{
 "engine": "engine_cards_v2c.json",
 "envelope": false,
 "dropped": [],
 "cards": [
  {
   "id": "P1",
   "include": "nn\\.Linear|torch\\.matmul|\\bmatmul\\(|torch\\.bmm|\\.bmm\\(|\\bbmm\\(|einsum|\\s@\\s|torch\\.mm\\(|F\\.linear|addmm|baddbmm",
   "exclude": "nn\\.Conv|nn\\.ConvTranspose|F\\.conv|conv[123]d|conv_transpose",
   "exclude_families": [
    "rnn"
   ],
   "text": "### Verified technique: fp32-accurate GEMM on bf16 tensor cores (3-product split)\nLiterature: Henry, Tang & Heinecke 2019 (bf16x3 split products); Ootomo & Yokota 2022 (tensor cores with error correction beat the fp32 SIMT peak on A100). Checked here by the protected evaluator; facts, not orders.\n- Physics of this A100 (strict fp32, measured on 3 GEMM shapes, x ~ U[0,1) and N(0,1)): eager fp32 cuBLAS (TF32 off) runs at 18-19 TFLOP/s. Plain TF32 tensor cores FAIL the tolerance (error 6-9x over it). Triton tl.dot(input_precision='tf32x3') passes but runs at only 21-22 TFLOP/s (1.12-1.20x). The 3-product bf16 split passes with max error <= 0.33 of the tolerance and runs at 64-79 TFLOP/s as ONE cuBLAS bf16 GEMM over a 3K-concatenated operand (3.5-4.2x), or 40-43 TFLOP/s as a Triton kernel that splits in registers.\n- Verified on training problems (protected evaluator, correct, fast_1): L1_017 Matmul_with_transposed_B: 3.87x (cuBLAS split, 7.7 -> 2.0 ms); L2_076 Gemm_Add_ReLU: 3.55x (cuBLAS split, 8.0 -> 2.2 ms); L2_059 Matmul_Swish_Scaling: 2.21x (fused Triton split, 16.1 -> 7.3 ms); L2_076 Gemm_Add_ReLU: 2.23x (fused Triton split, 8.0 -> 3.6 ms); L3_001 MLP: 1.76x (fused Triton split, 10.7 -> 6.1 ms). Exact-fp32 kernels on such problems had reached ~1.0-1.3x.\n- When it applies: the runtime is dominated by fp32 GEMM-like contractions (nn.Linear, matmul, bmm, einsum, attention QK^T and PV, 1x1 convolutions as GEMM) with K up to ~8192. Large M (>= ~1024): `linear_bf16x3` (cuBLAS over the concatenated operands); batched: the same concatenation with torch.bmm(a16, b16, out_dtype=torch.float32). Small M, or when an epilogue would cost an extra full pass: the fused Triton kernel. Keep the rest in fp32 and fuse epilogues (bias, activations, normalisation) into one pass over the GEMM output.\n- Numerics: x = xh + xl with xh = bf16(x), xl = bf16(x - xh); x.w ~= xh.wh + xh.wl + xl.wh with fp32 accumulation (the dropped xl.wl term is ~2^-16 relative). Error grows ~sqrt(K); for K >> 8192 or cancellation-heavy outputs, CHECK and read max_abs_diff. bf16 keeps the fp32 exponent range (no scaling needed). Dropping the cross terms is plain bf16 and fails.\n- Building block (verified; copy and adapt; `split3` is your own kernel launched every forward):\n```python\n# P1: fp32-accurate GEMM on bf16 tensor cores (3-product split, Henry/Tang/Heinecke 2019; Ootomo/Yokota 2022).\n# x = xh + xl with xh = bf16(x), xl = bf16(x - xh) (same for w); x.w ~= xh.wh + xh.wl + xl.wh (drops xl.wl ~ 2^-16).\n# Measured on A100: error <= 0.34 of atol=rtol=1e-4 at K <= 8192; cuBLAS fp32 = 19 TFLOP/s.\n#   linear_bf16x3(x, w)         one cuBLAS bf16 GEMM over a 3K-concatenated operand, fp32 out: 65-79 TFLOP/s (3.5-4.2x)\n#   linear_bf16x3_fused(x, w, b, ACT, s1, s2)  Triton, splits inside the kernel, fused bias+epilogue: 40-45 TFLOP/s\n#       (2.2-2.6x); use it when M is small (W re-read dominates) or the epilogue would cost an extra pass.\nimport torch\nimport triton\nimport triton.language as tl\n\n\n@triton.jit\ndef split3_kernel(src, dst, R, K, LO_MID: tl.constexpr, BR: tl.constexpr, BKK: tl.constexpr):\n    # src [R, K] fp32 contiguous -> dst [R, 3K] bf16: [hi | hi | lo] (LO_MID=0, left operand) or [hi | lo | hi] (right)\n    r = tl.program_id(0) * BR + tl.arange(0, BR)\n    k = tl.program_id(1) * BKK + tl.arange(0, BKK)\n    m = (r[:, None] < R) & (k[None, :] < K)\n    v = tl.load(src + r[:, None] * K + k[None, :], mask=m, other=0.0)\n    h = v.to(tl.bfloat16)\n    lo = (v - h.to(tl.float32)).to(tl.bfloat16)\n    base = dst + r[:, None] * (3 * K) + k[None, :]\n    tl.store(base, h, mask=m)\n    if LO_MID:\n        tl.store(base + K, lo, mask=m)\n        tl.store(base + 2 * K, h, mask=m)\n    else:\n        tl.store(base + K, h, mask=m)\n        tl.store(base + 2 * K, lo, mask=m)\n\n\ndef split3(t, lo_mid):\n    t = t.contiguous()\n    R, K = t.shape\n    out = torch.empty((R, 3 * K), device=t.device, dtype=torch.bfloat16)\n    split3_kernel[(triton.cdiv(R, 32), triton.cdiv(K, 128))](t, out, R, K, LO_MID=lo_mid, BR=32, BKK=128)\n    return out\n\n\ndef linear_bf16x3(x2d, w):\n    \"\"\"x2d [M, K] fp32, w [N, K] fp32 (nn.Linear layout) -> x2d @ w.T in fp32 (no bias). For A @ B with B [K, N],\n    pass w = B.t() (split3 makes it contiguous).\"\"\"\n    return torch.mm(split3(x2d, 0), split3(w, 1).t(), out_dtype=torch.float32)\n\n\n@triton.autotune(\n    configs=[\n        triton.Config({\"BM\": 128, \"BN\": 128, \"BK\": 32, \"GM\": 8}, num_stages=3, num_warps=8),\n        triton.Config({\"BM\": 128, \"BN\": 256, \"BK\": 32, \"GM\": 8}, num_stages=3, num_warps=8),\n        triton.Config({\"BM\": 64, \"BN\": 128, \"BK\": 32, \"GM\": 8}, num_stages=4, num_warps=4),\n        triton.Config({\"BM\": 64, \"BN\": 64, \"BK\": 64, \"GM\": 8}, num_stages=4, num_warps=4),\n    ],\n    key=[\"M\", \"N\", \"K\"],\n)\n@triton.jit\ndef gemm_bf16x3_fused_kernel(x_ptr, w_ptr, b_ptr, y_ptr, M, N, K, sxm, sxk, swn, swk, sym, syn, s1, s2,\n                             HAS_BIAS: tl.constexpr, ACT: tl.constexpr,\n                             BM: tl.constexpr, BN: tl.constexpr, BK: tl.constexpr, GM: tl.constexpr):\n    # y = act(x @ w.T + b); ACT: 0 none, 1 relu, 2 swish(v)*s1, 3 v*s1 + s2 (affine). Add your own epilogue here.\n    pid = tl.program_id(0)\n    npm = tl.cdiv(M, BM)\n    npn = tl.cdiv(N, BN)\n    width = GM * npn\n    gid = pid // width\n    fm = gid * GM\n    gsz = min(npm - fm, GM)\n    pm = fm + (pid % width) % gsz\n    pn = (pid % width) // gsz\n    rm = pm * BM + tl.arange(0, BM)\n    rn = pn * BN + tl.arange(0, BN)\n    rk = tl.arange(0, BK)\n    xp = x_ptr + rm[:, None] * sxm + rk[None, :] * sxk\n    wp = w_ptr + rn[None, :] * swn + rk[:, None] * swk\n    acc = tl.zeros((BM, BN), dtype=tl.float32)\n    acc2 = tl.zeros((BM, BN), dtype=tl.float32)\n    for k in range(0, tl.cdiv(K, BK)):\n        km = (k * BK + rk) < K\n        a = tl.load(xp, mask=(rm[:, None] < M) & km[None, :], other=0.0)\n        b = tl.load(wp, mask=km[:, None] & (rn[None, :] < N), other=0.0)\n        ah = a.to(tl.bfloat16)\n        al = (a - ah.to(tl.float32)).to(tl.bfloat16)\n        bh = b.to(tl.bfloat16)\n        bl = (b - bh.to(tl.float32)).to(tl.bfloat16)\n        acc2 = tl.dot(ah, bl, acc2)\n        acc2 = tl.dot(al, bh, acc2)\n        acc = tl.dot(ah, bh, acc)\n        xp += BK * sxk\n        wp += BK * swk\n    v = acc + acc2\n    if HAS_BIAS:\n        v = v + tl.load(b_ptr + rn, mask=rn < N, other=0.0)[None, :]\n    if ACT == 1:\n        v = tl.maximum(v, 0.0)\n    elif ACT == 2:\n        v = v * tl.sigmoid(v) * s1\n    elif ACT == 3:\n        v = v * s1 + s2\n    tl.store(y_ptr + rm[:, None] * sym + rn[None, :] * syn, v, mask=(rm[:, None] < M) & (rn[None, :] < N))\n\n\ndef linear_bf16x3_fused(x2d, w, b=None, act=0, s1=1.0, s2=0.0):\n    M, K = x2d.shape\n    N = w.shape[0]\n    y = torch.empty((M, N), device=x2d.device, dtype=torch.float32)\n    grid = lambda META: (triton.cdiv(M, META[\"BM\"]) * triton.cdiv(N, META[\"BN\"]),)\n    gemm_bf16x3_fused_kernel[grid](x2d, w, b if b is not None else y, y, M, N, K, x2d.stride(0), x2d.stride(1),\n                                   w.stride(0), w.stride(1), y.stride(0), y.stride(1), float(s1), float(s2),\n                                   HAS_BIAS=b is not None, ACT=act)\n    return y\n```\n",
   "demote": "allow_tf32\\s*=\\s*False|TF32 disabled|beats? cuBLAS|exact fp32 tl\\.dot|Keep (the |every )?(GEMM|Linear GEMM|matmul)[^.]{0,60}cuBLAS|Leave the GEMM to cuBLAS|hand-written fp32 Triton matmul|plain fp32 matmul with TF32 disabled",
   "provenance": {
    "origin": "literature-seeded, then verified",
    "literature": [
     "Henry, Tang, Heinecke 2019, Leveraging the bfloat16 AI datatype for higher-precision computations",
     "Ootomo, Yokota 2022, Recovering single precision accuracy from Tensor Cores while surpassing the FP32 theoretical peak performance (IJHPCA)"
    ],
    "verified_on_learn": "L1_017 3.87x, L2_076 3.55x/2.23x, L2_059 2.21x, L3_001 1.76x",
    "admission_trial": "F1: 10 learn problems, card vs bare, +2.17 mean, 9-0"
   },
   "example": {
    "pid": "L2_076",
    "source": "import torch\nimport torch.nn as nn\nimport torch.nn.functional as F\nimport triton\nimport triton.language as tl\n# P1: fp32-accurate GEMM on bf16 tensor cores (3-product split, Henry/Tang/Heinecke 2019; Ootomo/Yokota 2022).\n# x = xh + xl with xh = bf16(x), xl = bf16(x - xh) (same for w); x.w ~= xh.wh + xh.wl + xl.wh (drops xl.wl ~ 2^-16).\n# Measured on A100: error <= 0.34 of atol=rtol=1e-4 at K <= 8192; cuBLAS fp32 = 19 TFLOP/s.\n#   linear_bf16x3(x, w)         one cuBLAS bf16 GEMM over a 3K-concatenated operand, fp32 out: 65-79 TFLOP/s (3.5-4.2x)\n#   linear_bf16x3_fused(x, w, b, ACT, s1, s2)  Triton, splits inside the kernel, fused bias+epilogue: 40-45 TFLOP/s\n#       (2.2-2.6x); use it when M is small (W re-read dominates) or the epilogue would cost an extra pass.\n\n\n@triton.jit\ndef split3_kernel(src, dst, R, K, LO_MID: tl.constexpr, BR: tl.constexpr, BKK: tl.constexpr):\n    # src [R, K] fp32 contiguous -> dst [R, 3K] bf16: [hi | hi | lo] (LO_MID=0, left operand) or [hi | lo | hi] (right)\n    r = tl.program_id(0) * BR + tl.arange(0, BR)\n    k = tl.program_id(1) * BKK + tl.arange(0, BKK)\n    m = (r[:, None] < R) & (k[None, :] < K)\n    v = tl.load(src + r[:, None] * K + k[None, :], mask=m, other=0.0)\n    h = v.to(tl.bfloat16)\n    lo = (v - h.to(tl.float32)).to(tl.bfloat16)\n    base = dst + r[:, None] * (3 * K) + k[None, :]\n    tl.store(base, h, mask=m)\n    if LO_MID:\n        tl.store(base + K, lo, mask=m)\n        tl.store(base + 2 * K, h, mask=m)\n    else:\n        tl.store(base + K, h, mask=m)\n        tl.store(base + 2 * K, lo, mask=m)\n\n\ndef split3(t, lo_mid):\n    t = t.contiguous()\n    R, K = t.shape\n    out = torch.empty((R, 3 * K), device=t.device, dtype=torch.bfloat16)\n    split3_kernel[(triton.cdiv(R, 32), triton.cdiv(K, 128))](t, out, R, K, LO_MID=lo_mid, BR=32, BKK=128)\n    return out\n\n\ndef linear_bf16x3(x2d, w):\n    \"\"\"x2d [M, K] fp32, w [N, K] fp32 (nn.Linear layout) -> x2d @ w.T in fp32 (no bias). For A @ B with B [K, N],\n    pass w = B.t() (split3 makes it contiguous).\"\"\"\n    return torch.mm(split3(x2d, 0), split3(w, 1).t(), out_dtype=torch.float32)\n\n\n@triton.autotune(\n    configs=[\n        triton.Config({\"BM\": 128, \"BN\": 128, \"BK\": 32, \"GM\": 8}, num_stages=3, num_warps=8),\n        triton.Config({\"BM\": 128, \"BN\": 256, \"BK\": 32, \"GM\": 8}, num_stages=3, num_warps=8),\n        triton.Config({\"BM\": 64, \"BN\": 128, \"BK\": 32, \"GM\": 8}, num_stages=4, num_warps=4),\n        triton.Config({\"BM\": 64, \"BN\": 64, \"BK\": 64, \"GM\": 8}, num_stages=4, num_warps=4),\n    ],\n    key=[\"M\", \"N\", \"K\"],\n)\n@triton.jit\ndef gemm_bf16x3_fused_kernel(x_ptr, w_ptr, b_ptr, y_ptr, M, N, K, sxm, sxk, swn, swk, sym, syn, s1, s2,\n                             HAS_BIAS: tl.constexpr, ACT: tl.constexpr,\n                             BM: tl.constexpr, BN: tl.constexpr, BK: tl.constexpr, GM: tl.constexpr):\n    # y = act(x @ w.T + b); ACT: 0 none, 1 relu, 2 swish(v)*s1, 3 v*s1 + s2 (affine). Add your own epilogue here.\n    pid = tl.program_id(0)\n    npm = tl.cdiv(M, BM)\n    npn = tl.cdiv(N, BN)\n    width = GM * npn\n    gid = pid // width\n    fm = gid * GM\n    gsz = min(npm - fm, GM)\n    pm = fm + (pid % width) % gsz\n    pn = (pid % width) // gsz\n    rm = pm * BM + tl.arange(0, BM)\n    rn = pn * BN + tl.arange(0, BN)\n    rk = tl.arange(0, BK)\n    xp = x_ptr + rm[:, None] * sxm + rk[None, :] * sxk\n    wp = w_ptr + rn[None, :] * swn + rk[:, None] * swk\n    acc = tl.zeros((BM, BN), dtype=tl.float32)\n    acc2 = tl.zeros((BM, BN), dtype=tl.float32)\n    for k in range(0, tl.cdiv(K, BK)):\n        km = (k * BK + rk) < K\n        a = tl.load(xp, mask=(rm[:, None] < M) & km[None, :], other=0.0)\n        b = tl.load(wp, mask=km[:, None] & (rn[None, :] < N), other=0.0)\n        ah = a.to(tl.bfloat16)\n        al = (a - ah.to(tl.float32)).to(tl.bfloat16)\n        bh = b.to(tl.bfloat16)\n        bl = (b - bh.to(tl.float32)).to(tl.bfloat16)\n        acc2 = tl.dot(ah, bl, acc2)\n        acc2 = tl.dot(al, bh, acc2)\n        acc = tl.dot(ah, bh, acc)\n        xp += BK * sxk\n        wp += BK * swk\n    v = acc + acc2\n    if HAS_BIAS:\n        v = v + tl.load(b_ptr + rn, mask=rn < N, other=0.0)[None, :]\n    if ACT == 1:\n        v = tl.maximum(v, 0.0)\n    elif ACT == 2:\n        v = v * tl.sigmoid(v) * s1\n    elif ACT == 3:\n        v = v * s1 + s2\n    tl.store(y_ptr + rm[:, None] * sym + rn[None, :] * syn, v, mask=(rm[:, None] < M) & (rn[None, :] < N))\n\n\ndef linear_bf16x3_fused(x2d, w, b=None, act=0, s1=1.0, s2=0.0):\n    M, K = x2d.shape\n    N = w.shape[0]\n    y = torch.empty((M, N), device=x2d.device, dtype=torch.float32)\n    grid = lambda META: (triton.cdiv(M, META[\"BM\"]) * triton.cdiv(N, META[\"BN\"]),)\n    gemm_bf16x3_fused_kernel[grid](x2d, w, b if b is not None else y, y, M, N, K, x2d.stride(0), x2d.stride(1),\n                                   w.stride(0), w.stride(1), y.stride(0), y.stride(1), float(s1), float(s2),\n                                   HAS_BIAS=b is not None, ACT=act)\n    return y\n\n\n@triton.jit\ndef bias_act_kernel(y_ptr, b_ptr, out_ptr, N, total, ACT: tl.constexpr, BLOCK: tl.constexpr):\n    offs = tl.program_id(0) * BLOCK + tl.arange(0, BLOCK)\n    m = offs < total\n    v = tl.load(y_ptr + offs, mask=m, other=0.0) + tl.load(b_ptr + offs % N, mask=m, other=0.0)\n    if ACT == 1:\n        v = tl.maximum(v, 0.0)\n    tl.store(out_ptr + offs, v, mask=m)\n\n\nclass ModelNew(nn.Module):\n    def __init__(self, in_features, out_features, bias_shape):\n        super().__init__()\n        self.gemm = nn.Linear(in_features, out_features, bias=False)\n        self.bias = nn.Parameter(torch.randn(bias_shape))\n\n    def forward(self, x):\n        y = linear_bf16x3(x, self.gemm.weight)\n        out = torch.empty_like(y)\n        n = y.numel()\n        bias_act_kernel[(triton.cdiv(n, 2048),)](y, self.bias, out, y.shape[1], n, ACT=1, BLOCK=2048)\n        return out\n",
    "speedup": 3.5505
   }
  },
  {
   "id": "E-H1",
   "include": "nn\\.Conv[23]d\\(|F\\.conv[23]d\\(",
   "exclude": null,
   "exclude_families": [],
   "text": "### Verified technique (E-H1, admitted by the KFORGE engine): Dense Conv2d/Conv3d as a Triton implicit GEMM with an in-register bf16 3-product split and a fused epilogue\nFor dense convolutions with reduction length K = Ci*kh*kw (*kd) >= ~256 and Co >= 64, a Triton implicit-GEMM kernel should beat strict-fp32 cuDNN by 1.5-3x on the conv itself, and more once bias/activation/pool epilogues are fused. The kernel splits each loaded fp32 operand into bf16 hi+lo in registers and issues three bf16 tl.dot calls into one fp32 accumulator.\nPhysics: cuDNN under strict fp32 runs on the 19.5 TFLOP/s SIMT path. A100 bf16 mma peaks at 312 TFLOP/s, so even at 40% efficiency the three products deliver about 40 TFLOP/s of fp32-equivalent work. P1 already shows the split stays within about 0.34 of the tolerance for K <= 8192, and conv K is almost always below that. This is not the tf32x3 path that measured only 1.1-1.2x: bf16 mma has 2x the TF32 rate, and the hi/lo split is exact in fp32 adds. Implicit GEMM never materializes im2col, so it avoids the bandwidth loss of the AlexNet failure.\nHow: Map GEMM M = N*OH*OW (output pixels), GEMM N = Co, and K = (r, s, ci), with ci innermost. Use one cheap Triton prep kernel to convert x to channels_last and emit xh = bf16(x) and xl = bf16(x - float(xh)); these two bf16 tensors take the same bytes as fp32. Cache the weight reorganised as [kh*kw*Ci, Co] in hi/lo bf16 form in self.<x>, keyed on weight.data_ptr(). In the kernel, loop over (r, s), compute shifted input coordinates with padding masks, and load BK = 32 channel slabs of xh/xl and wh/wl. Accumulate acc += dot(xl, wh) + dot(xh, wl) first, then acc += dot(xh, wh), with fp32 accumulation. Use BM = 128, BN = 64 or 128, num_stages = 3-4, and a small autotune list. In the epilogue, add bias, apply the following elementwise ops (and pool if the window fits the tile), and store straight to NCHW with int64 offsets. Consecutive M indices are consecutive ow, so the stores coalesce. Validate against F.conv2d at atol = rtol = 1e-4 before timing, and fall back to cuDNN when K < 128 or Co < 32.\n- Verified on training problems (protected evaluator): L1_080 conv_standard_2D_square_input_asymmetric_kernel___dilated____padded__: 2.29x vs 0.44x with the previous library; L2_032 Conv2d_Scaling_Min: 2.50x vs 1.11x with the previous library.\n- Verified example implementing it (L2_032, 2.50x; adapt what applies):\n```python\nimport torch\nimport torch.nn as nn\nimport triton\nimport triton.language as tl\n\n\n@triton.autotune(\n    configs=[\n        triton.Config({\"BM\": 128, \"BK\": 32}, num_stages=3, num_warps=8),\n        triton.Config({\"BM\": 64, \"BK\": 32}, num_stages=4, num_warps=4),\n        triton.Config({\"BM\": 128, \"BK\": 64}, num_stages=2, num_warps=8),\n    ],\n    key=[\"M\", \"Ci\", \"Co\"],\n)\n@triton.jit\ndef conv_min_kernel(x, wh, wl, b, out, M, Ci, H, W, OH, OW, Co, scale,\n                    KH: tl.constexpr, KW: tl.constexpr, BN: tl.constexpr,\n                    BM: tl.constexpr, BK: tl.constexpr):\n    pid = tl.program_id(0)\n    rm = pid * BM + tl.arange(0, BM)\n    mm = rm < M\n    rmc = tl.where(mm, rm, 0)\n    ow = rmc % OW\n    t = rmc // OW\n    oh = t % OH\n    n = t // OH\n    base = n * (Ci * H * W) + oh * W + ow\n    rk = tl.arange(0, BK)\n    rn = tl.arange(0, BN)\n    nm = rn < Co\n    HW = H * W\n    nci = tl.cdiv(Ci, BK)\n    total = KH * KW * nci\n    acc = tl.zeros((BM, BN), dtype=tl.float32)\n    for kk in range(0, total):\n        tap = kk // nci\n        c0 = (kk - tap * nci) * BK\n        r = tap // KW\n        s = tap - r * KW\n        ci = c0 + rk\n        cm = ci < Ci\n        a = tl.load(x + base[:, None] + (r * W + s) + ci[None, :] * HW,\n                    mask=cm[None, :], other=0.0)\n        krow = tap * Ci + ci\n        boff = krow[:, None] * Co + rn[None, :]\n        bmask = cm[:, None] & nm[None, :]\n        bh = tl.load(wh + boff, mask=bmask, other=0.0)\n        bl = tl.load(wl + boff, mask=bmask, other=0.0)\n        ah = a.to(tl.bfloat16)\n        al = (a - ah.to(tl.float32)).to(tl.bfloat16)\n        acc = tl.dot(al, bh, acc)\n        acc = tl.dot(ah, bl, acc)\n        acc = tl.dot(ah, bh, acc)\n    bias = tl.load(b + rn, mask=nm, other=0.0)\n    v = (acc + bias[None, :]) * scale\n    v = tl.where(nm[None, :], v, float(\"inf\"))\n    mn = tl.min(v, axis=1)\n    tl.store(out + rm, mn, mask=mm)\n\n\nclass ModelNew(nn.Module):\n    def __init__(self, in_channels, out_channels, kernel_size, scale_factor):\n        super().__init__()\n        self.conv = nn.Conv2d(in_channels, out_channels, kernel_size)\n        self.scale_factor = scale_factor\n\n    def forward(self, x):\n        x = x.contiguous()\n        w = self.conv.weight\n        Co, Ci, KH, KW = w.shape\n        N, _, H, W = x.shape\n        OH = H - KH + 1\n        OW = W - KW + 1\n        wr = w.permute(2, 3, 1, 0).reshape(KH * KW * Ci, Co).contiguous()\n        whi = wr.to(torch.bfloat16)\n        wlo = (wr - whi.float()).to(torch.bfloat16)\n        out = torch.empty((N, 1, OH, OW), device=x.device, dtype=torch.float32)\n        M = N * OH * OW\n        BN = max(16, triton.next_power_of_2(Co))\n        grid = lambda META: (triton.cdiv(M, META[\"BM\"]),)\n        conv_min_kernel[grid](x, whi, wlo, self.conv.bias, out, M, Ci, H, W, OH, OW, Co,\n                              float(self.scale_factor), KH=KH, KW=KW, BN=BN)\n        return out\n```\n",
   "demote": "Keep the conv(olution)?[^.]{0,40}cuDNN|Leave the conv[^.]{0,30}cuDNN|benchmark against cuDNN|naive Triton or CUDA kernel rarely wins",
   "provenance": {
    "origin": "engine (EXTEND)",
    "round": "E",
    "hypothesis": "Dense Conv2d/Conv3d as a Triton implicit GEMM with an in-register bf16 3-product split and a fused epilogue",
    "trial_per_problem": {
     "L1_080": [
      0.4429,
      2.294
     ],
     "L2_032": [
      1.1083,
      2.4977
     ]
    },
    "gain": 1.62,
    "adoption": {
     "L1_080": "ADOPTED: yes - Implicit-GEMM Triton conv over (r,s,ci) using pre-split bf16 hi/lo inputs and weights, three bf16 dots into fp32 accumulators, and a fused bias epilogue storing NCHW; weights aren't cac",
     "L2_032": "ADOPTED: yes - Triton implicit GEMM over (r,s,ci), in-register bf16 hi/lo split, three dots into fp32 accumulator, fused bias/scale/min epilogue. Skips weight caching and channels_last prep."
    }
   },
   "example": {
    "pid": "L2_032",
    "source": "import torch\nimport torch.nn as nn\nimport triton\nimport triton.language as tl\n\n\n@triton.autotune(\n    configs=[\n        triton.Config({\"BM\": 128, \"BK\": 32}, num_stages=3, num_warps=8),\n        triton.Config({\"BM\": 64, \"BK\": 32}, num_stages=4, num_warps=4),\n        triton.Config({\"BM\": 128, \"BK\": 64}, num_stages=2, num_warps=8),\n    ],\n    key=[\"M\", \"Ci\", \"Co\"],\n)\n@triton.jit\ndef conv_min_kernel(x, wh, wl, b, out, M, Ci, H, W, OH, OW, Co, scale,\n                    KH: tl.constexpr, KW: tl.constexpr, BN: tl.constexpr,\n                    BM: tl.constexpr, BK: tl.constexpr):\n    pid = tl.program_id(0)\n    rm = pid * BM + tl.arange(0, BM)\n    mm = rm < M\n    rmc = tl.where(mm, rm, 0)\n    ow = rmc % OW\n    t = rmc // OW\n    oh = t % OH\n    n = t // OH\n    base = n * (Ci * H * W) + oh * W + ow\n    rk = tl.arange(0, BK)\n    rn = tl.arange(0, BN)\n    nm = rn < Co\n    HW = H * W\n    nci = tl.cdiv(Ci, BK)\n    total = KH * KW * nci\n    acc = tl.zeros((BM, BN), dtype=tl.float32)\n    for kk in range(0, total):\n        tap = kk // nci\n        c0 = (kk - tap * nci) * BK\n        r = tap // KW\n        s = tap - r * KW\n        ci = c0 + rk\n        cm = ci < Ci\n        a = tl.load(x + base[:, None] + (r * W + s) + ci[None, :] * HW,\n                    mask=cm[None, :], other=0.0)\n        krow = tap * Ci + ci\n        boff = krow[:, None] * Co + rn[None, :]\n        bmask = cm[:, None] & nm[None, :]\n        bh = tl.load(wh + boff, mask=bmask, other=0.0)\n        bl = tl.load(wl + boff, mask=bmask, other=0.0)\n        ah = a.to(tl.bfloat16)\n        al = (a - ah.to(tl.float32)).to(tl.bfloat16)\n        acc = tl.dot(al, bh, acc)\n        acc = tl.dot(ah, bl, acc)\n        acc = tl.dot(ah, bh, acc)\n    bias = tl.load(b + rn, mask=nm, other=0.0)\n    v = (acc + bias[None, :]) * scale\n    v = tl.where(nm[None, :], v, float(\"inf\"))\n    mn = tl.min(v, axis=1)\n    tl.store(out + rm, mn, mask=mm)\n\n\nclass ModelNew(nn.Module):\n    def __init__(self, in_channels, out_channels, kernel_size, scale_factor):\n        super().__init__()\n        self.conv = nn.Conv2d(in_channels, out_channels, kernel_size)\n        self.scale_factor = scale_factor\n\n    def forward(self, x):\n        x = x.contiguous()\n        w = self.conv.weight\n        Co, Ci, KH, KW = w.shape\n        N, _, H, W = x.shape\n        OH = H - KH + 1\n        OW = W - KW + 1\n        wr = w.permute(2, 3, 1, 0).reshape(KH * KW * Ci, Co).contiguous()\n        whi = wr.to(torch.bfloat16)\n        wlo = (wr - whi.float()).to(torch.bfloat16)\n        out = torch.empty((N, 1, OH, OW), device=x.device, dtype=torch.float32)\n        M = N * OH * OW\n        BN = max(16, triton.next_power_of_2(Co))\n        grid = lambda META: (triton.cdiv(M, META[\"BM\"]),)\n        conv_min_kernel[grid](x, whi, wlo, self.conv.bias, out, M, Ci, H, W, OH, OW, Co,\n                              float(self.scale_factor), KH=KH, KW=KW, BN=BN)\n        return out\n",
    "speedup": 2.4977,
    "status": "ok"
   }
  },
  {
   "id": "E-H3obs",
   "include": "ConvTranspose[123]d|conv_transpose[123]d",
   "exclude": null,
   "exclude_families": [],
   "text": "### Verified technique (E-H3obs, observed in a verified solution; admitted by the KFORGE engine): Split-bf16 implicit-GEMM transposed conv via padded channels-last taps\nMechanism: A stride-1 ConvTranspose3d is rewritten as an ordinary convolution, using a flipped and transposed weight and an input zero-padded by K\u22121\u2212p. A prep kernel writes the input once into a channels-last buffer that already contains the spatial halo, with channels padded to a power of two (at least 16). The main kernel is then an implicit GEMM: each program owns BM output voxels, loops over the KD\u00b7KH\u00b7KW taps, and for each tap does an unmasked contiguous [BM\u00d7CIP] activation load against a [CIP\u00d7COP] weight slice. It accumulates with tensor-core tl.dot and writes straight to NCDHW with the bias fused.\n\nNumerics: Every fp32 operand is split into a bf16 high part and a bf16 residual (lo = x \u2212 bf16(x)). Three bf16 MMAs are computed (hi\u00b7hi, hi\u00b7lo, lo\u00b7hi), dropping lo\u00b7lo, all accumulated in fp32. The small terms go into a separate accumulator that is added at the end, which recovers roughly 16 mantissa bits and passes atol/rtol 1e-4 while running on bf16 tensor cores. This avoids TF32, which would fail the tolerance, and avoids slow FP32 FMA.\n\nWhere it applies: The pattern suits any conv or transposed conv with modest channel counts (Cin and Cout small enough to fit one tile) and many spatial positions. It also fits any fp32 GEMM-like op that needs near-fp32 accuracy but wants tensor-core throughput.\n\nWhat to watch:\n- The pre-padded buffers cost extra memory and a separate prep pass.\n- Channel padding wastes MMA work when C is far from a power of two.\n- Large C or Co would need a K-loop and N-tiling rather than whole-channel tiles.\n- Stride > 1, groups, or output_padding fall back to PyTorch.\n- Error grows with the reduction length (taps \u00d7 channels), so recheck the tolerance for larger kernels or inputs with a large dynamic range.\n- Verified on a training problem (protected evaluator): L1_058 conv_transposed_3D__asymmetric_input__asymmetric_kernel: 4.93x vs 2.17x with the previous library.\n- Verified example (L1_058, 4.93x; adapt what applies):\n```python\nimport torch\nimport torch.nn as nn\nimport torch.nn.functional as F\nimport triton\nimport triton.language as tl\n\n\n@triton.jit\ndef prep_x_kernel(x_ptr, xh_ptr, xl_ptr, P, C, Di, Hi, Wi, Dp, Hp, Wp, pd, ph, pw,\n                  CIP: tl.constexpr, BP: tl.constexpr):\n    p = tl.program_id(0) * BP + tl.arange(0, BP)\n    pm = p < P\n    pc = tl.where(pm, p, 0)\n    wq = pc % Wp\n    t = pc // Wp\n    hq = t % Hp\n    t = t // Hp\n    dq = t % Dp\n    n = t // Dp\n    idd = dq - pd\n    ih = hq - ph\n    iw = wq - pw\n    valid = pm & (idd >= 0) & (idd < Di) & (ih >= 0) & (ih < Hi) & (iw >= 0) & (iw < Wi)\n    ci = tl.arange(0, CIP)\n    Sin = Di * Hi * Wi\n    sp = (idd * Hi + ih) * Wi + iw\n    addr = n[:, None] * (C * Sin) + ci[None, :] * Sin + sp[:, None]\n    m = valid[:, None] & (ci[None, :] < C)\n    v = tl.load(x_ptr + addr, mask=m, other=0.0)\n    h = v.to(tl.bfloat16)\n    lo = (v - h.to(tl.float32)).to(tl.bfloat16)\n    o = pc[:, None] * CIP + ci[None, :]\n    sm = pm[:, None] & (ci[None, :] < CIP)\n    tl.store(xh_ptr + o, h, mask=sm)\n    tl.store(xl_ptr + o, lo, mask=sm)\n\n\n@triton.autotune(\n    configs=[\n        triton.Config({\"BM\": 128}, num_stages=3, num_warps=4),\n        triton.Config({\"BM\": 128}, num_stages=4, num_warps=4),\n        triton.Config({\"BM\": 256}, num_stages=3, num_warps=8),\n        triton.Config({\"BM\": 64}, num_stages=4, num_warps=4),\n    ],\n    key=[\"M\", \"NT\", \"CO\"],\n)\n@triton.jit\ndef convt_kernel(xh_ptr, xl_ptr, wh_ptr, wl_ptr, b_ptr, y_ptr, M, S, CO, NT,\n                 Do, Ho, Wo, Dp, Hp, Wp, KH, KW,\n                 CIP: tl.constexpr, COP: tl.constexpr, HAS_BIAS: tl.constexpr, BM: tl.constexpr):\n    pid = tl.program_id(0)\n    m = pid * BM + tl.arange(0, BM)\n    mm = m < M\n    mc = tl.where(mm, m, 0)\n    ow = mc % Wo\n    t = mc // Wo\n    oh = t % Ho\n    t = t // Ho\n    od = t % Do\n    n = t // Do\n    base = ((n * Dp + od) * Hp + oh) * Wp + ow\n    ci = tl.arange(0, CIP)\n    co = tl.arange(0, COP)\n    acc = tl.zeros((BM, COP), dtype=tl.float32)\n    acc2 = tl.zeros((BM, COP), dtype=tl.float32)\n    for tap in range(NT):\n        kw = tap % KW\n        tt = tap // KW\n        kh = tt % KH\n        kd = tt // KH\n        off = (kd * Hp + kh) * Wp + kw\n        pa = (base + off)[:, None] * CIP + ci[None, :]\n        ah = tl.load(xh_ptr + pa)\n        al = tl.load(xl_ptr + pa)\n        pw_ = tap * (CIP * COP) + ci[:, None] * COP + co[None, :]\n        bh = tl.load(wh_ptr + pw_)\n        bl = tl.load(wl_ptr + pw_)\n        acc = tl.dot(ah, bh, acc)\n        acc2 = tl.dot(ah, bl, acc2)\n        acc2 = tl.dot(al, bh, acc2)\n    v = acc + acc2\n    if HAS_BIAS:\n        v = v + tl.load(b_ptr + co, mask=co < CO, other=0.0)[None, :]\n    s = mc - n * S\n    addr = n[:, None] * (CO * S) + co[None, :] * S + s[:, None]\n    tl.store(y_ptr + addr, v, mask=mm[:, None] & (co[None, :] < CO))\n\n\ndef _p2(v):\n    r = 16\n    while r < v:\n        r *= 2\n    return r\n\n\nclass ModelNew(nn.Module):\n    def __init__(self, in_channels: int, out_channels: int, kernel_size: tuple, stride: tuple = (1, 1, 1),\n                 padding: tuple = (0, 0, 0), output_padding: tuple = (0, 0, 0), groups: int = 1, bias: bool = False):\n        super(ModelNew, self).__init__()\n        self.conv_transpose3d = nn.ConvTranspose3d(in_channels, out_channels, kernel_size, stride=stride,\n                                                   padding=padding, output_padding=output_padding,\n                                                   groups=groups, bias=bias)\n        self.ks = tuple(kernel_size)\n        self.st = tuple(stride)\n        self.pad = tuple(padding)\n        self.opad = tuple(output_padding)\n        self.groups = groups\n\n    def forward(self, x):\n        KD, KH, KW = self.ks\n        pd0, ph0, pw0 = self.pad\n        ok = (self.st == (1, 1, 1) and self.groups == 1 and self.opad == (0, 0, 0)\n              and pd0 <= KD - 1 and ph0 <= KH - 1 and pw0 <= KW - 1)\n        if not ok:\n            return self.conv_transpose3d(x)\n        x = x.contiguous()\n        N, C, Di, Hi, Wi = x.shape\n        w = self.conv_transpose3d.weight\n        Co = w.shape[1]\n        CIP = _p2(C)\n        COP = _p2(Co)\n        pd, ph, pw = KD - 1 - pd0, KH - 1 - ph0, KW - 1 - pw0\n        Dp, Hp, Wp = Di + 2 * pd, Hi + 2 * ph, Wi + 2 * pw\n        Do, Ho, Wo = Dp - KD + 1, Hp - KH + 1, Wp - KW + 1\n        wf = w.flip(2, 3, 4).permute(2, 3, 4, 0, 1)\n        wpad = torch.zeros((KD, KH, KW, CIP, COP), device=x.device, dtype=torch.float32)\n        wpad[:, :, :, :C, :Co] = wf\n        wh = wpad.to(torch.bfloat16)\n        wl = (wpad - wh.float()).to(torch.bfloat16)\n        P = N * Dp * Hp * Wp\n        xh = torch.empty((P, CIP), device=x.device, dtype=torch.bfloat16)\n        xl = torch.empty((P, CIP), device=x.device, dtype=torch.bfloat16)\n        BP = 64\n        prep_x_kernel[(triton.cdiv(P, BP),)](x, xh, xl, P, C, Di, Hi, Wi, Dp, Hp, Wp, pd, ph, pw,\n                                             CIP=CIP, BP=BP)\n        S = Do * Ho * Wo\n        M = N * S\n        y = torch.empty((N, Co, Do, Ho, Wo), device=x.device, dtype=torch.float32)\n        b = self.conv_transpose3d.bias\n        NT = KD * KH * KW\n        grid = lambda META: (triton.cdiv(M, META[\"BM\"]),)\n        convt_kernel[grid](xh, xl, wh, wl, b if b is not None else y, y, M, S, Co, NT,\n                           Do, Ho, Wo, Dp, Hp, Wp, KH, KW,\n                           CIP=CIP, COP=COP, HAS_BIAS=b is not None)\n        return y\n```\n",
   "demote": null,
   "provenance": {
    "origin": "engine (observed in a verified solution)",
    "round": "E",
    "hypothesis": "ConvTranspose as GEMM: a stride == kernel transpose conv is one GEMM plus pixel shuffle, and general strides use sub-pixel decomposition (Shi et al. 2016)",
    "trial_per_problem": {
     "L1_058": [
      2.1678,
      4.9293
     ],
     "L2_015": [
      1.426,
      4.2432
     ]
    },
    "gain": 2.789,
    "adoption": {
     "L1_058": "ADOPTED: no - It handles only stride-1 transposed conv, as a flipped-kernel implicit-GEMM direct conv, and falls back otherwise; there is no s==k GEMM+pixel-shuffle or sub-pixel phase decomposition.",
     "L2_015": "ADOPTED: no - It is a hand-written CUDA direct-FMA kernel with parity-based tap selection. There are no GEMMs, no hi/lo bf16 torch.mm, and no per-phase stride-1 sub-convolutions."
    }
   },
   "example": {
    "pid": "L1_058",
    "source": "import torch\nimport torch.nn as nn\nimport torch.nn.functional as F\nimport triton\nimport triton.language as tl\n\n\n@triton.jit\ndef prep_x_kernel(x_ptr, xh_ptr, xl_ptr, P, C, Di, Hi, Wi, Dp, Hp, Wp, pd, ph, pw,\n                  CIP: tl.constexpr, BP: tl.constexpr):\n    p = tl.program_id(0) * BP + tl.arange(0, BP)\n    pm = p < P\n    pc = tl.where(pm, p, 0)\n    wq = pc % Wp\n    t = pc // Wp\n    hq = t % Hp\n    t = t // Hp\n    dq = t % Dp\n    n = t // Dp\n    idd = dq - pd\n    ih = hq - ph\n    iw = wq - pw\n    valid = pm & (idd >= 0) & (idd < Di) & (ih >= 0) & (ih < Hi) & (iw >= 0) & (iw < Wi)\n    ci = tl.arange(0, CIP)\n    Sin = Di * Hi * Wi\n    sp = (idd * Hi + ih) * Wi + iw\n    addr = n[:, None] * (C * Sin) + ci[None, :] * Sin + sp[:, None]\n    m = valid[:, None] & (ci[None, :] < C)\n    v = tl.load(x_ptr + addr, mask=m, other=0.0)\n    h = v.to(tl.bfloat16)\n    lo = (v - h.to(tl.float32)).to(tl.bfloat16)\n    o = pc[:, None] * CIP + ci[None, :]\n    sm = pm[:, None] & (ci[None, :] < CIP)\n    tl.store(xh_ptr + o, h, mask=sm)\n    tl.store(xl_ptr + o, lo, mask=sm)\n\n\n@triton.autotune(\n    configs=[\n        triton.Config({\"BM\": 128}, num_stages=3, num_warps=4),\n        triton.Config({\"BM\": 128}, num_stages=4, num_warps=4),\n        triton.Config({\"BM\": 256}, num_stages=3, num_warps=8),\n        triton.Config({\"BM\": 64}, num_stages=4, num_warps=4),\n    ],\n    key=[\"M\", \"NT\", \"CO\"],\n)\n@triton.jit\ndef convt_kernel(xh_ptr, xl_ptr, wh_ptr, wl_ptr, b_ptr, y_ptr, M, S, CO, NT,\n                 Do, Ho, Wo, Dp, Hp, Wp, KH, KW,\n                 CIP: tl.constexpr, COP: tl.constexpr, HAS_BIAS: tl.constexpr, BM: tl.constexpr):\n    pid = tl.program_id(0)\n    m = pid * BM + tl.arange(0, BM)\n    mm = m < M\n    mc = tl.where(mm, m, 0)\n    ow = mc % Wo\n    t = mc // Wo\n    oh = t % Ho\n    t = t // Ho\n    od = t % Do\n    n = t // Do\n    base = ((n * Dp + od) * Hp + oh) * Wp + ow\n    ci = tl.arange(0, CIP)\n    co = tl.arange(0, COP)\n    acc = tl.zeros((BM, COP), dtype=tl.float32)\n    acc2 = tl.zeros((BM, COP), dtype=tl.float32)\n    for tap in range(NT):\n        kw = tap % KW\n        tt = tap // KW\n        kh = tt % KH\n        kd = tt // KH\n        off = (kd * Hp + kh) * Wp + kw\n        pa = (base + off)[:, None] * CIP + ci[None, :]\n        ah = tl.load(xh_ptr + pa)\n        al = tl.load(xl_ptr + pa)\n        pw_ = tap * (CIP * COP) + ci[:, None] * COP + co[None, :]\n        bh = tl.load(wh_ptr + pw_)\n        bl = tl.load(wl_ptr + pw_)\n        acc = tl.dot(ah, bh, acc)\n        acc2 = tl.dot(ah, bl, acc2)\n        acc2 = tl.dot(al, bh, acc2)\n    v = acc + acc2\n    if HAS_BIAS:\n        v = v + tl.load(b_ptr + co, mask=co < CO, other=0.0)[None, :]\n    s = mc - n * S\n    addr = n[:, None] * (CO * S) + co[None, :] * S + s[:, None]\n    tl.store(y_ptr + addr, v, mask=mm[:, None] & (co[None, :] < CO))\n\n\ndef _p2(v):\n    r = 16\n    while r < v:\n        r *= 2\n    return r\n\n\nclass ModelNew(nn.Module):\n    def __init__(self, in_channels: int, out_channels: int, kernel_size: tuple, stride: tuple = (1, 1, 1),\n                 padding: tuple = (0, 0, 0), output_padding: tuple = (0, 0, 0), groups: int = 1, bias: bool = False):\n        super(ModelNew, self).__init__()\n        self.conv_transpose3d = nn.ConvTranspose3d(in_channels, out_channels, kernel_size, stride=stride,\n                                                   padding=padding, output_padding=output_padding,\n                                                   groups=groups, bias=bias)\n        self.ks = tuple(kernel_size)\n        self.st = tuple(stride)\n        self.pad = tuple(padding)\n        self.opad = tuple(output_padding)\n        self.groups = groups\n\n    def forward(self, x):\n        KD, KH, KW = self.ks\n        pd0, ph0, pw0 = self.pad\n        ok = (self.st == (1, 1, 1) and self.groups == 1 and self.opad == (0, 0, 0)\n              and pd0 <= KD - 1 and ph0 <= KH - 1 and pw0 <= KW - 1)\n        if not ok:\n            return self.conv_transpose3d(x)\n        x = x.contiguous()\n        N, C, Di, Hi, Wi = x.shape\n        w = self.conv_transpose3d.weight\n        Co = w.shape[1]\n        CIP = _p2(C)\n        COP = _p2(Co)\n        pd, ph, pw = KD - 1 - pd0, KH - 1 - ph0, KW - 1 - pw0\n        Dp, Hp, Wp = Di + 2 * pd, Hi + 2 * ph, Wi + 2 * pw\n        Do, Ho, Wo = Dp - KD + 1, Hp - KH + 1, Wp - KW + 1\n        wf = w.flip(2, 3, 4).permute(2, 3, 4, 0, 1)\n        wpad = torch.zeros((KD, KH, KW, CIP, COP), device=x.device, dtype=torch.float32)\n        wpad[:, :, :, :C, :Co] = wf\n        wh = wpad.to(torch.bfloat16)\n        wl = (wpad - wh.float()).to(torch.bfloat16)\n        P = N * Dp * Hp * Wp\n        xh = torch.empty((P, CIP), device=x.device, dtype=torch.bfloat16)\n        xl = torch.empty((P, CIP), device=x.device, dtype=torch.bfloat16)\n        BP = 64\n        prep_x_kernel[(triton.cdiv(P, BP),)](x, xh, xl, P, C, Di, Hi, Wi, Dp, Hp, Wp, pd, ph, pw,\n                                             CIP=CIP, BP=BP)\n        S = Do * Ho * Wo\n        M = N * S\n        y = torch.empty((N, Co, Do, Ho, Wo), device=x.device, dtype=torch.float32)\n        b = self.conv_transpose3d.bias\n        NT = KD * KH * KW\n        grid = lambda META: (triton.cdiv(M, META[\"BM\"]),)\n        convt_kernel[grid](xh, xl, wh, wl, b if b is not None else y, y, M, S, Co, NT,\n                           Do, Ho, Wo, Dp, Hp, Wp, KH, KW,\n                           CIP=CIP, COP=COP, HAS_BIAS=b is not None)\n        return y\n",
    "speedup": 4.9293,
    "status": "ok"
   }
  },
  {
   "id": "E-H4obs",
   "include": "nn\\.Linear\\(\\s*(in_features|\\w+)\\s*,|F\\.linear\\(|torch\\.matmul\\(",
   "exclude": null,
   "exclude_families": [],
   "text": "### Verified technique (E-H4obs, observed in a verified solution; admitted by the KFORGE engine): bf16x3 split-precision GEMM via K-concatenation on tensor cores\nEach fp32 input is split into a bf16 \"hi\" part plus a bf16 \"lo\" residual (lo = x \u2212 hi). The pieces are laid out along the K dimension so that one ordinary bf16 GEMM computes the cross terms hi\u00b7hi + hi\u00b7lo + lo\u00b7hi in a single pass. A's rows become [hi | hi | lo] and B's columns become [hi ; lo ; hi]. K is also cut into 4 chunks run as one batched bmm with fp32 output. A small elementwise kernel then sums the chunk partials, which shortens each accumulation chain and adds parallelism (split-K).\n\nNumerically, hi carries about 8 mantissa bits and lo the next 8, so the result has roughly 16 bits of effective precision. The dropped lo\u00b7lo term is about 2^-16 relative, and accumulation stays in fp32. That lands near 1e-5 relative error, inside a 1e-4 tolerance, while running on bf16 tensor cores (about 312 TFLOPS on A100). The alternative is fp32 FMA (about 19.5 TFLOPS) when TF32 is disallowed. Since bf16 shares fp32's exponent range, the split does not overflow.\n\nThe technique applies to any large, compute-bound fp32 matmul, conv-as-GEMM, or batched GEMM where strict fp32 accuracy is checked but TF32 is unavailable or too imprecise. Things to watch:\n- **Cost:** it does 3\u00d7 the FLOPs and writes 3\u00d7 bf16 copies of the operands, so it loses on small or memory-bound shapes.\n- **Tiny residuals:** lo values can underflow or flush to zero for very small or denormal inputs.\n- **Error growth:** error grows with K and with ill-conditioned or cancellation-heavy data, so validate on worst-case magnitudes.\n- **Dependency:** `torch.bmm(..., out_dtype=float32)` requires a recent PyTorch.\n- Verified on a training problem (protected evaluator): L1_002 Standard_matrix_multiplication_: 3.56x vs 3.25x with the previous library.\n- Verified example (L1_002, 3.56x; adapt what applies):\n```python\nimport torch\nimport torch.nn as nn\nimport triton\nimport triton.language as tl\n\n\n@triton.jit\ndef split_rows_kernel(src, dst, R, K, KC, BR: tl.constexpr, BKK: tl.constexpr):\n    # src [R, K] fp32 -> dst [C, R, 3KC] bf16, each chunk: [hi | hi | lo]\n    r = tl.program_id(0) * BR + tl.arange(0, BR)\n    k = tl.program_id(1) * BKK + tl.arange(0, BKK)\n    m = (r[:, None] < R) & (k[None, :] < K)\n    v = tl.load(src + r[:, None] * K + k[None, :], mask=m, other=0.0)\n    h = v.to(tl.bfloat16)\n    lo = (v - h.to(tl.float32)).to(tl.bfloat16)\n    c = k // KC\n    kk = k - c * KC\n    base = dst + c[None, :] * (R * 3 * KC) + r[:, None] * (3 * KC) + kk[None, :]\n    tl.store(base, h, mask=m)\n    tl.store(base + KC, h, mask=m)\n    tl.store(base + 2 * KC, lo, mask=m)\n\n\n@triton.jit\ndef split_blocks_kernel(src, dst, T, N, KC, BLOCK: tl.constexpr):\n    # src [K, N] fp32 flat -> dst [C, 3KC, N] bf16, each chunk: [hi; lo; hi]\n    i = tl.program_id(0) * BLOCK + tl.arange(0, BLOCK)\n    m = i < T\n    v = tl.load(src + i, mask=m, other=0.0)\n    h = v.to(tl.bfloat16)\n    lo = (v - h.to(tl.float32)).to(tl.bfloat16)\n    k = i // N\n    n = i - k * N\n    c = k // KC\n    kk = k - c * KC\n    o = c * (3 * KC * N) + kk * N + n\n    tl.store(dst + o, h, mask=m)\n    tl.store(dst + o + KC * N, lo, mask=m)\n    tl.store(dst + o + 2 * KC * N, h, mask=m)\n\n\n@triton.jit\ndef chunk_sum_kernel(src, dst, T, C: tl.constexpr, BLOCK: tl.constexpr):\n    i = tl.program_id(0) * BLOCK + tl.arange(0, BLOCK)\n    m = i < T\n    acc = tl.load(src + i, mask=m, other=0.0)\n    for c in tl.static_range(1, C):\n        acc += tl.load(src + c * T + i, mask=m, other=0.0)\n    tl.store(dst + i, acc, mask=m)\n\n\nclass ModelNew(nn.Module):\n    def __init__(self):\n        super(ModelNew, self).__init__()\n\n    def forward(self, A: torch.Tensor, B: torch.Tensor) -> torch.Tensor:\n        A = A.contiguous()\n        B = B.contiguous()\n        M, K = A.shape\n        N = B.shape[1]\n        C = 4 if K % 4 == 0 else 1\n        KC = K // C\n        a3 = torch.empty((C, M, 3 * KC), device=A.device, dtype=torch.bfloat16)\n        split_rows_kernel[(triton.cdiv(M, 32), triton.cdiv(K, 128))](A, a3, M, K, KC, BR=32, BKK=128)\n        b3 = torch.empty((C, 3 * KC, N), device=B.device, dtype=torch.bfloat16)\n        T = K * N\n        split_blocks_kernel[(triton.cdiv(T, 2048),)](B, b3, T, N, KC, BLOCK=2048)\n        part = torch.bmm(a3, b3, out_dtype=torch.float32)\n        if C == 1:\n            return part[0]\n        out = torch.empty((M, N), device=A.device, dtype=torch.float32)\n        TO = M * N\n        chunk_sum_kernel[(triton.cdiv(TO, 2048),)](part, out, TO, C=C, BLOCK=2048)\n        return out\n```\n",
   "demote": null,
   "provenance": {
    "origin": "engine (observed in a verified solution)",
    "round": "E",
    "hypothesis": "Skinny-M, huge-weight Linear with a fused Triton kernel that reads fp32 W once and splits it into bf16 hi/lo in registers",
    "trial_per_problem": {
     "L1_008": [
      4.0987,
      4.1289
     ],
     "L1_002": [
      3.2478,
      3.5639
     ]
    },
    "gain": 0.173,
    "adoption": {
     "L1_002": "ADOPTED: no - It pre-splits A and B into bf16 hi/lo copies in memory and runs torch.bmm. There is no fused Triton kernel splitting fp32 W tiles in registers before three tl.dot calls."
    }
   },
   "example": {
    "pid": "L1_002",
    "source": "import torch\nimport torch.nn as nn\nimport triton\nimport triton.language as tl\n\n\n@triton.jit\ndef split_rows_kernel(src, dst, R, K, KC, BR: tl.constexpr, BKK: tl.constexpr):\n    # src [R, K] fp32 -> dst [C, R, 3KC] bf16, each chunk: [hi | hi | lo]\n    r = tl.program_id(0) * BR + tl.arange(0, BR)\n    k = tl.program_id(1) * BKK + tl.arange(0, BKK)\n    m = (r[:, None] < R) & (k[None, :] < K)\n    v = tl.load(src + r[:, None] * K + k[None, :], mask=m, other=0.0)\n    h = v.to(tl.bfloat16)\n    lo = (v - h.to(tl.float32)).to(tl.bfloat16)\n    c = k // KC\n    kk = k - c * KC\n    base = dst + c[None, :] * (R * 3 * KC) + r[:, None] * (3 * KC) + kk[None, :]\n    tl.store(base, h, mask=m)\n    tl.store(base + KC, h, mask=m)\n    tl.store(base + 2 * KC, lo, mask=m)\n\n\n@triton.jit\ndef split_blocks_kernel(src, dst, T, N, KC, BLOCK: tl.constexpr):\n    # src [K, N] fp32 flat -> dst [C, 3KC, N] bf16, each chunk: [hi; lo; hi]\n    i = tl.program_id(0) * BLOCK + tl.arange(0, BLOCK)\n    m = i < T\n    v = tl.load(src + i, mask=m, other=0.0)\n    h = v.to(tl.bfloat16)\n    lo = (v - h.to(tl.float32)).to(tl.bfloat16)\n    k = i // N\n    n = i - k * N\n    c = k // KC\n    kk = k - c * KC\n    o = c * (3 * KC * N) + kk * N + n\n    tl.store(dst + o, h, mask=m)\n    tl.store(dst + o + KC * N, lo, mask=m)\n    tl.store(dst + o + 2 * KC * N, h, mask=m)\n\n\n@triton.jit\ndef chunk_sum_kernel(src, dst, T, C: tl.constexpr, BLOCK: tl.constexpr):\n    i = tl.program_id(0) * BLOCK + tl.arange(0, BLOCK)\n    m = i < T\n    acc = tl.load(src + i, mask=m, other=0.0)\n    for c in tl.static_range(1, C):\n        acc += tl.load(src + c * T + i, mask=m, other=0.0)\n    tl.store(dst + i, acc, mask=m)\n\n\nclass ModelNew(nn.Module):\n    def __init__(self):\n        super(ModelNew, self).__init__()\n\n    def forward(self, A: torch.Tensor, B: torch.Tensor) -> torch.Tensor:\n        A = A.contiguous()\n        B = B.contiguous()\n        M, K = A.shape\n        N = B.shape[1]\n        C = 4 if K % 4 == 0 else 1\n        KC = K // C\n        a3 = torch.empty((C, M, 3 * KC), device=A.device, dtype=torch.bfloat16)\n        split_rows_kernel[(triton.cdiv(M, 32), triton.cdiv(K, 128))](A, a3, M, K, KC, BR=32, BKK=128)\n        b3 = torch.empty((C, 3 * KC, N), device=B.device, dtype=torch.bfloat16)\n        T = K * N\n        split_blocks_kernel[(triton.cdiv(T, 2048),)](B, b3, T, N, KC, BLOCK=2048)\n        part = torch.bmm(a3, b3, out_dtype=torch.float32)\n        if C == 1:\n            return part[0]\n        out = torch.empty((M, N), device=A.device, dtype=torch.float32)\n        TO = M * N\n        chunk_sum_kernel[(triton.cdiv(TO, 2048),)](part, out, TO, C=C, BLOCK=2048)\n        return out\n",
    "speedup": 3.5639,
    "status": "ok"
   }
  },
  {
   "id": "E-H5",
   "include": "BatchNorm[123]d",
   "exclude": null,
   "exclude_families": [],
   "text": "### Verified technique (E-H5, admitted by the KFORGE engine): Training-mode BatchNorm with a following elementwise or pool tail: two Triton kernels (parallel Welford partials, then a fused apply) beat cuDNN BN plus eager activations\nThe recorded losses came from computing stats with several torch reductions and scalar ops (many launches and passes), not from replacing cuDNN BN as such. Under 1e-4 numerics, two kernels with at most one tiny torch op between them should give 1.2-1.6x on post-conv BN plus activation chains. The first kernel computes per-(channel, block) Welford partials and the second fuses the combine, normalize, affine, activation and pool.\nPhysics: Eager training BN reads the tensor twice and writes once, then each activation or pool adds a full read and write. The fused path reads the tensor twice and writes once, or writes 1/4 for a 2x2 pool. For tensors of 100 MB or more on 1.5 TB/s, that saves 0.1-0.5 ms per removed pass. Welford/Chan merging avoids the cancellation of E[x^2] - E[x]^2, so fp32 matches cuDNN to about 1e-6.\nHow: Kernel A uses a grid of (C, P) blocks, each streaming a contiguous HW chunk over all N. It writes count, mean and M2 partials to a [C, P] buffer. Kernel B has each program first merge its channel's P partials in registers with the Chan formula; this is cheap with P <= 64 and avoids a separate finalize launch. It then computes scale = gamma*rstd and shift = beta - mean*scale, applies the activation and any pool, and stores the result. Update running_mean and running_var (unbiased, factor M/(M-1)) with one in-place torch op under no_grad, using self.<x> buffers only. Keep the conv in cuDNN, or use H1/H2, and use int64 offsets throughout.\n- Verified on training problems (protected evaluator): L2_097 Matmul_BatchNorm_BiasAdd_Divide_Swish: 3.53x vs 3.47x with the previous library; L3_046 NetVladWithGhostClusters: 2.96x vs 2.91x with the previous library.\n- Verified example implementing it (L3_046, 2.96x; adapt what applies):\n```python\nimport math\nimport torch\nimport torch.nn as nn\nimport triton\nimport triton.language as tl\n\n\n@triton.jit\ndef assign_kernel(x_ptr, w_ptr, a_ptr, pmean_ptr, pm2_ptr, pcnt_ptr, M, C,\n                  D: tl.constexpr, BM: tl.constexpr, BN: tl.constexpr, BK: tl.constexpr):\n    pid = tl.program_id(0)\n    rm = pid * BM + tl.arange(0, BM)\n    rn = tl.arange(0, BN)\n    rk = tl.arange(0, BK)\n    rm64 = rm.to(tl.int64)\n    rmask = rm < M\n    nmask = rn < C\n    xp = x_ptr + rm64[:, None] * D + rk[None, :]\n    wp = w_ptr + rk[:, None] * C + rn[None, :]\n    acc = tl.zeros((BM, BN), dtype=tl.float32)\n    acc2 = tl.zeros((BM, BN), dtype=tl.float32)\n    for k in range(0, D, BK):\n        a = tl.load(xp, mask=rmask[:, None], other=0.0)\n        b = tl.load(wp, mask=nmask[None, :], other=0.0)\n        ah = a.to(tl.bfloat16)\n        al = (a - ah.to(tl.float32)).to(tl.bfloat16)\n        bh = b.to(tl.bfloat16)\n        bl = (b - bh.to(tl.float32)).to(tl.bfloat16)\n        acc2 = tl.dot(ah, bl, acc2)\n        acc2 = tl.dot(al, bh, acc2)\n        acc = tl.dot(ah, bh, acc)\n        xp += BK\n        wp += BK * C\n    v = acc + acc2\n    tl.store(a_ptr + rm64[:, None] * C + rn[None, :], v, mask=rmask[:, None] & nmask[None, :])\n    cnt = tl.minimum(M - pid * BM, BM).to(tl.float32)\n    vm = tl.where(rmask[:, None], v, 0.0)\n    mean = tl.sum(vm, 0) / cnt\n    dl = tl.where(rmask[:, None], v - mean[None, :], 0.0)\n    m2 = tl.sum(dl * dl, 0)\n    tl.store(pmean_ptr + pid * BN + rn, mean)\n    tl.store(pm2_ptr + pid * BN + rn, m2)\n    tl.store(pcnt_ptr + pid, cnt)\n\n\n@triton.jit\ndef stats_kernel(pmean_ptr, pm2_ptr, pcnt_ptr, g_ptr, b_ptr, sc_ptr, sh_ptr, mean_ptr, var_ptr,\n                 P, C, Mtot, eps, BP: tl.constexpr, BN: tl.constexpr):\n    rn = tl.arange(0, BN)\n    nm = rn < C\n    s = tl.zeros((BN,), dtype=tl.float32)\n    for p0 in range(0, P, BP):\n        rp = p0 + tl.arange(0, BP)\n        pk = rp < P\n        mu = tl.load(pmean_ptr + rp[:, None] * BN + rn[None, :], mask=pk[:, None], other=0.0)\n        c = tl.load(pcnt_ptr + rp, mask=pk, other=0.0)\n        s += tl.sum(mu * c[:, None], 0)\n    mean = s / Mtot\n    m2 = tl.zeros((BN,), dtype=tl.float32)\n    for p0 in range(0, P, BP):\n        rp = p0 + tl.arange(0, BP)\n        pk = rp < P\n        mu = tl.load(pmean_ptr + rp[:, None] * BN + rn[None, :], mask=pk[:, None], other=0.0)\n        q = tl.load(pm2_ptr + rp[:, None] * BN + rn[None, :], mask=pk[:, None], other=0.0)\n        c = tl.load(pcnt_ptr + rp, mask=pk, other=0.0)\n        d = mu - mean[None, :]\n        m2 += tl.sum(q + c[:, None] * d * d, 0)\n    var = m2 / Mtot\n    rstd = 1.0 / tl.sqrt(var + eps)\n    g = tl.load(g_ptr + rn, mask=nm, other=0.0)\n    bt = tl.load(b_ptr + rn, mask=nm, other=0.0)\n    sc = g * rstd\n    sh = bt - mean * sc\n    tl.store(sc_ptr + rn, sc)\n    tl.store(sh_ptr + rn, sh)\n    tl.store(mean_ptr + rn, mean)\n    tl.store(var_ptr + rn, var)\n\n\n@triton.jit\ndef vlad_kernel(x_ptr, a_ptr, sc_ptr, sh_ptr, c2_ptr, out_ptr, NS, C,\n                D: tl.constexpr, K: tl.constexpr, BNS: tl.constexpr, BC: tl.constexpr, BD: tl.constexpr):\n    b = tl.program_id(0).to(tl.int64)\n    rn = tl.arange(0, BNS)\n    nmask = rn < NS\n    rc = tl.arange(0, BC)\n    cmask = rc < C\n    rowbase = b * NS + rn\n    t = tl.load(a_ptr + rowbase[:, None] * C + rc[None, :], mask=nmask[:, None] & cmask[None, :], other=0.0)\n    sc = tl.load(sc_ptr + rc)\n    sh = tl.load(sh_ptr + rc)\n    z = t * sc[None, :] + sh[None, :]\n    z = tl.where(cmask[None, :], z, float(\"-inf\"))\n    mx = tl.max(z, 1)\n    e = tl.exp(z - mx[:, None])\n    den = tl.sum(e, 1)\n    rk = tl.arange(0, K)\n    t2 = tl.load(a_ptr + rowbase[:, None] * C + rk[None, :], mask=nmask[:, None], other=0.0)\n    sck = tl.load(sc_ptr + rk)\n    shk = tl.load(sh_ptr + rk)\n    A = tl.exp(t2 * sck[None, :] + shk[None, :] - mx[:, None]) / den[:, None]\n    A = tl.where(nmask[:, None], A, 0.0)\n    asum = tl.sum(A, 0)\n    At = tl.trans(A)\n    Ah = At.to(tl.bfloat16)\n    Al = (At - Ah.to(tl.float32)).to(tl.bfloat16)\n    ss = tl.zeros((K,), dtype=tl.float32)\n    xbase = x_ptr + b * NS * D\n    obase = out_ptr + b * D * K\n    for d0 in range(0, D, BD):\n        rd = d0 + tl.arange(0, BD)\n        xt = tl.load(xbase + rn[:, None] * D + rd[None, :], mask=nmask[:, None], other=0.0)\n        xh = xt.to(tl.bfloat16)\n        xl = (xt - xh.to(tl.float32)).to(tl.bfloat16)\n        v2 = tl.zeros((K, BD), dtype=tl.float32)\n        v2 = tl.dot(Ah, xl, v2)\n        v2 = tl.dot(Al, xh, v2)\n        v = tl.zeros((K, BD), dtype=tl.float32)\n        v = tl.dot(Ah, xh, v)\n        v = v + v2\n        c2 = tl.load(c2_ptr + rd[None, :] * K + rk[:, None])\n        v = v - asum[:, None] * c2\n        ss += tl.sum(v * v, 1)\n        tl.store(obase + rd[None, :] * K + rk[:, None], v)\n    kn = tl.maximum(tl.sqrt(ss), 1e-12)\n    inv = 1.0 / kn\n    tot = tl.sum(ss * inv * inv, 0)\n    g = tl.maximum(tl.sqrt(tot), 1e-12)\n    f = inv / g\n    tl.debug_barrier()\n    for d0 in range(0, D, BD):\n        rd = d0 + tl.arange(0, BD)\n        p = obase + rd[None, :] * K + rk[:, None]\n        v = tl.load(p)\n        tl.store(p, v * f[:, None])\n\n\nclass ModelNew(nn.Module):\n    def __init__(self, cluster_size, feature_size, ghost_clusters):\n        super(ModelNew, self).__init__()\n        self.feature_size = feature_size\n        self.cluster_size = cluster_size\n        self.ghost_clusters = ghost_clusters\n        init_sc = (1 / math.sqrt(feature_size))\n        clusters = cluster_size + ghost_clusters\n        self.clusters = nn.Parameter(init_sc * torch.randn(feature_size, clusters))\n        self.batch_norm = nn.BatchNorm1d(clusters)\n        self.clusters2 = nn.Parameter(init_sc * torch.randn(1, feature_size, cluster_size))\n        self.out_dim = self.cluster_size * feature_size\n\n    def forward(self, x, mask=None):\n        B, NS, D = x.shape\n        K = self.cluster_size\n        x2 = x.reshape(-1, D).contiguous()\n        M = x2.shape[0]\n        C = K + self.ghost_clusters\n        BN = max(16, triton.next_power_of_2(C))\n        dev = x.device\n        w = self```\n",
   "demote": "replace cuDNN BatchNorm",
   "provenance": {
    "origin": "engine (INVERT)",
    "round": "E",
    "hypothesis": "Training-mode BatchNorm with a following elementwise or pool tail: two Triton kernels (parallel Welford partials, then a fused apply) beat cuDNN BN plus eager activations",
    "trial_per_problem": {
     "L2_097": [
      3.4705,
      3.5266
     ],
     "L3_046": [
      2.9074,
      2.9605
     ]
    },
    "gain": 0.055,
    "adoption": {
     "L2_097": "ADOPTED: no - A single kernel computes each channel's stats with a two-pass sum/variance loop. There are no (C,P) Welford partials, no Chan merge, and no separate partials kernel.",
     "L3_046": "ADOPTED: yes - Per-block mean/M2 partials (GEMM epilogue), a Chan-formula combine producing scale/shift, and BN apply fused into the softmax/VLAD tail; finalize is separate."
    }
   },
   "example": {
    "pid": "L3_046",
    "source": "import math\nimport torch\nimport torch.nn as nn\nimport triton\nimport triton.language as tl\n\n\n@triton.jit\ndef assign_kernel(x_ptr, w_ptr, a_ptr, pmean_ptr, pm2_ptr, pcnt_ptr, M, C,\n                  D: tl.constexpr, BM: tl.constexpr, BN: tl.constexpr, BK: tl.constexpr):\n    pid = tl.program_id(0)\n    rm = pid * BM + tl.arange(0, BM)\n    rn = tl.arange(0, BN)\n    rk = tl.arange(0, BK)\n    rm64 = rm.to(tl.int64)\n    rmask = rm < M\n    nmask = rn < C\n    xp = x_ptr + rm64[:, None] * D + rk[None, :]\n    wp = w_ptr + rk[:, None] * C + rn[None, :]\n    acc = tl.zeros((BM, BN), dtype=tl.float32)\n    acc2 = tl.zeros((BM, BN), dtype=tl.float32)\n    for k in range(0, D, BK):\n        a = tl.load(xp, mask=rmask[:, None], other=0.0)\n        b = tl.load(wp, mask=nmask[None, :], other=0.0)\n        ah = a.to(tl.bfloat16)\n        al = (a - ah.to(tl.float32)).to(tl.bfloat16)\n        bh = b.to(tl.bfloat16)\n        bl = (b - bh.to(tl.float32)).to(tl.bfloat16)\n        acc2 = tl.dot(ah, bl, acc2)\n        acc2 = tl.dot(al, bh, acc2)\n        acc = tl.dot(ah, bh, acc)\n        xp += BK\n        wp += BK * C\n    v = acc + acc2\n    tl.store(a_ptr + rm64[:, None] * C + rn[None, :], v, mask=rmask[:, None] & nmask[None, :])\n    cnt = tl.minimum(M - pid * BM, BM).to(tl.float32)\n    vm = tl.where(rmask[:, None], v, 0.0)\n    mean = tl.sum(vm, 0) / cnt\n    dl = tl.where(rmask[:, None], v - mean[None, :], 0.0)\n    m2 = tl.sum(dl * dl, 0)\n    tl.store(pmean_ptr + pid * BN + rn, mean)\n    tl.store(pm2_ptr + pid * BN + rn, m2)\n    tl.store(pcnt_ptr + pid, cnt)\n\n\n@triton.jit\ndef stats_kernel(pmean_ptr, pm2_ptr, pcnt_ptr, g_ptr, b_ptr, sc_ptr, sh_ptr, mean_ptr, var_ptr,\n                 P, C, Mtot, eps, BP: tl.constexpr, BN: tl.constexpr):\n    rn = tl.arange(0, BN)\n    nm = rn < C\n    s = tl.zeros((BN,), dtype=tl.float32)\n    for p0 in range(0, P, BP):\n        rp = p0 + tl.arange(0, BP)\n        pk = rp < P\n        mu = tl.load(pmean_ptr + rp[:, None] * BN + rn[None, :], mask=pk[:, None], other=0.0)\n        c = tl.load(pcnt_ptr + rp, mask=pk, other=0.0)\n        s += tl.sum(mu * c[:, None], 0)\n    mean = s / Mtot\n    m2 = tl.zeros((BN,), dtype=tl.float32)\n    for p0 in range(0, P, BP):\n        rp = p0 + tl.arange(0, BP)\n        pk = rp < P\n        mu = tl.load(pmean_ptr + rp[:, None] * BN + rn[None, :], mask=pk[:, None], other=0.0)\n        q = tl.load(pm2_ptr + rp[:, None] * BN + rn[None, :], mask=pk[:, None], other=0.0)\n        c = tl.load(pcnt_ptr + rp, mask=pk, other=0.0)\n        d = mu - mean[None, :]\n        m2 += tl.sum(q + c[:, None] * d * d, 0)\n    var = m2 / Mtot\n    rstd = 1.0 / tl.sqrt(var + eps)\n    g = tl.load(g_ptr + rn, mask=nm, other=0.0)\n    bt = tl.load(b_ptr + rn, mask=nm, other=0.0)\n    sc = g * rstd\n    sh = bt - mean * sc\n    tl.store(sc_ptr + rn, sc)\n    tl.store(sh_ptr + rn, sh)\n    tl.store(mean_ptr + rn, mean)\n    tl.store(var_ptr + rn, var)\n\n\n@triton.jit\ndef vlad_kernel(x_ptr, a_ptr, sc_ptr, sh_ptr, c2_ptr, out_ptr, NS, C,\n                D: tl.constexpr, K: tl.constexpr, BNS: tl.constexpr, BC: tl.constexpr, BD: tl.constexpr):\n    b = tl.program_id(0).to(tl.int64)\n    rn = tl.arange(0, BNS)\n    nmask = rn < NS\n    rc = tl.arange(0, BC)\n    cmask = rc < C\n    rowbase = b * NS + rn\n    t = tl.load(a_ptr + rowbase[:, None] * C + rc[None, :], mask=nmask[:, None] & cmask[None, :], other=0.0)\n    sc = tl.load(sc_ptr + rc)\n    sh = tl.load(sh_ptr + rc)\n    z = t * sc[None, :] + sh[None, :]\n    z = tl.where(cmask[None, :], z, float(\"-inf\"))\n    mx = tl.max(z, 1)\n    e = tl.exp(z - mx[:, None])\n    den = tl.sum(e, 1)\n    rk = tl.arange(0, K)\n    t2 = tl.load(a_ptr + rowbase[:, None] * C + rk[None, :], mask=nmask[:, None], other=0.0)\n    sck = tl.load(sc_ptr + rk)\n    shk = tl.load(sh_ptr + rk)\n    A = tl.exp(t2 * sck[None, :] + shk[None, :] - mx[:, None]) / den[:, None]\n    A = tl.where(nmask[:, None], A, 0.0)\n    asum = tl.sum(A, 0)\n    At = tl.trans(A)\n    Ah = At.to(tl.bfloat16)\n    Al = (At - Ah.to(tl.float32)).to(tl.bfloat16)\n    ss = tl.zeros((K,), dtype=tl.float32)\n    xbase = x_ptr + b * NS * D\n    obase = out_ptr + b * D * K\n    for d0 in range(0, D, BD):\n        rd = d0 + tl.arange(0, BD)\n        xt = tl.load(xbase + rn[:, None] * D + rd[None, :], mask=nmask[:, None], other=0.0)\n        xh = xt.to(tl.bfloat16)\n        xl = (xt - xh.to(tl.float32)).to(tl.bfloat16)\n        v2 = tl.zeros((K, BD), dtype=tl.float32)\n        v2 = tl.dot(Ah, xl, v2)\n        v2 = tl.dot(Al, xh, v2)\n        v = tl.zeros((K, BD), dtype=tl.float32)\n        v = tl.dot(Ah, xh, v)\n        v = v + v2\n        c2 = tl.load(c2_ptr + rd[None, :] * K + rk[:, None])\n        v = v - asum[:, None] * c2\n        ss += tl.sum(v * v, 1)\n        tl.store(obase + rd[None, :] * K + rk[:, None], v)\n    kn = tl.maximum(tl.sqrt(ss), 1e-12)\n    inv = 1.0 / kn\n    tot = tl.sum(ss * inv * inv, 0)\n    g = tl.maximum(tl.sqrt(tot), 1e-12)\n    f = inv / g\n    tl.debug_barrier()\n    for d0 in range(0, D, BD):\n        rd = d0 + tl.arange(0, BD)\n        p = obase + rd[None, :] * K + rk[:, None]\n        v = tl.load(p)\n        tl.store(p, v * f[:, None])\n\n\nclass ModelNew(nn.Module):\n    def __init__(self, cluster_size, feature_size, ghost_clusters):\n        super(ModelNew, self).__init__()\n        self.feature_size = feature_size\n        self.cluster_size = cluster_size\n        self.ghost_clusters = ghost_clusters\n        init_sc = (1 / math.sqrt(feature_size))\n        clusters = cluster_size + ghost_clusters\n        self.clusters = nn.Parameter(init_sc * torch.randn(feature_size, clusters))\n        self.batch_norm = nn.BatchNorm1d(clusters)\n        self.clusters2 = nn.Parameter(init_sc * torch.randn(1, feature_size, cluster_size))\n        self.out_dim = self.cluster_size * feature_size\n\n    def forward(self, x, mask=None):\n        B, NS, D = x.shape\n        K = self.cluster_size\n        x2 = x.reshape(-1, D).contiguous()\n        M = x2.shape[0]\n        C = K + self.ghost_clusters\n        BN = max(16, triton.next_power_of_2(C))\n        dev = x.device\n        w = self.clusters.contiguous()\n        assign = torch.empty((M, C), device=dev, dtype=torch.float32)\n        BM = 128\n        P = triton.cdiv(M, BM)\n        pmean = torch.empty((P, BN), device=dev, dtype=torch.float32)\n        pm2 = torch.empty((P, BN), device=dev, dtype=torch.float32)\n        pcnt = torch.empty((P,), device=dev, dtype=torch.float32)\n        assign_kernel[(P,)](x2, w, assign, pmean, pm2, pcnt, M, C, D=D, BM=BM, BN=BN, BK=32,\n                            num_warps=4, num_stages=3)\n        bn = self.batch_norm\n        if self.training or bn.running_mean is None:\n            sc = torch.empty((BN,), device=dev, dtype=torch.float32)\n            sh = torch.empty((BN,), device=dev, dtype=torch.float32)\n            mean = torch.empty((BN,), device=dev, dtype=torch.float32)\n            var = torch.empty((BN,), device=dev, dtype=torch.float32)\n            stats_kernel[(1,)](pmean, pm2, pcnt, bn.weight, bn.bias, sc, sh, mean, var,\n                               P, C, float(M), float(bn.eps), BP=64, BN=BN, num_warps=4)\n            if self.training and bn.running_mean is not None:\n                with torch.no_grad():\n                    m = bn.momentum if bn.momentum is not None else 0.1\n                    bn.running_mean.mul_(1 - m).add_(mean[:C] * m)\n                    bn.running_var.mul_(1 - m).add_(var[:C] * (m * M / max(M - 1, 1)))\n                    bn.num_batches_tracked.add_(1)\n        else:\n            with torch.no_grad():\n                s = bn.weight / torch.sqrt(bn.running_var + bn.eps)\n                t = bn.bias - bn.running_mean * s\n                sc = torch.zeros((BN,), device=dev, dtype=torch.float32)\n                sh = torch.zeros((BN,), device=dev, dtype=torch.float32)\n                sc[:C] = s\n                sh[:C] = t\n        out = torch.empty((B, D * K), device=dev, dtype=torch.float32)\n        c2 = self.clusters2.contiguous()\n        BNS = max(16, triton.next_power_of_2(NS))\n        vlad_kernel[(B,)](x2, assign, sc, sh, c2, out, NS, C, D=D, K=K, BNS=BNS, BC=BN, BD=64,\n                          num_warps=8, num_stages=2)\n        return out\n",
    "speedup": 2.9605,
    "status": "ok"
   }
  },
  {
   "id": "E-H6",
   "include": "scaled_dot_product_attention|MultiheadAttention|torch\\.bmm\\(|softmax\\([^)]*\\)\\s*@|@\\s*\\w*\\.transpose\\(-2,\\s*-1\\)|torch\\.matmul\\([^)]*transpose",
   "exclude": null,
   "exclude_families": [],
   "text": "### Verified technique (E-H6, admitted by the KFORGE engine): Attention and batched matmuls on bf16x3: a flash-attention Triton kernel with in-register hi/lo split for QK^T and PV\nFor attention blocks and attention layers inside networks (ViT/Swin/transformer, head_dim 32-128), a FlashAttention-2-style Triton kernel should beat eager fp32 matmul -> softmax -> matmul by 2-4x. The kernel computes S = QK^T and O = PV with three bf16 tl.dot calls each, keeps online softmax in fp32, and never materializes the [L, L] scores.\nPhysics: The eager reference does two SIMT fp32 bmm calls plus a softmax that writes and reads an L x L fp32 tensor, which is HBM-bound at large L. The fused kernel removes that traffic and moves the flops to bf16 mma. Reduction lengths are tiny (K = head_dim for QK^T, block length for PV with fp32 rescaling), so the 3-product error sits far below 1e-4. Softmax exponent error stays around 1e-6 because the score error is about 2^-16 relative.\nHow: Use grid = (cdiv(L, BLOCK_M), B*H) with BLOCK_M = 64-128, BLOCK_N = 64, and head_dim held as a constexpr. Split the Q tile into hi/lo bf16 once per program and split each K/V tile on load. Compute s = (dot(qh, kl) + dot(ql, kh) + dot(qh, kh)) * scale, apply masks if present, update the running max m and sum l in fp32, and rescale acc. Split p = exp(s - m) into hi/lo and compute acc += dot(pl, vh) + dot(ph, vl) + dot(ph, vh). Write O/l in the layout the reference returns, handling the transpose and reshape via strides with no .contiguous(). If the reference adds a relative-position bias (as in Swin), cache the parameter-only bias tensor in self.<x> keyed on data_ptr and load it inside the kernel. Fall back to P1 torch.mm for non-attention bmm with a shared weight by reshaping to 2D.\n- Verified on training problems (protected evaluator): L3_043 MinGPTCausalAttention: 4.16x vs 3.74x with the previous library.\n- Verified example implementing it (L3_043, 4.16x; adapt what applies):\n```python\nimport math\nimport torch\nimport torch.nn as nn\nimport triton\nimport triton.language as tl\n\n\n@triton.jit\ndef split3_kernel(src, dst, R, K, LO_MID: tl.constexpr, BR: tl.constexpr, BKK: tl.constexpr):\n    r = tl.program_id(0) * BR + tl.arange(0, BR)\n    k = tl.program_id(1) * BKK + tl.arange(0, BKK)\n    m = (r[:, None] < R) & (k[None, :] < K)\n    v = tl.load(src + r[:, None] * K + k[None, :], mask=m, other=0.0)\n    h = v.to(tl.bfloat16)\n    lo = (v - h.to(tl.float32)).to(tl.bfloat16)\n    base = dst + r[:, None] * (3 * K) + k[None, :]\n    tl.store(base, h, mask=m)\n    if LO_MID:\n        tl.store(base + K, lo, mask=m)\n        tl.store(base + 2 * K, h, mask=m)\n    else:\n        tl.store(base + K, h, mask=m)\n        tl.store(base + 2 * K, lo, mask=m)\n\n\ndef split3(t, lo_mid):\n    t = t.contiguous()\n    R, K = t.shape\n    out = torch.empty((R, 3 * K), device=t.device, dtype=torch.bfloat16)\n    split3_kernel[(triton.cdiv(R, 32), triton.cdiv(K, 128))](t, out, R, K, LO_MID=lo_mid, BR=32, BKK=128)\n    return out\n\n\n@triton.autotune(\n    configs=[\n        triton.Config({\"BM\": 64, \"BN\": 64}, num_stages=2, num_warps=4),\n        triton.Config({\"BM\": 128, \"BN\": 64}, num_stages=2, num_warps=8),\n        triton.Config({\"BM\": 64, \"BN\": 32}, num_stages=2, num_warps=4),\n        triton.Config({\"BM\": 128, \"BN\": 32}, num_stages=2, num_warps=8),\n    ],\n    key=[\"T\", \"C\"],\n)\n@triton.jit\ndef attn_kernel(qkv, bias, out, T, H, C, scale_log2,\n                HS: tl.constexpr, D: tl.constexpr, BM: tl.constexpr, BN: tl.constexpr):\n    num_m = tl.cdiv(T, BM)\n    pid_m = num_m - 1 - tl.program_id(0)\n    bh = tl.program_id(1)\n    b = bh // H\n    h = bh % H\n    offs_m = pid_m * BM + tl.arange(0, BM)\n    offs_n0 = tl.arange(0, BN)\n    offs_d = tl.arange(0, D)\n    dmask = offs_d < HS\n    rs = 3 * C\n    base = qkv + b.to(tl.int64) * T * rs + h * HS\n    mrow = offs_m < T\n    q = tl.load(base + offs_m[:, None] * rs + offs_d[None, :],\n                mask=mrow[:, None] & dmask[None, :], other=0.0)\n    bq = tl.load(bias + h * HS + offs_d, mask=dmask, other=0.0)\n    bk = tl.load(bias + C + h * HS + offs_d, mask=dmask, other=0.0)\n    bv = tl.load(bias + 2 * C + h * HS + offs_d, mask=dmask, other=0.0)\n    q = q + bq[None, :]\n    qh = q.to(tl.bfloat16)\n    ql = (q - qh.to(tl.float32)).to(tl.bfloat16)\n    m_i = tl.full((BM,), float(\"-inf\"), tl.float32)\n    l_i = tl.zeros((BM,), tl.float32)\n    acc = tl.zeros((BM, D), tl.float32)\n    hi = tl.minimum((pid_m + 1) * BM, T)\n    for start_n in range(0, hi, BN):\n        offs_n = start_n + offs_n0\n        nmask = offs_n < T\n        kT = tl.load(base + C + offs_n[None, :] * rs + offs_d[:, None],\n                     mask=nmask[None, :] & dmask[:, None], other=0.0) + bk[:, None]\n        kh = kT.to(tl.bfloat16)\n        kl = (kT - kh.to(tl.float32)).to(tl.bfloat16)\n        s2 = tl.dot(qh, kl)\n        s2 = tl.dot(ql, kh, s2)\n        s = tl.dot(qh, kh)\n        s = (s + s2) * scale_log2\n        valid = (offs_m[:, None] >= offs_n[None, :]) & nmask[None, :]\n        s = tl.where(valid, s, float(\"-inf\"))\n        m_new = tl.maximum(m_i, tl.max(s, 1))\n        alpha = tl.exp2(m_i - m_new)\n        p = tl.exp2(s - m_new[:, None])\n        l_i = l_i * alpha + tl.sum(p, 1)\n        acc = acc * alpha[:, None]\n        v = tl.load(base + 2 * C + offs_n[:, None] * rs + offs_d[None, :],\n                    mask=nmask[:, None] & dmask[None, :], other=0.0) + bv[None, :]\n        vh = v.to(tl.bfloat16)\n        vl = (v - vh.to(tl.float32)).to(tl.bfloat16)\n        ph = p.to(tl.bfloat16)\n        pl = (p - ph.to(tl.float32)).to(tl.bfloat16)\n        acc = tl.dot(pl, vh, acc)\n        acc = tl.dot(ph, vl, acc)\n        acc = tl.dot(ph, vh, acc)\n        m_i = m_new\n    o = acc / l_i[:, None]\n    oh = o.to(tl.bfloat16)\n    ol = (o - oh.to(tl.float32)).to(tl.bfloat16)\n    obase = out + (b.to(tl.int64) * T + offs_m[:, None]) * rs + h * HS + offs_d[None, :]\n    om = mrow[:, None] & dmask[None, :]\n    tl.store(obase, oh, mask=om)\n    tl.store(obase + C, oh, mask=om)\n    tl.store(obase + 2 * C, ol, mask=om)\n\n\nclass ModelNew(nn.Module):\n    def __init__(self, n_embd, n_head, attn_pdrop, resid_pdrop, max_seqlen):\n        super().__init__()\n        assert n_embd % n_head == 0\n        self.c_attn = nn.Linear(n_embd, 3 * n_embd)\n        self.c_proj = nn.Linear(n_embd, n_embd)\n        self.attn_dropout = nn.Dropout(attn_pdrop)\n        self.resid_dropout = nn.Dropout(resid_pdrop)\n        self.register_buffer(\"bias\", torch.tril(torch.ones(max_seqlen, max_seqlen))\n                             .view(1, 1, max_seqlen, max_seqlen))\n        self.n_head = n_head\n        self.n_embd = n_embd\n\n    def forward(self, x):\n        B, T, C = x.size()\n        H = self.n_head\n        hs = C // H\n        x2 = x.reshape(B * T, C)\n        qkv = torch.mm(split3(x2, 0), split3(self.c_attn.weight, 1).t(), out_dtype=torch.float32)\n        ysplit = torch.empty((B * T, 3 * C), device=x.device, dtype=torch.bfloat16)\n        D = triton.next_power_of_2(hs)\n        scale_log2 = (1.0 / math.sqrt(hs)) * 1.4426950408889634\n        grid = lambda META: (triton.cdiv(T, META[\"BM\"]), B * H)\n        attn_kernel[grid](qkv, self.c_attn.bias, ysplit, T, H, C, scale_log2, HS=hs, D=D)\n        y = torch.mm(ysplit, split3(self.c_proj.weight, 1).t(), out_dtype=torch.float32)\n        y.add_(self.c_proj.bias)\n        return y.view(B, T, C)\n```\n",
   "demote": "flash kernel was about 9x SLOWER",
   "provenance": {
    "origin": "engine (EXTEND)",
    "round": "E",
    "hypothesis": "Attention and batched matmuls on bf16x3: a flash-attention Triton kernel with in-register hi/lo split for QK^T and PV",
    "trial_per_problem": {
     "L3_043": [
      3.7407,
      4.158
     ]
    },
    "gain": 0.417,
    "adoption": {
     "L3_043": "ADOPTED: yes - A flash-attention Triton kernel splits Q, K, and V into bf16 hi/lo in registers and uses three tl.dot calls each for QK^T and PV. It keeps online softmax in fp32 and never materializes "
    }
   },
   "example": {
    "pid": "L3_043",
    "source": "import math\nimport torch\nimport torch.nn as nn\nimport triton\nimport triton.language as tl\n\n\n@triton.jit\ndef split3_kernel(src, dst, R, K, LO_MID: tl.constexpr, BR: tl.constexpr, BKK: tl.constexpr):\n    r = tl.program_id(0) * BR + tl.arange(0, BR)\n    k = tl.program_id(1) * BKK + tl.arange(0, BKK)\n    m = (r[:, None] < R) & (k[None, :] < K)\n    v = tl.load(src + r[:, None] * K + k[None, :], mask=m, other=0.0)\n    h = v.to(tl.bfloat16)\n    lo = (v - h.to(tl.float32)).to(tl.bfloat16)\n    base = dst + r[:, None] * (3 * K) + k[None, :]\n    tl.store(base, h, mask=m)\n    if LO_MID:\n        tl.store(base + K, lo, mask=m)\n        tl.store(base + 2 * K, h, mask=m)\n    else:\n        tl.store(base + K, h, mask=m)\n        tl.store(base + 2 * K, lo, mask=m)\n\n\ndef split3(t, lo_mid):\n    t = t.contiguous()\n    R, K = t.shape\n    out = torch.empty((R, 3 * K), device=t.device, dtype=torch.bfloat16)\n    split3_kernel[(triton.cdiv(R, 32), triton.cdiv(K, 128))](t, out, R, K, LO_MID=lo_mid, BR=32, BKK=128)\n    return out\n\n\n@triton.autotune(\n    configs=[\n        triton.Config({\"BM\": 64, \"BN\": 64}, num_stages=2, num_warps=4),\n        triton.Config({\"BM\": 128, \"BN\": 64}, num_stages=2, num_warps=8),\n        triton.Config({\"BM\": 64, \"BN\": 32}, num_stages=2, num_warps=4),\n        triton.Config({\"BM\": 128, \"BN\": 32}, num_stages=2, num_warps=8),\n    ],\n    key=[\"T\", \"C\"],\n)\n@triton.jit\ndef attn_kernel(qkv, bias, out, T, H, C, scale_log2,\n                HS: tl.constexpr, D: tl.constexpr, BM: tl.constexpr, BN: tl.constexpr):\n    num_m = tl.cdiv(T, BM)\n    pid_m = num_m - 1 - tl.program_id(0)\n    bh = tl.program_id(1)\n    b = bh // H\n    h = bh % H\n    offs_m = pid_m * BM + tl.arange(0, BM)\n    offs_n0 = tl.arange(0, BN)\n    offs_d = tl.arange(0, D)\n    dmask = offs_d < HS\n    rs = 3 * C\n    base = qkv + b.to(tl.int64) * T * rs + h * HS\n    mrow = offs_m < T\n    q = tl.load(base + offs_m[:, None] * rs + offs_d[None, :],\n                mask=mrow[:, None] & dmask[None, :], other=0.0)\n    bq = tl.load(bias + h * HS + offs_d, mask=dmask, other=0.0)\n    bk = tl.load(bias + C + h * HS + offs_d, mask=dmask, other=0.0)\n    bv = tl.load(bias + 2 * C + h * HS + offs_d, mask=dmask, other=0.0)\n    q = q + bq[None, :]\n    qh = q.to(tl.bfloat16)\n    ql = (q - qh.to(tl.float32)).to(tl.bfloat16)\n    m_i = tl.full((BM,), float(\"-inf\"), tl.float32)\n    l_i = tl.zeros((BM,), tl.float32)\n    acc = tl.zeros((BM, D), tl.float32)\n    hi = tl.minimum((pid_m + 1) * BM, T)\n    for start_n in range(0, hi, BN):\n        offs_n = start_n + offs_n0\n        nmask = offs_n < T\n        kT = tl.load(base + C + offs_n[None, :] * rs + offs_d[:, None],\n                     mask=nmask[None, :] & dmask[:, None], other=0.0) + bk[:, None]\n        kh = kT.to(tl.bfloat16)\n        kl = (kT - kh.to(tl.float32)).to(tl.bfloat16)\n        s2 = tl.dot(qh, kl)\n        s2 = tl.dot(ql, kh, s2)\n        s = tl.dot(qh, kh)\n        s = (s + s2) * scale_log2\n        valid = (offs_m[:, None] >= offs_n[None, :]) & nmask[None, :]\n        s = tl.where(valid, s, float(\"-inf\"))\n        m_new = tl.maximum(m_i, tl.max(s, 1))\n        alpha = tl.exp2(m_i - m_new)\n        p = tl.exp2(s - m_new[:, None])\n        l_i = l_i * alpha + tl.sum(p, 1)\n        acc = acc * alpha[:, None]\n        v = tl.load(base + 2 * C + offs_n[:, None] * rs + offs_d[None, :],\n                    mask=nmask[:, None] & dmask[None, :], other=0.0) + bv[None, :]\n        vh = v.to(tl.bfloat16)\n        vl = (v - vh.to(tl.float32)).to(tl.bfloat16)\n        ph = p.to(tl.bfloat16)\n        pl = (p - ph.to(tl.float32)).to(tl.bfloat16)\n        acc = tl.dot(pl, vh, acc)\n        acc = tl.dot(ph, vl, acc)\n        acc = tl.dot(ph, vh, acc)\n        m_i = m_new\n    o = acc / l_i[:, None]\n    oh = o.to(tl.bfloat16)\n    ol = (o - oh.to(tl.float32)).to(tl.bfloat16)\n    obase = out + (b.to(tl.int64) * T + offs_m[:, None]) * rs + h * HS + offs_d[None, :]\n    om = mrow[:, None] & dmask[None, :]\n    tl.store(obase, oh, mask=om)\n    tl.store(obase + C, oh, mask=om)\n    tl.store(obase + 2 * C, ol, mask=om)\n\n\nclass ModelNew(nn.Module):\n    def __init__(self, n_embd, n_head, attn_pdrop, resid_pdrop, max_seqlen):\n        super().__init__()\n        assert n_embd % n_head == 0\n        self.c_attn = nn.Linear(n_embd, 3 * n_embd)\n        self.c_proj = nn.Linear(n_embd, n_embd)\n        self.attn_dropout = nn.Dropout(attn_pdrop)\n        self.resid_dropout = nn.Dropout(resid_pdrop)\n        self.register_buffer(\"bias\", torch.tril(torch.ones(max_seqlen, max_seqlen))\n                             .view(1, 1, max_seqlen, max_seqlen))\n        self.n_head = n_head\n        self.n_embd = n_embd\n\n    def forward(self, x):\n        B, T, C = x.size()\n        H = self.n_head\n        hs = C // H\n        x2 = x.reshape(B * T, C)\n        qkv = torch.mm(split3(x2, 0), split3(self.c_attn.weight, 1).t(), out_dtype=torch.float32)\n        ysplit = torch.empty((B * T, 3 * C), device=x.device, dtype=torch.bfloat16)\n        D = triton.next_power_of_2(hs)\n        scale_log2 = (1.0 / math.sqrt(hs)) * 1.4426950408889634\n        grid = lambda META: (triton.cdiv(T, META[\"BM\"]), B * H)\n        attn_kernel[grid](qkv, self.c_attn.bias, ysplit, T, H, C, scale_log2, HS=hs, D=D)\n        y = torch.mm(ysplit, split3(self.c_proj.weight, 1).t(), out_dtype=torch.float32)\n        y.add_(self.c_proj.bias)\n        return y.view(B, T, C)\n",
    "speedup": 4.158,
    "status": "ok"
   }
  }
 ]
}