examples: sub-pel interpolation = a BF16 16×16 tile op — measured (HEVC 8-tap)

Next probe from the HEVC amortization map: HEVC fractional-pel motion
compensation applies a separable 8-tap FIR (half-pel luma taps
[-1,4,-11,40,40,-11,4,-1]/64). An FIR is a linear operator, so filtering a
16×16 block is a matrix product — one bf16_tile_gemm_16x16 per pass, and the
full separable H+V is Hᵀ·(X·H): the SAME two-tile sandwich shape as the splat
covariance projection (Mᵀ·Σ·M, #233) and the codec transform (M·X, #232).

Measured on real AMX (this host: AMX TDPBF16PS, amx_available = true), vs a
direct 8-tap FIR f64 reference (same 16×16 band operator, reflected edges, so
only BF16 rounding differs):

  (1) horizontal 8-tap  out = X·H       frobenius_rel_err = 0.157%  (one tile op)
  (2) separable H+V     out = Hᵀ·(X·H)  frobenius_rel_err = 0.215%  (two tile ops)
  (3) throughput: 1.22 M sub-pel tile ops/s

Both asserts pass (rel err < BF16 tolerance). Fractional-pel motion is therefore
"just another tile op", closing the MC story past integer-pel. R-5 dispatch noted
(per-block 8-tap butterfly below the batch crossover, batched tile GEMM above).

x86-gated via the same `mod amx` / non-x86 fallback-main pattern as
gridlake_field_tile; canonical ndarray::simd imports; fmt + clippy clean.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01MLBnPuScZy6w9di2QEjsXM

Author: claude

Cargo.toml

@@ -95,6 +95,10 @@ required-features = ["codec"]
 name = "motion_transform_noop"
 required-features = ["codec"]
 
+[[example]]
+name = "subpel_tap_tile"
+required-features = ["std"]
+
 [[example]]
 name = "entropy_ladder_probe"
 required-features = ["std"]

examples/subpel_tap_tile.rs

@@ -0,0 +1,231 @@
+//! Sub-pel interpolation = a BF16 16×16 tile op — measured (HEVC 8-tap).
+//!
+//! Next probe from the HEVC amortization map (`.claude/knowledge/hevc-amortization-map.md`):
+//! HEVC fractional-pel motion compensation applies a **separable 8-tap FIR** to a
+//! block (half-pel luma taps `[-1, 4, -11, 40, 40, -11, 4, -1] / 64`). An FIR is a
+//! **linear operator**, so filtering a 16×16 block is a matrix product — one
+//! `bf16_tile_gemm_16x16` per pass, and the full separable H+V interpolation is
+//! `H_v · X · H_h`: **the same two-tile "sandwich" shape as the splat covariance
+//! projection** (`Mᵀ·Σ·M`, #233) and the codec transform (`M·X`, #232). This
+//! probe measures that the tile-GEMM form matches the direct 8-tap FIR to BF16
+//! precision, on the shipped kernel, and reports the tier.
+//!
+//! Fairness / honesty:
+//!   • Both the tile path and the reference use the SAME 16×16 band operator `H`
+//!     (8 taps per output column, reflected at the block edge), so this is an
+//!     equal-result comparison — only BF16 rounding differs.
+//!   • R-5 dispatch still applies: below the per-arch batch crossover (SPR 64,
+//!     ICX 32, …) a per-block 8-tap butterfly wins; the tile GEMM wins in the
+//!     batched regime. This probe shows the *equivalence*, not that the tile path
+//!     is always the right dispatch for one block.
+//!
+//! The x86-only AMX/tile path is wrapped in a `cfg(target_arch = "x86_64")`
+//! module with a non-x86 fallback `main` (see `gridlake_field_tile`).
+//!
+//! Run: `cargo run --release --example subpel_tap_tile --features std`
+
+#[cfg(target_arch = "x86_64")]
+mod amx {
+    use ndarray::hpc::quantized::{f32_to_bf16_rounded, BF16};
+    use ndarray::simd::{amx_available, bf16_tile_gemm_16x16_amx as bf16_tile_gemm_16x16, bf16_tile_gemm_tier};
+
+    const M: usize = 16; // block edge
+    const KPAD: usize = 32; // K padded to a multiple of 32 for the tile GEMM
+
+    /// HEVC half-pel luma 8-tap (sum = 64).
+    const TAPS: [f32; 8] = [-1.0, 4.0, -11.0, 40.0, 40.0, -11.0, 4.0, -1.0];
+
+    fn mix(mut z: u64) -> u64 {
+        z = z.wrapping_add(0x9E37_79B9_7F4A_7C15);
+        z = (z ^ (z >> 30)).wrapping_mul(0xBF58_476D_1CE4_E5B9);
+        z = (z ^ (z >> 27)).wrapping_mul(0x94D0_49BB_1331_11EB);
+        z ^ (z >> 31)
+    }
+
+    fn to_bf16(src: &[f32]) -> Vec<u16> {
+        let mut b = vec![BF16(0); src.len()];
+        f32_to_bf16_rounded(src, &mut b);
+        b.iter().map(|x| x.0).collect()
+    }
+
+    /// Reflect an index into `[0, M)` (block-edge boundary handling).
+    fn reflect(i: i32) -> usize {
+        let mut i = i;
+        while !(0..M as i32).contains(&i) {
+            if i < 0 {
+                i = -i - 1;
+            } else if i >= M as i32 {
+                i = 2 * M as i32 - 1 - i;
+            }
+        }
+        i as usize
+    }
+
+    /// The 16×16 band operator `H`: `out[·][j] = Σ_t TAPS[t]/64 · in[·][reflect(j-3+t)]`,
+    /// i.e. `H[k][j]` accumulates the tap weight of input column `k` for output
+    /// column `j`. `out = X · H`. This IS the sub-pel FIR as a fixed matrix.
+    fn band_operator() -> [f32; M * M] {
+        let mut h = [0.0f32; M * M];
+        for j in 0..M {
+            for (t, &w) in TAPS.iter().enumerate() {
+                let k = reflect(j as i32 - 3 + t as i32);
+                h[k * M + j] += w / 64.0;
+            }
+        }
+        h
+    }
+
+    /// `C[16×16] = A[16×16] · B[16×16]` via the shipped BF16 tile op (K padded 16→32).
+    fn tile_matmul(a: &[f32; M * M], b: &[f32; M * M]) -> [f32; M * M] {
+        let mut a_pad = [0.0f32; M * KPAD];
+        let mut b_pad = [0.0f32; KPAD * M];
+        for i in 0..M {
+            for k in 0..M {
+                a_pad[i * KPAD + k] = a[i * M + k];
+                b_pad[k * M + i] = b[k * M + i];
+            }
+        }
+        let (ab, bb) = (to_bf16(&a_pad), to_bf16(&b_pad));
+        let mut c = vec![0.0f32; M * M];
+        bf16_tile_gemm_16x16(&ab, &bb, &mut c, KPAD);
+        let mut out = [0.0f32; M * M];
+        out.copy_from_slice(&c);
+        out
+    }
+
+    fn direct_matmul(a: &[f32; M * M], b: &[f32; M * M]) -> [f64; M * M] {
+        let mut c = [0.0f64; M * M];
+        for i in 0..M {
+            for j in 0..M {
+                let mut s = 0.0f64;
+                for k in 0..M {
+                    s += a[i * M + k] as f64 * b[k * M + j] as f64;
+                }
+                c[i * M + j] = s;
+            }
+        }
+        c
+    }
+
+    fn transpose(a: &[f32; M * M]) -> [f32; M * M] {
+        let mut t = [0.0f32; M * M];
+        for i in 0..M {
+            for j in 0..M {
+                t[j * M + i] = a[i * M + j];
+            }
+        }
+        t
+    }
+
+    /// Direct separable H+V 8-tap FIR reference (`f64`), reflected edges — the
+    /// ground truth the tile-GEMM sandwich must match.
+    fn direct_hv(x: &[f32; M * M]) -> [f64; M * M] {
+        // horizontal
+        let mut h = [0.0f64; M * M];
+        for i in 0..M {
+            for j in 0..M {
+                let mut s = 0.0f64;
+                for (t, &w) in TAPS.iter().enumerate() {
+                    s += (w as f64 / 64.0) * x[i * M + reflect(j as i32 - 3 + t as i32)] as f64;
+                }
+                h[i * M + j] = s;
+            }
+        }
+        // vertical
+        let mut out = [0.0f64; M * M];
+        for i in 0..M {
+            for j in 0..M {
+                let mut s = 0.0f64;
+                for (t, &w) in TAPS.iter().enumerate() {
+                    s += (w as f64 / 64.0) * h[reflect(i as i32 - 3 + t as i32) * M + j];
+                }
+                out[i * M + j] = s;
+            }
+        }
+        out
+    }
+
+    fn frob_rel(tile: &[f32; M * M], direct: &[f64; M * M]) -> (f64, f64) {
+        let (mut max_abs, mut num, mut den) = (0.0f64, 0.0f64, 0.0f64);
+        for i in 0..M * M {
+            let e = tile[i] as f64 - direct[i];
+            max_abs = max_abs.max(e.abs());
+            num += e * e;
+            den += direct[i] * direct[i];
+        }
+        (max_abs, (num / den.max(1e-12)).sqrt())
+    }
+
+    pub fn run() {
+        println!("Sub-pel interpolation = BF16 16×16 tile op — measured (HEVC 8-tap half-pel)");
+        println!(
+            "  shipped kernel tier on THIS host: {}  (amx_available = {})\n",
+            bf16_tile_gemm_tier(),
+            amx_available()
+        );
+
+        // 7-bit luma block (fits u8 and i8; positive operand for the tile GEMM)
+        let mut x = [0.0f32; M * M];
+        for r in 0..M {
+            for c in 0..M {
+                x[r * M + c] = ((mix((r as u64) << 8 | c as u64) & 0x7F) as f32) * 0.7 + 20.0 * (r as f32 * 0.3).sin();
+            }
+        }
+        let h = band_operator();
+        let ht = transpose(&h);
+
+        // (1) horizontal half-pel = X · H  (one tile op)
+        let horiz_tile = tile_matmul(&x, &h);
+        let horiz_direct = direct_matmul(&x, &h);
+        let (ha, hr) = frob_rel(&horiz_tile, &horiz_direct);
+        println!("  (1) horizontal 8-tap   out = X·H            (one tile op)");
+        println!("      max_abs_err={ha:.4}  frobenius_rel_err={:.3}%", hr * 100.0);
+
+        // (2) full separable H+V = Hᵀ · (X · H)  (two tile ops — the sandwich shape)
+        let hv_tile = tile_matmul(&ht, &horiz_tile);
+        let hv_direct = direct_hv(&x);
+        let (va, vr) = frob_rel(&hv_tile, &hv_direct);
+        println!("  (2) separable H+V      out = Hᵀ·(X·H)       (two tile ops = Mᵀ·Σ·M sandwich)");
+        println!("      max_abs_err={va:.4}  frobenius_rel_err={:.3}%", vr * 100.0);
+
+        // (3) throughput of the sub-pel tile op
+        let iters = 200_000usize;
+        let t0 = std::time::Instant::now();
+        let mut acc = 0.0f32;
+        for it in 0..iters {
+            let mut xv = x;
+            xv[0] += (it & 0x7) as f32 * 0.01;
+            let c = tile_matmul(&xv, &h);
+            acc += c[0] + c[M * M - 1];
+        }
+        let dt = t0.elapsed().as_secs_f64();
+        println!(
+            "  (3) throughput: {iters} sub-pel tile ops in {:.1} ms → {:.2} M/s  (checksum {acc:.1})",
+            dt * 1000.0,
+            iters as f64 / dt / 1e6
+        );
+
+        assert!(hr < 0.05, "horizontal FIR tile-GEMM rel err too high for BF16 class");
+        assert!(vr < 0.05, "separable H+V tile-GEMM rel err too high for BF16 class");
+
+        println!(
+            "\n  MEASURED: HEVC 8-tap sub-pel interpolation IS a tile GEMM — X·H (one op)\n\
+             \x20 and the separable H+V is Hᵀ·(X·H), the SAME two-tile sandwich shape as the\n\
+             \x20 splat covariance projection (#233) and the codec transform (#232). Matches\n\
+             \x20 the direct 8-tap FIR to BF16 precision. Fractional-pel motion is therefore\n\
+             \x20 'just another tile op', closing the MC story past integer-pel. (Dispatch\n\
+             \x20 per R-5: per-block 8-tap butterfly below the batch crossover, batched tile\n\
+             \x20 GEMM above it.)"
+        );
+    }
+}
+
+#[cfg(target_arch = "x86_64")]
+fn main() {
+    amx::run();
+}
+
+#[cfg(not(target_arch = "x86_64"))]
+fn main() {
+    eprintln!("subpel_tap_tile requires x86_64 (AMX TDPBF16PS / AVX-512 VDPBF16PS BF16 tile GEMM).");
+}