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simplify.cpp
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#include "Halide.h"
using namespace Halide;
using namespace Halide::Internal;
using namespace Halide::ConciseCasts;
#define internal_assert _halide_user_assert
// Helper to wrap an expression in a statement using the expression
// that won't be simplified away.
Stmt not_no_op(Expr x) {
x = Call::make(x.type(), "not_no_op", {x}, Call::Extern);
return Evaluate::make(x);
}
void check_is_sio(const Expr &e) {
Expr simpler = simplify(e);
if (!Call::as_intrinsic(simpler, {Call::signed_integer_overflow})) {
std::cerr
<< "\nSimplification failure:\n"
<< "Input: " << e << "\n"
<< "Output: " << simpler << "\n"
<< "Expected output: signed_integer_overflow(n)\n";
abort();
}
}
void check(const Expr &a, const Expr &b, const Scope<ModulusRemainder> &alignment = Scope<ModulusRemainder>()) {
Expr simpler = simplify(a, true, Scope<Interval>(), alignment);
if (!equal(simpler, b)) {
std::cerr
<< "\nSimplification failure:\n"
<< "Input: " << a << "\n"
<< "Output: " << simpler << "\n"
<< "Expected output: " << b << "\n";
abort();
}
}
void check(const Stmt &a, const Stmt &b) {
Stmt simpler = simplify(a);
if (!equal(simpler, b)) {
std::cerr
<< "\nSimplification failure:\n"
<< "Input:\n"
<< a << "\n"
<< "Output:\n"
<< simpler << "\n"
<< "Expected output:\n"
<< b << "\n";
abort();
}
}
void check_in_bounds(const Expr &a, const Expr &b, const Scope<Interval> &bi) {
Expr simpler = simplify(a, true, bi);
if (!equal(simpler, b)) {
std::cerr
<< "\nSimplification failure:\n"
<< "Input: " << a << "\n"
<< "Output: " << simpler << "\n"
<< "Expected output: " << b << "\n";
abort();
}
}
// Helper functions to use in the tests below
Expr interleave_vectors(const std::vector<Expr> &e) {
return Shuffle::make_interleave(e);
}
Expr concat_vectors(const std::vector<Expr> &e) {
return Shuffle::make_concat(e);
}
Expr slice(const Expr &e, int begin, int stride, int w) {
return Shuffle::make_slice(e, begin, stride, w);
}
// An arbitrary fixed permutation of the lanes of a single vector that isn't one
// of the classes above. Requires a power of two number of lanes.
Expr permute_lanes(const Expr &e) {
std::vector<int> mask(e.type().lanes());
for (int i = 0; i < e.type().lanes(); i++) {
mask[i] = i;
// Some arbitrary permutation
if (i & 1) {
std::swap(mask[i], mask[i / 2]);
}
}
return Shuffle::make({e}, std::move(mask));
}
Expr ramp(const Expr &base, const Expr &stride, int w) {
return Ramp::make(base, stride, w);
}
Expr broadcast(const Expr &base, int w) {
return Broadcast::make(base, w);
}
void check_casts() {
Expr x = Var("x"), y = Var("y");
check(cast(Int(32), cast(Int(32), x)), x);
check(cast(Float(32), 3), 3.0f);
check(cast(Int(32), 5.0f), 5);
check(cast(Int(32), cast(Int(8), 3)), 3);
check(cast(Int(32), cast(Int(8), 1232)), -48);
// Check redundant casts
check(cast(Float(32), cast(Float(64), x)), cast(Float(32), x));
check(cast(Int(16), cast(Int(32), x)), cast(Int(16), x));
check(cast(Int(16), cast(UInt(32), x)), cast(Int(16), x));
check(cast(UInt(16), cast(Int(32), x)), cast(UInt(16), x));
check(cast(UInt(16), cast(UInt(32), x)), cast(UInt(16), x));
// Check evaluation of constant expressions involving casts
check(cast(UInt(16), 53) + cast(UInt(16), 87), make_const(UInt(16), 140));
check(cast(Int(8), 127) + cast(Int(8), 1), make_const(Int(8), -128));
check(cast(UInt(16), -1) - cast(UInt(16), 1), make_const(UInt(16), 65534));
check(cast(Int(16), 4) * cast(Int(16), -5), make_const(Int(16), -20));
check(cast(Int(16), 16) / cast(Int(16), 4), make_const(Int(16), 4));
check(cast(Int(16), 23) % cast(Int(16), 5), make_const(Int(16), 3));
check(min(cast(Int(16), 30000), cast(Int(16), -123)), make_const(Int(16), -123));
check(max(cast(Int(16), 30000), cast(Int(16), 65000)), make_const(Int(16), 30000));
check(cast(UInt(16), -1) == cast(UInt(16), 65535), const_true());
check(cast(UInt(16), 65) == cast(UInt(16), 66), const_false());
check(cast(UInt(16), -1) < cast(UInt(16), 65535), const_false());
check(cast(UInt(16), 65) < cast(UInt(16), 66), const_true());
check(cast(UInt(16), 123.4f), make_const(UInt(16), 123));
check(cast(Float(32), cast(UInt(16), 123456.0f)), 57920.0f);
// Specific checks for 32 bit unsigned expressions - ensure simplifications are actually unsigned.
// 4000000000 (4 billion) is less than 2^32 but more than 2^31. As an int, it is negative.
check(cast(UInt(32), (int)4000000000UL) + cast(UInt(32), 5), make_const(UInt(32), (int)4000000005UL));
check(make_const(UInt(32, 4), (int)4000000000UL) - make_const(UInt(32, 4), 5), make_const(UInt(32, 4), (int)3999999995UL));
check(cast(UInt(32), (int)4000000000UL) / cast(UInt(32), 5), make_const(UInt(32), 800000000));
check(cast(UInt(32), 800000000) * cast(UInt(32), 5), make_const(UInt(32), (int)4000000000UL));
check(make_const(UInt(32, 2), (int)4000000023UL) % make_const(UInt(32, 2), 100), make_const(UInt(32, 2), 23));
check(min(cast(UInt(32), (int)4000000023UL), cast(UInt(32), 1000)), make_const(UInt(32), (int)1000));
check(max(cast(UInt(32), (int)4000000023UL), cast(UInt(32), 1000)), make_const(UInt(32), (int)4000000023UL));
check(cast(UInt(32), (int)4000000023UL) < cast(UInt(32), 1000), const_false());
check(make_const(UInt(32, 3), (int)4000000023UL) == make_const(UInt(32, 3), 1000), const_false(3));
check(cast(Float(64), 0.5f), Expr(0.5));
check((x - cast(Float(64), 0.5f)) * (x - cast(Float(64), 0.5f)),
(x + Expr(-0.5)) * (x + Expr(-0.5)));
check(cast(Int(64, 3), ramp(5.5f, 2.0f, 3)),
cast(Int(64, 3), ramp(5.5f, 2.0f, 3)));
check(cast(Int(64, 3), ramp(x, 2, 3)),
ramp(cast(Int(64), x), cast(Int(64), 2), 3));
// We do not currently expect cancellations to occur through casts
// check(cast(Int(64), x + 1) - cast(Int(64), x), cast(Int(64), 1));
// check(cast(Int(64), 1 + x) - cast(Int(64), x), cast(Int(64), 1));
// But only when overflow is undefined for the type
check(cast(UInt(8), x + 1) - cast(UInt(8), x),
cast(UInt(8), x + 1) - cast(UInt(8), x));
// Overflow is well-defined for ints < 32 bits
check(cast(Int(8), make_const(UInt(8), 128)), make_const(Int(8), -128));
// Check that chains of widening casts don't lose the distinction
// between zero-extending and sign-extending.
check(cast(UInt(64), cast(UInt(32), cast(Int(8), -1))),
UIntImm::make(UInt(64), 0xffffffffULL));
// It's a good idea to pull widening casts outside of shuffles
// when the shuffle reduces the lane count (e.g. a slice_vector).
Expr some_vector = ramp(y, 2, 8) * ramp(x, 1, 8);
check(slice(cast(UInt(64, 8), some_vector), 2, 1, 3),
cast(UInt(64, 3), slice(some_vector, 2, 1, 3)));
// But we currently have no logic for pulling things outside of shuffles
// other than slices.
check(permute_lanes(some_vector) + permute_lanes(some_vector + 1),
permute_lanes(some_vector) + permute_lanes(some_vector + 1));
std::vector<int> indices(18);
for (int i = 0; i < 18; i++) {
indices[i] = i & 3;
}
check(Shuffle::make({cast(UInt(64, 8), some_vector)}, indices),
Shuffle::make({cast(UInt(64, 8), some_vector)}, indices));
// Interleaving simplifications can result in slices.
Expr var_vector = Variable::make(Int(32, 12), "v");
Expr even = Shuffle::make_slice(var_vector, 0, 2, 4);
Expr odd = Shuffle::make_slice(var_vector, 1, 2, 4);
check(Shuffle::make_interleave({even, odd}), Shuffle::make_slice(var_vector, 0, 1, 8));
}
void check_algebra() {
Expr x = Var("x"), y = Var("y"), z = Var("z"), w = Var("w"), v = Var("v");
Expr xf = cast<float>(x);
Expr yf = cast<float>(y);
Expr t = const_true(), f = const_false();
check(3 + x, x + 3);
check(x + 0, x);
check(0 + x, x);
check(Expr(ramp(x, 2, 3)) + Expr(ramp(y, 4, 3)), ramp(x + y, 6, 3));
check(Expr(broadcast(4.0f, 5)) + Expr(ramp(3.25f, 4.5f, 5)), ramp(7.25f, 4.5f, 5));
check(Expr(ramp(3.25f, 4.5f, 5)) + Expr(broadcast(4.0f, 5)), ramp(7.25f, 4.5f, 5));
check(Expr(broadcast(3, 3)) + Expr(broadcast(1, 3)), broadcast(4, 3));
check((x + 3) + 4, x + 7);
check(4 + (3 + x), x + 7);
check((x + 3) + y, (x + y) + 3);
check(y + (x + 3), (x + y) + 3);
check((3 - x) + x, 3);
check(x + (3 - x), 3);
check(x * y + x * z, (y + z) * x);
check(x * y + z * x, (y + z) * x);
check(y * x + x * z, (y + z) * x);
check(y * x + z * x, (y + z) * x);
check(x - 0, x);
check((x / y) - (x / y), 0);
check(x - 2, x + (-2));
check(Expr(ramp(x, 2, 3)) - Expr(ramp(y, 4, 3)), ramp(x - y, -2, 3));
check(Expr(broadcast(4.0f, 5)) - Expr(ramp(3.25f, 4.5f, 5)), ramp(0.75f, -4.5f, 5));
check(Expr(ramp(3.25f, 4.5f, 5)) - Expr(broadcast(4.0f, 5)), ramp(-0.75f, 4.5f, 5));
check(Expr(broadcast(3, 3)) - Expr(broadcast(1, 3)), broadcast(2, 3));
check((x + y) - x, y);
check((x + y) - y, x);
check(x - (x + y), 0 - y);
check(x - (y + x), 0 - y);
check((x + 3) - 2, x + 1);
check((x + 3) - y, (x - y) + 3);
check((x - 3) - y, (x - y) + (-3));
check(x - (y - 2), (x - y) + 2);
check(3 - (y - 2), 5 - y);
check(x - (0 - y), x + y);
check(x + (0 - y), x - y);
check((0 - x) + y, y - x);
check(x * y - x * z, (y - z) * x);
check(x * y - z * x, (y - z) * x);
check(y * x - x * z, (y - z) * x);
check(y * x - z * x, (y - z) * x);
check((x * 8) - (y * 4), (x * 2 - y) * 4);
check((x * 4) - (y * 8), (x - y * 2) * 4);
check((x * 2) % 6, (x % 3) * 2);
check(x - (x / 8) * 8, x % 8);
check((x / 8) * 8 - x, -(x % 8));
check((x / 8) * 8 < x + y, 0 < x % 8 + y);
check((x / 8) * 8 < x - y, y < x % 8);
check((x / 8) * 8 < x, x % 8 != 0);
check(((x + 3) / 8) * 8 < x + y, 3 < (x + 3) % 8 + y);
check(((x + 3) / 8) * 8 < x - y, y < (x + 3) % 8 + (-3));
check(((x + 3) / 8) * 8 < x, 3 < (x + 3) % 8);
check(x * 0, 0);
check(0 * x, 0);
check(x * 1, x);
check(1 * x, x);
check(Expr(2.0f) * 4.0f, 8.0f);
check(Expr(2) * 4, 8);
check((3 * x) * 4, x * 12);
check(4 * (3 + x), x * 4 + 12);
check(Expr(broadcast(4.0f, 5)) * Expr(ramp(3.0f, 4.0f, 5)), ramp(12.0f, 16.0f, 5));
check(Expr(ramp(3.0f, 4.0f, 5)) * Expr(broadcast(2.0f, 5)), ramp(6.0f, 8.0f, 5));
check(Expr(broadcast(3, 3)) * Expr(broadcast(2, 3)), broadcast(6, 3));
check(x * y + x, (y + 1) * x);
check(x * y - x, (y + -1) * x);
check(x + x * y, (y + 1) * x);
check(x - x * y, (1 - y) * x);
check(x * y + y, (x + 1) * y);
check(x * y - y, (x + -1) * y);
check(y + x * y, (x + 1) * y);
check(y - x * y, (1 - x) * y);
check(0 / max(x, 1), 0);
check(x / 1, x);
check(max(x, 1) / (max(x, 1)), 1);
check(min(x, -1) / (min(x, -1)), 1);
check((x * 2 + 1) / (x * 2 + 1), 1);
check((-1) / (x * 2 + 1), select(x < 0, 1, -1));
check(Expr(7) / 3, 2);
check(Expr(6.0f) / 2.0f, 3.0f);
check((x / 3) / 4, x / 12);
check((x * 4) / 2, x * 2);
check((x * 2) / 4, x / 2);
check((x * (-4)) / 2, x * (-2));
check((x * 4 + y) / 2, y / 2 + x * 2);
check((y + x * 4) / 2, y / 2 + x * 2);
check((x * 2 - y) / 2, (0 - y) / 2 + x);
check((x * -2 - y) / 2, (0 - y) / 2 - x);
check((y - x * 4) / 2, y / 2 - x * 2);
check((x + 3) / 2 + 7, (x + 17) / 2);
check((x / 2 + 3) / 5, (x + 6) / 10);
check((x + (y + 3) / 5) + 5, (y + 28) / 5 + x);
check((x + 8) / 2, x / 2 + 4);
check((x - y) * -2, (y - x) * 2);
check((xf - yf) * -2.0f, (yf - xf) * 2.0f);
check(x * 3 + y * 9, (y * 3 + x) * 3);
check(x * 9 + y * 3, (x * 3 + y) * 3);
// Pull terms that are a multiple of the divisor out of a ternary expression
check(((x * 4 + y) + z) / 2, (y + z) / 2 + x * 2);
check(((x * 4 - y) + z) / 2, (z - y) / 2 + x * 2);
check(((x * 4 + y) - z) / 2, (y - z) / 2 + x * 2);
check(((x * 2 - y) - z) / 2, (0 - y - z) / 2 + x);
check(((x * -2 - y) - z) / 2, (0 - y - z) / 2 - x);
check((x + (y * 4 + z)) / 2, (x + z) / 2 + y * 2);
check(((x + y * 4) + z) / 2, (x + z) / 2 + y * 2);
check((x + (y * 4 - z)) / 2, (x - z) / 2 + y * 2);
check((x - (y * 4 + z)) / 2, (x - z) / 2 + y * -2);
check((x - (y * 4 - z)) / 2, (x + z) / 2 - y * 2);
// Pull out the gcd of the numerator and divisor
check((x * 3 + 5) / 9, (x + 1) / 3);
// Cancellations in integer divisions.
check((7 * y) / 7, y);
check((y * 7) / 7, y);
check((7 * y + z) / 7, z / 7 + y);
check((y * 7 + z) / 7, z / 7 + y);
check((z + 7 * y) / 7, z / 7 + y);
check((z + y * 7) / 7, z / 7 + y);
check((7 * y - z) / 7, (-z) / 7 + y);
check((y * 7 - z) / 7, (-z) / 7 + y);
check((z - 7 * y) / 7, z / 7 - y);
check((z - y * 7) / 7, z / 7 - y);
check((7 + y) / 7, y / 7 + 1);
check((y + 7) / 7, y / 7 + 1);
check((7 - y) / 7, (-y) / 7 + 1);
check((y - 7) / 7, y / 7 + (-1));
Scope<ModulusRemainder> alignment;
alignment.push("x", ModulusRemainder(2, 0));
check((x + 0) / 2, x / 2, alignment);
check((x + 1) / 2, x / 2, alignment);
check((x + 2) / 2, x / 2 + 1, alignment);
check((x + 3) / 2, x / 2 + 1, alignment);
alignment.pop("x");
alignment.push("x", ModulusRemainder(2, 1));
check((x + 0) / 2, x / 2, alignment);
check((x + 1) / 2, x / 2 + 1, alignment);
check((x + 2) / 2, x / 2 + 1, alignment);
check((x + 3) / 2, x / 2 + 2, alignment);
alignment.pop("x");
alignment.push("x", ModulusRemainder(3, 0));
check((x + 0) / 3, x / 3, alignment);
check((x + 1) / 3, x / 3, alignment);
check((x + 2) / 3, x / 3, alignment);
check((x + 3) / 3, x / 3 + 1, alignment);
check((x + 4) / 3, x / 3 + 1, alignment);
check((x + 5) / 3, x / 3 + 1, alignment);
alignment.pop("x");
alignment.push("x", ModulusRemainder(3, 1));
check((x + 0) / 3, x / 3, alignment);
check((x + 1) / 3, x / 3, alignment);
check((x + 2) / 3, x / 3 + 1, alignment);
check((x + 3) / 3, x / 3 + 1, alignment);
check((x + 4) / 3, x / 3 + 1, alignment);
check((x + 5) / 3, x / 3 + 2, alignment);
alignment.pop("x");
alignment.push("x", ModulusRemainder(3, 2));
check((x + 0) / 3, x / 3, alignment);
check((x + 1) / 3, x / 3 + 1, alignment);
check((x + 2) / 3, x / 3 + 1, alignment);
check((x + 3) / 3, x / 3 + 1, alignment);
check((x + 4) / 3, x / 3 + 2, alignment);
check((x + 5) / 3, x / 3 + 2, alignment);
alignment.pop("x");
alignment.push("x", ModulusRemainder(4, 0));
check((x + 0) / 2, x / 2, alignment);
check((x + 1) / 2, x / 2, alignment);
check((x + 2) / 2, x / 2 + 1, alignment);
check((x + 3) / 2, x / 2 + 1, alignment);
alignment.pop("x");
alignment.push("x", ModulusRemainder(4, 1));
check((x + 0) / 2, x / 2, alignment);
check((x + 1) / 2, x / 2 + 1, alignment);
check((x + 2) / 2, x / 2 + 1, alignment);
check((x + 3) / 2, x / 2 + 2, alignment);
alignment.pop("x");
alignment.push("x", ModulusRemainder(2, 0));
check((x + 0) / 3, x / 3, alignment);
check((x + 1) / 3, (x + 1) / 3, alignment);
check((x + 2) / 3, (x + 2) / 3, alignment);
check((x + 3) / 3, x / 3 + 1, alignment);
check((x + 4) / 3, (x + 4) / 3, alignment);
check((x + 5) / 3, (x + 5) / 3, alignment);
alignment.pop("x");
check(((7 + y) + z) / 7, (y + z) / 7 + 1);
check(((y + 7) + z) / 7, (y + z) / 7 + 1);
check((y + (7 + z)) / 7, (y + z) / 7 + 1);
check((y + (z + 7)) / 7, (y + z) / 7 + 1);
check(xf / 4.0f, xf * 0.25f);
// Some quaternary rules with cancellations
check((x + y) - (y + z), x - z);
check((x + y) - (y + z), x - z);
check((y + x) - (y + z), x - z);
check((y + x) - (y + z), x - z);
check((x - y) - (z - y), x - z);
check((y - z) - (y - x), x - z);
check(((x + y) + z) - x, y + z);
check(((x + y) + z) - y, x + z);
check((x + (y + z)) - y, x + z);
check((x + (y + z)) - z, x + y);
check((x * 8) % 4, 0);
check((x * 8 + y) % 4, y % 4);
check((y + 8) % 4, y % 4);
check((y + x * 8) % 4, y % 4);
check((y * 16 - 13) % 2, 1);
check((x * y) % 1, 0);
check((y * 16 - 13) % 2, 1);
check((y - 8) % 4, y % 4);
check((y - x * 8) % 4, y % 4);
check((x * 8 - y) % 4, (-y) % 4);
// Check an optimization important for fusing dimensions
check((x / 3) * 3 + x % 3, x);
check(x % 3 + (x / 3) * 3, x);
check(((x / 3) * 3 + y) + x % 3, x + y);
check(((x / 3) + y) * 3 + x % 3, y * 3 + x);
check((x % 3 + y) + (x / 3) * 3, x + y);
check((y + x % 3) + (x / 3) * 3, x + y);
check((y + (x / 3 * 3)) + x % 3, x + y);
check((y + (x / 3)) * 3 + x % 3, y * 3 + x);
check(x / 2 + x % 2, (x + 1) / 2);
check(x % 2 + x / 2, (x + 1) / 2);
check(((x + 1) / 2) * 2 - x, x % 2);
check(((x + 2) / 3) * 3 - x, (-x) % 3);
check(x - ((x + 1) / 2) * 2, ((x + 1) % 2 + -1));
check(x - ((x + 2) / 3) * 3, ((x + 2) % 3 + -2));
check((x % 2 + 4) / 2, 2);
check((x % 2 + 5) / 2, x % 2 + 2);
// Almost-cancellations through integer divisions. These rules all
// deduplicate x and wrap it in a modulo operator, neutering it
// for the purposes of bounds inference. Patterns below look
// confusing, but were brute-force tested.
check((x + 17) / 3 - (x + 7) / 3, ((x + 1) % 3 + 10) / 3);
check((x + 17) / 3 - (x + y) / 3, (19 - y - (x + 2) % 3) / 3);
check((x + y) / 3 - (x + 7) / 3, ((x + 1) % 3 + y + -7) / 3);
check(x / 3 - (x + y) / 3, (2 - y - x % 3) / 3);
check((x + y) / 3 - x / 3, (x % 3 + y) / 3);
check(x / 3 - (x + 7) / 3, (-5 - x % 3) / 3);
check((x + 17) / 3 - x / 3, (x % 3 + 17) / 3);
check((x + 17) / 3 - (x - y) / 3, (y - (x + 2) % 3 + 19) / 3);
check((x - y) / 3 - (x + 7) / 3, ((x + 1) % 3 - y + (-7)) / 3);
check(x / 3 - (x - y) / 3, (y - x % 3 + 2) / 3);
check((x - y) / 3 - x / 3, (x % 3 - y) / 3);
// Check some specific expressions involving div and mod
check(Expr(23) / 4, Expr(5));
check(Expr(-23) / 4, Expr(-6));
check(Expr(-23) / -4, Expr(6));
check(Expr(23) / -4, Expr(-5));
check(Expr(-2000000000) / 1000000001, Expr(-2));
check(Expr(23) % 4, Expr(3));
check(Expr(-23) % 4, Expr(1));
check(Expr(-23) % -4, Expr(1));
check(Expr(23) % -4, Expr(3));
check(Expr(-2000000000) % 1000000001, Expr(2));
check(Expr(3) + Expr(8), 11);
check(Expr(3.25f) + Expr(7.75f), 11.0f);
check(Expr(7) % 2, 1);
check(Expr(7.25f) % 2.0f, 1.25f);
check(Expr(-7.25f) % 2.0f, 0.75f);
check(Expr(-7.25f) % -2.0f, -1.25f);
check(Expr(7.25f) % -2.0f, -0.75f);
check(2 * x + (2 * x + y) / 5, (x * 12 + y) / 5);
check(x + (x - y) / 4, (x * 5 - y) / 4);
check((x + z) + (y + (x + z)) / 3, ((x + z) * 4 + y) / 3);
check(x + ((y + w) - x) / 2, ((w + y) + x) / 2);
check((x + y) / 3 + x, (x * 4 + y) / 3);
check((x - y) / 4 + x, (x * 5 - y) / 4);
check((y + x) / 3 + x, (x * 4 + y) / 3);
check((y - x) / 3 + x, (x * 2 + y) / 3);
check(1 + (1 + y) / 2, (y + 3) / 2);
check((y + 1) / 2 + 1, (y + 3) / 2);
check((0 - y) / 5 + 1, (0 - y) / 5 + 1);
check(x - (x + y) / 3, (x * 2 - y + 2) / 3);
check((w + x) - ((w + x) - y * z) / 3, ((w + x) * 2 + y * z + 2) / 3);
check(x - (y + x) / 2, (x - y + 1) / 2);
check(x - (y - x) / 6, (x * 7 - y + 5) / 6);
check(x - (x + y) / -3, x - (x + y) / -3);
check((w + x) - ((w + x) - y * z) / -3, (w + x) - ((w + x) - y * z) / -3);
check(x - (y + x) / -2, x - (x + y) / -2);
check(x - (y - x) / -6, x - (y - x) / -6);
check((x + y) / 3 - x, (x * -2 + y) / 3);
check((x * y - w) / 4 - x * y, (x * y * (-3) - w) / 4);
check((y + x) / 5 - x, (x * -4 + y) / 5);
check((y - x) / 6 - x, (y - x * 7) / 6);
check(1 - (1 + y) / 2 - 1, (0 - y) / 2);
check(1 - (-y + 1) / 2 - 1, y / 2);
check(1 - (0 - y) / 5, (y + 9) / 5);
// Div/mod can't make things larger
check(5 / x < 6, const_true());
check(5 / x > -6, const_true());
check(5 / x < 5, 5 / x < 5);
check(5 / x > -5, -5 < 5 / x);
check(5 % x < 6, const_true());
check(5 % x < 5, 5 % x < 5);
check(5 % x >= 0, const_true());
check(5 % x > 0, 5 % x != 0);
// Test case with most negative 32-bit number, as constant to check that it is not negated.
check(((x * (int32_t)0x80000000) + (z * (int32_t)0x80000000 + y)),
((x * (int32_t)0x80000000) + (z * (int32_t)0x80000000 + y)));
// Use a require with no error message to test chains of reasoning
auto require = [](Expr cond, Expr val) {
return Internal::Call::make(val.type(),
Internal::Call::require,
{cond, val, 0},
Internal::Call::PureIntrinsic);
};
check(require(2 < x && x < 4, x),
require(2 < x && x < 4, 3));
check(require(2 < x && x < 5 && x % 4 == 0, x),
require(2 < x && x < 5 && x % 4 == 0, 4));
check(require(x % 4 == 3, x % 2),
require(x % 4 == 3, 1));
// Check modulo of expressions that are not-obviously a multiple of something
check(max(min(x * 8, 32), y * 16) % 4 == 0, const_true());
check(select(x > 4, x * 9 + 1, y * 6 - 2) % 3 == 1, const_true());
check(max(32, x * 4) % 16 < 13, const_true()); // After the %16 the max value is 12, not 15, due to alignment
Expr complex_cond = ((10 < y) && (y % 17 == 4) && (y < 30) && (x == y * 16 + 3));
// The condition is enough to imply that y == 21, x == 339
check(require(complex_cond, select(x % 2 == 0, 1237, y)),
require(complex_cond, 21));
}
void check_vectors() {
Expr x = Var("x"), y = Var("y"), z = Var("z");
check(Expr(broadcast(y, 4)) / Expr(broadcast(x, 4)),
Expr(broadcast(y / x, 4)));
check(Expr(ramp(x, 4, 4)) / 2, ramp(x / 2, 2, 4));
check(Expr(ramp(x, -4, 7)) / 2, ramp(x / 2, -2, 7));
check(Expr(ramp(x, 4, 5)) / -2, ramp(x / -2, -2, 5));
check(Expr(ramp(x, -8, 5)) / -2, ramp(x / -2, 4, 5));
check(Expr(ramp(4 * x, 1, 4)) / 4, broadcast(x, 4));
check(Expr(ramp(x * 4, 1, 3)) / 4, broadcast(x, 3));
check(Expr(ramp(x * 8, 2, 4)) / 8, broadcast(x, 4));
check(Expr(ramp(x * 8, 3, 3)) / 8, broadcast(x, 3));
check(Expr(ramp(0, 1, 8)) % 16, Expr(ramp(0, 1, 8)));
check(Expr(ramp(8, 1, 8)) % 16, Expr(ramp(8, 1, 8)));
check(Expr(ramp(9, 1, 8)) % 16, Expr(ramp(9, 1, 8)) % 16);
check(Expr(ramp(16, 1, 8)) % 16, Expr(ramp(0, 1, 8)));
check(Expr(ramp(0, 1, 8)) % 8, Expr(ramp(0, 1, 8)));
check(Expr(ramp(x * 8 + 17, 1, 4)) % 8, Expr(ramp(1, 1, 4)));
check(Expr(ramp(x * 8 + 17, 1, 8)) % 8, Expr(ramp(1, 1, 8) % 8));
check(Expr(broadcast(x, 4)) % Expr(broadcast(y, 4)),
Expr(broadcast(x % y, 4)));
check(Expr(ramp(x, 2, 4)) % (broadcast(2, 4)),
broadcast(x % 2, 4));
check(Expr(ramp(2 * x + 1, 4, 4)) % (broadcast(2, 4)),
broadcast(1, 4));
check(max(broadcast(24, 2), broadcast(x, 2) % ramp(-8, -33, 2)),
max(broadcast(x, 2) % ramp(-8, -33, 2), broadcast(24, 2)));
check(max(broadcast(41, 2), broadcast(x, 2) % ramp(-8, -33, 2)),
broadcast(41, 2));
check(ramp(0, 1, 4) == broadcast(2, 4),
ramp(-2, 1, 4) == broadcast(0, 4));
check(ramp(broadcast(0, 6), broadcast(6, 6), 4) + broadcast(ramp(0, 1, 3), 8) +
broadcast(ramp(broadcast(0, 3), broadcast(3, 3), 2), 4),
ramp(0, 1, 24));
// Any linear combination of simple ramps and broadcasts should
// reduce to a single ramp or broadcast.
std::mt19937 rng(0);
for (int i = 0; i < 50; i++) {
std::vector<Expr> leaves =
{ramp(x, 1, 4),
ramp(x, y, 4),
ramp(z, x, 4),
broadcast(x, 4),
broadcast(y, 4),
broadcast(z, 4)};
while (leaves.size() > 1) {
int idx1 = rng() % (int)leaves.size();
int idx2 = 0;
do {
idx2 = rng() % (int)leaves.size();
} while (idx2 == idx1);
switch (rng() % 4) {
case 0:
leaves[idx1] += leaves[idx2];
break;
case 1:
leaves[idx1] -= leaves[idx2];
break;
case 2:
leaves[idx1] += (int)(rng() % 8) * leaves[idx2];
break;
case 3:
leaves[idx1] -= (int)(rng() % 8) * leaves[idx2];
break;
}
std::swap(leaves[idx2], leaves.back());
leaves.pop_back();
}
Expr simpler = simplify(leaves[0]);
if (!simpler.as<Ramp>() && !simpler.as<Broadcast>()) {
std::cerr << "A linear combination of ramps and broadcasts should be a single ramp or broadcast:\n"
<< simpler << "\n";
abort();
}
}
{
Expr test = select(ramp(const_true(), const_true(), 2),
ramp(const_false(), const_true(), 2),
broadcast(const_false(), 2)) ==
broadcast(const_false(), 2);
Expr expected = !(ramp(const_true(), const_true(), 2) &&
ramp(const_false(), const_true(), 2));
check(test, expected);
}
{
Expr test = select(ramp(const_true(), const_true(), 2),
broadcast(const_true(), 2),
ramp(const_false(), const_true(), 2)) ==
broadcast(const_false(), 2);
Expr expected = !(ramp(const_true(), const_true(), 2) ||
ramp(const_false(), const_true(), 2));
check(test, expected);
}
// Collapse some vector interleaves
check(interleave_vectors({ramp(x, 2, 4), ramp(x + 1, 2, 4)}), ramp(x, 1, 8));
check(interleave_vectors({ramp(x, 4, 4), ramp(x + 2, 4, 4)}), ramp(x, 2, 8));
check(interleave_vectors({ramp(x - y, 2 * y, 4), ramp(x, 2 * y, 4)}), ramp(x - y, y, 8));
check(interleave_vectors({ramp(x, 3, 4), ramp(x + 1, 3, 4), ramp(x + 2, 3, 4)}), ramp(x, 1, 12));
{
Expr vec = ramp(x, 1, 16);
check(interleave_vectors({slice(vec, 0, 2, 8), slice(vec, 1, 2, 8)}), vec);
check(interleave_vectors({slice(vec, 0, 4, 4), slice(vec, 1, 4, 4), slice(vec, 2, 4, 4), slice(vec, 3, 4, 4)}), vec);
}
// Collapse some vector concats
check(concat_vectors({ramp(x, 2, 4), ramp(x + 8, 2, 4)}), ramp(x, 2, 8));
check(concat_vectors({ramp(x, 3, 2), ramp(x + 6, 3, 2), ramp(x + 12, 3, 2)}), ramp(x, 3, 6));
// Now some ones that can't work
{
Expr e = interleave_vectors({ramp(x, 2, 4), ramp(x, 2, 4)});
check(e, e);
e = interleave_vectors({ramp(x, 2, 4), ramp(x + 2, 2, 4)});
check(e, e);
e = interleave_vectors({ramp(x, 3, 4), ramp(x + 1, 3, 4)});
check(e, e);
e = interleave_vectors({ramp(x, 2, 4), ramp(y + 1, 2, 4)});
check(e, e);
e = interleave_vectors({ramp(x, 2, 4), ramp(x + 1, 3, 4)});
check(e, e);
e = concat_vectors({ramp(x, 1, 4), ramp(x + 4, 2, 4)});
check(e, e);
e = concat_vectors({ramp(x, 1, 4), ramp(x + 8, 1, 4)});
check(e, e);
e = concat_vectors({ramp(x, 1, 4), ramp(y + 4, 1, 4)});
check(e, e);
}
// Now check that an interleave of some collapsible loads collapses into a single dense load
{
Expr load1 = Load::make(Float(32, 4), "buf", ramp(x, 2, 4), Buffer<>(), Parameter(), const_true(4), ModulusRemainder());
Expr load2 = Load::make(Float(32, 4), "buf", ramp(x + 1, 2, 4), Buffer<>(), Parameter(), const_true(4), ModulusRemainder());
Expr load12 = Load::make(Float(32, 8), "buf", ramp(x, 1, 8), Buffer<>(), Parameter(), const_true(8), ModulusRemainder());
check(interleave_vectors({load1, load2}), load12);
// They don't collapse in the other order
Expr e = interleave_vectors({load2, load1});
check(e, e);
// Or if the buffers are different
Expr load3 = Load::make(Float(32, 4), "buf2", ramp(x + 1, 2, 4), Buffer<>(), Parameter(), const_true(4), ModulusRemainder());
e = interleave_vectors({load1, load3});
check(e, e);
}
// Check that concatenated loads of adjacent scalars collapse into a vector load.
{
int lanes = 4;
std::vector<Expr> loads;
for (int i = 0; i < lanes; i++) {
loads.push_back(Load::make(Float(32), "buf", 4 * x + i, Buffer<>(), Parameter(), const_true(), ModulusRemainder()));
}
check(concat_vectors(loads), Load::make(Float(32, lanes), "buf", ramp(x * 4, 1, lanes), Buffer<>(), Parameter(), const_true(lanes), ModulusRemainder(4, 0)));
}
// Check that concatenated loads of adjacent vectors collapse into a vector load, with appropriate alignment.
{
int lanes = 4;
int vectors = 4;
std::vector<Expr> loads;
for (int i = 0; i < vectors; i++) {
loads.push_back(Load::make(Float(32, lanes), "buf", ramp(i * lanes, 1, lanes), Buffer<>(), Parameter(), const_true(lanes), ModulusRemainder(4, 0)));
}
check(concat_vectors(loads), Load::make(Float(32, lanes * vectors), "buf", ramp(0, 1, lanes * vectors), Buffer<>(), Parameter(), const_true(vectors * lanes), ModulusRemainder(0, 0)));
}
{
Expr vx = Variable::make(Int(32, 32), "x");
Expr vy = Variable::make(Int(32, 32), "y");
Expr vz = Variable::make(Int(32, 8), "z");
Expr vw = Variable::make(Int(32, 16), "w");
// Check that vector slices are hoisted.
check(slice(vx, 0, 2, 8) + slice(vy, 0, 2, 8), slice(vx + vy, 0, 2, 8));
check(slice(vx, 0, 2, 8) + (slice(vy, 0, 2, 8) + vz), slice(vx + vy, 0, 2, 8) + vz);
check(slice(vx, 0, 2, 8) + (vz + slice(vy, 0, 2, 8)), slice(vx + vy, 0, 2, 8) + vz);
// Check that degenerate vector slices are not hoisted.
check(slice(vx, 0, 2, 1) + slice(vy, 0, 2, 1), slice(vx, 0, 2, 1) + slice(vy, 0, 2, 1));
check(slice(vx, 0, 2, 1) + (slice(vy, 0, 2, 1) + z), slice(vx, 0, 2, 1) + (slice(vy, 0, 2, 1) + z));
// Check slices are only hoisted when the lanes of the sliced vectors match.
check(slice(vx, 0, 2, 8) + slice(vw, 0, 2, 8), slice(vx, 0, 2, 8) + slice(vw, 0, 2, 8));
check(slice(vx, 0, 2, 8) + (slice(vw, 0, 2, 8) + vz), slice(vx, 0, 2, 8) + (slice(vw, 0, 2, 8) + vz));
}
{
// A predicated store with a provably-false predicate.
Expr pred = ramp(x * y + x * z, 2, 8) > 2;
Expr index = ramp(x + y, 1, 8);
Expr value = Load::make(index.type(), "f", index, Buffer<>(), Parameter(), const_true(index.type().lanes()), ModulusRemainder());
Stmt stmt = Store::make("f", value, index, Parameter(), pred, ModulusRemainder());
check(stmt, Evaluate::make(0));
}
auto make_allocation = [](const char *name, Type t, Stmt body) {
return Allocate::make(name, t.element_of(), MemoryType::Stack, {t.lanes()}, const_true(), body);
};
{
// A store completely out of bounds.
Expr index = ramp(-8, 1, 8);
Expr value = Broadcast::make(0, 8);
Stmt stmt = Store::make("f", value, index, Parameter(), const_true(8), ModulusRemainder(8, 0));
stmt = make_allocation("f", value.type(), stmt);
check(stmt, Evaluate::make(unreachable()));
}
{
// A store with one lane in bounds at the min.
Expr index = ramp(-7, 1, 8);
Expr value = Broadcast::make(0, 8);
Stmt stmt = Store::make("f", value, index, Parameter(), const_true(8), ModulusRemainder(0, -7));
stmt = make_allocation("f", value.type(), stmt);
check(stmt, stmt);
}
{
// A store with one lane in bounds at the max.
Expr index = ramp(7, 1, 8);
Expr value = Broadcast::make(0, 8);
Stmt stmt = Store::make("f", value, index, Parameter(), const_true(8), ModulusRemainder(0, 7));
stmt = make_allocation("f", value.type(), stmt);
check(stmt, stmt);
}
{
// A store completely out of bounds.
Expr index = ramp(8, 1, 8);
Expr value = Broadcast::make(0, 8);
Stmt stmt = Store::make("f", value, index, Parameter(), const_true(8), ModulusRemainder(8, 0));
stmt = make_allocation("f", value.type(), stmt);
check(stmt, Evaluate::make(unreachable()));
}
Expr bool_vector = Variable::make(Bool(4), "bool_vector");
Expr int_vector = Variable::make(Int(32, 4), "int_vector");
check(VectorReduce::make(VectorReduce::And, Broadcast::make(bool_vector, 4), 1),
VectorReduce::make(VectorReduce::And, bool_vector, 1));
check(VectorReduce::make(VectorReduce::Or, Broadcast::make(bool_vector, 4), 2),
VectorReduce::make(VectorReduce::Or, bool_vector, 2));
check(VectorReduce::make(VectorReduce::Min, Broadcast::make(int_vector, 4), 4),
int_vector);
check(VectorReduce::make(VectorReduce::Max, Broadcast::make(int_vector, 4), 8),
VectorReduce::make(VectorReduce::Max, Broadcast::make(int_vector, 4), 8));
}
void check_bounds() {
Expr x = Var("x"), y = Var("y"), z = Var("z"), w = Var("w");
check(min(Expr(7), 3), 3);
check(min(Expr(4.25f), 1.25f), 1.25f);
check(min(broadcast(x, 4), broadcast(y, 4)),
broadcast(min(x, y), 4));
check(min(x, x + 3), x);
check(min(x + 4, x), x);
check(min(x - 1, x + 2), x + (-1));
check(min(7, min(x, 3)), min(x, 3));
check(min(min(x, y), x), min(x, y));
check(min(min(x, y), y), min(x, y));
check(min(x, min(x, y)), min(x, y));
check(min(y, min(x, y)), min(x, y));
check(min(min(x, y) + 1, x), min(y + 1, x));
check(min(min(x, y) - (-1), x), min(y + 1, x));
check(min(min(x, y) + (-1), x), min(x, y) + (-1));
check(min(min(x, y) - 1, x), min(x, y) + (-1));
check(min(min(y, x) + 1, x), min(y + 1, x));
check(min(min(y, x) - (-1), x), min(y + 1, x));
check(min(min(y, x) + (-1), x), min(x, y) + (-1));
check(min(min(y, x) - 1, x), min(x, y) + (-1));
check(max(max(x, y) - 1, x), max(y + (-1), x));
check(max(max(x, y) + (-1), x), max(y + (-1), x));
check(max(max(x, y) + 1, x), max(x, y) + 1);
check(max(max(x, y) - (-1), x), max(x, y) + 1);
check(max(max(y, x) - 1, x), max(y + (-1), x));
check(max(max(y, x) + (-1), x), max(y + (-1), x));
check(max(max(y, x) + 1, x), max(x, y) + 1);
check(max(max(y, x) - (-1), x), max(x, y) + 1);
check(min(x, min(x, y) + 1), min(y + 1, x));
check(min(x, min(x, y) - (-1)), min(y + 1, x));
check(min(x, min(x, y) + (-1)), min(x, y) + (-1));
check(min(x, min(x, y) - 1), min(x, y) + (-1));
check(min(x, min(y, x) + 1), min(y + 1, x));
check(min(x, min(y, x) - (-1)), min(y + 1, x));
check(min(x, min(y, x) + (-1)), min(x, y) + (-1));
check(min(x, min(y, x) - 1), min(x, y) + (-1));
check(max(x, max(x, y) - 1), max(y + (-1), x));
check(max(x, max(x, y) + (-1)), max(y + (-1), x));
check(max(x, max(x, y) + 1), max(x, y) + 1);
check(max(x, max(x, y) - (-1)), max(x, y) + 1);
check(max(x, max(y, x) - 1), max(y + (-1), x));
check(max(x, max(y, x) + (-1)), max(y + (-1), x));
check(max(x, max(y, x) + 1), max(x, y) + 1);
check(max(x, max(y, x) - (-1)), max(x, y) + 1);
check(max(Expr(7), 3), 7);
check(max(Expr(4.25f), 1.25f), 4.25f);
check(max(broadcast(x, 4), broadcast(y, 4)),
broadcast(max(x, y), 4));
check(max(x, x + 3), x + 3);
check(max(x + 4, x), x + 4);
check(max(x - 1, x + 2), x + 2);
check(max(7, max(x, 3)), max(x, 7));
check(max(max(x, y), x), max(x, y));
check(max(max(x, y), y), max(x, y));
check(max(x, max(x, y)), max(x, y));
check(max(y, max(x, y)), max(x, y));
// Check that simplifier can recognise instances where the extremes of the
// datatype appear as constants in comparisons, Min and Max expressions.
// The result of min/max with extreme is known to be either the extreme or
// the other expression. The result of < or > comparison is known to be true or false.
check(x <= Int(32).max(), const_true());
check(cast(Int(16), x) >= Int(16).min(), const_true());
check(x < Int(32).min(), const_false());
check(min(cast(UInt(16), x), cast(UInt(16), 65535)), cast(UInt(16), x));
check(min(x, Int(32).max()), x);
check(min(Int(32).min(), x), Int(32).min());
check(max(cast(Int(8), x), cast(Int(8), -128)), cast(Int(8), x));
check(max(x, Int(32).min()), x);
check(max(x, Int(32).max()), Int(32).max());
// Check that non-extremes do not lead to incorrect simplification
check(max(cast(Int(8), x), cast(Int(8), -127)), max(cast(Int(8), x), make_const(Int(8), -127)));
// Some quaternary rules with cancellations
check((x + y) - (y + z), x - z);
check((x + y) - (y + z), x - z);
check((y + x) - (y + z), x - z);
check((y + x) - (y + z), x - z);
check((x - y) - (z - y), x - z);
check((y - z) - (y - x), x - z);
check((x + 3) / 4 - (x + 2) / 4, ((x + 2) % 4 + 1) / 4);
check(min(x + y, y + z), min(x, z) + y);
check(min(y + x, y + z), min(x, z) + y);
check(min(x + y, y + z), min(x, z) + y);
check(min(y + x, y + z), min(x, z) + y);
check(min(x, y) - min(y, x), 0);
check(max(x, y) - max(y, x), 0);
check(min(123 - x, 1 - x), 1 - x);
check(max(123 - x, 1 - x), 123 - x);
check(min(x * 43, y * 43), min(x, y) * 43);
check(max(x * 43, y * 43), max(x, y) * 43);
check(min(x * -43, y * -43), max(x, y) * -43);
check(max(x * -43, y * -43), min(x, y) * -43);
check(min(min(x, 4), y), min(min(x, y), 4));
check(max(max(x, 4), y), max(max(x, y), 4));
check(min(x * 8, 24), min(x, 3) * 8);
check(max(x * 8, 24), max(x, 3) * 8);
check(min(x * -8, 24), max(x, -3) * -8);
check(max(x * -8, 24), min(x, -3) * -8);
check(min(clamp(x, -10, 14), clamp(y, -10, 14)), clamp(min(x, y), -10, 14));
check(min(x / 4, y / 4), min(x, y) / 4);
check(max(x / 4, y / 4), max(x, y) / 4);
check(min(x / (-4), y / (-4)), max(x, y) / (-4));
check(max(x / (-4), y / (-4)), min(x, y) / (-4));
check(min(x / 4 + 2, y / 4), min(x + 8, y) / 4);
check(max(x / 4 + 2, y / 4), max(x + 8, y) / 4);
check(min(x / 4, y / 4 + 2), min(y + 8, x) / 4);
check(max(x / 4, y / 4 + 2), max(y + 8, x) / 4);
check(min(x / (-4) + 2, y / (-4)), max(x + -8, y) / (-4));
check(max(x / (-4) + 2, y / (-4)), min(x + -8, y) / (-4));
check(min(x / (-4), y / (-4) + 2), max(y + -8, x) / (-4));
check(max(x / (-4), y / (-4) + 2), min(y + -8, x) / (-4));
check(min(x * 4 + 8, y * 4), min(x + 2, y) * 4);
check(max(x * 4 + 8, y * 4), max(x + 2, y) * 4);
check(min(x * 4, y * 4 + 8), min(y + 2, x) * 4);
check(max(x * 4, y * 4 + 8), max(y + 2, x) * 4);
check(min(x * (-4) + 8, y * (-4)), max(x + -2, y) * (-4));
check(max(x * (-4) + 8, y * (-4)), min(x + -2, y) * (-4));
check(min(x * (-4), y * (-4) + 8), max(y + -2, x) * (-4));
check(max(x * (-4), y * (-4) + 8), min(y + -2, x) * (-4));
// Min and max of clamped expressions
check(min(clamp(x + 1, y, z), clamp(x - 1, y, z)), clamp(x + (-1), y, z));
check(max(clamp(x + 1, y, z), clamp(x - 1, y, z)), clamp(x + 1, y, z));
// Additions that cancel a term inside a min or max
check(x + min(y - x, z), min(x + z, y));
check(x + max(y - x, z), max(x + z, y));
check(min(y + (-2), z) + 2, min(z + 2, y));
check(max(y + (-2), z) + 2, max(z + 2, y));
// Min/Max distributive law
check(max(max(x, y), max(x, z)), max(max(y, z), x));
check(min(max(x, y), max(x, z)), max(min(y, z), x));
check(min(min(x, y), min(x, z)), min(min(y, z), x));
check(max(min(x, y), min(x, z)), min(max(y, z), x));
// Mins of expressions and rounded up versions of them
check(min(((x + 7) / 8) * 8, x), x);
check(min(x, ((x + 7) / 8) * 8), x);
check(max(((x + 7) / 8) * 8, x), ((x + 7) / 8) * 8);
check(max(x, ((x + 7) / 8) * 8), ((x + 7) / 8) * 8);
// And rounded down...
check(max((x / 8) * 8, x), x);
check(max(x, (x / 8) * 8), x);
check(min((x / 8) * 8, x), (x / 8) * 8);
check(min(x, (x / 8) * 8), (x / 8) * 8);
// "likely" marks which side of a containing min/max/select is the
// one to optimize for, so if the min/max/select gets simplified
// away, the likely should be stripped too.
check(min(x, likely(x)), x);
check(min(likely(x), x), x);
check(max(x, likely(x)), x);
check(max(likely(x), x), x);
check(select(x > y, likely(x), x), x);
check(select(x > y, x, likely(x)), x);
// Check constant-bounds reasoning works through likelies
check(min(4, likely(5)), 4);
check(min(7, likely(5)), 5);