1 # bend-ml-tensor-array: tensors over a flat Array, with the shape in the type. 2 # 3 # import bend-ml-tensor-array@0.1.15.0/main.bend as TA 4 # 5 # Mat stores r*c numbers in an Array, row by row (index i*c + j). The 6 # dimensions are erased type parameters: the same guarantee as bend-ml-tensor 7 # (a product with wrong dimensions does not compile), but ~50x faster than lists: 8 # Array.get/set by index, and products in parallel blocks of rows. 9 # 10 # An Array is affine (a single owner): every operation that reads an operand returns it 11 # together with the result (as Array.get does), in records MM, AR, ... Use Mat.clone 12 # when you need two copies. 13 # 14 # What is NOT proved: the invariant "the Array has capacity >= r*c" holds because 15 # the constructors (Mat.zeros, Mat.of_list) allocate the right size; and F32 is 16 # validated by tests against PyTorch (reference/test_tensor_array.py), not by proof. 17 18 import Base 19 20 # --------------------------------------------------------------------- 21 # helper types 22 # --------------------------------------------------------------------- 23 24 type R is Type: 25 R{a: Array, b: Array, x: F32} 26 27 type G is Type: 28 G{a: Array, b: Array, c: Array} 29 30 # A dot product with eight partial sums (0.1.15.0; four before). Each step adds its product to a0 and rotates the sums 31 # (a1, ..., a7, a0), so each sum gets every eighth product and the processor can work on the next additions before the 32 # last one has finished; at the end the sums are added as a balanced tree of pairs. The order of the additions is fixed, 33 # so the result is the same on every run and every thread count, but it is not the order of a single running sum 34 # (NOTES.md, exp. 17 and 2026-10-25). The cells of 8 below keep exactly this order: sum q is lane q of every cell. 35 def dotr(k: Nat, +ia: U32, +ib: U32, +sa: U32, +sb: U32, a0: F32, a1: F32, a2: F32, a3: F32, a4: F32, a5: F32, a6: F32, a7: F32, ra: Array & F32, rb: Array & F32) -> R: 36 match k ra rb: 37 case 0n Tuple{a, x} Tuple{b, y}: 38 R{a, b, (((a0 + a1 : F32) + (a2 + a3 : F32) : F32) + ((a4 + a5 : F32) + (a6 + a7 : F32) : F32) : F32)} 39 case 1n+p Tuple{a, +x} Tuple{b, +y}: 40 dotr(p, (ia + sa : U32), (ib + sb : U32), sa, sb, a1, a2, a3, a4, a5, a6, a7, (a0 + (x * y : F32) : F32), Array.get(F32, a, (ia + sa : U32)), Array.get(F32, b, (ib + sb : U32))) 41 42 def dotg(k: Nat, +ia: U32, +ib: U32, +sa: U32, +sb: U32, acc: F32, ra: Array & F32, rb: Array & F32) -> R: 43 dotr(k, ia, ib, sa, sb, acc, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, ra, rb) 44 45 # writes one cell of C: c[idx] = x, or c[idx] = c[idx] + x when ac (an accumulating product, 0.1.10.0). x is computed 46 # exactly as without ac and added once, so C + A·B has the bits of adding a separately computed A·B to C. 47 48 def cwr_add(+idx: U32, +x: F32, r: Array & F32) -> Array: 49 match r: 50 case Tuple{c, +v}: 51 Array.set(F32, c, idx, (v + x : F32)) 52 53 def cwr(ac: Bool, +idx: U32, +x: F32, c: Array) -> Array: 54 match ac: 55 case True{}: 56 cwr_add(idx, x, Array.get(F32, c, idx)) 57 case False{}: 58 Array.set(F32, c, idx, x) 59 60 # row i of C, columns j.. ; r is the product of the current column (already computed) 61 62 def gcols(+ac: Bool, mleft: Nat, +kn: Nat, +ia0: U32, +ib0: U32, +sa: U32, +sb: U32, +bcol: U32, +cidx: U32, c: Array, r: R) -> G: 63 match mleft r: 64 case 0n R{a, b, +x}: 65 G{a, b, cwr(ac, cidx, x, c)} 66 case 1n+p R{a, b, +x}: 67 gcols(ac, p, kn, ia0, (ib0 + bcol : U32), sa, sb, bcol, (cidx + 1 : U32), cwr(ac, cidx, x, c), dotg(kn, ia0, (ib0 + bcol : U32), sa, sb, 0.0, Array.get(F32, a, ia0), Array.get(F32, b, (ib0 + bcol : U32)))) 68 69 def grows(+ac: Bool, nleft: Nat, +mm1: Nat, +kn: Nat, +i: U32, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, +mu: U32, g: G) -> G: 70 match nleft g: 71 case 0n G{a, b, c}: 72 G{a, b, c} 73 case 1n+p G{a, b, c}: 74 grows(ac, p, mm1, kn, (i + 1 : U32), arow, sa, sb, bcol, mu, gcols(ac, mm1, kn, (i * arow : U32), 0, sa, sb, bcol, (i * mu : U32), c, dotg(kn, (i * arow : U32), 0, sa, sb, 0.0, Array.get(F32, a, (i * arow : U32)), Array.get(F32, b, 0)))) 75 76 # C (n x m) = A · B with k steps, or C + A · B when ac; c is the output array (with capacity n*m) 77 78 def gemm_ac(+ac: Bool, +n: Nat, +m: Nat, +k: Nat, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, a: Array, b: Array, c: Array) -> G: 79 grows(ac, n, Nat.sub(m, 1n), k, 0, arow, sa, sb, bcol, U32.from_nat(m), G{a, b, c}) 80 81 def gemm(+n: Nat, +m: Nat, +k: Nat, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, a: Array, b: Array, c: Array) -> G: 82 gemm_ac(False{}, n, m, k, arow, sa, sb, bcol, a, b, c) 83 84 85 # --------------------------------------------------------------------- 86 # parallel gemm: 2^d blocks of rows of C, each with its own copy of A and B, 87 # returning lists that are written into C at the end 88 # --------------------------------------------------------------------- 89 90 type S3 is Type: 91 S3{a: Array, b: Array, out: List<&2, F32>} 92 93 def lcols(mleft: Nat, +kn: Nat, +ia0: U32, +ib0: U32, +sa: U32, +sb: U32, +bcol: U32, out: List<&2, F32>, r: R) -> S3: 94 match mleft r: 95 case 0n R{a, b, +x}: 96 S3{a, b, x <> out} 97 case 1n+p R{a, b, +x}: 98 lcols(p, kn, ia0, (ib0 + bcol : U32), sa, sb, bcol, x <> out, dotg(kn, ia0, (ib0 + bcol : U32), sa, sb, 0.0, Array.get(F32, a, ia0), Array.get(F32, b, (ib0 + bcol : U32)))) 99 100 def lrows(nleft: Nat, +mm1: Nat, +kn: Nat, +i: U32, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, st: S3) -> S3: 101 match nleft st: 102 case 0n S3{a, b, out}: 103 S3{a, b, out} 104 case 1n+p S3{a, b, out}: 105 lrows(p, mm1, kn, (i + 1 : U32), arow, sa, sb, bcol, lcols(mm1, kn, (i * arow : U32), 0, sa, sb, bcol, out, dotg(kn, (i * arow : U32), 0, sa, sb, 0.0, Array.get(F32, a, (i * arow : U32)), Array.get(F32, b, 0)))) 106 107 def lblock_fin(st: S3) -> List<&2, F32>: 108 match st: 109 case S3{a, b, out}: 110 List.reverse(&2, F32, out) 111 112 def lblock(cnt: Nat, +mm1: Nat, +kn: Nat, +i: U32, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, a: Array, b: Array) -> List<&2, F32>: 113 lblock_fin(lrows(cnt, mm1, kn, i, arow, sa, sb, bcol, S3{a, b, Nil{}})) 114 115 def lpar(d: Nat, +cnt: Nat, +mm1: Nat, +kn: Nat, +i: U32, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, ra: Array & Array, rb: Array & Array) -> List<&2, F32>: 116 match d ra rb: 117 case 0n Tuple{a, a2} Tuple{b, b2}: 118 lblock(cnt, mm1, kn, i, arow, sa, sb, bcol, a, b) 119 case 1n++q Tuple{a, a2} Tuple{b, b2}: 120 x y = lpar(q, Nat.div(cnt, 2n), mm1, kn, i, arow, sa, sb, bcol, Array.clone(F32, a), Array.clone(F32, b)) lpar(q, Nat.sub(cnt, Nat.div(cnt, 2n)), mm1, kn, (i + U32.from_nat(Nat.div(cnt, 2n)) : U32), arow, sa, sb, bcol, Array.clone(F32, a2), Array.clone(F32, b2)) 121 List.append(&2, F32, x, y) 122 123 def write_l(xs: List<&2, F32>, +idx: U32, c: Array) -> Array: 124 match xs: 125 case Nil{}: 126 c 127 case Con{h, t}: 128 write_l(t, (idx + 1 : U32), Array.set(F32, c, idx, h)) 129 130 # c[idx..] = list, or c[idx..] += list when ac 131 132 def write_la(+ac: Bool, xs: List<&2, F32>, +idx: U32, c: Array) -> Array: 133 match xs: 134 case Nil{}: 135 c 136 case Con{+h, t}: 137 write_la(ac, t, (idx + 1 : U32), cwr(ac, idx, h, c)) 138 139 def gp3(+ac: Bool, a: Array, b: Array, c: Array, xs: List<&2, F32>) -> G: 140 G{a, b, write_la(ac, xs, 0, c)} 141 142 def gp2(d: Nat, +n: Nat, +mm1: Nat, +kn: Nat, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, c: Array, pa: Array & Array, pb: Array & Array) -> G: 143 match pa pb: 144 case Tuple{a1, a2} Tuple{b1, b2}: 145 gp3(False{}, a1, b1, c, lpar(d, n, mm1, kn, 0, arow, sa, sb, bcol, Array.clone(F32, a2), Array.clone(F32, b2))) 146 147 # C (n x m) = A · B, with 2^d blocks of rows in parallel; returns A, B (intact) and C 148 149 def gemm_par(d: Nat, +n: Nat, +m: Nat, +k: Nat, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, a: Array, b: Array, c: Array) -> G: 150 gp2(d, n, Nat.sub(m, 1n), k, arow, sa, sb, bcol, c, Array.clone(F32, a), Array.clone(F32, b)) 151 152 153 # --------------------------------------------------------------------- 154 # parallel gemm by COLUMNS, for n = 1 (matrix · vector): each block computes a 155 # range of columns of C, with its own copy of the operands 156 # --------------------------------------------------------------------- 157 158 def cblock_fin(st: S3) -> List<&2, F32>: 159 match st: 160 case S3{a, b, out}: 161 List.reverse(&2, F32, out) 162 163 # cnt columns starting at j0 (cnt >= 1); row 0 of A is at ia0 164 165 def cblock_go(cnt1: Nat, +kn: Nat, +ia0: U32, +j0: U32, +sa: U32, +sb: U32, +bcol: U32, a: Array, b: Array) -> List<&2, F32>: 166 cblock_fin(lcols(cnt1, kn, ia0, (j0 * bcol : U32), sa, sb, bcol, Nil{}, dotg(kn, ia0, (j0 * bcol : U32), sa, sb, 0.0, Array.get(F32, a, ia0), Array.get(F32, b, (j0 * bcol : U32))))) 167 168 def cblock(cnt: Nat, +kn: Nat, +ia0: U32, +j0: U32, +sa: U32, +sb: U32, +bcol: U32, a: Array, b: Array) -> List<&2, F32>: 169 match cnt: 170 case 0n: 171 Nil{} 172 case 1n+p: 173 cblock_go(p, kn, ia0, j0, sa, sb, bcol, a, b) 174 175 def cpar(d: Nat, +cnt: Nat, +kn: Nat, +ia0: U32, +j0: U32, +sa: U32, +sb: U32, +bcol: U32, ra: Array & Array, rb: Array & Array) -> List<&2, F32>: 176 match d ra rb: 177 case 0n Tuple{a, a2} Tuple{b, b2}: 178 cblock(cnt, kn, ia0, j0, sa, sb, bcol, a, b) 179 case 1n++q Tuple{a, a2} Tuple{b, b2}: 180 x y = cpar(q, Nat.div(cnt, 2n), kn, ia0, j0, sa, sb, bcol, Array.clone(F32, a), Array.clone(F32, b)) cpar(q, Nat.sub(cnt, Nat.div(cnt, 2n)), kn, ia0, (j0 + U32.from_nat(Nat.div(cnt, 2n)) : U32), sa, sb, bcol, Array.clone(F32, a2), Array.clone(F32, b2)) 181 List.append(&2, F32, x, y) 182 183 def cp2(+ac: Bool, d: Nat, +m: Nat, +kn: Nat, +sa: U32, +sb: U32, +bcol: U32, c: Array, pa: Array & Array, pb: Array & Array) -> G: 184 match pa pb: 185 case Tuple{a1, a2} Tuple{b1, b2}: 186 gp3(ac, a1, b1, c, cpar(d, m, kn, 0, 0, sa, sb, bcol, Array.clone(F32, a2), Array.clone(F32, b2))) 187 188 # C (1 x m) = A (1 x k) · B, with 2^d blocks of columns in parallel. Since 0.1.12.0 no product calls it (one row runs 189 # the scalar loop, see gemm_any3): it clones both operands at every node. Kept for callers of earlier versions. 190 191 def gemm_par_cols(+ac: Bool, d: Nat, +m: Nat, +k: Nat, +sa: U32, +sb: U32, +bcol: U32, a: Array, b: Array, c: Array) -> G: 192 cp2(ac, d, m, k, sa, sb, bcol, c, Array.clone(F32, a), Array.clone(F32, b)) 193 194 # --------------------------------------------------------------------- 195 # element-wise operations (in place, by index) 196 # --------------------------------------------------------------------- 197 198 # --------------------------------------------------------------------- 199 # gemm: C[i][j] = sum_p A[i*arow + p*sa] * B[p*sb + j*bcol] 200 # (nn: arow=K, sa=1, sb=M, bcol=1 | nt: B stored m x k: sb=1, bcol=K | tn: A stored k x n: arow=1, sa=N) 201 # --------------------------------------------------------------------- 202 203 # acc += A[ia + t*sa] * B[ib + t*sb] for k steps; ra, rb already carry the first pair 204 205 type P2 is Type: 206 P2{a: Array, b: Array} 207 208 type AL is Type: 209 AL{c: Array, l: List<&2, F32>} 210 211 # list of m elements starting at idx (reads with Array.get, returning the array) 212 213 def read_l(mleft: Nat, +idx: U32, acc: List<&2, F32>, rc: Array & F32) -> AL: 214 match mleft rc: 215 case 0n Tuple{c, +v}: 216 AL{c, List.reverse(&2, F32, v <> acc)} 217 case 1n+p Tuple{c, +v}: 218 read_l(p, (idx + 1 : U32), v <> acc, Array.get(F32, c, (idx + 1 : U32))) 219 220 # writes the list starting at idx 221 222 def addl(xs: List<&2, F32>, +idx: U32, rc: Array & F32) -> Array: 223 match xs rc: 224 case Nil{} Tuple{c, v}: 225 c 226 case Con{+h, t} Tuple{c, +v}: 227 addl(t, (idx + 1 : U32), Array.get(F32, Array.set(F32, c, idx, (v + h : F32)), (idx + 1 : U32))) 228 229 # adds a bias (a list of m) to each of the n rows of c (n x m) 230 231 def bias_rows(nleft: Nat, +i: U32, +mu: U32, +bl: List<&2, F32>, c: Array) -> Array: 232 match nleft: 233 case 0n: 234 c 235 case 1n+p: 236 bias_rows(p, (i + 1 : U32), mu, bl, addl(bl, (i * mu : U32), Array.get(F32, c, (i * mu : U32)))) 237 238 # in-place relu on the first n elements 239 240 def relu_l(nleft: Nat, +t: U32, rc: Array & F32) -> Array: 241 match nleft rc: 242 case 0n Tuple{c, v}: 243 c 244 case 1n+p Tuple{c, +v}: 245 relu_l(p, (t + 1 : U32), Array.get(F32, Array.set(F32, c, t, F32.max(v, 0.0)), (t + 1 : U32))) 246 247 def pos2(b: Bool) -> F32: 248 match b: 249 case True{}: 250 1.0 251 case False{}: 252 0.0 253 254 def pos(+y: F32) -> F32: 255 pos2(F32.is_gt(y, 0.0)) 256 257 # d[t] *= (h[t] > 0): the relu gradient; returns both arrays 258 259 def mask_l(nleft: Nat, +t: U32, rd: Array & F32, rh: Array & F32) -> P2: 260 match nleft rd rh: 261 case 0n Tuple{d, x} Tuple{h, y}: 262 P2{d, h} 263 case 1n+p Tuple{d, +x} Tuple{h, +y}: 264 mask_l(p, (t + 1 : U32), Array.get(F32, Array.set(F32, d, t, (x * pos(y) : F32)), (t + 1 : U32)), Array.get(F32, h, (t + 1 : U32))) 265 266 # w[t] -= lr * dw[t] 267 268 def maxl2(xs: List<&2, F32>, cur: F32) -> F32: 269 match xs: 270 case Nil{}: 271 cur 272 case Con{+h, t}: 273 maxl2(t, F32.max(cur, h)) 274 275 def maxl(xs: List<&2, F32>) -> F32: 276 match xs: 277 case Nil{}: 278 0.0 279 case Con{+h, t}: 280 maxl2(t, h) 281 282 def expl(xs: List<&2, F32>, +m: F32) -> List<&2, F32>: 283 match xs: 284 case Nil{}: 285 Nil{} 286 case Con{+h, t}: 287 F32.exp((h - m : F32)) <> expl(t, m) 288 289 def suml(xs: List<&2, F32>, acc: F32) -> F32: 290 match xs: 291 case Nil{}: 292 acc 293 case Con{+h, t}: 294 suml(t, (acc + h : F32)) 295 296 def divl(xs: List<&2, F32>, +s: F32) -> List<&2, F32>: 297 match xs: 298 case Nil{}: 299 Nil{} 300 case Con{+h, t}: 301 (h / s : F32) <> divl(t, s) 302 303 def softmax2(+es: List<&2, F32>) -> List<&2, F32>: 304 divl(es, suml(es, 0.0)) 305 306 def softmax(+xs: List<&2, F32>) -> List<&2, F32>: 307 softmax2(expl(xs, maxl(xs))) 308 309 # p[label] 310 311 def sgd_l(nleft: Nat, +t: U32, +lr: F32, rw: Array & F32, rd: Array & F32) -> P2: 312 match nleft rw rd: 313 case 0n Tuple{w, x} Tuple{d, y}: 314 P2{w, d} 315 case 1n+p Tuple{w, +x} Tuple{d, +y}: 316 sgd_l(p, (t + 1 : U32), lr, Array.get(F32, Array.set(F32, w, t, (x - (lr * y : F32) : F32)), (t + 1 : U32)), Array.get(F32, d, (t + 1 : U32))) 317 318 # --------------------------------------------------------------------- 319 # softmax + cross-entropy per row (lists of 10 elements: negligible cost) 320 # --------------------------------------------------------------------- 321 322 def pick(ps: List<&2, F32>, n: Nat) -> F32: 323 match ps n: 324 case Con{+h, t} 0n: 325 h 326 case Con{h, t} 1n+q: 327 pick(t, q) 328 case Nil{} _: 329 0.0 330 331 # (p - onehot(label)) / n 332 333 def hit(b: Bool) -> F32: 334 match b: 335 case True{}: 336 1.0 337 case False{}: 338 0.0 339 340 def grad_l(ps: List<&2, F32>, +label: Nat, +j: Nat, +nf: F32) -> List<&2, F32>: 341 match ps: 342 case Nil{}: 343 Nil{} 344 case Con{+h, t}: 345 (((h - hit(Nat.is_eq(label, j)) : F32) / nf) : F32) <> grad_l(t, label, 1n+j, nf) 346 347 type L2 is Type: 348 L2{loss: F32, c: Array} 349 350 def ce_row2(+ps: List<&2, F32>, +label: Nat, +nf: F32, +idx: U32, loss: F32, c: Array) -> L2: 351 L2{(loss - F32.log(F32.max(pick(ps, label), 0.000000000001)) : F32), write_l(grad_l(ps, label, 0n, nf), idx, c)} 352 353 def ce_row(al: AL, +label: Nat, +nf: F32, +idx: U32, loss: F32) -> L2: 354 match al: 355 case AL{c, l}: 356 ce_row2(softmax(l), label, nf, idx, loss, c) 357 358 # walks the rows of z2 (n x m): r carries the accumulated loss and the array; at the end the array has become the gradient 359 360 type Mat<-r: Nat, -c: Nat> is Type: 361 Mat{d: Array} 362 363 def ce_rows(labels: List<&2, Nat>, +mm1: Nat, +mu: U32, +nf: F32, +i: U32, r: L2) -> L2: 364 match labels r: 365 case Nil{} L2{loss, c}: 366 L2{loss, c} 367 case Con{+y, t} L2{loss, c}: 368 ce_rows(t, mm1, mu, nf, (i + 1 : U32), ce_row(read_l(mm1, (i * mu : U32), Nil{}, Array.get(F32, c, (i * mu : U32))), y, nf, (i * mu : U32), loss)) 369 370 # ---- helper: hits per row ---- 371 372 type CR is Type: 373 CR{c: Array, n: Nat} 374 375 def gt_head(xs: List<&2, F32>, +v: F32) -> Bool: 376 match xs: 377 case Nil{}: 378 False{} 379 case Con{+h, t}: 380 F32.is_gt(h, v) 381 382 def argmax2(xs: List<&2, F32>, g: Bool, +i: Nat, +bi: Nat, +bv: F32) -> Nat: 383 match xs g: 384 case Nil{} _: 385 bi 386 case Con{+h, +t} True{}: 387 argmax2(t, gt_head(t, h), 1n+i, i, h) 388 case Con{h, +t} False{}: 389 argmax2(t, gt_head(t, bv), 1n+i, bi, bv) 390 391 def argmax(xs: List<&2, F32>) -> Nat: 392 match xs: 393 case Nil{}: 394 0n 395 case Con{+h, +t}: 396 argmax2(t, gt_head(t, h), 1n, 0n, h) 397 398 def one(b: Bool) -> Nat: 399 match b: 400 case True{}: 401 1n 402 case False{}: 403 0n 404 405 def count_rows3(+y: Nat, n: Nat, al: AL) -> CR: 406 match al: 407 case AL{c, l}: 408 CR{c, Nat.add(n, one(Nat.is_eq(argmax(l), y)))} 409 410 def count_rows(labels: List<&2, Nat>, +mm1: Nat, +mu: U32, +i: U32, r: CR) -> CR: 411 match labels r: 412 case Nil{} CR{c, n}: 413 CR{c, n} 414 case Con{+y, t} CR{c, n}: 415 count_rows(t, mm1, mu, (i + 1 : U32), count_rows3(y, n, read_l(mm1, (i * mu : U32), Nil{}, Array.get(F32, c, (i * mu : U32))))) 416 417 418 # ===================================================================== 419 # Typed API: Mat (the dimensions enter the type) 420 # ===================================================================== 421 422 # ===================================================================== 423 # Array capacity: a PROVED depth for the number of slots 424 # ===================================================================== 425 # 426 # cap_depth(n) is the d such that an Array with 2^d slots holds n elements. It is computed by 427 # structural recursion on n (not with U32.log2) so that the law cap_ok below can prove 428 # `n <= 2^cap_depth(n)`. What stays trusted is only that Array.new(T, d, v) allocates 2^d slots. 429 430 # a type chosen by a Bool: rewriting through it refutes True == False 431 def BD(b: Bool, t: Type, f: Type) -> Type: 432 match b: 433 case True{}: 434 t 435 case False{}: 436 f 437 438 # a <= b, by direct recursion 439 def leb(a: Nat, b: Nat) -> Bool: 440 match a b: 441 case 0n _: 442 True{} 443 case 1n+p 0n: 444 False{} 445 case 1n+p 1n+q: 446 leb(p, q) 447 448 def pow2(d: Nat) -> Nat: 449 match d: 450 case 0n: 451 1n 452 case 1n++q: 453 Nat.add(pow2(q), pow2(q)) 454 455 # smallest power of two >= n, by structural recursion on n: returns (depth, 2^depth) 456 def cap_pick(ok: Bool, d: Nat, +p: Nat) -> Nat & Nat: 457 match ok: 458 case True{}: 459 (d, p) 460 case False{}: 461 (1n+d, Nat.add(p, p)) 462 463 def cap_step(r: Nat & Nat, +n: Nat) -> Nat & Nat: 464 match r: 465 case Tuple{+d, +p}: 466 cap_pick(leb(n, p), d, p) 467 468 def cap_pair(n: Nat) -> Nat & Nat: 469 match n: 470 case 0n: 471 (0n, 1n) 472 case 1n++m: 473 cap_step(cap_pair(m), 1n+m) 474 475 def fst_nat(r: Nat & Nat) -> Nat: 476 match r: 477 case Tuple{d, p}: 478 d 479 480 def snd_nat(r: Nat & Nat) -> Nat: 481 match r: 482 case Tuple{d, p}: 483 p 484 485 # leb is Base's Nat.is_le, case for case. leb walks its arguments one step at a time in the compiled program; Base's 486 # comparison is one native comparison there (0.1.13.0). 487 def le_leb(a: Nat, b: Nat) -> {Nat.is_le(a, b) == leb(a, b) : Bool}: 488 match a b: 489 case 0n 0n: 490 {==} 491 case 0n 1n+q: 492 {==} 493 case 1n+p 0n: 494 {==} 495 case 1n+p 1n+q: 496 le_leb(p, q) 497 498 def leb_of(+a: Nat, +b: Nat, e: {True{} == Nat.is_le(a, b) : Bool}) -> {True{} == leb(a, b) : Bool}: 499 %le_leb(a, b) : {True{} == _ : Bool} 500 e 501 502 # the same pair by doubling (0.1.10.0): p doubles until n <= p, log2(n) steps in a tail loop, where cap_pair takes n 503 # steps of a recursion that is not a tail call, each with a comparison of up to n steps. Its fuel is n, which always 504 # suffices (2^n >= n); if it ran out, the answer would be cap_pair's, so the law below needs no arithmetic about how 505 # many doublings there are. ok is Nat.is_le(n, p), computed by the caller (match only inspects a parameter); with leb 506 # it was n steps in all, 152 ms for n = 786,432 on one GPU lane (0.1.13.0). 507 def cap_dbl(f: Nat, ok: Bool, +n: Nat, +d: Nat, +p: Nat) -> Nat & Nat: 508 match f ok: 509 case _ True{}: 510 (d, p) 511 case 0n False{}: 512 cap_pair(n) 513 case 1n+g False{}: 514 cap_dbl(g, Nat.is_le(n, Nat.add(p, p)), n, 1n+d, Nat.add(p, p)) 515 516 # the depth d such that an Array with 2^d slots holds n elements (the smallest such d) 517 def cap_depth(+n: Nat) -> Nat: 518 fst_nat(cap_dbl(n, Nat.is_le(n, 1n), n, 0n, 1n)) 519 520 # ---- lemmas ---- 521 522 # a <= b -> a <= 1 + b 523 def leb_succ(a: Nat, b: Nat, e: {True{} == leb(a, b) : Bool}) -> {True{} == leb(a, 1n+b) : Bool}: 524 match a b: 525 case 0n _: 526 {==} 527 case 1n+p 0n: 528 %e : BD(_, Unit, {True{} == leb(1n+p, 1n) : Bool}) 529 Unit{} 530 case 1n+p 1n+q: 531 leb_succ(p, q, e) 532 533 # a <= b -> a <= c + b 534 def leb_up_l(+a: Nat, +b: Nat, c: Nat, +e: {True{} == leb(a, b) : Bool}) -> {True{} == leb(a, Nat.add(c, b)) : Bool}: 535 match c: 536 case 0n: 537 e 538 case 1n++c2: 539 leb_succ(a, Nat.add(c2, b), leb_up_l(a, b, c2, e)) 540 541 # a <= b -> a <= b + c 542 def leb_up(a: Nat, b: Nat, c: Nat, e: {True{} == leb(a, b) : Bool}) -> {True{} == leb(a, Nat.add(b, c)) : Bool}: 543 match a b: 544 case 0n _: 545 {==} 546 case 1n+p 0n: 547 %e : BD(_, Unit, {True{} == leb(1n+p, Nat.add(0n, c)) : Bool}) 548 Unit{} 549 case 1n+p 1n+q: 550 leb_up(p, q, c, e) 551 552 # a <= b and c <= d -> a + c <= b + d 553 def leb_add(a: Nat, b: Nat, c: Nat, d: Nat, e1: {True{} == leb(a, b) : Bool}, e2: {True{} == leb(c, d) : Bool}) -> {True{} == leb(Nat.add(a, c), Nat.add(b, d)) : Bool}: 554 match a b: 555 case 0n _: 556 leb_up_l(c, d, b, e2) 557 case 1n+p 0n: 558 %e1 : BD(_, Unit, {True{} == leb(Nat.add(1n+p, c), Nat.add(0n, d)) : Bool}) 559 Unit{} 560 case 1n+p 1n+q: 561 leb_add(p, q, c, d, e1, e2) 562 563 # 1 <= 2^d 564 def pow2_pos(d: Nat) -> {True{} == leb(1n, pow2(d)) : Bool}: 565 match d: 566 case 0n: 567 {==} 568 case 1n++q: 569 leb_up(1n, pow2(q), pow2(q), pow2_pos(q)) 570 571 # 1 <= p when p = 2^d 572 def pos_p(d: Nat, p: Nat, i1: {p == pow2(d) : Nat}) -> {True{} == leb(1n, p) : Bool}: 573 %Equal.sym(Nat, p, pow2(d), i1) : {True{} == leb(1n, _) : Bool} 574 pow2_pos(d) 575 576 # p + p == 2^(1+d) when p == 2^d 577 def step_eq(d: Nat, p: Nat, i1: {p == pow2(d) : Nat}) -> {Nat.add(p, p) == pow2(1n+d) : Nat}: 578 %i1 : {Nat.add(p, p) == Nat.add(_, _) : Nat} 579 {==} 580 581 # one step: given the invariant for m, the invariant for 1+m holds whichever way the comparison goes 582 def step_ok(ok: Bool, +d: Nat, +p: Nat, +m: Nat, +e: {ok == leb(1n+m, p) : Bool}, +i1: {p == pow2(d) : Nat}, +i2: {True{} == leb(m, p) : Bool}) -> {snd_nat(cap_pick(ok, d, p)) == pow2(fst_nat(cap_pick(ok, d, p))) : Nat} & {True{} == leb(1n+m, snd_nat(cap_pick(ok, d, p))) : Bool}: 583 match ok: 584 case True{}: 585 (i1, e) 586 case False{}: 587 (step_eq(d, p, i1), leb_add(1n, p, m, p, pos_p(d, p, i1), i2)) 588 589 590 # the invariant for cap_step on a pair, opened 591 def step_pair_ok(+m: Nat, r: Nat & Nat, i1: {snd_nat(r) == pow2(fst_nat(r)) : Nat}, i2: {True{} == leb(m, snd_nat(r)) : Bool}) -> {snd_nat(cap_step(r, 1n+m)) == pow2(fst_nat(cap_step(r, 1n+m))) : Nat} & {True{} == leb(1n+m, snd_nat(cap_step(r, 1n+m))) : Bool}: 592 match r: 593 case Tuple{+d, +p}: 594 step_ok(leb(1n+m, p), d, p, m, {==}, i1, i2) 595 596 def cap_succ(+m: Nat, ih: {snd_nat(cap_pair(m)) == pow2(fst_nat(cap_pair(m))) : Nat} & {True{} == leb(m, snd_nat(cap_pair(m))) : Bool}) -> {snd_nat(cap_pair(1n+m)) == pow2(fst_nat(cap_pair(1n+m))) : Nat} & {True{} == leb(1n+m, snd_nat(cap_pair(1n+m))) : Bool}: 597 match ih: 598 case Tuple{i1, i2}: 599 step_pair_ok(m, cap_pair(m), i1, i2) 600 601 # the invariant of cap_pair: the second component is 2^(first) and is >= n 602 def cap_inv(n: Nat) -> {snd_nat(cap_pair(n)) == pow2(fst_nat(cap_pair(n))) : Nat} & {True{} == leb(n, snd_nat(cap_pair(n))) : Bool}: 603 match n: 604 case 0n: 605 ({==}, {==}) 606 case 1n++m: 607 cap_succ(m, cap_inv(m)) 608 609 # the invariant of cap_dbl: whichever way it ends (n <= p found, or the fuel out and cap_pair's answer), the second 610 # component is 2^(first) and is >= n 611 def dbl_inv(f: Nat, ok: Bool, +n: Nat, +d: Nat, +p: Nat, e: {ok == Nat.is_le(n, p) : Bool}, i1: {p == pow2(d) : Nat}) -> {snd_nat(cap_dbl(f, ok, n, d, p)) == pow2(fst_nat(cap_dbl(f, ok, n, d, p))) : Nat} & {True{} == leb(n, snd_nat(cap_dbl(f, ok, n, d, p))) : Bool}: 612 match f ok: 613 case _ True{}: 614 (i1, leb_of(n, p, e)) 615 case 0n False{}: 616 cap_inv(n) 617 case 1n+g False{}: 618 dbl_inv(g, Nat.is_le(n, Nat.add(p, p)), n, 1n+d, Nat.add(p, p), {==}, step_eq(d, p, i1)) 619 620 def dbl_final(n: Nat, pr: {snd_nat(cap_dbl(n, Nat.is_le(n, 1n), n, 0n, 1n)) == pow2(fst_nat(cap_dbl(n, Nat.is_le(n, 1n), n, 0n, 1n))) : Nat} & {True{} == leb(n, snd_nat(cap_dbl(n, Nat.is_le(n, 1n), n, 0n, 1n))) : Bool}) -> {True{} == leb(n, pow2(cap_depth(n))) : Bool}: 621 match pr: 622 case Tuple{i1, i2}: 623 %i1 : {True{} == leb(n, _) : Bool} 624 i2 625 626 # LAW (cap_ok): an Array with 2^cap_depth(n) slots always has room for n elements. 627 law cap_ok: 628 for +n: Nat 629 {True{} == leb(n, pow2(cap_depth(n))) : Bool} 630 631 def cap_ok(n): 632 dbl_final(n, dbl_inv(n, Nat.is_le(n, 1n), n, 0n, 1n, {==}, {==})) 633 634 # r x c of zeros, with the exact capacity (smallest power of 2) 635 636 def Mat.zeros(+r: Nat, +c: Nat) -> Mat: 637 Mat{Array.new(F32, cap_depth(Nat.mul(r, c)), 0.0)} 638 639 def Mat.fill(+r: Nat, +c: Nat, +v: F32) -> Mat: 640 Mat{Array.new(F32, cap_depth(Nat.mul(r, c)), v)} 641 642 def fill_list(xs: List<&2, F32>, +i: U32, a: Array) -> Array: 643 match xs: 644 case Nil{}: 645 a 646 case Con{h, t}: 647 fill_list(t, (i + 1 : U32), Array.set(F32, a, i, h)) 648 649 def len_is(xs: List<&2, F32>, n: Nat) -> Bool: 650 match xs n: 651 case Nil{} 0n: 652 True{} 653 case Con{h, t} 1n+p: 654 len_is(t, p) 655 case _ _: 656 False{} 657 658 def Mat.of_list2(-r: Nat, -c: Nat, +rr: Nat, +cc: Nat, ok: Bool, xs: List<&2, F32>) -> Maybe<&1, Mat>: 659 match ok: 660 case True{}: 661 Some{Mat{fill_list(xs, 0, Array.new(F32, cap_depth(Nat.mul(rr, cc)), 0.0))}} 662 case False{}: 663 None{} 664 665 # Mat from r*c numbers in row order; None if the size does not match 666 667 def Mat.of_list(+r: Nat, +c: Nat, +xs: List<&2, F32>) -> Maybe<&1, Mat>: 668 Mat.of_list2(r, c, r, c, len_is(xs, Nat.mul(r, c)), xs) 669 670 # reads the first n numbers, returning the matrix 671 672 # Mat without checking the list size (a short list leaves the rest at zero, a long one is cut 673 # by the capacity): only for when the size is known by construction. Prefer Mat.of_list. 674 675 def Mat.from_list(+r: Nat, +c: Nat, xs: List<&2, F32>) -> Mat: 676 Mat{fill_list(xs, 0, Array.new(F32, cap_depth(Nat.mul(r, c)), 0.0))} 677 678 # writes the numbers of xs into m starting at flat index i (row-major); numbers past the 679 # capacity are dropped. For loading a large matrix block by block without building one big list. 680 def Mat.fill_at(-r: Nat, -c: Nat, +i: U32, xs: List<&2, F32>, m: Mat) -> Mat: 681 match m: 682 case Mat{d}: 683 Mat{fill_list(xs, i, d)} 684 685 type ML<-r: Nat, -c: Nat> is Type: 686 ML{m: Mat, l: List<&2, F32>} 687 688 def Mat.to_list2(-r: Nat, -c: Nat, al: AL) -> ML: 689 match al: 690 case AL{d, l}: 691 ML{Mat{d}, l} 692 693 # two independent copies 694 695 def Mat.to_list(+r: Nat, +c: Nat, m: Mat) -> ML: 696 match m: 697 case Mat{d}: 698 Mat.to_list2(r, c, read_l(Nat.sub(Nat.mul(r, c), 1n), 0, Nil{}, Array.get(F32, d, 0))) 699 700 type MM2<-r: Nat, -c: Nat> is Type: 701 MM2{a: Mat, b: Mat} 702 703 def Mat.clone2(-r: Nat, -c: Nat, p: Array & Array) -> MM2: 704 match p: 705 case Tuple{x, y}: 706 MM2{Mat{x}, Mat{y}} 707 708 # ---- products: C (n x m) = A · B, with 2^par blocks of rows in parallel ---- 709 710 # C = A(n x k) · B(k x m) 711 712 def Mat.clone(-r: Nat, -c: Nat, m: Mat) -> MM2: 713 match m: 714 case Mat{d}: 715 Mat.clone2(r, c, Array.clone(F32, d)) 716 717 type MMul<-n: Nat, -k: Nat, -m: Nat> is Type: 718 MMul{a: Mat, b: Mat, c: Mat} 719 720 # C = A(n x k) · Bᵀ, with B stored m x k 721 722 type MMulNT<-n: Nat, -k: Nat, -m: Nat> is Type: 723 MMulNT{a: Mat, b: Mat, c: Mat} 724 725 # C = Aᵀ · B, with A stored k x n 726 727 type MMulTN<-n: Nat, -k: Nat, -m: Nat> is Type: 728 MMulTN{a: Mat, b: Mat, c: Mat} 729 730 # --------------------------------------------------------------------- 731 # gemm in tiles of 8 (0.1.8.0): for products with 2 or more rows, A's rows and B's columns are packed once into cells of 732 # 8 F32 (a row of k numbers is ceil(k/8) cells, zeros after the last), so the inner loop reads 8 numbers per array 733 # access instead of one. The order of the additions is that of `dotr`: product p of a dot goes to partial sum p mod 8 734 # (lane p mod 8 of its cell), in order, and the eight sums are combined as dotr combines them after k steps (its 735 # rotation depends on k mod 8). The padding adds +0 to a sum that starts at +0, which changes nothing, so the results are 736 # the same bits as the scalar loop's. Since 0.1.15.0 a pass over k computes two cells of C (columns j and j + 1), so each 737 # A cell is read once for both: with one sum a lane, the two dots' sums are independent chains (23% fewer cycles over 738 # SmolLM2's three products on one thread; NOTES.md, 2026-10-25). Products with one row (matrix · vector) keep the scalar loop: packing B would cost as much as the product. 739 # --------------------------------------------------------------------- 740 741 type T8 is Data: 742 T8{v0: F32, v1: F32, v2: F32, v3: F32, v4: F32, v5: F32, v6: F32, v7: F32} 743 744 def t8_zero() -> T8: 745 T8{0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0} 746 747 def pick0(b: Bool, +v: F32) -> F32: 748 match b: 749 case True{}: 750 v 751 case False{}: 752 0.0 753 754 type PK is Type: 755 PK{a: Array, t: Array} 756 757 # one cell: the 8 numbers of a at idx, idx + st, ..., those at positions q0 + q >= lim (past the end of the line) as 0 758 def pk_g7(+v0: F32, +v1: F32, +v2: F32, +v3: F32, +v4: F32, +v5: F32, +v6: F32, +q0: U32, +lim: U32, +idx: U32, +st: U32, +o: U32, t: Array, r: Array & F32) -> PK: 759 match r: 760 case Tuple{a, +v}: 761 PK{a, Array.set(T8, t, o, T8{v0, v1, v2, v3, v4, v5, v6, pick0(U32.is_lt((q0 + 7 : U32), lim), v)})} 762 763 def pk_g6(+v0: F32, +v1: F32, +v2: F32, +v3: F32, +v4: F32, +v5: F32, +q0: U32, +lim: U32, +idx: U32, +st: U32, +o: U32, t: Array, r: Array & F32) -> PK: 764 match r: 765 case Tuple{a, +v}: 766 pk_g7(v0, v1, v2, v3, v4, v5, pick0(U32.is_lt((q0 + 6 : U32), lim), v), q0, lim, idx, st, o, t, Array.get(F32, a, (idx + (st * 7 : U32) : U32))) 767 768 def pk_g5(+v0: F32, +v1: F32, +v2: F32, +v3: F32, +v4: F32, +q0: U32, +lim: U32, +idx: U32, +st: U32, +o: U32, t: Array, r: Array & F32) -> PK: 769 match r: 770 case Tuple{a, +v}: 771 pk_g6(v0, v1, v2, v3, v4, pick0(U32.is_lt((q0 + 5 : U32), lim), v), q0, lim, idx, st, o, t, Array.get(F32, a, (idx + (st * 6 : U32) : U32))) 772 773 def pk_g4(+v0: F32, +v1: F32, +v2: F32, +v3: F32, +q0: U32, +lim: U32, +idx: U32, +st: U32, +o: U32, t: Array, r: Array & F32) -> PK: 774 match r: 775 case Tuple{a, +v}: 776 pk_g5(v0, v1, v2, v3, pick0(U32.is_lt((q0 + 4 : U32), lim), v), q0, lim, idx, st, o, t, Array.get(F32, a, (idx + (st * 5 : U32) : U32))) 777 778 def pk_g3(+v0: F32, +v1: F32, +v2: F32, +q0: U32, +lim: U32, +idx: U32, +st: U32, +o: U32, t: Array, r: Array & F32) -> PK: 779 match r: 780 case Tuple{a, +v}: 781 pk_g4(v0, v1, v2, pick0(U32.is_lt((q0 + 3 : U32), lim), v), q0, lim, idx, st, o, t, Array.get(F32, a, (idx + (st * 4 : U32) : U32))) 782 783 def pk_g2(+v0: F32, +v1: F32, +q0: U32, +lim: U32, +idx: U32, +st: U32, +o: U32, t: Array, r: Array & F32) -> PK: 784 match r: 785 case Tuple{a, +v}: 786 pk_g3(v0, v1, pick0(U32.is_lt((q0 + 2 : U32), lim), v), q0, lim, idx, st, o, t, Array.get(F32, a, (idx + (st * 3 : U32) : U32))) 787 788 def pk_g1(+v0: F32, +q0: U32, +lim: U32, +idx: U32, +st: U32, +o: U32, t: Array, r: Array & F32) -> PK: 789 match r: 790 case Tuple{a, +v}: 791 pk_g2(v0, pick0(U32.is_lt((q0 + 1 : U32), lim), v), q0, lim, idx, st, o, t, Array.get(F32, a, (idx + (st * 2 : U32) : U32))) 792 793 def pk_g0(+q0: U32, +lim: U32, +idx: U32, +st: U32, +o: U32, t: Array, r: Array & F32) -> PK: 794 match r: 795 case Tuple{a, +v}: 796 pk_g1(pick0(U32.is_lt((q0 + 0 : U32), lim), v), q0, lim, idx, st, o, t, Array.get(F32, a, (idx + (st * 1 : U32) : U32))) 797 798 # the cells of one line, cell o.. of t; q0 counts the line's numbers before the cell 799 def pk_line(tl: Nat, +q0: U32, +lim: U32, +base: U32, +st: U32, +o: U32, r: PK) -> PK: 800 match tl r: 801 case 0n r: 802 r 803 case 1n+q PK{a, t}: 804 pk_line(q, (q0 + 8 : U32), lim, (base + (st * 8 : U32) : U32), st, (o + 1 : U32), pk_g0(q0, lim, base, st, o, t, Array.get(F32, a, base))) 805 806 # lines count lines; source line i starts at i * lstep (in a), and goes to cell row o (cells o * tku .. in t) 807 def pk_lines(cnt: Nat, +tk: Nat, +tku: U32, +ku: U32, +i: U32, +o: U32, +lstep: U32, +st: U32, r: PK) -> PK: 808 match cnt: 809 case 0n: 810 r 811 case 1n+p: 812 pk_lines(p, tk, tku, ku, (i + 1 : U32), (o + 1 : U32), lstep, st, pk_line(tk, 0, ku, (i * lstep : U32), st, (o * tku : U32), r)) 813 814 def pk_pack(+cnt: Nat, +tk: Nat, +k: Nat, +lstep: U32, +st: U32, a: Array) -> PK: 815 pk_lines(cnt, tk, U32.from_nat(tk), U32.from_nat(k), 0, 0, lstep, st, PK{a, Array.new(T8, cap_depth(Nat.mul(cnt, tk)), t8_zero())}) 816 817 # the dot of two lines of cells: tk cells from ia (in a) and ib (in b); ra, rb hold the arrays and the cells at ia, ib, 818 # already read. Product q of a cell goes to sum q, as dotr adds them. 819 def t8_dot(tk: Nat, +ia: U32, +ib: U32, s0: F32, s1: F32, s2: F32, s3: F32, s4: F32, s5: F32, s6: F32, s7: F32, ra: Array & T8, rb: Array & T8) -> TD: 820 match tk ra rb: 821 case 0n Tuple{a, x} Tuple{b, y}: 822 TD{a, b, s0, s1, s2, s3, s4, s5, s6, s7} 823 case 1n+p Tuple{a, T8{x0, x1, x2, x3, x4, x5, x6, x7}} Tuple{b, T8{y0, y1, y2, y3, y4, y5, y6, y7}}: 824 t8_dot(p, (ia + 1 : U32), (ib + 1 : U32), (s0 + (x0 * y0 : F32) : F32), (s1 + (x1 * y1 : F32) : F32), (s2 + (x2 * y2 : F32) : F32), (s3 + (x3 * y3 : F32) : F32), (s4 + (x4 * y4 : F32) : F32), (s5 + (x5 * y5 : F32) : F32), (s6 + (x6 * y6 : F32) : F32), (s7 + (x7 * y7 : F32) : F32), Array.get(T8, a, (ia + 1 : U32)), Array.get(T8, b, (ib + 1 : U32))) 825 826 type TD is Type: 827 TD{a: Array, b: Array, s0: F32, s1: F32, s2: F32, s3: F32, s4: F32, s5: F32, s6: F32, s7: F32} 828 829 # dotr's final sum after k products: the sums sit rotated by k mod 8 when its loop ends 830 def t8_comb(+rot: Nat, +s0: F32, +s1: F32, +s2: F32, +s3: F32, +s4: F32, +s5: F32, +s6: F32, +s7: F32) -> F32: 831 match rot: 832 case 0n: 833 (((s0 + s1 : F32) + (s2 + s3 : F32) : F32) + ((s4 + s5 : F32) + (s6 + s7 : F32) : F32) : F32) 834 case 1n: 835 (((s1 + s2 : F32) + (s3 + s4 : F32) : F32) + ((s5 + s6 : F32) + (s7 + s0 : F32) : F32) : F32) 836 case 2n: 837 (((s2 + s3 : F32) + (s4 + s5 : F32) : F32) + ((s6 + s7 : F32) + (s0 + s1 : F32) : F32) : F32) 838 case 3n: 839 (((s3 + s4 : F32) + (s5 + s6 : F32) : F32) + ((s7 + s0 : F32) + (s1 + s2 : F32) : F32) : F32) 840 case 4n: 841 (((s4 + s5 : F32) + (s6 + s7 : F32) : F32) + ((s0 + s1 : F32) + (s2 + s3 : F32) : F32) : F32) 842 case 5n: 843 (((s5 + s6 : F32) + (s7 + s0 : F32) : F32) + ((s1 + s2 : F32) + (s3 + s4 : F32) : F32) : F32) 844 case 6n: 845 (((s6 + s7 : F32) + (s0 + s1 : F32) : F32) + ((s2 + s3 : F32) + (s4 + s5 : F32) : F32) : F32) 846 case _: 847 (((s7 + s0 : F32) + (s1 + s2 : F32) : F32) + ((s3 + s4 : F32) + (s5 + s6 : F32) : F32) : F32) 848 849 type TR is Type: 850 TR{a: Array, b: Array, x: F32} 851 852 def t8_fin(+rot: Nat, d: TD) -> TR: 853 match d: 854 case TD{a, b, +s0, +s1, +s2, +s3, +s4, +s5, +s6, +s7}: 855 TR{a, b, t8_comb(rot, s0, s1, s2, s3, s4, s5, s6, s7)} 856 857 def t8_cell(+tk: Nat, +tku: U32, +rot: Nat, +i: U32, +j: U32, a: Array, b: Array) -> TR: 858 t8_fin(rot, t8_dot(tk, (i * tku : U32), (j * tku : U32), 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, Array.get(T8, a, (i * tku : U32)), Array.get(T8, b, (j * tku : U32)))) 859 860 # two cells of C at once: row i against columns j and j + 1 (cells ib and ic of b), each A cell read once for both; each 861 # dot keeps t8_dot's order, so the two cells are t8_cell's bits 862 type B2 is Type: 863 B2{b: Array, y: T8, z: T8} 864 865 def b2_mk(y: T8, r: Array & T8) -> B2: 866 match r: 867 case Tuple{b, z}: 868 B2{b, y, z} 869 870 def b2_get(+i2: U32, r: Array & T8) -> B2: 871 match r: 872 case Tuple{b, y}: 873 b2_mk(y, Array.get(T8, b, i2)) 874 875 def t8_get2(b: Array, +i1: U32, +i2: U32) -> B2: 876 b2_get(i2, Array.get(T8, b, i1)) 877 878 type TD2 is Type: 879 TD2{a: Array, b: Array, s0: F32, s1: F32, s2: F32, s3: F32, s4: F32, s5: F32, s6: F32, s7: F32, u0: F32, u1: F32, u2: F32, u3: F32, u4: F32, u5: F32, u6: F32, u7: F32} 880 881 def t8_dot2(tk: Nat, +ia: U32, +ib: U32, +ic: U32, s0: F32, s1: F32, s2: F32, s3: F32, s4: F32, s5: F32, s6: F32, s7: F32, u0: F32, u1: F32, u2: F32, u3: F32, u4: F32, u5: F32, u6: F32, u7: F32, ra: Array & T8, rb: B2) -> TD2: 882 match tk ra rb: 883 case 0n Tuple{a, x} B2{b, y, z}: 884 TD2{a, b, s0, s1, s2, s3, s4, s5, s6, s7, u0, u1, u2, u3, u4, u5, u6, u7} 885 case 1n+p Tuple{a, T8{+x0, +x1, +x2, +x3, +x4, +x5, +x6, +x7}} B2{b, T8{y0, y1, y2, y3, y4, y5, y6, y7}, T8{z0, z1, z2, z3, z4, z5, z6, z7}}: 886 t8_dot2(p, (ia + 1 : U32), (ib + 1 : U32), (ic + 1 : U32), (s0 + (x0 * y0 : F32) : F32), (s1 + (x1 * y1 : F32) : F32), (s2 + (x2 * y2 : F32) : F32), (s3 + (x3 * y3 : F32) : F32), (s4 + (x4 * y4 : F32) : F32), (s5 + (x5 * y5 : F32) : F32), (s6 + (x6 * y6 : F32) : F32), (s7 + (x7 * y7 : F32) : F32), (u0 + (x0 * z0 : F32) : F32), (u1 + (x1 * z1 : F32) : F32), (u2 + (x2 * z2 : F32) : F32), (u3 + (x3 * z3 : F32) : F32), (u4 + (x4 * z4 : F32) : F32), (u5 + (x5 * z5 : F32) : F32), (u6 + (x6 * z6 : F32) : F32), (u7 + (x7 * z7 : F32) : F32), Array.get(T8, a, (ia + 1 : U32)), t8_get2(b, (ib + 1 : U32), (ic + 1 : U32))) 887 888 type TR2 is Type: 889 TR2{a: Array, b: Array, x: F32, y: F32} 890 891 def t8_fin2(+rot: Nat, d: TD2) -> TR2: 892 match d: 893 case TD2{a, b, +s0, +s1, +s2, +s3, +s4, +s5, +s6, +s7, +u0, +u1, +u2, +u3, +u4, +u5, +u6, +u7}: 894 TR2{a, b, t8_comb(rot, s0, s1, s2, s3, s4, s5, s6, s7), t8_comb(rot, u0, u1, u2, u3, u4, u5, u6, u7)} 895 896 def t8_cell2(+tk: Nat, +tku: U32, +rot: Nat, +i: U32, +j: U32, a: Array, b: Array) -> TR2: 897 t8_fin2(rot, t8_dot2(tk, (i * tku : U32), (j * tku : U32), ((j + 1 : U32) * tku : U32), 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, Array.get(T8, a, (i * tku : U32)), t8_get2(b, (j * tku : U32), ((j + 1 : U32) * tku : U32)))) 898 899 type TG is Type: 900 TG{a: Array, b: Array, c: Array} 901 902 def t8_set(+ac: Bool, +idx: U32, c: Array, r: TR) -> TG: 903 match r: 904 case TR{a, b, +x}: 905 TG{a, b, cwr(ac, idx, x, c)} 906 907 def t8_set2(+ac: Bool, +idx: U32, c: Array, r: TR2) -> TG: 908 match r: 909 case TR2{a, b, +x, +y}: 910 TG{a, b, cwr(ac, (idx + 1 : U32), y, cwr(ac, idx, x, c))} 911 912 # cells j.. of row i, two at a time, the last one alone when mleft is odd 913 def t8_cols(+ac: Bool, mleft: Nat, +tk: Nat, +tku: U32, +rot: Nat, +i: U32, +j: U32, +cidx: U32, g: TG) -> TG: 914 match mleft g: 915 case 0n g: 916 g 917 case 1n TG{a, b, c}: 918 t8_set(ac, cidx, c, t8_cell(tk, tku, rot, i, j, a, b)) 919 case 2n+p TG{a, b, c}: 920 t8_cols(ac, p, tk, tku, rot, i, (j + 2 : U32), (cidx + 2 : U32), t8_set2(ac, cidx, c, t8_cell2(tk, tku, rot, i, j, a, b))) 921 922 def t8_rows(+ac: Bool, nleft: Nat, +m: Nat, +tk: Nat, +tku: U32, +rot: Nat, +i: U32, +mu: U32, g: TG) -> TG: 923 match nleft: 924 case 0n: 925 g 926 case 1n+p: 927 t8_rows(ac, p, m, tk, tku, rot, (i + 1 : U32), mu, t8_cols(ac, m, tk, tku, rot, i, 0, (i * mu : U32), g)) 928 929 # rows i0 .. i0+cnt-1 of C as a list, in order (the parallel version's leaves) 930 type TL is Type: 931 TL{a: Array, b: Array, out: List<&2, F32>} 932 933 def tl_one(out: List<&2, F32>, r: TR) -> TL: 934 match r: 935 case TR{a, b, +x}: 936 TL{a, b, x <> out} 937 938 def tl_two(out: List<&2, F32>, r: TR2) -> TL: 939 match r: 940 case TR2{a, b, +x, +y}: 941 TL{a, b, y <> (x <> out)} 942 943 # cells j.. of row i, mleft of them, onto out (newest first), two at a time as t8_cols 944 def t8_lcols(mleft: Nat, +tk: Nat, +tku: U32, +rot: Nat, +i: U32, +j: U32, l: TL) -> TL: 945 match mleft l: 946 case 0n l: 947 l 948 case 1n TL{a, b, out}: 949 tl_one(out, t8_cell(tk, tku, rot, i, j, a, b)) 950 case 2n+p TL{a, b, out}: 951 t8_lcols(p, tk, tku, rot, i, (j + 2 : U32), tl_two(out, t8_cell2(tk, tku, rot, i, j, a, b))) 952 953 def t8_lrows(nleft: Nat, +m: Nat, +tk: Nat, +tku: U32, +rot: Nat, +i: U32, l: TL) -> TL: 954 match nleft: 955 case 0n: 956 l 957 case 1n+p: 958 t8_lrows(p, m, tk, tku, rot, (i + 1 : U32), t8_lcols(m, tk, tku, rot, i, 0, l)) 959 960 def t8_lfin(l: TL) -> List<&2, F32>: 961 match l: 962 case TL{a, b, out}: 963 List.reverse(&2, F32, out) 964 965 type PKK is Type: 966 PKK{a: Array, b: Array, ta: Array, tb: Array} 967 968 def t8_pk2(pa: PK, pb: PK) -> PKK: 969 match pa pb: 970 case PK{a, ta} PK{b, tb}: 971 PKK{a, b, ta, tb} 972 973 def t8_g(a: Array, b: Array, g: TG) -> G: 974 match g: 975 case TG{ta, tb, c}: 976 G{a, b, c} 977 978 def t8_out_seq(+ac: Bool, +n: Nat, +m: Nat, +tk: Nat, +tku: U32, +rot: Nat, c: Array, p: PKK) -> G: 979 match p: 980 case PKK{a, b, ta, tb}: 981 t8_g(a, b, t8_rows(ac, n, m, tk, tku, rot, 0, U32.from_nat(m), TG{ta, tb, c})) 982 983 # ---- the parallel path (0.1.9.0): the larger operand is packed into one piece per block, so a block owns its piece; 984 # the other operand is copied once per node of the fork tree (2^d - 1 copies), instead of both at every node ---- 985 986 type PT is Type: 987 PLeaf{t: Array} 988 PNode{l: PT, r: PT} 989 990 type PKT is Type: 991 PKT{a: Array, t: PT} 992 993 def pkt_leaf(r: PK) -> PKT: 994 match r: 995 case PK{a, t}: 996 PKT{a, PLeaf{t}} 997 998 def pkt_then(r: PKT, k: Array -> PT -> PKT) -> PKT: 999 match r: 1000 case PKT{a, t}: 1001 k(a, t) 1002 1003 # lines i .. i+cnt-1 of a, packed into 2^d pieces (halves of the line count, as the fork tree splits them) 1004 def pk_tree(+d: Nat, +cnt: Nat, +i: U32, +tk: Nat, +tku: U32, +ku: U32, +lstep: U32, +st: U32, a: Array) -> PKT: 1005 match d: 1006 case 0n: 1007 pkt_leaf(pk_lines(cnt, tk, tku, ku, i, 0, lstep, st, PK{a, Array.new(T8, cap_depth(Nat.mul(cnt, tk)), t8_zero())})) 1008 case 1n+q: 1009 pkt_then(pk_tree(q, Nat.div(cnt, 2n), i, tk, tku, ku, lstep, st, a), a2 => l => pkt_then(pk_tree(q, Nat.sub(cnt, Nat.div(cnt, 2n)), (i + U32.from_nat(Nat.div(cnt, 2n)) : U32), tk, tku, ku, lstep, st, a2), a3 => r => PKT{a3, PNode{l, r}})) 1010 1011 def t8_with2(-R: Type, p: Array & Array, k: Array -> Array -> R) -> R: 1012 match p: 1013 case Tuple{x, y}: 1014 k(x, y) 1015 1016 # split by rows of C: the pieces are A's rows, B is shared; the result is C's rows in order 1017 def t8_rpar(t: PT, +cnt: Nat, +m: Nat, +tk: Nat, +tku: U32, +rot: Nat, b: Array) -> List<&2, F32>: 1018 match t: 1019 case PLeaf{a}: 1020 t8_lfin(t8_lrows(cnt, m, tk, tku, rot, 0, TL{a, b, Nil{}})) 1021 case PNode{l, r}: 1022 t8_with2(List<&2, F32>, Array.clone(T8, b), b1 => b2 => ( 1023 x y = t8_rpar(l, Nat.div(cnt, 2n), m, tk, tku, rot, b1) t8_rpar(r, Nat.sub(cnt, Nat.div(cnt, 2n)), m, tk, tku, rot, b2) 1024 List.append(&2, F32, x, y))) 1025 1026 # a block of C's columns j0 .. j0+w-1 for every row, row by row 1027 type CB is Data: 1028 CB{j0: U32, w: Nat, xs: List<&2, F32>} 1029 1030 # split by columns of C: the pieces are B's columns, A is shared 1031 def t8_cpar(t: PT, +cnt: Nat, +j0: U32, +n: Nat, +tk: Nat, +tku: U32, +rot: Nat, a: Array) -> List<&2, CB>: 1032 match t: 1033 case PLeaf{b}: 1034 [CB{j0, cnt, t8_lfin(t8_lrows(n, cnt, tk, tku, rot, 0, TL{a, b, Nil{}}))}] 1035 case PNode{l, r}: 1036 t8_with2(List<&2, CB>, Array.clone(T8, a), a1 => a2 => ( 1037 x y = t8_cpar(l, Nat.div(cnt, 2n), j0, n, tk, tku, rot, a1) t8_cpar(r, Nat.sub(cnt, Nat.div(cnt, 2n)), (j0 + U32.from_nat(Nat.div(cnt, 2n)) : U32), n, tk, tku, rot, a2) 1038 List.append(&2, CB, x, y))) 1039 1040 type RW is Type: 1041 RW{c: Array, xs: List<&2, F32>} 1042 1043 # w numbers of xs into c from idx; what is left of xs comes back 1044 def cb_row(+ac: Bool, w: Nat, +idx: U32, c: Array, xs: List<&2, F32>) -> RW: 1045 match w xs: 1046 case 1n+p Con{+h, t}: 1047 cb_row(ac, p, (idx + 1 : U32), cwr(ac, idx, h, c), t) 1048 case _ _: 1049 RW{c, xs} 1050 1051 def cb_rows(+ac: Bool, n: Nat, +i: U32, +j0: U32, +w: Nat, +mu: U32, r: RW) -> Array: 1052 match n r: 1053 case 0n RW{c, xs}: 1054 c 1055 case 1n+p RW{c, xs}: 1056 cb_rows(ac, p, (i + 1 : U32), j0, w, mu, cb_row(ac, w, ((i * mu : U32) + j0 : U32), c, xs)) 1057 1058 def cb_write(+ac: Bool, bs: List<&2, CB>, +n: Nat, +mu: U32, c: Array) -> Array: 1059 match bs: 1060 case Nil{}: 1061 c 1062 case Con{CB{+j0, +w, xs}, t}: 1063 cb_write(ac, t, n, mu, cb_rows(ac, n, 0, j0, w, mu, RW{c, xs})) 1064 1065 def t8_pr2(+ac: Bool, +n: Nat, +m: Nat, +tk: Nat, +tku: U32, +rot: Nat, c: Array, ra: PKT, rb: PK) -> G: 1066 match ra rb: 1067 case PKT{a, t} PK{b, tb}: 1068 G{a, b, write_la(ac, t8_rpar(t, n, m, tk, tku, rot, tb), 0, c)} 1069 1070 def t8_pc2(+ac: Bool, +n: Nat, +m: Nat, +tk: Nat, +tku: U32, +rot: Nat, c: Array, ra: PK, rb: PKT) -> G: 1071 match ra rb: 1072 case PK{a, ta} PKT{b, t}: 1073 G{a, b, cb_write(ac, t8_cpar(t, m, 0, n, tk, tku, rot, ta), n, U32.from_nat(m), c)} 1074 1075 # 2^d blocks along the larger of n (rows of C, pieces of A) and m (columns of C, pieces of B) 1076 def t8_par2(+ac: Bool, byrows: Bool, d: Nat, +n: Nat, +m: Nat, +k: Nat, +tk: Nat, +tku: U32, +rot: Nat, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, a: Array, b: Array, c: Array) -> G: 1077 match byrows: 1078 case True{}: 1079 t8_pr2(ac, n, m, tk, tku, rot, c, pk_tree(d, n, 0, tk, tku, U32.from_nat(k), arow, sa, a), pk_pack(m, tk, k, bcol, sb, b)) 1080 case False{}: 1081 t8_pc2(ac, n, m, tk, tku, rot, c, pk_pack(n, tk, k, arow, sa, a), pk_tree(d, m, 0, tk, tku, U32.from_nat(k), bcol, sb, b)) 1082 1083 def gemm_t8b(+ac: Bool, d: Nat, +n: Nat, +m: Nat, +k: Nat, +tk: Nat, +tku: U32, +rot: Nat, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, a: Array, b: Array, c: Array) -> G: 1084 match d: 1085 case 0n: 1086 t8_out_seq(ac, n, m, tk, tku, rot, c, t8_pk2(pk_pack(n, tk, k, arow, sa, a), pk_pack(m, tk, k, bcol, sb, b))) 1087 case 1n+q: 1088 t8_par2(ac, Nat.is_le(m, n), 1n+q, n, m, k, tk, tku, rot, arow, sa, sb, bcol, a, b, c) 1089 1090 # C (n x m) = A · B (or C + A · B when ac) in cells of 8, sequential (d = 0) or with 2^d blocks of rows in parallel; A 1091 # and B come back intact 1092 def gemm_t8_ac(+ac: Bool, d: Nat, +n: Nat, +m: Nat, +k: Nat, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, a: Array, b: Array, c: Array) -> G: 1093 gemm_t8b(ac, d, n, m, k, Nat.div(Nat.add(k, 7n), 8n), U32.from_nat(Nat.div(Nat.add(k, 7n), 8n)), Nat.mod(k, 8n), arow, sa, sb, bcol, a, b, c) 1094 1095 def gemm_t8(d: Nat, +n: Nat, +m: Nat, +k: Nat, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, a: Array, b: Array, c: Array) -> G: 1096 gemm_t8_ac(False{}, d, n, m, k, arow, sa, sb, bcol, a, b, c) 1097 1098 # one row (matrix · vector): the scalar loop at every depth; two or more rows: cells of 8. Up to 0.1.11.0 one row in 1099 # parallel ran gemm_par_cols, which clones both operands at every node of the fork tree: a 50,257 x 768 weight matrix at 1100 # depth 4 ran out of the default heap (5.1 GB) and 8192 x 2048 took 124 ms against 10 ms sequentially. Packing the 1101 # matrix into cells of 8 instead costs more than the product, which reads each weight once (64 ms against 24 at 1102 # 50,257 x 768). Matrix · vector in parallel is Bands.matvec. Measured 2026-10-16 (NOTES.md). 1103 def gemm_any3(+ac: Bool, one: Bool, d: Nat, +n: Nat, +m: Nat, +k: Nat, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, a: Array, b: Array, c: Array) -> G: 1104 match one: 1105 case True{}: 1106 gemm_ac(ac, n, m, k, arow, sa, sb, bcol, a, b, c) 1107 case False{}: 1108 gemm_t8_ac(ac, d, n, m, k, arow, sa, sb, bcol, a, b, c) 1109 1110 def gemm_any2(+ac: Bool, d: Nat, +n: Nat, +m: Nat, +k: Nat, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, a: Array, b: Array, c: Array) -> G: 1111 gemm_any3(ac, Nat.is_le(n, 1n), d, n, m, k, arow, sa, sb, bcol, a, b, c) 1112 1113 def gemm_any(+ac: Bool, d: Nat, +n: Nat, +m: Nat, +k: Nat, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, a: Array, b: Array, c: Array) -> G: 1114 gemm_any2(ac, d, n, m, k, arow, sa, sb, bcol, a, b, c) 1115 1116 def mmul_fin(-n: Nat, -k: Nat, -m: Nat, g: G) -> MMul: 1117 match g: 1118 case G{a, b, c}: 1119 MMul{Mat{a}, Mat{b}, Mat{c}} 1120 1121 def mmul_nt_fin(-n: Nat, -k: Nat, -m: Nat, g: G) -> MMulNT: 1122 match g: 1123 case G{a, b, c}: 1124 MMulNT{Mat{a}, Mat{b}, Mat{c}} 1125 1126 def mmul_tn_fin(-n: Nat, -k: Nat, -m: Nat, g: G) -> MMulTN: 1127 match g: 1128 case G{a, b, c}: 1129 MMulTN{Mat{a}, Mat{b}, Mat{c}} 1130 1131 # ---- the split depth of a product (0.1.11.0): par is an upper bound, and a product too small to gain splits 0 times ---- 1132 # 1133 # A product of n x k by k x m splits into 2^par blocks only when it does at least 2^23 multiply-adds and each cell of C 1134 # sums at least 64 of them; otherwise it runs sequentially. Measured on 16 threads (2026-10-16, 18 shapes): below 2^23 the 1135 # split costs more than it saves, and with a short inner dimension (a weight gradient over a few tokens) every split 1136 # was 2-3.6x slower, because each block hands its rows of C back as a list that is then written into C, a cost per cell 1137 # of C that k multiply-adds per cell do not pay for. The result does not depend on the depth: the same bits at every 1138 # depth (tests/t8.bend, tests/acc.bend, tests/split.bend). 1139 def split_pick(big: Bool, d: Nat) -> Nat: 1140 match big: 1141 case True{}: 1142 d 1143 case False{}: 1144 0n 1145 1146 def split_depth(d: Nat, +n: Nat, +m: Nat, +k: Nat) -> Nat: 1147 split_pick(Bool.and(Nat.is_le(64n, k), Nat.is_le(8388608n, Nat.mul(Nat.mul(n, m), k))), d) 1148 1149 def mm_a(+n: Nat, +k: Nat, +m: Nat, d: Nat, a: Array, b: Array) -> MMul: 1150 mmul_fin(n, k, m, gemm_any(False{}, d, n, m, k, U32.from_nat(k), 1, U32.from_nat(m), 1, a, b, Array.new(F32, cap_depth(Nat.mul(n, m)), 0.0))) 1151 1152 # C = A · B (n x k) · (k x m); par = log2 of the most parallel blocks (0 = sequential); split_depth picks 0 or par 1153 1154 def Mat.matmul(+n: Nat, +k: Nat, +m: Nat, par: Nat, a: Mat, b: Mat) -> MMul: 1155 match a b: 1156 case Mat{x} Mat{y}: 1157 mm_a(n, k, m, split_depth(par, n, m, k), x, y) 1158 1159 # C + A · B, into C (n x m); A and B come back with the result, as Mat.matmul returns them. Each cell of A · B is 1160 # computed as Mat.matmul computes it and added once to C's cell: the same bits as Mat.add(c, Mat.matmul(a, b)'s c), 1161 # without allocating the product or walking it a second time. 1162 1163 def Mat.matmul_acc(+n: Nat, +k: Nat, +m: Nat, par: Nat, a: Mat, b: Mat, c: Mat) -> MMul: 1164 match a b c: 1165 case Mat{x} Mat{y} Mat{z}: 1166 mmul_fin(n, k, m, gemm_any(True{}, split_depth(par, n, m, k), n, m, k, U32.from_nat(k), 1, U32.from_nat(m), 1, x, y, z)) 1167 1168 def mm_nt(+n: Nat, +k: Nat, +m: Nat, d: Nat, a: Array, b: Array) -> MMulNT: 1169 mmul_nt_fin(n, k, m, gemm_any(False{}, d, n, m, k, U32.from_nat(k), 1, 1, U32.from_nat(k), a, b, Array.new(F32, cap_depth(Nat.mul(n, m)), 0.0))) 1170 1171 # C = A · Bᵀ (n x k) · (m x k)ᵀ: for weights stored with one row per output 1172 1173 def Mat.matmul_nt(+n: Nat, +k: Nat, +m: Nat, par: Nat, a: Mat, b: Mat) -> MMulNT: 1174 match a b: 1175 case Mat{x} Mat{y}: 1176 mm_nt(n, k, m, split_depth(par, n, m, k), x, y) 1177 1178 # C + A · Bᵀ, into C (n x m) 1179 1180 def Mat.matmul_nt_acc(+n: Nat, +k: Nat, +m: Nat, par: Nat, a: Mat, b: Mat, c: Mat) -> MMulNT: 1181 match a b c: 1182 case Mat{x} Mat{y} Mat{z}: 1183 mmul_nt_fin(n, k, m, gemm_any(True{}, split_depth(par, n, m, k), n, m, k, U32.from_nat(k), 1, 1, U32.from_nat(k), x, y, z)) 1184 1185 def mm_tn(+n: Nat, +k: Nat, +m: Nat, d: Nat, a: Array, b: Array) -> MMulTN: 1186 mmul_tn_fin(n, k, m, gemm_any(False{}, d, n, m, k, 1, U32.from_nat(n), U32.from_nat(m), 1, a, b, Array.new(F32, cap_depth(Nat.mul(n, m)), 0.0))) 1187 1188 # C = Aᵀ · B (k x n)ᵀ · (k x m): weight gradient, X^T · dY 1189 1190 def Mat.matmul_tn(+n: Nat, +k: Nat, +m: Nat, par: Nat, a: Mat, b: Mat) -> MMulTN: 1191 match a b: 1192 case Mat{x} Mat{y}: 1193 mm_tn(n, k, m, split_depth(par, n, m, k), x, y) 1194 1195 # C + Aᵀ · B, into C (n x m): a weight gradient added into the gradient it accumulates, dW += Xᵀ · dY 1196 1197 def Mat.matmul_tn_acc(+n: Nat, +k: Nat, +m: Nat, par: Nat, a: Mat, b: Mat, c: Mat) -> MMulTN: 1198 match a b c: 1199 case Mat{x} Mat{y} Mat{z}: 1200 mmul_tn_fin(n, k, m, gemm_any(True{}, split_depth(par, n, m, k), n, m, k, 1, U32.from_nat(n), U32.from_nat(m), 1, x, y, z)) 1201 1202 # ---- a precise product for long dots (0.1.14.0) ---- 1203 # 1204 # Mat.matmul_precise is Mat.matmul with k summed in blocks of 1024 (the last one 1..1024): block b's columns of A and 1205 # rows of B are copied out, the first block's product written into C, each next one's added to it as matmul_acc adds, 1206 # so every cell is ((x0 + x1) + x2) .., x_b the block's dot with the usual four partial sums. A dot of 49,152 products 1207 # as four running sums of 12,288 lost the small terms (SmolLM2's head: its gradient 1.8x further from float64 than 1208 # PyTorch's float32, 0.03x in blocks; BendTorch NOTES, 2026-10-18). Opt-in: it copies the slices and runs one product 1209 # a block, and the default products keep their speed and bits. Up to 1024 steps it is Mat.matmul itself. 1210 1211 type SL is Type: 1212 SL{a: Array, s: Array} 1213 1214 # d[di..] = src[si..], k numbers; r holds src[si], already read 1215 def sl_row(k: Nat, +si: U32, +di: U32, d: Array, r: Array & F32) -> SL: 1216 match k r: 1217 case 0n Tuple{a, x}: SL{a, d} 1218 case 1n+p Tuple{a, +x}: sl_row(p, (si + 1 : U32), (di + 1 : U32), Array.set(F32, d, di, x), Array.get(F32, a, (si + 1 : U32))) 1219 1220 # rows i.. of a slice: w numbers a row from src[i*st + o], into d[i*w] 1221 def sl_rows(r: Nat, +i: U32, +w: Nat, +w32: U32, +st: U32, +o: U32, sl: SL) -> SL: 1222 match r sl: 1223 case 0n sl: sl 1224 case 1n+p SL{a, d}: sl_rows(p, (i + 1 : U32), w, w32, st, o, sl_row(w, ((i * st : U32) + o : U32), (i * w32 : U32), d, Array.get(F32, a, ((i * st : U32) + o : U32)))) 1225 1226 def blk_len(+bi: U32, +k: U32) -> Nat: 1227 U32.to_nat(Bool.pick(U32, U32.is_lt((k - (bi * 1024 : U32) : U32), 1024), (k - (bi * 1024 : U32) : U32), 1024)) 1228 1229 def mp_s4(a: Array, b: Array, g: G) -> G: 1230 match g: 1231 case G{x, y, c}: G{a, b, c} 1232 1233 def mp_s3(+ac: Bool, +par: Nat, +n: Nat, +m: Nat, +w: Nat, +w32: U32, +m32: U32, a: Array, as: Array, c: Array, sb: SL) -> G: 1234 match sb: 1235 case SL{b, bs}: mp_s4(a, b, gemm_any(ac, split_depth(par, n, m, w), n, m, w, w32, 1, m32, 1, as, bs, c)) 1236 1237 def mp_s2(+ac: Bool, +par: Nat, +n: Nat, +m: Nat, +w: Nat, +w32: U32, +m32: U32, +o: U32, c: Array, b: Array, sa: SL) -> G: 1238 match sa: 1239 case SL{a, as}: mp_s3(ac, par, n, m, w, w32, m32, a, as, c, sl_row(Nat.mul(w, m), (o * m32 : U32), 0, Array.new(F32, cap_depth(Nat.mul(w, m)), 0.0), Array.get(F32, b, (o * m32 : U32)))) 1240 1241 # block bi: A's columns o..o+w (A is n x k) and B's rows o..o+w (B is k x m), multiplied into C (added when ac) 1242 def mp_step(+ac: Bool, +par: Nat, +n: Nat, +m: Nat, +k32: U32, +bi: U32, a: Array, b: Array, c: Array) -> G: 1243 mp_s2(ac, par, n, m, blk_len(bi, k32), U32.from_nat(blk_len(bi, k32)), U32.from_nat(m), (bi * 1024 : U32), c, b, sl_rows(n, 0, blk_len(bi, k32), U32.from_nat(blk_len(bi, k32)), k32, (bi * 1024 : U32), SL{a, Array.new(F32, cap_depth(Nat.mul(n, blk_len(bi, k32))), 0.0)})) 1244 1245 def mp_go(nb: Nat, +bi: U32, +par: Nat, +n: Nat, +m: Nat, +k32: U32, g: G) -> G: 1246 match nb g: 1247 case 0n g: g 1248 case 1n++q G{a, b, c}: mp_go(q, (bi + 1 : U32), par, n, m, k32, mp_step(U32.is_lt(0, bi), par, n, m, k32, bi, a, b, c)) 1249 1250 def mm_precise2(long: Bool, +n: Nat, +k: Nat, +m: Nat, par: Nat, a: Mat, b: Mat) -> MMul: 1251 match long a b: 1252 case True{} Mat{x} Mat{y}: mmul_fin(n, k, m, mp_go(U32.to_nat(U32.div((U32.from_nat(k) + 1023 : U32), 1024)), 0, par, n, m, U32.from_nat(k), G{x, y, Array.new(F32, cap_depth(Nat.mul(n, m)), 0.0)})) 1253 case False{} a b: Mat.matmul(n, k, m, par, a, b) 1254 1255 # C = A · B (n x k) · (k x m), k summed in blocks of 1024 (above); opt-in 1256 def Mat.matmul_precise(+n: Nat, +k: Nat, +m: Nat, +par: Nat, a: Mat, b: Mat) -> MMul: 1257 mm_precise2(Nat.is_lt(1024n, k), n, k, m, par, a, b) 1258 1259 # ---- element-wise and training operations ---- 1260 1261 # adds the bias (one row, m numbers) to each row of C (n x m) 1262 1263 type MBias<-n: Nat, -m: Nat> is Type: 1264 MBias{c: Mat, b: Mat<1n, m>} 1265 1266 def add_row2(-n: Nat, -m: Nat, +nn: Nat, +mm: Nat, cd: Array, al: AL) -> MBias: 1267 match al: 1268 case AL{bd, bl}: 1269 MBias{Mat{bias_rows(nn, 0, U32.from_nat(mm), bl, cd)}, Mat{bd}} 1270 1271 def Mat.add_row(+n: Nat, +m: Nat, c: Mat, b: Mat<1n, m>) -> MBias: 1272 match c b: 1273 case Mat{cd} Mat{bd}: 1274 add_row2(n, m, n, m, cd, read_l(Nat.sub(m, 1n), 0, Nil{}, Array.get(F32, bd, 0))) 1275 1276 # pair of matrices of the same shape (the result of an operation that reads two operands) 1277 1278 type MPair<-r: Nat, -c: Nat> is Type: 1279 MPair{a: Mat, b: Mat} 1280 1281 def mpair_fin(-r: Nat, -c: Nat, p: P2) -> MPair: 1282 match p: 1283 case P2{a, b}: 1284 MPair{Mat{a}, Mat{b}} 1285 1286 def relu_fin(-r: Nat, -c: Nat, d: Array) -> Mat: 1287 Mat{d} 1288 1289 def Mat.relu(+r: Nat, +c: Nat, a: Mat) -> Mat: 1290 match a: 1291 case Mat{d}: 1292 relu_fin(r, c, relu_l(Nat.mul(r, c), 0, Array.get(F32, d, 0))) 1293 1294 # (dy * (h > 0), h): the gradient that passes through the relu 1295 1296 def Mat.relu_bwd(+r: Nat, +c: Nat, dy: Mat, h: Mat) -> MPair: 1297 match dy h: 1298 case Mat{d} Mat{x}: 1299 mpair_fin(r, c, mask_l(Nat.mul(r, c), 0, Array.get(F32, d, 0), Array.get(F32, x, 0))) 1300 1301 # (w - lr * dw, dw): one SGD step 1302 1303 def Mat.sgd(+r: Nat, +c: Nat, +lr: F32, w: Mat, dw: Mat) -> MPair: 1304 match w dw: 1305 case Mat{x} Mat{y}: 1306 mpair_fin(r, c, sgd_l(Nat.mul(r, c), 0, lr, Array.get(F32, x, 0), Array.get(F32, y, 0))) 1307 1308 # (a + b, b): element-wise sum (a - (-1)*b is exactly a + b) 1309 1310 def Mat.add(+r: Nat, +c: Nat, a: Mat, b: Mat) -> MPair: 1311 match a b: 1312 case Mat{x} Mat{y}: 1313 mpair_fin(r, c, sgd_l(Nat.mul(r, c), 0, (0.0 - 1.0 : F32), Array.get(F32, x, 0), Array.get(F32, y, 0))) 1314 1315 # sum of each column (n x m -> 1 x m), returning the matrix 1316 1317 type MColSum<-n: Nat, -m: Nat> is Type: 1318 MColSum{a: Mat, s: Mat<1n, m>} 1319 1320 def colsum_fin(-n: Nat, -m: Nat, g: G) -> MColSum: 1321 match g: 1322 case G{ones, a, s}: 1323 MColSum{Mat{a}, Mat{s}} 1324 1325 def Mat.col_sums(+n: Nat, +m: Nat, a: Mat) -> MColSum: 1326 match a: 1327 case Mat{d}: 1328 colsum_fin(n, m, gemm(1n, m, n, 0, 1, U32.from_nat(m), 1, Array.new(F32, cap_depth(n), 1.0), d, Array.new(F32, cap_depth(m), 0.0))) 1329 1330 # s + the sum of each column, into s (1 x m): a bias gradient added into its accumulator, with the bits of 1331 # Mat.add(s, Mat.col_sums(a)'s s) 1332 1333 def Mat.col_sums_acc(+n: Nat, +m: Nat, a: Mat, s: Mat<1n, m>) -> MColSum: 1334 match a s: 1335 case Mat{d} Mat{z}: 1336 colsum_fin(n, m, gemm_ac(True{}, 1n, m, n, 0, 1, U32.from_nat(m), 1, Array.new(F32, cap_depth(n), 1.0), d, z)) 1337 1338 # mean cross-entropy and its gradient on the logits: (softmax - one-hot) / n. 1339 # labels has one entry per row (n); the gradient replaces the logits. 1340 1341 type MCE<-n: Nat, -c: Nat> is Type: 1342 MCE{loss: F32, grad: Mat} 1343 1344 def ce_fin(-n: Nat, -c: Nat, +nf: F32, r: L2) -> MCE: 1345 match r: 1346 case L2{loss, d}: 1347 MCE{(loss / nf : F32), Mat{d}} 1348 1349 def Mat.softmax_ce(+n: Nat, +c: Nat, z: Mat, labels: List<&2, Nat>) -> MCE: 1350 match z: 1351 case Mat{d}: 1352 ce_fin(n, c, F32.from_nat(n), ce_rows(labels, Nat.sub(c, 1n), U32.from_nat(c), F32.from_nat(n), 0, L2{0.0, d})) 1353 1354 # argmax hits per row against labels 1355 1356 type MHits<-n: Nat, -c: Nat> is Type: 1357 MHits{z: Mat, hits: Nat} 1358 1359 def hits_fin(-n: Nat, -c: Nat, r: CR) -> MHits: 1360 match r: 1361 case CR{d, k}: 1362 MHits{Mat{d}, k} 1363 1364 def Mat.count_correct(+n: Nat, +c: Nat, z: Mat, labels: List<&2, Nat>) -> MHits: 1365 match z: 1366 case Mat{d}: 1367 hits_fin(n, c, count_rows(labels, Nat.sub(c, 1n), U32.from_nat(c), 0, CR{d, 0n})) 1368 1369 # a row as a list (returning the matrix) and the other way around 1370 1371 type MRow<-r: Nat, -c: Nat> is Type: 1372 MRow{m: Mat, row: List<&2, F32>} 1373 1374 def row_fin(-r: Nat, -c: Nat, al: AL) -> MRow: 1375 match al: 1376 case AL{d, l}: 1377 MRow{Mat{d}, l} 1378 1379 def Mat.read_row(+r: Nat, +c: Nat, m: Mat, +i: U32) -> MRow: 1380 match m: 1381 case Mat{d}: 1382 row_fin(r, c, read_l(Nat.sub(c, 1n), (i * U32.from_nat(c) : U32), Nil{}, Array.get(F32, d, (i * U32.from_nat(c) : U32)))) 1383 1384 def Mat.write_row(-r: Nat, +c: Nat, m: Mat, +i: U32, xs: List<&2, F32>) -> Mat: 1385 match m: 1386 case Mat{d}: 1387 Mat{write_l(xs, (i * U32.from_nat(c) : U32), d)} 1388 1389 # ---- variants that check the number of labels ---- 1390 1391 # does the list of labels have exactly n entries (one per row)? 1392 def labels_len_is(xs: List<&2, Nat>, n: Nat) -> Bool: 1393 match xs n: 1394 case Nil{} 0n: 1395 True{} 1396 case Con{h, t} 1n+p: 1397 labels_len_is(t, p) 1398 case _ _: 1399 False{} 1400 1401 def ce_checked(ok: Bool, +n: Nat, +c: Nat, z: Mat, labels: List<&2, Nat>) -> Maybe<&1, MCE>: 1402 match ok: 1403 case True{}: 1404 Some{Mat.softmax_ce(n, c, z, labels)} 1405 case False{}: 1406 None{} 1407 1408 # like Mat.softmax_ce, but returns None unless labels has exactly n entries 1409 def Mat.softmax_ce_checked(+n: Nat, +c: Nat, z: Mat, +labels: List<&2, Nat>) -> Maybe<&1, MCE>: 1410 ce_checked(labels_len_is(labels, n), n, c, z, labels) 1411 1412 def hits_checked(ok: Bool, +n: Nat, +c: Nat, z: Mat, labels: List<&2, Nat>) -> Maybe<&1, MHits>: 1413 match ok: 1414 case True{}: 1415 Some{Mat.count_correct(n, c, z, labels)} 1416 case False{}: 1417 None{} 1418 1419 # like Mat.count_correct, but returns None unless labels has exactly n entries 1420 def Mat.count_correct_checked(+n: Nat, +c: Nat, z: Mat, +labels: List<&2, Nat>) -> Maybe<&1, MHits>: 1421 hits_checked(labels_len_is(labels, n), n, c, z, labels) 1422 1423 # --------------------------------------------------------------------- 1424 # Bands: a matrix cut into bands of rows, for matrix · vector in parallel 1425 # without copying the weights. 1426 # 1427 # Mat.matmul_nt with n = 1 can only run in parallel by cloning the whole weight 1428 # matrix for every task, and in matrix · vector each weight is read once, so the 1429 # copies cost more than the products (NOTES.md, exp. 9). Bands keeps each band in 1430 # its own Array: a task takes its band and nothing is copied except the input vector. 1431 # 1432 # A node with r rows has children with half(r) and r - half(r) rows. The split is 1433 # part of the type, so the row count of every band follows from r, and the law 1434 # half_cover says that the two halves always cover the r rows exactly. 1435 # --------------------------------------------------------------------- 1436 1437 # n / 2 rounded down. A tail loop with an accumulator: a non-tail recursion here would put a 1438 # continuation on every node of the band tree and keep the bands from running in parallel. 1439 def half.go(n: Nat, acc: Nat) -> Nat: 1440 match n: 1441 case 0n: 1442 acc 1443 case 1n+p: 1444 match p: 1445 case 0n: 1446 acc 1447 case 1n+q: 1448 half.go(q, 1n+acc) 1449 1450 def half(n: Nat) -> Nat: 1451 half.go(n, 0n) 1452 1453 # a <= a 1454 def leb_refl(+a: Nat) -> {True{} == leb(a, a) : Bool}: 1455 match a: 1456 case 0n: 1457 {==} 1458 case 1n+p: 1459 leb_refl(p) 1460 1461 # 1 + (a + b) == a + (1 + b) 1462 def add_succ_r(+a: Nat, -b: Nat) -> {1n+Nat.add(a, b) == Nat.add(a, 1n+b) : Nat}: 1463 match a: 1464 case 0n: 1465 {==} 1466 case 1n+p: 1467 %add_succ_r(p, b) : {2n+Nat.add(p, b) == 1n+_ : Nat} 1468 {==} 1469 1470 # a + 0 == a 1471 def add_zero_r(+a: Nat) -> {Nat.add(a, 0n) == a : Nat}: 1472 match a: 1473 case 0n: 1474 {==} 1475 case 1n+p: 1476 %add_zero_r(p) : {1n+Nat.add(p, 0n) == 1n+_ : Nat} 1477 {==} 1478 1479 # half.go(n, acc) <= n + acc 1480 def half_go_le(+n: Nat, +acc: Nat) -> {True{} == leb(half.go(n, acc), Nat.add(n, acc)) : Bool}: 1481 match n: 1482 case 0n: 1483 leb_refl(acc) 1484 case 1n+p: 1485 match p: 1486 case 0n: 1487 leb_succ(acc, acc, leb_refl(acc)) 1488 case 1n+q: 1489 %Equal.sym(Nat, 1n+Nat.add(q, acc), Nat.add(q, 1n+acc), add_succ_r(q, acc)) : {True{} == leb(half.go(q, 1n+acc), 1n+_) : Bool} 1490 leb_succ(half.go(q, 1n+acc), Nat.add(q, 1n+acc), half_go_le(q, 1n+acc)) 1491 1492 # half(n) <= n 1493 def half_le(+n: Nat) -> {True{} == leb(half(n), n) : Bool}: 1494 %add_zero_r(n) : {True{} == leb(half(n), _) : Bool} 1495 half_go_le(n, 0n) 1496 1497 # a <= r -> r == a + (r - a) 1498 def add_sub(+a: Nat, +r: Nat, e: {True{} == leb(a, r) : Bool}) -> {r == Nat.add(a, Nat.sub(r, a)) : Nat}: 1499 match a r: 1500 case 0n 0n: 1501 {==} 1502 case 0n 1n+q: 1503 {==} 1504 case 1n+p 0n: 1505 %e : BD(_, Unit, {0n == Nat.add(1n+p, Nat.sub(0n, 1n+p)) : Nat}) 1506 Unit{} 1507 case 1n+p 1n+q: 1508 %add_sub(p, q, e) : {1n+q == 1n+_ : Nat} 1509 {==} 1510 1511 # LAW (half_cover): the two bands of a node, half(r) rows and r - half(r) rows, 1512 # add up to exactly r rows: no row is lost and none is counted twice. 1513 law half_cover: 1514 for +n: Nat 1515 {n == Nat.add(half(n), Nat.sub(n, half(n))) : Nat} 1516 1517 def half_cover(n): 1518 add_sub(half(n), n, half_le(n)) 1519 1520 type Bands<-r: Nat, -c: Nat> is Type: 1521 BLeaf{m: Mat} 1522 BNode{x: Bands, y: Bands} 1523 1524 # r x c of zeros in 2^d bands (d = 0: a single band, the same as a Mat) 1525 def Bands.zeros(+d: Nat, +r: Nat, +c: Nat) -> Bands: 1526 match d: 1527 case 0n: 1528 BLeaf{Mat.zeros(r, c)} 1529 case 1n+q: 1530 BNode{Bands.zeros(q, half(r), c), Bands.zeros(q, Nat.sub(r, half(r)), c)} 1531 1532 # the row counts of the two bands follow the row count of the node: if rr == r at run 1533 # time, the same holds for both halves (the bands carry r only in their type) 1534 def eq_half(-a: Nat, -b: Nat, e: {a == b : Nat}) -> {half(a) == half(b) : Nat}: 1535 %e : {half(a) == half(_) : Nat} 1536 {==} 1537 1538 def eq_rest(-a: Nat, -b: Nat, e: {a == b : Nat}) -> {Nat.sub(a, half(a)) == Nat.sub(b, half(b)) : Nat}: 1539 %e : {Nat.sub(a, half(a)) == Nat.sub(_, half(_)) : Nat} 1540 {==} 1541 1542 # does a block of n numbers starting at flat index i end before the second band of a 1543 # node with r rows? does it start in the second band? 1544 def bfill_l(+r: Nat, +c: Nat, +i: Nat, +n: Nat) -> Bool: 1545 Nat.is_le(Nat.add(i, n), Nat.mul(half(r), c)) 1546 1547 def bfill_r(+r: Nat, +c: Nat, +i: Nat) -> Bool: 1548 Nat.is_le(Nat.mul(half(r), c), i) 1549 1550 # writes the first n numbers of xs at index i (and nothing past them) 1551 def fill_n(n: Nat, xs: List<&2, F32>, +i: U32, a: Array) -> Array: 1552 match n xs: 1553 case 1n+p Con{h, t}: 1554 fill_n(p, t, (i + 1 : U32), Array.set(F32, a, i, h)) 1555 case _ _: 1556 a 1557 1558 def fill_band(-r: Nat, -c: Nat, n: Nat, xs: List<&2, F32>, +i: U32, m: Mat) -> Mat: 1559 match m: 1560 case Mat{d}: 1561 Mat{fill_n(n, xs, i, d)} 1562 1563 # writes the first n numbers of xs at flat index i; l and rt say which bands they touch 1564 # (bfill_l, bfill_r); rr is the row count r at run time. A block that crosses into the second 1565 # band is not copied: the first band writes only its part, the second gets the rest by List.drop. 1566 def bfill(+c: Nat, -r: Nat, b: Bands, l: Bool, rt: Bool, +rr: Nat, -e: {rr == r : Nat}, +i: Nat, +n: Nat, +xs: List<&2, F32>) -> Bands: 1567 match b l rt: 1568 case BLeaf{m} _ _: 1569 BLeaf{fill_band(r, c, n, xs, U32.from_nat(i), m)} 1570 case BNode{x, y} True{} _: 1571 BNode{bfill(c, half(r), x, bfill_l(half(rr), c, i, n), bfill_r(half(rr), c, i), half(rr), eq_half(rr, r, e), i, n, xs), y} 1572 case BNode{x, y} False{} True{}: 1573 BNode{x, bfill(c, Nat.sub(r, half(r)), y, bfill_l(Nat.sub(rr, half(rr)), c, Nat.sub(i, Nat.mul(half(rr), c)), n), bfill_r(Nat.sub(rr, half(rr)), c, Nat.sub(i, Nat.mul(half(rr), c))), Nat.sub(rr, half(rr)), eq_rest(rr, r, e), Nat.sub(i, Nat.mul(half(rr), c)), n, xs)} 1574 case BNode{x, y} False{} False{}: 1575 BNode{bfill(c, half(r), x, bfill_l(half(rr), c, i, Nat.sub(Nat.mul(half(rr), c), i)), bfill_r(half(rr), c, i), half(rr), eq_half(rr, r, e), i, Nat.sub(Nat.mul(half(rr), c), i), xs), bfill(c, Nat.sub(r, half(r)), y, bfill_l(Nat.sub(rr, half(rr)), c, 0n, Nat.sub(n, Nat.sub(Nat.mul(half(rr), c), i))), bfill_r(Nat.sub(rr, half(rr)), c, 0n), Nat.sub(rr, half(rr)), eq_rest(rr, r, e), 0n, Nat.sub(n, Nat.sub(Nat.mul(half(rr), c), i)), List.drop(&2, F32, xs, Nat.sub(Nat.mul(half(rr), c), i)))} 1576 1577 def bfill_top(ok: Bool, +r: Nat, +c: Nat, b: Bands, +i: Nat, +n: Nat, +xs: List<&2, F32>) -> Bands: 1578 match ok: 1579 case True{}: 1580 bfill(c, r, b, bfill_l(r, c, i, n), bfill_r(r, c, i), r, {==}, i, n, xs) 1581 case False{}: 1582 bfill(c, r, b, bfill_l(r, c, i, Nat.sub(Nat.mul(r, c), i)), bfill_r(r, c, i), r, {==}, i, Nat.sub(Nat.mul(r, c), i), xs) 1583 1584 # writes the n numbers of xs into b starting at flat index i (row-major, as Mat.fill_at); 1585 # numbers past r*c are dropped. For loading a large matrix block by block. 1586 def Bands.fill_at(+r: Nat, +c: Nat, +i: Nat, +n: Nat, +xs: List<&2, F32>, b: Bands) -> Bands: 1587 bfill_top(Nat.is_le(Nat.add(i, n), Nat.mul(r, c)), r, c, b, i, n, xs) 1588 1589 # ---- y = B · x for B with r rows of c numbers (a weight matrix stored out x in) ---- 1590 1591 # B (intact) and y = B · x, typed like Mat.matmul_nt: x is 1 x c, y is 1 x r 1592 type BV<-r: Nat, -c: Nat> is Type: 1593 BV{b: Bands, y: Mat<1n, r>} 1594 1595 # the bands and y as a list in row order (the result of Bands.matvec_l) 1596 type BL<-r: Nat, -c: Nat> is Type: 1597 BL{b: Bands, y: List<&2, F32>} 1598 1599 def bv_g2(-r: Nat, -c: Nat, w: Array, al: AL) -> BL: 1600 match al: 1601 case AL{d, l}: 1602 BL{BLeaf{Mat{w}}, l} 1603 1604 # the band's 1 + m rows came out in an Array (one number per row); read them back as a list 1605 def bv_g(-r: Nat, -c: Nat, +m: Nat, g: G) -> BL: 1606 match g: 1607 case G{a, b, y}: 1608 bv_g2(r, c, b, read_l(m, 0, Nil{}, Array.get(F32, y, 0))) 1609 1610 # the rows of one band, one dot product each, with the gemm of Mat.matmul_nt (n = 1), so the 1611 # results are identical. Writing the results into an Array and reading them back is faster than 1612 # building the list row by row (NOTES.md, exp. 14). 1613 def bv_gemm(-r: Nat, +c: Nat, rows: Nat, +rr: Nat, x: Array, w: Array) -> BL: 1614 match rows: 1615 case 0n: 1616 BL{BLeaf{Mat{w}}, Nil{}} 1617 case 1n+p: 1618 bv_g(r, c, p, gemm(1n, rr, c, U32.from_nat(c), 1, 1, U32.from_nat(c), x, w, Array.new(F32, cap_depth(rr), 0.0))) 1619 1620 def bv_mat(-r: Nat, +c: Nat, +rr: Nat, x: Array, m: Mat) -> BL: 1621 match m: 1622 case Mat{w}: 1623 bv_gemm(r, c, rr, rr, x, w) 1624 1625 def bv_join(-r: Nat, -c: Nat, p: BL, q: BL) -> BL: 1626 match p q: 1627 case BL{bx, yx} BL{by, yy}: 1628 BL{BNode{bx, by}, List.append(&2, F32, yx, yy)} 1629 1630 # the two bands of a node run in parallel, each with its own copy of x (c numbers); 1631 # xp is a pair of copies of x (a leaf uses one). xp comes after the erased e: up to Bend 2.0.36 an erased 1632 # parameter at the end of the list turned the parallel let into two sequential calls (NOTES.md, exp. 11; fixed in 1633 # 2.0.37 by bendlang/bend#1377). The order is kept, and harmless. 1634 def bmv(+c: Nat, -r: Nat, b: Bands, +rr: Nat, -e: {rr == r : Nat}, xp: Array & Array) -> BL: 1635 match b xp: 1636 case BLeaf{m} Tuple{x, x2}: 1637 bv_mat(r, c, rr, x, m) 1638 case BNode{bx, by} Tuple{x1, x2}: 1639 p q = bmv(c, half(r), bx, half(rr), eq_half(rr, r, e), Array.clone(F32, x1)) bmv(c, Nat.sub(r, half(r)), by, Nat.sub(rr, half(rr)), eq_rest(rr, r, e), Array.clone(F32, x2)) 1640 bv_join(r, c, p, q) 1641 1642 def bv_fin(+r: Nat, -c: Nat, bl: BL) -> BV: 1643 match bl: 1644 case BL{b, y}: 1645 BV{b, Mat.from_list(1n, r, y)} 1646 1647 # y = B · x for B (r x c, one row per output) and x (1 x c); returns B intact and y (1 x r). 1648 # The same product as Mat.matmul_nt(1n, c, r, ..), and the same numbers, bit for bit. 1649 def Bands.matvec(+r: Nat, +c: Nat, b: Bands, x: Mat<1n, c>) -> BV: 1650 match x: 1651 case Mat{xa}: 1652 bv_fin(r, c, bmv(c, r, b, r, {==}, Array.clone(F32, xa))) 1653 1654 # the same product with x and y as lists (x: c numbers, y: r numbers in row order), without the 1655 # Mat conversions of Bands.matvec: for callers that already hold the vector as a list 1656 def Bands.matvec_l(+r: Nat, +c: Nat, b: Bands, +x: List<&2, F32>) -> BL: 1657 bmv(c, r, b, r, {==}, Array.clone(F32, fill_list(x, 0, Array.new(F32, cap_depth(c), 0.0)))) 1658 1659 # ---- reading rows and the whole matrix ---- 1660 1661 type BRow<-r: Nat, -c: Nat> is Type: 1662 BRow{b: Bands, row: List<&2, F32>} 1663 1664 def brow_fin(-r: Nat, -c: Nat, al: AL) -> BRow: 1665 match al: 1666 case AL{d, l}: 1667 BRow{BLeaf{Mat{d}}, l} 1668 1669 # n numbers starting at flat index i of one band (a band may have no rows) 1670 def brow_leaf2(-r: Nat, -c: Nat, n: Nat, +i: U32, d: Array) -> BRow: 1671 match n: 1672 case 0n: 1673 BRow{BLeaf{Mat{d}}, Nil{}} 1674 case 1n+p: 1675 brow_fin(r, c, read_l(p, i, Nil{}, Array.get(F32, d, i))) 1676 1677 def brow_leaf(-r: Nat, -c: Nat, m: Mat, +i: U32, n: Nat) -> BRow: 1678 match m: 1679 case Mat{d}: 1680 brow_leaf2(r, c, n, i, d) 1681 1682 def brow_l(-r: Nat, -c: Nat, p: BRow, y: Bands) -> BRow: 1683 match p: 1684 case BRow{bx, row}: 1685 BRow{BNode{bx, y}, row} 1686 1687 def brow_r(-r: Nat, -c: Nat, x: Bands, q: BRow) -> BRow: 1688 match q: 1689 case BRow{by, row}: 1690 BRow{BNode{x, by}, row} 1691 1692 # row i; left says whether i is in the first band of the node (i < half(rr)) 1693 def brow(+c: Nat, -r: Nat, b: Bands, left: Bool, +rr: Nat, -e: {rr == r : Nat}, +i: Nat) -> BRow: 1694 match b left: 1695 case BLeaf{m} _: 1696 brow_leaf(r, c, m, U32.from_nat(Nat.mul(i, c)), c) 1697 case BNode{x, y} True{}: 1698 brow_l(r, c, brow(c, half(r), x, Nat.is_lt(i, half(half(rr))), half(rr), eq_half(rr, r, e), i), y) 1699 case BNode{x, y} False{}: 1700 brow_r(r, c, x, brow(c, Nat.sub(r, half(r)), y, Nat.is_lt(Nat.sub(i, half(rr)), half(Nat.sub(rr, half(rr)))), Nat.sub(rr, half(rr)), eq_rest(rr, r, e), Nat.sub(i, half(rr)))) 1701 1702 # row i (c numbers), for example an embedding lookup 1703 def Bands.read_row(+r: Nat, +c: Nat, b: Bands, +i: Nat) -> BRow: 1704 brow(c, r, b, Nat.is_lt(i, half(r)), r, {==}, i) 1705 1706 def bl_join(-r: Nat, -c: Nat, p: BRow, q: BRow) -> BRow: 1707 match p q: 1708 case BRow{bx, xs} BRow{by, ys}: 1709 BRow{BNode{bx, by}, List.append(&2, F32, xs, ys)} 1710 1711 def blist(+c: Nat, -r: Nat, b: Bands, +rr: Nat, -e: {rr == r : Nat}) -> BRow: 1712 match b: 1713 case BLeaf{m}: 1714 brow_leaf(r, c, m, 0, Nat.mul(rr, c)) 1715 case BNode{x, y}: 1716 bl_join(r, c, blist(c, half(r), x, half(rr), eq_half(rr, r, e)), blist(c, Nat.sub(r, half(r)), y, Nat.sub(rr, half(rr)), eq_rest(rr, r, e))) 1717 1718 # all r*c numbers in row order (in BRow.row) 1719 def Bands.to_list(+r: Nat, +c: Nat, b: Bands) -> BRow: 1720 blist(c, r, b, r, {==}) 1721 1722 # ---- the list path returns exactly r numbers ---- 1723 # 1724 # The numbers do not matter here, only how many there are: each lemma below counts the list a 1725 # function builds, following the function's own recursion. 1726 1727 def bl_len(-r: Nat, -c: Nat, bl: BL) -> Nat: 1728 match bl: 1729 case BL{b, y}: 1730 List.length(&2, F32, y) 1731 1732 # s == a + (1 + b) -> s == 1 + (a + b) 1733 def succ_out(-s: Nat, +a: Nat, +b: Nat, e: {s == Nat.add(a, 1n+b) : Nat}) -> {s == 1n+Nat.add(a, b) : Nat}: 1734 %Equal.sym(Nat, 1n+Nat.add(a, b), Nat.add(a, 1n+b), add_succ_r(a, b)) : {s == _ : Nat} 1735 e 1736 1737 # s == 1 + (p + (1 + l)) -> s == 2 + (p + l) 1738 def len_step(-s: Nat, +p: Nat, +l: Nat, e: {s == 1n+Nat.add(p, 1n+l) : Nat}) -> {s == 2n+Nat.add(p, l) : Nat}: 1739 %Equal.sym(Nat, 1n+Nat.add(p, l), Nat.add(p, 1n+l), add_succ_r(p, l)) : {s == 1n+_ : Nat} 1740 e 1741 1742 # s == n + 0 -> s == n 1743 def zero_out(-s: Nat, +n: Nat, e: {s == Nat.add(n, 0n) : Nat}) -> {s == n : Nat}: 1744 %add_zero_r(n) : {s == _ : Nat} 1745 e 1746 1747 # reversing onto acc adds the lengths 1748 def rev_len(+xs: List<&2, F32>, +acc: List<&2, F32>) -> {List.length(&2, F32, List.reverse.go(&2, F32, xs, acc)) == Nat.add(List.length(&2, F32, xs), List.length(&2, F32, acc)) : Nat}: 1749 match xs: 1750 case Nil{}: 1751 {==} 1752 case Con{h, t}: 1753 succ_out(List.length(&2, F32, List.reverse.go(&2, F32, t, h <> acc)), List.length(&2, F32, t), List.length(&2, F32, acc), rev_len(t, h <> acc)) 1754 1755 # appending adds the lengths 1756 def app_len(+xs: List<&2, F32>, +ys: List<&2, F32>) -> {List.length(&2, F32, List.append(&2, F32, xs, ys)) == Nat.add(List.length(&2, F32, xs), List.length(&2, F32, ys)) : Nat}: 1757 match xs: 1758 case Nil{}: 1759 {==} 1760 case Con{h, t}: 1761 %app_len(t, ys) : {1n+List.length(&2, F32, List.append(&2, F32, t, ys)) == 1n+_ : Nat} 1762 {==} 1763 1764 # a leaf's list: read_l reads mleft + 1 numbers onto acc and reverses them. By induction on 1765 # mleft: with 0 left it reads one number (rev_len counts the reversed list); with 1 + p left it 1766 # reads one and recurses, and len_step moves the 1 out of the sum. 1767 law leaf_len: 1768 for -r: Nat 1769 for -c: Nat 1770 for w: Array 1771 for +mleft: Nat 1772 for +idx: U32 1773 for +acc: List<&2, F32> 1774 for rc: Array & F32 1775 {bl_len(r, c, bv_g2(r, c, w, read_l(mleft, idx, acc, rc))) == Nat.add(1n+mleft, List.length(&2, F32, acc)) : Nat} 1776 1777 def leaf_len(r, c, w, mleft, idx, acc, rc): 1778 match mleft rc: 1779 case 0n Tuple{d, +v}: 1780 zero_out(List.length(&2, F32, List.reverse.go(&2, F32, v <> acc, Nil{})), 1n+List.length(&2, F32, acc), rev_len(v <> acc, Nil{})) 1781 case 1n+p Tuple{d, +v}: 1782 len_step(bl_len(r, c, bv_g2(r, c, w, read_l(p, (idx + 1 : U32), v <> acc, Array.get(F32, d, (idx + 1 : U32))))), p, List.length(&2, F32, acc), leaf_len(r, c, w, p, (idx + 1 : U32), v <> acc, Array.get(F32, d, (idx + 1 : U32)))) 1783 1784 # whatever the gemm wrote, reading 1 + p rows back gives 1 + p numbers (G has one constructor, so 1785 # matching g opens it, and leaf_len counts the list) 1786 def g_len(-r: Nat, -c: Nat, +p: Nat, g: G) -> {bl_len(r, c, bv_g(r, c, p, g)) == 1n+p : Nat}: 1787 match g: 1788 case G{a, b, y}: 1789 zero_out(bl_len(r, c, bv_g2(r, c, b, read_l(p, 0, Nil{}, Array.get(F32, y, 0)))), 1n+p, leaf_len(r, c, b, p, 0, Nil{}, Array.get(F32, y, 0))) 1790 1791 # LAW (band_len): the product of one band with `rows` rows gives exactly `rows` numbers. 1792 # 0 rows give the empty list; 1 + p rows go through the gemm, and g_len counts what is read back. 1793 law band_len: 1794 for -r: Nat 1795 for +c: Nat 1796 for +rows: Nat 1797 for x: Array 1798 for w: Array 1799 {bl_len(r, c, bv_gemm(r, c, rows, rows, x, w)) == rows : Nat} 1800 1801 def band_len(r, c, rows, x, w): 1802 match rows: 1803 case 0n: 1804 {==} 1805 case 1n+p: 1806 g_len(r, c, p, gemm(1n, 1n+p, c, U32.from_nat(c), 1, 1, U32.from_nat(c), x, w, Array.new(F32, cap_depth(1n+p), 0.0))) 1807 1808 # joining two halves of h1 and h2 numbers gives h1 + h2 numbers. (The statement for the whole tree, 1809 # "Bands.matvec_l gives r numbers", would apply this to the results of the two recursive calls and 1810 # to the induction hypotheses about the same calls; that uses each band's Array twice, which Bend's 1811 # affine rules refuse. It stays open, covered by tests: NOTES.md, v3.1.) 1812 def join_len(-r: Nat, -c: Nat, -h1: Nat, -h2: Nat, p: BL, q: BL, hp: {bl_len(half(r), c, p) == h1 : Nat}, hq: {bl_len(Nat.sub(r, half(r)), c, q) == h2 : Nat}) -> {bl_len(r, c, bv_join(r, c, p, q)) == Nat.add(h1, h2) : Nat}: 1813 match p q: 1814 case BL{bx, +yx} BL{by, +yy}: 1815 %hp : {List.length(&2, F32, List.append(&2, F32, yx, yy)) == Nat.add(_, h2) : Nat} 1816 %hq : {List.length(&2, F32, List.append(&2, F32, yx, yy)) == Nat.add(List.length(&2, F32, yx), _) : Nat} 1817 app_len(yx, yy) 1818