1 # BendTorch's layers with their work on the GPU (bend-ml-tensor-gpu): the same layers, types and bits as 2 # torch/main.bend's and torch/fast.bend's. 3 # 4 # import ../torch/main.bend as T 5 # import ../torch/gpu.bend as F 6 # F.LinearFirstF(100n, 784n, 128n, 4n) # this layer's products and optimizer step run as bangs 7 # 8 # Every layer here runs its work as bangs (one `!` call each), which run on the GPU in a binary built with CUDA and on 9 # the CPU threads otherwise: 10 # - LinearF, LinearFirstF and LinearNBF: their three products (x·Wᵀ, dY·W, dYᵀ·X added into dW) from tensor-gpu, and 11 # their optimizer step. 12 # - SiLU, Mul2, Add2, Residual, Mat.dup and Mat.sum2: their element-wise loops. 13 # - RMSNorm, RoPE and Attention (main.bend's names; fast.bend's are main.bend's, on the CPU). 14 # - MSE, CrossEntropy and Embedding (its rows, its gradient and its step), so that a training step's work, the loss 15 # included, stays on the device; the host reads back a few numbers a row for the loss. 16 # The names are fast.bend's, so a program switches by its import line alone (the Makefile builds the GPU demos so). 17 # 18 # Each bang is one of tensor-gpu's run1..run4 (outputs in place) or run1_new..run4_new (outputs made on the device): 19 # a tree of leaves, every node sharing the arrays. The code of a leaf is below, in plain Bend, and computes each number 20 # with main.bend's own expression or row code (rmsf_row, ssq, rmsb_dot, rope_one, dot0, psmc, g_set2, adam_m..), so the bits 21 # are main.bend's, except where an exponential is taken on the device (SiLU, attention's softmax, cross-entropy): CUDA 22 # takes the float overload of exp there and its last bit can differ from the host's (the known 23 # limit of bendlang/bend#1305); 1 ulp is accepted there, my decision of 2026-10-18 (tests: demo/gpu_test.bend 24 # and demo/gpu_layers_test.bend). Sizes, depths and capacities are 25 # computed on the host and passed in. The arrays stay on the device from one bang to the next as long as the host only 26 # passes their handles; a bang that follows the host's parallel lets in one evaluation runs on the CPU threads. 27 # A sum whose order fixes the bits (a row's softmax, a row's squares) is a row bang: a leaf a row, a tail loop along 28 # it. Everything else is a cell bang: a leaf takes cells t, t + l, .., each one number (a dot product, or one 29 # expression of a row's sums), so a bang has as many independent pieces of work as the output has numbers. 30 # 31 # Outside the proofs, on purpose, like fast.bend: tensor-gpu shares arrays between the leaves of a bang through Base's 32 # @unsafe Array.fork, so `bend` names the defs below that reach it, and no file with laws imports this one (`make laws` 33 # checks both). This file writes no @unsafe of its own. All of it is Bend. 34 35 import Base 36 import bend-ml-tensor-array@0.1.14.0/main.bend as TA 37 import bend-ml-tensor-gpu@0.1.2.1/main.bend as TG 38 import bendtorch@1.2.0.0/main.bend as T 39 40 # ---- Linear: y = x·Wᵀ + b ---- 41 42 def lin_fwd(+n: Nat, +i: Nat, +o: Nat, p: T.Lin, x: TA.Mat) -> T.Fwd, TA.Mat, TA.Mat>: 43 match p: 44 case T.Lin{w, b}: 45 T.lin_f2(n, i, o, b, TG.matmul_nt(n, i, o, 0n, x, w)) 46 47 def lin_b2(+n: Nat, +i: Nat, +o: Nat, b: TA.Mat<1n, o>, x: TA.Mat, g: T.Lin, r: TA.MMul) -> T.Bwd, TA.Mat>: 48 match g r: 49 case T.Lin{gw, gb} TA.MMul{dy, w, dx}: 50 T.lin_b3(n, i, o, w, b, dx, gb, TG.matmul_tn_acc(o, n, i, 0n, dy, x, gw)) 51 52 def lin_bwd(+n: Nat, +i: Nat, +o: Nat, p: T.Lin, x: TA.Mat, dy: TA.Mat, g: T.Lin) -> T.Bwd, TA.Mat>: 53 match p: 54 case T.Lin{w, b}: 55 lin_b2(n, i, o, b, x, g, TG.matmul_precise(n, o, i, 0n, dy, w)) 56 57 # ===================================================================================================== 58 # The leaves' shared pieces. A leaf gets its index t among l leaves, a record of scalars and four input arrays, and 59 # returns its outputs; an input it does not use is tensor-gpu's none(Unit{}). Element-wise leaves take the cells t, 60 # t + l, .. (TG.share), so neighbouring lanes read neighbouring cells; row and column leaves take one row or column 61 # each and leave their outputs unchanged past the last one. 62 # ===================================================================================================== 63 64 type EN is Data: 65 EN{n: U32} 66 67 def nn(u: Unit) -> Array: 68 TG.none(u) 69 70 # out[i] = f(x[i]) for the leaf's cells 71 def ew1_loop(~f: @+a: F32 -> F32, k: Nat, +i: U32, +l: U32, o: Array, r: Array & F32) -> Array: 72 match k r: 73 case 0n Tuple{a, +x}: o 74 case 1n+q Tuple{a, +x}: ew1_loop(~f, q, (i + l : U32), l, Array.set(F32, o, i, f(x)), Array.get(F32, a, (i + l : U32))) 75 76 def ew1(~f: @+a: F32 -> F32, +t: U32, +l: U32, +e: EN, x: Array, o: Array) -> Array: 77 match e: 78 case EN{+n}: ew1_loop(~f, TG.share(t, n, l), t, l, o, Array.get(F32, x, t)) 79 80 # out[i] = f(x[i], y[i]) for the leaf's cells 81 def ew2_loop(~f: @+a: F32 -> @+b: F32 -> F32, k: Nat, +i: U32, +l: U32, o: Array, ra: Array & F32, rb: Array & F32) -> Array: 82 match k ra rb: 83 case 0n Tuple{a, +x} Tuple{b, +y}: o 84 case 1n+q Tuple{a, +x} Tuple{b, +y}: ew2_loop(~f, q, (i + l : U32), l, Array.set(F32, o, i, f(x, y)), Array.get(F32, a, (i + l : U32)), Array.get(F32, b, (i + l : U32))) 85 86 def ew2(~f: @+a: F32 -> @+b: F32 -> F32, +t: U32, +l: U32, +e: EN, x: Array, y: Array, o: Array) -> Array: 87 match e: 88 case EN{+n}: ew2_loop(~f, TG.share(t, n, l), t, l, o, Array.get(F32, x, t), Array.get(F32, y, t)) 89 90 # o[i] = f(o[i], y[i]) for the leaf's cells, in place (o's old value is the first argument) 91 def ewi_loop(~f: @+a: F32 -> @+b: F32 -> F32, k: Nat, +i: U32, +l: U32, ro: Array & F32, ry: Array & F32) -> Array: 92 match k ro ry: 93 case 0n Tuple{o, +x} Tuple{y, +v}: o 94 case 1n+q Tuple{o, +x} Tuple{y, +v}: ewi_loop(~f, q, (i + l : U32), l, Array.get(F32, Array.set(F32, o, i, f(x, v)), (i + l : U32)), Array.get(F32, y, (i + l : U32))) 95 96 def ewi(~f: @+a: F32 -> @+b: F32 -> F32, +t: U32, +l: U32, +e: EN, o: Array, y: Array) -> Array: 97 match e: 98 case EN{+n}: ewi_loop(~f, TG.share(t, n, l), t, l, Array.get(F32, o, t), Array.get(F32, y, t)) 99 100 def f32_id(+x: F32) -> F32: 101 x 102 103 def lf_copy(+t: U32, +l: U32, +e: EN, x1: Array, x2: Array, x3: Array, x4: Array, o: Array) -> Array: 104 ew1(~f32_id, t, l, e, x1, o) 105 106 def lf_silu(+t: U32, +l: U32, +e: EN, x1: Array, x2: Array, x3: Array, x4: Array, o: Array) -> Array: 107 ew1(~T.silu_v, t, l, e, x1, o) 108 109 # o = dy, written in place, x1 = x: main.bend's silu_d(dy, x) 110 def lf_silud(+t: U32, +l: U32, +e: EN, x1: Array, x2: Array, x3: Array, x4: Array, o: Array) -> Array: 111 ewi(~T.silu_d, t, l, e, o, x1) 112 113 # o = a, written in place, x1 = b: a + b 114 def lf_add(+t: U32, +l: U32, +e: EN, x1: Array, x2: Array, x3: Array, x4: Array, o: Array) -> Array: 115 ewi(~T.f32_add, t, l, e, o, x1) 116 117 def lf_mul(+t: U32, +l: U32, +e: EN, x1: Array, x2: Array, x3: Array, x4: Array, o: Array) -> Array: 118 ew2(~T.f32_mul, t, l, e, x1, x2, o) 119 120 # x1 = dy; the saved a and b, written in place: a becomes db = dy * a and b becomes da = dy * b, in one bang 121 def mulb_loop(k: Nat, +i: U32, +l: U32, rd: Array & F32, ra: Array & F32, rb: Array & F32) -> TG.O2: 122 match k rd ra rb: 123 case 0n Tuple{d, +x} Tuple{a, +y} Tuple{b, +z}: TG.O2{a, b} 124 case 1n+q Tuple{d, +x} Tuple{a, +y} Tuple{b, +z}: 125 +u = (i + l : U32) 126 mulb_loop(q, u, l, Array.get(F32, d, u), Array.get(F32, Array.set(F32, a, i, (x * y : F32)), u), Array.get(F32, Array.set(F32, b, i, (x * z : F32)), u)) 127 128 def lf_mulb(+t: U32, +l: U32, +e: EN, x1: Array, x2: Array, x3: Array, x4: Array, o: TG.O2) -> TG.O2: 129 match e o: 130 case EN{+n} TG.O2{a, b}: mulb_loop(TG.share(t, n, l), t, l, Array.get(F32, x1, t), Array.get(F32, a, t), Array.get(F32, b, t)) 131 132 def en(+cnt: Nat) -> EN: 133 EN{U32.from_nat(cnt)} 134 135 def r1_a(r: TG.R1) -> Array: 136 match r: 137 case TG.R1{x1, x2, x3, x4, a}: a 138 139 # ---- element-wise layers ---- 140 141 def dup_fin(-r: Nat, -c: Nat, q: TG.R1) -> T.Two, TA.Mat>: 142 match q: 143 case TG.R1{x1, x2, x3, x4, a}: T.Two{TA.Mat{x1}, TA.Mat{a}} 144 145 # two copies of a matrix, the second made on the device 146 def Mat.dup(+r: Nat, +c: Nat, m: TA.Mat) -> T.Two, TA.Mat>: 147 match m: 148 case TA.Mat{a}: dup_fin(r, c, TG.run1_new(~EN, ~lf_copy, TG.depth(Nat.mul(r, c)), en(Nat.mul(r, c)), a, nn(Unit{}), nn(Unit{}), nn(Unit{}), TA.cap_depth(Nat.mul(r, c)))) 149 150 # a + b, as main.bend's Mat.sum2 adds each cell, written over a (both are consumed) 151 def Mat.sum2(+r: Nat, +c: Nat, a: TA.Mat, b: TA.Mat) -> TA.Mat: 152 match a b: 153 case TA.Mat{x} TA.Mat{y}: TA.Mat{r1_a(TG.run1(~EN, ~lf_add, TG.depth(Nat.mul(r, c)), en(Nat.mul(r, c)), y, nn(Unit{}), nn(Unit{}), nn(Unit{}), x))} 154 155 def silu_f2(-n: Nat, -c: Nat, q: TG.R1) -> T.Fwd, TA.Mat>: 156 match q: 157 case TG.R1{x1, x2, x3, x4, a}: T.Fwd{Unit{}, TA.Mat{a}, TA.Mat{x1}} 158 159 def silu_f(+n: Nat, +c: Nat, x: TA.Mat) -> T.Fwd, TA.Mat>: 160 match x: 161 case TA.Mat{a}: silu_f2(n, c, TG.run1_new(~EN, ~lf_silu, TG.depth(Nat.mul(n, c)), en(Nat.mul(n, c)), a, nn(Unit{}), nn(Unit{}), nn(Unit{}), TA.cap_depth(Nat.mul(n, c)))) 162 163 def silu_b(+n: Nat, +c: Nat, s: TA.Mat, dy: TA.Mat) -> T.Bwd>: 164 match s dy: 165 case TA.Mat{x} TA.Mat{d}: T.Bwd{Unit{}, TA.Mat{r1_a(TG.run1(~EN, ~lf_silud, TG.depth(Nat.mul(n, c)), en(Nat.mul(n, c)), x, nn(Unit{}), nn(Unit{}), nn(Unit{}), d))}, Unit{}} 166 167 # SiLU: y = silu(x), the input saved; dx = silu'(x) * dy 168 def SiLU(+n: Nat, +c: Nat) -> T.Layer, TA.Mat, TA.Mat>: 169 T.Layer{p => x => silu_f(n, c, x), p => s => dy => g => silu_b(n, c, s, dy), op => p => g => st => T.no_step(Unit, op, p, g, st), u => Unit{}, p => T.PL{p, Nil{}}, p => xs => off => T.PA{p, xs, off}, p => xs => off => T.PA{p, xs, off}, u => 0} 170 171 def mul2_f2(-n: Nat, -c: Nat, q: TG.R1) -> T.Fwd, T.Two, TA.Mat>>: 172 match q: 173 case TG.R1{x1, x2, x3, x4, y}: T.Fwd{Unit{}, TA.Mat{y}, T.Two{TA.Mat{x1}, TA.Mat{x2}}} 174 175 def mul2_f(+n: Nat, +c: Nat, x: T.Two, TA.Mat>) -> T.Fwd, T.Two, TA.Mat>>: 176 match x: 177 case T.Two{TA.Mat{a}, TA.Mat{b}}: mul2_f2(n, c, TG.run1_new(~EN, ~lf_mul, TG.depth(Nat.mul(n, c)), en(Nat.mul(n, c)), a, b, nn(Unit{}), nn(Unit{}), TA.cap_depth(Nat.mul(n, c)))) 178 179 def mul2_b2(-n: Nat, -c: Nat, q: TG.R2) -> T.Bwd, TA.Mat>>: 180 match q: 181 case TG.R2{x1, x2, x3, x4, TG.O2{db, da}}: T.Bwd{Unit{}, T.Two{TA.Mat{da}, TA.Mat{db}}, Unit{}} 182 183 def mul2_b(+n: Nat, +c: Nat, s: T.Two, TA.Mat>, dy: TA.Mat) -> T.Bwd, TA.Mat>>: 184 match s dy: 185 case T.Two{TA.Mat{a}, TA.Mat{b}} TA.Mat{d}: mul2_b2(n, c, TG.run2(~EN, ~lf_mulb, TG.depth(Nat.mul(n, c)), en(Nat.mul(n, c)), d, nn(Unit{}), nn(Unit{}), nn(Unit{}), a, b)) 186 187 # Mul2: y = a * b, the inputs saved; da = dy * b, db = dy * a 188 def Mul2(+n: Nat, +c: Nat) -> T.Layer, TA.Mat>, TA.Mat, T.Two, TA.Mat>>: 189 T.Layer{p => x => mul2_f(n, c, x), p => s => dy => g => mul2_b(n, c, s, dy), op => p => g => st => T.no_step(Unit, op, p, g, st), u => Unit{}, p => T.PL{p, Nil{}}, p => xs => off => T.PA{p, xs, off}, p => xs => off => T.PA{p, xs, off}, u => 0} 190 191 def add2_f(+n: Nat, +c: Nat, x: T.Two, TA.Mat>) -> T.Fwd, Unit>: 192 match x: 193 case T.Two{a, b}: T.Fwd{Unit{}, Mat.sum2(n, c, a, b), Unit{}} 194 195 def Add2(+n: Nat, +c: Nat) -> T.Layer, TA.Mat>, TA.Mat, Unit>: 196 T.Layer{p => x => add2_f(n, c, x), p => s => dy => g => T.Bwd{p, Mat.dup(n, c, dy), g}, op => p => g => st => T.no_step(Unit, op, p, g, st), u => Unit{}, p => T.PL{p, Nil{}}, p => xs => off => T.PA{p, xs, off}, p => xs => off => T.PA{p, xs, off}, u => 0} 197 198 # T.Residual with these copies and sums 199 def Residual(~n: Nat, ~c: Nat, ~P: Type, ~S: Type, ~f: T.Layer, TA.Mat, S>) -> T.Layer, Unit>, TA.Mat, TA.Mat, T.Two, Unit>>: 200 T.Seq(~T.Two, ~Unit, ~TA.Mat, ~T.Two, TA.Mat>, ~TA.Mat, ~T.Two, ~Unit, ~T.Fork(~Unit, ~P, ~TA.Mat, ~TA.Mat, ~TA.Mat, ~Unit, ~S, ~(m => Mat.dup(n, c, m)), ~(a => b => Mat.sum2(n, c, a, b)), ~T.Id(~TA.Mat), ~f), ~Add2(n, c)) 201 202 # ---- the optimizer step in place, each cell as main.bend's sgd_l, mom_l and adam_l compute it ---- 203 204 type ESGD is Data: 205 ESGD{n: U32, lr: F32} 206 207 def sgd_loop(k: Nat, +i: U32, +l: U32, +lr: F32, rw: Array & F32, rg: Array & F32) -> TG.O2: 208 match k rw rg: 209 case 0n Tuple{w, +x} Tuple{g, +y}: TG.O2{w, g} 210 case 1n+q Tuple{w, +x} Tuple{g, +y}: 211 sgd_loop(q, (i + l : U32), l, lr, Array.get(F32, Array.set(F32, w, i, (x + (F32.neg(lr) * y : F32) : F32)), (i + l : U32)), Array.get(F32, Array.set(F32, g, i, 0.0), (i + l : U32))) 212 213 def lf_sgd(+t: U32, +l: U32, +e: ESGD, x1: Array, x2: Array, x3: Array, x4: Array, o: TG.O2) -> TG.O2: 214 match e o: 215 case ESGD{+n, +lr} TG.O2{w, g}: sgd_loop(TG.share(t, n, l), t, l, lr, Array.get(F32, w, t), Array.get(F32, g, t)) 216 217 type EMOM is Data: 218 EMOM{n: U32, lr: F32, mu: F32} 219 220 def mom_loop(k: Nat, +i: U32, +l: U32, +lr: F32, +mu: F32, rw: Array & F32, rg: Array & F32, rm: Array & F32) -> TG.O3: 221 match k rw rg rm: 222 case 0n Tuple{w, +x} Tuple{g, +y} Tuple{m, +b}: TG.O3{w, g, m} 223 case 1n+q Tuple{w, +x} Tuple{g, +y} Tuple{m, +b}: 224 +nb = ((b * mu : F32) + y : F32) 225 +u = (i + l : U32) 226 mom_loop(q, u, l, lr, mu, Array.get(F32, Array.set(F32, w, i, (x + (F32.neg(lr) * nb : F32) : F32)), u), Array.get(F32, Array.set(F32, g, i, 0.0), u), Array.get(F32, Array.set(F32, m, i, nb), u)) 227 228 def lf_mom(+t: U32, +l: U32, +e: EMOM, x1: Array, x2: Array, x3: Array, x4: Array, o: TG.O3) -> TG.O3: 229 match e o: 230 case EMOM{+n, +lr, +mu} TG.O3{w, g, m}: mom_loop(TG.share(t, n, l), t, l, lr, mu, Array.get(F32, w, t), Array.get(F32, g, t), Array.get(F32, m, t)) 231 232 type EADAM is Data: 233 EADAM{n: U32, lr: F32, c1: F32, b2: F32, c2: F32, eps: F32, wd: F32, dec: Bool, step: F32, bc2: F32} 234 235 def adam_loop(k: Nat, +i: U32, +l: U32, +lr: F32, +c1: F32, +b2: F32, +c2: F32, +eps: F32, +wd: F32, +dec: Bool, +step: F32, +bc2: F32, rw: Array & F32, rg: Array & F32, rm: Array & F32, rv: Array & F32) -> TG.O4: 236 match k rw rg rm rv: 237 case 0n Tuple{w, +x} Tuple{g, +y} Tuple{m, +a} Tuple{v, +b}: TG.O4{w, g, m, v} 238 case 1n+q Tuple{w, +x} Tuple{g, +y} Tuple{m, +a} Tuple{v, +b}: 239 +gg = T.adam_grad(dec, wd, x, y) 240 +nm = T.adam_m(c1, a, gg) 241 +nv = T.adam_v(b2, c2, b, gg) 242 +nw = T.adam_w(step, bc2, eps, T.adam_decay(dec, lr, wd, x), nm, nv) 243 +u = (i + l : U32) 244 adam_loop(q, u, l, lr, c1, b2, c2, eps, wd, dec, step, bc2, Array.get(F32, Array.set(F32, w, i, nw), u), Array.get(F32, Array.set(F32, g, i, 0.0), u), Array.get(F32, Array.set(F32, m, i, nm), u), Array.get(F32, Array.set(F32, v, i, nv), u)) 245 246 def lf_adam(+t: U32, +l: U32, +e: EADAM, x1: Array, x2: Array, x3: Array, x4: Array, o: TG.O4) -> TG.O4: 247 match e o: 248 case EADAM{+n, +lr, +c1, +b2, +c2, +eps, +wd, +dec, +step, +bc2} TG.O4{w, g, m, v}: adam_loop(TG.share(t, n, l), t, l, lr, c1, b2, c2, eps, wd, dec, step, bc2, Array.get(F32, w, t), Array.get(F32, g, t), Array.get(F32, m, t), Array.get(F32, v, t)) 249 250 def sgd_fin(-r: Nat, -c: Nat, q: TG.R2) -> T.MatStep: 251 match q: 252 case TG.R2{x1, x2, x3, x4, TG.O2{w, g}}: T.MatStep{TA.Mat{w}, TA.Mat{g}, T.NoMom{}} 253 254 def mom_fin(-r: Nat, -c: Nat, q: TG.R3) -> T.MatStep: 255 match q: 256 case TG.R3{x1, x2, x3, x4, TG.O3{w, g, m}}: T.MatStep{TA.Mat{w}, TA.Mat{g}, T.OneMom{TA.Mat{m}}} 257 258 def opt_fin(-r: Nat, -c: Nat, q: TG.R4) -> T.MatStep: 259 match q: 260 case TG.R4{x1, x2, x3, x4, TG.O4{w, g, m, v}}: T.MatStep{TA.Mat{w}, TA.Mat{g}, T.TwoMom{TA.Mat{m}, TA.Mat{v}}} 261 262 # one matrix's step, in place: SGD writes two arrays, momentum three, Adam four 263 def opt_mat(+r: Nat, +c: Nat, +o: T.Opt, w: TA.Mat, g: TA.Mat, st: T.OptState>) -> T.MatStep: 264 match o w g st: 265 case T.SGD{+lr} TA.Mat{wa} TA.Mat{ga} _: 266 sgd_fin(r, c, TG.run2(~ESGD, ~lf_sgd, TG.depth(Nat.mul(r, c)), ESGD{U32.from_nat(Nat.mul(r, c)), lr}, nn(Unit{}), nn(Unit{}), nn(Unit{}), nn(Unit{}), wa, ga)) 267 case T.Momentum{+lr, +mu} TA.Mat{wa} TA.Mat{ga} T.OneMom{TA.Mat{ma}}: 268 mom_fin(r, c, TG.run3(~EMOM, ~lf_mom, TG.depth(Nat.mul(r, c)), EMOM{U32.from_nat(Nat.mul(r, c)), lr, mu}, nn(Unit{}), nn(Unit{}), nn(Unit{}), nn(Unit{}), wa, ga, ma)) 269 case T.Adam{+lr, +b1, +b2, +eps, +wd, +dec, +step, +bc2, +c1, +c2} TA.Mat{wa} TA.Mat{ga} T.TwoMom{TA.Mat{ma}, TA.Mat{va}}: 270 opt_fin(r, c, TG.run4(~EADAM, ~lf_adam, TG.depth(Nat.mul(r, c)), EADAM{U32.from_nat(Nat.mul(r, c)), lr, c1, b2, c2, eps, wd, dec, step, bc2}, nn(Unit{}), nn(Unit{}), nn(Unit{}), nn(Unit{}), wa, ga, ma, va)) 271 # packed BF16 moments (CAdamWBF16): main.bend's step, in one block 272 case o w g T.Packed{t}: T.Opt.mat(r, c, o, w, g, T.Packed{t}) 273 case _ w g st: T.MatStep{w, g, st} 274 275 def mat_step(+r: Nat, +c: Nat, +op: T.Opt, p: TA.Mat, g: TA.Mat, st: T.OptState>) -> T.Step>: 276 T.mat_step_fin(r, c, opt_mat(r, c, op, p, g, st)) 277 278 def lin_step2(+i: Nat, +o: Nat, +op: T.Opt, p: T.Lin, g: T.Lin, st: T.Two>, T.OptState>>) -> T.Step>: 279 match p g st: 280 case T.Lin{w, b} T.Lin{gw, gb} T.Two{sw, sb}: 281 T.lin_s2(i, o, opt_mat(o, i, op, w, gw, sw), opt_mat(1n, o, op, b, gb, sb)) 282 283 def lin_step(+i: Nat, +o: Nat, +op: T.Opt, p: T.Lin, g: T.Lin, st: T.OptState>) -> T.Step>: 284 lin_step2(i, o, op, p, g, T.OptState.split(TA.Mat, TA.Mat<1n, o>, T.OptState.map(T.Lin, T.Two, TA.Mat<1n, o>>, st, x => T.lin_pair(i, o, x), x => T.lin_pair(i, o, x)))) 285 286 # ---- the linear layers ---- 287 288 def LinearF(+n: Nat, +i: Nat, +o: Nat, +par: Nat) -> T.Layer, TA.Mat, TA.Mat, TA.Mat>: 289 T.Layer{p => x => lin_fwd(n, i, o, p, x), p => s => dy => g => lin_bwd(n, i, o, p, s, dy, g), op => p => g => st => lin_step(i, o, op, p, g, st), u => T.lin_zeros(i, o), p => T.lin_dump(i, o, p), p => xs => off => T.lin_load(i, o, p, xs, off), p => xs => off => T.lin_store(i, o, p, xs, off), u => U32.from_nat(Nat.mul(o, 1n+i))} 290 291 # LinearFirst: no dX (the first layer's input needs no gradient) 292 def linf_bwd(+n: Nat, +i: Nat, +o: Nat, p: T.Lin, x: TA.Mat, dy: TA.Mat, g: T.Lin) -> T.Bwd, TA.Mat>: 293 match p g: 294 case T.Lin{w, b} T.Lin{gw, gb}: 295 T.linf_b3(n, i, o, w, b, gb, TG.matmul_tn_acc(o, n, i, 0n, dy, x, gw)) 296 297 def LinearFirstF(+n: Nat, +i: Nat, +o: Nat, +par: Nat) -> T.Layer, TA.Mat, TA.Mat, TA.Mat>: 298 T.Layer{p => x => lin_fwd(n, i, o, p, x), p => s => dy => g => linf_bwd(n, i, o, p, s, dy, g), op => p => g => st => lin_step(i, o, op, p, g, st), u => T.lin_zeros(i, o), p => T.lin_dump(i, o, p), p => xs => off => T.lin_load(i, o, p, xs, off), p => xs => off => T.lin_store(i, o, p, xs, off), u => U32.from_nat(Nat.mul(o, 1n+i))} 299 300 # LinearNB: y = x·Wᵀ, no bias (Llama's projections) 301 def lnb_b2(+n: Nat, +i: Nat, +o: Nat, x: TA.Mat, g: TA.Mat, r: TA.MMul) -> T.Bwd, TA.Mat>: 302 match r: 303 case TA.MMul{dy, w, dx}: 304 T.lnb_b3(n, i, o, w, dx, TG.matmul_tn_acc(o, n, i, 0n, dy, x, g)) 305 306 def LinearNBF(+n: Nat, +i: Nat, +o: Nat, +par: Nat) -> T.Layer, TA.Mat, TA.Mat, TA.Mat>: 307 T.Layer{w => x => T.lnb_f2(n, i, o, TG.matmul_nt(n, i, o, 0n, x, w)), w => s => dy => g => lnb_b2(n, i, o, s, g, TG.matmul_precise(n, o, i, 0n, dy, w)), op => p => g => st => mat_step(o, i, op, p, g, st), u => TA.Mat.zeros(o, i), p => T.Mat.dump(o, i, p), p => xs => off => T.Mat.restore(o, i, p, xs, off), p => xs => off => T.Mat.store(o, i, p, xs, off), u => U32.from_nat(Nat.mul(o, i))} 308 309 # ---- RMSNorm: a leaf a row with main.bend's own row code (rmsf_row) ---- 310 311 type ERF is Data: 312 ERF{n: U32, d: Nat, d32: U32, df: F32, eps: F32} 313 314 def r3_y(s: T.R3) -> Array: 315 match s: 316 case T.R3{x, g, y}: y 317 318 def rmsf_if(ok: Bool, +t: U32, +d: Nat, +d32: U32, +df: F32, +eps: F32, x: Array, g: Array, y: Array) -> Array: 319 match ok: 320 case True{}: r3_y(T.rmsf_row(d, (t * d32 : U32), df, eps, T.R3{x, g, y})) 321 case False{}: y 322 323 # x1 = x, x2 = the gain g; y a row a leaf 324 def lf_rmsf(+t: U32, +l: U32, +e: ERF, x1: Array, x2: Array, x3: Array, x4: Array, y: Array) -> Array: 325 match e: 326 case ERF{+n, +d, +d32, +df, +eps}: rmsf_if(U32.is_lt(t, n), t, d, d32, df, eps, x1, x2, y) 327 328 def erf(+n: Nat, +d: Nat, +eps: F32) -> ERF: 329 ERF{U32.from_nat(n), d, U32.from_nat(d), F32.from_nat(d), eps} 330 331 def rms_f2(-n: Nat, -d: Nat, q: TG.R1) -> T.Fwd, TA.Mat, TA.Mat>: 332 match q: 333 case TG.R1{x1, x2, x3, x4, y}: T.Fwd{TA.Mat{x2}, TA.Mat{y}, TA.Mat{x1}} 334 335 def rms_f(+n: Nat, +d: Nat, +eps: F32, p: TA.Mat<1n, d>, x: TA.Mat) -> T.Fwd, TA.Mat, TA.Mat>: 336 match p x: 337 case TA.Mat{g} TA.Mat{a}: rms_f2(n, d, TG.run1_new(~ERF, ~lf_rmsf, TA.cap_depth(n), erf(n, d, eps), a, g, nn(Unit{}), nn(Unit{}), TA.cap_depth(Nat.mul(n, d)))) 338 339 # the backward pass in two bangs: a leaf a row writes dx and the row's 1/rms (main.bend's rmsb_row without dg), then a 340 # leaf a column sums dg down the rows in order from +0.0 and adds it to the gradient it is given, as main.bend adds each 341 # row's term into dg 342 343 def gdx_w(k: Nat, +j: U32, +off: U32, +r: F32, +m: F32, dx: Array, rx: Array & F32, rd: Array & F32, rg: Array & F32) -> Array: 344 match k rx rd rg: 345 case 0n Tuple{x, +a} Tuple{dy, +b} Tuple{g, +c}: 346 +xh = (a * r : F32) 347 Array.set(F32, dx, (off + j : U32), (((b * c : F32) - (xh * m : F32) : F32) * r : F32)) 348 case 1n+p Tuple{x, +a} Tuple{dy, +b} Tuple{g, +c}: 349 +xh = (a * r : F32) 350 gdx_w(p, (j + 1 : U32), off, r, m, Array.set(F32, dx, (off + j : U32), (((b * c : F32) - (xh * m : F32) : F32) * r : F32)), Array.get(F32, x, ((off + j : U32) + 1 : U32)), Array.get(F32, dy, ((off + j : U32) + 1 : U32)), Array.get(F32, g, (j + 1 : U32))) 351 352 def gdx_3(+d: Nat, +off: U32, +df: F32, +r: F32, +t: U32, dx: Array, rr: Array, q: T.R3S) -> TG.O2: 353 match q: 354 case T.R3S{x, dy, g, +s}: 355 TG.O2{gdx_w(Nat.sub(d, 1n), 0, off, r, (s / df : F32), dx, Array.get(F32, x, off), Array.get(F32, dy, off), Array.get(F32, g, 0)), Array.set(F32, rr, t, r)} 356 357 def gdx_2(+d: Nat, +off: U32, +df: F32, +eps: F32, +t: U32, dy: Array, g: Array, dx: Array, rr: Array, q: T.AS) -> TG.O2: 358 match q: 359 case T.AS{x, +s}: 360 +r = T.rms_inv(s, df, eps) 361 gdx_3(d, off, df, r, t, dx, rr, T.rmsb_dot(Nat.sub(d, 1n), 0, off, r, 0.0, Array.get(F32, x, off), Array.get(F32, dy, off), Array.get(F32, g, 0))) 362 363 def gdx_if(ok: Bool, +t: U32, +d: Nat, +d32: U32, +df: F32, +eps: F32, x: Array, dy: Array, g: Array, dx: Array, rr: Array) -> TG.O2: 364 match ok: 365 case True{}: gdx_2(d, (t * d32 : U32), df, eps, t, dy, g, dx, rr, T.ssq(Nat.sub(d, 1n), (t * d32 : U32), 0.0, Array.get(F32, x, (t * d32 : U32)))) 366 case False{}: TG.O2{dx, rr} 367 368 # x1 = x, x2 = dy, x3 = g; outputs dx and 1/rms of each row 369 def lf_gdx(+t: U32, +l: U32, +e: ERF, x1: Array, x2: Array, x3: Array, x4: Array, o: TG.O2) -> TG.O2: 370 match e o: 371 case ERF{+n, +d, +d32, +df, +eps} TG.O2{dx, rr}: gdx_if(U32.is_lt(t, n), t, d, d32, df, eps, x1, x2, x3, dx, rr) 372 373 # acc + dy[i]·(x[i]·r[row]) down a column, i from j by d 374 def gdg_loop(k: Nat, +i: U32, +d: U32, +row: U32, acc: F32, rx: Array & F32, rd: Array & F32, rr: Array & F32) -> F32: 375 match k rx rd rr: 376 case 0n Tuple{x, +a} Tuple{dy, +b} Tuple{r, +c}: (acc + (b * (a * c : F32) : F32) : F32) 377 case 1n+p Tuple{x, +a} Tuple{dy, +b} Tuple{r, +c}: 378 gdg_loop(p, (i + d : U32), d, (row + 1 : U32), (acc + (b * (a * c : F32) : F32) : F32), Array.get(F32, x, (i + d : U32)), Array.get(F32, dy, (i + d : U32)), Array.get(F32, r, (row + 1 : U32))) 379 380 def gdg_put(+j: U32, +s: F32, r: Array & F32) -> Array: 381 match r: 382 case Tuple{ga, +v}: Array.set(F32, ga, j, (v + s : F32)) 383 384 type EDG is Data: 385 EDG{d: U32, n: Nat} 386 387 def gdg_if(ok: Bool, +j: U32, +n: Nat, +d: U32, x: Array, dy: Array, rr: Array, ga: Array) -> Array: 388 match ok: 389 case True{}: gdg_put(j, gdg_loop(Nat.sub(n, 1n), j, d, 0, 0.0, Array.get(F32, x, j), Array.get(F32, dy, j), Array.get(F32, rr, 0)), Array.get(F32, ga, j)) 390 case False{}: ga 391 392 # x1 = x, x2 = dy, x3 = 1/rms; the gradient of the gain, in place: a leaf a column 393 def lf_gdg(+t: U32, +l: U32, +e: EDG, x1: Array, x2: Array, x3: Array, x4: Array, ga: Array) -> Array: 394 match e: 395 case EDG{+d, +n}: gdg_if(U32.is_lt(t, d), t, n, d, x1, x2, x3, ga) 396 397 def rms_b3(-n: Nat, -d: Nat, g: Array, dx: Array, q: TG.R1) -> T.Bwd, TA.Mat>: 398 match q: 399 case TG.R1{x1, x2, x3, x4, ga}: T.Bwd{TA.Mat{g}, TA.Mat{dx}, TA.Mat{ga}} 400 401 def rms_b2(+n: Nat, +d: Nat, ga: Array, q: TG.R2) -> T.Bwd, TA.Mat>: 402 match q: 403 case TG.R2{x, dy, g, x4, TG.O2{dx, rr}}: rms_b3(n, d, g, dx, TG.run1(~EDG, ~lf_gdg, TA.cap_depth(d), EDG{U32.from_nat(d), n}, x, dy, rr, x4, ga)) 404 405 def rms_b(+n: Nat, +d: Nat, +eps: F32, p: TA.Mat<1n, d>, x: TA.Mat, dy: TA.Mat, gacc: TA.Mat<1n, d>) -> T.Bwd, TA.Mat>: 406 match p x dy gacc: 407 case TA.Mat{g} TA.Mat{xa} TA.Mat{da} TA.Mat{ga}: rms_b2(n, d, ga, TG.run2_new(~ERF, ~lf_gdx, TA.cap_depth(n), erf(n, d, eps), xa, da, g, nn(Unit{}), TA.cap_depth(Nat.mul(n, d)), TA.cap_depth(n))) 408 409 # RMSNorm over the last dimension, its gain trained (main.bend's RMSNorm) 410 def RMSNorm(+n: Nat, +d: Nat, +eps: F32) -> T.Layer, TA.Mat, TA.Mat, TA.Mat>: 411 T.Layer{p => x => rms_f(n, d, eps, p, x), p => s => dy => g => rms_b(n, d, eps, p, s, dy, g), op => p => g => st => mat_step(1n, d, op, p, g, st), u => TA.Mat.zeros(1n, d), p => T.Mat.dump(1n, d, p), p => xs => off => T.Mat.restore(1n, d, p, xs, off), p => xs => off => T.Mat.store(1n, d, p, xs, off), u => U32.from_nat(d)} 412 413 # ---- RoPE: the tables on the host (main.bend's rope_tab, so the host's cos and sin), each pair rotated in place by 414 # main.bend's rope_one, the pairs dealt to the leaves round-robin ---- 415 416 type ERO is Data: 417 ERO{n: U32, fwd: Bool, hh: U32, half: U32, w: U32, t: U32} 418 419 def rope_loop(k: Nat, +e: U32, +l: U32, +fwd: Bool, +hh: U32, +half: U32, +w: U32, +t: U32, st: T.RS3) -> T.RS3: 420 match k: 421 case 0n: st 422 case 1n+q: rope_loop(q, (e + l : U32), l, fwd, hh, half, w, t, T.rope_one(e, fwd, hh, half, w, t, st)) 423 424 def rs3_x(st: T.RS3) -> Array: 425 match st: 426 case T.RS3{x, c, s}: x 427 428 # x1 = cos, x2 = sin; x rotated in place 429 def lf_rope(+t: U32, +l: U32, +e: ERO, x1: Array, x2: Array, x3: Array, x4: Array, x: Array) -> Array: 430 match e: 431 case ERO{+n, +fwd, +hh, +half, +w, +tt}: rs3_x(rope_loop(TG.share(t, n, l), t, l, fwd, hh, half, w, tt, T.RS3{x, x1, x2})) 432 433 def rope_go2(+n: Nat, +w: Nat, +fwd: Bool, +t: Nat, +h: Nat, +half: Nat, x: Array, tb: T.CS) -> TA.Mat: 434 match tb: 435 case T.CS{cs, sn}: TA.Mat{r1_a(TG.run1(~ERO, ~lf_rope, TG.depth(Nat.mul(n, Nat.mul(h, half))), ERO{U32.from_nat(Nat.mul(n, Nat.mul(h, half))), fwd, U32.from_nat(Nat.mul(h, half)), U32.from_nat(half), U32.from_nat(w), U32.from_nat(t)}, cs, sn, nn(Unit{}), nn(Unit{}), x))} 436 437 def rope_go(+n: Nat, +w: Nat, +fwd: Bool, +t: Nat, +h: Nat, +half: Nat, +base: F32, m: TA.Mat) -> TA.Mat: 438 match m: 439 case TA.Mat{x}: rope_go2(n, w, fwd, t, h, half, x, T.rope_tab(Nat.sub(Nat.mul(t, half), 1n), 0, U32.from_nat(half), base, T.CS{T.zeros_a(Nat.mul(t, half)), T.zeros_a(Nat.mul(t, half))})) 440 441 # RoPE on n = b·t rows of h heads of 2·half (main.bend's RoPE) 442 def RoPE(+n: Nat, +w: Nat, +t: Nat, +h: Nat, +half: Nat, +base: F32, -e: {w == Nat.mul(h, Nat.add(half, half)) : Nat}) -> T.Layer, TA.Mat, Unit>: 443 T.Layer{p => x => T.Fwd{p, rope_go(n, w, True{}, t, h, half, base, x), Unit{}}, p => s => dy => g => T.Bwd{p, rope_go(n, w, False{}, t, h, half, base, dy), g}, op => p => g => st => T.no_step(Unit, op, p, g, st), u => Unit{}, p => T.PL{p, Nil{}}, p => xs => off => T.PA{p, xs, off}, p => xs => off => T.PA{p, xs, off}, u => 0} 444 445 type S4 is Type: 446 S4{a: Array, b: Array, c: Array, d: Array} 447 448 # ---- attention in cell bangs and row bangs. A leaf handles cells or rows t, t + l, .. of one bang. A cell is one number 449 # of an output: a score, a dP, a number of o, dQ, dK or dV, each one dot product with main.bend's own code (dot0, or 450 # tdot, dot0's sum down a column of the packed triangle below; tsum_h adds the group's heads as main.bend's pacc). A 451 # row is one row of P (and of G), whose softmax and its backward pass are sums along the row in order: a leaf a row with 452 # main.bend's row code (psmc, g_set2), a tail loop over the row's i + 1 positions. Every number is computed by 453 # main.bend's expression in main.bend's order, so the bits are main.bend's (but CUDA's exp in the softmax). 454 # forward: 0. cells: kᵀ 1. cells: P's scores (q·k scaled, j <= i) 2. rows: the softmax over the i + 1 455 # 3. cells: o = P·V over the query's i + 1 positions 456 # backward: 0. cells: kᵀ and vᵀ 1. cells: P's scores again and dP = dO·V into G 2. rows: P's softmax, then 457 # G = (P * (dP - s)) * sc with s = dP·P over the row, as gr3 3. cells: dQ = G·K, dK = Σ Gᵀ·Q and 458 # dV = Σ Pᵀ·dO over the group's heads, dK written over kᵀ and dV over V (neither read any more) 459 # The old form (a leaf per row doing the scores, the softmax and the row's hd outputs, ~8,000 steps in a row at 460 # SmolLM2's t = 128, and dK and dV a leaf per key position) took 56 ms a layer's forward and backward at SmolLM2-135M's 461 # sizes on the RTX 4050; this one takes 17 ms, about half of it filling the new arrays (one lane, Array.new). ---- 462 463 # the sizes; n is how many cells (or rows) the bang has, nw and nk the cells of dQ and of dK, tr = t(t + 1)/2 the cells 464 # of a head's P 465 type EAC is Data: 466 EAC{n: U32, t: Nat, t32: U32, h: U32, g: U32, gn: Nat, hd: Nat, hd32: U32, w: U32, kvw: U32, sc: F32, nw: U32, nk: U32, tr: U32} 467 468 # where row i of head bh starts in P and G, which keep only the lower triangle (j <= i), row after row: main.bend's P 469 # and G are t·t with zeros above the diagonal, and a product by one of those zeros adds +0.0 or -0.0 to a sum that 470 # started at +0.0, which leaves it as it is; so the sums below skip them and give the same bits (for finite numbers: 471 # an infinity or a NaN in a masked key's row would reach main.bend's sums as 0·inf = NaN, and does not reach these) 472 def toff(+bh: U32, +i: U32, +tr: U32) -> U32: 473 ((bh * tr : U32) + U32.div((i * (i + 1 : U32) : U32), 2) : U32) 474 475 def eac_n(+c: EAC) -> U32: 476 match c: 477 case EAC{+n, +t, +t32, +h, +g, +gn, +hd, +hd32, +w, +kvw, +sc, +nw, +nk, +tr}: n 478 479 type S3 is Type: 480 S3{a: Array, b: Array, c: Array} 481 482 def s3_c(s: S3) -> Array: 483 match s: 484 case S3{a, b, c}: c 485 486 def sc_fin(y: Array, s: S3) -> TG.O2: 487 match s: 488 case S3{q, k, p}: TG.O2{p, y} 489 490 # the cell e = row·w + col of o (a = P, b = V) or of dQ (a = G, b = K): row i of P (or G) for the query's head, its 491 # i + 1 positions, · V's (or K's) column of the head's key/value head 492 def pv_val(+e: U32, +c: EAC, a: Array, b: Array) -> T.D2: 493 match c: 494 case EAC{+n, +t, +t32, +h, +g, +gn, +hd, +hd32, +w, +kvw, +sc, +nw, +nk, +tr}: 495 +row = U32.div(e, w) 496 +col = U32.mod(e, w) 497 +bi = U32.div(row, t32) 498 +hi = U32.div(col, hd32) 499 +i = U32.mod(row, t32) 500 T.dot0(U32.to_nat((i + 1 : U32)), toff(((bi * h : U32) + hi : U32), i, tr), ((((bi * t32 : U32) * kvw : U32) + (U32.div(hi, g) * hd32 : U32) : U32) + U32.mod(col, hd32) : U32), 1, kvw, a, b) 501 502 # acc + a[ia]·b[ib] + .. down column j of a head's packed triangle (row i + 1's cell is i + 1 after row i's) and b's 503 # rows (sb apart), from row i on: dot0's sum, but with the step in a growing by one a row 504 def tdot(k: Nat, +ia: U32, +ib: U32, +i: U32, +sb: U32, acc: F32, ra: Array & F32, rb: Array & F32) -> T.D2: 505 match k ra rb: 506 case 0n Tuple{a, x} Tuple{b, y}: T.D2{a, b, acc} 507 case 1n+p Tuple{a, +x} Tuple{b, +y}: tdot(p, (ia + (i + 1 : U32) : U32), (ib + sb : U32), (i + 1 : U32), sb, (acc + (x * y : F32) : F32), Array.get(F32, a, (ia + (i + 1 : U32) : U32)), Array.get(F32, b, (ib + sb : U32))) 508 509 # acc + the group's heads' sums, in head order (main.bend's pacc): a's column j (tr apart a head) · b's column (hd 510 # apart a head), rows j..t-1 511 def tsum_h(k: Nat, +ia: U32, +ib: U32, +j: U32, +tn: Nat, +w: U32, +sa: U32, +sb: U32, acc: F32, r: T.D2) -> T.D2: 512 match k r: 513 case 0n T.D2{a, b, +s}: T.D2{a, b, (acc + s : F32)} 514 case 1n+q T.D2{a, b, +s}: tsum_h(q, (ia + sa : U32), (ib + sb : U32), j, tn, w, sa, sb, (acc + s : F32), tdot(tn, (ia + sa : U32), (ib + sb : U32), j, w, 0.0, Array.get(F32, a, (ia + sa : U32)), Array.get(F32, b, (ib + sb : U32)))) 515 516 # the cell e = row·kvw + col of dK (a = G, b = Q) or of dV (a = P, b = dO): key j = row % t, over the group's g query 517 # heads: the head's column j of a (rows j..t-1) · the head's column of b 518 def kv_val(+e: U32, +c: EAC, a: Array, b: Array) -> T.D2: 519 match c: 520 case EAC{+n, +t, +t32, +h, +g, +gn, +hd, +hd32, +w, +kvw, +sc, +nw, +nk, +tr}: 521 +row = U32.div(e, kvw) 522 +col = U32.mod(e, kvw) 523 +bi = U32.div(row, t32) 524 +jj = U32.div(col, hd32) 525 +j = U32.mod(row, t32) 526 +ia = (toff(((bi * h : U32) + (jj * g : U32) : U32), j, tr) + j : U32) 527 +ib = ((((bi * t32 : U32) + j : U32) * w : U32) + (((jj * g : U32) * hd32 : U32) + U32.mod(col, hd32) : U32) : U32) 528 tsum_h(Nat.sub(gn, 1n), ia, ib, j, U32.to_nat((t32 - j : U32)), w, tr, hd32, 0.0, tdot(U32.to_nat((t32 - j : U32)), ia, ib, j, w, 0.0, Array.get(F32, a, ia), Array.get(F32, b, ib))) 529 530 # K and V transposed for the score cells (kᵀ: b·kv·hd·t, by sequence, key/value head, column c, position j), so 531 # that neighbouring lanes, which take neighbouring keys j, read neighbouring cells. dot0 reads kᵀ with stride t: the same 532 # numbers in the same order as reading k's row, so the same bits. 533 def tk_src(+e: U32, +t32: U32, +kv: U32, +hd32: U32, +kvw: U32) -> U32: 534 +r = U32.div(e, t32) 535 +bj = U32.div(r, hd32) 536 +row = ((U32.div(bj, kv) * t32 : U32) + U32.mod(e, t32) : U32) 537 ((row * kvw : U32) + ((U32.mod(bj, kv) * hd32 : U32) + U32.mod(r, hd32) : U32) : U32) 538 539 def tk_put(+e: U32, o: Array, r: Array & F32) -> TG.O2: 540 match r: 541 case Tuple{a, +x}: TG.O2{a, Array.set(F32, o, e, x)} 542 543 def tk_loop(k: Nat, +e: U32, +l: U32, +t32: U32, +kv: U32, +hd32: U32, +kvw: U32, s: TG.O2) -> TG.O2: 544 match k s: 545 case 0n r: r 546 case 1n+q TG.O2{a, o}: tk_loop(q, (e + l : U32), l, t32, kv, hd32, kvw, tk_put(e, o, Array.get(F32, a, tk_src(e, t32, kv, hd32, kvw)))) 547 548 def o2_b(s: TG.O2) -> Array: 549 match s: 550 case TG.O2{a, b}: b 551 552 # x1 = k; kᵀ written (made here) 553 def lf_tk(+t: U32, +l: U32, +c: EAC, x1: Array, x2: Array, x3: Array, x4: Array, o: Array) -> Array: 554 match c: 555 case EAC{+n, +tt, +t32, +h, +g, +gn, +hd, +hd32, +w, +kvw, +sc, +nw, +nk, +tr}: o2_b(tk_loop(TG.share(t, n, l), t, l, t32, U32.div(h, g), hd32, kvw, TG.O2{x1, o})) 556 557 # both: cells [0, nk) are kᵀ's from k, the next nk vᵀ's from v (S4{k, v, kᵀ, vᵀ}) 558 def tkv_k(v: Array, vt: Array, r: TG.O2) -> S4: 559 match r: 560 case TG.O2{k, kt}: S4{k, v, kt, vt} 561 562 def tkv_v(k: Array, kt: Array, r: TG.O2) -> S4: 563 match r: 564 case TG.O2{v, vt}: S4{k, v, kt, vt} 565 566 def tkv_sel(lo: Bool, +e: U32, +nk: U32, +t32: U32, +kv: U32, +hd32: U32, +kvw: U32, s: S4) -> S4: 567 match lo s: 568 case True{} S4{k, v, kt, vt}: tkv_k(v, vt, tk_put(e, kt, Array.get(F32, k, tk_src(e, t32, kv, hd32, kvw)))) 569 case False{} S4{k, v, kt, vt}: tkv_v(k, kt, tk_put((e - nk : U32), vt, Array.get(F32, v, tk_src((e - nk : U32), t32, kv, hd32, kvw)))) 570 571 def tkv_loop(k: Nat, +e: U32, +l: U32, +nk: U32, +t32: U32, +kv: U32, +hd32: U32, +kvw: U32, s: S4) -> S4: 572 match k: 573 case 0n: s 574 case 1n+q: tkv_loop(q, (e + l : U32), l, nk, t32, kv, hd32, kvw, tkv_sel(U32.is_lt(e, nk), e, nk, t32, kv, hd32, kvw, s)) 575 576 def tkv_o(dq: Array, s: S4) -> TG.O3: 577 match s: 578 case S4{k, v, kt, vt}: TG.O3{kt, vt, dq} 579 580 # x1 = k, x2 = v; outputs kᵀ and vᵀ (written), dQ (made here, written by backward 3) 581 def lf_tkv(+t: U32, +l: U32, +c: EAC, x1: Array, x2: Array, x3: Array, x4: Array, o: TG.O3) -> TG.O3: 582 match c o: 583 case EAC{+n, +tt, +t32, +h, +g, +gn, +hd, +hd32, +w, +kvw, +sc, +nw, +nk, +tr} TG.O3{kt, vt, dq}: tkv_o(dq, tkv_loop(TG.share(t, n, l), t, l, nk, t32, U32.div(h, g), hd32, kvw, S4{x1, x2, kt, vt})) 584 585 # the score cells are P's lower triangle (j <= i), dealt in pairs of rows i and t - 1 - i (t + 1 cells a pair), so 586 # every lane gets real work: cell e is pair p of head bh, place c; c <= p is (p, c), the rest (t - 1 - p, c - p - 1); 587 # in the middle row of an odd t the second half repeats the first and is skipped (ok false) 588 type IJ is Data: 589 IJ{ok: Bool, bh: U32, i: U32, j: U32} 590 591 def ij_of(+e: U32, +t32: U32) -> IJ: 592 +np = U32.div((t32 + 1 : U32), 2) 593 +c = U32.mod(e, (t32 + 1 : U32)) 594 +pr = U32.div(e, (t32 + 1 : U32)) 595 +p = U32.mod(pr, np) 596 +lo = U32.is_le(c, p) 597 IJ{Bool.or(lo, U32.is_ne(((t32 - 1 : U32) - p : U32), p)), U32.div(pr, np), Bool.pick(U32, lo, p, ((t32 - 1 : U32) - p : U32)), Bool.pick(U32, lo, c, ((c - p : U32) - 1 : U32))} 598 599 # forward 1: P's cell (i, j) = (q's row i · k's row j) * sc, as qrow's dot and pw's product (S3{q, kᵀ, p}) 600 def sc_put(+e: U32, +sc: F32, p: Array, r: T.D2) -> S3: 601 match r: 602 case T.D2{q, k, +s}: S3{q, k, Array.set(F32, p, e, (s * sc : F32))} 603 604 def sc_if(ok: Bool, +e: U32, +qa: U32, +ka: U32, +hd: Nat, +t32: U32, +sc: F32, s: S3) -> S3: 605 match ok s: 606 case True{} S3{q, k, p}: sc_put(e, sc, p, T.dot0(hd, qa, ka, 1, t32, q, k)) 607 case False{} S3{q, k, p}: S3{q, k, p} 608 609 def sc_ij(+a: IJ, +t32: U32, +h: U32, +g: U32, +hd: Nat, +hd32: U32, +w: U32, +sc: F32, +tr: U32, s: S3) -> S3: 610 match a: 611 case IJ{+ok, +bh, +i, +j}: sc_if(ok, (toff(bh, i, tr) + j : U32), ((((U32.div(bh, h) * t32 : U32) + i : U32) * w : U32) + (U32.mod(bh, h) * hd32 : U32) : U32), (((U32.div(bh, g) * hd32 : U32) * t32 : U32) + j : U32), hd, t32, sc, s) 612 613 def sc_cell(+e: U32, +c: EAC, s: S3) -> S3: 614 match c: 615 case EAC{+n, +t, +t32, +h, +g, +gn, +hd, +hd32, +w, +kvw, +sc, +nw, +nk, +tr}: sc_ij(ij_of(e, t32), t32, h, g, hd, hd32, w, sc, tr, s) 616 617 def sc_loop(k: Nat, +e: U32, +l: U32, +c: EAC, s: S3) -> S3: 618 match k: 619 case 0n: s 620 case 1n+q: sc_loop(q, (e + l : U32), l, c, sc_cell(e, c, s)) 621 622 # x1 = q, x2 = k; outputs P (written) and o (made here, written by forward 3) 623 def lf_sc(+t: U32, +l: U32, +c: EAC, x1: Array, x2: Array, x3: Array, x4: Array, o: TG.O2) -> TG.O2: 624 match o: 625 case TG.O2{p, y}: sc_fin(y, sc_loop(TG.share(t, eac_n(c), l), t, l, c, S3{x1, x2, p})) 626 627 # forward 2: P's row r (query i = r % t) by main.bend's softmax over its i + 1 positions 628 def sm_loop(k: Nat, +r: U32, +l: U32, +t32: U32, +tr: U32, p: Array) -> Array: 629 match k: 630 case 0n: p 631 case 1n+q: sm_loop(q, (r + l : U32), l, t32, tr, T.pc_a(T.psmc(U32.to_nat((U32.mod(r, t32) + 1 : U32)), toff(U32.div(r, t32), U32.mod(r, t32), tr), p))) 632 633 def lf_sm(+t: U32, +l: U32, +c: EAC, x1: Array, x2: Array, x3: Array, x4: Array, p: Array) -> Array: 634 match c: 635 case EAC{+n, +tt, +t32, +h, +g, +gn, +hd, +hd32, +w, +kvw, +sc, +nw, +nk, +tr}: sm_loop(TG.share(t, n, l), t, l, t32, tr, p) 636 637 # forward 3 (and backward 3's dQ): o[e] from pv_val (S3{a, b, o}) 638 def put_c(+e: U32, o: Array, r: T.D2) -> S3: 639 match r: 640 case T.D2{a, b, +s}: S3{a, b, Array.set(F32, o, e, s)} 641 642 def pv_loop(k: Nat, +e: U32, +l: U32, +c: EAC, s: S3) -> S3: 643 match k s: 644 case 0n r: r 645 case 1n+q S3{a, b, o}: pv_loop(q, (e + l : U32), l, c, put_c(e, o, pv_val(e, c, a, b))) 646 647 # x1 = P, x2 = v; o written in place 648 def lf_pv(+t: U32, +l: U32, +c: EAC, x1: Array, x2: Array, x3: Array, x4: Array, o: Array) -> Array: 649 s3_c(pv_loop(TG.share(t, eac_n(c), l), t, l, c, S3{x1, x2, o})) 650 651 # backward 1: P's score cell as forward 1, and G's cell dP = dO's row · v's row, as drow's dot (S6{q, k, v, d, p, g}) 652 type S6 is Type: 653 S6{q: Array, k: Array, v: Array, d: Array, p: Array, g: Array} 654 655 def b1_g(+e: U32, q: Array, k: Array, p: Array, g: Array, r: T.D2) -> S6: 656 match r: 657 case T.D2{d, v, +s}: S6{q, k, v, d, p, Array.set(F32, g, e, s)} 658 659 def b1_p(+e: U32, +sc: F32, +qa: U32, +ka: U32, +hd: Nat, +t32: U32, v: Array, d: Array, p: Array, g: Array, r: T.D2) -> S6: 660 match r: 661 case T.D2{q, k, +s}: b1_g(e, q, k, Array.set(F32, p, e, (s * sc : F32)), g, T.dot0(hd, qa, ka, 1, t32, d, v)) 662 663 def b1_if(ok: Bool, +e: U32, +qa: U32, +ka: U32, +hd: Nat, +t32: U32, +sc: F32, s: S6) -> S6: 664 match ok s: 665 case True{} S6{q, k, v, d, p, g}: b1_p(e, sc, qa, ka, hd, t32, v, d, p, g, T.dot0(hd, qa, ka, 1, t32, q, k)) 666 case False{} S6{q, k, v, d, p, g}: S6{q, k, v, d, p, g} 667 668 def b1_ij(+a: IJ, +t32: U32, +h: U32, +g: U32, +hd: Nat, +hd32: U32, +w: U32, +sc: F32, +tr: U32, s: S6) -> S6: 669 match a: 670 case IJ{+ok, +bh, +i, +j}: b1_if(ok, (toff(bh, i, tr) + j : U32), ((((U32.div(bh, h) * t32 : U32) + i : U32) * w : U32) + (U32.mod(bh, h) * hd32 : U32) : U32), (((U32.div(bh, g) * hd32 : U32) * t32 : U32) + j : U32), hd, t32, sc, s) 671 672 def b1_cell(+e: U32, +c: EAC, s: S6) -> S6: 673 match c: 674 case EAC{+n, +t, +t32, +h, +g, +gn, +hd, +hd32, +w, +kvw, +sc, +nw, +nk, +tr}: b1_ij(ij_of(e, t32), t32, h, g, hd, hd32, w, sc, tr, s) 675 676 def b1_loop(k: Nat, +e: U32, +l: U32, +c: EAC, s: S6) -> S6: 677 match k: 678 case 0n: s 679 case 1n+q: b1_loop(q, (e + l : U32), l, c, b1_cell(e, c, s)) 680 681 def b1_o(s: S6) -> TG.O2: 682 match s: 683 case S6{q, k, v, d, p, g}: TG.O2{p, g} 684 685 # x1 = q, x2 = kᵀ, x3 = vᵀ, x4 = dO; outputs P and G (made here, written) 686 def lf_b1(+t: U32, +l: U32, +c: EAC, x1: Array, x2: Array, x3: Array, x4: Array, o: TG.O2) -> TG.O2: 687 match o: 688 case TG.O2{p, g}: b1_o(b1_loop(TG.share(t, eac_n(c), l), t, l, c, S6{x1, x2, x3, x4, p, g})) 689 690 # backward 2: row r of P by the softmax, then G's row: s = dP·P over the i + 1 positions and G = (P * (dP - s)) * sc, 691 # main.bend's gr3 and g_set2 692 def gr_2(+n: Nat, +off: U32, +sc: F32, g: Array, r: T.PC) -> T.A2: 693 match r: 694 case T.PC{p, c}: T.g_set2(n, sc, off, T.dot0(n, off, off, 1, 1, g, p)) 695 696 def gr_loop(k: Nat, +r: U32, +l: U32, +t32: U32, +tr: U32, +sc: F32, s: T.A2) -> T.A2: 697 match k s: 698 case 0n r: r 699 case 1n+q T.A2{p, g}: gr_loop(q, (r + l : U32), l, t32, tr, sc, gr_2(U32.to_nat((U32.mod(r, t32) + 1 : U32)), toff(U32.div(r, t32), U32.mod(r, t32), tr), sc, g, T.psmc(U32.to_nat((U32.mod(r, t32) + 1 : U32)), toff(U32.div(r, t32), U32.mod(r, t32), tr), p))) 700 701 def a2_o2(s: T.A2) -> TG.O2: 702 match s: 703 case T.A2{p, g}: TG.O2{p, g} 704 705 def lf_gr(+t: U32, +l: U32, +c: EAC, x1: Array, x2: Array, x3: Array, x4: Array, o: TG.O2) -> TG.O2: 706 match c o: 707 case EAC{+n, +tt, +t32, +h, +g, +gn, +hd, +hd32, +w, +kvw, +sc, +nw, +nk, +tr} TG.O2{p, gb}: a2_o2(gr_loop(TG.share(t, n, l), t, l, t32, tr, sc, T.A2{p, gb})) 708 709 # backward 3: cells [0, nw) are dQ's, [nw, nw + nk) dK's, the next nk dV's (S8{q, k, d, g, p, dq, dk, v}) 710 type S8 is Type: 711 S8{q: Array, k: Array, d: Array, g: Array, p: Array, dq: Array, dk: Array, v: Array} 712 713 def b3_q(+e: U32, q: Array, d: Array, p: Array, dq: Array, dk: Array, v: Array, r: T.D2) -> S8: 714 match r: 715 case T.D2{g, k, +s}: S8{q, k, d, g, p, Array.set(F32, dq, e, s), dk, v} 716 717 def b3_k(+e: U32, k: Array, d: Array, p: Array, dq: Array, dk: Array, v: Array, r: T.D2) -> S8: 718 match r: 719 case T.D2{g, q, +s}: S8{q, k, d, g, p, dq, Array.set(F32, dk, e, s), v} 720 721 def b3_v(+e: U32, q: Array, k: Array, g: Array, dq: Array, dk: Array, v: Array, r: T.D2) -> S8: 722 match r: 723 case T.D2{p, d, +s}: S8{q, k, d, g, p, dq, dk, Array.set(F32, v, e, s)} 724 725 def b3_sel(a: Bool, b: Bool, +e: U32, +nw: U32, +nk: U32, +c: EAC, s: S8) -> S8: 726 match a b s: 727 case True{} _ S8{q, k, d, g, p, dq, dk, v}: b3_q(e, q, d, p, dq, dk, v, pv_val(e, c, g, k)) 728 case False{} True{} S8{q, k, d, g, p, dq, dk, v}: b3_k((e - nw : U32), k, d, p, dq, dk, v, kv_val((e - nw : U32), c, g, q)) 729 case False{} False{} S8{q, k, d, g, p, dq, dk, v}: b3_v(((e - nw : U32) - nk : U32), q, k, g, dq, dk, v, kv_val(((e - nw : U32) - nk : U32), c, p, d)) 730 731 def b3_cell(+e: U32, +c: EAC, s: S8) -> S8: 732 match c: 733 case EAC{+n, +t, +t32, +h, +g, +gn, +hd, +hd32, +w, +kvw, +sc, +nw, +nk, +tr}: b3_sel(U32.is_lt(e, nw), U32.is_lt(e, (nw + nk : U32)), e, nw, nk, c, s) 734 735 def b3_loop(k: Nat, +e: U32, +l: U32, +c: EAC, s: S8) -> S8: 736 match k: 737 case 0n: s 738 case 1n+q: b3_loop(q, (e + l : U32), l, c, b3_cell(e, c, s)) 739 740 def b3_o(s: S8) -> TG.O4: 741 match s: 742 case S8{q, k, d, g, p, dq, dk, v}: TG.O4{p, dq, dk, v} 743 744 # x1 = q, x2 = k, x3 = dO, x4 = G; P read from the first output (unchanged), dQ, dK and dV (over V) written 745 def lf_b3(+t: U32, +l: U32, +c: EAC, x1: Array, x2: Array, x3: Array, x4: Array, o: TG.O4) -> TG.O4: 746 match o: 747 case TG.O4{p, dq, dk, v}: b3_o(b3_loop(TG.share(t, eac_n(c), l), t, l, c, S8{x1, x2, x3, x4, p, dq, dk, v})) 748 749 # the sizes of a bang of cnt cells (or rows) 750 def eac(+cnt: Nat, +n: Nat, +w: Nat, +kvw: Nat, +t: Nat, +kv: Nat, +g: Nat, +hd: Nat, +sc: F32) -> EAC: 751 EAC{U32.from_nat(cnt), t, U32.from_nat(t), U32.from_nat(Nat.mul(kv, g)), U32.from_nat(g), g, hd, U32.from_nat(hd), U32.from_nat(w), U32.from_nat(kvw), sc, U32.from_nat(Nat.mul(n, w)), U32.from_nat(Nat.mul(n, kvw)), U32.div((U32.from_nat(t) * (U32.from_nat(t) + 1 : U32) : U32), 2)} 752 753 # P's cells (b·h·t(t + 1)/2), the score cells dealt in pairs of rows (b·h·ceil(t/2)·(t + 1)) and P's rows (b·h·t) 754 def pcells(+b: Nat, +t: Nat, +kv: Nat, +g: Nat) -> Nat: 755 Nat.mul(Nat.mul(b, Nat.mul(kv, g)), Nat.div(Nat.mul(t, 1n+t), 2n)) 756 757 def tcells(+b: Nat, +t: Nat, +kv: Nat, +g: Nat) -> Nat: 758 Nat.mul(Nat.mul(Nat.mul(b, Nat.mul(kv, g)), Nat.div(1n+t, 2n)), 1n+t) 759 760 def prows(+b: Nat, +t: Nat, +kv: Nat, +g: Nat) -> Nat: 761 Nat.mul(Nat.mul(b, Nat.mul(kv, g)), t) 762 763 def af3(-n: Nat, -w: Nat, -kvw: Nat, q: Array, k: Array, r: TG.R1) -> T.Fwd, T.AttnS>: 764 match r: 765 case TG.R1{p, v, x3, x4, o}: T.Fwd{Unit{}, TA.Mat{o}, T.AttnS{q, k, v}} 766 767 def af2(+n: Nat, +w: Nat, +kvw: Nat, +b: Nat, +t: Nat, +kv: Nat, +g: Nat, +hd: Nat, +sc: F32, q: Array, k: Array, v: Array, o: Array, r: TG.R1) -> T.Fwd, T.AttnS>: 768 match r: 769 case TG.R1{x1, x2, x3, x4, p}: af3(n, w, kvw, q, k, TG.run1(~EAC, ~lf_pv, TG.depth(Nat.mul(n, w)), eac(Nat.mul(n, w), n, w, kvw, t, kv, g, hd, sc), p, v, nn(Unit{}), nn(Unit{}), o)) 770 771 def af1(+n: Nat, +w: Nat, +kvw: Nat, +b: Nat, +t: Nat, +kv: Nat, +g: Nat, +hd: Nat, +sc: F32, k: Array, v: Array, r: TG.R2) -> T.Fwd, T.AttnS>: 772 match r: 773 case TG.R2{q, kt, x3, x4, TG.O2{p, o}}: af2(n, w, kvw, b, t, kv, g, hd, sc, q, k, v, o, TG.run1(~EAC, ~lf_sm, TG.depth(prows(b, t, kv, g)), eac(prows(b, t, kv, g), n, w, kvw, t, kv, g, hd, sc), nn(Unit{}), nn(Unit{}), nn(Unit{}), nn(Unit{}), p)) 774 775 def af0(+n: Nat, +w: Nat, +kvw: Nat, +b: Nat, +t: Nat, +kv: Nat, +g: Nat, +hd: Nat, +sc: F32, q: Array, v: Array, r: TG.R1) -> T.Fwd, T.AttnS>: 776 match r: 777 case TG.R1{k, x2, x3, x4, kt}: af1(n, w, kvw, b, t, kv, g, hd, sc, k, v, TG.run2_new(~EAC, ~lf_sc, TG.depth(tcells(b, t, kv, g)), eac(tcells(b, t, kv, g), n, w, kvw, t, kv, g, hd, sc), q, kt, nn(Unit{}), nn(Unit{}), TA.cap_depth(pcells(b, t, kv, g)), TA.cap_depth(Nat.mul(n, w)))) 778 779 def attn_f(+n: Nat, +w: Nat, +kvw: Nat, +b: Nat, +t: Nat, +kv: Nat, +g: Nat, +hd: Nat, +sc: F32, x: T.Two, T.Two, TA.Mat>>) -> T.Fwd, T.AttnS>: 780 match x: 781 case T.Two{TA.Mat{q}, T.Two{TA.Mat{k}, TA.Mat{v}}}: af0(n, w, kvw, b, t, kv, g, hd, sc, q, v, TG.run1_new(~EAC, ~lf_tk, TG.depth(Nat.mul(n, kvw)), eac(Nat.mul(n, kvw), n, w, kvw, t, kv, g, hd, sc), k, nn(Unit{}), nn(Unit{}), nn(Unit{}), TA.cap_depth(Nat.mul(n, kvw)))) 782 783 def ab4(-n: Nat, -w: Nat, -kvw: Nat, r: TG.R4) -> T.Bwd, T.Two, TA.Mat>>>: 784 match r: 785 case TG.R4{x1, x2, x3, x4, TG.O4{p, dq, dk, dv}}: T.Bwd{Unit{}, T.Two{TA.Mat{dq}, T.Two{TA.Mat{dk}, TA.Mat{dv}}}, Unit{}} 786 787 # backward 3 writes dK over kᵀ and dV over v (neither read any more) 788 def ab3(+n: Nat, +w: Nat, +kvw: Nat, +b: Nat, +t: Nat, +kv: Nat, +g: Nat, +hd: Nat, +sc: F32, q: Array, k: Array, v: Array, d: Array, dq: Array, kt: Array, r: TG.R2) -> T.Bwd, T.Two, TA.Mat>>>: 789 match r: 790 case TG.R2{x1, x2, x3, x4, TG.O2{p, gb}}: ab4(n, w, kvw, TG.run4(~EAC, ~lf_b3, TG.depth(Nat.add(Nat.mul(n, w), Nat.mul(2n, Nat.mul(n, kvw)))), eac(Nat.add(Nat.mul(n, w), Nat.mul(2n, Nat.mul(n, kvw))), n, w, kvw, t, kv, g, hd, sc), q, k, d, gb, p, dq, kt, v)) 791 792 def ab2(+n: Nat, +w: Nat, +kvw: Nat, +b: Nat, +t: Nat, +kv: Nat, +g: Nat, +hd: Nat, +sc: F32, k: Array, v: Array, dq: Array, r: TG.R2) -> T.Bwd, T.Two, TA.Mat>>>: 793 match r: 794 case TG.R2{q, kt, vt, d, TG.O2{p, gb}}: ab3(n, w, kvw, b, t, kv, g, hd, sc, q, k, v, d, dq, kt, TG.run2(~EAC, ~lf_gr, TG.depth(prows(b, t, kv, g)), eac(prows(b, t, kv, g), n, w, kvw, t, kv, g, hd, sc), nn(Unit{}), nn(Unit{}), nn(Unit{}), nn(Unit{}), p, gb)) 795 796 def ab1(+n: Nat, +w: Nat, +kvw: Nat, +b: Nat, +t: Nat, +kv: Nat, +g: Nat, +hd: Nat, +sc: F32, q: Array, d: Array, r: TG.R3) -> T.Bwd, T.Two, TA.Mat>>>: 797 match r: 798 case TG.R3{k, v, x3, x4, TG.O3{kt, vt, dq}}: ab2(n, w, kvw, b, t, kv, g, hd, sc, k, v, dq, TG.run2_new(~EAC, ~lf_b1, TG.depth(tcells(b, t, kv, g)), eac(tcells(b, t, kv, g), n, w, kvw, t, kv, g, hd, sc), q, kt, vt, d, TA.cap_depth(pcells(b, t, kv, g)), TA.cap_depth(pcells(b, t, kv, g)))) 799 800 def attn_b(+n: Nat, +w: Nat, +kvw: Nat, +b: Nat, +t: Nat, +kv: Nat, +g: Nat, +hd: Nat, +sc: F32, s: T.AttnS, dy: TA.Mat) -> T.Bwd, T.Two, TA.Mat>>>: 801 match s dy: 802 case T.AttnS{q, k, v} TA.Mat{d}: ab1(n, w, kvw, b, t, kv, g, hd, sc, q, d, TG.run3_new(~EAC, ~lf_tkv, TG.depth(Nat.mul(2n, Nat.mul(n, kvw))), eac(Nat.mul(2n, Nat.mul(n, kvw)), n, w, kvw, t, kv, g, hd, sc), k, v, nn(Unit{}), nn(Unit{}), TA.cap_depth(Nat.mul(n, kvw)), TA.cap_depth(Nat.mul(n, kvw)), TA.cap_depth(Nat.mul(n, w)))) 803 804 # causal attention with grouped heads over b sequences of t positions (main.bend's Attention) 805 def Attention(+n: Nat, +w: Nat, +kvw: Nat, +b: Nat, +t: Nat, +kv: Nat, +g: Nat, +hd: Nat, +sc: F32, -e1: {n == Nat.mul(b, t) : Nat}, -e2: {w == Nat.mul(Nat.mul(kv, g), hd) : Nat}, -e3: {kvw == Nat.mul(kv, hd) : Nat}) -> T.Layer, T.Two, TA.Mat>>, TA.Mat, T.AttnS>: 806 T.Layer{p => x => attn_f(n, w, kvw, b, t, kv, g, hd, sc, x), p => s => dy => g2 => attn_b(n, w, kvw, b, t, kv, g, hd, sc, s, dy), op => p => g2 => st => T.no_step(Unit, op, p, g2, st), u => Unit{}, p => T.PL{p, Nil{}}, p => xs => off => T.PA{p, xs, off}, p => xs => off => T.PA{p, xs, off}, u => 0} 807 808 # ---- MSE on the device. main.bend's MSE writes each gradient cell 2 (y - t) / (n·c) over y and sums the squares with 809 # tree_sum, whose passes add neighbouring pairs; its first p passes carry no odd cell when 2^p divides the count, so a 810 # leaf that sums a block of 2^p squares as a balanced tree of pairs gets exactly the number those passes leave at the 811 # block's place. The leaves do that on the device (y written in place), and the host finishes tree_sum over the 812 # count / 2^p block sums: the same bits as main.bend's MSE, with only the block sums read back. ---- 813 814 type EMS is Data: 815 EMS{nb: U32, p: Nat, bs: U32, kk: F32} 816 817 type MS is Type: 818 MS{y: Array, t: Array, s: F32} 819 820 def ms_add(+s: F32, r: MS) -> MS: 821 match r: 822 case MS{y, t, +s2}: MS{y, t, (s + s2 : F32)} 823 824 def ms_then(r: MS, k: @+s: F32 -> Array -> Array -> MS) -> MS: 825 match r: 826 case MS{y, t, +s}: k(s, y, t) 827 828 def ms_cell(+i: U32, +kk: F32, ry: Array & F32, rt: Array & F32) -> MS: 829 match ry rt: 830 case Tuple{y, +a} Tuple{t, +b}: MS{Array.set(F32, y, i, ((a - b : F32) * kk : F32)), t, ((a - b : F32) * (a - b : F32) : F32)} 831 832 # the squares of the block of 2^p cells from i summed as a balanced tree of pairs (h = 2^(p-1)), gradients written 833 def ms_blk(p: Nat, +i: U32, +h: U32, +kk: F32, y: Array, t: Array) -> MS: 834 match p: 835 case 0n: ms_cell(i, kk, Array.get(F32, y, i), Array.get(F32, t, i)) 836 case 1n++q: ms_then(ms_blk(q, i, U32.shr(h), kk, y, t), +s => y2 => t2 => ms_add(s, ms_blk(q, (i + h : U32), U32.shr(h), kk, y2, t2))) 837 838 def ms_put(+t: U32, ps: Array, r: MS) -> TG.O2: 839 match r: 840 case MS{y, tt, +s}: TG.O2{y, Array.set(F32, ps, t, s)} 841 842 def ms_if(ok: Bool, +t: U32, +p: Nat, +bs: U32, +kk: F32, tg: Array, y: Array, ps: Array) -> TG.O2: 843 match ok: 844 case True{}: ms_put(t, ps, ms_blk(p, (t * bs : U32), U32.shr(bs), kk, y, tg)) 845 case False{}: TG.O2{y, ps} 846 847 # x1 = the target; y written in place with the gradient, ps[t] = leaf t's block sum (one leaf a block) 848 def lf_mse(+t: U32, +l: U32, +e: EMS, x1: Array, x2: Array, x3: Array, x4: Array, o: TG.O2) -> TG.O2: 849 match e o: 850 case EMS{+nb, +p, +bs, +kk} TG.O2{y, ps}: ms_if(U32.is_lt(t, nb), t, p, bs, kk, x1, y, ps) 851 852 # the largest p <= k with 2^p dividing m (m > 0) 853 def ms_p(k: Nat, +m: U32) -> Nat: 854 match k: 855 case 0n: 0n 856 case 1n+q: Bool.pick(Nat, U32.is_eq(U32.mod(m, 2), 0), 1n+ms_p(q, U32.div(m, 2)), 0n) 857 858 def mse_fin(-n: Nat, -c: Nat, +nb: U32, +nc: F32, q: TG.R2) -> T.LossOut>: 859 match q: 860 case TG.R2{x1, x2, x3, x4, TG.O2{y, ps}}: T.LossOut{(T.tree_sum(nb, ps) / nc : F32), TA.Mat{y}} 861 862 def mse_go(+n: Nat, +c: Nat, +m: U32, +p: Nat, y: Array, t: Array) -> T.LossOut>: 863 mse_fin(n, c, U32.shrn(m, p), F32.from_nat(Nat.mul(n, c)), TG.run2(~EMS, ~lf_mse, TA.cap_depth(U32.to_nat(U32.shrn(m, p))), EMS{U32.shrn(m, p), p, U32.pow(2, p), (2.0 / F32.from_nat(Nat.mul(n, c)) : F32)}, t, nn(Unit{}), nn(Unit{}), nn(Unit{}), y, T.zeros_a(U32.to_nat(U32.shrn(m, p))))) 864 865 # mean squared error over every number (main.bend's MSE, the same bits) 866 def MSE(+n: Nat, +c: Nat, y: TA.Mat, t: TA.Mat) -> T.LossOut>: 867 match y t: 868 case TA.Mat{ya} TA.Mat{ta}: mse_go(n, c, U32.from_nat(Nat.mul(n, c)), ms_p(6n, U32.from_nat(Nat.mul(n, c))), ya, ta) 869 870 # ---- ids and labels on the device: a list of n Nats as an array of n cells, each holding the id's U32 word as an F32's 871 # bits (a bit copy, not a conversion, so every U32 id is exact); the count is how many of the list's first n there are ---- 872 873 def u2f(x: U32) -> F32: 874 match x: 875 case U32{w}: F32{w} 876 877 def f2u(x: F32) -> U32: 878 match x: 879 case F32{w}: U32{w} 880 881 type IA is Type: 882 IA{a: Array, k: Nat} 883 884 def ia_go(left: Nat, ids: List<&2, Nat>, +i: U32, k: Nat, a: Array) -> IA: 885 match left ids: 886 case 1n+q Con{+id, t}: ia_go(q, t, (i + 1 : U32), 1n+k, Array.set(F32, a, i, u2f(U32.from_nat(id)))) 887 case _ _: IA{a, k} 888 889 def ids_arr(+n: Nat, ids: List<&2, Nat>) -> IA: 890 ia_go(n, ids, 0, 0n, T.zeros_a(n)) 891 892 # ---- Embedding: a leaf a column j of the table (d of them), each looping over the ids in order: the forward pass 893 # copies e[id_i, j] into row i, the backward adds dy[i, j] into g[id_i, j] as main.bend's emb_add does (g + dy), one id 894 # after the other, so every cell of the gradient gets its rows added in main.bend's order, with no two leaves writing 895 # one cell. Neighbouring lanes read neighbouring cells. ---- 896 897 type EEM is Data: 898 EEM{k: Nat, d: U32} 899 900 type ES3 is Type: 901 ES3{ids: Array, a: Array, o: Array} 902 903 def ef_3(+i: U32, +j: U32, +d: U32, ids: Array, o: Array, r: Array & F32) -> ES3: 904 match r: 905 case Tuple{e, +x}: ES3{ids, e, Array.set(F32, o, ((i * d : U32) + j : U32), x)} 906 907 def ef_2(+i: U32, +j: U32, +d: U32, e: Array, o: Array, r: Array & F32) -> ES3: 908 match r: 909 case Tuple{ids, +v}: ef_3(i, j, d, ids, o, Array.get(F32, e, ((f2u(v) * d : U32) + j : U32))) 910 911 # o[i, j] = e[ids[i], j] 912 def ef_cell(+i: U32, +j: U32, +d: U32, st: ES3) -> ES3: 913 match st: 914 case ES3{ids, e, o}: ef_2(i, j, d, e, o, Array.get(F32, ids, i)) 915 916 def ef_loop(k: Nat, +i: U32, +j: U32, +d: U32, st: ES3) -> ES3: 917 match k: 918 case 0n: st 919 case 1n+q: ef_loop(q, (i + 1 : U32), j, d, ef_cell(i, j, d, st)) 920 921 def es3_o(st: ES3) -> Array: 922 match st: 923 case ES3{ids, a, o}: o 924 925 def ef_if(ok: Bool, +t: U32, +k: Nat, +d: U32, e: Array, ids: Array, o: Array) -> Array: 926 match ok: 927 case True{}: es3_o(ef_loop(k, 0, t, d, ES3{ids, e, o})) 928 case False{}: o 929 930 # x1 = the table, x2 = the ids; o (made on the device, zeros) gets the rows 931 def lf_embf(+t: U32, +l: U32, +e: EEM, x1: Array, x2: Array, x3: Array, x4: Array, o: Array) -> Array: 932 match e: 933 case EEM{+k, +d}: ef_if(U32.is_lt(t, d), t, k, d, x1, x2, o) 934 935 def eb_4(+gi: U32, ids: Array, g: Array, +a: F32, r: Array & F32) -> ES3: 936 match r: 937 case Tuple{dy, +b}: ES3{ids, dy, Array.set(F32, g, gi, (a + b : F32))} 938 939 def eb_3(+i: U32, +j: U32, +d: U32, +gi: U32, ids: Array, dy: Array, r: Array & F32) -> ES3: 940 match r: 941 case Tuple{g, +a}: eb_4(gi, ids, g, a, Array.get(F32, dy, ((i * d : U32) + j : U32))) 942 943 def eb_2(+i: U32, +j: U32, +d: U32, dy: Array, g: Array, r: Array & F32) -> ES3: 944 match r: 945 case Tuple{ids, +v}: eb_3(i, j, d, ((f2u(v) * d : U32) + j : U32), ids, dy, Array.get(F32, g, ((f2u(v) * d : U32) + j : U32))) 946 947 # g[ids[i], j] = g[ids[i], j] + dy[i, j] 948 def eb_cell(+i: U32, +j: U32, +d: U32, st: ES3) -> ES3: 949 match st: 950 case ES3{ids, dy, g}: eb_2(i, j, d, dy, g, Array.get(F32, ids, i)) 951 952 def eb_loop(k: Nat, +i: U32, +j: U32, +d: U32, st: ES3) -> ES3: 953 match k: 954 case 0n: st 955 case 1n+q: eb_loop(q, (i + 1 : U32), j, d, eb_cell(i, j, d, st)) 956 957 def eb_if(ok: Bool, +t: U32, +k: Nat, +d: U32, dy: Array, ids: Array, g: Array) -> Array: 958 match ok: 959 case True{}: es3_o(eb_loop(k, 0, t, d, ES3{ids, dy, g})) 960 case False{}: g 961 962 # x1 = dy, x2 = the ids; the gradient g, in place 963 def lf_embb(+t: U32, +l: U32, +e: EEM, x1: Array, x2: Array, x3: Array, x4: Array, g: Array) -> Array: 964 match e: 965 case EEM{+k, +d}: eb_if(U32.is_lt(t, d), t, k, d, x1, x2, g) 966 967 def emb_f3(-n: Nat, -v: Nat, -d: Nat, +ids: List<&2, Nat>, q: TG.R1) -> T.Fwd, TA.Mat, List<&2, Nat>>: 968 match q: 969 case TG.R1{x1, x2, x3, x4, o}: T.Fwd{TA.Mat{x1}, TA.Mat{o}, ids} 970 971 def emb_f2(+n: Nat, +v: Nat, +d: Nat, +ids: List<&2, Nat>, ea: Array, r: IA) -> T.Fwd, TA.Mat, List<&2, Nat>>: 972 match r: 973 case IA{ia, +k}: emb_f3(n, v, d, ids, TG.run1_new(~EEM, ~lf_embf, TA.cap_depth(d), EEM{k, U32.from_nat(d)}, ea, ia, nn(Unit{}), nn(Unit{}), TA.cap_depth(Nat.mul(n, d)))) 974 975 def emb_f(+n: Nat, +v: Nat, +d: Nat, +ids: List<&2, Nat>, e: TA.Mat) -> T.Fwd, TA.Mat, List<&2, Nat>>: 976 match e: 977 case TA.Mat{ea}: emb_f2(n, v, d, ids, ea, ids_arr(n, ids)) 978 979 def emb_b3(-v: Nat, -d: Nat, e: TA.Mat, q: TG.R1) -> T.Bwd, List<&2, Nat>>: 980 match q: 981 case TG.R1{x1, x2, x3, x4, g}: T.Bwd{e, Nil{}, TA.Mat{g}} 982 983 def emb_b2(+v: Nat, +d: Nat, e: TA.Mat, da: Array, ga: Array, r: IA) -> T.Bwd, List<&2, Nat>>: 984 match r: 985 case IA{ia, +k}: emb_b3(v, d, e, TG.run1(~EEM, ~lf_embb, TA.cap_depth(d), EEM{k, U32.from_nat(d)}, da, ia, nn(Unit{}), nn(Unit{}), ga)) 986 987 def emb_bwd(+n: Nat, +v: Nat, +d: Nat, e: TA.Mat, ids: List<&2, Nat>, dy: TA.Mat, g: TA.Mat) -> T.Bwd, List<&2, Nat>>: 988 match dy g: 989 case TA.Mat{da} TA.Mat{ga}: emb_b2(v, d, e, da, ga, ids_arr(n, ids)) 990 991 # a table of v rows of d numbers, looked up by n ids (main.bend's Embedding, the same bits); its step on the device 992 def Embedding(+n: Nat, +v: Nat, +d: Nat) -> T.Layer, List<&2, Nat>, TA.Mat, List<&2, Nat>>: 993 T.Layer{p => x => emb_f(n, v, d, x, p), p => s => dy => g => emb_bwd(n, v, d, p, s, dy, g), op => p => g => st => mat_step(v, d, op, p, g, st), u => TA.Mat.zeros(v, d), p => T.Mat.dump(v, d, p), p => xs => off => T.Mat.restore(v, d, p, xs, off), p => xs => off => T.Mat.store(v, d, p, xs, off), u => U32.from_nat(Nat.mul(v, d))} 994 995 # ---- cross-entropy on the device, the same numbers as main.bend's CrossEntropy (but CUDA's exp on the GPU), in four 996 # bangs; only 3 numbers a row come back to the host, and the gradient stays on the device: 997 # 1. a leaf a row: its max m by main.bend's pmax0 (the same order of comparisons) and x[y] - m; 998 # 2. a leaf for each block of 2^p cells of a row (2^p dividing c, p <= 9): e = exp(x - m) written over each cell, as 999 # pexp writes it, and the block's sum as a balanced tree of pairs. tree_sum's first p passes over a row add exactly 1000 # these pairs (no odd cell is carried while 2^p divides the count), so a block's sum is the number they leave there; 1001 # 3. a leaf a row: tree_sum's remaining passes over its c / 2^p block sums (tnode below), which is s, cesm's sum; 1002 # 4. a leaf for 2^14 cells at a time: ((e / s) - [j == y]) / n, pdiv's and ce_g's two expressions. 1003 # The host takes log(s) and adds the rows' terms in order from 0.0, as ce_rows does. Rows past the labels are left as 1004 # they are, as in main.bend; labels past the n rows are not read (main.bend would write past the matrix). ---- 1005 1006 type TS is Type: 1007 TS{a: Array, s: F32} 1008 1009 def ts_leaf(r: Array & F32) -> TS: 1010 match r: 1011 case Tuple{a, +x}: TS{a, x} 1012 1013 def ts_plus(+s: F32, r: TS) -> TS: 1014 match r: 1015 case TS{a, +s2}: TS{a, (s + s2 : F32)} 1016 1017 def ts_then(r: TS, k: @+s: F32 -> Array -> TS) -> TS: 1018 match r: 1019 case TS{a, +s}: k(s, a) 1020 1021 # tree_sum of the len cells from lo, read in place: a node l passes above the cells sums its left child (the first h = 1022 # 2^(l-1) cells) and its right child (the rest) when there is a rest (big = h < len), and is its left child otherwise 1023 # (tree_sum's odd last cell carried up a pass) 1024 def tnode(l: Nat, big: Bool, +lo: U32, +len: U32, +h: U32, a: Array) -> TS: 1025 match l big: 1026 case 0n _: ts_leaf(Array.get(F32, a, lo)) 1027 case 1n++q True{}: ts_then(tnode(q, U32.is_lt(U32.shr(h), h), lo, h, U32.shr(h), a), +s => a2 => ts_plus(s, tnode(q, U32.is_lt(U32.shr(h), (len - h : U32)), (lo + h : U32), (len - h : U32), U32.shr(h), a2))) 1028 case 1n+q False{}: tnode(q, U32.is_lt(U32.shr(h), len), lo, len, U32.shr(h), a) 1029 1030 # the passes tree_sum makes over m cells: the smallest l with 2^l >= m (f is fuel) 1031 def ts_lv(f: Nat, +m: U32, +p: U32) -> Nat: 1032 match f: 1033 case 0n: 0n 1034 case 1n+q: Bool.pick(Nat, U32.is_lt(p, m), 1n+ts_lv(q, m, (p * 2 : U32)), 0n) 1035 1036 # the host's part, row by row in order: loss - ((x[y] - m) - log(s)), or main.bend's clamped term for a label past the 1037 # last class 1038 type LS is Type: 1039 LS{a: Array, l: F32} 1040 1041 def ce_t2(ok: Bool, +loss: F32, +xm: F32, r: Array & F32) -> LS: 1042 match ok r: 1043 case True{} Tuple{a, +s}: LS{a, (loss - (xm - F32.log(s) : F32) : F32)} 1044 case False{} Tuple{a, s}: LS{a, (loss - F32.log(F32.max(0.0, 0.000000000001)) : F32)} 1045 1046 def ce_g4(+n: Nat, -c: Nat, z: Array, r: LS) -> T.LossOut>: 1047 match r: 1048 case LS{a, +l}: T.LossOut{(l / F32.from_nat(n) : F32), TA.Mat{z}} 1049 1050 # 1. a leaf a row: st[3t] = x[y] - m, st[3t + 1] = m (x[y] read only when y is a class) 1051 type EC1 is Data: 1052 EC1{k: U32, c1: Nat, c32: U32} 1053 1054 def c1_3(+t: U32, +m: F32, st: Array, r: Array & F32) -> Array: 1055 match r: 1056 case Tuple{z, +xy}: Array.set(F32, Array.set(F32, st, (t * 3 : U32), (xy - m : F32)), ((t * 3 : U32) + 1 : U32), m) 1057 1058 def c1_2(ok: Bool, +t: U32, +off: U32, +y: U32, st: Array, r: T.AS) -> Array: 1059 match ok r: 1060 case True{} T.AS{z, +m}: c1_3(t, m, st, Array.get(F32, z, (off + y : U32))) 1061 case False{} T.AS{z, +m}: c1_3(t, m, st, Tuple{z, 0.0}) 1062 1063 def c1_1(+t: U32, +off: U32, +c1: Nat, +c32: U32, z: Array, st: Array, r: Array & F32) -> Array: 1064 match r: 1065 case Tuple{la, +yv}: c1_2(U32.is_lt(f2u(yv), c32), t, off, f2u(yv), st, T.pmax0(c1, off, Array.get(F32, z, off))) 1066 1067 def c1_if(ok: Bool, +t: U32, +c1: Nat, +c32: U32, lab: Array, z: Array, st: Array) -> Array: 1068 match ok: 1069 case True{}: c1_1(t, (t * c32 : U32), c1, c32, z, st, Array.get(F32, lab, t)) 1070 case False{}: st 1071 1072 def lf_c1(+t: U32, +l: U32, +e: EC1, x1: Array, x2: Array, x3: Array, x4: Array, st: Array) -> Array: 1073 match e: 1074 case EC1{+k, +c1, +c32}: c1_if(U32.is_lt(t, k), t, c1, c32, x1, x2, st) 1075 1076 # 2. the 2^l cells from lo: e = exp(x - m) written over each, and their sum as a balanced tree of pairs (h = 2^(l-1)) 1077 def tsx_leaf(+lo: U32, +m: F32, r: Array & F32) -> TS: 1078 match r: 1079 case Tuple{a, +x}: 1080 +e = F32.exp((x - m : F32)) 1081 TS{Array.set(F32, a, lo, e), e} 1082 1083 def tsx(l: Nat, +lo: U32, +h: U32, +m: F32, a: Array) -> TS: 1084 match l: 1085 case 0n: tsx_leaf(lo, m, Array.get(F32, a, lo)) 1086 case 1n++q: ts_then(tsx(q, lo, U32.shr(h), m, a), +s => a2 => ts_plus(s, tsx(q, (lo + h : U32), U32.shr(h), m, a2))) 1087 1088 # the same over 8 cells in straight-line code (no closure a node): e0..e7 written, then ((e0 + e1) + (e2 + e3)) + 1089 # ((e4 + e5) + (e6 + e7)), the balanced tree's three levels 1090 def x8_7(+lo: U32, +m: F32, +e0: F32, +e1: F32, +e2: F32, +e3: F32, +e4: F32, +e5: F32, +e6: F32, r: Array & F32) -> TS: 1091 match r: 1092 case Tuple{a, +x}: 1093 +e7 = F32.exp((x - m : F32)) 1094 TS{Array.set(F32, a, (lo + 7 : U32), e7), ((((e0 + e1 : F32) + (e2 + e3 : F32) : F32) + ((e4 + e5 : F32) + (e6 + e7 : F32) : F32)) : F32)} 1095 1096 def x8_6(+lo: U32, +m: F32, +e0: F32, +e1: F32, +e2: F32, +e3: F32, +e4: F32, +e5: F32, r: Array & F32) -> TS: 1097 match r: 1098 case Tuple{a, +x}: 1099 +e = F32.exp((x - m : F32)) 1100 x8_7(lo, m, e0, e1, e2, e3, e4, e5, e, Array.get(F32, Array.set(F32, a, (lo + 6 : U32), e), (lo + 7 : U32))) 1101 1102 def x8_5(+lo: U32, +m: F32, +e0: F32, +e1: F32, +e2: F32, +e3: F32, +e4: F32, r: Array & F32) -> TS: 1103 match r: 1104 case Tuple{a, +x}: 1105 +e = F32.exp((x - m : F32)) 1106 x8_6(lo, m, e0, e1, e2, e3, e4, e, Array.get(F32, Array.set(F32, a, (lo + 5 : U32), e), (lo + 6 : U32))) 1107 1108 def x8_4(+lo: U32, +m: F32, +e0: F32, +e1: F32, +e2: F32, +e3: F32, r: Array & F32) -> TS: 1109 match r: 1110 case Tuple{a, +x}: 1111 +e = F32.exp((x - m : F32)) 1112 x8_5(lo, m, e0, e1, e2, e3, e, Array.get(F32, Array.set(F32, a, (lo + 4 : U32), e), (lo + 5 : U32))) 1113 1114 def x8_3(+lo: U32, +m: F32, +e0: F32, +e1: F32, +e2: F32, r: Array & F32) -> TS: 1115 match r: 1116 case Tuple{a, +x}: 1117 +e = F32.exp((x - m : F32)) 1118 x8_4(lo, m, e0, e1, e2, e, Array.get(F32, Array.set(F32, a, (lo + 3 : U32), e), (lo + 4 : U32))) 1119 1120 def x8_2(+lo: U32, +m: F32, +e0: F32, +e1: F32, r: Array & F32) -> TS: 1121 match r: 1122 case Tuple{a, +x}: 1123 +e = F32.exp((x - m : F32)) 1124 x8_3(lo, m, e0, e1, e, Array.get(F32, Array.set(F32, a, (lo + 2 : U32), e), (lo + 3 : U32))) 1125 1126 def x8_1(+lo: U32, +m: F32, +e0: F32, r: Array & F32) -> TS: 1127 match r: 1128 case Tuple{a, +x}: 1129 +e = F32.exp((x - m : F32)) 1130 x8_2(lo, m, e0, e, Array.get(F32, Array.set(F32, a, (lo + 1 : U32), e), (lo + 2 : U32))) 1131 1132 def x8_0(+lo: U32, +m: F32, r: Array & F32) -> TS: 1133 match r: 1134 case Tuple{a, +x}: 1135 +e = F32.exp((x - m : F32)) 1136 x8_1(lo, m, e, Array.get(F32, Array.set(F32, a, lo, e), (lo + 1 : U32))) 1137 1138 # tsx over 2^(l+3) cells, its last three levels in blocks of 8 1139 def tsx8(l: Nat, +lo: U32, +h: U32, +m: F32, a: Array) -> TS: 1140 match l: 1141 case 0n: x8_0(lo, m, Array.get(F32, a, lo)) 1142 case 1n++q: ts_then(tsx8(q, lo, U32.shr(h), m, a), +s => a2 => ts_plus(s, tsx8(q, (lo + h : U32), U32.shr(h), m, a2))) 1143 1144 # a leaf takes the blocks b, b + l, .. of the nb = rows * per; block b is row b / per, cells (b % per) * 2^p on 1145 type EC2 is Data: 1146 EC2{nb: U32, per: U32, p: Nat, bsz: U32, c32: U32} 1147 1148 type C3S is Type: 1149 C3S{a: Array, b: Array, c: Array} 1150 1151 def c2_put(+b: U32, st: Array, bs: Array, r: TS) -> C3S: 1152 match r: 1153 case TS{z, +s}: C3S{st, z, Array.set(F32, bs, b, s)} 1154 1155 def c2_m(b8: Bool, +b: U32, +lo: U32, +p: Nat, +h: U32, z: Array, bs: Array, r: Array & F32) -> C3S: 1156 match b8 r: 1157 case True{} Tuple{st, +m}: c2_put(b, st, bs, tsx8(Nat.sub(p, 3n), lo, h, m, z)) 1158 case False{} Tuple{st, +m}: c2_put(b, st, bs, tsx(p, lo, h, m, z)) 1159 1160 def c2_cell(+b: U32, +per: U32, +p: Nat, +bsz: U32, +c32: U32, s: C3S) -> C3S: 1161 match s: 1162 case C3S{st, z, bs}: c2_m(Nat.is_le(3n, p), b, ((U32.div(b, per) * c32 : U32) + (U32.mod(b, per) * bsz : U32) : U32), p, U32.shr(bsz), z, bs, Array.get(F32, st, ((U32.div(b, per) * 3 : U32) + 1 : U32))) 1163 1164 def c2_loop(k: Nat, +b: U32, +l: U32, +per: U32, +p: Nat, +bsz: U32, +c32: U32, s: C3S) -> C3S: 1165 match k: 1166 case 0n: s 1167 case 1n+q: c2_loop(q, (b + l : U32), l, per, p, bsz, c32, c2_cell(b, per, p, bsz, c32, s)) 1168 1169 def c2_o(s: C3S) -> TG.O2: 1170 match s: 1171 case C3S{st, z, bs}: TG.O2{z, bs} 1172 1173 def lf_c2(+t: U32, +l: U32, +e: EC2, x1: Array, x2: Array, x3: Array, x4: Array, o: TG.O2) -> TG.O2: 1174 match e o: 1175 case EC2{+nb, +per, +p, +bsz, +c32} TG.O2{z, bs}: c2_o(c2_loop(TG.share(t, nb, l), t, l, per, p, bsz, c32, C3S{x1, z, bs})) 1176 1177 # 3. a leaf a row: st[3t + 2] = s, tree_sum's remaining passes over the row's per block sums 1178 type EC3 is Data: 1179 EC3{k: U32, per: U32, tl: Nat, th: U32} 1180 1181 def c3_put(+t: U32, st: Array, r: TS) -> Array: 1182 match r: 1183 case TS{bs, +s}: Array.set(F32, st, ((t * 3 : U32) + 2 : U32), s) 1184 1185 def c3_if(ok: Bool, +t: U32, +per: U32, +tl: Nat, +th: U32, bs: Array, st: Array) -> Array: 1186 match ok: 1187 case True{}: c3_put(t, st, tnode(tl, U32.is_lt(th, per), (t * per : U32), per, th, bs)) 1188 case False{}: st 1189 1190 def lf_c3(+t: U32, +l: U32, +e: EC3, x1: Array, x2: Array, x3: Array, x4: Array, st: Array) -> Array: 1191 match e: 1192 case EC3{+k, +per, +tl, +th}: c3_if(U32.is_lt(t, k), t, per, tl, th, x1, st) 1193 1194 # 4. element-wise over the first k rows: z[i] = ((e / s) - [j == y]) / nf, j = i % c, s and y of row i / c 1195 type EC4 is Data: 1196 EC4{n: U32, c32: U32, nf: F32} 1197 1198 def c4_4(+i: U32, +j: U32, +y: U32, +s: F32, +nf: F32, lab: Array, st: Array, r: Array & F32) -> C3S: 1199 match r: 1200 case Tuple{z, +x}: C3S{lab, st, Array.set(F32, z, i, (((x / s : F32) - T.hit(U32.is_eq(j, y)) : F32) / nf : F32))} 1201 1202 def c4_3(+i: U32, +j: U32, +y: U32, +nf: F32, lab: Array, z: Array, r: Array & F32) -> C3S: 1203 match r: 1204 case Tuple{st, +s}: c4_4(i, j, y, s, nf, lab, st, Array.get(F32, z, i)) 1205 1206 def c4_2(+i: U32, +row: U32, +j: U32, +nf: F32, st: Array, z: Array, r: Array & F32) -> C3S: 1207 match r: 1208 case Tuple{lab, +yv}: c4_3(i, j, f2u(yv), nf, lab, z, Array.get(F32, st, ((row * 3 : U32) + 2 : U32))) 1209 1210 def c4_cell(+i: U32, +c32: U32, +nf: F32, s: C3S) -> C3S: 1211 match s: 1212 case C3S{lab, st, z}: c4_2(i, U32.div(i, c32), (i - (U32.div(i, c32) * c32 : U32) : U32), nf, st, z, Array.get(F32, lab, U32.div(i, c32))) 1213 1214 def c4_loop(k: Nat, +i: U32, +l: U32, +c32: U32, +nf: F32, s: C3S) -> C3S: 1215 match k: 1216 case 0n: s 1217 case 1n+q: c4_loop(q, (i + l : U32), l, c32, nf, c4_cell(i, c32, nf, s)) 1218 1219 def c3s_c(s: C3S) -> Array: 1220 match s: 1221 case C3S{a, b, c}: c 1222 1223 def lf_c4(+t: U32, +l: U32, +e: EC4, x1: Array, x2: Array, x3: Array, x4: Array, z: Array) -> Array: 1224 match e: 1225 case EC4{+n, +c32, +nf}: c3s_c(c4_loop(TG.share(t, n, l), t, l, c32, nf, C3S{x1, x2, z})) 1226 1227 # the host's part over st, row by row 1228 def cx_t1(ok: Bool, +i: U32, +loss: F32, r: Array & F32) -> LS: 1229 match r: 1230 case Tuple{a, +xm}: ce_t2(ok, loss, xm, Array.get(F32, a, ((i * 3 : U32) + 2 : U32))) 1231 1232 def cx_ls(left: Nat, ls: List<&2, Nat>, +i: U32, +c: Nat, r: LS) -> LS: 1233 match left ls r: 1234 case 1n+q Con{+y, t} LS{a, +loss}: cx_ls(q, t, (i + 1 : U32), c, cx_t1(Nat.is_le(1n+y, c), i, loss, Array.get(F32, a, (i * 3 : U32)))) 1235 case _ _ r: r 1236 1237 def cx_5(+n: Nat, +c: Nat, +labels: List<&2, Nat>, +k: Nat, q: TG.R1) -> T.LossOut>: 1238 match q: 1239 case TG.R1{x1, x2, x3, x4, z}: ce_g4(n, c, z, cx_ls(k, labels, 0, c, LS{x2, 0.0})) 1240 1241 def cx_4(+n: Nat, +c: Nat, +labels: List<&2, Nat>, +k: Nat, la: Array, z: Array, q: TG.R1) -> T.LossOut>: 1242 match q: 1243 case TG.R1{x1, x2, x3, x4, st}: cx_5(n, c, labels, k, TG.run1(~EC4, ~lf_c4, TG.depth(Nat.mul(k, c)), EC4{U32.from_nat(Nat.mul(k, c)), U32.from_nat(c), F32.from_nat(n)}, la, st, nn(Unit{}), nn(Unit{}), z)) 1244 1245 def cx_3(+n: Nat, +c: Nat, +labels: List<&2, Nat>, +k: Nat, +per: U32, la: Array, q: TG.R2) -> T.LossOut>: 1246 match q: 1247 case TG.R2{st, x2, x3, x4, TG.O2{z, bs}}: cx_4(n, c, labels, k, la, z, TG.run1(~EC3, ~lf_c3, TA.cap_depth(k), EC3{U32.from_nat(k), per, ts_lv(32n, per, 1), U32.shr(U32.pow(2, ts_lv(32n, per, 1)))}, bs, nn(Unit{}), nn(Unit{}), nn(Unit{}), st)) 1248 1249 def cx_2(+n: Nat, +c: Nat, +labels: List<&2, Nat>, +k: Nat, +p: Nat, q: TG.R1) -> T.LossOut>: 1250 match q: 1251 case TG.R1{la, z, x3, x4, st}: cx_3(n, c, labels, k, U32.shrn(U32.from_nat(c), p), la, TG.run2(~EC2, ~lf_c2, TG.depth(Nat.mul(k, U32.to_nat(U32.shrn(U32.from_nat(c), p)))), EC2{(U32.from_nat(k) * U32.shrn(U32.from_nat(c), p) : U32), U32.shrn(U32.from_nat(c), p), p, U32.pow(2, p), U32.from_nat(c)}, st, nn(Unit{}), nn(Unit{}), nn(Unit{}), z, T.zeros_a(Nat.mul(k, U32.to_nat(U32.shrn(U32.from_nat(c), p)))))) 1252 1253 def cx_1(+n: Nat, +c: Nat, +labels: List<&2, Nat>, z: Array, r: IA) -> T.LossOut>: 1254 match r: 1255 case IA{la, +k}: cx_2(n, c, labels, k, ms_p(9n, U32.from_nat(c)), TG.run1(~EC1, ~lf_c1, TA.cap_depth(n), EC1{U32.from_nat(k), Nat.sub(c, 1n), U32.from_nat(c)}, la, z, nn(Unit{}), nn(Unit{}), T.zeros_a(Nat.mul(3n, n)))) 1256 1257 # cross-entropy over n rows of c classes against the labels (main.bend's CrossEntropy; its bits but CUDA's exp) 1258 def CrossEntropy(+n: Nat, +c: Nat, z: TA.Mat, +labels: List<&2, Nat>) -> T.LossOut>: 1259 match z: 1260 case TA.Mat{d}: cx_1(n, c, labels, d, ids_arr(n, labels)) 1261 1262 # ---- main.bend's names for the same layers (#13) ---- 1263 # 1264 # A program written against torch/main.bend switches to this lane by its import line alone: 1265 # `import ../torch/gpu.bend as F` and F.Linear / F.LinearFirst / F.LinearNB, no renames. 1266 1267 def Linear(+n: Nat, +i: Nat, +o: Nat, +par: Nat) -> T.Layer, TA.Mat, TA.Mat, TA.Mat>: 1268 LinearF(n, i, o, par) 1269 1270 def LinearFirst(+n: Nat, +i: Nat, +o: Nat, +par: Nat) -> T.Layer, TA.Mat, TA.Mat, TA.Mat>: 1271 LinearFirstF(n, i, o, par) 1272 1273 def LinearNB(+n: Nat, +i: Nat, +o: Nat, +par: Nat) -> T.Layer, TA.Mat, TA.Mat, TA.Mat>: 1274 LinearNBF(n, i, o, par)