1 # bend-ml-tensor-gpu: tensor-array's products on the GPU, with tensor-array's bits. 2 # 3 # import ../tensor-array/main.bend as TA 4 # import bend-ml-tensor-gpu@0.1.5.0/main.bend as TG 5 # TG.matmul(n, k, m, par, a, b) # the same arguments, result type and bits as TA.Mat.matmul 6 # 7 # The six products (matmul, matmul_nt, matmul_tn and their accumulating forms) take tensor-array's Mat values and return 8 # its result types; par is ignored (the GPU decides the split). Each product is one `!` call (a bang): a balanced fork 9 # tree whose leaves each compute a 4 x 4 tile of C, every node sharing A, B and C through Base's Array.fork (two 10 # handles to one block, no copy), as upstream's one-block pattern does (bendlang/bend tests/run/stencil3d.bend). On a 11 # machine without a GPU, or with `+RTS --gpu off -RTS`, the same bang runs on the CPU threads. 12 # 13 # The bits: a cell of C is computed in tensor-array's order. Product p of a dot goes to partial sum p mod 4, in 14 # increasing p, each sum starting at +0.0, and the four sums are combined by TA.t8_comb with rot = k mod 4, as 15 # tensor-array's dotr and t8_dot do; an accumulating product adds the cell to C once with TA.cwr. The loop takes four 16 # steps of k a turn; the steps past k in the last turn add +0.0, which leaves a sum unchanged (a sum that starts at +0.0 17 # is never -0.0). Bend builds the device code with --fmad=false, so no multiply-add is fused. tests/bits.bend checks the 18 # bits against tensor-array on many shapes, on the GPU and the CPU. 19 # 20 # Counter-based random numbers (fill_uniform, fill_gauss, uniform_mat, gauss_mat and their _at forms; the last section): 21 # n uniforms or Gaussians of a counter made in one bang, each a pure function of (counter, index), with the same bits on 22 # every lane. 23 # 24 # Not proved, and marked so: Array.fork and Array.join are @unsafe in Base, so the defs below that call them, or take 25 # a shared array, are @unsafe, and `bend main.bend` names every def that relies on them. Nothing checks that a leaf 26 # writes only its own tile of C, or never writes A or B; the tiles are disjoint by construction (leaf t writes tile t). 27 # All of it is Bend: no foreign code. Written by bend-ml's scripts/gen_tensor_gpu.py (4 x 4 tiles). 28 import Base 29 import bend-ml-tensor-array@0.1.18.0/main.bend as TA 30 31 # the sizes and strides of one product: C[i][j] = sum_p A[oa + i*arow + p*sa] * B[ob + p*sb + j*bcol], n x m, k steps 32 # (oa, ob: 0 except for matmul_precise's blocks) 33 type E is Data: 34 E{n: U32, m: U32, k: U32, tcols: U32, tiles: U32, arow: U32, sa: U32, sb: U32, bcol: U32, kg: Nat, rot: Nat, rem: U32, ac: Bool, oa: U32, ob: U32} 35 36 # a product that counts only when its step is inside k (the last turn's steps past k add +0.0) 37 def pk(ok: Bool, +x: F32) -> F32: 38 match ok: 39 case True{}: x 40 case False{}: 0.0 41 42 # the values of one turn: A's 4 rows and B's 4 columns at four consecutive steps, read through the arrays in order 43 type GR is Type: 44 GR{a: Array, b: Array, x0_0: F32, x0_1: F32, x0_2: F32, x0_3: F32, x1_0: F32, x1_1: F32, x1_2: F32, x1_3: F32, x2_0: F32, x2_1: F32, x2_2: F32, x2_3: F32, x3_0: F32, x3_1: F32, x3_2: F32, x3_3: F32, y0_0: F32, y0_1: F32, y0_2: F32, y0_3: F32, y1_0: F32, y1_1: F32, y1_2: F32, y1_3: F32, y2_0: F32, y2_1: F32, y2_2: F32, y2_3: F32, y3_0: F32, y3_1: F32, y3_2: F32, y3_3: F32} 45 46 def gr_31(+ia: U32, +ib: U32, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, +x0_0: F32, +x0_1: F32, +x0_2: F32, +x0_3: F32, +x1_0: F32, +x1_1: F32, +x1_2: F32, +x1_3: F32, +x2_0: F32, +x2_1: F32, +x2_2: F32, +x2_3: F32, +x3_0: F32, +x3_1: F32, +x3_2: F32, +x3_3: F32, +y0_0: F32, +y0_1: F32, +y0_2: F32, +y0_3: F32, +y1_0: F32, +y1_1: F32, +y1_2: F32, +y1_3: F32, +y2_0: F32, +y2_1: F32, +y2_2: F32, +y2_3: F32, +y3_0: F32, +y3_1: F32, +y3_2: F32, a: Array, rv: Array & F32) -> GR: 47 match rv: 48 case Tuple{b, +y3_3}: 49 GR{a, b, x0_0, x0_1, x0_2, x0_3, x1_0, x1_1, x1_2, x1_3, x2_0, x2_1, x2_2, x2_3, x3_0, x3_1, x3_2, x3_3, y0_0, y0_1, y0_2, y0_3, y1_0, y1_1, y1_2, y1_3, y2_0, y2_1, y2_2, y2_3, y3_0, y3_1, y3_2, y3_3} 50 def gr_30(+ia: U32, +ib: U32, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, +x0_0: F32, +x0_1: F32, +x0_2: F32, +x0_3: F32, +x1_0: F32, +x1_1: F32, +x1_2: F32, +x1_3: F32, +x2_0: F32, +x2_1: F32, +x2_2: F32, +x2_3: F32, +x3_0: F32, +x3_1: F32, +x3_2: F32, +x3_3: F32, +y0_0: F32, +y0_1: F32, +y0_2: F32, +y0_3: F32, +y1_0: F32, +y1_1: F32, +y1_2: F32, +y1_3: F32, +y2_0: F32, +y2_1: F32, +y2_2: F32, +y2_3: F32, +y3_0: F32, +y3_1: F32, a: Array, rv: Array & F32) -> GR: 51 match rv: 52 case Tuple{b, +y3_2}: 53 gr_31(ia, ib, arow, sa, sb, bcol, x0_0, x0_1, x0_2, x0_3, x1_0, x1_1, x1_2, x1_3, x2_0, x2_1, x2_2, x2_3, x3_0, x3_1, x3_2, x3_3, y0_0, y0_1, y0_2, y0_3, y1_0, y1_1, y1_2, y1_3, y2_0, y2_1, y2_2, y2_3, y3_0, y3_1, y3_2, a, Array.get(F32, b, (ib + ((sb * 3 : U32) + (bcol * 3 : U32) : U32) : U32))) 54 def gr_29(+ia: U32, +ib: U32, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, +x0_0: F32, +x0_1: F32, +x0_2: F32, +x0_3: F32, +x1_0: F32, +x1_1: F32, +x1_2: F32, +x1_3: F32, +x2_0: F32, +x2_1: F32, +x2_2: F32, +x2_3: F32, +x3_0: F32, +x3_1: F32, +x3_2: F32, +x3_3: F32, +y0_0: F32, +y0_1: F32, +y0_2: F32, +y0_3: F32, +y1_0: F32, +y1_1: F32, +y1_2: F32, +y1_3: F32, +y2_0: F32, +y2_1: F32, +y2_2: F32, +y2_3: F32, +y3_0: F32, a: Array, rv: Array & F32) -> GR: 55 match rv: 56 case Tuple{b, +y3_1}: 57 gr_30(ia, ib, arow, sa, sb, bcol, x0_0, x0_1, x0_2, x0_3, x1_0, x1_1, x1_2, x1_3, x2_0, x2_1, x2_2, x2_3, x3_0, x3_1, x3_2, x3_3, y0_0, y0_1, y0_2, y0_3, y1_0, y1_1, y1_2, y1_3, y2_0, y2_1, y2_2, y2_3, y3_0, y3_1, a, Array.get(F32, b, (ib + ((sb * 3 : U32) + (bcol * 2 : U32) : U32) : U32))) 58 def gr_28(+ia: U32, +ib: U32, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, +x0_0: F32, +x0_1: F32, +x0_2: F32, +x0_3: F32, +x1_0: F32, +x1_1: F32, +x1_2: F32, +x1_3: F32, +x2_0: F32, +x2_1: F32, +x2_2: F32, +x2_3: F32, +x3_0: F32, +x3_1: F32, +x3_2: F32, +x3_3: F32, +y0_0: F32, +y0_1: F32, +y0_2: F32, +y0_3: F32, +y1_0: F32, +y1_1: F32, +y1_2: F32, +y1_3: F32, +y2_0: F32, +y2_1: F32, +y2_2: F32, +y2_3: F32, a: Array, rv: Array & F32) -> GR: 59 match rv: 60 case Tuple{b, +y3_0}: 61 gr_29(ia, ib, arow, sa, sb, bcol, x0_0, x0_1, x0_2, x0_3, x1_0, x1_1, x1_2, x1_3, x2_0, x2_1, x2_2, x2_3, x3_0, x3_1, x3_2, x3_3, y0_0, y0_1, y0_2, y0_3, y1_0, y1_1, y1_2, y1_3, y2_0, y2_1, y2_2, y2_3, y3_0, a, Array.get(F32, b, (ib + ((sb * 3 : U32) + (bcol * 1 : U32) : U32) : U32))) 62 def gr_27(+ia: U32, +ib: U32, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, +x0_0: F32, +x0_1: F32, +x0_2: F32, +x0_3: F32, +x1_0: F32, +x1_1: F32, +x1_2: F32, +x1_3: F32, +x2_0: F32, +x2_1: F32, +x2_2: F32, +x2_3: F32, +x3_0: F32, +x3_1: F32, +x3_2: F32, +x3_3: F32, +y0_0: F32, +y0_1: F32, +y0_2: F32, +y0_3: F32, +y1_0: F32, +y1_1: F32, +y1_2: F32, +y1_3: F32, +y2_0: F32, +y2_1: F32, +y2_2: F32, a: Array, rv: Array & F32) -> GR: 63 match rv: 64 case Tuple{b, +y2_3}: 65 gr_28(ia, ib, arow, sa, sb, bcol, x0_0, x0_1, x0_2, x0_3, x1_0, x1_1, x1_2, x1_3, x2_0, x2_1, x2_2, x2_3, x3_0, x3_1, x3_2, x3_3, y0_0, y0_1, y0_2, y0_3, y1_0, y1_1, y1_2, y1_3, y2_0, y2_1, y2_2, y2_3, a, Array.get(F32, b, (ib + ((sb * 3 : U32) + (bcol * 0 : U32) : U32) : U32))) 66 def gr_26(+ia: U32, +ib: U32, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, +x0_0: F32, +x0_1: F32, +x0_2: F32, +x0_3: F32, +x1_0: F32, +x1_1: F32, +x1_2: F32, +x1_3: F32, +x2_0: F32, +x2_1: F32, +x2_2: F32, +x2_3: F32, +x3_0: F32, +x3_1: F32, +x3_2: F32, +x3_3: F32, +y0_0: F32, +y0_1: F32, +y0_2: F32, +y0_3: F32, +y1_0: F32, +y1_1: F32, +y1_2: F32, +y1_3: F32, +y2_0: F32, +y2_1: F32, a: Array, rv: Array & F32) -> GR: 67 match rv: 68 case Tuple{b, +y2_2}: 69 gr_27(ia, ib, arow, sa, sb, bcol, x0_0, x0_1, x0_2, x0_3, x1_0, x1_1, x1_2, x1_3, x2_0, x2_1, x2_2, x2_3, x3_0, x3_1, x3_2, x3_3, y0_0, y0_1, y0_2, y0_3, y1_0, y1_1, y1_2, y1_3, y2_0, y2_1, y2_2, a, Array.get(F32, b, (ib + ((sb * 2 : U32) + (bcol * 3 : U32) : U32) : U32))) 70 def gr_25(+ia: U32, +ib: U32, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, +x0_0: F32, +x0_1: F32, +x0_2: F32, +x0_3: F32, +x1_0: F32, +x1_1: F32, +x1_2: F32, +x1_3: F32, +x2_0: F32, +x2_1: F32, +x2_2: F32, +x2_3: F32, +x3_0: F32, +x3_1: F32, +x3_2: F32, +x3_3: F32, +y0_0: F32, +y0_1: F32, +y0_2: F32, +y0_3: F32, +y1_0: F32, +y1_1: F32, +y1_2: F32, +y1_3: F32, +y2_0: F32, a: Array, rv: Array & F32) -> GR: 71 match rv: 72 case Tuple{b, +y2_1}: 73 gr_26(ia, ib, arow, sa, sb, bcol, x0_0, x0_1, x0_2, x0_3, x1_0, x1_1, x1_2, x1_3, x2_0, x2_1, x2_2, x2_3, x3_0, x3_1, x3_2, x3_3, y0_0, y0_1, y0_2, y0_3, y1_0, y1_1, y1_2, y1_3, y2_0, y2_1, a, Array.get(F32, b, (ib + ((sb * 2 : U32) + (bcol * 2 : U32) : U32) : U32))) 74 def gr_24(+ia: U32, +ib: U32, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, +x0_0: F32, +x0_1: F32, +x0_2: F32, +x0_3: F32, +x1_0: F32, +x1_1: F32, +x1_2: F32, +x1_3: F32, +x2_0: F32, +x2_1: F32, +x2_2: F32, +x2_3: F32, +x3_0: F32, +x3_1: F32, +x3_2: F32, +x3_3: F32, +y0_0: F32, +y0_1: F32, +y0_2: F32, +y0_3: F32, +y1_0: F32, +y1_1: F32, +y1_2: F32, +y1_3: F32, a: Array, rv: Array & F32) -> GR: 75 match rv: 76 case Tuple{b, +y2_0}: 77 gr_25(ia, ib, arow, sa, sb, bcol, x0_0, x0_1, x0_2, x0_3, x1_0, x1_1, x1_2, x1_3, x2_0, x2_1, x2_2, x2_3, x3_0, x3_1, x3_2, x3_3, y0_0, y0_1, y0_2, y0_3, y1_0, y1_1, y1_2, y1_3, y2_0, a, Array.get(F32, b, (ib + ((sb * 2 : U32) + (bcol * 1 : U32) : U32) : U32))) 78 def gr_23(+ia: U32, +ib: U32, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, +x0_0: F32, +x0_1: F32, +x0_2: F32, +x0_3: F32, +x1_0: F32, +x1_1: F32, +x1_2: F32, +x1_3: F32, +x2_0: F32, +x2_1: F32, +x2_2: F32, +x2_3: F32, +x3_0: F32, +x3_1: F32, +x3_2: F32, +x3_3: F32, +y0_0: F32, +y0_1: F32, +y0_2: F32, +y0_3: F32, +y1_0: F32, +y1_1: F32, +y1_2: F32, a: Array, rv: Array & F32) -> GR: 79 match rv: 80 case Tuple{b, +y1_3}: 81 gr_24(ia, ib, arow, sa, sb, bcol, x0_0, x0_1, x0_2, x0_3, x1_0, x1_1, x1_2, x1_3, x2_0, x2_1, x2_2, x2_3, x3_0, x3_1, x3_2, x3_3, y0_0, y0_1, y0_2, y0_3, y1_0, y1_1, y1_2, y1_3, a, Array.get(F32, b, (ib + ((sb * 2 : U32) + (bcol * 0 : U32) : U32) : U32))) 82 def gr_22(+ia: U32, +ib: U32, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, +x0_0: F32, +x0_1: F32, +x0_2: F32, +x0_3: F32, +x1_0: F32, +x1_1: F32, +x1_2: F32, +x1_3: F32, +x2_0: F32, +x2_1: F32, +x2_2: F32, +x2_3: F32, +x3_0: F32, +x3_1: F32, +x3_2: F32, +x3_3: F32, +y0_0: F32, +y0_1: F32, +y0_2: F32, +y0_3: F32, +y1_0: F32, +y1_1: F32, a: Array, rv: Array & F32) -> GR: 83 match rv: 84 case Tuple{b, +y1_2}: 85 gr_23(ia, ib, arow, sa, sb, bcol, x0_0, x0_1, x0_2, x0_3, x1_0, x1_1, x1_2, x1_3, x2_0, x2_1, x2_2, x2_3, x3_0, x3_1, x3_2, x3_3, y0_0, y0_1, y0_2, y0_3, y1_0, y1_1, y1_2, a, Array.get(F32, b, (ib + ((sb * 1 : U32) + (bcol * 3 : U32) : U32) : U32))) 86 def gr_21(+ia: U32, +ib: U32, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, +x0_0: F32, +x0_1: F32, +x0_2: F32, +x0_3: F32, +x1_0: F32, +x1_1: F32, +x1_2: F32, +x1_3: F32, +x2_0: F32, +x2_1: F32, +x2_2: F32, +x2_3: F32, +x3_0: F32, +x3_1: F32, +x3_2: F32, +x3_3: F32, +y0_0: F32, +y0_1: F32, +y0_2: F32, +y0_3: F32, +y1_0: F32, a: Array, rv: Array & F32) -> GR: 87 match rv: 88 case Tuple{b, +y1_1}: 89 gr_22(ia, ib, arow, sa, sb, bcol, x0_0, x0_1, x0_2, x0_3, x1_0, x1_1, x1_2, x1_3, x2_0, x2_1, x2_2, x2_3, x3_0, x3_1, x3_2, x3_3, y0_0, y0_1, y0_2, y0_3, y1_0, y1_1, a, Array.get(F32, b, (ib + ((sb * 1 : U32) + (bcol * 2 : U32) : U32) : U32))) 90 def gr_20(+ia: U32, +ib: U32, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, +x0_0: F32, +x0_1: F32, +x0_2: F32, +x0_3: F32, +x1_0: F32, +x1_1: F32, +x1_2: F32, +x1_3: F32, +x2_0: F32, +x2_1: F32, +x2_2: F32, +x2_3: F32, +x3_0: F32, +x3_1: F32, +x3_2: F32, +x3_3: F32, +y0_0: F32, +y0_1: F32, +y0_2: F32, +y0_3: F32, a: Array, rv: Array & F32) -> GR: 91 match rv: 92 case Tuple{b, +y1_0}: 93 gr_21(ia, ib, arow, sa, sb, bcol, x0_0, x0_1, x0_2, x0_3, x1_0, x1_1, x1_2, x1_3, x2_0, x2_1, x2_2, x2_3, x3_0, x3_1, x3_2, x3_3, y0_0, y0_1, y0_2, y0_3, y1_0, a, Array.get(F32, b, (ib + ((sb * 1 : U32) + (bcol * 1 : U32) : U32) : U32))) 94 def gr_19(+ia: U32, +ib: U32, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, +x0_0: F32, +x0_1: F32, +x0_2: F32, +x0_3: F32, +x1_0: F32, +x1_1: F32, +x1_2: F32, +x1_3: F32, +x2_0: F32, +x2_1: F32, +x2_2: F32, +x2_3: F32, +x3_0: F32, +x3_1: F32, +x3_2: F32, +x3_3: F32, +y0_0: F32, +y0_1: F32, +y0_2: F32, a: Array, rv: Array & F32) -> GR: 95 match rv: 96 case Tuple{b, +y0_3}: 97 gr_20(ia, ib, arow, sa, sb, bcol, x0_0, x0_1, x0_2, x0_3, x1_0, x1_1, x1_2, x1_3, x2_0, x2_1, x2_2, x2_3, x3_0, x3_1, x3_2, x3_3, y0_0, y0_1, y0_2, y0_3, a, Array.get(F32, b, (ib + ((sb * 1 : U32) + (bcol * 0 : U32) : U32) : U32))) 98 def gr_18(+ia: U32, +ib: U32, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, +x0_0: F32, +x0_1: F32, +x0_2: F32, +x0_3: F32, +x1_0: F32, +x1_1: F32, +x1_2: F32, +x1_3: F32, +x2_0: F32, +x2_1: F32, +x2_2: F32, +x2_3: F32, +x3_0: F32, +x3_1: F32, +x3_2: F32, +x3_3: F32, +y0_0: F32, +y0_1: F32, a: Array, rv: Array & F32) -> GR: 99 match rv: 100 case Tuple{b, +y0_2}: 101 gr_19(ia, ib, arow, sa, sb, bcol, x0_0, x0_1, x0_2, x0_3, x1_0, x1_1, x1_2, x1_3, x2_0, x2_1, x2_2, x2_3, x3_0, x3_1, x3_2, x3_3, y0_0, y0_1, y0_2, a, Array.get(F32, b, (ib + ((sb * 0 : U32) + (bcol * 3 : U32) : U32) : U32))) 102 def gr_17(+ia: U32, +ib: U32, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, +x0_0: F32, +x0_1: F32, +x0_2: F32, +x0_3: F32, +x1_0: F32, +x1_1: F32, +x1_2: F32, +x1_3: F32, +x2_0: F32, +x2_1: F32, +x2_2: F32, +x2_3: F32, +x3_0: F32, +x3_1: F32, +x3_2: F32, +x3_3: F32, +y0_0: F32, a: Array, rv: Array & F32) -> GR: 103 match rv: 104 case Tuple{b, +y0_1}: 105 gr_18(ia, ib, arow, sa, sb, bcol, x0_0, x0_1, x0_2, x0_3, x1_0, x1_1, x1_2, x1_3, x2_0, x2_1, x2_2, x2_3, x3_0, x3_1, x3_2, x3_3, y0_0, y0_1, a, Array.get(F32, b, (ib + ((sb * 0 : U32) + (bcol * 2 : U32) : U32) : U32))) 106 def gr_16(+ia: U32, +ib: U32, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, +x0_0: F32, +x0_1: F32, +x0_2: F32, +x0_3: F32, +x1_0: F32, +x1_1: F32, +x1_2: F32, +x1_3: F32, +x2_0: F32, +x2_1: F32, +x2_2: F32, +x2_3: F32, +x3_0: F32, +x3_1: F32, +x3_2: F32, +x3_3: F32, a: Array, rv: Array & F32) -> GR: 107 match rv: 108 case Tuple{b, +y0_0}: 109 gr_17(ia, ib, arow, sa, sb, bcol, x0_0, x0_1, x0_2, x0_3, x1_0, x1_1, x1_2, x1_3, x2_0, x2_1, x2_2, x2_3, x3_0, x3_1, x3_2, x3_3, y0_0, a, Array.get(F32, b, (ib + ((sb * 0 : U32) + (bcol * 1 : U32) : U32) : U32))) 110 def gr_15(+ia: U32, +ib: U32, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, +x0_0: F32, +x0_1: F32, +x0_2: F32, +x0_3: F32, +x1_0: F32, +x1_1: F32, +x1_2: F32, +x1_3: F32, +x2_0: F32, +x2_1: F32, +x2_2: F32, +x2_3: F32, +x3_0: F32, +x3_1: F32, +x3_2: F32, rv: Array & F32, b: Array) -> GR: 111 match rv: 112 case Tuple{a, +x3_3}: 113 gr_16(ia, ib, arow, sa, sb, bcol, x0_0, x0_1, x0_2, x0_3, x1_0, x1_1, x1_2, x1_3, x2_0, x2_1, x2_2, x2_3, x3_0, x3_1, x3_2, x3_3, a, Array.get(F32, b, (ib + ((sb * 0 : U32) + (bcol * 0 : U32) : U32) : U32))) 114 def gr_14(+ia: U32, +ib: U32, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, +x0_0: F32, +x0_1: F32, +x0_2: F32, +x0_3: F32, +x1_0: F32, +x1_1: F32, +x1_2: F32, +x1_3: F32, +x2_0: F32, +x2_1: F32, +x2_2: F32, +x2_3: F32, +x3_0: F32, +x3_1: F32, rv: Array & F32, b: Array) -> GR: 115 match rv: 116 case Tuple{a, +x3_2}: 117 gr_15(ia, ib, arow, sa, sb, bcol, x0_0, x0_1, x0_2, x0_3, x1_0, x1_1, x1_2, x1_3, x2_0, x2_1, x2_2, x2_3, x3_0, x3_1, x3_2, Array.get(F32, a, (ia + ((sa * 3 : U32) + (arow * 3 : U32) : U32) : U32)), b) 118 def gr_13(+ia: U32, +ib: U32, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, +x0_0: F32, +x0_1: F32, +x0_2: F32, +x0_3: F32, +x1_0: F32, +x1_1: F32, +x1_2: F32, +x1_3: F32, +x2_0: F32, +x2_1: F32, +x2_2: F32, +x2_3: F32, +x3_0: F32, rv: Array & F32, b: Array) -> GR: 119 match rv: 120 case Tuple{a, +x3_1}: 121 gr_14(ia, ib, arow, sa, sb, bcol, x0_0, x0_1, x0_2, x0_3, x1_0, x1_1, x1_2, x1_3, x2_0, x2_1, x2_2, x2_3, x3_0, x3_1, Array.get(F32, a, (ia + ((sa * 3 : U32) + (arow * 2 : U32) : U32) : U32)), b) 122 def gr_12(+ia: U32, +ib: U32, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, +x0_0: F32, +x0_1: F32, +x0_2: F32, +x0_3: F32, +x1_0: F32, +x1_1: F32, +x1_2: F32, +x1_3: F32, +x2_0: F32, +x2_1: F32, +x2_2: F32, +x2_3: F32, rv: Array & F32, b: Array) -> GR: 123 match rv: 124 case Tuple{a, +x3_0}: 125 gr_13(ia, ib, arow, sa, sb, bcol, x0_0, x0_1, x0_2, x0_3, x1_0, x1_1, x1_2, x1_3, x2_0, x2_1, x2_2, x2_3, x3_0, Array.get(F32, a, (ia + ((sa * 3 : U32) + (arow * 1 : U32) : U32) : U32)), b) 126 def gr_11(+ia: U32, +ib: U32, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, +x0_0: F32, +x0_1: F32, +x0_2: F32, +x0_3: F32, +x1_0: F32, +x1_1: F32, +x1_2: F32, +x1_3: F32, +x2_0: F32, +x2_1: F32, +x2_2: F32, rv: Array & F32, b: Array) -> GR: 127 match rv: 128 case Tuple{a, +x2_3}: 129 gr_12(ia, ib, arow, sa, sb, bcol, x0_0, x0_1, x0_2, x0_3, x1_0, x1_1, x1_2, x1_3, x2_0, x2_1, x2_2, x2_3, Array.get(F32, a, (ia + ((sa * 3 : U32) + (arow * 0 : U32) : U32) : U32)), b) 130 def gr_10(+ia: U32, +ib: U32, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, +x0_0: F32, +x0_1: F32, +x0_2: F32, +x0_3: F32, +x1_0: F32, +x1_1: F32, +x1_2: F32, +x1_3: F32, +x2_0: F32, +x2_1: F32, rv: Array & F32, b: Array) -> GR: 131 match rv: 132 case Tuple{a, +x2_2}: 133 gr_11(ia, ib, arow, sa, sb, bcol, x0_0, x0_1, x0_2, x0_3, x1_0, x1_1, x1_2, x1_3, x2_0, x2_1, x2_2, Array.get(F32, a, (ia + ((sa * 2 : U32) + (arow * 3 : U32) : U32) : U32)), b) 134 def gr_9(+ia: U32, +ib: U32, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, +x0_0: F32, +x0_1: F32, +x0_2: F32, +x0_3: F32, +x1_0: F32, +x1_1: F32, +x1_2: F32, +x1_3: F32, +x2_0: F32, rv: Array & F32, b: Array) -> GR: 135 match rv: 136 case Tuple{a, +x2_1}: 137 gr_10(ia, ib, arow, sa, sb, bcol, x0_0, x0_1, x0_2, x0_3, x1_0, x1_1, x1_2, x1_3, x2_0, x2_1, Array.get(F32, a, (ia + ((sa * 2 : U32) + (arow * 2 : U32) : U32) : U32)), b) 138 def gr_8(+ia: U32, +ib: U32, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, +x0_0: F32, +x0_1: F32, +x0_2: F32, +x0_3: F32, +x1_0: F32, +x1_1: F32, +x1_2: F32, +x1_3: F32, rv: Array & F32, b: Array) -> GR: 139 match rv: 140 case Tuple{a, +x2_0}: 141 gr_9(ia, ib, arow, sa, sb, bcol, x0_0, x0_1, x0_2, x0_3, x1_0, x1_1, x1_2, x1_3, x2_0, Array.get(F32, a, (ia + ((sa * 2 : U32) + (arow * 1 : U32) : U32) : U32)), b) 142 def gr_7(+ia: U32, +ib: U32, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, +x0_0: F32, +x0_1: F32, +x0_2: F32, +x0_3: F32, +x1_0: F32, +x1_1: F32, +x1_2: F32, rv: Array & F32, b: Array) -> GR: 143 match rv: 144 case Tuple{a, +x1_3}: 145 gr_8(ia, ib, arow, sa, sb, bcol, x0_0, x0_1, x0_2, x0_3, x1_0, x1_1, x1_2, x1_3, Array.get(F32, a, (ia + ((sa * 2 : U32) + (arow * 0 : U32) : U32) : U32)), b) 146 def gr_6(+ia: U32, +ib: U32, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, +x0_0: F32, +x0_1: F32, +x0_2: F32, +x0_3: F32, +x1_0: F32, +x1_1: F32, rv: Array & F32, b: Array) -> GR: 147 match rv: 148 case Tuple{a, +x1_2}: 149 gr_7(ia, ib, arow, sa, sb, bcol, x0_0, x0_1, x0_2, x0_3, x1_0, x1_1, x1_2, Array.get(F32, a, (ia + ((sa * 1 : U32) + (arow * 3 : U32) : U32) : U32)), b) 150 def gr_5(+ia: U32, +ib: U32, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, +x0_0: F32, +x0_1: F32, +x0_2: F32, +x0_3: F32, +x1_0: F32, rv: Array & F32, b: Array) -> GR: 151 match rv: 152 case Tuple{a, +x1_1}: 153 gr_6(ia, ib, arow, sa, sb, bcol, x0_0, x0_1, x0_2, x0_3, x1_0, x1_1, Array.get(F32, a, (ia + ((sa * 1 : U32) + (arow * 2 : U32) : U32) : U32)), b) 154 def gr_4(+ia: U32, +ib: U32, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, +x0_0: F32, +x0_1: F32, +x0_2: F32, +x0_3: F32, rv: Array & F32, b: Array) -> GR: 155 match rv: 156 case Tuple{a, +x1_0}: 157 gr_5(ia, ib, arow, sa, sb, bcol, x0_0, x0_1, x0_2, x0_3, x1_0, Array.get(F32, a, (ia + ((sa * 1 : U32) + (arow * 1 : U32) : U32) : U32)), b) 158 def gr_3(+ia: U32, +ib: U32, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, +x0_0: F32, +x0_1: F32, +x0_2: F32, rv: Array & F32, b: Array) -> GR: 159 match rv: 160 case Tuple{a, +x0_3}: 161 gr_4(ia, ib, arow, sa, sb, bcol, x0_0, x0_1, x0_2, x0_3, Array.get(F32, a, (ia + ((sa * 1 : U32) + (arow * 0 : U32) : U32) : U32)), b) 162 def gr_2(+ia: U32, +ib: U32, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, +x0_0: F32, +x0_1: F32, rv: Array & F32, b: Array) -> GR: 163 match rv: 164 case Tuple{a, +x0_2}: 165 gr_3(ia, ib, arow, sa, sb, bcol, x0_0, x0_1, x0_2, Array.get(F32, a, (ia + ((sa * 0 : U32) + (arow * 3 : U32) : U32) : U32)), b) 166 def gr_1(+ia: U32, +ib: U32, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, +x0_0: F32, rv: Array & F32, b: Array) -> GR: 167 match rv: 168 case Tuple{a, +x0_1}: 169 gr_2(ia, ib, arow, sa, sb, bcol, x0_0, x0_1, Array.get(F32, a, (ia + ((sa * 0 : U32) + (arow * 2 : U32) : U32) : U32)), b) 170 def gr_0(+ia: U32, +ib: U32, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, rv: Array & F32, b: Array) -> GR: 171 match rv: 172 case Tuple{a, +x0_0}: 173 gr_1(ia, ib, arow, sa, sb, bcol, x0_0, Array.get(F32, a, (ia + ((sa * 0 : U32) + (arow * 1 : U32) : U32) : U32)), b) 174 # the turn whose A values start at ia and B values at ib 175 def turn(+ia: U32, +ib: U32, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, a: Array, b: Array) -> GR: 176 gr_0(ia, ib, arow, sa, sb, bcol, Array.get(F32, a, ia), b) 177 178 # the tile's cells after the four sums are combined, row by row 179 type T is Type: 180 T{a: Array, b: Array, c0_0: F32, c0_1: F32, c0_2: F32, c0_3: F32, c1_0: F32, c1_1: F32, c1_2: F32, c1_3: F32, c2_0: F32, c2_1: F32, c2_2: F32, c2_3: F32, c3_0: F32, c3_1: F32, c3_2: F32, c3_3: F32} 181 182 # kg more full turns; g holds the current turn's values. The last call (kg = 0) adds the turn that holds the steps 183 # past the last full turn (rem = k mod 4 of them), counting only those, and combines the sums. 184 @unsafe 185 def mm(kg: Nat, +ia: U32, +ib: U32, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, +rem: U32, +rot: Nat, +s0_0_0: F32, +s1_0_0: F32, +s2_0_0: F32, +s3_0_0: F32, +s0_0_1: F32, +s1_0_1: F32, +s2_0_1: F32, +s3_0_1: F32, +s0_0_2: F32, +s1_0_2: F32, +s2_0_2: F32, +s3_0_2: F32, +s0_0_3: F32, +s1_0_3: F32, +s2_0_3: F32, +s3_0_3: F32, +s0_1_0: F32, +s1_1_0: F32, +s2_1_0: F32, +s3_1_0: F32, +s0_1_1: F32, +s1_1_1: F32, +s2_1_1: F32, +s3_1_1: F32, +s0_1_2: F32, +s1_1_2: F32, +s2_1_2: F32, +s3_1_2: F32, +s0_1_3: F32, +s1_1_3: F32, +s2_1_3: F32, +s3_1_3: F32, +s0_2_0: F32, +s1_2_0: F32, +s2_2_0: F32, +s3_2_0: F32, +s0_2_1: F32, +s1_2_1: F32, +s2_2_1: F32, +s3_2_1: F32, +s0_2_2: F32, +s1_2_2: F32, +s2_2_2: F32, +s3_2_2: F32, +s0_2_3: F32, +s1_2_3: F32, +s2_2_3: F32, +s3_2_3: F32, +s0_3_0: F32, +s1_3_0: F32, +s2_3_0: F32, +s3_3_0: F32, +s0_3_1: F32, +s1_3_1: F32, +s2_3_1: F32, +s3_3_1: F32, +s0_3_2: F32, +s1_3_2: F32, +s2_3_2: F32, +s3_3_2: F32, +s0_3_3: F32, +s1_3_3: F32, +s2_3_3: F32, +s3_3_3: F32, g: GR) -> T: 186 match kg g: 187 case 0n GR{a, b, +x0_0, +x0_1, +x0_2, +x0_3, +x1_0, +x1_1, +x1_2, +x1_3, +x2_0, +x2_1, +x2_2, +x2_3, +x3_0, +x3_1, +x3_2, +x3_3, +y0_0, +y0_1, +y0_2, +y0_3, +y1_0, +y1_1, +y1_2, +y1_3, +y2_0, +y2_1, +y2_2, +y2_3, +y3_0, +y3_1, +y3_2, +y3_3}: 188 T{a, b, TA.t8_comb(rot, (s0_0_0 + pk(U32.is_lt(0, rem), (x0_0 * y0_0 : F32)) : F32), (s1_0_0 + pk(U32.is_lt(1, rem), (x1_0 * y1_0 : F32)) : F32), (s2_0_0 + pk(U32.is_lt(2, rem), (x2_0 * y2_0 : F32)) : F32), (s3_0_0 + pk(U32.is_lt(3, rem), (x3_0 * y3_0 : F32)) : F32)), TA.t8_comb(rot, (s0_0_1 + pk(U32.is_lt(0, rem), (x0_0 * y0_1 : F32)) : F32), (s1_0_1 + pk(U32.is_lt(1, rem), (x1_0 * y1_1 : F32)) : F32), (s2_0_1 + pk(U32.is_lt(2, rem), (x2_0 * y2_1 : F32)) : F32), (s3_0_1 + pk(U32.is_lt(3, rem), (x3_0 * y3_1 : F32)) : F32)), TA.t8_comb(rot, (s0_0_2 + pk(U32.is_lt(0, rem), (x0_0 * y0_2 : F32)) : F32), (s1_0_2 + pk(U32.is_lt(1, rem), (x1_0 * y1_2 : F32)) : F32), (s2_0_2 + pk(U32.is_lt(2, rem), (x2_0 * y2_2 : F32)) : F32), (s3_0_2 + pk(U32.is_lt(3, rem), (x3_0 * y3_2 : F32)) : F32)), TA.t8_comb(rot, (s0_0_3 + pk(U32.is_lt(0, rem), (x0_0 * y0_3 : F32)) : F32), (s1_0_3 + pk(U32.is_lt(1, rem), (x1_0 * y1_3 : F32)) : F32), (s2_0_3 + pk(U32.is_lt(2, rem), (x2_0 * y2_3 : F32)) : F32), (s3_0_3 + pk(U32.is_lt(3, rem), (x3_0 * y3_3 : F32)) : F32)), TA.t8_comb(rot, (s0_1_0 + pk(U32.is_lt(0, rem), (x0_1 * y0_0 : F32)) : F32), (s1_1_0 + pk(U32.is_lt(1, rem), (x1_1 * y1_0 : F32)) : F32), (s2_1_0 + pk(U32.is_lt(2, rem), (x2_1 * y2_0 : F32)) : F32), (s3_1_0 + pk(U32.is_lt(3, rem), (x3_1 * y3_0 : F32)) : F32)), TA.t8_comb(rot, (s0_1_1 + pk(U32.is_lt(0, rem), (x0_1 * y0_1 : F32)) : F32), (s1_1_1 + pk(U32.is_lt(1, rem), (x1_1 * y1_1 : F32)) : F32), (s2_1_1 + pk(U32.is_lt(2, rem), (x2_1 * y2_1 : F32)) : F32), (s3_1_1 + pk(U32.is_lt(3, rem), (x3_1 * y3_1 : F32)) : F32)), TA.t8_comb(rot, (s0_1_2 + pk(U32.is_lt(0, rem), (x0_1 * y0_2 : F32)) : F32), (s1_1_2 + pk(U32.is_lt(1, rem), (x1_1 * y1_2 : F32)) : F32), (s2_1_2 + pk(U32.is_lt(2, rem), (x2_1 * y2_2 : F32)) : F32), (s3_1_2 + pk(U32.is_lt(3, rem), (x3_1 * y3_2 : F32)) : F32)), TA.t8_comb(rot, (s0_1_3 + pk(U32.is_lt(0, rem), (x0_1 * y0_3 : F32)) : F32), (s1_1_3 + pk(U32.is_lt(1, rem), (x1_1 * y1_3 : F32)) : F32), (s2_1_3 + pk(U32.is_l 189 case 1n+q GR{a, b, +x0_0, +x0_1, +x0_2, +x0_3, +x1_0, +x1_1, +x1_2, +x1_3, +x2_0, +x2_1, +x2_2, +x2_3, +x3_0, +x3_1, +x3_2, +x3_3, +y0_0, +y0_1, +y0_2, +y0_3, +y1_0, +y1_1, +y1_2, +y1_3, +y2_0, +y2_1, +y2_2, +y2_3, +y3_0, +y3_1, +y3_2, +y3_3}: 190 mm(q, (ia + (sa * 4 : U32) : U32), (ib + (sb * 4 : U32) : U32), arow, sa, sb, bcol, rem, rot, (s0_0_0 + (x0_0 * y0_0 : F32) : F32), (s1_0_0 + (x1_0 * y1_0 : F32) : F32), (s2_0_0 + (x2_0 * y2_0 : F32) : F32), (s3_0_0 + (x3_0 * y3_0 : F32) : F32), (s0_0_1 + (x0_0 * y0_1 : F32) : F32), (s1_0_1 + (x1_0 * y1_1 : F32) : F32), (s2_0_1 + (x2_0 * y2_1 : F32) : F32), (s3_0_1 + (x3_0 * y3_1 : F32) : F32), (s0_0_2 + (x0_0 * y0_2 : F32) : F32), (s1_0_2 + (x1_0 * y1_2 : F32) : F32), (s2_0_2 + (x2_0 * y2_2 : F32) : F32), (s3_0_2 + (x3_0 * y3_2 : F32) : F32), (s0_0_3 + (x0_0 * y0_3 : F32) : F32), (s1_0_3 + (x1_0 * y1_3 : F32) : F32), (s2_0_3 + (x2_0 * y2_3 : F32) : F32), (s3_0_3 + (x3_0 * y3_3 : F32) : F32), (s0_1_0 + (x0_1 * y0_0 : F32) : F32), (s1_1_0 + (x1_1 * y1_0 : F32) : F32), (s2_1_0 + (x2_1 * y2_0 : F32) : F32), (s3_1_0 + (x3_1 * y3_0 : F32) : F32), (s0_1_1 + (x0_1 * y0_1 : F32) : F32), (s1_1_1 + (x1_1 * y1_1 : F32) : F32), (s2_1_1 + (x2_1 * y2_1 : F32) : F32), (s3_1_1 + (x3_1 * y3_1 : F32) : F32), (s0_1_2 + (x0_1 * y0_2 : F32) : F32), (s1_1_2 + (x1_1 * y1_2 : F32) : F32), (s2_1_2 + (x2_1 * y2_2 : F32) : F32), (s3_1_2 + (x3_1 * y3_2 : F32) : F32), (s0_1_3 + (x0_1 * y0_3 : F32) : F32), (s1_1_3 + (x1_1 * y1_3 : F32) : F32), (s2_1_3 + (x2_1 * y2_3 : F32) : F32), (s3_1_3 + (x3_1 * y3_3 : F32) : F32), (s0_2_0 + (x0_2 * y0_0 : F32) : F32), (s1_2_0 + (x1_2 * y1_0 : F32) : F32), (s2_2_0 + (x2_2 * y2_0 : F32) : F32), (s3_2_0 + (x3_2 * y3_0 : F32) : F32), (s0_2_1 + (x0_2 * y0_1 : F32) : F32), (s1_2_1 + (x1_2 * y1_1 : F32) : F32), (s2_2_1 + (x2_2 * y2_1 : F32) : F32), (s3_2_1 + (x3_2 * y3_1 : F32) : F32), (s0_2_2 + (x0_2 * y0_2 : F32) : F32), (s1_2_2 + (x1_2 * y1_2 : F32) : F32), (s2_2_2 + (x2_2 * y2_2 : F32) : F32), (s3_2_2 + (x3_2 * y3_2 : F32) : F32), (s0_2_3 + (x0_2 * y0_3 : F32) : F32), (s1_2_3 + (x1_2 * y1_3 : F32) : F32), (s2_2_3 + (x2_2 * y2_3 : F32) : F32), (s3_2_3 + (x3_2 * y3_3 : F32) : F32), (s0_3_0 + (x0_3 * y0_0 : F32) : F32), (s1_3_0 + (x1_3 * y1_0 : F32) : F32), 191 192 # writes cell (i, j) of C when it is inside n x m 193 def wr(ok: Bool, +ac: Bool, +idx: U32, +x: F32, c: Array) -> Array: 194 match ok: 195 case True{}: TA.cwr(ac, idx, x, c) 196 case False{}: c 197 198 # the tile at rows i0.., columns j0.. of C, from the loop's result; A and B are dropped (the caller holds them) 199 @unsafe 200 def put(+ac: Bool, +n: U32, +m: U32, +i0: U32, +j0: U32, c: Array, t: T) -> Array: 201 match t: 202 case T{a, b, +c0_0, +c0_1, +c0_2, +c0_3, +c1_0, +c1_1, +c1_2, +c1_3, +c2_0, +c2_1, +c2_2, +c2_3, +c3_0, +c3_1, +c3_2, +c3_3}: 203 wr(Bool.and(U32.is_lt((i0 + 3 : U32), n), U32.is_lt((j0 + 3 : U32), m)), ac, (((i0 + 3 : U32) * m : U32) + (j0 + 3 : U32) : U32), c3_3, wr(Bool.and(U32.is_lt((i0 + 3 : U32), n), U32.is_lt((j0 + 2 : U32), m)), ac, (((i0 + 3 : U32) * m : U32) + (j0 + 2 : U32) : U32), c3_2, wr(Bool.and(U32.is_lt((i0 + 3 : U32), n), U32.is_lt((j0 + 1 : U32), m)), ac, (((i0 + 3 : U32) * m : U32) + (j0 + 1 : U32) : U32), c3_1, wr(Bool.and(U32.is_lt((i0 + 3 : U32), n), U32.is_lt((j0 + 0 : U32), m)), ac, (((i0 + 3 : U32) * m : U32) + (j0 + 0 : U32) : U32), c3_0, wr(Bool.and(U32.is_lt((i0 + 2 : U32), n), U32.is_lt((j0 + 3 : U32), m)), ac, (((i0 + 2 : U32) * m : U32) + (j0 + 3 : U32) : U32), c2_3, wr(Bool.and(U32.is_lt((i0 + 2 : U32), n), U32.is_lt((j0 + 2 : U32), m)), ac, (((i0 + 2 : U32) * m : U32) + (j0 + 2 : U32) : U32), c2_2, wr(Bool.and(U32.is_lt((i0 + 2 : U32), n), U32.is_lt((j0 + 1 : U32), m)), ac, (((i0 + 2 : U32) * m : U32) + (j0 + 1 : U32) : U32), c2_1, wr(Bool.and(U32.is_lt((i0 + 2 : U32), n), U32.is_lt((j0 + 0 : U32), m)), ac, (((i0 + 2 : U32) * m : U32) + (j0 + 0 : U32) : U32), c2_0, wr(Bool.and(U32.is_lt((i0 + 1 : U32), n), U32.is_lt((j0 + 3 : U32), m)), ac, (((i0 + 1 : U32) * m : U32) + (j0 + 3 : U32) : U32), c1_3, wr(Bool.and(U32.is_lt((i0 + 1 : U32), n), U32.is_lt((j0 + 2 : U32), m)), ac, (((i0 + 1 : U32) * m : U32) + (j0 + 2 : U32) : U32), c1_2, wr(Bool.and(U32.is_lt((i0 + 1 : U32), n), U32.is_lt((j0 + 1 : U32), m)), ac, (((i0 + 1 : U32) * m : U32) + (j0 + 1 : U32) : U32), c1_1, wr(Bool.and(U32.is_lt((i0 + 1 : U32), n), U32.is_lt((j0 + 0 : U32), m)), ac, (((i0 + 1 : U32) * m : U32) + (j0 + 0 : U32) : U32), c1_0, wr(Bool.and(U32.is_lt((i0 + 0 : U32), n), U32.is_lt((j0 + 3 : U32), m)), ac, (((i0 + 0 : U32) * m : U32) + (j0 + 3 : U32) : U32), c0_3, wr(Bool.and(U32.is_lt((i0 + 0 : U32), n), U32.is_lt((j0 + 2 : U32), m)), ac, (((i0 + 0 : U32) * m : U32) + (j0 + 2 : U32) : U32), c0_2, wr(Bool.and(U32.is_lt((i0 + 0 : U32), n), U32.is_lt((j0 + 1 : U32), m)), ac, (((i0 + 0 : U 204 205 @unsafe 206 def tile(+e: E, +i0: U32, +j0: U32, +a: Array, +b: Array, c: Array) -> Array: 207 match e: 208 case E{+n, +m, +k, +tcols, +tiles, +arow, +sa, +sb, +bcol, +kg, +rot, +rem, +ac, +oa, +ob}: 209 put(ac, n, m, i0, j0, c, mm(kg, (oa + (i0 * arow : U32) : U32), (ob + (j0 * bcol : U32) : U32), arow, sa, sb, bcol, rem, rot, 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, 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, 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, 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, turn((oa + (i0 * arow : U32) : U32), (ob + (j0 * bcol : U32) : U32), arow, sa, sb, bcol, a, b))) 210 211 # leaf t computes tile t (row-major over the tiles); leaves past the last tile write nothing 212 @unsafe 213 def leaf_if(ok: Bool, +e: E, +t: U32, +tcols: U32, +a: Array, +b: Array, c: Array) -> Array: 214 match ok: 215 case True{}: tile(e, (U32.div(t, tcols) * 4 : U32), (U32.mod(t, tcols) * 4 : U32), a, b, c) 216 case False{}: c 217 218 @unsafe 219 def leaf(+e: E, +t: U32, +a: Array, +b: Array, c: Array) -> Array: 220 match e: 221 case E{+n, +m, +k, +tcols, +tiles, +arow, +sa, +sb, +bcol, +kg, +rot, +rem, +ac, +oa, +ob}: 222 leaf_if(U32.is_lt(t, tiles), e, t, tcols, a, b, c) 223 224 @unsafe 225 def tree(+e: E, d: Nat, +t: U32, +a: Array, +b: Array, +c: Array) -> Array: 226 match d: 227 case 0n: leaf(e, t, a, b, c) 228 case 1n++q: 229 l r = tree(e, q, (t * 2 : U32), a, b, c) tree(e, q, ((t * 2 : U32) + 1 : U32), a, b, c) 230 Array.join(F32, l, r) 231 232 # C made inside the bang: the root of the ! call allocates it on the device. Made on the host instead, a new C often 233 # reuses a block whose pages sit on the device, so zeroing it on the host pulls them back over PCIe and the bang moves 234 # them again (a 256 x 1536 product: 9.2 ms against 6.0, measured 2026-10-18). The leaves write every cell of C. 235 @unsafe 236 def tree_new(+e: E, d: Nat, +a: Array, +b: Array, +cd: Nat) -> Array: 237 tree(e, d, 0, a, b, Array.new(F32, cd, 0.0)) 238 239 @unsafe 240 def bang_new(+e: E, d: Nat, +a: Array, +b: Array, +cd: Nat) -> Array: 241 tree_new!(e, d, a, b, cd) 242 243 # one bang over the tiles; A and B are shared with every leaf and come back unchanged 244 @unsafe 245 def bang(+e: E, d: Nat, +a: Array, +b: Array, c: Array) -> Array: 246 tree!(e, d, 0, a, b, c) 247 248 type ABC is Type: 249 ABC{a: Array, b: Array, c: Array} 250 251 @unsafe 252 def run_e(+e: E, +tiles: U32, +a: Array, +b: Array, c: Array) -> ABC: 253 ABC{a, b, bang(e, TA.cap_depth(U32.to_nat(tiles)), a, b, c)} 254 255 # the product's sizes and strides 256 def e_ofo(+ac: Bool, +n: Nat, +m: Nat, +k: Nat, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, +oa: U32, +ob: U32) -> E: 257 E{U32.from_nat(n), U32.from_nat(m), U32.from_nat(k), U32.from_nat(Nat.div(Nat.add(m, 3n), 4n)), U32.from_nat(Nat.mul(Nat.div(Nat.add(n, 3n), 4n), Nat.div(Nat.add(m, 3n), 4n))), arow, sa, sb, bcol, Nat.div(k, 4n), Nat.mod(k, 4n), U32.from_nat(Nat.mod(k, 4n)), ac, oa, ob} 258 259 def e_of(+ac: Bool, +n: Nat, +m: Nat, +k: Nat, +arow: U32, +sa: U32, +sb: U32, +bcol: U32) -> E: 260 e_ofo(ac, n, m, k, arow, sa, sb, bcol, 0, 0) 261 262 def tiles(+n: Nat, +m: Nat) -> Nat: 263 Nat.mul(Nat.div(Nat.add(n, 3n), 4n), Nat.div(Nat.add(m, 3n), 4n)) 264 265 # C (n x m) = A · B with k steps into a new C made on the device 266 @unsafe 267 def gemm_new(+n: Nat, +m: Nat, +k: Nat, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, +a: Array, +b: Array) -> ABC: 268 ABC{a, b, bang_new(e_of(False{}, n, m, k, arow, sa, sb, bcol), TA.cap_depth(tiles(n, m)), a, b, TA.cap_depth(Nat.mul(n, m)))} 269 270 # C (n x m) = A · B with k steps, or C + A · B when ac, C[i][j] = sum_p A[i*arow + p*sa] * B[p*sb + j*bcol] 271 @unsafe 272 def gemm(+ac: Bool, +n: Nat, +m: Nat, +k: Nat, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, a: Array, b: Array, c: Array) -> ABC: 273 run_e(e_of(ac, n, m, k, arow, sa, sb, bcol), U32.from_nat(tiles(n, m)), a, b, c) 274 275 def mm_fin(-n: Nat, -k: Nat, -m: Nat, g: ABC) -> TA.MMul: 276 match g: 277 case ABC{a, b, c}: TA.MMul{TA.Mat{a}, TA.Mat{b}, TA.Mat{c}} 278 279 def mm_nt_fin(-n: Nat, -k: Nat, -m: Nat, g: ABC) -> TA.MMulNT: 280 match g: 281 case ABC{a, b, c}: TA.MMulNT{TA.Mat{a}, TA.Mat{b}, TA.Mat{c}} 282 283 def mm_tn_fin(-n: Nat, -k: Nat, -m: Nat, g: ABC) -> TA.MMulTN: 284 match g: 285 case ABC{a, b, c}: TA.MMulTN{TA.Mat{a}, TA.Mat{b}, TA.Mat{c}} 286 287 # C = A · B (n x k) · (k x m), tensor-array's TA.Mat.matmul bits 288 @unsafe 289 def matmul(+n: Nat, +k: Nat, +m: Nat, par: Nat, a: TA.Mat, b: TA.Mat) -> TA.MMul: 290 match a b: 291 case TA.Mat{x} TA.Mat{y}: 292 mm_fin(n, k, m, gemm_new(n, m, k, U32.from_nat(k), 1, U32.from_nat(m), 1, x, y)) 293 294 # C + A · B, into C (n x m) 295 @unsafe 296 def matmul_acc(+n: Nat, +k: Nat, +m: Nat, par: Nat, a: TA.Mat, b: TA.Mat, c: TA.Mat) -> TA.MMul: 297 match a b c: 298 case TA.Mat{x} TA.Mat{y} TA.Mat{z}: 299 mm_fin(n, k, m, gemm(True{}, n, m, k, U32.from_nat(k), 1, U32.from_nat(m), 1, x, y, z)) 300 301 # C = A · Bᵀ (n x k) · (m x k)ᵀ: for weights stored with one row per output 302 @unsafe 303 def matmul_nt(+n: Nat, +k: Nat, +m: Nat, par: Nat, a: TA.Mat, b: TA.Mat) -> TA.MMulNT: 304 match a b: 305 case TA.Mat{x} TA.Mat{y}: 306 mm_nt_fin(n, k, m, gemm_new(n, m, k, U32.from_nat(k), 1, 1, U32.from_nat(k), x, y)) 307 308 # C + A · Bᵀ, into C (n x m) 309 @unsafe 310 def matmul_nt_acc(+n: Nat, +k: Nat, +m: Nat, par: Nat, a: TA.Mat, b: TA.Mat, c: TA.Mat) -> TA.MMulNT: 311 match a b c: 312 case TA.Mat{x} TA.Mat{y} TA.Mat{z}: 313 mm_nt_fin(n, k, m, gemm(True{}, n, m, k, U32.from_nat(k), 1, 1, U32.from_nat(k), x, y, z)) 314 315 # C = Aᵀ · B (k x n)ᵀ · (k x m): weight gradient, Xᵀ · dY 316 @unsafe 317 def matmul_tn(+n: Nat, +k: Nat, +m: Nat, par: Nat, a: TA.Mat, b: TA.Mat) -> TA.MMulTN: 318 match a b: 319 case TA.Mat{x} TA.Mat{y}: 320 mm_tn_fin(n, k, m, gemm_new(n, m, k, 1, U32.from_nat(n), U32.from_nat(m), 1, x, y)) 321 322 # C + Aᵀ · B, into C (n x m): dW += Xᵀ · dY 323 @unsafe 324 def matmul_tn_acc(+n: Nat, +k: Nat, +m: Nat, par: Nat, a: TA.Mat, b: TA.Mat, c: TA.Mat) -> TA.MMulTN: 325 match a b c: 326 case TA.Mat{x} TA.Mat{y} TA.Mat{z}: 327 mm_tn_fin(n, k, m, gemm(True{}, n, m, k, 1, U32.from_nat(n), U32.from_nat(m), 1, x, y, z)) 328 329 # ===================================================================================================================== 330 # Bangs over any leaf code (0.1.1.0). runK and runK_new (K = 1..4 outputs) run one bang over 2^d leaves: leaf t of 331 # l = 2^d gets t, l, a record e of scalars (the same for every leaf), four input arrays and the K outputs, and returns 332 # the outputs. The leaf code is a template (~f), written by the caller in plain Bend: it takes every array as an ordinary 333 # argument (it reads them with Array.get and may write the outputs), so it needs no @unsafe; the sharing is here, where 334 # every node of the tree holds a handle to each array (Array.fork in Base, upstream's one-block pattern) and joins the 335 # outputs (Array.join). runK takes the outputs from the caller and writes them in place (an optimizer's weights and 336 # moments); runK_new makes them inside the bang with 2^c1.. cells, since an array made on the host pulls device pages 337 # back (see bang_new above). Both return the inputs unchanged with the outputs (RK). An unused input is any array, such 338 # as none(Unit{}). 339 # 340 # Not checked, as for the products: that a leaf writes only its own cells of the outputs (two leaves writing one cell 341 # race) and never an input. Leaves past the work must return their outputs unchanged; share(t, n, l) is how many of the 342 # cells t, t + l, t + 2l, .. lie below n (leaf t's cells when n cells are dealt round-robin, so neighbouring lanes read 343 # neighbouring cells), and depth(n) is a tree for n cells: 2^14 leaves once there are that many, else the fewest that 344 # cover them. The leaves run in a run-time order inside each group of 128 (bendlang/bend's CUDA lowering), so a leaf's 345 # result must not depend on another's. Every number a leaf computes is the caller's own code, so the bits are the 346 # caller's: the same on the GPU as on the CPU threads, except where CUDA's math functions differ from the host's (an exp 347 # on the device takes the float overload; the known limit of bendlang/bend#1305). 348 # ===================================================================================================================== 349 350 def depth2(big: Bool, +n: Nat) -> Nat: 351 match big: 352 case True{}: 14n 353 case False{}: TA.cap_depth(n) 354 355 # leaves for n cells: 2^14 when there are that many, else the fewest powers of two that cover them 356 def depth(+n: Nat) -> Nat: 357 depth2(Nat.is_le(16384n, n), n) 358 359 # how many of the cells t, t + l, t + 2l, .. lie below n 360 def share(+t: U32, +n: U32, +l: U32) -> Nat: 361 U32.to_nat(Bool.pick(U32, U32.is_lt(t, n), U32.div(((n - t : U32) + (l - 1 : U32) : U32), l), 0)) 362 363 # an array of one cell, for an unused input 364 def none(u: Unit) -> Array: 365 Array.new(F32, 0n, 0.0) 366 367 type O2 is Type: 368 O2{a: Array, b: Array} 369 370 type O3 is Type: 371 O3{a: Array, b: Array, c: Array} 372 373 type O4 is Type: 374 O4{a: Array, b: Array, c: Array, d: Array} 375 376 type R1 is Type: 377 R1{x1: Array, x2: Array, x3: Array, x4: Array, a: Array} 378 379 type R2 is Type: 380 R2{x1: Array, x2: Array, x3: Array, x4: Array, o: O2} 381 382 type R3 is Type: 383 R3{x1: Array, x2: Array, x3: Array, x4: Array, o: O3} 384 385 type R4 is Type: 386 R4{x1: Array, x2: Array, x3: Array, x4: Array, o: O4} 387 388 @unsafe 389 def join2(x: O2, y: O2) -> O2: 390 match x y: 391 case O2{a, b} O2{a2, b2}: O2{Array.join(F32, a, a2), Array.join(F32, b, b2)} 392 393 @unsafe 394 def join3(x: O3, y: O3) -> O3: 395 match x y: 396 case O3{a, b, c} O3{a2, b2, c2}: O3{Array.join(F32, a, a2), Array.join(F32, b, b2), Array.join(F32, c, c2)} 397 398 @unsafe 399 def join4(x: O4, y: O4) -> O4: 400 match x y: 401 case O4{a, b, c, d} O4{a2, b2, c2, d2}: O4{Array.join(F32, a, a2), Array.join(F32, b, b2), Array.join(F32, c, c2), Array.join(F32, d, d2)} 402 403 @unsafe 404 def tree1(~E: Data, ~f: @+t: U32 -> @+l: U32 -> @+e: E -> Array -> Array -> Array -> Array -> Array -> Array, +d: Nat, +t: U32, +l: U32, +e: E, +x1: Array, +x2: Array, +x3: Array, +x4: Array, +oa: Array) -> Array: 405 match d: 406 case 0n: f(t, l, e, x1, x2, x3, x4, oa) 407 case 1n+q: 408 x y = tree1(~E, ~f, q, (t * 2 : U32), l, e, x1, x2, x3, x4, oa) tree1(~E, ~f, q, ((t * 2 : U32) + 1 : U32), l, e, x1, x2, x3, x4, oa) 409 Array.join(F32, x, y) 410 411 @unsafe 412 def root1(~E: Data, ~f: @+t: U32 -> @+l: U32 -> @+e: E -> Array -> Array -> Array -> Array -> Array -> Array, +d: Nat, +e: E, +x1: Array, +x2: Array, +x3: Array, +x4: Array, oa: Array) -> Array: 413 tree1(~E, ~f, d, 0, U32.pow(2, d), e, x1, x2, x3, x4, oa) 414 415 @unsafe 416 def root1_new(~E: Data, ~f: @+t: U32 -> @+l: U32 -> @+e: E -> Array -> Array -> Array -> Array -> Array -> Array, +d: Nat, +e: E, +x1: Array, +x2: Array, +x3: Array, +x4: Array, +ca: Nat) -> Array: 417 root1(~E, ~f, d, e, x1, x2, x3, x4, Array.new(F32, ca, 0.0)) 418 419 @unsafe 420 def go1(~E: Data, ~f: @+t: U32 -> @+l: U32 -> @+e: E -> Array -> Array -> Array -> Array -> Array -> Array, +d: Nat, +e: E, +x1: Array, +x2: Array, +x3: Array, +x4: Array, oa: Array) -> R1: 421 R1{x1, x2, x3, x4, root1!(~E, ~f, d, e, x1, x2, x3, x4, oa)} 422 423 @unsafe 424 def go1_new(~E: Data, ~f: @+t: U32 -> @+l: U32 -> @+e: E -> Array -> Array -> Array -> Array -> Array -> Array, +d: Nat, +e: E, +x1: Array, +x2: Array, +x3: Array, +x4: Array, +ca: Nat) -> R1: 425 R1{x1, x2, x3, x4, root1_new!(~E, ~f, d, e, x1, x2, x3, x4, ca)} 426 427 # one bang over 2^d leaves with 1 output written in place 428 @unsafe 429 def run1(~E: Data, ~f: @+t: U32 -> @+l: U32 -> @+e: E -> Array -> Array -> Array -> Array -> Array -> Array, +d: Nat, +e: E, x1: Array, x2: Array, x3: Array, x4: Array, oa: Array) -> R1: 430 go1(~E, ~f, d, e, x1, x2, x3, x4, oa) 431 432 # one bang over 2^d leaves with 1 new output of 2^c.. cells, made on the device 433 @unsafe 434 def run1_new(~E: Data, ~f: @+t: U32 -> @+l: U32 -> @+e: E -> Array -> Array -> Array -> Array -> Array -> Array, +d: Nat, +e: E, x1: Array, x2: Array, x3: Array, x4: Array, +ca: Nat) -> R1: 435 go1_new(~E, ~f, d, e, x1, x2, x3, x4, ca) 436 437 @unsafe 438 def tree2(~E: Data, ~f: @+t: U32 -> @+l: U32 -> @+e: E -> Array -> Array -> Array -> Array -> O2 -> O2, +d: Nat, +t: U32, +l: U32, +e: E, +x1: Array, +x2: Array, +x3: Array, +x4: Array, +oa: Array, +ob: Array) -> O2: 439 match d: 440 case 0n: f(t, l, e, x1, x2, x3, x4, O2{oa, ob}) 441 case 1n+q: 442 x y = tree2(~E, ~f, q, (t * 2 : U32), l, e, x1, x2, x3, x4, oa, ob) tree2(~E, ~f, q, ((t * 2 : U32) + 1 : U32), l, e, x1, x2, x3, x4, oa, ob) 443 join2(x, y) 444 445 @unsafe 446 def root2(~E: Data, ~f: @+t: U32 -> @+l: U32 -> @+e: E -> Array -> Array -> Array -> Array -> O2 -> O2, +d: Nat, +e: E, +x1: Array, +x2: Array, +x3: Array, +x4: Array, oa: Array, ob: Array) -> O2: 447 tree2(~E, ~f, d, 0, U32.pow(2, d), e, x1, x2, x3, x4, oa, ob) 448 449 @unsafe 450 def root2_new(~E: Data, ~f: @+t: U32 -> @+l: U32 -> @+e: E -> Array -> Array -> Array -> Array -> O2 -> O2, +d: Nat, +e: E, +x1: Array, +x2: Array, +x3: Array, +x4: Array, +ca: Nat, +cb: Nat) -> O2: 451 root2(~E, ~f, d, e, x1, x2, x3, x4, Array.new(F32, ca, 0.0), Array.new(F32, cb, 0.0)) 452 453 @unsafe 454 def go2(~E: Data, ~f: @+t: U32 -> @+l: U32 -> @+e: E -> Array -> Array -> Array -> Array -> O2 -> O2, +d: Nat, +e: E, +x1: Array, +x2: Array, +x3: Array, +x4: Array, oa: Array, ob: Array) -> R2: 455 R2{x1, x2, x3, x4, root2!(~E, ~f, d, e, x1, x2, x3, x4, oa, ob)} 456 457 @unsafe 458 def go2_new(~E: Data, ~f: @+t: U32 -> @+l: U32 -> @+e: E -> Array -> Array -> Array -> Array -> O2 -> O2, +d: Nat, +e: E, +x1: Array, +x2: Array, +x3: Array, +x4: Array, +ca: Nat, +cb: Nat) -> R2: 459 R2{x1, x2, x3, x4, root2_new!(~E, ~f, d, e, x1, x2, x3, x4, ca, cb)} 460 461 # one bang over 2^d leaves with 2 outputs written in place 462 @unsafe 463 def run2(~E: Data, ~f: @+t: U32 -> @+l: U32 -> @+e: E -> Array -> Array -> Array -> Array -> O2 -> O2, +d: Nat, +e: E, x1: Array, x2: Array, x3: Array, x4: Array, oa: Array, ob: Array) -> R2: 464 go2(~E, ~f, d, e, x1, x2, x3, x4, oa, ob) 465 466 # one bang over 2^d leaves with 2 new outputs of 2^c.. cells, made on the device 467 @unsafe 468 def run2_new(~E: Data, ~f: @+t: U32 -> @+l: U32 -> @+e: E -> Array -> Array -> Array -> Array -> O2 -> O2, +d: Nat, +e: E, x1: Array, x2: Array, x3: Array, x4: Array, +ca: Nat, +cb: Nat) -> R2: 469 go2_new(~E, ~f, d, e, x1, x2, x3, x4, ca, cb) 470 471 @unsafe 472 def tree3(~E: Data, ~f: @+t: U32 -> @+l: U32 -> @+e: E -> Array -> Array -> Array -> Array -> O3 -> O3, +d: Nat, +t: U32, +l: U32, +e: E, +x1: Array, +x2: Array, +x3: Array, +x4: Array, +oa: Array, +ob: Array, +oc: Array) -> O3: 473 match d: 474 case 0n: f(t, l, e, x1, x2, x3, x4, O3{oa, ob, oc}) 475 case 1n+q: 476 x y = tree3(~E, ~f, q, (t * 2 : U32), l, e, x1, x2, x3, x4, oa, ob, oc) tree3(~E, ~f, q, ((t * 2 : U32) + 1 : U32), l, e, x1, x2, x3, x4, oa, ob, oc) 477 join3(x, y) 478 479 @unsafe 480 def root3(~E: Data, ~f: @+t: U32 -> @+l: U32 -> @+e: E -> Array -> Array -> Array -> Array -> O3 -> O3, +d: Nat, +e: E, +x1: Array, +x2: Array, +x3: Array, +x4: Array, oa: Array, ob: Array, oc: Array) -> O3: 481 tree3(~E, ~f, d, 0, U32.pow(2, d), e, x1, x2, x3, x4, oa, ob, oc) 482 483 @unsafe 484 def root3_new(~E: Data, ~f: @+t: U32 -> @+l: U32 -> @+e: E -> Array -> Array -> Array -> Array -> O3 -> O3, +d: Nat, +e: E, +x1: Array, +x2: Array, +x3: Array, +x4: Array, +ca: Nat, +cb: Nat, +cc: Nat) -> O3: 485 root3(~E, ~f, d, e, x1, x2, x3, x4, Array.new(F32, ca, 0.0), Array.new(F32, cb, 0.0), Array.new(F32, cc, 0.0)) 486 487 @unsafe 488 def go3(~E: Data, ~f: @+t: U32 -> @+l: U32 -> @+e: E -> Array -> Array -> Array -> Array -> O3 -> O3, +d: Nat, +e: E, +x1: Array, +x2: Array, +x3: Array, +x4: Array, oa: Array, ob: Array, oc: Array) -> R3: 489 R3{x1, x2, x3, x4, root3!(~E, ~f, d, e, x1, x2, x3, x4, oa, ob, oc)} 490 491 @unsafe 492 def go3_new(~E: Data, ~f: @+t: U32 -> @+l: U32 -> @+e: E -> Array -> Array -> Array -> Array -> O3 -> O3, +d: Nat, +e: E, +x1: Array, +x2: Array, +x3: Array, +x4: Array, +ca: Nat, +cb: Nat, +cc: Nat) -> R3: 493 R3{x1, x2, x3, x4, root3_new!(~E, ~f, d, e, x1, x2, x3, x4, ca, cb, cc)} 494 495 # one bang over 2^d leaves with 3 outputs written in place 496 @unsafe 497 def run3(~E: Data, ~f: @+t: U32 -> @+l: U32 -> @+e: E -> Array -> Array -> Array -> Array -> O3 -> O3, +d: Nat, +e: E, x1: Array, x2: Array, x3: Array, x4: Array, oa: Array, ob: Array, oc: Array) -> R3: 498 go3(~E, ~f, d, e, x1, x2, x3, x4, oa, ob, oc) 499 500 # one bang over 2^d leaves with 3 new outputs of 2^c.. cells, made on the device 501 @unsafe 502 def run3_new(~E: Data, ~f: @+t: U32 -> @+l: U32 -> @+e: E -> Array -> Array -> Array -> Array -> O3 -> O3, +d: Nat, +e: E, x1: Array, x2: Array, x3: Array, x4: Array, +ca: Nat, +cb: Nat, +cc: Nat) -> R3: 503 go3_new(~E, ~f, d, e, x1, x2, x3, x4, ca, cb, cc) 504 505 @unsafe 506 def tree4(~E: Data, ~f: @+t: U32 -> @+l: U32 -> @+e: E -> Array -> Array -> Array -> Array -> O4 -> O4, +d: Nat, +t: U32, +l: U32, +e: E, +x1: Array, +x2: Array, +x3: Array, +x4: Array, +oa: Array, +ob: Array, +oc: Array, +od: Array) -> O4: 507 match d: 508 case 0n: f(t, l, e, x1, x2, x3, x4, O4{oa, ob, oc, od}) 509 case 1n+q: 510 x y = tree4(~E, ~f, q, (t * 2 : U32), l, e, x1, x2, x3, x4, oa, ob, oc, od) tree4(~E, ~f, q, ((t * 2 : U32) + 1 : U32), l, e, x1, x2, x3, x4, oa, ob, oc, od) 511 join4(x, y) 512 513 @unsafe 514 def root4(~E: Data, ~f: @+t: U32 -> @+l: U32 -> @+e: E -> Array -> Array -> Array -> Array -> O4 -> O4, +d: Nat, +e: E, +x1: Array, +x2: Array, +x3: Array, +x4: Array, oa: Array, ob: Array, oc: Array, od: Array) -> O4: 515 tree4(~E, ~f, d, 0, U32.pow(2, d), e, x1, x2, x3, x4, oa, ob, oc, od) 516 517 @unsafe 518 def root4_new(~E: Data, ~f: @+t: U32 -> @+l: U32 -> @+e: E -> Array -> Array -> Array -> Array -> O4 -> O4, +d: Nat, +e: E, +x1: Array, +x2: Array, +x3: Array, +x4: Array, +ca: Nat, +cb: Nat, +cc: Nat, +cd: Nat) -> O4: 519 root4(~E, ~f, d, e, x1, x2, x3, x4, Array.new(F32, ca, 0.0), Array.new(F32, cb, 0.0), Array.new(F32, cc, 0.0), Array.new(F32, cd, 0.0)) 520 521 @unsafe 522 def go4(~E: Data, ~f: @+t: U32 -> @+l: U32 -> @+e: E -> Array -> Array -> Array -> Array -> O4 -> O4, +d: Nat, +e: E, +x1: Array, +x2: Array, +x3: Array, +x4: Array, oa: Array, ob: Array, oc: Array, od: Array) -> R4: 523 R4{x1, x2, x3, x4, root4!(~E, ~f, d, e, x1, x2, x3, x4, oa, ob, oc, od)} 524 525 @unsafe 526 def go4_new(~E: Data, ~f: @+t: U32 -> @+l: U32 -> @+e: E -> Array -> Array -> Array -> Array -> O4 -> O4, +d: Nat, +e: E, +x1: Array, +x2: Array, +x3: Array, +x4: Array, +ca: Nat, +cb: Nat, +cc: Nat, +cd: Nat) -> R4: 527 R4{x1, x2, x3, x4, root4_new!(~E, ~f, d, e, x1, x2, x3, x4, ca, cb, cc, cd)} 528 529 # one bang over 2^d leaves with 4 outputs written in place 530 @unsafe 531 def run4(~E: Data, ~f: @+t: U32 -> @+l: U32 -> @+e: E -> Array -> Array -> Array -> Array -> O4 -> O4, +d: Nat, +e: E, x1: Array, x2: Array, x3: Array, x4: Array, oa: Array, ob: Array, oc: Array, od: Array) -> R4: 532 go4(~E, ~f, d, e, x1, x2, x3, x4, oa, ob, oc, od) 533 534 # one bang over 2^d leaves with 4 new outputs of 2^c.. cells, made on the device 535 @unsafe 536 def run4_new(~E: Data, ~f: @+t: U32 -> @+l: U32 -> @+e: E -> Array -> Array -> Array -> Array -> O4 -> O4, +d: Nat, +e: E, x1: Array, x2: Array, x3: Array, x4: Array, +ca: Nat, +cb: Nat, +cc: Nat, +cd: Nat) -> R4: 537 go4_new(~E, ~f, d, e, x1, x2, x3, x4, ca, cb, cc, cd) 538 539 # ---- tensor-array 0.1.14.0's precise product on the GPU (opt-in) ---- 540 # TA.Mat.matmul_precise's sums, a bang a block of 1024 steps that starts at the block's place in A and B (oa, ob; no 541 # copy): the first block's bang makes C on the device, each next one adds its cells into C as the accumulating kernel 542 # does (once, as tensor-array's cwr), so every cell is ((x0 + x1) + x2) .., the same bits as tensor-array's 543 # (tests/precise.bend). Only plain (not accumulating) products, as tensor-array's matmul_precise. 544 545 def blk_len(+bi: U32, +k: U32) -> Nat: 546 U32.to_nat(Bool.pick(U32, U32.is_lt((k - (bi * 1024 : U32) : U32), 1024), (k - (bi * 1024 : U32) : U32), 1024)) 547 548 # the blocks after the first: nb more, from block bi, each added into C 549 @unsafe 550 def blocks_acc(nb: Nat, +bi: U32, +n: Nat, +m: Nat, +k: U32, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, +a: Array, +b: Array, c: Array) -> Array: 551 match nb: 552 case 0n: c 553 case 1n++q: blocks_acc(q, (bi + 1 : U32), n, m, k, arow, sa, sb, bcol, a, b, bang(e_ofo(True{}, n, m, blk_len(bi, k), arow, sa, sb, bcol, ((bi * 1024 : U32) * sa : U32), ((bi * 1024 : U32) * sb : U32)), TA.cap_depth(tiles(n, m)), a, b, c)) 554 555 @unsafe 556 def precise_c(+n: Nat, +m: Nat, +k: Nat, +arow: U32, +sa: U32, +sb: U32, +bcol: U32, +a: Array, +b: Array) -> Array: 557 blocks_acc(Nat.sub(U32.to_nat(U32.div((U32.from_nat(k) + 1023 : U32), 1024)), 1n), 1, n, m, U32.from_nat(k), arow, sa, sb, bcol, a, b, bang_new(e_ofo(False{}, n, m, 1024n, arow, sa, sb, bcol, 0, 0), TA.cap_depth(tiles(n, m)), a, b, TA.cap_depth(Nat.mul(n, m)))) 558 559 @unsafe 560 def mm_precise3(+n: Nat, +k: Nat, +m: Nat, +a: Array, +b: Array) -> TA.MMul: 561 mm_fin(n, k, m, ABC{a, b, precise_c(n, m, k, U32.from_nat(k), 1, U32.from_nat(m), 1, a, b)}) 562 563 @unsafe 564 def mm_precise2(long: Bool, +n: Nat, +k: Nat, +m: Nat, +par: Nat, a: TA.Mat, b: TA.Mat) -> TA.MMul: 565 match long a b: 566 case True{} TA.Mat{x} TA.Mat{y}: mm_precise3(n, k, m, x, y) 567 case False{} a b: matmul(n, k, m, par, a, b) 568 569 # C = A · B with k summed in blocks of 1024: TA.Mat.matmul_precise's bits (opt-in) 570 @unsafe 571 def matmul_precise(+n: Nat, +k: Nat, +m: Nat, +par: Nat, a: TA.Mat, b: TA.Mat) -> TA.MMul: 572 mm_precise2(Nat.is_lt(1024n, k), n, k, m, par, a, b) 573 574 # ===================================================================================================================== 575 # Counter-based random fills (prototype). fill_uniform and fill_gauss make n numbers of a counter s in one bang, each 576 # number a pure function of (s, its index), so a lane computes its cells without reading anything; uniform_mat and 577 # gauss_mat give them as a tensor-array Mat. The _at forms (fill_uniform_at, fill_gauss_at, uniform_mat_at, gauss_mat_at) 578 # make the cells i0 .. i0 + n - 1 of the same counter, so a batch can be made in slices: their cell k is the cell 579 # function at i0 + k, and the Gaussians' pairs stay global (2p, 2p + 1), so an odd i0 starts with the sine half of the 580 # pair (i0 - 1, i0). The plain forms are the case i0 = 0. Indexes are U32 and wrap: i0 + k is taken mod 2^32. The 581 # functions behind them (rng_uniform, rng_gauss2, rng_gauss) are plain 582 # Bend that a caller's own leaf (runK) can call too, to make noise where it is used (x_t = a * x_0 + b * noise in one 583 # bang) instead of in an array of its own. 584 # 585 # The counter hash and the uniform are mlbend-generative 0.1.0.0's (its mix32 over s * 0x9E3779B9 + i + 0x1234567A, 586 # the low 24 bits plus one over 2^24, a uniform in (0, 1]), so rng_uniform gives that package's words. A Gaussian is 587 # Box-Muller over the pair (2p, 2p + 1): cell 2p is r cos(2 pi u_b), cell 2p + 1 is r sin(2 pi u_b), with 588 # r = sqrt(-2 ln u_a), u_a the uniform of index 2p and u_b of 2p + 1, so each pair costs one ln, one sqrt and one sine 589 # and cosine. ln, sin and cos are written here with + - * / and the exponent's bits, not F32.log/sin/cos: CUDA's float 590 # functions give other bits than the host's (bendlang/bend#1305), while + - * / and sqrt are 591 # IEEE's on every lane and Bend fuses no multiply-add, so these numbers have the same bits on the GPU, on the CPU 592 # threads and on the JS lane. They are within a few ulp of the libm values (tests/fill.bend measures it), not equal. 593 # ===================================================================================================================== 594 595 def rng_word(x: F32) -> U32: 596 match x: 597 case F32{w}: U32{w} 598 599 def rng_f32(x: U32) -> F32: 600 match x: 601 case U32{w}: F32{w} 602 603 # mlbend-generative's mix32: xor-shift 16, multiply, xor-shift 13, multiply, xor-shift 16, over U32 that wraps 604 def rng_mix(+z: U32) -> U32: 605 +a = U32.xor(z, U32.shrn(z, 16n)) 606 +b = (a * 2246822519 : U32) 607 +c = U32.xor(b, U32.shrn(b, 13n)) 608 +e = (c * 3266489917 : U32) 609 U32.xor(e, U32.shrn(e, 16n)) 610 611 # the hash of index i of counter s 612 def rng_hash(+s: U32, +i: U32) -> U32: 613 rng_mix(((s * 2654435769 : U32) + (i + 305419898 : U32) : U32)) 614 615 # the uniform of index i of counter s, in (0, 1]: the hash's low 24 bits plus one, over 2^24 (exact) 616 def rng_uniform(+s: U32, +i: U32) -> F32: 617 (U32.to_f32((U32.and(rng_hash(s, i), 16777215) + 1 : U32)) / 16777216.0 : F32) 618 619 # ln m for m in [sqrt(1/2), sqrt(2)], from s = (m - 1) / (m + 1): ln m = 2 (s + s^3/3 + s^5/5 + s^7/7 + s^9/9) 620 def rng_ln_m(+ex: F32, +m: F32) -> F32: 621 +s = ((m - 1.0 : F32) / (m + 1.0 : F32) : F32) 622 +z = (s * s : F32) 623 +c3 = (0.14285715 + (z * 0.11111111 : F32) : F32) 624 +c2 = (0.2 + (z * c3 : F32) : F32) 625 +p = (z * (0.33333334 + (z * c2 : F32) : F32) : F32) 626 +t = (s + s : F32) 627 ((ex * 0.693145751953125 : F32) + ((t + (t * p : F32) : F32) + (ex * 0.0000014286067653 : F32) : F32) : F32) 628 629 # ln x for a positive normal x: x = m * 2^ex with m in [sqrt(1/2), sqrt(2)) read from the bits (a mantissa above 630 # sqrt(2)'s, 0x3504F3, is halved and the exponent raised by one); ex * ln 2 in two parts, the first exact 631 def rng_ln_w(+w: U32) -> F32: 632 +big = U32.is_lt(3474675, U32.and(w, 8388607)) 633 +mb = U32.or(U32.and(w, 8388607), Bool.pick(U32, big, 1056964608, 1065353216)) 634 rng_ln_m((U32.to_f32(U32.shrn(w, 23n)) - Bool.pick(F32, big, 126.0, 127.0) : F32), rng_f32(mb)) 635 636 def rng_ln(x: F32) -> F32: 637 rng_ln_w(rng_word(x)) 638 639 # sin x and cos x for x in [0, pi/4], Taylor to x^9 and x^10 (the next terms are below 2^-28) 640 def rng_sin(+x: F32) -> F32: 641 +z = (x * x : F32) 642 +c4 = (0.0001984127 - (z * 0.0000027557319 : F32) : F32) 643 +c3 = (0.008333334 - (z * c4 : F32) : F32) 644 +c2 = (0.16666667 - (z * c3 : F32) : F32) 645 (x - ((x * z : F32) * c2 : F32) : F32) 646 647 def rng_cos(+x: F32) -> F32: 648 +z = (x * x : F32) 649 +c5 = (0.0000248015873 - (z * 0.00000027557319 : F32) : F32) 650 +c4 = (0.0013888889 - (z * c5 : F32) : F32) 651 +c3 = (0.041666668 - (z * c4 : F32) : F32) 652 +c2 = (0.5 - (z * c3 : F32) : F32) 653 (1.0 - (z * c2 : F32) : F32) 654 655 # the result of rng_sc: sin and cos of one angle 656 type SC is Data: 657 SC{s: F32, c: F32} 658 659 # sin and cos of a quarter turn's first eighth (x) or second (pi/2 - x, the two swapped), then turned by q quarters 660 def rng_sc_q(+q: U32, +sx: F32, +cx: F32) -> SC: 661 SC{Bool.pick(F32, U32.is_eq(q, 0), sx, Bool.pick(F32, U32.is_eq(q, 1), cx, Bool.pick(F32, U32.is_eq(q, 2), F32.neg(sx), F32.neg(cx)))), Bool.pick(F32, U32.is_eq(q, 0), cx, Bool.pick(F32, U32.is_eq(q, 1), F32.neg(sx), Bool.pick(F32, U32.is_eq(q, 2), F32.neg(cx), sx)))} 662 663 def rng_sc_x(+q: U32, +hi: Bool, +x: F32) -> SC: 664 +a = rng_sin(x) 665 +b = rng_cos(x) 666 rng_sc_q(q, Bool.pick(F32, hi, b, a), Bool.pick(F32, hi, a, b)) 667 668 # sin and cos of 2 pi v / 2^24 (v < 2^24): the quarter q is v's top 2 bits, w the rest; past the eighth (2^21) the angle 669 # is taken from the quarter's end (2^22 - w, exact), so the polynomial only sees [0, pi/4] 670 def rng_sc_w(+q: U32, +w: U32) -> SC: 671 +hi = U32.is_lt(2097152, w) 672 rng_sc_x(q, hi, (U32.to_f32(Bool.pick(U32, hi, (4194304 - w : U32), w)) * 0.00000037450702 : F32)) 673 674 def rng_sc(+v: U32) -> SC: 675 rng_sc_w(U32.shrn(v, 22n), U32.and(v, 4194303)) 676 677 def rng_pair(+r: F32, a: SC) -> F32 & F32: 678 match a: 679 case SC{+s, +c}: Tuple{(r * c : F32), (r * s : F32)} 680 681 # the Gaussians of cells 2p and 2p + 1 of counter s (Box-Muller). The angle's 24 bits are u_b's (its hash plus one, 682 # mod 2^24), so the angle is 2 pi u_b exactly before rounding 683 def rng_gauss2(+s: U32, +p: U32) -> F32 & F32: 684 +i = (p + p : U32) 685 rng_pair(F32.sqrt((0.0 - (2.0 * rng_ln(rng_uniform(s, i)) : F32) : F32)), rng_sc(U32.and((rng_hash(s, (i + 1 : U32)) + 1 : U32), 16777215))) 686 687 def rng_pick(+odd: Bool, g: F32 & F32) -> F32: 688 match g: 689 case Tuple{+a, +b}: Bool.pick(F32, odd, b, a) 690 691 # the Gaussian of cell i alone (fill_gauss's cell i) 692 def rng_gauss(+s: U32, +i: U32) -> F32: 693 rng_pick(U32.is_eq(U32.and(i, 1), 1), rng_gauss2(s, U32.shrn(i, 1n))) 694 695 # the counter, the first index and the number of cells, the same for every leaf 696 type FE is Data: 697 FE{s: U32, i0: U32, n: U32} 698 699 def fu_go(k: Nat, +i: U32, +l: U32, +s: U32, +i0: U32, o: Array) -> Array: 700 match k: 701 case 0n: o 702 case 1n+q: fu_go(q, (i + l : U32), l, s, i0, Array.set(F32, o, i, rng_uniform(s, (i0 + i : U32)))) 703 704 # leaf t writes the cells t, t + l, .. below n, cell k the uniform of index i0 + k 705 def fu_leaf(+t: U32, +l: U32, +e: FE, x1: Array, x2: Array, x3: Array, x4: Array, o: Array) -> Array: 706 match e: 707 case FE{+s, +i0, +n}: fu_go(share(t, n, l), t, l, s, i0, o) 708 709 # writes cell i when it is below n (the cell before the first of an odd i0, which wraps to 2^32 - 1, and the cell after 710 # the last are not written) 711 def fg_if(ok: Bool, +i: U32, +x: F32, o: Array) -> Array: 712 match ok: 713 case True{}: Array.set(F32, o, i, x) 714 case False{}: o 715 716 def fg_put(+i: U32, +n: U32, g: F32 & F32, o: Array) -> Array: 717 match g: 718 case Tuple{+a, +b}: fg_if(U32.is_lt((i + 1 : U32), n), (i + 1 : U32), b, fg_if(U32.is_lt(i, n), i, a, o)) 719 720 # pair j of the slice is the global pair q + j (q = i0 / 2), cells 2j - h and 2j - h + 1 of the output (h = i0 mod 2) 721 def fg_go(k: Nat, +j: U32, +l: U32, +s: U32, +q: U32, +h: U32, +n: U32, o: Array) -> Array: 722 match k: 723 case 0n: o 724 case 1n+r: fg_go(r, (j + l : U32), l, s, q, h, n, fg_put(((j + j : U32) - h : U32), n, rng_gauss2(s, (q + j : U32)), o)) 725 726 # the number of global pairs that cells i0 .. i0 + n - 1 touch: (i0 mod 2 + n + 1) / 2 727 def fg_pairs(+i0: U32, +n: U32) -> U32: 728 U32.shrn(((U32.and(i0, 1) + n : U32) + 1 : U32), 1n) 729 730 # leaf t writes the pairs t, t + l, .. of the slice's pairs, each into the cells of it that lie in the slice 731 def fg_leaf(+t: U32, +l: U32, +e: FE, x1: Array, x2: Array, x3: Array, x4: Array, o: Array) -> Array: 732 match e: 733 case FE{+s, +i0, +n}: fg_go(share(t, fg_pairs(i0, n), l), t, l, s, U32.shrn(i0, 1n), U32.and(i0, 1), n, o) 734 735 def r1_out(r: R1) -> Array: 736 match r: 737 case R1{x1, x2, x3, x4, a}: a 738 739 # n uniforms of counter s from index i0, cell k = rng_uniform(s, i0 + k) (mod 2^32), in a new array of 2^c cells made 740 # on the device (one bang) 741 @unsafe 742 def fill_uniform_at(+s: U32, +i0: U32, +n: Nat, +c: Nat) -> Array: 743 r1_out(run1_new(~FE, ~fu_leaf, depth(n), FE{s, i0, U32.from_nat(n)}, none(Unit{}), none(Unit{}), none(Unit{}), none(Unit{}), c)) 744 745 # n Gaussians of counter s from index i0, cell k = rng_gauss(s, i0 + k) (mod 2^32), in a new array of 2^c cells made 746 # on the device (one bang, a lane a global pair of cells; an odd i0 starts with the sine half of the pair (i0 - 1, i0)) 747 @unsafe 748 def fill_gauss_at(+s: U32, +i0: U32, +n: Nat, +c: Nat) -> Array: 749 r1_out(run1_new(~FE, ~fg_leaf, depth(U32.to_nat(fg_pairs(i0, U32.from_nat(n)))), FE{s, i0, U32.from_nat(n)}, none(Unit{}), none(Unit{}), none(Unit{}), none(Unit{}), c)) 750 751 # n uniforms of counter s, cell i = rng_uniform(s, i), in a new array of 2^c cells made on the device (one bang) 752 @unsafe 753 def fill_uniform(+s: U32, +n: Nat, +c: Nat) -> Array: 754 fill_uniform_at(s, 0, n, c) 755 756 # n Gaussians of counter s, cell i = rng_gauss(s, i), in a new array of 2^c cells made on the device (one bang, a lane a 757 # pair of cells) 758 @unsafe 759 def fill_gauss(+s: U32, +n: Nat, +c: Nat) -> Array: 760 fill_gauss_at(s, 0, n, c) 761 762 # an r x m Mat of uniforms of counter s, row-major (cell i * m + j = rng_uniform(s, i * m + j)) 763 @unsafe 764 def uniform_mat(+s: U32, +r: Nat, +m: Nat) -> TA.Mat: 765 TA.Mat{fill_uniform(s, Nat.mul(r, m), TA.cap_depth(Nat.mul(r, m)))} 766 767 # an r x m Mat of Gaussians of counter s, row-major (cell i * m + j = rng_gauss(s, i * m + j)) 768 @unsafe 769 def gauss_mat(+s: U32, +r: Nat, +m: Nat) -> TA.Mat: 770 TA.Mat{fill_gauss(s, Nat.mul(r, m), TA.cap_depth(Nat.mul(r, m)))} 771 772 # an r x m Mat of uniforms of counter s from index i0 (cell i * m + j = rng_uniform(s, i0 + i * m + j)); rows b .. b + r - 1 773 # of an R x m uniform_mat are uniform_mat_at(s, b * m, r, m) 774 @unsafe 775 def uniform_mat_at(+s: U32, +i0: U32, +r: Nat, +m: Nat) -> TA.Mat: 776 TA.Mat{fill_uniform_at(s, i0, Nat.mul(r, m), TA.cap_depth(Nat.mul(r, m)))} 777 778 # an r x m Mat of Gaussians of counter s from index i0 (cell i * m + j = rng_gauss(s, i0 + i * m + j)); rows 779 # b .. b + r - 1 of an R x m gauss_mat are gauss_mat_at(s, b * m, r, m) 780 @unsafe 781 def gauss_mat_at(+s: U32, +i0: U32, +r: Nat, +m: Nat) -> TA.Mat: 782 TA.Mat{fill_gauss_at(s, i0, Nat.mul(r, m), TA.cap_depth(Nat.mul(r, m)))}