python/triton/__init__.py
+++ b/python/triton/__init__.py
@@ -1,5 +1,4 @@
 from .kernel import *
-from .function import *
 from .utils import *
 import triton.ops
 
diff --git a/python/triton/function.py b/python/triton/function.py
deleted file mode 100644
index 2c9cb60fe..000000000
--- a/python/triton/function.py
+++ /dev/null
@@ -1,127 +0,0 @@
-import triton.frameworks as fw
-import triton.utils as utils
-
-class OpContext(object):
-
-    def __init__(self):
-      self.to_save = []
-
-    def save_for_backward(self, *tensors):
-      self.to_save = [x.to_tensor() if isinstance(x, utils.tf_empty_proxy) else x 
-                      for x in tensors]
-    
-    @property
-    def saved_tensors(self):
-      return self.to_save
-
-class function_meta(type):
-
-    def __init__(cls, name, bases, attrs):
-        cls.registered = False
-        return super(function_meta, cls).__init__(name, bases, attrs)
-
-ctx_registry = utils.id_dict()
-
-class function(metaclass = function_meta):
-  
-  @staticmethod
-  def forward(ctx, *args, **kwargs):
-    raise NotImplementedError
-
-  @staticmethod
-  def backward(ctx, grad_output):
-    raise NotImplementedError
-
-  @classmethod
-  def apply_torch(cls, *args, **kwargs):
-    class TorchFunction(fw.torch.autograd.Function):
-      @staticmethod
-      def forward(ctx, *targs):
-        y = cls.forward(ctx, *targs, **cls.torch_kwargs)
-        ctx_registry[y] = ctx
-        return y
-      @staticmethod
-      def backward(ctx, grad_output):
-        return cls.backward(ctx, grad_output)
-    cls.torch_kwargs = kwargs
-    return TorchFunction.apply(*args)
-  torch_kwargs = 0
-
-  @classmethod
-  def extract_tf_tensors(cls, lst, err):
-    ret = []
-    for x in lst:
-      if x is None:
-        ret += [None]
-      elif isinstance(x, fw.tensorflow.Tensor):
-        ret += [x]
-      elif isinstance(x, utils.tf_empty_proxy):
-        if x.tensor is None:
-          raise ValueError('Empty tensor never filled during ' + err)
-        else:
-          ret += [x.tensor]
-      else:
-        raise ValueError('Unsupported return type', type(x))
-    return ret
-  
-  @classmethod
-  def map_in_to_args(cls, op, args):
-    ret = dict()
-    for i, ix in enumerate(op.inputs):
-      for j, jx in enumerate(args):
-        if ix is jx:
-          ret[j] = i
-    return ret
-  
-  @classmethod
-  def map_res_to_out(cls, op, result):
-    ret = []
-    for i, ix in enumerate(result):
-      for j, jx in enumerate(op.outputs):
-        if ix is jx:
-          ret.append(j)
-    return ret
-    
-  @classmethod
-  def apply_tensorflow(cls, *args, **kwargs):
-    ctx = OpContext()
-
-    # run forward pass
-    result = cls.forward(ctx, *args, **kwargs)
-    result = result if isinstance(result, tuple) else (result, )
-    result = function.extract_tf_tensors(result, 'forward')
-    
-    # Register backward pass
-    op = result[0].op
-    ctx_registry[op] = ctx
-    if not cls.registered:
-      remap_in = cls.map_in_to_args(op, args)
-      remap_out = cls.map_res_to_out(op, result)
-      @fw.tensorflow.RegisterGradient(op.op_def.name)
-      def gradient(op, *dy):
-        # Remap gradient inputs in the right order
-        dy = [dy[i] for i in remap_out]
-        dy = dy if len(dy) > 1 else dy[0]
-        # Execute gradient function
-        grad = cls.backward(ctx_registry[op], dy)
-        grad = function.extract_tf_tensors(grad, 'backward')
-        # Remap gradient in the right order
-        ret = [None] * len(op.inputs)
-        for i in range(len(grad)):
-          if i in remap_in:
-            ret[remap_in[i]] = grad[i]
-        # Return
-        return ret
-      cls.registered = True
-
-    # Return tensor
-    return result[0] if len(result)==1 else result
-
-  @classmethod
-  def apply(cls, *args, **kwargs):
-    if fw.has_tensorflow():
-        return cls.apply_tensorflow(*args, **kwargs)
-    elif fw.has_torch():
-        return cls.apply_torch(*args, **kwargs)
-    else:
-        assert False
diff --git a/python/triton/kernel.py b/python/triton/kernel.py
index ab7c3f49e..fe5e0aabd 100644
--- a/python/triton/kernel.py
+++ b/python/triton/kernel.py
@@ -17,43 +17,6 @@ import triton.frameworks as fw
 import triton.utils
 import triton._C.libtriton as libtriton
 
-def _make_framework_src(src, grid):
-  if fw.has_torch:
-    return libtriton.make_torch_src(src, grid)
-  else:
-    assert False
-
-def _make_cache_path(src):
-  md5 = hashlib.sha1(src.encode())
-  hexhash = md5.hexdigest()
-  home = os.path.expanduser('~')
-  cacheroot = os.path.join(home, '.triton', 'cache')
-  cachepath = os.path.join(cacheroot, str(hexhash))
-  if not os.path.exists(cachepath):
-    os.makedirs(cachepath)
-  return cachepath
-
-def _write_bindings(src, root):
-  if fw.has_torch():
-    name = 'torch'
-  else:
-    assert False
-  cpp = os.path.join(root, '{name}.cpp'.format(name=name))
-  suffix = sysconfig.get_config_var('EXT_SUFFIX')
-  so = os.path.join(root, '{name}{suffix}'.format(name=name, suffix=suffix))
-  recompile = False
-  # recompile if .so does not exist
-  if not os.path.exists(cpp) or not os.path.exists(so):
-    recompile = True
-  # recompile if cpp was modified after .so
-  elif max(cpp, so, key=os.path.getctime) == cpp:
-    recompile = True
-  # write cpp file
-  if recompile:
-    with open(cpp, 'w+') as handle:
-      handle.writelines(src)
-  # return path of cpp file
-  return (cpp, so)
 
 @contextlib.contextmanager
 def quiet():
@@ -64,7 +27,7 @@ def quiet():
     finally:
         sys.stdout, sys.stderr = old_stdout, old_stderr
 
-def _build(src, path):
+def _build(src, path, name):
   ccdir = os.path.join(libtriton.__file__, os.path.pardir)
   ccdir = os.path.realpath(ccdir)
   # include directories
@@ -88,7 +51,6 @@ def _build(src, path):
     libraries += ['torch']
     abi = fw.torch._C._GLIBCXX_USE_CXX11_ABI
     extra_compile_args += ['-D_GLIBCXX_USE_CXX11_ABI={abi}'.format(abi=abi)]
-    name = 'torch'
   else:
     assert False
   # extra arguments
@@ -142,30 +104,47 @@ def _cvt_to_def_str(obj):
   return str(obj)
 
 
-def _make_framework_op(src, options):
-  src, name = _make_framework_src(src, options)
-  cache_path = _make_cache_path(src)
-  cpp, so = _write_bindings(src, cache_path)
-  _build(cpp, cache_path)
-  if fw.has_torch():
-    fw.torch.ops.load_library(so)
-    return getattr(fw.torch.ops.triton, name)
-  else:
-    assert False
+def _encode(arg_types):
+  codes = {
+    libtriton.arg_type.int1:   'i1',
+    libtriton.arg_type.int8:   'i8',
+    libtriton.arg_type.int32:  'i32',
+    libtriton.arg_type.int64:  'i64',
+    libtriton.arg_type.half:   'f16',
+    libtriton.arg_type.float:  'f32',
+    libtriton.arg_type.double: 'f64',
+    libtriton.arg_type.buffer: 'buf'
+  }
+  ret = '_'.join(map(codes.get, arg_types))
+  return ret
 
-def _make_grid(grid, args) :
-  scalars = [x for x in args if isinstance(x, triton.utils.scalar)]
-  def grid(opt):
-    for x in scalars:
-      x.set_assume_initialized()
-    result = grid(opt)
-    for x in scalars:
-      x.unset_assume_initialized()
-    return result
-  return grid
-
-
-bench_registry = triton.utils.id_dict()
+def _make_framework_op(arg_types):
+  name = _encode(arg_types)
+  # path of .cpp and .so file
+  home = os.path.expanduser('~')
+  root = os.path.join(home, '.triton', 'torch', name)
+  if not os.path.exists(root):
+    os.makedirs(root)
+  suffix = sysconfig.get_config_var('EXT_SUFFIX')
+  so = os.path.join(root, f'op{suffix}')
+  cpp = os.path.join(root, f'op.cpp')
+  # handle cached .so file
+  if os.path.exists(so):
+    tt_mtime = os.stat(os.path.realpath(libtriton.__file__)).st_mtime
+    so_mtime = os.stat(so).st_mtime
+    # can use cached if libtriton is older than the .so
+    if tt_mtime < so_mtime:
+      fw.torch.ops.load_library(so)
+      return getattr(fw.torch.ops.triton, name)
+  # create torch source code
+  src, _ = libtriton.make_torch_src(name, arg_types)
+  with open(cpp, 'w+') as handle:
+      handle.writelines(src)
+  # compile torch source code
+  _build(cpp, root, 'op')
+  fw.torch.ops.load_library(so)
+  return getattr(fw.torch.ops.triton, name)
+  
 
 class kernel:
 
@@ -180,9 +159,8 @@ class kernel:
     self.cst[name] = value
 
   def __call__(self, *args, **kwargs):
-
     ########################
-    # keyword arguments
+    # JIT Options
     ########################
     num_warps = kwargs['num_warps'] if 'num_warps' in kwargs else [2, 4, 8]
     defines = kwargs['defines']     if 'defines'   in kwargs else dict()
@@ -195,7 +173,6 @@ class kernel:
     #########################
     # cache
     ########################
-
     # create a new framework op when defines are different
     key = '-'.join(['{key}-{val}'.format(key=key, val=val) for key, val in defines.items()])
     if key not in self.fw_id.keys():
@@ -211,15 +188,16 @@ class kernel:
       opt = libtriton.options_space()
       opt.defines = macros
       opt.num_warps = num_warps
-      # create unique id for this op
+      # create triton function for this op
       op_id = libtriton.make_op_id()
       self.fw_id[key] = op_id
-      # register function
-      libtriton.register_fn(op_id, self.src, opt)
+      libtriton.register_fn(op_id, self.src, opt, os.path.realpath(libtriton.__file__))
       for name, value in self.cst.items():
         libtriton.register_cst(op_id, name, value)
+      # create pytorch hook for this op
+      arg_types = libtriton.get_fn_signature(self.src, opt)
       if self.fw_op is None:
-        self.fw_op = _make_framework_op(self.src, opt)
+        self.fw_op = _make_framework_op(arg_types)
 
     ########################
     # initialize
diff --git a/python/triton/ops/batchnorm.py b/python/triton/ops/batchnorm.py
index 3d415c7bb..5b1cc806d 100644
--- a/python/triton/ops/batchnorm.py
+++ b/python/triton/ops/batchnorm.py
@@ -1,7 +1,8 @@
 import triton
+import torch
 import math
 
-class _batchnorm(triton.function):
+class _batchnorm(torch.autograd.Function):
 
   fwd_src = """
 void fwdbatchnorm(float *Y, float *M, float *V,
diff --git a/python/triton/ops/einsum.py b/python/triton/ops/einsum.py
index 1bd318b13..6c74dc92c 100644
--- a/python/triton/ops/einsum.py
+++ b/python/triton/ops/einsum.py
@@ -73,7 +73,7 @@ class _einsum(torch.autograd.Function):
                     stride_a_last, stride_b_last, stride_c_last,
                     lut_mode_a, lut_mode_b,
                     delta_a, delta_b,
-                    subscripted):
+                    subscripted, varnames):
 
         use_lut_a = True
         use_lut_b = True
@@ -123,6 +123,8 @@ __global__ void {name}(
         src += "\n"
         for ptr in subscripted:
             src += f", int* {ptr}"
+        for name in varnames:
+            src += f", int {name}"
         src += """) {
 
     // re-order outer program ids
@@ -274,6 +276,9 @@ __global__ void {name}(
     TYPE c[TM, TN, TB] = acc;
 
     // re-materialize ranges
+    pid_mn = get_program_id(0) / div_m;
+    pid_n = pid_mn % grid_n;
+    pid_m = (pid_mn / grid_n)*div_m + (get_program_id(0) % div_m);
 """
         for axes, tile, off in zip([axes_m, axes_n, axes_b],
                                    ['TM', 'TN', 'TB'],
@@ -410,11 +415,8 @@ __global__ void {name}(
         batch = [d for d in sym_a if d in sym_b and d in sym_c]
         outer = [d for d in sym_a if d not in sym_b and d in sym_c]
         inner = [d for d in sym_a if d in sym_b and d not in sym_c]
-        illegal = [d for d in sym_a if d not in sym_b and d not in sym_c]
-        if illegal:
-            raise ValueError(f"einsum labels {illegal} ({expr_a}) "\
-                             f"not present in {expr_b} or {expr_c}")
-        return _einsum.uniq(batch), _einsum.uniq(outer), _einsum.uniq(inner)
+        variables = [d for d in sym_a if d not in sym_b and d not in sym_c]
+        return _einsum.uniq(batch), _einsum.uniq(outer), _einsum.uniq(inner), variables
 
 
     def replace_subscript(expr, arrays):
@@ -467,7 +469,33 @@ __global__ void {name}(
         locks = None
         kernel_cache = dict()
 
-        def __init__(self, einsum, dtype, stride_a, stride_b, stride_c, shape_a, shape_b, arrays, mask, shape_c):
+        @staticmethod
+        def _tile(M, N, B, TMs, TNs, TBs, TZs, TK):
+            smp = 15
+            # occupancy estimation
+            grid = lambda TM, TN, TB, TZ:   \
+                        triton.cdiv(M, TM)* \
+                        triton.cdiv(N, TN)* \
+                        triton.cdiv(B, TB)* \
+                        TZ
+            occupancy = lambda TM, TN, TB, TZ: \
+                           min(grid(TM, TN, TB, TZ), 4*smp)
+            # arithmetic intensity estimation
+            intensity = lambda TM, TN: \
+                           TM * TN * TK / (TM*TK + TK*TN)
+            # occupancy/intensity for all configurations
+            estimates = {(TM, TN, TB, TZ): (occupancy(TM, TN, TB, TZ), intensity(TM, TN)) \
+                        for TM in TMs \
+                        for TN in TNs \
+                        for TB in TBs \
+                        for TZ in TZs }
+            # returns configuration that maximizes occupancy subject to maximizing intensity
+            estimates = sorted(estimates.items(), 
+                               key=lambda item: item[1], 
+                               reverse=True)
+            return estimates[0][0]
+
+        def __init__(self, einsum, dtype, stride_a, stride_b, stride_c, shape_a, shape_b, arrays, mask, shape_c, varnames):
             # parse symbols
             expr_a, expr_bc = einsum.split(",")
             expr_b, expr_c  = expr_bc.split("->")
@@ -476,9 +504,13 @@ __global__ void {name}(
             sym_b = _einsum.parse_expr(expr_b, subscripted)
             sym_c = _einsum.parse_expr(expr_c, subscripted)
             # parse axes
-            axes_b, axes_m, axes_k = _einsum.parse_axes(sym_a, sym_b, sym_c, subscripted)
-            _, axes_n, _           = _einsum.parse_axes(sym_b, sym_a, sym_c, subscripted)
+            axes_b, axes_m, axes_k, var = _einsum.parse_axes(sym_a, sym_b, sym_c, subscripted)
+            _, axes_n, _, _           = _einsum.parse_axes(sym_b, sym_a, sym_c, subscripted)
             axes = axes_b + axes_m + axes_n + axes_k
+            # unresolved symbols
+            unresolved = [x for x in map(str, var) if x not in varnames]
+            if unresolved:
+                raise ValueError(f'unresolved symbols: {unresolved}')
             # check dimensions
             dims_a  = dict(zip(sym_a, shape_a))
             dims_b  = dict(zip(sym_b, shape_b))
@@ -520,7 +552,7 @@ __global__ void {name}(
                                                         stride_a_last, stride_b_last, stride_c_last,
                                                         lut_mode_a, lut_mode_b,
                                                         delta_a, delta_b,
-                                                        subscripted)
+                                                        subscripted, varnames)
             self.kernel = cache[name]
             # Initialize locks
             if _einsum.instance.locks is None:
@@ -565,19 +597,21 @@ __global__ void {name}(
             self.pos_a = 0
             self.pos_b = 1
             self.pos_c = 2
-            # pre-processor macros
-            TM = [16] + [x for x in [32, 64, 128] if x <= M]
-            TN = [16] + [x for x in [32, 64, 128] if x <= N]
-            TB = [x for x in [1, 2, 4] if x <= B]
-            MAX_GZ = K // 2048
-            MIN_GM = M // max(TM)
-            MIN_GN = N // max(TN)
-            MIN_GB = B // max(TB)
-            TZ = [x for x in [1, 2, 4, 8, 16, 32] \
-                    if x < MAX_GZ and x*MIN_GM*MIN_GN*MIN_GB < 256]
-            TZ = [1] if not TZ else [TZ[-1], TZ[-1]*2]
-            TM, TN, TB, TZ = 64, 64, 1, 1
-            self.macros = {  'TM': TM, 'TN': TN, 'TB': TB, 'TK': TK, 'TZ': TZ, 'TYPE': dtype}
+            # user-provided variables
+            self.pos_vars = len(self.args)
+            self.varnames = varnames
+            self.args += [None] * len(varnames)
+            # tile size ranges
+            MAX_GZ = triton.cdiv(K, 2048)
+            TMs = [16] + [x for x in [32, 64, 128] if x <= M]
+            TNs = [16] + [x for x in [32, 64, 128] if x <= N]
+            TBs = [x for x in [1, 2, 4, 8] if x <= B]
+            TZs = [x for x in [1, 2, 4, 8, 16, 32] if x <= MAX_GZ]
+            # tile sizes
+            TM, TN, TB, TZ = _einsum.instance._tile(M, N, B, TMs, TNs, TBs, TZs, TK)
+            TM, TN, TB, TZ = 64, 128, 1, 1
+            self.macros =  {'TM': TM, 'TN': TN, 'TB': TB, 'TK': TK, 'TZ': TZ, 'TYPE': dtype}
+            self.num_warps = [4]
             if mask:
                 self.macros['MASK'] = '{0:#0{1}x}'.format(mask, 10)
             # save information on the operation
@@ -589,12 +623,15 @@ __global__ void {name}(
             self.matmul_N = N
             self.matmul_K = K
             self.is_extended = any([not x.is_symbol for x in sym_a + sym_b])
+
                     
-        def run(self, a, b, c, bench):
+        def run(self, a, b, c, values, bench):
             self.args[self.pos_a] = a
             self.args[self.pos_b] = b
             self.args[self.pos_c] = c
-            return self.kernel(*self.args, grid=self.grid, bench=bench, defines=self.macros)
+            for i, name in enumerate(self.varnames):
+                self.args[self.pos_vars + i] = values[name]
+            return self.kernel(*self.args, grid=self.grid, bench=bench, defines=self.macros, num_warps=self.num_warps)
 
 
 
@@ -604,8 +641,9 @@ __global__ void {name}(
     ############################
 
     instance_cache = dict()
+    registry = triton.utils.id_dict()
     @staticmethod
-    def forward(ctx, expr, a, b, output, mask=None, arrays=dict(), bench=False):
+    def forward(ctx, expr, a, b, output, mask, arrays, bench, values):
         # compile einsum instance
         cache = _einsum.instance_cache
         key = (expr, a.dtype, 
@@ -615,10 +653,10 @@ __global__ void {name}(
             cache[key] = _einsum.instance(expr, a.dtype, 
                                           a.stride(), b.stride(), output.stride(),
                                           a.shape, b.shape, arrays,
-                                          mask, output.shape)
+                                          mask, output.shape, values.keys())
         instance = cache[key]
         # run and mark as dirty output modified in-place
-        perf = instance.run(a, b, output, bench)
+        perf = instance.run(a, b, output, values, bench)
         ctx.mark_dirty(output)
         # save information in context
         ctx.is_extended = instance.is_extended
@@ -629,8 +667,9 @@ __global__ void {name}(
         ctx.matmul_M = instance.matmul_M
         ctx.matmul_N = instance.matmul_N
         ctx.matmul_K = instance.matmul_K
-        ctx.perf = perf
+        ctx.forward_ms = perf
         ctx.save_for_backward(a, b)
+        _einsum.registry[output] = ctx
         return output
 
 
@@ -662,5 +701,5 @@ __global__ void {name}(
 
 def einsum(expr, a, b, output, 
            mask=None, arrays=dict(), 
-           bench=False):
-    return _einsum.apply(expr, a, b, output, mask, arrays, bench)
\ No newline at end of file
+           bench=False, values=dict()):
+    return _einsum.apply(expr, a, b, output, mask, arrays, bench, values)
\ No newline at end of file
diff --git a/tests/bench/dot.cc b/tests/bench/dot.cc
index 131083e72..67a09cfef 100644
--- a/tests/bench/dot.cc
+++ b/tests/bench/dot.cc
@@ -10,10 +10,10 @@ int main() {
   typedef std::tuple<std::vector<int>, bool, bool, int, int, int> config_t;
   std::vector<config_t> configs;
   for(auto ord: std::vector<std::vector<int>>{{1, 0}})
-    for(auto x: std::vector<std::array<bool, 2>>{{false, false}, {true, false}}){
+    for(auto x: std::vector<std::array<bool, 2>>{{false, false}}){
     std::vector<config_t> tmp = {
 //      config_t{ord, x[0], x[1], 512, 512, 512},
-      config_t{ord, x[0], x[1], 8192, 8192, 8192},
+      config_t{ord, x[0], x[1], 2048, 2048, 2048},
 //      config_t{ord, x[0], x[1], 127008, 768, 576},
 //      config_t{ord, x[0], x[1], 8192, 8192, 8192}
 //      config_t{ord, x[0], x[1], 16, 2048, 2048},
@@ -36,7 +36,7 @@ int main() {
   for(const auto& c: configs){
     std::tie(ord, AT, BT, M, N, K) = c;
     std::cout << "// " << c ;
-    for(auto perf: bench_dot(stream, HALF, AT, BT, M, N, K, ord, ord))
+    for(auto perf: bench_dot(stream, FLOAT, AT, BT, M, N, K, ord, ord))
       std::cout << ", " << perf << std::flush;
     std::cout << std::endl;
   }