https://github.com/halide/Halide
Tip revision: 92e43859923e2d63738ace1514c9ca201ec9f416 authored by Steven Johnson on 21 September 2022, 23:20:58 UTC
Merge branch 'main' into srj/tls-4
Merge branch 'main' into srj/tls-4
Tip revision: 92e4385
Bounds.cpp
#include <iostream>
#include <utility>
#include "Bounds.h"
#include "CSE.h"
#include "ConciseCasts.h"
#include "Debug.h"
#include "Deinterleave.h"
#include "ExprUsesVar.h"
#include "FindIntrinsics.h"
#include "Func.h"
#include "IR.h"
#include "IREquality.h"
#include "IRMutator.h"
#include "IROperator.h"
#include "IRPrinter.h"
#include "IRVisitor.h"
#include "InlineReductions.h"
#include "Param.h"
#include "PurifyIndexMath.h"
#include "Simplify.h"
#include "SimplifyCorrelatedDifferences.h"
#include "Solve.h"
#include "Util.h"
#include "Var.h"
#ifndef DO_TRACK_BOUNDS_INTERVALS
#define DO_TRACK_BOUNDS_INTERVALS 0
#endif
#ifndef DO_DUMP_BOXES_TOUCHED
#define DO_DUMP_BOXES_TOUCHED 0
#endif
namespace Halide {
namespace Internal {
using std::map;
using std::pair;
using std::set;
using std::string;
using std::vector;
namespace {
int static_sign(const Expr &x) {
if (is_positive_const(x)) {
return 1;
} else if (is_negative_const(x)) {
return -1;
} else {
Expr zero = make_zero(x.type());
if (equal(const_true(), simplify(x > zero))) {
return 1;
} else if (equal(const_true(), simplify(x < zero))) {
return -1;
}
}
return 0;
}
} // anonymous namespace
const FuncValueBounds &empty_func_value_bounds() {
static FuncValueBounds empty;
return empty;
}
Expr find_constant_bound(const Expr &e, Direction d, const Scope<Interval> &scope) {
Interval interval = find_constant_bounds(e, scope);
Expr bound;
if (interval.has_lower_bound() && (d == Direction::Lower)) {
bound = interval.min;
} else if (interval.has_upper_bound() && (d == Direction::Upper)) {
bound = interval.max;
}
return bound;
}
Interval find_constant_bounds(const Expr &e, const Scope<Interval> &scope) {
Expr expr = bound_correlated_differences(simplify(remove_likelies(e)));
Interval interval = bounds_of_expr_in_scope(expr, scope, FuncValueBounds(), true);
interval.min = simplify(interval.min);
interval.max = simplify(interval.max);
// Note that we can get non-const but well-defined results (e.g. signed_integer_overflow);
// for our purposes here, treat anything non-const as no-bound.
if (!is_const(interval.min)) {
interval.min = Interval::neg_inf();
}
if (!is_const(interval.max)) {
interval.max = Interval::pos_inf();
}
return interval;
}
bool Box::maybe_unused() const {
return used.defined() && !is_const_one(used);
}
std::ostream &operator<<(std::ostream &stream, const Box &b) {
stream << "{";
for (size_t dim = 0; dim < b.size(); dim++) {
if (dim > 0) {
stream << ", ";
}
stream << "[" << b[dim].min << ", " << b[dim].max << "]";
}
stream << "}";
if (b.used.defined()) {
stream << " if " << b.used;
}
return stream;
}
namespace {
class Bounds : public IRVisitor {
public:
Interval interval;
Scope<Interval> scope;
const FuncValueBounds &func_bounds;
// If set to true, attempt to return an interval with constant upper
// and lower bounds. If the bound is not constant, it is set to
// unbounded.
bool const_bound;
Bounds(const Scope<Interval> *s, const FuncValueBounds &fb, bool const_bound)
: func_bounds(fb), const_bound(const_bound) {
scope.set_containing_scope(s);
// Find any points that are single_points but fail is_single_point due to
// pointer equality checks and replace with single_points.
for (auto item = s->cbegin(); item != s->cend(); ++item) {
const Interval &item_interval = item.value();
if (!item_interval.is_single_point() &&
equal(item_interval.min, item_interval.max)) {
scope.push(item.name(), Interval::single_point(item_interval.min));
}
}
}
#if DO_TRACK_BOUNDS_INTERVALS
public:
int log_indent = 0;
private:
std::string log_spaces() const {
return std::string(log_indent, ' ');
}
struct BoundsLogger final {
Bounds *const self;
BoundsLogger(Bounds *self, const char *pretty_function)
: self(self) {
string name = replace_all(pretty_function, "(anonymous namespace)::", "");
name = replace_all(name, "virtual void Halide::Internal::", "");
name = replace_all(name, "(const Halide::Internal::", "(");
name = replace_all(name, "::visit", "");
name = replace_all(name, " *)", ")");
log_line(name, " {");
self->log_indent++;
}
template<typename... Args>
void log_line(Args &&...args) {
debug(0) << self->log_spaces();
// C++17 right fold
(debug(0) << ... << args) << "\n";
}
~BoundsLogger() {
log_line("mn=", self->interval.min);
log_line("mx=", self->interval.max);
self->log_indent--;
log_line('}');
}
};
#define TRACK_BOUNDS_INTERVAL BoundsLogger log_me_here_(this, __PRETTY_FUNCTION__)
#define TRACK_BOUNDS_INFO(...) \
do { \
log_me_here_.log_line(__VA_ARGS__); \
} while (0)
#else
#define TRACK_BOUNDS_INTERVAL \
do { \
} while (0)
#define TRACK_BOUNDS_INFO(...) \
do { \
} while (0)
#endif // DO_TRACK_BOUNDS_INTERVALS
private:
// Compute the intrinsic bounds of a function.
void bounds_of_func(const string &name, int value_index, Type t) {
// if we can't get a good bound from the function, fall back to the bounds of the type.
bounds_of_type(t);
pair<string, int> key = {name, value_index};
FuncValueBounds::const_iterator iter = func_bounds.find(key);
if (iter != func_bounds.end()) {
if (iter->second.has_lower_bound()) {
interval.min = iter->second.min;
}
if (iter->second.has_upper_bound()) {
interval.max = iter->second.max;
}
}
}
void bounds_of_type(Type t) {
t = t.element_of();
if ((t.is_uint() || t.is_int()) && t.bits() <= 16) {
interval = Interval(t.min(), t.max());
} else {
interval = Interval::everything();
}
}
using IRVisitor::visit;
void visit(const IntImm *op) override {
TRACK_BOUNDS_INTERVAL;
interval = Interval::single_point(op);
}
void visit(const UIntImm *op) override {
TRACK_BOUNDS_INTERVAL;
interval = Interval::single_point(op);
}
void visit(const FloatImm *op) override {
TRACK_BOUNDS_INTERVAL;
interval = Interval::single_point(op);
}
void visit(const StringImm *op) override {
TRACK_BOUNDS_INTERVAL;
interval = Interval::single_point(op);
}
void visit(const Reinterpret *op) override {
TRACK_BOUNDS_INTERVAL;
Type t = op->type.element_of();
if (t.is_handle()) {
interval = Interval::everything();
return;
}
// Just use the bounds of the type
bounds_of_type(t);
}
void visit(const Cast *op) override {
TRACK_BOUNDS_INTERVAL;
op->value.accept(this);
Interval a = interval;
if (a.is_single_point(op->value)) {
interval = Interval::single_point(op);
return;
}
Type to = op->type.element_of();
Type from = op->value.type().element_of();
if (a.is_single_point()) {
interval = Interval::single_point(Cast::make(to, a.min));
return;
}
// If overflow is impossible, cast the min and max. If it's
// possible, use the bounds of the destination type.
bool could_overflow = true;
if (to.can_represent(from) || to.is_float()) {
could_overflow = false;
} else if (to.is_int() && to.bits() >= 32) {
// If we cast to an int32 or greater, assume that it won't
// overflow. Signed 32-bit integer overflow is undefined.
could_overflow = false;
} else if (a.is_bounded()) {
if (from.can_represent(to)) {
// The other case to consider is narrowing where the
// bounds of the original fit into the narrower type. We
// can only really prove that this is the case if they're
// constants, so try to make the constants first.
// First constant-fold
a.min = simplify(a.min);
a.max = simplify(a.max);
// Then try to strip off junk mins and maxes.
bool old_constant_bound = const_bound;
const_bound = true;
a.min.accept(this);
Expr lower_bound = interval.has_lower_bound() ? interval.min : Expr();
a.max.accept(this);
Expr upper_bound = interval.has_upper_bound() ? interval.max : Expr();
const_bound = old_constant_bound;
if (lower_bound.defined() && upper_bound.defined()) {
// Cast them to the narrow type and back and see if
// they're provably unchanged.
Expr test =
(cast(from, cast(to, lower_bound)) == lower_bound &&
cast(from, cast(to, upper_bound)) == upper_bound);
if (can_prove(test)) {
could_overflow = false;
// Relax the bounds to the constants we found. Not
// strictly necessary, but probably helpful to
// keep the expressions small.
a = Interval(lower_bound, upper_bound);
}
}
} else {
// a is bounded, but from and to can't necessarily represent
// each other; however, if the bounds can be simplified to
// constants, they might fit regardless of types.
a.min = simplify(a.min);
a.max = simplify(a.max);
const auto *umin = as_const_uint(a.min);
const auto *umax = as_const_uint(a.max);
if (umin && umax && to.can_represent(*umin) && to.can_represent(*umax)) {
could_overflow = false;
} else {
const auto *imin = as_const_int(a.min);
const auto *imax = as_const_int(a.max);
if (imin && imax && to.can_represent(*imin) && to.can_represent(*imax)) {
could_overflow = false;
} else {
const auto *fmin = as_const_float(a.min);
const auto *fmax = as_const_float(a.max);
if (fmin && fmax && to.can_represent(*fmin) && to.can_represent(*fmax)) {
could_overflow = false;
}
}
}
}
}
if (!could_overflow) {
// Start with the bounds of the narrow type.
bounds_of_type(from);
// If we have a better min or max for the arg use that.
if (a.has_lower_bound()) {
interval.min = a.min;
}
if (a.has_upper_bound()) {
interval.max = a.max;
}
// Then cast those bounds to the wider type.
if (interval.has_lower_bound()) {
interval.min = Cast::make(to, interval.min);
}
if (interval.has_upper_bound()) {
interval.max = Cast::make(to, interval.max);
}
} else {
// This might overflow, so use the bounds of the destination type.
bounds_of_type(to);
}
}
void visit(const Variable *op) override {
TRACK_BOUNDS_INTERVAL;
TRACK_BOUNDS_INFO("name:", op->name);
if (const_bound) {
bounds_of_type(op->type);
if (scope.contains(op->name)) {
const Interval &scope_interval = scope.get(op->name);
if (scope_interval.has_upper_bound() && is_const(scope_interval.max)) {
interval.max = Interval::make_min(interval.max, scope_interval.max);
}
if (scope_interval.has_lower_bound() && is_const(scope_interval.min)) {
interval.min = Interval::make_max(interval.min, scope_interval.min);
}
}
if (op->param.defined() &&
!op->param.is_buffer() &&
(op->param.min_value().defined() ||
op->param.max_value().defined())) {
if (op->param.max_value().defined() && is_const(op->param.max_value())) {
interval.max = Interval::make_min(interval.max, op->param.max_value());
}
if (op->param.min_value().defined() && is_const(op->param.min_value())) {
interval.min = Interval::make_max(interval.min, op->param.min_value());
}
}
} else {
if (scope.contains(op->name)) {
interval = scope.get(op->name);
} else if (op->type.is_vector()) {
// Uh oh, we need to take the min/max lane of some unknown vector. Treat as unbounded.
bounds_of_type(op->type);
} else {
interval = Interval::single_point(op);
}
}
}
void visit(const Add *op) override {
TRACK_BOUNDS_INTERVAL;
op->a.accept(this);
Interval a = interval;
op->b.accept(this);
Interval b = interval;
if (a.is_single_point(op->a) && b.is_single_point(op->b)) {
interval = Interval::single_point(op);
} else if (a.is_single_point() && b.is_single_point()) {
interval = Interval::single_point(a.min + b.min);
} else {
bounds_of_type(op->type);
if (a.has_lower_bound() && b.has_lower_bound()) {
interval.min = a.min + b.min;
}
if (a.has_upper_bound() && b.has_upper_bound()) {
interval.max = a.max + b.max;
}
// Assume no overflow for float, int32, and int64
if (op->type.can_overflow()) {
if (!interval.is_bounded()) {
// Possibly infinite things that wrap can be anything.
bounds_of_type(op->type);
return;
}
// TODO(5682): Can't catch overflow of UInt(64) currently.
Type t = op->type.is_uint() ? UInt(64) : Int(32);
Expr no_overflow_max = (cast(t, a.max) + cast(t, b.max) == cast(t, interval.max));
Expr no_overflow_min = (cast(t, a.min) + cast(t, b.min) == cast(t, interval.min));
if (!can_prove(no_overflow_max && no_overflow_min)) {
bounds_of_type(op->type);
return;
}
}
}
}
void visit(const Sub *op) override {
TRACK_BOUNDS_INTERVAL;
op->a.accept(this);
Interval a = interval;
op->b.accept(this);
Interval b = interval;
if (a.is_single_point(op->a) && b.is_single_point(op->b)) {
interval = Interval::single_point(op);
} else if (a.is_single_point() && b.is_single_point()) {
interval = Interval::single_point(a.min - b.min);
} else {
bounds_of_type(op->type);
if (a.has_lower_bound() && b.has_upper_bound()) {
interval.min = a.min - b.max;
}
if (a.has_upper_bound() && b.has_lower_bound()) {
interval.max = a.max - b.min;
}
// Assume no overflow for float, int32, and int64
if (op->type.can_overflow()) {
if (!interval.is_bounded()) {
// Possibly infinite things that wrap can be anything.
bounds_of_type(op->type);
return;
}
Expr no_overflow_max = (cast<int>(a.max) - cast<int>(b.min) == cast<int>(interval.max));
Expr no_overflow_min = (cast<int>(a.min) - cast<int>(b.max) == cast<int>(interval.min));
if (!can_prove(no_overflow_max && no_overflow_min)) {
bounds_of_type(op->type);
return;
}
}
// Check underflow for uint
if (op->type.is_uint() &&
interval.has_lower_bound() &&
!can_prove(b.max <= a.min)) {
bounds_of_type(op->type);
}
}
}
void visit(const Mul *op) override {
TRACK_BOUNDS_INTERVAL;
op->a.accept(this);
Interval a = interval;
op->b.accept(this);
Interval b = interval;
// Move constants to the right
if (a.is_single_point() && !b.is_single_point()) {
std::swap(a, b);
}
if (a.is_single_point(op->a) && b.is_single_point(op->b)) {
interval = Interval::single_point(op);
return;
} else if (a.is_single_point() && b.is_single_point()) {
interval = Interval::single_point(a.min * b.min);
return;
} else if (b.is_single_point()) {
Expr e1 = a.has_lower_bound() ? a.min * b.min : a.min;
Expr e2 = a.has_upper_bound() ? a.max * b.min : a.max;
if (is_const_zero(b.min)) {
interval = b;
} else if (is_positive_const(b.min) || op->type.is_uint()) {
interval = Interval(e1, e2);
} else if (is_negative_const(b.min)) {
if (e1.same_as(Interval::neg_inf())) {
e1 = Interval::pos_inf();
}
if (e2.same_as(Interval::pos_inf())) {
e2 = Interval::neg_inf();
}
interval = Interval(e2, e1);
} else if (a.is_bounded()) {
// Sign of b is unknown
Expr cmp = b.min >= make_zero(b.min.type().element_of());
interval = Interval(select(cmp, e1, e2), select(cmp, e2, e1));
} else {
bounds_of_type(op->type);
}
} else if (a.is_bounded() && b.is_bounded()) {
interval = Interval::nothing();
interval.include(a.min * b.min);
interval.include(a.min * b.max);
interval.include(a.max * b.min);
interval.include(a.max * b.max);
} else {
bounds_of_type(op->type);
}
// Assume no overflow for float, int32, and int64
if (op->type.can_overflow()) {
if (a.is_bounded() && b.is_bounded()) {
// Try to prove it can't overflow. (Be sure to use uint32 for unsigned
// types so that the case of 65535*65535 won't misleadingly fail.)
// TODO(5682): Can't catch overflow of UInt(64) currently.
Type t = op->type.is_uint() ? UInt(64) : Int(32);
Expr test1 = (cast(t, a.min) * cast(t, b.min) == cast(t, a.min * b.min));
Expr test2 = (cast(t, a.min) * cast(t, b.max) == cast(t, a.min * b.max));
Expr test3 = (cast(t, a.max) * cast(t, b.min) == cast(t, a.max * b.min));
Expr test4 = (cast(t, a.max) * cast(t, b.max) == cast(t, a.max * b.max));
if (!can_prove(test1 && test2 && test3 && test4)) {
bounds_of_type(op->type);
}
} else {
bounds_of_type(op->type);
}
}
}
bool div_cannot_overflow(const Interval &a, const Interval &b, Type t) {
// No overflow if: not an allowed overflow int type, or `a` cannot be t.min() or
// `b` cannot be -1, because t.min() / -1 overflows for int16 and int8.
Expr neg_one = make_const(t, -1);
return !t.can_overflow_int() ||
(a.has_lower_bound() && can_prove(a.min != t.min())) ||
(b.has_upper_bound() && can_prove(b.max < neg_one)) ||
(b.has_lower_bound() && can_prove(b.min > neg_one));
}
void visit(const Div *op) override {
TRACK_BOUNDS_INTERVAL;
op->a.accept(this);
Interval a = interval;
op->b.accept(this);
Interval b = interval;
if (!b.is_bounded()) {
// Integer division can only make things smaller in
// magnitude (but can flip the sign).
if (a.is_bounded() && op->type.is_int() && op->type.bits() >= 32) {
// Restrict to no-overflow types to avoid worrying
// about overflow due to negating the most negative int.
if (can_prove(a.min >= 0)) {
interval.min = -a.max;
interval.max = a.max;
} else if (can_prove(a.max <= 0)) {
interval.min = a.min;
interval.max = -a.min;
} else if (a.is_single_point()) {
// The following case would also be correct, but
// would duplicate the expression, which is
// generally a bad thing for any later interval
// arithmetic.
interval.min = -cast(a.min.type(), abs(a.min));
interval.max = cast(a.min.type(), abs(a.max));
} else {
// div by 0 is 0 and the magnitude cannot increase by integer division
interval.min = min(-a.max, a.min);
interval.max = max(-a.min, a.max);
}
} else {
bounds_of_type(op->type);
}
} else if (a.is_single_point(op->a) && b.is_single_point(op->b)) {
interval = Interval::single_point(op);
} else if (can_prove(b.min == b.max)) {
Expr e1 = a.has_lower_bound() ? a.min / b.min : a.min;
Expr e2 = a.has_upper_bound() ? a.max / b.max : a.max;
Type t = op->type.element_of();
if (div_cannot_overflow(a, b, t)) {
// TODO: handle real numbers with can_prove(b.min > 0) and can_prove(b.min < 0) as well - treating floating point as
// reals can be error prone when dealing with division near 0, so for now we only consider integers in the can_prove() path
if (op->type.is_uint() || is_positive_const(b.min) || (op->type.is_int() && can_prove(b.min >= 0))) {
interval = Interval(e1, e2);
} else if (is_negative_const(b.min) || (op->type.is_int() && can_prove(b.min <= 0))) {
if (e1.same_as(Interval::neg_inf())) {
e1 = Interval::pos_inf();
}
if (e2.same_as(Interval::pos_inf())) {
e2 = Interval::neg_inf();
}
interval = Interval(e2, e1);
} else if (a.is_bounded()) {
// Sign of b is unknown.
Expr cmp = b.min > make_zero(b.min.type().element_of());
interval = Interval(select(cmp, e1, e2), select(cmp, e2, e1));
} else {
bounds_of_type(op->type);
}
} else {
// Overflow is possible because a can be min value of type t and
// b can be -1.
bounds_of_type(op->type);
}
} else if (a.is_bounded()) {
// if we can't statically prove that the divisor can't span zero, then we're unbounded
int min_sign = static_sign(b.min);
int max_sign = static_sign(b.max);
if (min_sign != max_sign || min_sign == 0 || max_sign == 0) {
if (op->type.is_int() && op->type.bits() >= 32) {
// Division can't make signed integers larger
// Restricted to 32-bits or greater to ensure the
// negation can't overflow.
interval = Interval::nothing();
interval.include(a.min);
interval.include(a.max);
interval.include(-a.min);
interval.include(-a.max);
} else if (op->type.is_uint()) {
// Division can't make unsigned integers large,
// but could make them arbitrarily small.
interval.min = make_zero(a.min.type());
interval.max = a.max;
} else {
// Division can make floats arbitrarily large, and
// we can't easily negate narrow bit-width signed
// integers because they just wrap.
bounds_of_type(op->type);
}
} else {
Type t = op->type.element_of();
if (div_cannot_overflow(a, b, t)) {
// Divisor is either strictly positive or strictly
// negative, so we can just take the extrema.
interval = Interval::nothing();
interval.include(a.min / b.min);
interval.include(a.max / b.min);
interval.include(a.min / b.max);
interval.include(a.max / b.max);
} else {
// Overflow is possible because a can be min value of type t and
// b can be -1.
bounds_of_type(op->type);
}
}
} else {
bounds_of_type(op->type);
}
}
void visit(const Mod *op) override {
TRACK_BOUNDS_INTERVAL;
op->a.accept(this);
Interval a = interval;
op->b.accept(this);
Interval b = interval;
if (a.is_single_point(op->a) && b.is_single_point(op->b)) {
interval = Interval::single_point(op);
return;
}
Type t = op->type.element_of();
// Mod is always positive
interval.min = make_zero(t);
interval.max = Interval::pos_inf();
if (!b.is_bounded()) {
if (a.has_lower_bound() && can_prove(a.min >= 0)) {
// Mod cannot make positive values larger
interval.max = a.max;
}
} else {
// b is bounded
if (b.max.type().is_int_or_uint() && is_positive_const(b.min)) {
// If the RHS is >= 1, the result is in [0, max_b-1]
interval.max = b.max - make_one(t);
} else if (b.max.type().is_uint()) {
// if b.max = 0 then result is [0, 0], else [0, b.max - 1]
interval.max = select(b.max == make_zero(t), make_zero(t), b.max - make_one(t));
} else if (b.max.type().is_int()) {
// x % [4,10] -> [0,9]
// x % [-8,-3] -> [0,7]
// x % [-8, 10] -> [0,9]
interval.max = Max::make(interval.min, b.max - make_one(t));
interval.max = Max::make(interval.max, make_const(t, -1) - b.min);
} else if (b.max.type().is_float()) {
// The floating point version has the same sign rules,
// but can reach all the way up to the original value,
// so there's no -1.
interval.max = Max::make(b.max, -b.min);
}
}
}
void visit(const Min *op) override {
TRACK_BOUNDS_INTERVAL;
op->a.accept(this);
Interval a = interval;
op->b.accept(this);
Interval b = interval;
if (a.is_single_point(op->a) && b.is_single_point(op->b)) {
interval = Interval::single_point(op);
} else {
interval = Interval(Interval::make_min(a.min, b.min),
Interval::make_min(a.max, b.max));
}
}
void visit(const Max *op) override {
TRACK_BOUNDS_INTERVAL;
op->a.accept(this);
Interval a = interval;
op->b.accept(this);
Interval b = interval;
if (a.is_single_point(op->a) && b.is_single_point(op->b)) {
interval = Interval::single_point(op);
} else {
interval = Interval(Interval::make_max(a.min, b.min),
Interval::make_max(a.max, b.max));
}
}
// only used for LT and LE - GT and GE normalize to LT and LTE
template<typename Cmp>
void visit_compare(const Expr &a_expr, const Expr &b_expr) {
a_expr.accept(this);
if (!interval.has_upper_bound() && !interval.has_lower_bound()) {
bounds_of_type(Bool());
return;
}
Interval a = interval;
b_expr.accept(this);
if (!interval.has_upper_bound() && !interval.has_lower_bound()) {
bounds_of_type(Bool());
return;
}
Interval b = interval;
bounds_of_type(Bool());
// The returned interval should have the property that min <=
// val <= max. For integers it's clear what this means. For
// bools, treating false < true, '<=' is in fact
// implication. So we want conditions min and max such that
// min implies val implies max. So min should be a sufficient
// condition, and max should be a necessary condition.
// a.max <(=) b.min implies a <(=) b, so a <(=) b is at least
// as true as a.max <(=) b.min. This does not depend on a's
// lower bound or b's upper bound.
if (a.has_upper_bound() && b.has_lower_bound()) {
interval.min = Cmp::make(a.max, b.min);
}
// a <(=) b implies a.min <(=) b.max, so a <(=) b is at most
// as true as a.min <(=) b.max. This does not depend on a's
// upper bound or b's lower bound.
if (a.has_lower_bound() && b.has_upper_bound()) {
interval.max = Cmp::make(a.min, b.max);
}
}
void visit(const LT *op) override {
TRACK_BOUNDS_INTERVAL;
visit_compare<LT>(op->a, op->b);
}
void visit(const LE *op) override {
TRACK_BOUNDS_INTERVAL;
visit_compare<LE>(op->a, op->b);
}
void visit(const GT *op) override {
TRACK_BOUNDS_INTERVAL;
visit_compare<LT>(op->b, op->a);
}
void visit(const GE *op) override {
TRACK_BOUNDS_INTERVAL;
visit_compare<LE>(op->b, op->a);
}
void visit(const EQ *op) override {
TRACK_BOUNDS_INTERVAL;
op->a.accept(this);
Interval a = interval;
op->b.accept(this);
Interval b = interval;
if (a.is_single_point(op->a) && b.is_single_point(op->b)) {
interval = Interval::single_point(op);
} else if (a.is_single_point() && b.is_single_point()) {
interval = Interval::single_point(a.min == b.min);
} else {
// If either vary, it could always be false, so we have no
// good sufficient condition.
bounds_of_type(op->type);
// But could it be true? A necessary condition is that the
// ranges overlap.
if (a.is_bounded() && b.is_bounded()) {
interval.max = a.min <= b.max && b.min <= a.max;
} else if (a.has_upper_bound() && b.has_lower_bound()) {
// a.min <= b.max is implied if a.min = -inf or b.max = +inf.
interval.max = b.min <= a.max;
} else if (a.has_lower_bound() && b.has_upper_bound()) {
// b.min <= a.max is implied if a.max = +inf or b.min = -inf.
interval.max = a.min <= b.max;
}
}
}
void visit(const NE *op) override {
TRACK_BOUNDS_INTERVAL;
op->a.accept(this);
Interval a = interval;
op->b.accept(this);
Interval b = interval;
if (a.is_single_point(op->a) && b.is_single_point(op->b)) {
interval = Interval::single_point(op);
} else if (a.is_single_point() && b.is_single_point()) {
interval = Interval::single_point(a.min != b.min);
} else {
// If either vary, it could always be true that they're
// not equal, so we have no good necessary condition.
bounds_of_type(op->type);
// But we do have a sufficient condition. If the ranges of
// a and b do not overlap, then they must be not equal.
if (a.is_bounded() && b.is_bounded()) {
interval.min = a.min > b.max || b.min > a.max;
} else if (a.has_upper_bound() && b.has_lower_bound()) {
// a.min > b.max is false if a.min = -inf or b.max = +inf.
// a does not need a lower bound nor does b need
// an upper bound for this condition.
interval.min = b.min > a.max;
} else if (a.has_lower_bound() && b.has_upper_bound()) {
// b.min > a.max is false if a.max = +inf or b.min = -inf.
// a does not need an upper bound nor does b need
// a lower bound for this condition.
interval.min = a.min > b.max;
}
}
}
Expr make_and(Expr a, Expr b) {
if (is_const_one(a)) {
return b;
}
if (is_const_one(b)) {
return a;
}
if (is_const_zero(a)) {
return a;
}
if (is_const_zero(b)) {
return b;
}
return a && b;
}
void visit(const And *op) override {
TRACK_BOUNDS_INTERVAL;
op->a.accept(this);
Interval a = interval;
op->b.accept(this);
Interval b = interval;
if (a.is_single_point(op->a) && b.is_single_point(op->b)) {
interval = Interval::single_point(op);
} else if (a.is_single_point() && b.is_single_point()) {
interval = Interval::single_point(a.min && b.min);
} else {
// And is monotonic increasing in both args
interval.min = make_and(a.min, b.min);
interval.max = make_and(a.max, b.max);
}
}
Expr make_or(Expr a, Expr b) {
if (is_const_one(a)) {
return a;
}
if (is_const_one(b)) {
return b;
}
if (is_const_zero(a)) {
return b;
}
if (is_const_zero(b)) {
return a;
}
return a || b;
}
void visit(const Or *op) override {
TRACK_BOUNDS_INTERVAL;
op->a.accept(this);
Interval a = interval;
op->b.accept(this);
Interval b = interval;
if (a.is_single_point(op->a) && b.is_single_point(op->b)) {
interval = Interval::single_point(op);
} else if (a.is_single_point() && b.is_single_point()) {
interval = Interval::single_point(a.min || b.min);
} else {
// Or is monotonic increasing in both args
interval.min = make_or(a.min, b.min);
interval.max = make_or(a.max, b.max);
}
}
Expr make_not(const Expr &e) {
if (is_const_one(e)) {
return make_zero(e.type());
}
if (is_const_zero(e)) {
return make_one(e.type());
}
return !e;
}
void visit(const Not *op) override {
TRACK_BOUNDS_INTERVAL;
op->a.accept(this);
Interval a = interval;
if (a.is_single_point(op->a)) {
interval = Interval::single_point(op);
} else if (a.is_single_point()) {
interval = Interval::single_point(!a.min);
} else {
interval.min = make_not(a.max);
interval.max = make_not(a.min);
}
}
void visit(const Select *op) override {
TRACK_BOUNDS_INTERVAL;
op->true_value.accept(this);
Interval a = interval;
op->false_value.accept(this);
Interval b = interval;
op->condition.accept(this);
Interval cond = interval;
if (cond.is_single_point()) {
if (is_const_one(cond.min)) {
interval = a;
return;
} else if (is_const_zero(cond.min)) {
interval = b;
return;
}
}
Type t = op->type.element_of();
if (!a.has_lower_bound() || !b.has_lower_bound()) {
interval.min = Interval::neg_inf();
} else if (a.min.same_as(b.min)) {
interval.min = a.min;
} else if (cond.is_single_point()) {
interval.min = select(cond.min, a.min, b.min);
} else if (is_const_zero(cond.min) && is_const_one(cond.max)) {
interval.min = Interval::make_min(a.min, b.min);
} else if (is_const_one(cond.max)) {
// cond.min is non-trivial
string var_name = unique_name('t');
Expr var = Variable::make(t, var_name);
interval.min = Interval::make_min(select(cond.min, var, b.min), var);
interval.min = Let::make(var_name, a.min, interval.min);
} else if (is_const_zero(cond.min)) {
// cond.max is non-trivial
string var_name = unique_name('t');
Expr var = Variable::make(t, var_name);
interval.min = Interval::make_min(select(cond.max, a.min, var), var);
interval.min = Let::make(var_name, b.min, interval.min);
} else {
string a_var_name = unique_name('t'), b_var_name = unique_name('t');
Expr a_var = Variable::make(t, a_var_name);
Expr b_var = Variable::make(t, b_var_name);
interval.min = Interval::make_min(select(cond.min, a_var, b_var),
select(cond.max, a_var, b_var));
interval.min = Let::make(a_var_name, a.min, interval.min);
interval.min = Let::make(b_var_name, b.min, interval.min);
}
if (!a.has_upper_bound() || !b.has_upper_bound()) {
interval.max = Interval::pos_inf();
} else if (a.max.same_as(b.max)) {
interval.max = a.max;
} else if (cond.is_single_point()) {
interval.max = select(cond.min, a.max, b.max);
} else if (is_const_zero(cond.min) && is_const_one(cond.max)) {
interval.max = Interval::make_max(a.max, b.max);
} else if (is_const_one(cond.max)) {
// cond.min is non-trivial
string var_name = unique_name('t');
Expr var = Variable::make(t, var_name);
interval.max = Interval::make_max(select(cond.min, var, b.max), var);
interval.max = Let::make(var_name, a.max, interval.max);
} else if (is_const_zero(cond.min)) {
// cond.max is non-trivial
string var_name = unique_name('t');
Expr var = Variable::make(t, var_name);
interval.max = Interval::make_max(select(cond.max, a.max, var), var);
interval.max = Let::make(var_name, b.max, interval.max);
} else {
string a_var_name = unique_name('t'), b_var_name = unique_name('t');
Expr a_var = Variable::make(t, a_var_name);
Expr b_var = Variable::make(t, b_var_name);
interval.max = Interval::make_max(select(cond.min, a_var, b_var),
select(cond.max, a_var, b_var));
interval.max = Let::make(a_var_name, a.max, interval.max);
interval.max = Let::make(b_var_name, b.max, interval.max);
}
}
void visit(const Load *op) override {
TRACK_BOUNDS_INTERVAL;
op->index.accept(this);
if (!const_bound && interval.is_single_point() && is_const_one(op->predicate)) {
// If the index is const and it is not a predicated load,
// we can return the load of that index
Expr load_min =
Load::make(op->type.element_of(), op->name, interval.min,
op->image, op->param, const_true(), ModulusRemainder());
interval = Interval::single_point(load_min);
} else {
// Otherwise use the bounds of the type
bounds_of_type(op->type);
}
}
void visit(const Ramp *op) override {
TRACK_BOUNDS_INTERVAL;
// Treat the ramp lane as a free variable
string var_name = unique_name('t');
Expr var = Variable::make(op->base.type().element_of(), var_name);
Expr lane = op->base + var * op->stride;
Expr min_value = make_const(var.type(), 0);
Expr max_value = make_const(var.type(), op->lanes - 1);
if (!var.type().can_represent((int64_t)(op->lanes - 1))) {
// max_value will overflow.
min_value = var.type().min();
max_value = var.type().max();
}
ScopedBinding<Interval> p(scope, var_name, Interval(min_value, max_value));
lane.accept(this);
}
void visit(const Broadcast *op) override {
TRACK_BOUNDS_INTERVAL;
op->value.accept(this);
}
void visit(const Call *op) override {
TRACK_BOUNDS_INTERVAL;
TRACK_BOUNDS_INFO("name:", op->name);
// Tags are hints that don't affect the results of the expression,
// and can be very deeply nested in the case of strict_float. The
// bounds of this call are *always* exactly that of its first argument,
// so short circuit it here before checking for const_args. This is
// important because evaluating const_args for such a deeply nested case
// essentially becomes O(n^2) doing work that is unnecessary, making
// otherwise simple pipelines take several minutes to compile.
if (op->is_tag()) {
internal_assert(op->args.size() == 1);
op->args[0].accept(this);
return;
}
Type t = op->type.element_of();
if (t.is_handle()) {
interval = Interval::everything();
return;
}
if (!const_bound &&
(op->call_type == Call::PureExtern ||
op->call_type == Call::Image)) {
// If the args are const we can return the call of those args
// for pure functions. For other types of functions, the same
// call in two different places might produce different
// results (e.g. during the update step of a reduction), so we
// can't move around call nodes.
//
// Note: Only evaluate new_args if we know the call is a candidate;
// otherwise we can get n^2 evaluation time for deeply-nested
// Expr trees.
std::vector<Expr> new_args(op->args.size());
bool const_args = true;
for (size_t i = 0; i < op->args.size() && const_args; i++) {
op->args[i].accept(this);
if (interval.is_single_point()) {
new_args[i] = interval.min;
} else {
const_args = false;
}
}
if (const_args) {
Expr call = Call::make(t, op->name, new_args, op->call_type,
op->func, op->value_index, op->image, op->param);
interval = Interval::single_point(call);
return;
}
// else fall thru and continue
}
if (op->is_intrinsic(Call::abs)) {
op->args[0].accept(this);
Interval a = interval;
interval.min = make_zero(t);
if (a.is_bounded()) {
if (equal(a.min, a.max)) {
interval = Interval::single_point(Call::make(t, Call::abs, {a.max}, Call::PureIntrinsic));
} else if (op->args[0].type().is_int() && op->args[0].type().bits() >= 32) {
interval.max = Max::make(Cast::make(t, -a.min), Cast::make(t, a.max));
} else {
a.min = Call::make(t, Call::abs, {a.min}, Call::PureIntrinsic);
a.max = Call::make(t, Call::abs, {a.max}, Call::PureIntrinsic);
interval.max = Max::make(a.min, a.max);
}
} else {
// If the argument is unbounded on one side, then the max is unbounded.
interval.max = Interval::pos_inf();
}
} else if (op->is_intrinsic(Call::absd)) {
internal_assert(!t.is_handle());
if (t.is_float()) {
Expr e = abs(op->args[0] - op->args[1]);
e.accept(this);
} else {
// absd() for int types will always produce a uint result
internal_assert(t.is_uint());
Expr a = op->args[0];
Expr b = op->args[1];
internal_assert(a.type() == b.type());
a.accept(this);
Interval a_interval = interval;
b.accept(this);
Interval b_interval = interval;
if (a_interval.is_bounded() && b_interval.is_bounded()) {
interval.min = make_zero(t);
interval.max = max(absd(a_interval.max, b_interval.min), absd(a_interval.min, b_interval.max));
} else {
bounds_of_type(t);
}
}
} else if (op->is_intrinsic(Call::saturating_cast)) {
internal_assert(op->args.size() == 1);
Expr a = lower_saturating_cast(op->type, op->args[0]);
a.accept(this);
return;
} else if (op->is_intrinsic(Call::unsafe_promise_clamped) ||
op->is_intrinsic(Call::promise_clamped)) {
// Unlike an explicit clamp, we are also permitted to
// assume the upper bound is greater than the lower bound.
op->args[1].accept(this);
Interval lower = interval;
op->args[2].accept(this);
Interval upper = interval;
op->args[0].accept(this);
if (op->is_intrinsic(Call::promise_clamped) &&
interval.is_single_point()) {
// It's not safe to lift a promise_clamped
// intrinsic. They make a claim that holds true at
// that specific point in the IR. But if it's a single
// point we're probably inside the scope over which
// the thing varies, so we don't want to needlessly
// complicate the IR by injecting the min/max. For now
// we just drop the annotation and return the bounds
// of the first arg.
return;
}
if (op->is_intrinsic(Call::unsafe_promise_clamped) &&
interval.is_single_point(op->args[0]) &&
lower.is_single_point(op->args[1]) &&
upper.is_single_point(op->args[2])) {
// It *is* safe to lift an
// unsafe_promise_clamped. Those are injected by the
// user and represent a promise that holds globally
// across the entire program. So in the case that
// nothing varies we return the full Expr, not just
// the first arg. In the case where things are varying
// we resolve to min/max (i.e. we exploit the promise).
interval = Interval::single_point(op);
return;
}
interval.min = Interval::make_max(interval.min, lower.min);
interval.max = Interval::make_min(interval.max, upper.max);
} else if (Call::as_tag(op)) {
op->args[0].accept(this);
} else if (op->is_intrinsic(Call::return_second)) {
internal_assert(op->args.size() == 2);
op->args[1].accept(this);
} else if (op->is_intrinsic(Call::if_then_else)) {
internal_assert(op->args.size() == 2 || op->args.size() == 3);
// Probably more conservative than necessary
Expr false_value = op->args.size() == 2 ? op->args[1] : op->args[2];
Expr equivalent_select = Select::make(op->args[0], op->args[1], false_value);
equivalent_select.accept(this);
} else if (op->is_intrinsic(Call::require)) {
internal_assert(op->args.size() == 3);
op->args[1].accept(this);
} else if (op->is_intrinsic(Call::shift_left) ||
op->is_intrinsic(Call::shift_right) ||
op->is_intrinsic(Call::bitwise_xor) ||
op->is_intrinsic(Call::bitwise_and) ||
op->is_intrinsic(Call::bitwise_or)) {
Expr a = op->args[0], b = op->args[1];
a.accept(this);
Interval a_interval = interval;
b.accept(this);
Interval b_interval = interval;
if (a_interval.is_single_point(a) && b_interval.is_single_point(b)) {
interval = Interval::single_point(op);
} else if (a_interval.is_single_point() && b_interval.is_single_point()) {
interval = Interval::single_point(Call::make(t, op->name, {a_interval.min, b_interval.min}, op->call_type));
} else {
bounds_of_type(t);
// For some of these intrinsics applied to integer
// types we can go a little further.
if (t.is_int() || t.is_uint()) {
if (op->is_intrinsic(Call::shift_left)) {
if (t.is_int() && t.bits() >= 32) {
// Overflow is UB
if (a_interval.has_lower_bound() &&
b_interval.has_lower_bound() &&
can_prove(b_interval.min >= 0 &&
b_interval.min < t.bits())) {
interval.min = a_interval.min << b_interval.min;
} else if (a_interval.has_lower_bound() &&
b_interval.has_lower_bound() &&
!b_interval.min.type().is_uint() &&
(a_interval.min.type().is_uint() ||
can_prove(a_interval.min >= 0)) &&
can_prove(b_interval.min < 0 &&
b_interval.min > -t.bits())) {
interval.min = a_interval.min >> abs(b_interval.min);
} else if (a_interval.has_lower_bound() &&
a_interval.min.type().is_int() &&
can_prove(a_interval.min < 0) &&
b_interval.has_upper_bound()) {
// If a can be negative, then we split a_interval into
// two ranges, [a.min, 0) and [0, a.max]. Note that the
// second range may not exist, if a's range is fully
// negative, but that doesn't matter - a positive value
// cannot be shifted to produce a negative, so the min
// of the operation is produced in the negative range.
if (!b_interval.max.type().is_uint() &&
can_prove(b_interval.max <= 0)) {
// If b is strictly non-positive, then the magnitude can only decrease.
interval.min = a_interval.min;
} else {
// If b could be positive, then the magnitude might increase.
interval.min = min(a_interval.min, a_interval.min << b_interval.max);
}
} else if (a_interval.has_lower_bound() &&
(a_interval.min.type().is_uint() ||
can_prove(a_interval.min >= 0))) {
// A positive value shifted cannot change sign.
interval.min = make_zero(t);
}
// TODO: Are there any other cases we can handle for interval.min?
if (a_interval.has_upper_bound() &&
b_interval.has_upper_bound() &&
can_prove(b_interval.max >= 0 &&
b_interval.max < t.bits())) {
interval.max = a_interval.max << b_interval.max;
} else if (a_interval.has_upper_bound() &&
b_interval.has_upper_bound() &&
!b_interval.max.type().is_uint() &&
can_prove(b_interval.max < 0 &&
b_interval.max > -t.bits())) {
interval.max = a_interval.max >> abs(b_interval.max);
}
} else if (is_const(b)) {
// We can normalize to multiplication
Expr equiv = a * (make_const(t, 1) << b);
equiv.accept(this);
}
} else if (op->is_intrinsic(Call::shift_right)) {
// Only try to improve on bounds-of-type if we can prove 0 <= b < t.bits,
// as shift_right(a, b) is UB for b outside that range.
if (b_interval.is_bounded()) {
bool b_min_ok_positive = can_prove(b_interval.min >= 0 &&
b_interval.min < t.bits());
bool b_max_ok_positive = can_prove(b_interval.max >= 0 &&
b_interval.max < t.bits());
bool b_min_ok_negative =
!b_interval.min.type().is_uint() &&
can_prove(b_interval.min < 0 && b_interval.min > -t.bits());
bool b_max_ok_negative =
!b_interval.max.type().is_uint() &&
can_prove(b_interval.max < 0 && b_interval.max > -t.bits());
if (a_interval.has_lower_bound()) {
if (b_max_ok_positive && (a_interval.min.type().is_uint() ||
can_prove(a_interval.min >= 0))) {
interval.min = a_interval.min >> b_interval.max;
} else if (can_prove(a_interval.min < 0) && b_max_ok_negative) {
interval.min = a_interval.min << abs(b_interval.max);
} else if (b_min_ok_positive && b_max_ok_positive) {
// if a < 0, the smallest value will be a >> b.min
// if a > 0, the smallest value will be a >> b.max
interval.min = min(a_interval.min >> b_interval.min,
a_interval.min >> b_interval.max);
} else if (b_min_ok_negative && b_max_ok_negative) {
// if a < 0, the smallest value will be a << abs(b.min)
// if a > 0, the smallest value will be a << abs(b.max)
interval.min = min(a_interval.min << abs(b_interval.min),
a_interval.min << abs(b_interval.max));
}
}
if (a_interval.has_upper_bound()) {
if (can_prove(a_interval.max >= 0) && b_min_ok_positive) {
interval.max = a_interval.max >> b_interval.min;
} else if (can_prove(a_interval.max < 0) && b_min_ok_negative) {
interval.max = a_interval.max << abs(b_interval.min);
} else if (b_min_ok_positive && b_max_ok_positive) {
// if a < 0, the largest value will be a >> b.max
// if a > 0, the largest value will be a >> b.min
interval.max = max(a_interval.max >> b_interval.max,
a_interval.max >> b_interval.min);
} else if (b_min_ok_negative && b_max_ok_negative) {
// if a < 0, the largest value will be a << abs(b.max)
// if a > 0, the largest value will be a << abs(b.min)
interval.max = max(a_interval.max << abs(b_interval.max),
a_interval.max << abs(b_interval.min));
}
}
}
} else if (op->is_intrinsic(Call::bitwise_and) &&
a_interval.has_upper_bound() &&
b_interval.has_upper_bound()) {
bool a_positive = a_interval.has_lower_bound() && can_prove(a_interval.min >= 0);
bool b_positive = b_interval.has_lower_bound() && can_prove(b_interval.min >= 0);
if (a_positive && b_positive) {
// Positive and smaller than both args
interval.max = min(a_interval.max, b_interval.max);
interval.min = make_zero(t);
} else if (t.is_int()) {
if (b_positive) {
interval.min = make_zero(t);
interval.max = b_interval.max;
} else if (a_positive) {
interval.min = make_zero(t);
interval.max = a_interval.max;
} else {
// Smaller than the larger of the two args
interval.max = max(a_interval.max, b_interval.max);
}
}
} else if (op->is_intrinsic(Call::bitwise_or) &&
a_interval.has_lower_bound() &&
b_interval.has_lower_bound()) {
if (t.is_int()) {
// Larger than the smaller arg
interval.min = min(a_interval.min, b_interval.min);
} else if (t.is_uint()) {
// Larger than both args
interval.min = max(a_interval.min, b_interval.min);
}
}
}
}
} else if (op->is_intrinsic(Call::bitwise_not)) {
// In 2's complement bitwise not inverts the ordering of
// the space, without causing overflow (unlike negation),
// so bitwise not is monotonic decreasing.
op->args[0].accept(this);
Interval a_interval = interval;
if (a_interval.is_single_point(op->args[0])) {
interval = Interval::single_point(op);
} else if (a_interval.is_single_point()) {
interval = Interval::single_point(~a_interval.min);
} else {
bounds_of_type(t);
if (t.is_int() || t.is_uint()) {
if (a_interval.has_upper_bound()) {
interval.min = ~a_interval.max;
}
if (a_interval.has_lower_bound()) {
interval.max = ~a_interval.min;
}
}
}
} else if (op->args.size() == 1 && interval.is_bounded() &&
(op->name == "ceil_f32" || op->name == "ceil_f64" ||
op->name == "floor_f32" || op->name == "floor_f64" ||
op->name == "round_f32" || op->name == "round_f64" ||
op->name == "exp_f32" || op->name == "exp_f64" ||
op->name == "log_f32" || op->name == "log_f64")) {
// For monotonic, pure, single-argument functions, we can
// make two calls for the min and the max.
interval = Interval(
Call::make(t, op->name, {interval.min}, op->call_type,
op->func, op->value_index, op->image, op->param),
Call::make(t, op->name, {interval.max}, op->call_type,
op->func, op->value_index, op->image, op->param));
} else if (op->is_intrinsic(Call::popcount) ||
op->is_intrinsic(Call::count_leading_zeros) ||
op->is_intrinsic(Call::count_trailing_zeros)) {
internal_assert(op->args.size() == 1);
const Type &t = op->type.element_of();
Expr min = make_zero(t);
Expr max = make_const(t, op->args[0].type().bits());
if (op->is_intrinsic(Call::count_leading_zeros)) {
// clz treats signed and unsigned ints the same way;
// cast all ints to uint to simplify this.
cast(op->type.with_code(halide_type_uint), op->args[0]).accept(this);
Interval a = interval;
if (a.has_lower_bound()) {
max = cast(t, count_leading_zeros(a.min));
}
if (a.has_upper_bound()) {
min = cast(t, count_leading_zeros(a.max));
}
}
interval = Interval(min, max);
} else if (op->is_intrinsic(Call::memoize_expr)) {
internal_assert(!op->args.empty());
op->args[0].accept(this);
} else if (op->is_intrinsic(Call::scatter_gather)) {
// Take the union of the args
Interval result = Interval::nothing();
for (const Expr &e : op->args) {
e.accept(this);
result.include(interval);
}
interval = result;
} else if (op->is_intrinsic(Call::mux)) {
// Take the union of all args but the first
Interval result = Interval::nothing();
for (size_t i = 1; i < op->args.size(); i++) {
op->args[i].accept(this);
result.include(interval);
}
interval = result;
} else if (op->is_intrinsic(Call::widen_right_add)) {
Expr add = Add::make(op->args[0], cast(op->args[0].type(), op->args[1]));
add.accept(this);
} else if (op->is_intrinsic(Call::widen_right_sub)) {
Expr sub = Sub::make(op->args[0], cast(op->args[0].type(), op->args[1]));
sub.accept(this);
} else if (op->is_intrinsic(Call::widen_right_mul)) {
Expr mul = Mul::make(op->args[0], cast(op->args[0].type(), op->args[1]));
mul.accept(this);
} else if (op->call_type == Call::Halide) {
bounds_of_func(op->name, op->value_index, op->type);
} else {
// Just use the bounds of the type
bounds_of_type(t);
}
}
void visit(const Let *op) override {
TRACK_BOUNDS_INTERVAL;
op->value.accept(this);
Interval val = interval;
// We'll either substitute the values in directly, or pass
// them in as variables and add an outer let (to avoid
// combinatorial explosion).
Interval var;
const string min_name = unique_name(op->name + ".min");
const string max_name = unique_name(op->name + ".max");
if (val.has_lower_bound()) {
if (is_const(val.min)) {
// Substitute it in
var.min = val.min;
val.min = Expr();
} else {
var.min = Variable::make(op->value.type().element_of(), min_name);
}
}
if (val.has_upper_bound()) {
if (is_const(val.max)) {
// Substitute it in
var.max = val.max;
val.max = Expr();
} else if (val.is_single_point()) {
var.max = var.min;
} else {
var.max = Variable::make(op->value.type().element_of(), max_name);
}
}
{
ScopedBinding<Interval> p(scope, op->name, var);
op->body.accept(this);
}
bool single_point = interval.is_single_point();
if (interval.has_lower_bound()) {
if (val.min.defined() &&
expr_uses_var(interval.min, min_name)) {
interval.min = Let::make(min_name, val.min, interval.min);
}
if (val.max.defined() &&
!val.is_single_point() &&
expr_uses_var(interval.min, max_name)) {
interval.min = Let::make(max_name, val.max, interval.min);
}
}
if (single_point) {
interval.max = interval.min;
} else if (interval.has_upper_bound()) {
if (val.min.defined() &&
expr_uses_var(interval.max, min_name)) {
interval.max = Let::make(min_name, val.min, interval.max);
}
if (val.max.defined() &&
!val.is_single_point() &&
expr_uses_var(interval.max, max_name)) {
interval.max = Let::make(max_name, val.max, interval.max);
}
}
}
void visit(const Shuffle *op) override {
TRACK_BOUNDS_INTERVAL;
Interval result = Interval::nothing();
for (const Expr &i : op->vectors) {
i.accept(this);
result.include(interval);
}
interval = result;
}
void visit(const VectorReduce *op) override {
TRACK_BOUNDS_INTERVAL;
op->value.accept(this);
int factor = op->value.type().lanes() / op->type.lanes();
switch (op->op) {
case VectorReduce::Add:
if (interval.has_upper_bound()) {
interval.max *= factor;
}
if (interval.has_lower_bound()) {
interval.min *= factor;
}
break;
case VectorReduce::SaturatingAdd:
case VectorReduce::Mul:
// Technically there are some things we could say
// here. E.g. if all the lanes are positive then we're
// bounded by the upper bound raised to the factor
// power. However it's extremely unlikely that a mul
// reduce will ever make it into a bounds expression, so
// for now we bail.
bounds_of_type(op->value.type());
break;
case VectorReduce::Min:
case VectorReduce::Max:
case VectorReduce::And:
case VectorReduce::Or:
// The bounds of a single lane are sufficient
break;
}
}
void visit(const LetStmt *) override {
internal_error << "Bounds of statement\n";
}
void visit(const AssertStmt *) override {
internal_error << "Bounds of statement\n";
}
void visit(const ProducerConsumer *) override {
internal_error << "Bounds of statement\n";
}
void visit(const For *) override {
internal_error << "Bounds of statement\n";
}
void visit(const Store *) override {
internal_error << "Bounds of statement\n";
}
void visit(const Provide *) override {
internal_error << "Bounds of statement\n";
}
void visit(const Allocate *) override {
internal_error << "Bounds of statement\n";
}
void visit(const Realize *) override {
internal_error << "Bounds of statement\n";
}
void visit(const Block *) override {
internal_error << "Bounds of statement\n";
}
};
// Version that exposes 'indent' is for internal use only
Interval bounds_of_expr_in_scope_with_indent(const Expr &expr, const Scope<Interval> &scope, const FuncValueBounds &fb, bool const_bound, int indent) {
#if DO_TRACK_BOUNDS_INTERVALS
const string spaces(indent, ' ');
debug(0) << spaces << "BoundsOfExprInScope {\n"
<< spaces << " expr: " << expr << "\n";
#endif
Bounds b(&scope, fb, const_bound);
#if DO_TRACK_BOUNDS_INTERVALS
b.log_indent = indent + 1;
#endif
expr.accept(&b);
#if DO_TRACK_BOUNDS_INTERVALS
debug(0) << spaces << " mn=" << simplify(b.interval.min) << "\n"
<< spaces << " mx=" << simplify(b.interval.max) << "\n"
<< spaces << "}\n";
#endif
Type expected = expr.type().element_of();
if (b.interval.has_lower_bound()) {
internal_assert(b.interval.min.type() == expected)
<< "Min of " << expr
<< " should have been a scalar of type " << expected
<< ": " << b.interval.min << "\n";
}
if (b.interval.has_upper_bound()) {
internal_assert(b.interval.max.type() == expected)
<< "Max of " << expr
<< " should have been a scalar of type " << expected
<< ": " << b.interval.max << "\n";
}
return b.interval;
}
} // namespace
Interval bounds_of_expr_in_scope(const Expr &expr, const Scope<Interval> &scope, const FuncValueBounds &fb, bool const_bound) {
return bounds_of_expr_in_scope_with_indent(expr, scope, fb, const_bound, 0);
}
Region region_union(const Region &a, const Region &b) {
internal_assert(a.size() == b.size()) << "Mismatched dimensionality in region union\n";
Region result;
for (size_t i = 0; i < a.size(); i++) {
Expr min = Min::make(a[i].min, b[i].min);
Expr max_a = a[i].min + a[i].extent;
Expr max_b = b[i].min + b[i].extent;
Expr max_plus_one = Max::make(max_a, max_b);
Expr extent = max_plus_one - min;
result.push_back(Range(simplify(min), simplify(extent)));
}
return result;
}
void merge_boxes(Box &a, const Box &b) {
if (b.empty()) {
return;
}
if (a.empty()) {
a = b;
return;
}
internal_assert(a.size() == b.size());
bool a_maybe_unused = a.maybe_unused();
bool b_maybe_unused = b.maybe_unused();
bool complementary = a_maybe_unused && b_maybe_unused &&
(equal(a.used, !b.used) || equal(!a.used, b.used));
for (size_t i = 0; i < a.size(); i++) {
if (!a[i].min.same_as(b[i].min)) {
if (a[i].has_lower_bound() && b[i].has_lower_bound()) {
if (a_maybe_unused && b_maybe_unused) {
if (complementary) {
a[i].min = select(a.used, a[i].min, b[i].min);
} else {
a[i].min = select(a.used && b.used, Interval::make_min(a[i].min, b[i].min),
a.used, a[i].min,
b[i].min);
}
} else if (a_maybe_unused) {
a[i].min = select(a.used, Interval::make_min(a[i].min, b[i].min), b[i].min);
} else if (b_maybe_unused) {
a[i].min = select(b.used, Interval::make_min(a[i].min, b[i].min), a[i].min);
} else {
a[i].min = Interval::make_min(a[i].min, b[i].min);
}
a[i].min = simplify(a[i].min);
} else {
a[i].min = Interval::neg_inf();
}
}
if (!a[i].max.same_as(b[i].max)) {
if (a[i].has_upper_bound() && b[i].has_upper_bound()) {
if (a_maybe_unused && b_maybe_unused) {
if (complementary) {
a[i].max = select(a.used, a[i].max, b[i].max);
} else {
a[i].max = select(a.used && b.used, Interval::make_max(a[i].max, b[i].max),
a.used, a[i].max,
b[i].max);
}
} else if (a_maybe_unused) {
a[i].max = select(a.used, Interval::make_max(a[i].max, b[i].max), b[i].max);
} else if (b_maybe_unused) {
a[i].max = select(b.used, Interval::make_max(a[i].max, b[i].max), a[i].max);
} else {
a[i].max = Interval::make_max(a[i].max, b[i].max);
}
a[i].max = simplify(a[i].max);
} else {
a[i].max = Interval::pos_inf();
}
}
}
if (a_maybe_unused && b_maybe_unused) {
if (!equal(a.used, b.used)) {
a.used = simplify(a.used || b.used);
if (is_const_one(a.used)) {
a.used = Expr();
}
}
} else {
a.used = Expr();
}
}
Box box_union(const Box &a, const Box &b) {
Box result = a;
merge_boxes(result, b);
return result;
}
Box box_intersection(const Box &a, const Box &b) {
Box result;
if (a.empty() || b.empty()) {
return result;
}
internal_assert(a.size() == b.size());
result.resize(a.size());
for (size_t i = 0; i < a.size(); i++) {
result[i].min = simplify(max(a[i].min, b[i].min));
result[i].max = simplify(min(a[i].max, b[i].max));
}
// The intersection is only used if both a and b are used.
if (a.maybe_unused() && b.maybe_unused()) {
result.used = a.used && b.used;
} else if (a.maybe_unused()) {
result.used = a.used;
} else if (b.maybe_unused()) {
result.used = b.used;
}
return result;
}
bool boxes_overlap(const Box &a, const Box &b) {
// If one box is scalar and the other is not, the boxes cannot
// overlap.
if (a.size() != b.size() && (a.empty() || b.empty())) {
return false;
}
internal_assert(a.size() == b.size());
bool a_maybe_unused = a.maybe_unused();
bool b_maybe_unused = b.maybe_unused();
// Overlapping requires both boxes to be used.
Expr overlap = ((a_maybe_unused ? a.used : const_true()) &&
(b_maybe_unused ? b.used : const_true()));
for (size_t i = 0; i < a.size(); i++) {
if (a[i].has_upper_bound() && b[i].has_lower_bound()) {
overlap = overlap && b[i].max >= a[i].min;
}
if (a[i].has_lower_bound() && b[i].has_upper_bound()) {
overlap = overlap && a[i].max >= b[i].min;
}
}
// Conservatively, assume they overlap if we can't prove there's no overlap
return !can_prove(simplify(!overlap));
}
bool box_contains(const Box &outer, const Box &inner) {
// If the inner box has more dimensions than the outer box, the
// inner box cannot fit in the outer box.
if (inner.size() > outer.size()) {
return false;
}
Expr condition = const_true();
for (size_t i = 0; i < inner.size(); i++) {
if ((outer[i].has_lower_bound() && !inner[i].has_lower_bound()) ||
(outer[i].has_upper_bound() && !inner[i].has_upper_bound())) {
return false;
}
if (outer[i].has_lower_bound()) {
condition = condition && (outer[i].min <= inner[i].min);
}
if (outer[i].has_upper_bound()) {
condition = condition && (outer[i].max >= inner[i].max);
}
}
if (outer.maybe_unused()) {
if (inner.maybe_unused()) {
// inner condition must imply outer one
condition = condition && ((outer.used && inner.used) == inner.used);
} else {
// outer box is conditional, but inner is not
return false;
}
}
return can_prove(condition);
}
namespace {
class FindInnermostVar : public IRVisitor {
public:
const Scope<int> &vars_depth;
string innermost_var;
FindInnermostVar(const Scope<int> &vars_depth)
: vars_depth(vars_depth) {
}
private:
using IRVisitor::visit;
int innermost_depth = -1;
void visit(const Variable *op) override {
if (vars_depth.contains(op->name)) {
int depth = vars_depth.get(op->name);
if (depth > innermost_depth) {
innermost_var = op->name;
innermost_depth = depth;
}
}
}
};
// Place innermost vars in an IfThenElse's condition as far to the left as possible.
class SolveIfThenElse : public IRMutator {
// Scope of variable names and their depths. Higher depth indicates
// variable defined more innermost.
Scope<int> vars_depth;
int depth = -1;
using IRMutator::visit;
void push_var(const string &var) {
depth += 1;
vars_depth.push(var, depth);
}
void pop_var(const string &var) {
depth -= 1;
vars_depth.pop(var);
}
Stmt visit(const LetStmt *op) override {
Stmt orig = op;
vector<const LetStmt *> frames;
Stmt body;
do {
frames.push_back(op);
push_var(op->name);
body = op->body;
op = body.as<LetStmt>();
} while (op);
Stmt s = mutate(body);
if (s.same_as(body)) {
for (auto it = frames.rbegin(); it != frames.rend(); it++) {
pop_var((*it)->name);
}
return orig;
} else {
for (auto it = frames.rbegin(); it != frames.rend(); it++) {
pop_var((*it)->name);
s = LetStmt::make((*it)->name, (*it)->value, s);
}
return s;
}
}
Stmt visit(const For *op) override {
push_var(op->name);
Stmt stmt = IRMutator::visit(op);
pop_var(op->name);
return stmt;
}
Stmt visit(const IfThenElse *op) override {
Stmt stmt = IRMutator::visit(op);
op = stmt.as<IfThenElse>();
internal_assert(op);
FindInnermostVar find(vars_depth);
op->condition.accept(&find);
if (!find.innermost_var.empty()) {
Expr condition = solve_expression(op->condition, find.innermost_var).result;
if (!condition.same_as(op->condition)) {
stmt = IfThenElse::make(condition, op->then_case, op->else_case);
}
}
return stmt;
}
};
// Collect all variables referenced in an expr or statement
// (excluding 'skipped_var')
class CollectVars : public IRGraphVisitor {
public:
string skipped_var;
set<string> vars;
CollectVars(const string &v)
: skipped_var(v) {
}
private:
using IRGraphVisitor::visit;
void visit(const Variable *op) override {
if (op->name != skipped_var) {
vars.insert(op->name);
}
}
};
// Compute the box produced by a statement
class BoxesTouched : public IRGraphVisitor {
public:
BoxesTouched(bool calls, bool provides, string fn, const Scope<Interval> *s, const FuncValueBounds &fb)
: func(std::move(fn)), consider_calls(calls), consider_provides(provides), func_bounds(fb) {
scope.set_containing_scope(s);
}
map<string, Box> boxes;
#if DO_TRACK_BOUNDS_INTERVALS
private:
int log_indent = 0;
HALIDE_ALWAYS_INLINE
Interval bounds_of_expr_in_scope(const Expr &expr,
const Scope<Interval> &scope,
const FuncValueBounds &func_bounds = empty_func_value_bounds(),
bool const_bound = false) {
return bounds_of_expr_in_scope_with_indent(expr, scope, func_bounds, const_bound, log_indent);
}
std::string log_spaces() const {
return std::string(log_indent, ' ');
}
struct BoxesTouchedLogger final {
BoxesTouched *const self;
BoxesTouchedLogger *const parent_logger;
map<string, Box> boxes;
template<typename... Args>
void log_line(Args &&...args) {
debug(0) << self->log_spaces();
// C++17 right fold
(debug(0) << ... << args) << "\n";
}
BoxesTouchedLogger(BoxesTouched *self, const char *pretty_function)
: self(self), parent_logger(self->current_logger), boxes(self->boxes) {
string name = replace_all(pretty_function, "(anonymous namespace)::", "");
name = replace_all(name, "virtual void Halide::Internal::", "");
name = replace_all(name, "(const Halide::Internal::", "(");
name = replace_all(name, "::visit", "");
name = replace_all(name, " *)", ")");
if (self->consider_calls && !self->consider_provides) {
name = replace_all(name, "BoxesTouched", "BoxesRequired");
} else if (!self->consider_calls && self->consider_provides) {
name = replace_all(name, "BoxesTouched", "BoxesProvided");
}
log_line(name, " {");
self->log_indent++;
self->current_logger = this;
}
static bool boxes_equal(const Box &a, const Box &b) {
if (!equal(a.used, b.used)) {
return false;
}
if (a.bounds.size() != b.bounds.size()) {
return false;
}
for (size_t i = 0; i < a.bounds.size(); i++) {
if (!equal(a.bounds[i].min, a.bounds[i].min)) {
return false;
}
if (!equal(a.bounds[i].max, a.bounds[i].max)) {
return false;
}
}
return true;
}
void log_box_diffs(const map<string, Box> &before, const map<string, Box> &after) {
const std::string spaces = self->log_spaces();
for (const auto &it : after) {
const auto &key = it.first;
const auto &new_box = it.second;
const auto old_box_it = before.find(key);
if (old_box_it == before.end()) {
// Added.
debug(0) << spaces << "Added: " << key << " = " << new_box << "\n";
} else {
const auto &old_box = old_box_it->second;
if (!boxes_equal(old_box, new_box)) {
debug(0) << spaces << "Changed: " << key << " = " << old_box << " -> " << new_box << "\n";
}
}
}
}
~BoxesTouchedLogger() {
log_box_diffs(this->boxes, self->boxes);
self->log_indent--;
log_line('}');
internal_assert(self->current_logger == this);
if (parent_logger) {
// Propagate changes to our parent.
// This isn't efficient at all, but it's usually-disabled debugging code.
for (const auto &it : self->boxes) {
parent_logger->boxes[it.first] = it.second;
}
}
self->current_logger = parent_logger;
}
};
BoxesTouchedLogger *current_logger = nullptr;
#define TRACK_BOXES_TOUCHED BoxesTouchedLogger log_me_here_(this, __PRETTY_FUNCTION__)
#define TRACK_BOXES_TOUCHED_INFO(...) \
do { \
log_me_here_.log_line(__VA_ARGS__); \
} while (0)
#else
HALIDE_ALWAYS_INLINE
Interval bounds_of_expr_in_scope(const Expr &expr,
const Scope<Interval> &scope,
const FuncValueBounds &func_bounds = empty_func_value_bounds(),
bool const_bound = false) {
return ::Halide::Internal::bounds_of_expr_in_scope(expr, scope, func_bounds, const_bound);
}
#define TRACK_BOXES_TOUCHED \
do { \
} while (0)
#define TRACK_BOXES_TOUCHED_INFO(...) \
do { \
} while (0)
#endif // DO_TRACK_BOUNDS_INTERVALS
private:
struct VarInstance {
string var;
int instance;
VarInstance(const string &v, int i)
: var(v), instance(i) {
}
VarInstance() = default;
bool operator==(const VarInstance &other) const {
return (var == other.var) && (instance == other.instance);
}
bool operator<(const VarInstance &other) const {
if (var == other.var) {
return (instance < other.instance);
}
return (var < other.var);
}
};
string func;
bool consider_calls, consider_provides;
Scope<Interval> scope;
const FuncValueBounds &func_bounds;
// Scope containing the current value definition of let stmts.
Scope<Expr> let_stmts;
// Keep track of variable renaming. Map variable name to instantiation number
// (0 for the first variable to be defined, 1 for the 1st redefinition, etc.).
map<string, int> vars_renaming;
// Map variable name to all other vars which values depend on that variable.
map<VarInstance, set<VarInstance>> children;
bool in_producer{false};
map<std::string, Expr> buffer_lets;
using IRGraphVisitor::visit;
bool box_from_extended_crop(const Expr &e, Box &b) {
const Call *call_expr = e.as<Call>();
if (call_expr != nullptr) {
if (call_expr->name == Call::buffer_crop) {
internal_assert(call_expr->args.size() == 5)
<< "Call::buffer_crop call with unexpected number of arguments.\n";
const Variable *in_buf = call_expr->args[2].as<Variable>();
const Call *mins_struct = call_expr->args[3].as<Call>();
const Call *extents_struct = call_expr->args[4].as<Call>();
// Ignore crops that apply to a different buffer than the one being looked for.
if (in_buf != nullptr && (in_buf->name == (func + ".buffer"))) {
internal_assert(mins_struct != nullptr && extents_struct != nullptr &&
mins_struct->is_intrinsic(Call::make_struct) &&
extents_struct->is_intrinsic(Call::make_struct))
<< "BoxesTouched::box_from_extended_crop -- unexpected buffer_crop form.\n";
b.resize(mins_struct->args.size());
b.used = const_true();
for (size_t i = 0; i < mins_struct->args.size(); i++) {
Interval min_interval = bounds_of_expr_in_scope(mins_struct->args[i], scope, func_bounds);
Interval max_interval = bounds_of_expr_in_scope(mins_struct->args[i] + extents_struct->args[i] - 1, scope, func_bounds);
b[i] = Interval(min_interval.min, max_interval.max);
}
return true;
}
} else if (call_expr->name == Call::buffer_set_bounds) {
internal_assert(call_expr->args.size() == 4)
<< "Call::buffer_set_bounds call with unexpected number of arguments.\n";
const IntImm *dim = call_expr->args[1].as<IntImm>();
if (dim != nullptr && box_from_extended_crop(call_expr->args[0], b)) {
internal_assert(dim->value >= 0 && dim->value < (int64_t)b.size())
<< "box_from_extended_crop setting bounds for out of range dim.\n";
Interval min_interval = bounds_of_expr_in_scope(call_expr->args[2], scope, func_bounds);
Interval max_interval = bounds_of_expr_in_scope(call_expr->args[2] + call_expr->args[3] - 1, scope, func_bounds);
b[dim->value] = Interval(min_interval.min, max_interval.max);
return true;
}
}
}
return false;
}
void visit(const Call *op) override {
TRACK_BOXES_TOUCHED;
TRACK_BOXES_TOUCHED_INFO("name:", op->name);
if (op->is_intrinsic(Call::declare_box_touched)) {
internal_assert(!op->args.empty());
const Variable *handle = op->args[0].as<Variable>();
const string &func = handle->name;
Box b(op->args.size() / 2);
for (size_t i = 0; i < b.size(); i++) {
b[i].min = op->args[2 * i + 1];
b[i].max = op->args[2 * i + 2];
}
merge_boxes(boxes[func], b);
}
if (consider_calls) {
if (op->is_intrinsic(Call::if_then_else)) {
// We wrap 'then_case' and 'else_case' inside 'dummy' call since IfThenElse
// only takes Stmts as arguments.
Stmt then_case = Evaluate::make(op->args[1]);
Stmt equivalent_if;
if (op->args.size() == 3) {
Stmt else_case = Evaluate::make(op->args[2]);
equivalent_if = IfThenElse::make(op->args[0], then_case, else_case);
} else {
internal_assert(op->args.size() == 2);
equivalent_if = IfThenElse::make(op->args[0], then_case);
}
equivalent_if.accept(this);
return;
}
IRGraphVisitor::visit(op);
if (op->call_type == Call::Halide ||
op->call_type == Call::Image) {
for (const Expr &e : op->args) {
e.accept(this);
}
if (op->name == func || func.empty()) {
Box b(op->args.size());
b.used = const_true();
for (size_t i = 0; i < op->args.size(); i++) {
b[i] = bounds_of_expr_in_scope(op->args[i], scope, func_bounds);
}
merge_boxes(boxes[op->name], b);
}
}
}
if (op->is_extern() && (in_producer || consider_calls)) {
if (op->name == "halide_buffer_copy") {
// Call doesn't yet have user_context inserted, so size is 3.
internal_assert(op->args.size() == 3) << "Unexpected arg list size for halide_buffer_copy\n";
for (int i = 0; i < 2; i++) {
// If considering calls, merge in the source bounds.
// If considering provides, merge in the destination bounds.
int var_index;
if (i == 0 && consider_calls) {
var_index = 0;
} else if (i == 1 && consider_provides && in_producer) {
var_index = 2;
} else {
continue;
}
const Variable *var = op->args[var_index].as<Variable>();
if (var != nullptr && var->type == type_of<halide_buffer_t *>()) {
if (func.empty() || starts_with(var->name, func)) {
const auto iter = buffer_lets.find(var->name);
if (iter != buffer_lets.end()) {
Box b;
if (box_from_extended_crop(iter->second, b)) {
merge_boxes(boxes[func], b);
}
}
}
}
}
}
}
}
class CountVars : public IRVisitor {
using IRVisitor::visit;
void visit(const Variable *var) override {
count++;
}
public:
int count = 0;
CountVars() = default;
};
// We get better simplification if we directly substitute mins
// and maxes in, but this can also cause combinatorial code
// explosion. Here we manage this by only substituting in
// reasonably-sized expressions. We determine the size by
// counting the number of var nodes.
bool is_small_enough_to_substitute(const Expr &e) {
CountVars c;
e.accept(&c);
return c.count < 10;
}
void push_var(const string &name) {
auto iter = vars_renaming.find(name);
if (iter == vars_renaming.end()) {
vars_renaming.emplace(name, 0);
} else {
iter->second += 1;
}
}
void pop_var(const string &name) {
auto iter = vars_renaming.find(name);
internal_assert(iter != vars_renaming.end());
iter->second -= 1;
if (iter->second < 0) {
vars_renaming.erase(iter);
}
}
VarInstance get_var_instance(const string &name) {
// It is possible for the variable to be not in 'vars_renaming', e.g.
// the output buffer min/max. In this case, we just add the variable
// to the renaming map and assign it to instance 0.
return VarInstance(name, vars_renaming[name]);
}
template<typename LetOrLetStmt>
void visit_let(const LetOrLetStmt *op) {
using is_let_stmt = typename std::is_same<LetOrLetStmt, LetStmt>;
// LetStmts can be deeply stacked, and this visitor is called
// before dead lets are eliminated, so we move all the
// internal state off the call stack into an explicit stack on
// the heap.
struct Frame {
set<string> old_let_vars;
const LetOrLetStmt *op;
VarInstance vi;
CollectVars collect;
string max_name, min_name;
Interval value_bounds;
Frame(const LetOrLetStmt *op)
: op(op), collect(op->name) {
}
};
vector<Frame> frames;
decltype(op->body) result;
while (op) {
frames.emplace_back(op);
Frame &f = frames.back();
push_var(op->name);
if (op->value.type() == type_of<struct halide_buffer_t *>()) {
buffer_lets[op->name] = op->value;
}
if (is_let_stmt::value) {
f.vi = get_var_instance(op->name);
// Update the 'children' map.
op->value.accept(&f.collect);
for (const auto &v : f.collect.vars) {
children[get_var_instance(v)].insert(f.vi);
}
// If this let stmt is a redefinition of a previous one, we should
// remove the old let stmt from the 'children' map since it is
// no longer valid at this point.
if ((f.vi.instance > 0) && let_stmts.contains(op->name)) {
const Expr &val = let_stmts.get(op->name);
CollectVars collect(op->name);
val.accept(&collect);
f.old_let_vars = collect.vars;
VarInstance old_vi = VarInstance(f.vi.var, f.vi.instance - 1);
for (const auto &v : f.old_let_vars) {
internal_assert(vars_renaming.count(v));
children[get_var_instance(v)].erase(old_vi);
}
}
let_stmts.push(op->name, op->value);
}
op->value.accept(this);
f.value_bounds = bounds_of_expr_in_scope(op->value, scope, func_bounds);
bool fixed = f.value_bounds.min.same_as(f.value_bounds.max);
f.value_bounds.min = simplify(f.value_bounds.min);
f.value_bounds.max = fixed ? f.value_bounds.min : simplify(f.value_bounds.max);
if (is_small_enough_to_substitute(f.value_bounds.min) &&
(fixed || is_small_enough_to_substitute(f.value_bounds.max))) {
scope.push(op->name, f.value_bounds);
} else {
f.max_name = unique_name('t');
f.min_name = unique_name('t');
scope.push(op->name, Interval(Variable::make(op->value.type(), f.min_name),
Variable::make(op->value.type(), f.max_name)));
}
result = op->body;
op = result.template as<LetOrLetStmt>();
}
result.accept(this);
for (auto it = frames.rbegin(); it != frames.rend(); it++) {
// Pop the value bounds
scope.pop(it->op->name);
if (it->op->value.type() == type_of<struct halide_buffer_t *>()) {
buffer_lets.erase(it->op->name);
}
if (!it->min_name.empty()) {
// We made up new names for the bounds of the
// value, and need to rewrap any boxes we're
// returning with appropriate lets.
for (pair<const string, Box> &i : boxes) {
Box &box = i.second;
for (size_t i = 0; i < box.size(); i++) {
if (box[i].has_lower_bound()) {
if (expr_uses_var(box[i].min, it->max_name)) {
box[i].min = Let::make(it->max_name, it->value_bounds.max, box[i].min);
}
if (expr_uses_var(box[i].min, it->min_name)) {
box[i].min = Let::make(it->min_name, it->value_bounds.min, box[i].min);
}
}
if (box[i].has_upper_bound()) {
if (expr_uses_var(box[i].max, it->max_name)) {
box[i].max = Let::make(it->max_name, it->value_bounds.max, box[i].max);
}
if (expr_uses_var(box[i].max, it->min_name)) {
box[i].max = Let::make(it->min_name, it->value_bounds.min, box[i].max);
}
}
}
}
}
if (is_let_stmt::value) {
let_stmts.pop(it->op->name);
// If this let stmt shadowed an outer one, we need
// to re-insert the children from the previous let
// stmt into the map.
if (!it->old_let_vars.empty()) {
internal_assert(it->vi.instance > 0);
VarInstance old_vi = VarInstance(it->vi.var, it->vi.instance - 1);
for (const auto &v : it->old_let_vars) {
internal_assert(vars_renaming.count(v));
children[get_var_instance(v)].insert(old_vi);
}
}
// Remove the children from the current let stmt.
for (const auto &v : it->collect.vars) {
internal_assert(vars_renaming.count(v));
children[get_var_instance(v)].erase(it->vi);
}
}
pop_var(it->op->name);
}
}
void visit(const Let *op) override {
TRACK_BOXES_TOUCHED;
TRACK_BOXES_TOUCHED_INFO("name:", op->name);
visit_let(op);
}
void visit(const LetStmt *op) override {
TRACK_BOXES_TOUCHED;
TRACK_BOXES_TOUCHED_INFO("name:", op->name);
visit_let(op);
}
struct LetBound {
string var, min_name, max_name;
LetBound(const string &v, const string &min, const string &max)
: var(v), min_name(min), max_name(max) {
}
};
void trim_scope_push(const string &name, const Interval &bound, vector<LetBound> &let_bounds) {
// We want to add all the children of 'name' to 'let_bounds',
// but avoiding duplicates (in some cases the dupes can
// explode the list size by ~80x); note that the exact order
// isn't important, as long as children are still visited
// after parents. So we want to do a topological traversal of
// the dependent lets.
// A recursive version of a topological traversal looks like:
// 1) if node already visited, return
// 2) mark node as visited
// 3) recursively visit children
// 4) prepend node to output list.
// Step 4 means that the node is the first thing in the output
// list, and step 3 means all of the children have been
// visited for sure and are in the output list somewhere
// else. No future operations after this recursive step
// returns ever move things around in the output list - we
// only ever prepend things. This means we have the
// topological sort property that every node is guaranteed to
// be before all of its children.
// For an example of doing this the recursive way, see
// realization_order_dfs in RealizationOrder.cpp. It uses two
// different senses of 'visited' to check for cycles, but we
// don't need that here. We'll assume there are no cycles.
// There could be many dependent lets, so we're going to do it
// non-recursively with an explicit stack of Task structs
// instead. Note that there's work to do (step 4) after the
// recursive step (step 3), so we can't just discard nodes at
// the same time as we enqueue their children. We need to
// consider every node in the stack twice - once just before
// pushing its children, and once again when we reach it again
// after dealing with all children and it's time to pop it
// (our pending stack is effectively a stack frame from the
// recursive version).
// As a minor optimization we'll also do the visited
// insert/check (steps 1 and 2) before pushing, so that
// already-visited nodes don't even make it into the
// stack. Finally, we actually want reverse topological order,
// so we'll append nodes to the output instead of prepending.
struct Task {
string var;
bool visited_children_already;
};
vector<Task> pending;
set<string> visited;
scope.push(name, bound);
visited.insert(name);
pending.push_back(Task{name, false});
// We don't want our root node 'name' in the let_bounds list,
// so we'll stop when there's only one thing left in the
// pending stack.
do {
Task &next = pending.back();
if (!next.visited_children_already) {
next.visited_children_already = true;
// Note that pushing may invalidate the reference to next.
for (const auto &v : children[get_var_instance(next.var)]) {
if (visited.insert(v.var).second) {
pending.push_back(Task{v.var, false});
}
}
} else {
string max_name = unique_name('t');
string min_name = unique_name('t');
let_bounds.emplace_back(next.var, min_name, max_name);
Type t = let_stmts.get(next.var).type();
Interval b = Interval(Variable::make(t, min_name), Variable::make(t, max_name));
scope.push(next.var, b);
pending.pop_back();
}
} while (pending.size() > 1);
}
void trim_scope_pop(const string &name, vector<LetBound> &let_bounds) {
for (const LetBound &l : let_bounds) {
scope.pop(l.var);
for (pair<const string, Box> &i : boxes) {
Box &box = i.second;
for (size_t i = 0; i < box.size(); i++) {
Interval v_bound;
if ((box[i].has_lower_bound() && (expr_uses_var(box[i].min, l.max_name) ||
expr_uses_var(box[i].min, l.min_name))) ||
(box[i].has_upper_bound() && (expr_uses_var(box[i].max, l.max_name) ||
expr_uses_var(box[i].max, l.min_name)))) {
internal_assert(let_stmts.contains(l.var));
const Expr &val = let_stmts.get(l.var);
v_bound = bounds_of_expr_in_scope(val, scope, func_bounds);
bool fixed = v_bound.min.same_as(v_bound.max);
v_bound.min = simplify(v_bound.min);
v_bound.max = fixed ? v_bound.min : simplify(v_bound.max);
internal_assert(scope.contains(l.var));
const Interval &old_bound = scope.get(l.var);
v_bound.max = simplify(min(v_bound.max, old_bound.max));
v_bound.min = simplify(max(v_bound.min, old_bound.min));
}
if (box[i].has_lower_bound()) {
if (expr_uses_var(box[i].min, l.max_name)) {
box[i].min = Let::make(l.max_name, v_bound.max, box[i].min);
}
if (expr_uses_var(box[i].min, l.min_name)) {
box[i].min = Let::make(l.min_name, v_bound.min, box[i].min);
}
}
if (box[i].has_upper_bound()) {
if (expr_uses_var(box[i].max, l.max_name)) {
box[i].max = Let::make(l.max_name, v_bound.max, box[i].max);
}
if (expr_uses_var(box[i].max, l.min_name)) {
box[i].max = Let::make(l.min_name, v_bound.min, box[i].max);
}
}
}
}
}
scope.pop(name);
let_bounds.clear();
}
vector<const Variable *> find_free_vars(const Expr &e) {
class FindFreeVars : public IRVisitor {
using IRVisitor::visit;
void visit(const Variable *op) override {
if (scope.contains(op->name)) {
result.push_back(op);
}
}
public:
const Scope<Interval> &scope;
vector<const Variable *> result;
FindFreeVars(const Scope<Interval> &s)
: scope(s) {
}
} finder(scope);
e.accept(&finder);
return finder.result;
}
void visit(const IfThenElse *op) override {
TRACK_BOXES_TOUCHED;
op->condition.accept(this);
if (expr_uses_vars(op->condition, scope) || !is_pure(op->condition)) {
// We need to simplify the condition to get it into a
// canonical form (e.g. (a < b) instead of !(a >= b))
vector<pair<Expr, Stmt>> cases;
{
Expr c = simplify(op->condition);
cases.emplace_back(c, op->then_case);
if (op->else_case.defined() && !is_no_op(op->else_case)) {
cases.emplace_back(simplify(!c), op->else_case);
}
}
for (const auto &pair : cases) {
Expr c = pair.first;
Stmt body = pair.second;
const Call *call = Call::as_tag(c);
if (call) {
c = call->args[0];
}
// Find the vars that vary, and solve for each in turn
// in order to bound it using the RHS. Maintain a list
// of the things we need to pop from scope once we're
// done.
struct RestrictedVar {
// This variable
const Variable *v;
// Takes on this range
Interval i;
// Implying that these other variables also have a restricted range
vector<LetBound> let_bounds;
};
vector<RestrictedVar> to_pop;
auto vars = find_free_vars(op->condition);
for (const auto *v : vars) {
auto result = solve_expression(c, v->name);
if (!result.fully_solved) {
continue;
}
Expr solved = result.result;
// Trim the scope down to represent the fact that the
// condition is true. We only understand certain types
// of conditions for now.
const LT *lt = solved.as<LT>();
const LE *le = solved.as<LE>();
const GT *gt = solved.as<GT>();
const GE *ge = solved.as<GE>();
const EQ *eq = solved.as<EQ>();
Expr lhs, rhs;
if (lt) {
lhs = lt->a;
rhs = lt->b;
} else if (le) {
lhs = le->a;
rhs = le->b;
} else if (gt) {
lhs = gt->a;
rhs = gt->b;
} else if (ge) {
lhs = ge->a;
rhs = ge->b;
} else if (eq) {
lhs = eq->a;
rhs = eq->b;
}
if (!rhs.defined() || rhs.type() != Int(32)) {
continue;
}
if (!equal(lhs, v)) {
continue;
}
Expr inner_min, inner_max;
Interval i = scope.get(v->name);
// If the original condition is likely, then
// the additional trimming of the domain due
// to the condition is probably unnecessary,
// which means the mins/maxes below should
// probably just be the LHS.
Interval likely_i = i;
if (call && call->is_intrinsic(Call::likely)) {
likely_i.min = likely(i.min);
likely_i.max = likely(i.max);
} else if (call && call->is_intrinsic(Call::likely_if_innermost)) {
likely_i.min = likely_if_innermost(i.min);
likely_i.max = likely_if_innermost(i.max);
}
Interval bi = bounds_of_expr_in_scope(rhs, scope, func_bounds);
if (bi.has_upper_bound() && i.has_upper_bound()) {
if (lt) {
i.max = min(likely_i.max, bi.max - 1);
}
if (le || eq) {
i.max = min(likely_i.max, bi.max);
}
}
if (bi.has_lower_bound() && i.has_lower_bound()) {
if (gt) {
i.min = max(likely_i.min, bi.min + 1);
}
if (ge || eq) {
i.min = max(likely_i.min, bi.min);
}
}
RestrictedVar p;
p.v = v;
p.i = i;
to_pop.emplace_back(std::move(p));
}
for (auto &p : to_pop) {
trim_scope_push(p.v->name, p.i, p.let_bounds);
}
body.accept(this);
while (!to_pop.empty()) {
trim_scope_pop(to_pop.back().v->name, to_pop.back().let_bounds);
to_pop.pop_back();
}
}
} else {
// If the condition is based purely on params, then we'll only
// ever go one way in a given run, so we should conditionalize
// the boxes touched on the condition.
// Fork the boxes touched and go down each path
map<string, Box> then_boxes, else_boxes;
then_boxes.swap(boxes);
op->then_case.accept(this);
then_boxes.swap(boxes);
if (op->else_case.defined()) {
else_boxes.swap(boxes);
op->else_case.accept(this);
else_boxes.swap(boxes);
}
// Make sure all the then boxes have an entry on the else
// side so that the merge doesn't skip them.
for (pair<const string, Box> &i : then_boxes) {
else_boxes[i.first];
}
// Merge
for (pair<const string, Box> &i : else_boxes) {
Box &else_box = i.second;
Box &then_box = then_boxes[i.first];
Box &orig_box = boxes[i.first];
if (then_box.maybe_unused()) {
then_box.used = then_box.used && op->condition;
} else {
then_box.used = op->condition;
}
if (else_box.maybe_unused()) {
else_box.used = else_box.used && !op->condition;
} else {
else_box.used = !op->condition;
}
merge_boxes(then_box, else_box);
merge_boxes(orig_box, then_box);
}
}
}
void visit(const For *op) override {
TRACK_BOXES_TOUCHED;
TRACK_BOXES_TOUCHED_INFO("var:", op->name);
if (consider_calls) {
op->min.accept(this);
op->extent.accept(this);
}
Expr min_val, max_val;
if (scope.contains(op->name + ".loop_min")) {
min_val = scope.get(op->name + ".loop_min").min;
} else {
min_val = bounds_of_expr_in_scope(op->min, scope, func_bounds).min;
}
if (scope.contains(op->name + ".loop_max")) {
max_val = scope.get(op->name + ".loop_max").max;
} else {
max_val = bounds_of_expr_in_scope(op->extent, scope, func_bounds).max;
max_val += bounds_of_expr_in_scope(op->min, scope, func_bounds).max;
max_val -= 1;
}
push_var(op->name);
{
ScopedBinding<Interval> p(scope, op->name, Interval(min_val, max_val));
op->body.accept(this);
}
pop_var(op->name);
}
void visit(const Provide *op) override {
TRACK_BOXES_TOUCHED;
TRACK_BOXES_TOUCHED_INFO("name:", op->name);
if (consider_provides) {
if (op->name == func || func.empty()) {
if (!is_const_one(op->predicate)) {
// Don't visit the RHS inside the if. This is handled below instead.
ScopedValue<bool> save_consider_calls(consider_calls, false);
Stmt equiv = IfThenElse::make(op->predicate, Provide::make(op->name, op->values, op->args, const_true()));
equiv.accept(this);
} else {
Box b(op->args.size());
for (size_t i = 0; i < op->args.size(); i++) {
b[i] = bounds_of_expr_in_scope(op->args[i], scope, func_bounds);
}
merge_boxes(boxes[op->name], b);
}
}
}
if (consider_calls) {
for (const auto &arg : op->args) {
arg.accept(this);
}
for (const auto &value : op->values) {
value.accept(this);
}
}
}
void visit(const ProducerConsumer *op) override {
TRACK_BOXES_TOUCHED;
TRACK_BOXES_TOUCHED_INFO("name:", op->name);
if (op->is_producer && (op->name == func || func.empty())) {
ScopedValue<bool> save_in_producer(in_producer, true);
IRGraphVisitor::visit(op);
} else {
IRGraphVisitor::visit(op);
}
}
};
} // namespace
map<string, Box> boxes_touched(const Expr &e, Stmt s, bool consider_calls, bool consider_provides,
const string &fn, const Scope<Interval> &scope, const FuncValueBounds &fb) {
if (!fn.empty() && s.defined()) {
// Filter things down to the relevant sub-Stmts, so we don't spend a
// long time reasoning about lets and ifs that don't surround an
// access to the buffer in question.
class Filter : public IRMutator {
using IRMutator::mutate;
using IRMutator::visit;
bool relevant = false;
Expr visit(const Call *op) override {
if (op->name == fn) {
relevant = true;
return op;
} else {
return IRMutator::visit(op);
}
}
Stmt visit(const Provide *op) override {
if (op->name == fn) {
relevant = true;
return op;
} else {
return IRMutator::visit(op);
}
}
Expr visit(const Variable *op) override {
if (op->name == fn_buffer || op->name == fn) {
relevant = true;
}
return op;
}
Stmt visit(const LetStmt *op) override {
// Walk eagerly through an entire let chain and either
// accept or reject all of them, not worrying about
// the case where some outer lets are relevant and
// some inner lets are not.
vector<const LetStmt *> frames;
Stmt orig = op;
Stmt body;
do {
// Visit the value just to check relevance. We
// don't expect Exprs to be mutated, so no need to
// keep the result.
mutate(op->value);
frames.push_back(op);
body = op->body;
op = body.as<LetStmt>();
} while (op);
Stmt s = mutate(body);
if (s.same_as(body)) {
return orig;
} else if (!relevant) {
// All the lets were irrelevant and so was the body
internal_assert(s.same_as(no_op));
return s;
} else {
// Rewrap the lets around the mutated body
for (auto it = frames.rbegin(); it != frames.rend(); it++) {
s = LetStmt::make((*it)->name, (*it)->value, s);
}
return s;
}
}
public:
Stmt mutate(const Stmt &s) override {
bool old = relevant;
relevant = false;
Stmt s_new = IRMutator::mutate(s);
if (!relevant) {
relevant = old;
return no_op;
} else {
return s_new;
}
}
const string &fn;
const string fn_buffer;
Stmt no_op;
Filter(const string &fn)
: fn(fn), fn_buffer(fn + ".buffer"), no_op(Evaluate::make(0)) {
}
} filter(fn);
s = filter.mutate(s);
}
// Move the innermost vars in an IfThenElse's condition as far to the left
// as possible, so that BoxesTouched can prune the variable scope tighter
// when encountering the IfThenElse.
if (s.defined()) {
s = SolveIfThenElse().mutate(s);
}
// Do calls and provides separately, for better simplification.
BoxesTouched calls(consider_calls, false, fn, &scope, fb);
BoxesTouched provides(false, consider_provides, fn, &scope, fb);
if (consider_calls) {
if (e.defined()) {
e.accept(&calls);
}
if (s.defined()) {
s.accept(&calls);
}
}
if (consider_provides) {
if (e.defined()) {
e.accept(&provides);
}
if (s.defined()) {
s.accept(&provides);
}
}
#if DO_DUMP_BOXES_TOUCHED
if (consider_calls && consider_provides) {
debug(0) << "boxes_touched:\n";
} else if (consider_calls && !consider_provides) {
debug(0) << "boxes_required:\n";
} else if (!consider_calls && consider_provides) {
debug(0) << "boxes_provided:\n";
} else {
internal_error;
}
for (const auto &it : calls.boxes) {
debug(0) << "calls.boxes[" << it.first << "] ->:\n";
for (size_t j = 0; j < it.second.size(); j++) {
debug(0) << " " << j << ": " << it.second[j].min
<< " .. "
<< it.second[j].max
<< "\n";
}
}
for (const auto &it : provides.boxes) {
debug(0) << "provides.boxes[" << it.first << "] ->:\n";
for (size_t j = 0; j < it.second.size(); j++) {
debug(0) << " " << j << ": " << it.second[j].min
<< " .. "
<< it.second[j].max
<< "\n";
}
}
#endif // DO_DUMP_BOXES_TOUCHED
if (!consider_calls) {
return provides.boxes;
}
if (!consider_provides) {
return calls.boxes;
}
// Combine the two maps.
for (pair<const string, Box> &i : provides.boxes) {
merge_boxes(calls.boxes[i.first], i.second);
}
// Make evaluating these boxes side-effect-free
for (auto &p : calls.boxes) {
auto &box = p.second;
box.used = purify_index_math(box.used);
for (Interval &i : box.bounds) {
i.min = purify_index_math(i.min);
i.max = purify_index_math(i.max);
}
}
return calls.boxes;
}
Box box_touched(const Expr &e, Stmt s, bool consider_calls, bool consider_provides,
const string &fn, const Scope<Interval> &scope, const FuncValueBounds &fb) {
map<string, Box> boxes = boxes_touched(e, std::move(s), consider_calls, consider_provides, fn, scope, fb);
internal_assert(boxes.size() <= 1);
return boxes[fn];
}
map<string, Box> boxes_required(const Expr &e, const Scope<Interval> &scope, const FuncValueBounds &fb) {
return boxes_touched(e, Stmt(), true, false, "", scope, fb);
}
Box box_required(const Expr &e, const string &fn, const Scope<Interval> &scope, const FuncValueBounds &fb) {
return box_touched(e, Stmt(), true, false, fn, scope, fb);
}
map<string, Box> boxes_required(Stmt s, const Scope<Interval> &scope, const FuncValueBounds &fb) {
return boxes_touched(Expr(), std::move(s), true, false, "", scope, fb);
}
Box box_required(Stmt s, const string &fn, const Scope<Interval> &scope, const FuncValueBounds &fb) {
return box_touched(Expr(), std::move(s), true, false, fn, scope, fb);
}
map<string, Box> boxes_provided(const Expr &e, const Scope<Interval> &scope, const FuncValueBounds &fb) {
return boxes_touched(e, Stmt(), false, true, "", scope, fb);
}
Box box_provided(const Expr &e, const string &fn, const Scope<Interval> &scope, const FuncValueBounds &fb) {
return box_touched(e, Stmt(), false, true, fn, scope, fb);
}
map<string, Box> boxes_provided(Stmt s, const Scope<Interval> &scope, const FuncValueBounds &fb) {
return boxes_touched(Expr(), std::move(s), false, true, "", scope, fb);
}
Box box_provided(Stmt s, const string &fn, const Scope<Interval> &scope, const FuncValueBounds &fb) {
return box_touched(Expr(), std::move(s), false, true, fn, scope, fb);
}
map<string, Box> boxes_touched(const Expr &e, const Scope<Interval> &scope, const FuncValueBounds &fb) {
return boxes_touched(e, Stmt(), true, true, "", scope, fb);
}
Box box_touched(const Expr &e, const string &fn, const Scope<Interval> &scope, const FuncValueBounds &fb) {
return box_touched(e, Stmt(), true, true, fn, scope, fb);
}
map<string, Box> boxes_touched(Stmt s, const Scope<Interval> &scope, const FuncValueBounds &fb) {
return boxes_touched(Expr(), std::move(s), true, true, "", scope, fb);
}
Box box_touched(Stmt s, const string &fn, const Scope<Interval> &scope, const FuncValueBounds &fb) {
return box_touched(Expr(), std::move(s), true, true, fn, scope, fb);
}
// Compute interval of all possible function's values (default + specialized values)
Interval compute_pure_function_definition_value_bounds(
const Definition &def, const Scope<Interval> &scope, const FuncValueBounds &fb, int dim) {
Interval result = bounds_of_expr_in_scope(def.values()[dim], scope, fb);
// Pure function might have different values due to specialization.
// We need to take the union of min and max bounds of all those possible values.
for (const Specialization &s : def.specializations()) {
Interval s_interval = compute_pure_function_definition_value_bounds(s.definition, scope, fb, dim);
result.include(s_interval);
}
return result;
}
FuncValueBounds compute_function_value_bounds(const vector<string> &order,
const map<string, Function> &env) {
FuncValueBounds fb;
for (const auto &func_name : order) {
Function f = env.find(func_name)->second;
const vector<string> f_args = f.args();
for (int j = 0; j < f.outputs(); j++) {
pair<string, int> key = {f.name(), j};
Interval result;
if (f.is_pure()) {
// Make a scope that says the args could be anything.
Scope<Interval> arg_scope;
for (size_t k = 0; k < f.args().size(); k++) {
arg_scope.push(f_args[k], Interval::everything());
}
result = compute_pure_function_definition_value_bounds(f.definition(), arg_scope, fb, j);
// These can expand combinatorially as we go down the
// pipeline if we don't run CSE on them.
if (result.has_lower_bound()) {
result.min = simplify(common_subexpression_elimination(result.min));
}
if (result.has_upper_bound()) {
result.max = simplify(common_subexpression_elimination(result.max));
}
fb[key] = result;
} else {
// If the Func is impure, we may still be able to specify a bounds-of-type here
Type t = f.output_types()[j].element_of();
if ((t.is_uint() || t.is_int()) && t.bits() <= 16) {
result = Interval(t.min(), t.max());
} else {
result = Interval::everything();
}
fb[key] = result;
// TODO: if a Function is impure, but the RDoms used by the update functions
// are all constant, it may be profitable to calculate the bounds here too
}
debug(2) << "Bounds on value " << j
<< " for func " << func_name
<< " are: " << result.min << ", " << result.max << "\n";
}
}
return fb;
}
namespace {
void check(const Scope<Interval> &scope, const Expr &e, const Expr &correct_min, const Expr &correct_max) {
FuncValueBounds fb;
Interval result = bounds_of_expr_in_scope(e, scope, fb);
result.min = simplify(result.min);
result.max = simplify(result.max);
if (!equal(result.min, correct_min)) {
internal_error << "In bounds of " << e << ":\n"
<< "Incorrect min: " << result.min << "\n"
<< "Should have been: " << correct_min << "\n";
}
if (!equal(result.max, correct_max)) {
internal_error << "In bounds of " << e << ":\n"
<< "Incorrect max: " << result.max << "\n"
<< "Should have been: " << correct_max << "\n";
}
}
void check_constant_bound(const Scope<Interval> &scope, const Expr &e, const Expr &correct_min, const Expr &correct_max) {
FuncValueBounds fb;
Interval result = bounds_of_expr_in_scope(e, scope, fb, true);
result.min = simplify(result.min);
result.max = simplify(result.max);
if (!equal(result.min, correct_min)) {
internal_error << "In find constant bound of " << e << ":\n"
<< "Incorrect min constant bound: " << result.min << "\n"
<< "Should have been: " << correct_min << "\n";
}
if (!equal(result.max, correct_max)) {
internal_error << "In find constant bound of " << e << ":\n"
<< "Incorrect max constant bound: " << result.max << "\n"
<< "Should have been: " << correct_max << "\n";
}
}
void check_constant_bound(const Expr &e, const Expr &correct_min, const Expr &correct_max) {
Scope<Interval> scope;
check_constant_bound(scope, e, correct_min, correct_max);
}
void constant_bound_test() {
using namespace ConciseCasts;
{
Param<int16_t> a;
Param<uint16_t> b;
check_constant_bound(a >> b, i16(-32768), i16(32767));
}
{
Param<int> x("x"), y("y");
x.set_range(10, 20);
y.set_range(5, 30);
check_constant_bound(clamp(x, 5, 30), 10, 20);
check_constant_bound(clamp(x, 15, 30), 15, 20);
check_constant_bound(clamp(x, 15, 17), 15, 17);
check_constant_bound(clamp(x, 5, 15), 10, 15);
check_constant_bound(x + y, 15, 50);
check_constant_bound(x - y, -20, 15);
check_constant_bound(x * y, 50, 600);
check_constant_bound(x / y, 0, 4);
check_constant_bound(select(x > 4, 3 * x - y / 2, max(x + y + 2, x - 20)), 15, 58);
check_constant_bound(select(x < 4, 3 * x - y / 2, max(x + y + 2, x - 20)), 17, 52);
check_constant_bound(select(x >= 11, 3 * x - y / 2, max(x + y + 2, x - 20)), 15, 58);
}
{
Param<uint8_t> x("x"), y("y");
x.set_range(Expr((uint8_t)10), Expr((uint8_t)20));
y.set_range(Expr((uint8_t)5), Expr((uint8_t)30));
check_constant_bound(clamp(x, 5, 30), Expr((uint8_t)10), Expr((uint8_t)20));
check_constant_bound(clamp(x, 15, 30), Expr((uint8_t)15), Expr((uint8_t)20));
check_constant_bound(clamp(x, 15, 17), Expr((uint8_t)15), Expr((uint8_t)17));
check_constant_bound(clamp(x, 5, 15), Expr((uint8_t)10), Expr((uint8_t)15));
check_constant_bound(x + y, Expr((uint8_t)15), Expr((uint8_t)50));
check_constant_bound(x / y, Expr((uint8_t)0), Expr((uint8_t)4));
check_constant_bound(select(x > 4, 3 * x - y / 2, max(x + y + 2, x + 20)),
Expr((uint8_t)15), Expr((uint8_t)58));
check_constant_bound(select(x < 4, 3 * x - y / 2, max(x + y + 2, x + 20)),
Expr((uint8_t)30), Expr((uint8_t)52));
check_constant_bound(select(x >= 11, 3 * x - y / 2, max(x + y + 2, x + 20)),
Expr((uint8_t)15), Expr((uint8_t)58));
// These two overflow
check_constant_bound(x - y, Expr((uint8_t)0), Expr((uint8_t)255));
check_constant_bound(x * y, Expr((uint8_t)0), Expr((uint8_t)255));
check_constant_bound(absd(x, y), Expr((uint8_t)0), Expr((uint8_t)20));
check_constant_bound(absd(cast<int16_t>(x), cast<int16_t>(y)), Expr((uint16_t)0), Expr((uint16_t)20));
}
{
Param<float> x("x"), y("y");
x.set_range(Expr((float)10), Expr((float)20));
y.set_range(Expr((float)5), Expr((float)30));
check_constant_bound(absd(x, y), Expr((float)0), Expr((float)20));
}
{
Param<int8_t> i("i"), x("x"), y("y"), d("d");
Expr cl = i16(i);
Expr cr1 = i16(x);
Expr cr2 = i16(y);
Expr fraction = (d & (int16_t)((1 << 7) - 1));
Expr cr = i16((((cr2 - cr1) * fraction) >> 7) + cr1);
check_constant_bound(absd(cr, cl), Expr((uint16_t)0), Expr((uint16_t)509));
check_constant_bound(i16(absd(cr, cl)), Expr((int16_t)0), Expr((int16_t)509));
}
check_constant_bound(Load::make(Int(32), "buf", 0, Buffer<>(), Parameter(), const_true(), ModulusRemainder()) * 20,
Interval::neg_inf(), Interval::pos_inf());
{
// Ensure that unnecessary integer overflow doesn't happen
// in cases involving unsigned integer math
Param<uint16_t> e1("e1"); // range 0..0xffff, type=uint16
Expr e2 = cast<uint32_t>(e1); // range 0..0xffff, type=uint32
Expr e3 = e2 * e2; // range 0..0xfffe0001, type=uint32
check_constant_bound(e3, Expr((uint32_t)0), Expr((uint32_t)0xfffe0001));
}
{
RDom r(0, 4);
// bounds of an expression with impure >= 32 bit expr will be unbounded
Expr e32 = sum(cast<int32_t>(r.x));
check_constant_bound(e32, Interval::neg_inf(), Interval::pos_inf());
// bounds of an expression with impure < 32 bit expr will be bounds-of-type
Expr e16 = sum(cast<int16_t>(r.x));
check_constant_bound(e16, Int(16).min(), Int(16).max());
}
{
Param<int32_t> x("x"), y("y");
x.set_range(2, 10);
check_constant_bound(count_leading_zeros(x), i32(28), i32(30));
check_constant_bound(count_leading_zeros(cast<int16_t>(x)), i16(12), i16(14));
check_constant_bound(count_leading_zeros(y), i32(0), i32(32));
check_constant_bound(count_leading_zeros(cast<int16_t>(y)), i16(0), i16(16));
}
}
void boxes_touched_test() {
Type t = Int(32);
Expr x = Variable::make(t, "x");
Expr y = Variable::make(t, "y");
Expr z = Variable::make(t, "z");
Expr w = Variable::make(t, "w");
Scope<Interval> scope;
scope.push("y", Interval(Expr(0), Expr(10)));
Stmt stmt = Provide::make("f", {10}, {x, y, z, w}, const_true());
stmt = IfThenElse::make(y > 4, stmt, Stmt());
stmt = IfThenElse::make(z > 18, stmt, Stmt());
stmt = LetStmt::make("w", z + 3, stmt);
stmt = LetStmt::make("z", x + 2, stmt);
stmt = LetStmt::make("x", y + 10, stmt);
Box expected({Interval(15, 20), Interval(5, 10), Interval(19, 22), Interval(22, 25)});
Box result = box_provided(stmt, "f", scope);
internal_assert(expected.size() == result.size())
<< "Expect dim size of " << expected.size()
<< ", got " << result.size() << " instead\n";
for (size_t i = 0; i < result.size(); ++i) {
const Interval &correct = expected[i];
Interval b = result[i];
b.min = simplify(b.min);
b.max = simplify(b.max);
if (!equal(correct.min, b.min)) {
internal_error << "In bounds of dim " << i << ":\n"
<< "Incorrect min: " << b.min << "\n"
<< "Should have been: " << correct.min << "\n";
}
if (!equal(correct.max, b.max)) {
internal_error << "In bounds of dim " << i << ":\n"
<< "Incorrect max: " << b.max << "\n"
<< "Should have been: " << correct.max << "\n";
}
}
}
} // anonymous namespace
void bounds_test() {
using namespace Halide::ConciseCasts;
constant_bound_test();
Scope<Interval> scope;
Var x("x"), y("y");
scope.push("x", Interval(Expr(0), Expr(10)));
check(scope, x, 0, 10);
check(scope, x + 1, 1, 11);
check(scope, (x + 1) * 2, 2, 22);
check(scope, x * x, 0, 100);
check(scope, 5 - x, -5, 5);
check(scope, x * (5 - x), -50, 50); // We don't expect bounds analysis to understand correlated terms
check(scope, Select::make(x < 4, x, x + 100), 0, 110);
check(scope, x + y, y, y + 10);
check(scope, x * y, min(y, 0) * 10, max(y, 0) * 10);
check(scope, x / (x + y), -10, 10);
check(scope, 11 / (x + 1), 1, 11);
check(scope, Load::make(Int(8), "buf", x, Buffer<>(), Parameter(), const_true(), ModulusRemainder()),
i8(-128), i8(127));
check(scope, y + (Let::make("y", x + 3, y - x + 10)), y + 3, y + 23); // Once again, we don't know that y is correlated with x
check(scope, clamp(1000 / (x - 2), x - 10, x + 10), -10, 20);
check(scope, cast<uint16_t>(x / 2), u16(0), u16(5));
check(scope, cast<uint16_t>((x + 10) / 2), u16(5), u16(10));
check(scope, x < 20, make_bool(true), make_bool(true));
check(scope, x < 5, make_bool(false), make_bool(true));
check(scope, Broadcast::make(x >= 11, 3), make_bool(false), make_bool(false));
check(scope, Ramp::make(x + 5, 1, 5) > Broadcast::make(2, 5), make_bool(true), make_bool(true));
check(scope, print(x, y), 0, 10);
check(scope, print_when(x > y, x, y), 0, 10);
check(scope, select(y == 5, 0, 3), select(y == 5, 0, 3), select(y == 5, 0, 3));
check(scope, select(y == 5, x, -3 * x + 8), select(y == 5, 0, -22), select(y == 5, 10, 8));
check(scope, select(y == x, x, -3 * x + 8), -22, select(y <= 10 && 0 <= y, 10, 8));
check(scope, cast<int32_t>(abs(cast<int16_t>(x ^ y))), 0, 32768);
check(scope, cast<float>(x), 0.0f, 10.0f);
check(scope, cast<int32_t>(abs(cast<float>(x))), 0, 10);
// Check some vectors
check(scope, Ramp::make(x * 2, 5, 5), 0, 40);
check(scope, Broadcast::make(x * 2, 5), 0, 20);
check(scope, Broadcast::make(3, 4), 3, 3);
// Check some operations that may overflow
check(scope, (cast<uint8_t>(x) + 250), u8(0), u8(255));
check(scope, (cast<uint8_t>(x) + 10) * 20, u8(0), u8(255));
check(scope, (cast<uint8_t>(x) + 10) * (cast<uint8_t>(x) + 5), u8(0), u8(255));
check(scope, (cast<uint8_t>(x) + 10) - (cast<uint8_t>(x) + 5), u8(0), u8(255));
// Check some operations that we should be able to prove do not overflow
check(scope, (cast<uint8_t>(x) + 240), u8(240), u8(250));
check(scope, (cast<uint8_t>(x) + 10) * 10, u8(100), u8(200));
check(scope, (cast<uint8_t>(x) + 10) * (cast<uint8_t>(x)), u8(0), u8(200));
check(scope, (cast<uint8_t>(x) + 20) - (cast<uint8_t>(x) + 5), u8(5), u8(25));
// Check div/mod by unbounded unknowns. div and mod can only ever
// make things smaller in magnitude.
scope.push("x", Interval::everything());
check(scope, -3 / x, -3, 3);
check(scope, 3 / x, -3, 3);
check(scope, y / x, -cast<int>(abs(y)), cast<int>(abs(y)));
check(scope, -3 % x, 0, Interval::pos_inf());
check(scope, 3 % x, 0, 3);
// Mod can't make values negative
check(scope, y % x, 0, Interval::pos_inf());
// Mod can't make positive values larger
check(scope, max(y, 0) % x, 0, max(y, 0));
scope.pop("x");
// Check some bitwise ops.
check(scope, (cast<uint8_t>(x) & cast<uint8_t>(7)), u8(0), u8(7));
check(scope, (cast<uint8_t>(3) & cast<uint8_t>(2)), u8(2), u8(2));
check(scope, (cast<uint8_t>(1) | cast<uint8_t>(2)), u8(3), u8(3));
check(scope, (cast<uint8_t>(3) ^ cast<uint8_t>(2)), u8(1), u8(1));
check(scope, (~cast<uint8_t>(3)), u8(0xfc), u8(0xfc));
check(scope, cast<uint8_t>(x + 5) & cast<uint8_t>(x + 3), u8(0), u8(13));
check(scope, cast<int8_t>(x - 5) & cast<int8_t>(x + 3), i8(0), i8(13));
check(scope, cast<int8_t>(2 * x - 5) & cast<int8_t>(x - 3), i8(-128), i8(15));
check(scope, cast<uint8_t>(x + 5) | cast<uint8_t>(x + 3), u8(5), u8(255));
check(scope, cast<int8_t>(x + 5) | cast<int8_t>(x + 3), i8(3), i8(127));
check(scope, ~cast<uint8_t>(x), u8(-11), u8(-1));
check(scope, (cast<uint8_t>(x) >> cast<uint8_t>(1)), u8(0), u8(5));
check(scope, (cast<uint8_t>(10) >> cast<uint8_t>(1)), u8(5), u8(5));
check(scope, (cast<uint8_t>(x + 3) << cast<uint8_t>(1)), u8(6), u8(26));
check(scope, (cast<uint8_t>(x + 3) << cast<uint8_t>(7)), u8(0), u8(255)); // Overflows
check(scope, (cast<uint8_t>(5) << cast<uint8_t>(1)), u8(10), u8(10));
check(scope, (x << 12), 0, 10 << 12);
check(scope, x & 4095, 0, 10); // LHS known to be positive
check(scope, x & 123, 0, 10); // Doesn't have to be a precise bitmask
check(scope, (x - 1) & 4095, 0, 4095); // LHS could be -1
// Regression tests on shifts (produced by z3).
{
ScopedBinding<Interval> xb(scope, "x", Interval(-123, Interval::pos_inf()));
ScopedBinding<Interval> yb(scope, "y", Interval(-6, 0));
// -123 << 0 = -123
check(scope, x << y, -123, Interval::pos_inf());
}
{
ScopedBinding<Interval> xb(scope, "x", Interval(-123, Interval::pos_inf()));
ScopedBinding<Interval> yb(scope, "y", Interval(-6, Interval::pos_inf()));
// A negative value can increase in magnitude if the rhs is positive.
check(scope, x << y, Interval::neg_inf(), Interval::pos_inf());
}
{
ScopedBinding<Interval> xb(scope, "x", Interval(-123, Interval::pos_inf()));
Var c("c");
ScopedBinding<Interval> yb(scope, "y", Interval(-6, c));
// Can't prove anything about the upper bound of y.
check(scope, x << y, min((-123) << c, -123), Interval::pos_inf());
}
{
ScopedBinding<Interval> xb(scope, "x", Interval(-123, Interval::pos_inf()));
ScopedBinding<Interval> yb(scope, "y", Interval(-6, 4));
// -123 << 4 = -1968
check(scope, x << y, -1968, Interval::pos_inf());
}
{
ScopedBinding<Interval> xb(scope, "x", Interval(24, Interval::pos_inf()));
ScopedBinding<Interval> yb(scope, "y", Interval(Interval::neg_inf(), -1));
// Cannot change sign, only can decrease magnitude.
check(scope, x << y, 0, Interval::pos_inf());
}
// Overflow testing (for types with defined overflow).
{
Type uint32 = UInt(32);
Expr a = Variable::make(uint32, "a");
Expr b = Variable::make(uint32, "b");
ScopedBinding<Interval> ab(scope, "a", Interval(UIntImm::make(uint32, 0), simplify(uint32.max() / 4 + 2)));
ScopedBinding<Interval> bb(scope, "b", Interval(UIntImm::make(uint32, 0), uint32.max()));
// Overflow should be detected
check(scope, a + b, Interval::neg_inf(), Interval::pos_inf());
check(scope, a * b, Interval::neg_inf(), Interval::pos_inf());
}
{
Type int16 = Int(16);
Expr a = Variable::make(int16, "a");
Expr b = Variable::make(int16, "b");
ScopedBinding<Interval> ab(scope, "a", Interval(int16.min(), int16.max()));
ScopedBinding<Interval> bb(scope, "b", Interval(IntImm::make(int16, -4), IntImm::make(int16, -1)));
check(scope, a * -1, int16.min(), int16.max());
// int16.min() / -1 should be caught as overflow.
check(scope, a / -1, int16.min(), int16.max());
check(scope, a / b, int16.min(), int16.max());
}
{
Expr zero = UIntImm::make(UInt(1), 0);
Expr one = UIntImm::make(UInt(1), 1);
check(scope, Ramp::make(zero, one, 3), zero, one);
}
// If we clamp something unbounded as one type, the bounds should
// propagate through casts whenever the cast can be proved to not
// overflow.
check(scope,
cast<uint16_t>(clamp(cast<float>(x ^ y), 0.0f, 4095.0f)),
u16(0), u16(4095));
check(scope,
cast<uint8_t>(clamp(cast<uint16_t>(x ^ y), cast<uint16_t>(0), cast<uint16_t>(128))),
u8(0), u8(128));
Expr u8_1 = cast<uint8_t>(Load::make(Int(8), "buf", x, Buffer<>(), Parameter(), const_true(), ModulusRemainder()));
Expr u8_2 = cast<uint8_t>(Load::make(Int(8), "buf", x + 17, Buffer<>(), Parameter(), const_true(), ModulusRemainder()));
check(scope, cast<uint16_t>(u8_1) + cast<uint16_t>(u8_2),
u16(0), u16(255 * 2));
check(scope, saturating_cast<uint8_t>(clamp(x, 5, 10)), cast<uint8_t>(5), cast<uint8_t>(10));
{
scope.push("x", Interval(UInt(32).min(), UInt(32).max()));
check(scope, saturating_cast<int32_t>(max(cast<uint32_t>(x), cast<uint32_t>(5))), cast<int32_t>(5), Int(32).max());
scope.pop("x");
}
{
Expr z = Variable::make(Float(32), "z");
scope.push("z", Interval(cast<float>(-1), cast<float>(1)));
check(scope, saturating_cast<int32_t>(z), cast<int32_t>(-1), cast<int32_t>(1));
check(scope, saturating_cast<double>(z), cast<double>(-1), cast<double>(1));
check(scope, saturating_cast<float16_t>(z), cast<float16_t>(-1), cast<float16_t>(1));
check(scope, saturating_cast<uint8_t>(z), cast<uint8_t>(0), cast<uint8_t>(1));
scope.pop("z");
}
{
Expr z = Variable::make(UInt(32), "z");
scope.push("z", Interval(UInt(32).max(), UInt(32).max()));
check(scope, saturating_cast<int32_t>(z), Int(32).max(), Int(32).max());
scope.pop("z");
}
{
Scope<Interval> scope;
Expr x = Variable::make(UInt(16), "x");
Expr y = Variable::make(UInt(16), "y");
scope.push("x", Interval(u16(0), u16(10)));
scope.push("y", Interval(u16(2), u16(4)));
Expr e = clamp(x / y, u16(0), u16(128));
check(scope, e, u16(0), u16(5));
check_constant_bound(scope, e, u16(0), u16(5));
}
{
Param<int16_t> x("x");
Param<uint16_t> y("y");
x.set_range(i16(-32), i16(-16));
y.set_range(i16(0), i16(4));
check_constant_bound((x >> y), i16(-32), i16(-1));
}
{
Param<uint16_t> x("x"), y("y");
x.set_range(u16(10), u16(20));
y.set_range(u16(0), u16(30));
Scope<Interval> scope;
scope.push("y", Interval(u16(2), u16(4)));
check_constant_bound(scope, x + y, u16(12), u16(24));
}
{
Scope<Interval> scope;
Interval i = Interval::everything();
i.min = 17;
internal_assert(i.has_lower_bound());
internal_assert(!i.has_upper_bound());
scope.push("y", i);
Var x("x"), y("y");
check(scope, select(x == y * 2, y, y - 10),
7, Interval::pos_inf());
check(scope, select(x == y * 2, y - 10, y),
select(x < 34, 17, 7), Interval::pos_inf());
}
vector<Expr> input_site_1 = {2 * x};
vector<Expr> input_site_2 = {2 * x + 1};
vector<Expr> output_site = {x + 1};
Buffer<int32_t> in(10);
in.set_name("input");
Stmt loop = For::make("x", 3, 10, ForType::Serial, DeviceAPI::Host,
Provide::make("output",
{Add::make(Call::make(in, input_site_1),
Call::make(in, input_site_2))},
output_site,
const_true()));
map<string, Box> r;
r = boxes_required(loop);
internal_assert(r.find("output") == r.end());
internal_assert(r.find("input") != r.end());
internal_assert(equal(simplify(r["input"][0].min), 6));
internal_assert(equal(simplify(r["input"][0].max), 25));
r = boxes_provided(loop);
internal_assert(r.find("output") != r.end());
internal_assert(equal(simplify(r["output"][0].min), 4));
internal_assert(equal(simplify(r["output"][0].max), 13));
Box r2({Interval(Expr(5), Expr(19))});
merge_boxes(r2, r["output"]);
internal_assert(equal(simplify(r2[0].min), 4));
internal_assert(equal(simplify(r2[0].max), 19));
boxes_touched_test();
// Check a deeply-nested bitwise expr to ensure it doesn't take n^2 time
// (this clause took ~30s on a typical laptop before the fix, ~10ms after)
{
Expr a = Variable::make(UInt(16), "t42");
Expr b = Variable::make(UInt(16), "t43");
Expr c = Variable::make(UInt(16), "t44");
Expr d = Variable::make(Int(32), "d");
Expr x = Variable::make(Int(32), "x");
Expr y = Variable::make(Int(32), "y");
Expr e1 = select(c >= Expr((uint16_t)128), c - Expr((uint16_t)128), c);
Expr e2 = Let::make("t44", (((((((((((((((((u16(0) << u16(1)) | u16((u8(d) & u8(1)))) << u16(1)) | u16(((u8(d) >> u8(1)) & u8(1)))) << u16(1)) | (u16(x) & u16(1))) << u16(1)) | (u16(y) & u16(1))) << u16(1)) | (a & u16(1))) << u16(1)) | (b & u16(1))) << u16(1)) | ((a >> u16(1)) & u16(1))) << u16(1)) | ((b >> u16(1)) & u16(1))) >> u16(1)), e1);
Expr e3 = Let::make("t43", u16(y) >> u16(1), e2);
Expr e4 = Let::make("t42", u16(x) >> u16(1), e3);
check_constant_bound(e4, u16(0), u16(65535));
}
std::cout << "Bounds test passed" << std::endl;
}
} // namespace Internal
} // namespace Halide
