https://github.com/halide/Halide
Tip revision: 346c37ba046f9b8c5b1b65b9194113b33afb9bac authored by Andrew Adams on 07 December 2017, 21:54:28 UTC
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Tip revision: 346c37b
Bounds.cpp
#include <iostream>
#include "Bounds.h"
#include "IRVisitor.h"
#include "IR.h"
#include "IROperator.h"
#include "IREquality.h"
#include "Simplify.h"
#include "IRPrinter.h"
#include "Util.h"
#include "Var.h"
#include "Debug.h"
#include "ExprUsesVar.h"
#include "IRMutator.h"
#include "CSE.h"
namespace Halide {
namespace Internal {
using std::make_pair;
using std::map;
using std::vector;
using std::string;
using std::pair;
namespace {
int static_sign(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;
}
}
Expr find_constant_bound(Expr e, Direction d) {
// We look through casts, so we only handle ops that can't
// overflow. E.g. if A >= a and B >= b, then you can't assume that
// (A + B) >= (a + b) in a world with overflow.
if (is_const(e)) {
return e;
} else if (const Min *min = e.as<Min>()) {
Expr a = find_constant_bound(min->a, d);
Expr b = find_constant_bound(min->b, d);
if (a.defined() && b.defined()) {
return simplify(Min::make(a, b));
} else if (a.defined() && d == Direction::Upper) {
return a;
} else if (b.defined() && d == Direction::Upper) {
return b;
}
} else if (const Max *max = e.as<Max>()) {
Expr a = find_constant_bound(max->a, d);
Expr b = find_constant_bound(max->b, d);
if (a.defined() && b.defined()) {
return simplify(Max::make(a, b));
} else if (a.defined() && d == Direction::Lower) {
return a;
} else if (b.defined() && d == Direction::Lower) {
return b;
}
} else if (const Cast *cast = e.as<Cast>()) {
Expr a = find_constant_bound(cast->value, d);
if (a.defined()) {
return simplify(Cast::make(cast->type, a));
}
}
return Expr();
}
class Bounds : public IRVisitor {
public:
Interval interval;
Scope<Interval> scope;
const FuncValueBounds &func_bounds;
Bounds(const Scope<Interval> *s, const FuncValueBounds &fb) :
func_bounds(fb) {
scope.set_containing_scope(s);
}
private:
// Compute the intrinsic bounds of a function.
void bounds_of_func(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 = make_pair(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) {
interval = Interval::single_point(op);
}
void visit(const UIntImm *op) {
interval = Interval::single_point(op);
}
void visit(const FloatImm *op) {
interval = Interval::single_point(op);
}
void visit(const Cast *op) {
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() && 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.
Expr lower_bound = find_constant_bound(a.min, Direction::Lower);
Expr upper_bound = find_constant_bound(a.max, Direction::Upper);
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);
}
}
}
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) {
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) {
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 {
interval = Interval::everything();
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;
}
// Check for overflow for (u)int8 and (u)int16
if (!op->type.is_float() && op->type.bits() < 32) {
if (interval.has_upper_bound()) {
Expr no_overflow = (cast<int>(a.max) + cast<int>(b.max) == cast<int>(interval.max));
if (!can_prove(no_overflow)) {
bounds_of_type(op->type);
return;
}
}
if (interval.has_lower_bound()) {
Expr no_overflow = (cast<int>(a.min) + cast<int>(b.min) == cast<int>(interval.min));
if (!can_prove(no_overflow)) {
bounds_of_type(op->type);
return;
}
}
}
}
}
void visit(const Sub *op) {
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 {
interval = Interval::everything();
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;
}
// Check for overflow for (u)int8 and (u)int16
if (!op->type.is_float() && op->type.bits() < 32) {
if (interval.has_upper_bound()) {
Expr no_overflow = (cast<int>(a.max) - cast<int>(b.min) == cast<int>(interval.max));
if (!can_prove(no_overflow)) {
bounds_of_type(op->type);
return;
}
}
if (interval.has_lower_bound()) {
Expr no_overflow = (cast<int>(a.min) - cast<int>(b.max) == cast<int>(interval.min));
if (!can_prove(no_overflow)) {
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) {
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);
} else if (a.is_single_point() && b.is_single_point()) {
interval = Interval::single_point(a.min * b.min);
} 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_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)) {
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 {
interval = Interval::everything();
}
} 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 {
interval = Interval::everything();
}
if (op->type.bits() < 32 && !op->type.is_float()) {
if (a.is_bounded() && b.is_bounded()) {
// Try to prove it can't overflow
Expr test1 = (cast<int>(a.min) * cast<int>(b.min) == cast<int>(a.min * b.min));
Expr test2 = (cast<int>(a.min) * cast<int>(b.max) == cast<int>(a.min * b.max));
Expr test3 = (cast<int>(a.max) * cast<int>(b.min) == cast<int>(a.max * b.min));
Expr test4 = (cast<int>(a.max) * cast<int>(b.max) == cast<int>(a.max * b.max));
if (!can_prove(test1 && test2 && test3 && test4)) {
bounds_of_type(op->type);
}
} else {
bounds_of_type(op->type);
}
}
}
void visit(const Div *op) {
op->a.accept(this);
Interval a = interval;
op->b.accept(this);
Interval b = interval;
if (!b.is_bounded()) {
interval = Interval::everything();
} 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;
if (is_positive_const(b.min) || op->type.is_uint()) {
interval = Interval(e1, e2);
} else if (is_negative_const(b.min)) {
interval = Interval(e2, e1);
} else if (a.is_bounded()) {
// Sign of b is unknown. Note that this might divide
// by zero, but only in cases where the original code
// divides by zero.
Expr cmp = b.min > make_zero(b.min.type().element_of());
interval = Interval(select(cmp, e1, e2), select(cmp, e2, e1));
} else {
interval = Interval::everything();
}
} 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) {
interval = Interval::everything();
} else {
// 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 {
interval = Interval::everything();
}
}
void visit(const Mod *op) {
op->a.accept(this);
Interval a = interval;
op->b.accept(this);
if (!interval.is_bounded()) {
return;
}
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();
if (a.is_single_point() && b.is_single_point()) {
interval = Interval::single_point(a.min % b.min);
} else {
// Only consider B (so A can be unbounded)
if (b.max.type().is_uint() || (b.max.type().is_int() && is_positive_const(b.min))) {
// If the RHS is a positive integer, the result is in [0, max_b-1]
interval = Interval(make_zero(t), b.max - make_one(t));
} else if (b.max.type().is_int()) {
// mod is always positive
// x % [4,10] -> [0,9]
// x % [-8,-3] -> [0,7]
// x % [-8, 10] -> [0,9]
interval = Interval(make_zero(t), Max::make(abs(b.min), abs(b.max)) - make_one(t));
} else {
// 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 = Interval(make_zero(t), Max::make(abs(b.min), abs(b.max)));
}
}
}
void visit(const Min *op) {
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) {
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));
}
}
void visit(const EQ *op) {
bounds_of_type(op->type);
}
void visit(const NE *op) {
bounds_of_type(op->type);
}
void visit(const LT *op) {
bounds_of_type(op->type);
}
void visit(const LE *op) {
bounds_of_type(op->type);
}
void visit(const GT *op) {
bounds_of_type(op->type);
}
void visit(const GE *op) {
bounds_of_type(op->type);
}
void visit(const And *op) {
bounds_of_type(op->type);
}
void visit(const Or *op) {
bounds_of_type(op->type);
}
void visit(const Not *op) {
bounds_of_type(op->type);
}
void visit(const Select *op) {
op->true_value.accept(this);
if (!interval.is_bounded()) {
return;
}
Interval a = interval;
op->false_value.accept(this);
if (!interval.is_bounded()) {
return;
}
Interval b = interval;
bool const_scalar_condition =
(op->condition.type().is_scalar() &&
!expr_uses_vars(op->condition, scope));
if (a.min.same_as(b.min)) {
interval.min = a.min;
} else if (const_scalar_condition) {
interval.min = select(op->condition, a.min, b.min);
} else {
interval.min = Interval::make_min(a.min, b.min);
}
if (a.max.same_as(b.max)) {
interval.max = a.max;
} else if (const_scalar_condition) {
interval.max = select(op->condition, a.max, b.max);
} else {
interval.max = Interval::make_max(a.max, b.max);
}
}
void visit(const Load *op) {
op->index.accept(this);
if (interval.is_single_point()) {
// If the index is const 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);
interval = Interval::single_point(load_min);
} else {
// Otherwise use the bounds of the type
bounds_of_type(op->type);
}
}
void visit(const Ramp *op) {
// Treat the ramp lane as a free variable
string var_name = unique_name('t');
Expr var = Variable::make(op->base.type(), var_name);
Expr lane = op->base + var * op->stride;
scope.push(var_name, Interval(make_const(var.type(), 0),
make_const(var.type(), op->lanes-1)));
lane.accept(this);
scope.pop(var_name);
}
void visit(const Broadcast *op) {
op->value.accept(this);
}
void visit(const Call *op) {
// 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.
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;
}
}
Type t = op->type.element_of();
if (t.is_handle()) {
interval = Interval::everything();
return;
}
if (const_args &&
(op->call_type == Call::PureExtern ||
op->call_type == Call::Image)) {
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);
} else if (op->is_intrinsic(Call::abs)) {
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::likely) ||
op->is_intrinsic(Call::likely_if_innermost)) {
assert(op->args.size() == 1);
op->args[0].accept(this);
} else if (op->is_intrinsic(Call::return_second)) {
assert(op->args.size() == 2);
op->args[1].accept(this);
} else if (op->is_intrinsic(Call::if_then_else)) {
assert(op->args.size() == 3);
// Probably more conservative than necessary
Expr equivalent_select = Select::make(op->args[0], op->args[1], op->args[2]);
equivalent_select.accept(this);
} else if (op->is_intrinsic(Call::shift_left) ||
op->is_intrinsic(Call::shift_right) ||
op->is_intrinsic(Call::bitwise_and)) {
Expr simplified = simplify(op);
if (!equal(simplified, op)) {
simplified.accept(this);
} else {
// Just use the bounds of the type
bounds_of_type(t);
}
} 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::extract_buffer_min) ||
op->is_intrinsic(Call::extract_buffer_max)) {
// Bounds query results should have perfect nesting. Their
// max over a loop is just the same bounds query call at
// an outer loop level. This requires that the query is
// also done at the outer loop level so that the buffer
// arg is still valid, which it is, so it is.
//
// TODO: There should be an assert injected in the inner
// loop to check perfect nesting.
interval = Interval(
Call::make(Int(32), Call::extract_buffer_min, op->args, Call::PureIntrinsic),
Call::make(Int(32), Call::extract_buffer_max, op->args, Call::PureIntrinsic));
} else if (op->is_intrinsic(Call::memoize_expr)) {
internal_assert(op->args.size() >= 1);
op->args[0].accept(this);
} else if (op->is_intrinsic(Call::trace_expr)) {
// trace_expr returns argument 4
internal_assert(op->args.size() >= 5);
op->args[4].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) {
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;
string min_name = op->name + ".min";
string max_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 {
var.max = Variable::make(op->value.type().element_of(), max_name);
}
}
scope.push(op->name, var);
op->body.accept(this);
scope.pop(op->name);
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() && expr_uses_var(interval.min, max_name)) {
interval.min = Let::make(max_name, val.max, interval.min);
}
}
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() && expr_uses_var(interval.max, max_name)) {
interval.max = Let::make(max_name, val.max, interval.max);
}
}
}
void visit(const LetStmt *) {
internal_error << "Bounds of statement\n";
}
void visit(const AssertStmt *) {
internal_error << "Bounds of statement\n";
}
void visit(const ProducerConsumer *) {
internal_error << "Bounds of statement\n";
}
void visit(const For *) {
internal_error << "Bounds of statement\n";
}
void visit(const Store *) {
internal_error << "Bounds of statement\n";
}
void visit(const Provide *) {
internal_error << "Bounds of statement\n";
}
void visit(const Allocate *) {
internal_error << "Bounds of statement\n";
}
void visit(const Realize *) {
internal_error << "Bounds of statement\n";
}
void visit(const Block *) {
internal_error << "Bounds of statement\n";
}
};
Interval bounds_of_expr_in_scope(Expr expr, const Scope<Interval> &scope, const FuncValueBounds &fb) {
//debug(3) << "computing bounds_of_expr_in_scope " << expr << "\n";
Bounds b(&scope, fb);
expr.accept(&b);
//debug(3) << "bounds_of_expr_in_scope " << expr << " = " << simplify(b.min) << ", " << simplify(b.max) << "\n";
if (b.interval.has_lower_bound()) {
internal_assert(b.interval.min.type().is_scalar())
<< "Min of " << expr
<< " should have been a scalar: " << b.interval.min << "\n";
}
if (b.interval.has_upper_bound()) {
internal_assert(b.interval.max.type().is_scalar())
<< "Max of " << expr
<< " should have been a scalar: " << b.interval.max << "\n";
}
return b.interval;
}
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)));
//result.push_back(Range(min, 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_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.size() == 0 || b.size() == 0)) {
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++) {
condition = (condition &&
(outer[i].min <= inner[i].min) &&
(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 is_one(simplify(condition));
}
// 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(fn), consider_calls(calls), consider_provides(provides), func_bounds(fb) {
scope.set_containing_scope(s);
}
map<string, Box> boxes;
private:
string func;
bool consider_calls, consider_provides;
Scope<Interval> scope;
const FuncValueBounds &func_bounds;
using IRGraphVisitor::visit;
void visit(const Call *op) {
if (!consider_calls) return;
// Calls inside of an address_of aren't touched, because no
// actual memory access takes place.
if (op->is_intrinsic(Call::address_of)) {
// Visit the args of the inner call
internal_assert(op->args.size() == 1);
const Call *c = op->args[0].as<Call>();
if (c) {
for (size_t i = 0; i < c->args.size(); i++) {
c->args[i].accept(this);
}
} else {
const Load *l = op->args[0].as<Load>();
internal_assert(l);
l->index.accept(this);
}
return;
}
if (op->is_intrinsic(Call::if_then_else)) {
assert(op->args.size() == 3);
// 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 else_case = Evaluate::make(op->args[2]);
Stmt equivalent_if = IfThenElse::make(op->args[0], then_case, else_case);
equivalent_if.accept(this);
return;
}
IRVisitor::visit(op);
if (op->call_type == Call::Halide ||
op->call_type == Call::Image) {
for (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);
}
}
}
class CountVars : public IRVisitor {
using IRVisitor::visit;
void visit(const Variable *var) {
count++;
}
public:
int count;
CountVars() : count(0) {}
};
// 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(Expr e) {
CountVars c;
e.accept(&c);
return c.count < 10;
}
template<typename LetOrLetStmt>
void visit_let(const LetOrLetStmt *op) {
if (consider_calls) {
op->value.accept(this);
}
Interval value_bounds = bounds_of_expr_in_scope(op->value, scope, func_bounds);
bool fixed = value_bounds.min.same_as(value_bounds.max);
value_bounds.min = simplify(value_bounds.min);
value_bounds.max = fixed ? value_bounds.min : simplify(value_bounds.max);
if (is_small_enough_to_substitute(value_bounds.min) &&
(fixed || is_small_enough_to_substitute(value_bounds.max))) {
scope.push(op->name, value_bounds);
op->body.accept(this);
scope.pop(op->name);
} else {
string max_name = unique_name('t');
string min_name = unique_name('t');
scope.push(op->name, Interval(Variable::make(op->value.type(), min_name),
Variable::make(op->value.type(), max_name)));
op->body.accept(this);
scope.pop(op->name);
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, max_name)) {
box[i].min = Let::make(max_name, value_bounds.max, box[i].min);
}
if (expr_uses_var(box[i].min, min_name)) {
box[i].min = Let::make(min_name, value_bounds.min, box[i].min);
}
}
if (box[i].has_upper_bound()) {
if (expr_uses_var(box[i].max, max_name)) {
box[i].max = Let::make(max_name, value_bounds.max, box[i].max);
}
if (expr_uses_var(box[i].max, min_name)) {
box[i].max = Let::make(min_name, value_bounds.min, box[i].max);
}
}
}
}
}
}
void visit(const Let *op) {
visit_let(op);
}
void visit(const LetStmt *op) {
visit_let(op);
}
void visit(const IfThenElse *op) {
op->condition.accept(this);
if (expr_uses_vars(op->condition, scope)) {
if (!op->else_case.defined() || is_no_op(op->else_case)) {
// Trim the scope down to represent the fact that the
// condition is true. We only understand certain types
// of conditions for now.
Expr c = op->condition;
const Call *call = c.as<Call>();
if (call && (call->is_intrinsic(Call::likely) ||
call->is_intrinsic(Call::likely_if_innermost))) {
c = call->args[0];
}
const LT *lt = c.as<LT>();
const LE *le = c.as<LE>();
const GT *gt = c.as<GT>();
const GE *ge = c.as<GE>();
const EQ *eq = c.as<EQ>();
Expr a, b;
if (lt) {a = lt->a; b = lt->b;}
if (le) {a = le->a; b = le->b;}
if (gt) {a = gt->a; b = gt->b;}
if (ge) {a = ge->a; b = ge->b;}
if (eq) {a = eq->a; b = eq->b;}
const Variable *var_a = a.as<Variable>();
const Variable *var_b = b.as<Variable>();
string var_to_pop;
if (a.defined() && b.defined() && a.type() == Int(32)) {
Expr inner_min, inner_max;
if (var_a && scope.contains(var_a->name)) {
Interval i = scope.get(var_a->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(b, scope, func_bounds);
if (bi.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()) {
if (gt) {
i.min = max(likely_i.min, bi.min + 1);
}
if (ge || eq) {
i.min = max(likely_i.min, bi.min);
}
}
scope.push(var_a->name, i);
var_to_pop = var_a->name;
} else if (var_b && scope.contains(var_b->name)) {
Interval i = scope.get(var_b->name);
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 ai = bounds_of_expr_in_scope(a, scope, func_bounds);
if (ai.has_upper_bound()) {
if (gt) {
i.max = min(likely_i.max, ai.max - 1);
}
if (ge || eq) {
i.max = min(likely_i.max, ai.max);
}
}
if (ai.has_lower_bound()) {
if (lt) {
i.min = max(likely_i.min, ai.min + 1);
}
if (le || eq) {
i.min = max(likely_i.min, ai.min);
}
}
scope.push(var_b->name, i);
var_to_pop = var_b->name;
}
}
op->then_case.accept(this);
if (!var_to_pop.empty()) {
scope.pop(var_to_pop);
}
} else {
// Just take the union over the branches
op->then_case.accept(this);
op->else_case.accept(this);
}
} 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) {
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;
}
scope.push(op->name, Interval(min_val, max_val));
op->body.accept(this);
scope.pop(op->name);
}
void visit(const Provide *op) {
if (consider_provides) {
if (op->name == func || func.empty()) {
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 (size_t i = 0; i < op->args.size(); i++) {
op->args[i].accept(this);
}
for (size_t i = 0; i < op->values.size(); i++) {
op->values[i].accept(this);
}
}
}
};
map<string, Box> boxes_touched(Expr e, Stmt s, bool consider_calls, bool consider_provides,
string fn, const Scope<Interval> &scope, const FuncValueBounds &fb) {
// 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 (!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);
}
return calls.boxes;
}
Box box_touched(Expr e, Stmt s, bool consider_calls, bool consider_provides,
string fn, const Scope<Interval> &scope, const FuncValueBounds &fb) {
map<string, Box> boxes = boxes_touched(e, s, consider_calls, consider_provides, fn, scope, fb);
internal_assert(boxes.size() <= 1);
return boxes[fn];
}
map<string, Box> boxes_required(Expr e, const Scope<Interval> &scope, const FuncValueBounds &fb) {
return boxes_touched(e, Stmt(), true, false, "", scope, fb);
}
Box box_required(Expr e, 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(), s, true, false, "", scope, fb);
}
Box box_required(Stmt s, string fn, const Scope<Interval> &scope, const FuncValueBounds &fb) {
return box_touched(Expr(), s, true, false, fn, scope, fb);
}
map<string, Box> boxes_provided(Expr e, const Scope<Interval> &scope, const FuncValueBounds &fb) {
return boxes_touched(e, Stmt(), false, true, "", scope, fb);
}
Box box_provided(Expr e, 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(), s, false, true, "", scope, fb);
}
Box box_provided(Stmt s, string fn, const Scope<Interval> &scope, const FuncValueBounds &fb) {
return box_touched(Expr(), s, false, true, fn, scope, fb);
}
map<string, Box> boxes_touched(Expr e, const Scope<Interval> &scope, const FuncValueBounds &fb) {
return boxes_touched(e, Stmt(), true, true, "", scope, fb);
}
Box box_touched(Expr e, 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(), s, true, true, "", scope, fb);
}
Box box_touched(Stmt s, string fn, const Scope<Interval> &scope, const FuncValueBounds &fb) {
return box_touched(Expr(), 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 (size_t i = 0; i < order.size(); i++) {
Function f = env.find(order[i])->second;
const vector<string> f_args = f.args();
for (int j = 0; j < f.outputs(); j++) {
pair<string, int> key = make_pair(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;
}
debug(2) << "Bounds on value " << j
<< " for func " << order[i]
<< " are: " << result.min << ", " << result.max << "\n";
}
}
return fb;
}
void check(const Scope<Interval> &scope, Expr e, Expr correct_min, 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 bounds_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, select(y < 0, y*10, 0), select(y < 0, 0, y*10));
check(scope, x/(x+y), Interval::neg_inf, Interval::pos_inf);
check(scope, 11/(x+1), 1, 11);
check(scope, Load::make(Int(8), "buf", x, BufferPtr(), Parameter()), make_const(Int(8), -128), make_const(Int(8), 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(1/(x-2), x-10, x+10), -10, 20);
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, 10);
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), make_const(UInt(8), 0), make_const(UInt(8), 255));
check(scope, (cast<uint8_t>(x)+10)*20, make_const(UInt(8), 0), make_const(UInt(8), 255));
check(scope, (cast<uint8_t>(x)+10)*(cast<uint8_t>(x)+5), make_const(UInt(8), 0), make_const(UInt(8), 255));
check(scope, (cast<uint8_t>(x)+10)-(cast<uint8_t>(x)+5), make_const(UInt(8), 0), make_const(UInt(8), 255));
// Check some operations that we should be able to prove do not overflow
check(scope, (cast<uint8_t>(x)+240), make_const(UInt(8), 240), make_const(UInt(8), 250));
check(scope, (cast<uint8_t>(x)+10)*10, make_const(UInt(8), 100), make_const(UInt(8), 200));
check(scope, (cast<uint8_t>(x)+10)*(cast<uint8_t>(x)), make_const(UInt(8), 0), make_const(UInt(8), 200));
check(scope, (cast<uint8_t>(x)+20)-(cast<uint8_t>(x)+5), make_const(UInt(8), 5), make_const(UInt(8), 25));
check(scope,
cast<uint16_t>(clamp(cast<float>(x/y), 0.0f, 4095.0f)),
make_const(UInt(16), 0), make_const(UInt(16), 4095));
check(scope,
cast<uint8_t>(clamp(cast<uint16_t>(x/y), cast<uint16_t>(0), cast<uint16_t>(128))),
make_const(UInt(8), 0), make_const(UInt(8), 128));
Expr u8_1 = cast<uint8_t>(Load::make(Int(8), "buf", x, BufferPtr(), Parameter()));
Expr u8_2 = cast<uint8_t>(Load::make(Int(8), "buf", x + 17, BufferPtr(), Parameter()));
check(scope, cast<uint16_t>(u8_1) + cast<uint16_t>(u8_2),
make_const(UInt(16), 0), make_const(UInt(16), 255*2));
vector<Expr> input_site_1 = {2*x};
vector<Expr> input_site_2 = {2*x+1};
vector<Expr> output_site = {x+1};
Buffer<int32_t, 1> in(10);
BufferPtr in_buf(in, "input");
Stmt loop = For::make("x", 3, 10, ForType::Serial, DeviceAPI::Host,
Provide::make("output",
{Add::make(Call::make(in_buf, input_site_1),
Call::make(in_buf, input_site_2))},
output_site));
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));
std::cout << "Bounds test passed" << std::endl;
}
}
}
