blob: 74662a9bec6b9a890508e0b1763ed3298d8cc1eb [file] [log] [blame]
/*
* Copyright (C) 2021 Collabora, Ltd.
*
* Permission is hereby granted, free of charge, to any person obtaining a
* copy of this software and associated documentation files (the "Software"),
* to deal in the Software without restriction, including without limitation
* the rights to use, copy, modify, merge, publish, distribute, sublicense,
* and/or sell copies of the Software, and to permit persons to whom the
* Software is furnished to do so, subject to the following conditions:
*
* The above copyright notice and this permission notice (including the next
* paragraph) shall be included in all copies or substantial portions of the
* Software.
*
* THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
* IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
* FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
* THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
* LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
* OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
* SOFTWARE.
*/
#include "bi_builder.h"
#include "bi_test.h"
#include "compiler.h"
#include <gtest/gtest.h>
static void
bi_optimizer(bi_context *ctx)
{
bi_opt_mod_prop_forward(ctx);
bi_opt_mod_prop_backward(ctx);
bi_opt_dce(ctx, true);
}
/* Define reg first so it has a consistent variable index, and pass it to an
* instruction that cannot be dead code eliminated so the program is nontrivial.
*/
#define CASE(instr, expected) \
INSTRUCTION_CASE( \
{ \
UNUSED bi_index reg = bi_temp(b->shader); \
instr; \
bi_kaboom(b, reg); \
}, \
{ \
UNUSED bi_index reg = bi_temp(b->shader); \
expected; \
bi_kaboom(b, reg); \
}, \
bi_optimizer);
#define NEGCASE(instr) CASE(instr, instr)
class Optimizer : public testing::Test {
protected:
Optimizer()
{
mem_ctx = ralloc_context(NULL);
x = bi_register(1);
y = bi_register(2);
negabsx = bi_neg(bi_abs(x));
}
~Optimizer()
{
ralloc_free(mem_ctx);
}
void *mem_ctx;
bi_index x;
bi_index y;
bi_index negabsx;
};
TEST_F(Optimizer, FusedFABSNEG)
{
CASE(bi_fadd_f32_to(b, reg, bi_fabsneg_f32(b, bi_abs(x)), y),
bi_fadd_f32_to(b, reg, bi_abs(x), y));
CASE(bi_fadd_f32_to(b, reg, bi_fabsneg_f32(b, bi_neg(x)), y),
bi_fadd_f32_to(b, reg, bi_neg(x), y));
CASE(bi_fadd_f32_to(b, reg, bi_fabsneg_f32(b, negabsx), y),
bi_fadd_f32_to(b, reg, negabsx, y));
CASE(bi_fadd_f32_to(b, reg, bi_fabsneg_f32(b, x), y),
bi_fadd_f32_to(b, reg, x, y));
CASE(bi_fmin_f32_to(b, reg, bi_fabsneg_f32(b, negabsx), bi_neg(y)),
bi_fmin_f32_to(b, reg, negabsx, bi_neg(y)));
}
TEST_F(Optimizer, FusedFABSNEGForFP16)
{
CASE(bi_fadd_v2f16_to(b, reg, bi_fabsneg_v2f16(b, negabsx), y),
bi_fadd_v2f16_to(b, reg, negabsx, y));
CASE(bi_fmin_v2f16_to(b, reg, bi_fabsneg_v2f16(b, negabsx), bi_neg(y)),
bi_fmin_v2f16_to(b, reg, negabsx, bi_neg(y)));
}
TEST_F(Optimizer, FuseFADD_F32WithEqualSourcesAbsAbsAndClamp)
{
CASE(
{
bi_instr *I =
bi_fadd_f32_to(b, reg, bi_fabsneg_f32(b, bi_abs(x)), bi_abs(x));
I->clamp = BI_CLAMP_CLAMP_0_1;
},
{
bi_instr *I = bi_fadd_f32_to(b, reg, bi_abs(x), bi_abs(x));
I->clamp = BI_CLAMP_CLAMP_0_1;
});
CASE(
{
bi_instr *I =
bi_fadd_f32_to(b, reg, bi_abs(x), bi_fabsneg_f32(b, bi_abs(x)));
I->clamp = BI_CLAMP_CLAMP_0_1;
},
{
bi_instr *I = bi_fadd_f32_to(b, reg, bi_abs(x), bi_abs(x));
I->clamp = BI_CLAMP_CLAMP_0_1;
});
CASE(
{
bi_instr *I =
bi_fclamp_f32_to(b, reg, bi_fadd_f32(b, bi_abs(x), bi_abs(x)));
I->clamp = BI_CLAMP_CLAMP_0_INF;
},
{
bi_instr *I = bi_fadd_f32_to(b, reg, bi_abs(x), bi_abs(x));
I->clamp = BI_CLAMP_CLAMP_0_INF;
});
}
TEST_F(Optimizer, FuseFADD_V2F16WithDifferentSourcesAbsAbsAndClamp)
{
CASE(
{
bi_instr *I =
bi_fadd_v2f16_to(b, reg, bi_fabsneg_v2f16(b, bi_abs(x)), bi_abs(y));
I->clamp = BI_CLAMP_CLAMP_0_1;
},
{
bi_instr *I = bi_fadd_v2f16_to(b, reg, bi_abs(x), bi_abs(y));
I->clamp = BI_CLAMP_CLAMP_0_1;
});
CASE(
{
bi_instr *I =
bi_fadd_v2f16_to(b, reg, bi_abs(x), bi_fabsneg_v2f16(b, bi_abs(y)));
I->clamp = BI_CLAMP_CLAMP_0_1;
},
{
bi_instr *I = bi_fadd_v2f16_to(b, reg, bi_abs(x), bi_abs(y));
I->clamp = BI_CLAMP_CLAMP_0_1;
});
CASE(
{
bi_instr *I =
bi_fclamp_v2f16_to(b, reg, bi_fadd_v2f16(b, bi_abs(x), bi_abs(y)));
I->clamp = BI_CLAMP_CLAMP_0_INF;
},
{
bi_instr *I = bi_fadd_v2f16_to(b, reg, bi_abs(x), bi_abs(y));
I->clamp = BI_CLAMP_CLAMP_0_INF;
});
}
TEST_F(Optimizer, AvoidFADD_V2F16WithEqualSourcesAbsAbsAndClamp)
{
NEGCASE({
bi_instr *I =
bi_fadd_v2f16_to(b, reg, bi_fabsneg_v2f16(b, bi_abs(x)), bi_abs(x));
I->clamp = BI_CLAMP_CLAMP_0_1;
});
NEGCASE({
bi_instr *I =
bi_fadd_v2f16_to(b, reg, bi_abs(x), bi_fabsneg_v2f16(b, bi_abs(x)));
I->clamp = BI_CLAMP_CLAMP_0_1;
});
NEGCASE({
bi_instr *I =
bi_fclamp_v2f16_to(b, reg, bi_fadd_v2f16(b, bi_abs(x), bi_abs(x)));
I->clamp = BI_CLAMP_CLAMP_0_INF;
});
}
TEST_F(Optimizer, SwizzlesComposedForFP16)
{
CASE(bi_fadd_v2f16_to(
b, reg, bi_fabsneg_v2f16(b, bi_swz_16(negabsx, true, false)), y),
bi_fadd_v2f16_to(b, reg, bi_swz_16(negabsx, true, false), y));
CASE(bi_fadd_v2f16_to(
b, reg, bi_swz_16(bi_fabsneg_v2f16(b, negabsx), true, false), y),
bi_fadd_v2f16_to(b, reg, bi_swz_16(negabsx, true, false), y));
CASE(bi_fadd_v2f16_to(
b, reg,
bi_swz_16(bi_fabsneg_v2f16(b, bi_swz_16(negabsx, true, false)), true,
false),
y),
bi_fadd_v2f16_to(b, reg, negabsx, y));
CASE(bi_fadd_v2f16_to(
b, reg,
bi_swz_16(bi_fabsneg_v2f16(b, bi_half(negabsx, false)), true, false),
y),
bi_fadd_v2f16_to(b, reg, bi_half(negabsx, false), y));
CASE(bi_fadd_v2f16_to(
b, reg,
bi_swz_16(bi_fabsneg_v2f16(b, bi_half(negabsx, true)), true, false),
y),
bi_fadd_v2f16_to(b, reg, bi_half(negabsx, true), y));
}
TEST_F(Optimizer, PreserveWidens)
{
/* Check that widens are passed through */
CASE(bi_fadd_f32_to(b, reg, bi_fabsneg_f32(b, bi_half(negabsx, false)), y),
bi_fadd_f32_to(b, reg, bi_half(negabsx, false), y));
CASE(bi_fadd_f32_to(b, reg, bi_fabsneg_f32(b, bi_half(negabsx, true)), y),
bi_fadd_f32_to(b, reg, bi_half(negabsx, true), y));
CASE(bi_fadd_f32_to(b, reg, bi_fabsneg_f32(b, bi_half(x, true)),
bi_fabsneg_f32(b, bi_half(x, false))),
bi_fadd_f32_to(b, reg, bi_half(x, true), bi_half(x, false)));
}
TEST_F(Optimizer, DoNotMixSizesForFABSNEG)
{
/* Refuse to mix sizes for fabsneg, that's wrong */
NEGCASE(bi_fadd_f32_to(b, reg, bi_fabsneg_v2f16(b, negabsx), y));
NEGCASE(bi_fadd_v2f16_to(b, reg, bi_fabsneg_f32(b, negabsx), y));
}
TEST_F(Optimizer, AvoidZeroAndFABSNEGFootguns)
{
/* It's tempting to use addition by 0.0 as the absneg primitive, but that
* has footguns around signed zero and round modes. Check we don't
* incorrectly fuse these rules. */
bi_index zero = bi_zero();
NEGCASE(bi_fadd_f32_to(b, reg, bi_fadd_f32(b, bi_abs(x), zero), y));
NEGCASE(bi_fadd_f32_to(b, reg, bi_fadd_f32(b, bi_neg(x), zero), y));
NEGCASE(bi_fadd_f32_to(b, reg, bi_fadd_f32(b, bi_neg(bi_abs(x)), zero), y));
NEGCASE(bi_fadd_f32_to(b, reg, bi_fadd_f32(b, x, zero), y));
}
TEST_F(Optimizer, ClampsPropagated)
{
CASE(
{
bi_instr *I = bi_fclamp_f32_to(b, reg, bi_fadd_f32(b, x, y));
I->clamp = BI_CLAMP_CLAMP_0_INF;
},
{
bi_instr *I = bi_fadd_f32_to(b, reg, x, y);
I->clamp = BI_CLAMP_CLAMP_0_INF;
});
CASE(
{
bi_instr *I = bi_fclamp_v2f16_to(b, reg, bi_fadd_v2f16(b, x, y));
I->clamp = BI_CLAMP_CLAMP_0_1;
},
{
bi_instr *I = bi_fadd_v2f16_to(b, reg, x, y);
I->clamp = BI_CLAMP_CLAMP_0_1;
});
}
TEST_F(Optimizer, ClampsComposed)
{
CASE(
{
bi_instr *I = bi_fadd_f32_to(b, bi_temp(b->shader), x, y);
bi_instr *J = bi_fclamp_f32_to(b, reg, I->dest[0]);
I->clamp = BI_CLAMP_CLAMP_M1_1;
J->clamp = BI_CLAMP_CLAMP_0_INF;
},
{
bi_instr *I = bi_fadd_f32_to(b, reg, x, y);
I->clamp = BI_CLAMP_CLAMP_0_1;
});
CASE(
{
bi_instr *I = bi_fadd_f32_to(b, bi_temp(b->shader), x, y);
bi_instr *J = bi_fclamp_f32_to(b, reg, I->dest[0]);
I->clamp = BI_CLAMP_CLAMP_0_1;
J->clamp = BI_CLAMP_CLAMP_0_INF;
},
{
bi_instr *I = bi_fadd_f32_to(b, reg, x, y);
I->clamp = BI_CLAMP_CLAMP_0_1;
});
CASE(
{
bi_instr *I = bi_fadd_f32_to(b, bi_temp(b->shader), x, y);
bi_instr *J = bi_fclamp_f32_to(b, reg, I->dest[0]);
I->clamp = BI_CLAMP_CLAMP_0_INF;
J->clamp = BI_CLAMP_CLAMP_0_INF;
},
{
bi_instr *I = bi_fadd_f32_to(b, reg, x, y);
I->clamp = BI_CLAMP_CLAMP_0_INF;
});
CASE(
{
bi_instr *I = bi_fadd_v2f16_to(b, bi_temp(b->shader), x, y);
bi_instr *J = bi_fclamp_v2f16_to(b, reg, I->dest[0]);
I->clamp = BI_CLAMP_CLAMP_M1_1;
J->clamp = BI_CLAMP_CLAMP_0_INF;
},
{
bi_instr *I = bi_fadd_v2f16_to(b, reg, x, y);
I->clamp = BI_CLAMP_CLAMP_0_1;
});
CASE(
{
bi_instr *I = bi_fadd_v2f16_to(b, bi_temp(b->shader), x, y);
bi_instr *J = bi_fclamp_v2f16_to(b, reg, I->dest[0]);
I->clamp = BI_CLAMP_CLAMP_0_1;
J->clamp = BI_CLAMP_CLAMP_0_INF;
},
{
bi_instr *I = bi_fadd_v2f16_to(b, reg, x, y);
I->clamp = BI_CLAMP_CLAMP_0_1;
});
CASE(
{
bi_instr *I = bi_fadd_v2f16_to(b, bi_temp(b->shader), x, y);
bi_instr *J = bi_fclamp_v2f16_to(b, reg, I->dest[0]);
I->clamp = BI_CLAMP_CLAMP_0_INF;
J->clamp = BI_CLAMP_CLAMP_0_INF;
},
{
bi_instr *I = bi_fadd_v2f16_to(b, reg, x, y);
I->clamp = BI_CLAMP_CLAMP_0_INF;
});
}
TEST_F(Optimizer, DoNotMixSizesWhenClamping)
{
NEGCASE({
bi_instr *I = bi_fclamp_f32_to(b, reg, bi_fadd_v2f16(b, x, y));
I->clamp = BI_CLAMP_CLAMP_0_1;
});
NEGCASE({
bi_instr *I = bi_fclamp_v2f16_to(b, reg, bi_fadd_f32(b, x, y));
I->clamp = BI_CLAMP_CLAMP_0_1;
});
}
TEST_F(Optimizer, DoNotUseAdditionByZeroForClamps)
{
bi_index zero = bi_zero();
/* We can't use addition by 0.0 for clamps due to signed zeros. */
NEGCASE({
bi_instr *I = bi_fadd_f32_to(b, reg, bi_fadd_f32(b, x, y), zero);
I->clamp = BI_CLAMP_CLAMP_M1_1;
});
NEGCASE({
bi_instr *I = bi_fadd_v2f16_to(b, reg, bi_fadd_v2f16(b, x, y), zero);
I->clamp = BI_CLAMP_CLAMP_0_1;
});
}
TEST_F(Optimizer, FuseComparisonsWithDISCARD)
{
CASE(bi_discard_b32(b, bi_fcmp_f32(b, x, y, BI_CMPF_LE, BI_RESULT_TYPE_F1)),
bi_discard_f32(b, x, y, BI_CMPF_LE));
CASE(bi_discard_b32(b, bi_fcmp_f32(b, x, y, BI_CMPF_NE, BI_RESULT_TYPE_I1)),
bi_discard_f32(b, x, y, BI_CMPF_NE));
CASE(bi_discard_b32(b, bi_fcmp_f32(b, x, y, BI_CMPF_EQ, BI_RESULT_TYPE_M1)),
bi_discard_f32(b, x, y, BI_CMPF_EQ));
for (unsigned h = 0; h < 2; ++h) {
CASE(bi_discard_b32(
b, bi_half(bi_fcmp_v2f16(b, x, y, BI_CMPF_LE, BI_RESULT_TYPE_F1),
h)),
bi_discard_f32(b, bi_half(x, h), bi_half(y, h), BI_CMPF_LE));
CASE(bi_discard_b32(
b, bi_half(bi_fcmp_v2f16(b, x, y, BI_CMPF_NE, BI_RESULT_TYPE_I1),
h)),
bi_discard_f32(b, bi_half(x, h), bi_half(y, h), BI_CMPF_NE));
CASE(bi_discard_b32(
b, bi_half(bi_fcmp_v2f16(b, x, y, BI_CMPF_EQ, BI_RESULT_TYPE_M1),
h)),
bi_discard_f32(b, bi_half(x, h), bi_half(y, h), BI_CMPF_EQ));
}
}
TEST_F(Optimizer, DoNotFuseSpecialComparisons)
{
NEGCASE(
bi_discard_b32(b, bi_fcmp_f32(b, x, y, BI_CMPF_GTLT, BI_RESULT_TYPE_F1)));
NEGCASE(bi_discard_b32(
b, bi_fcmp_f32(b, x, y, BI_CMPF_TOTAL, BI_RESULT_TYPE_F1)));
}
TEST_F(Optimizer, FuseResultType)
{
CASE(bi_mux_i32_to(b, reg, bi_imm_f32(0.0), bi_imm_f32(1.0),
bi_fcmp_f32(b, x, y, BI_CMPF_LE, BI_RESULT_TYPE_M1),
BI_MUX_INT_ZERO),
bi_fcmp_f32_to(b, reg, x, y, BI_CMPF_LE, BI_RESULT_TYPE_F1));
CASE(bi_mux_i32_to(
b, reg, bi_imm_f32(0.0), bi_imm_f32(1.0),
bi_fcmp_f32(b, bi_abs(x), bi_neg(y), BI_CMPF_LE, BI_RESULT_TYPE_M1),
BI_MUX_INT_ZERO),
bi_fcmp_f32_to(b, reg, bi_abs(x), bi_neg(y), BI_CMPF_LE,
BI_RESULT_TYPE_F1));
CASE(bi_mux_i32_to(
b, reg, bi_imm_u32(0), bi_imm_u32(1),
bi_fcmp_f32(b, bi_abs(x), bi_neg(y), BI_CMPF_LE, BI_RESULT_TYPE_M1),
BI_MUX_INT_ZERO),
bi_fcmp_f32_to(b, reg, bi_abs(x), bi_neg(y), BI_CMPF_LE,
BI_RESULT_TYPE_I1));
CASE(bi_mux_v2i16_to(b, reg, bi_imm_f16(0.0), bi_imm_f16(1.0),
bi_fcmp_v2f16(b, bi_abs(x), bi_neg(y), BI_CMPF_LE,
BI_RESULT_TYPE_M1),
BI_MUX_INT_ZERO),
bi_fcmp_v2f16_to(b, reg, bi_abs(x), bi_neg(y), BI_CMPF_LE,
BI_RESULT_TYPE_F1));
CASE(bi_mux_v2i16_to(b, reg, bi_imm_u16(0), bi_imm_u16(1),
bi_fcmp_v2f16(b, bi_abs(x), bi_neg(y), BI_CMPF_LE,
BI_RESULT_TYPE_M1),
BI_MUX_INT_ZERO),
bi_fcmp_v2f16_to(b, reg, bi_abs(x), bi_neg(y), BI_CMPF_LE,
BI_RESULT_TYPE_I1));
CASE(bi_mux_i32_to(b, reg, bi_imm_u32(0), bi_imm_u32(1),
bi_icmp_u32(b, x, y, BI_CMPF_LE, BI_RESULT_TYPE_M1),
BI_MUX_INT_ZERO),
bi_icmp_u32_to(b, reg, x, y, BI_CMPF_LE, BI_RESULT_TYPE_I1));
CASE(bi_mux_v2i16_to(b, reg, bi_imm_u16(0), bi_imm_u16(1),
bi_icmp_v2u16(b, x, y, BI_CMPF_LE, BI_RESULT_TYPE_M1),
BI_MUX_INT_ZERO),
bi_icmp_v2u16_to(b, reg, x, y, BI_CMPF_LE, BI_RESULT_TYPE_I1));
CASE(bi_mux_v4i8_to(b, reg, bi_imm_u8(0), bi_imm_u8(1),
bi_icmp_v4u8(b, x, y, BI_CMPF_LE, BI_RESULT_TYPE_M1),
BI_MUX_INT_ZERO),
bi_icmp_v4u8_to(b, reg, x, y, BI_CMPF_LE, BI_RESULT_TYPE_I1));
CASE(bi_mux_i32_to(b, reg, bi_imm_u32(0), bi_imm_u32(1),
bi_icmp_s32(b, x, y, BI_CMPF_LE, BI_RESULT_TYPE_M1),
BI_MUX_INT_ZERO),
bi_icmp_s32_to(b, reg, x, y, BI_CMPF_LE, BI_RESULT_TYPE_I1));
CASE(bi_mux_v2i16_to(b, reg, bi_imm_u16(0), bi_imm_u16(1),
bi_icmp_v2s16(b, x, y, BI_CMPF_LE, BI_RESULT_TYPE_M1),
BI_MUX_INT_ZERO),
bi_icmp_v2s16_to(b, reg, x, y, BI_CMPF_LE, BI_RESULT_TYPE_I1));
CASE(bi_mux_v4i8_to(b, reg, bi_imm_u8(0), bi_imm_u8(1),
bi_icmp_v4s8(b, x, y, BI_CMPF_LE, BI_RESULT_TYPE_M1),
BI_MUX_INT_ZERO),
bi_icmp_v4s8_to(b, reg, x, y, BI_CMPF_LE, BI_RESULT_TYPE_I1));
}
TEST_F(Optimizer, DoNotFuseMixedSizeResultType)
{
NEGCASE(bi_mux_i32_to(
b, reg, bi_imm_f32(0.0), bi_imm_f32(1.0),
bi_fcmp_v2f16(b, bi_abs(x), bi_neg(y), BI_CMPF_LE, BI_RESULT_TYPE_M1),
BI_MUX_INT_ZERO));
NEGCASE(bi_mux_v2i16_to(
b, reg, bi_imm_f16(0.0), bi_imm_f16(1.0),
bi_fcmp_f32(b, bi_abs(x), bi_neg(y), BI_CMPF_LE, BI_RESULT_TYPE_M1),
BI_MUX_INT_ZERO));
}
TEST_F(Optimizer, VarTexCoord32)
{
CASE(
{
bi_index ld =
bi_ld_var_imm(b, bi_null(), BI_REGISTER_FORMAT_F32,
BI_SAMPLE_CENTER, BI_UPDATE_STORE, BI_VECSIZE_V2, 0);
bi_index x = bi_temp(b->shader);
bi_index y = bi_temp(b->shader);
bi_instr *split = bi_split_i32_to(b, 2, ld);
split->dest[0] = x;
split->dest[1] = y;
bi_texs_2d_f32_to(b, reg, x, y, false, 0, 0);
},
{
bi_var_tex_f32_to(b, reg, false, BI_SAMPLE_CENTER, BI_UPDATE_STORE, 0,
0);
});
}
TEST_F(Optimizer, Int8ToFloat32)
{
for (unsigned i = 0; i < 4; ++i) {
CASE(bi_s32_to_f32_to(b, reg, bi_s8_to_s32(b, bi_byte(x, i))),
bi_s8_to_f32_to(b, reg, bi_byte(x, i)));
CASE(bi_s32_to_f32_to(b, reg, bi_u8_to_u32(b, bi_byte(x, i))),
bi_u8_to_f32_to(b, reg, bi_byte(x, i)));
CASE(bi_u32_to_f32_to(b, reg, bi_u8_to_u32(b, bi_byte(x, i))),
bi_u8_to_f32_to(b, reg, bi_byte(x, i)));
}
}
TEST_F(Optimizer, Int16ToFloat32)
{
for (unsigned i = 0; i < 2; ++i) {
CASE(bi_s32_to_f32_to(b, reg, bi_s16_to_s32(b, bi_half(x, i))),
bi_s16_to_f32_to(b, reg, bi_half(x, i)));
CASE(bi_s32_to_f32_to(b, reg, bi_u16_to_u32(b, bi_half(x, i))),
bi_u16_to_f32_to(b, reg, bi_half(x, i)));
CASE(bi_u32_to_f32_to(b, reg, bi_u16_to_u32(b, bi_half(x, i))),
bi_u16_to_f32_to(b, reg, bi_half(x, i)));
}
}