Simplify_LT.cpp
#include "Simplify_Internal.h"
namespace Halide {
namespace Internal {
Expr Simplify::visit(const LT *op, ExprInfo *bounds) {
ExprInfo a_bounds, b_bounds;
Expr a = mutate(op->a, &a_bounds);
Expr b = mutate(op->b, &b_bounds);
const int lanes = op->type.lanes();
Type ty = a.type();
if (truths.count(op)) {
return const_true(lanes);
} else if (falsehoods.count(op)) {
return const_false(lanes);
}
if (may_simplify(ty)) {
// Prove or disprove using bounds analysis
if (a_bounds.max_defined && b_bounds.min_defined && a_bounds.max < b_bounds.min) {
return const_true(lanes);
}
if (a_bounds.min_defined && b_bounds.max_defined && a_bounds.min >= b_bounds.max) {
return const_false(lanes);
}
auto rewrite = IRMatcher::rewriter(IRMatcher::lt(a, b), op->type, ty);
// clang-format off
if (EVAL_IN_LAMBDA
(rewrite(c0 < c1, fold(c0 < c1)) ||
rewrite(x < x, false) ||
rewrite(x < ty.min(), false) ||
rewrite(ty.max() < x, false) ||
rewrite(max(x, y) < x, false) ||
rewrite(max(y, x) < x, false) ||
rewrite(x < min(x, y), false) ||
rewrite(x < min(y, x), false) ||
// From the simplifier synthesis project
rewrite((max(y, z) < min(x, y)), false) ||
rewrite((max(y, z) < min(y, x)), false) ||
rewrite((max(z, y) < min(x, y)), false) ||
rewrite((max(z, y) < min(y, x)), false) ||
// Comparisons of ramps and broadcasts. If the first
// and last lanes are provably < or >= the broadcast
// we can collapse the comparison.
(no_overflow(op->type) &&
(rewrite(ramp(x, c1) < broadcast(z), true, can_prove(x + fold(max(0, c1 * (lanes - 1))) < z, this)) ||
rewrite(ramp(x, c1) < broadcast(z), false, can_prove(x + fold(min(0, c1 * (lanes - 1))) >= z, this)) ||
rewrite(broadcast(z) < ramp(x, c1), true, can_prove(z < x + fold(min(0, c1 * (lanes - 1))), this)) ||
rewrite(broadcast(z) < ramp(x, c1), false, can_prove(z >= x + fold(max(0, c1 * (lanes - 1))), this)))))) {
return rewrite.result;
}
// clang-format on
// clang-format off
if (rewrite(broadcast(x) < broadcast(y), broadcast(x < y, lanes)) ||
(no_overflow(ty) && EVAL_IN_LAMBDA
(rewrite(ramp(x, y) < ramp(z, y), broadcast(x < z, lanes)) ||
// Move constants to the RHS
rewrite(x + c0 < y, x < y + fold(-c0)) ||
// Merge RHS constant additions with a constant LHS
rewrite(c0 < x + c1, fold(c0 - c1) < x) ||
// Normalize subtractions to additions to cut down on cases to consider
rewrite(x - y < z, x < z + y) ||
rewrite(z < x - y, z + y < x) ||
rewrite((x - y) + z < w, x + z < y + w) ||
rewrite(z + (x - y) < w, x + z < y + w) ||
rewrite(w < (x - y) + z, w + y < x + z) ||
rewrite(w < z + (x - y), w + y < x + z) ||
rewrite(((x - y) + z) + u < w, x + z + u < w + y) ||
rewrite((z + (x - y)) + u < w, x + z + u < w + y) ||
rewrite(u + ((x - y) + z) < w, x + z + u < w + y) ||
rewrite(u + (z + (x - y)) < w, x + z + u < w + y) ||
rewrite(w < ((x - y) + z) + u, w + y < x + z + u) ||
rewrite(w < (z + (x - y)) + u, w + y < x + z + u) ||
rewrite(w < u + ((x - y) + z), w + y < x + z + u) ||
rewrite(w < u + (z + (x - y)), w + y < x + z + u) ||
// Cancellations in linear expressions
// 1 < 2
rewrite(x < x + y, 0 < y) ||
rewrite(x < y + x, 0 < y) ||
// 2 < 1
rewrite(x + y < x, y < 0) ||
rewrite(y + x < x, y < 0) ||
// 1 < 3
rewrite(x < (x + y) + z, 0 < y + z) ||
rewrite(x < (y + x) + z, 0 < y + z) ||
rewrite(x < z + (x + y), 0 < z + y) ||
rewrite(x < z + (y + x), 0 < z + y) ||
// 3 < 1
rewrite((x + y) + z < x, y + z < 0) ||
rewrite((y + x) + z < x, y + z < 0) ||
rewrite(z + (x + y) < x, z + y < 0) ||
rewrite(z + (y + x) < x, z + y < 0) ||
// 2 < 2
rewrite(x + y < x + z, y < z) ||
rewrite(x + y < z + x, y < z) ||
rewrite(y + x < x + z, y < z) ||
rewrite(y + x < z + x, y < z) ||
// 3 < 2
rewrite((x + y) + w < x + z, y + w < z) ||
rewrite((y + x) + w < x + z, y + w < z) ||
rewrite(w + (x + y) < x + z, y + w < z) ||
rewrite(w + (y + x) < x + z, y + w < z) ||
rewrite((x + y) + w < z + x, y + w < z) ||
rewrite((y + x) + w < z + x, y + w < z) ||
rewrite(w + (x + y) < z + x, y + w < z) ||
rewrite(w + (y + x) < z + x, y + w < z) ||
// 2 < 3
rewrite(x + z < (x + y) + w, z < y + w) ||
rewrite(x + z < (y + x) + w, z < y + w) ||
rewrite(x + z < w + (x + y), z < y + w) ||
rewrite(x + z < w + (y + x), z < y + w) ||
rewrite(z + x < (x + y) + w, z < y + w) ||
rewrite(z + x < (y + x) + w, z < y + w) ||
rewrite(z + x < w + (x + y), z < y + w) ||
rewrite(z + x < w + (y + x), z < y + w) ||
// 3 < 3
rewrite((x + y) + w < (x + z) + u, y + w < z + u) ||
rewrite((y + x) + w < (x + z) + u, y + w < z + u) ||
rewrite((x + y) + w < (z + x) + u, y + w < z + u) ||
rewrite((y + x) + w < (z + x) + u, y + w < z + u) ||
rewrite(w + (x + y) < (x + z) + u, y + w < z + u) ||
rewrite(w + (y + x) < (x + z) + u, y + w < z + u) ||
rewrite(w + (x + y) < (z + x) + u, y + w < z + u) ||
rewrite(w + (y + x) < (z + x) + u, y + w < z + u) ||
rewrite((x + y) + w < u + (x + z), y + w < z + u) ||
rewrite((y + x) + w < u + (x + z), y + w < z + u) ||
rewrite((x + y) + w < u + (z + x), y + w < z + u) ||
rewrite((y + x) + w < u + (z + x), y + w < z + u) ||
rewrite(w + (x + y) < u + (x + z), y + w < z + u) ||
rewrite(w + (y + x) < u + (x + z), y + w < z + u) ||
rewrite(w + (x + y) < u + (z + x), y + w < z + u) ||
rewrite(w + (y + x) < u + (z + x), y + w < z + u) ||
// Cancel a multiplication
rewrite(x * c0 < y * c0, x < y, c0 > 0) ||
rewrite(x * c0 < y * c0, y < x, c0 < 0) ||
(ty.is_int() && rewrite(x * c0 < c1, x < fold((c1 + c0 - 1) / c0), c0 > 0)) ||
(ty.is_float() && rewrite(x * c0 < c1, x < fold(c1 / c0), c0 > 0)) ||
rewrite(c1 < x * c0, fold(c1 / c0) < x, c0 > 0) ||
// Multiply-out a division
rewrite(x / c0 < c1, x < c1 * c0, c0 > 0) ||
(ty.is_int() && rewrite(c0 < x / c1, fold((c0 + 1) * c1 - 1) < x, c1 > 0)) ||
(ty.is_float() && rewrite(c0 < x / c1, fold(c0 * c1) < x, c1 > 0)) ||
// We want to break max(x, y) < z into x < z && y <
// z in cases where one of those two terms is going
// to fold.
rewrite(min(x + c0, y) < x + c1, y < x + c1 || fold(c0 < c1)) ||
rewrite(min(y, x + c0) < x + c1, y < x + c1 || fold(c0 < c1)) ||
rewrite(max(x + c0, y) < x + c1, y < x + c1 && fold(c0 < c1)) ||
rewrite(max(y, x + c0) < x + c1, y < x + c1 && fold(c0 < c1)) ||
rewrite(x < min(x + c0, y) + c1, x < y + c1 && fold(0 < c0 + c1)) ||
rewrite(x < min(y, x + c0) + c1, x < y + c1 && fold(0 < c0 + c1)) ||
rewrite(x < max(x + c0, y) + c1, x < y + c1 || fold(0 < c0 + c1)) ||
rewrite(x < max(y, x + c0) + c1, x < y + c1 || fold(0 < c0 + c1)) ||
// Special cases where c0 == 0
rewrite(min(x, y) < x + c1, y < x + c1 || fold(0 < c1)) ||
rewrite(min(y, x) < x + c1, y < x + c1 || fold(0 < c1)) ||
rewrite(max(x, y) < x + c1, y < x + c1 && fold(0 < c1)) ||
rewrite(max(y, x) < x + c1, y < x + c1 && fold(0 < c1)) ||
rewrite(x < min(x, y) + c1, x < y + c1 && fold(0 < c1)) ||
rewrite(x < min(y, x) + c1, x < y + c1 && fold(0 < c1)) ||
rewrite(x < max(x, y) + c1, x < y + c1 || fold(0 < c1)) ||
rewrite(x < max(y, x) + c1, x < y + c1 || fold(0 < c1)) ||
// Special cases where c1 == 0
rewrite(min(x + c0, y) < x, y < x || fold(c0 < 0)) ||
rewrite(min(y, x + c0) < x, y < x || fold(c0 < 0)) ||
rewrite(max(x + c0, y) < x, y < x && fold(c0 < 0)) ||
rewrite(max(y, x + c0) < x, y < x && fold(c0 < 0)) ||
rewrite(x < min(x + c0, y), x < y && fold(0 < c0)) ||
rewrite(x < min(y, x + c0), x < y && fold(0 < c0)) ||
rewrite(x < max(x + c0, y), x < y || fold(0 < c0)) ||
rewrite(x < max(y, x + c0), x < y || fold(0 < c0)) ||
// Special cases where c0 == c1 == 0
rewrite(min(x, y) < x, y < x) ||
rewrite(min(y, x) < x, y < x) ||
rewrite(x < max(x, y), x < y) ||
rewrite(x < max(y, x), x < y) ||
// Special case where x is constant
rewrite(min(y, c0) < c1, y < c1 || fold(c0 < c1)) ||
rewrite(max(y, c0) < c1, y < c1 && fold(c0 < c1)) ||
rewrite(c1 < min(y, c0), c1 < y && fold(c1 < c0)) ||
rewrite(c1 < max(y, c0), c1 < y || fold(c1 < c0)) ||
// Cases where we can remove a min on one side because
// one term dominates another. These rules were
// synthesized then extended by hand.
rewrite(min(z, y) < min(x, y), z < min(x, y)) ||
rewrite(min(z, y) < min(y, x), z < min(y, x)) ||
rewrite(min(z, y) < min(x, y + c0), min(z, y) < x, c0 > 0) ||
rewrite(min(z, y) < min(y + c0, x), min(z, y) < x, c0 > 0) ||
rewrite(min(z, y + c0) < min(x, y), min(z, y + c0) < x, c0 < 0) ||
rewrite(min(z, y + c0) < min(y, x), min(z, y + c0) < x, c0 < 0) ||
rewrite(min(y, z) < min(x, y), z < min(x, y)) ||
rewrite(min(y, z) < min(y, x), z < min(y, x)) ||
rewrite(min(y, z) < min(x, y + c0), min(z, y) < x, c0 > 0) ||
rewrite(min(y, z) < min(y + c0, x), min(z, y) < x, c0 > 0) ||
rewrite(min(y + c0, z) < min(x, y), min(z, y + c0) < x, c0 < 0) ||
rewrite(min(y + c0, z) < min(y, x), min(z, y + c0) < x, c0 < 0) ||
// Equivalents with max
rewrite(max(z, y) < max(x, y), max(z, y) < x) ||
rewrite(max(z, y) < max(y, x), max(z, y) < x) ||
rewrite(max(z, y) < max(x, y + c0), max(z, y) < x, c0 < 0) ||
rewrite(max(z, y) < max(y + c0, x), max(z, y) < x, c0 < 0) ||
rewrite(max(z, y + c0) < max(x, y), max(z, y + c0) < x, c0 > 0) ||
rewrite(max(z, y + c0) < max(y, x), max(z, y + c0) < x, c0 > 0) ||
rewrite(max(y, z) < max(x, y), max(z, y) < x) ||
rewrite(max(y, z) < max(y, x), max(z, y) < x) ||
rewrite(max(y, z) < max(x, y + c0), max(z, y) < x, c0 < 0) ||
rewrite(max(y, z) < max(y + c0, x), max(z, y) < x, c0 < 0) ||
rewrite(max(y + c0, z) < max(x, y), max(z, y + c0) < x, c0 > 0) ||
rewrite(max(y + c0, z) < max(y, x), max(z, y + c0) < x, c0 > 0) ||
// Comparisons with selects:
// x < select(c, t, f) == c && (x < t) || !c && (x < f)
// This is profitable when x < t or x < f is statically provable
rewrite(x < select(y, x + c0, z), !y && (x < z), c0 <= 0) ||
rewrite(x < select(y, x + c0, z), y || (x < z), c0 > 0) ||
rewrite(x < select(y, z, x + c0), y && (x < z), c0 <= 0) ||
rewrite(x < select(y, z, x + c0), !y || (x < z), c0 > 0) ||
rewrite(x < select(y, x + c0, z) + c1, !y && (x < z + c1), c0 + c1 <= 0) ||
rewrite(x < select(y, x + c0, z) + c1, y || (x < z + c1), c0 + c1 > 0) ||
rewrite(x < select(y, z, x + c0) + c1, y && (x < z + c1), c0 + c1 <= 0) ||
rewrite(x < select(y, z, x + c0) + c1, !y || (x < z + c1), c0 + c1 > 0) ||
rewrite(select(y, x + c0, z) < x, !y && (z < x), c0 >= 0) ||
rewrite(select(y, x + c0, z) < x, y || (z < x), c0 < 0) ||
rewrite(select(y, z, x + c0) < x, y && (z < x), c0 >= 0) ||
rewrite(select(y, z, x + c0) < x, !y || (z < x), c0 < 0) ||
rewrite(select(y, x + c0, z) < x + c1, !y && (z < x + c1), c0 >= c1) ||
rewrite(select(y, x + c0, z) < x + c1, y || (z < x + c1), c0 < c1) ||
rewrite(select(y, z, x + c0) < x + c1, y && (z < x + c1), c0 >= c1) ||
rewrite(select(y, z, x + c0) < x + c1, !y || (z < x + c1), c0 < c1) ||
// Normalize comparison of ramps to a comparison of a ramp and a broadacst
rewrite(ramp(x, y) < ramp(z, w), ramp(x - z, y - w, lanes) < 0))) ||
(no_overflow_int(ty) && EVAL_IN_LAMBDA
(rewrite(x * c0 < y * c1, x < y * fold(c1 / c0), c1 % c0 == 0 && c0 > 0) ||
rewrite(x * c0 < y * c1, x * fold(c0 / c1) < y, c0 % c1 == 0 && c1 > 0) ||
rewrite(x * c0 < y * c0 + c1, x < y + fold((c1 + c0 - 1)/c0), c0 > 0) ||
rewrite(x * c0 + c1 < y * c0, x + fold(c1/c0) < y, c0 > 0) ||
// Comparison of stair-step functions. The basic transformation is:
// ((x + y)/c1)*c1 < x
// = (x + y) - (x + y) % c1 < x (when c1 > 0)
// = y - (x + y) % c1 < 0
// = y < (x + y) % c1
// This cancels x but duplicates y, so we only do it when y is a constant.
// A more general version with extra terms w and z
rewrite(((x + c0)/c1)*c1 + w < x + z, (w + c0) < ((x + c0) % c1) + z, c1 > 0) ||
rewrite(w + ((x + c0)/c1)*c1 < x + z, (w + c0) < ((x + c0) % c1) + z, c1 > 0) ||
rewrite(((x + c0)/c1)*c1 + w < z + x, (w + c0) < ((x + c0) % c1) + z, c1 > 0) ||
rewrite(w + ((x + c0)/c1)*c1 < z + x, (w + c0) < ((x + c0) % c1) + z, c1 > 0) ||
rewrite(x + z < ((x + c0)/c1)*c1 + w, ((x + c0) % c1) + z < w + c0, c1 > 0) ||
rewrite(x + z < w + ((x + c0)/c1)*c1, ((x + c0) % c1) + z < w + c0, c1 > 0) ||
rewrite(z + x < ((x + c0)/c1)*c1 + w, ((x + c0) % c1) + z < w + c0, c1 > 0) ||
rewrite(z + x < w + ((x + c0)/c1)*c1, ((x + c0) % c1) + z < w + c0, c1 > 0) ||
// w = 0
rewrite(((x + c0)/c1)*c1 < x + z, c0 < ((x + c0) % c1) + z, c1 > 0) ||
rewrite(((x + c0)/c1)*c1 < z + x, c0 < ((x + c0) % c1) + z, c1 > 0) ||
rewrite(x + z < ((x + c0)/c1)*c1, ((x + c0) % c1) + z < c0, c1 > 0) ||
rewrite(z + x < ((x + c0)/c1)*c1, ((x + c0) % c1) + z < c0, c1 > 0) ||
// z = 0
rewrite(((x + c0)/c1)*c1 + w < x, (w + c0) < ((x + c0) % c1), c1 > 0) ||
rewrite(w + ((x + c0)/c1)*c1 < x, (w + c0) < ((x + c0) % c1), c1 > 0) ||
rewrite(x < ((x + c0)/c1)*c1 + w, ((x + c0) % c1) < w + c0, c1 > 0) ||
rewrite(x < w + ((x + c0)/c1)*c1, ((x + c0) % c1) < w + c0, c1 > 0) ||
// c0 = 0
rewrite((x/c1)*c1 + w < x + z, w < (x % c1) + z, c1 > 0) ||
rewrite(w + (x/c1)*c1 < x + z, w < (x % c1) + z, c1 > 0) ||
rewrite((x/c1)*c1 + w < z + x, w < (x % c1) + z, c1 > 0) ||
rewrite(w + (x/c1)*c1 < z + x, w < (x % c1) + z, c1 > 0) ||
rewrite(x + z < (x/c1)*c1 + w, (x % c1) + z < w, c1 > 0) ||
rewrite(x + z < w + (x/c1)*c1, (x % c1) + z < w, c1 > 0) ||
rewrite(z + x < (x/c1)*c1 + w, (x % c1) + z < w, c1 > 0) ||
rewrite(z + x < w + (x/c1)*c1, (x % c1) + z < w, c1 > 0) ||
// w = 0, z = 0
rewrite(((x + c0)/c1)*c1 < x, c0 < ((x + c0) % c1), c1 > 0) ||
rewrite(x < ((x + c0)/c1)*c1, ((x + c0) % c1) < c0, c1 > 0) ||
// w = 0, c0 = 0
rewrite((x/c1)*c1 < x + z, 0 < (x % c1) + z, c1 > 0) ||
rewrite((x/c1)*c1 < z + x, 0 < (x % c1) + z, c1 > 0) ||
rewrite(x + z < (x/c1)*c1, (x % c1) + z < 0, c1 > 0) ||
rewrite(z + x < (x/c1)*c1, (x % c1) + z < 0, c1 > 0) ||
// z = 0, c0 = 0
rewrite((x/c1)*c1 + w < x, w < (x % c1), c1 > 0) ||
rewrite(w + (x/c1)*c1 < x, w < (x % c1), c1 > 0) ||
rewrite(x < (x/c1)*c1 + w, (x % c1) < w, c1 > 0) ||
rewrite(x < w + (x/c1)*c1, (x % c1) < w, c1 > 0) ||
// z = 0, c0 = 0, w = 0
rewrite((x/c1)*c1 < x, (x % c1) != 0, c1 > 0) ||
rewrite(x < (x/c1)*c1, false, c1 > 0) ||
// Cancel a division
rewrite((x + c1)/c0 < (x + c2)/c0, false, c0 > 0 && c1 >= c2) ||
rewrite((x + c1)/c0 < (x + c2)/c0, true, c0 > 0 && c1 <= c2 - c0) ||
// c1 == 0
rewrite(x/c0 < (x + c2)/c0, false, c0 > 0 && 0 >= c2) ||
rewrite(x/c0 < (x + c2)/c0, true, c0 > 0 && 0 <= c2 - c0) ||
// c2 == 0
rewrite((x + c1)/c0 < x/c0, false, c0 > 0 && c1 >= 0) ||
rewrite((x + c1)/c0 < x/c0, true, c0 > 0 && c1 <= 0 - c0) ||
// The addition on the right could be outside
rewrite((x + c1)/c0 < x/c0 + c2, false, c0 > 0 && c1 >= c2 * c0) ||
rewrite((x + c1)/c0 < x/c0 + c2, true, c0 > 0 && c1 <= c2 * c0 - c0) ||
// With a confounding max or min
rewrite((x + c1)/c0 < (min(x/c0, y) + c2), false, c0 > 0 && c1 >= c2 * c0) ||
rewrite((x + c1)/c0 < (max(x/c0, y) + c2), true, c0 > 0 && c1 <= c2 * c0 - c0) ||
rewrite((x + c1)/c0 < min((x + c2)/c0, y), false, c0 > 0 && c1 >= c2) ||
rewrite((x + c1)/c0 < max((x + c2)/c0, y), true, c0 > 0 && c1 <= c2 - c0) ||
rewrite((x + c1)/c0 < min(x/c0, y), false, c0 > 0 && c1 >= 0) ||
rewrite((x + c1)/c0 < max(x/c0, y), true, c0 > 0 && c1 <= 0 - c0) ||
rewrite((x + c1)/c0 < (min(y, x/c0) + c2), false, c0 > 0 && c1 >= c2 * c0) ||
rewrite((x + c1)/c0 < (max(y, x/c0) + c2), true, c0 > 0 && c1 <= c2 * c0 - c0) ||
rewrite((x + c1)/c0 < min(y, (x + c2)/c0), false, c0 > 0 && c1 >= c2) ||
rewrite((x + c1)/c0 < max(y, (x + c2)/c0), true, c0 > 0 && c1 <= c2 - c0) ||
rewrite((x + c1)/c0 < min(y, x/c0), false, c0 > 0 && c1 >= 0) ||
rewrite((x + c1)/c0 < max(y, x/c0), true, c0 > 0 && c1 <= 0 - c0) ||
rewrite(max((x + c2)/c0, y) < (x + c1)/c0, false, c0 > 0 && c2 >= c1) ||
rewrite(min((x + c2)/c0, y) < (x + c1)/c0, true, c0 > 0 && c2 <= c1 - c0) ||
rewrite(max(x/c0, y) < (x + c1)/c0, false, c0 > 0 && 0 >= c1) ||
rewrite(min(x/c0, y) < (x + c1)/c0, true, c0 > 0 && 0 <= c1 - c0) ||
rewrite(max(y, (x + c2)/c0) < (x + c1)/c0, false, c0 > 0 && c2 >= c1) ||
rewrite(min(y, (x + c2)/c0) < (x + c1)/c0, true, c0 > 0 && c2 <= c1 - c0) ||
rewrite(max(y, x/c0) < (x + c1)/c0, false, c0 > 0 && 0 >= c1) ||
rewrite(min(y, x/c0) < (x + c1)/c0, true, c0 > 0 && 0 <= c1 - c0) ||
// Same as above with c1 outside the division, with redundant cases removed.
rewrite(max((x + c2)/c0, y) < x/c0 + c1, false, c0 > 0 && c2 >= c1 * c0) ||
rewrite(min((x + c2)/c0, y) < x/c0 + c1, true, c0 > 0 && c2 <= c1 * c0 - c0) ||
rewrite(max(y, (x + c2)/c0) < x/c0 + c1, false, c0 > 0 && c2 >= c1 * c0) ||
rewrite(min(y, (x + c2)/c0) < x/c0 + c1, true, c0 > 0 && c2 <= c1 * c0 - c0) ||
// Same as above with c1 = 0 and the predicates and redundant cases simplified accordingly.
rewrite(x/c0 < min((x + c2)/c0, y), false, c0 > 0 && c2 < 0) ||
rewrite(x/c0 < max((x + c2)/c0, y), true, c0 > 0 && c0 <= c2) ||
rewrite(x/c0 < min(y, (x + c2)/c0), false, c0 > 0 && c2 < 0) ||
rewrite(x/c0 < max(y, (x + c2)/c0), true, c0 > 0 && c0 <= c2) ||
rewrite(max((x + c2)/c0, y) < x/c0, false, c0 > 0 && c2 >= 0) ||
rewrite(min((x + c2)/c0, y) < x/c0, true, c0 > 0 && c2 + c0 <= 0) ||
rewrite(max(y, (x + c2)/c0) < x/c0, false, c0 > 0 && c2 >= 0) ||
rewrite(min(y, (x + c2)/c0) < x/c0, true, c0 > 0 && c2 + c0 <= 0) ||
// Comparison of two mins/maxes that don't cancel when subtracted
rewrite(min(x, c0) < min(x, c1), false, c0 >= c1) ||
rewrite(min(x, c0) < min(x, c1) + c2, false, c0 >= c1 + c2 && c2 <= 0) ||
rewrite(max(x, c0) < max(x, c1), false, c0 >= c1) ||
rewrite(max(x, c0) < max(x, c1) + c2, false, c0 >= c1 + c2 && c2 <= 0) ||
// Comparison of aligned ramps can simplify to a comparison of the base
rewrite(ramp(x * c3 + c2, c1) < broadcast(z * c0),
broadcast(x * fold(c3/c0) + fold(c2/c0) < z, lanes),
c0 > 0 && (c3 % c0 == 0) &&
(c2 % c0) + c1 * (lanes - 1) < c0 &&
(c2 % c0) + c1 * (lanes - 1) >= 0) ||
// c2 = 0
rewrite(ramp(x * c3, c1) < broadcast(z * c0),
broadcast(x * fold(c3/c0) < z, lanes),
c0 > 0 && (c3 % c0 == 0) &&
c1 * (lanes - 1) < c0 &&
c1 * (lanes - 1) >= 0)))) {
return mutate(rewrite.result, bounds);
}
// clang-format on
}
if (a.same_as(op->a) && b.same_as(op->b)) {
return op;
} else {
return LT::make(a, b);
}
}
// The other comparison operators redirect to the less-than operator
Expr Simplify::visit(const LE *op, ExprInfo *bounds) {
if (!may_simplify(op->a.type())) {
Expr a = mutate(op->a, nullptr);
Expr b = mutate(op->b, nullptr);
if (a.same_as(op->a) && b.same_as(op->b)) {
return op;
} else {
return LE::make(a, b);
}
}
Expr mutated = mutate(!(op->b < op->a), bounds);
if (const LE *le = mutated.as<LE>()) {
Expr a = le->a, b = le->b;
// Synthesized rules
#if USE_SYNTHESIZED_RULES
if (no_overflow_int(a.type())) {
auto rewrite = IRMatcher::rewriter(IRMatcher::le(a, b), op->type, a.type());
if (rewrite((x <= max(max(y, x), z)), true) ||
rewrite((x <= max(max(x, y), z)), true) ||
rewrite((x <= select((y < z), w, x)), x <= w || z <= y) ||
rewrite(((x + y) <= min(z, (w + y))), (x <= min((z - y), w))) ||
rewrite(((x + y) <= min(z, (y + w))), (x <= min((z - y), w))) ||
rewrite(((x + y) <= min((y + z), w)), (x <= min((w - y), z))) ||
rewrite(((x + y) <= min((z + y), w)), (x <= min((w - y), z))) ||
rewrite(((x + y) <= max(z, (w + x))), (y <= max((z - x), w))) ||
rewrite(((x + y) <= max(z, (w + y))), (x <= max((z - y), w))) ||
rewrite(((x + y) <= max((z + y), w)), (x <= max((w - y), z))) ||
rewrite(((x + y) <= (max(z, (w + y)) + u)), (x <= (max((z - y), w) + u))) ||
rewrite(((x + (y * z)) <= (w * z)), (x <= ((w - y) * z))) ||
rewrite((((x + y) + z) <= y), ((x + z) <= 0)) ||
rewrite((min(x, y) <= min(x, z)), (min(x, y) <= z)) ||
rewrite((min(x, y) <= min(z, y)), (min(x, y) <= z)) ||
rewrite((min(x, (y + z)) <= (y + w)), (min((x - y), z) <= w)) ||
rewrite((min(x, (y + z)) <= (z + w)), (min((x - z), y) <= w)) ||
rewrite((min((x + y), z) <= (x + w)), (min((z - x), y) <= w)) ||
rewrite((min((x + y), z) <= (y + w)), (min((z - y), x) <= w)) ||
rewrite((min(min(x, y), z) <= x), true) ||
rewrite((min(min(x, y), z) <= y), true) ||
rewrite((max(x, y) <= max(z, y)), (x <= max(y, z))) ||
rewrite((select((x < y), z, w) <= w), z <= w || y <= x) ||
rewrite((min(x, (y + z)) <= (w + z)), (min((x - z), y) <= w)) ||
rewrite((min((x + y), z) <= (w + y)), (min((z - y), x) <= w)) ||
rewrite((min(min(x, y), z) <= min(w, y)), (min(min(x, y), z) <= w)) ||
rewrite(((min(x, y) + z) <= (max(w, z) + y)), const_true()) ||
rewrite(min(x, y) <= min(y, x), true) ||
rewrite((x <= select((y < z), x, w)), ((x <= w) || (y < z))) ||
rewrite((((x + y) + z) <= min((y + w), u)), ((x + z) <= min((u - y), w))) ||
rewrite((((x + y) + z) <= max((y + w), u)), ((x + z) <= max((u - y), w))) ||
rewrite(((((x + y) + z) + w) <= (y + u)), (((x + w) + z) <= u)) ||
rewrite(((min(x, y) + z) <= (min(z, w) + x)), (z <= (max((x - y), 0) + w))) ||
rewrite(((x * y) <= ((y * z) + w)), (((x - z) * y) <= w)) ||
rewrite((min(x, y) <= min(y, z)), (min(x, y) <= z)) ||
rewrite((min(x, y) <= min(min(x, z), w)), (min(x, y) <= min(z, w))) ||
rewrite((min(x, y) <= max(x, y)), true) ||
//rewrite((min((x + y), z) <= ((x + y) + w)), (min((x + y), z) <= ((x + y) + w))) ||
rewrite((min(min(x, y), z) <= min(x, w)), (min(min(x, z), y) <= w)) ||
rewrite((min(max(x, y), z) <= max(x, w)), (min(y, z) <= max(x, w))) ||
rewrite((((x + (y + z)) + c0) <= (w + z)), (((x + y) + c0) <= w)) ||
rewrite(((min((min(x, c0) + y), z) + c1) <= y), true, ((c0 + c1) == 0)) ||
rewrite((((x + y) * c1) <= (z + (x * c1))), ((y * c1) <= z)) ||
rewrite((((x + y) * z) <= (w + (y * z))), ((x * z) <= w)) ||
rewrite((min(x, y) <= max(y, x)), true) ||
rewrite((min((x * y), c0) <= (min(x, c1) * y)), true, ((0 <= c1) && (c0 <= 0))) ||
rewrite((max(x, c0) <= max(y, 0)), (x <= max(y, 0)), (c0 <= 0)) ||
rewrite((select((x < y), max(z, w), z) <= max(z, w)), true) ||
rewrite((min((min(x, y) * c1), z) <= (x * c1)), true, (0 <= c1)) ||
rewrite((min(min((min(x, c0) + y), z), w) <= y), true, (c0 <= 0)) ||
rewrite(((min((min(x, y) + c0), z) + c1) <= y), true, ((c0 + c1) <= 0)) ||
rewrite((x <= (max(y, c0) + min(x, z))), (x <= (max(y, c0) + z)), (0 <= c0)) ||
rewrite(((x + c0) <= (min(x, c1) + y)), ((max(x, c1) + fold((c0 - c1))) <= y)) ||
// c0 unbound. Shouldn't have eq constraints
// rewrite((((max(x, 0) + y) + c1) <= x), (y < (min(x, c0) + 2)), ((c0 == 0) && (c1 == -1))) ||
rewrite((((max(x, 0) + y) + z) <= x), ((y + z) <= min(x, 0))) ||
rewrite(((max(x, c0) + c1) <= max(y, c2)), ((x + c1) <= max(y, c2)), (((c0 != 0) && ((c1 / c0) == 0)) && ((c0 <= 0) && (0 <= c2)))) || // weird one
rewrite(((x * c1) <= ((y * c1) + z)), (((x - y) * c1) <= z)) ||
rewrite((min(x, y) <= min(min(y, z), c0)), (min(x, y) <= min(z, c0))) ||
rewrite((min((x + y), z) <= (w + x)), (min((z - x), y) <= w)) ||
rewrite((min(((x * c0) + y), z) <= y), ((z <= y) || (x < 1)), (0 < c0)) ||
rewrite((min(((x * c0) + y), z) <= y), (select((y < z), 1, 0) <= select((0 < x), 0, 1)), (0 < c0)) || // crappy form
rewrite(((x + (y * z)) <= ((w + z) * y)), (x <= (w * y))) ||
rewrite(((max(x, 0) + min(y, 1)) <= y), (x < max(y, 1))) || // Too specific?
rewrite((((x + c0) / c1) <= y), (x <= (y * c1)), (((0 < c1) && (c1 < 16)) && (c0 == (c1 + -1)))) ||
rewrite((((c0 - x) / c3) <= ((c2 - x) / c3)), true, (((0 < c3) && (c3 < 16)) && (c0 <= c2))) ||
rewrite(((min(x, y) / c0) <= (y / c1)), true, ((((c1 != 0) && ((c0 % c1) == 0)) && (c0 <= c1)) && ((((0 < c0) && (c0 < 16)) && (0 < c1)) && (c1 < 16)))) || // Crazy predicate. c0 % c1 == 0 && c0 <= c1 should just be c0 == c1. TODO: throw all the predicates into the system too
rewrite((min(x, y) <= max(x, z)), true) ||
rewrite((min(x, y) <= max(y, z)), true) ||
rewrite((min((min(x, c0) + y), z) <= y), true, (c0 <= 0)) ||
rewrite((min((x * y), c0) <= (min(x, c1) * y)), true, ((c0 <= c1) && (0 <= c0))) ||
rewrite((x <= ((y + c0) / c1)), ((x * c1) < y), (((-1 <= c0) && (c0 < 0)) && ((0 < c1) && (c1 < 16)))) || // Predicate implies c0 == 1
rewrite(((x * c1) <= max((y * c1), c2)), (x <= max(y, fold((c2 / c1)))), ((c1 < 16) && (0 < c1))) || // < 16
rewrite((min((x * c2), c1) <= (y * c2)), (min(x, fold((((c1 + -1) / c2) + 1))) <= y), ((c2 < 16) && (0 < c2))) || // < 16
rewrite(((min(x, c0) + c1) <= min(y, 0)), (min(x, c0) < (y + fold((c0 - c1)))), ((c1 < 0) && (c0 == 1))) || // Should have subs'd in c0
rewrite(((max(x, c0) + c1) <= max(y, c0)), ((x + c1) <= max(y, c0)), (c1 <= 0)) ||
// From google list
rewrite((min(x, y) <= max(z, y)), true) ||
rewrite((max(x, y) <= max(x, z)), (y <= max(x, z))) ||
rewrite((min(x, y) <= min(z, x)), (min(x, y) <= z)) ||
rewrite(((min(x, y) + z) <= max(w, (z + y))), true) ||
rewrite((min(max(x, y), z) <= max(min(y, z), w)), (min(x, z) <= max(w, y))) ||
rewrite(((x + 1) <= y), (x < y)) ||
false) {
return mutate(rewrite.result, bounds);
}
}
#endif
if (a.same_as(op->a) && b.same_as(op->b)) {
return op;
}
}
return mutated;
}
Expr Simplify::visit(const GT *op, ExprInfo *bounds) {
if (!may_simplify(op->a.type())) {
Expr a = mutate(op->a, nullptr);
Expr b = mutate(op->b, nullptr);
if (a.same_as(op->a) && b.same_as(op->b)) {
return op;
} else {
return GT::make(a, b);
}
}
return mutate(op->b < op->a, bounds);
}
Expr Simplify::visit(const GE *op, ExprInfo *bounds) {
if (!may_simplify(op->a.type())) {
Expr a = mutate(op->a, nullptr);
Expr b = mutate(op->b, nullptr);
if (a.same_as(op->a) && b.same_as(op->b)) {
return op;
} else {
return GE::make(a, b);
}
}
return mutate(!(op->a < op->b), bounds);
}
} // namespace Internal
} // namespace Halide
