main.bend
bendtorch-gpu@1.0.1.0 · 1,258 lines · 0 definitions
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 (bend-ml
23# docs/upstream/gpu-math-overloads.md); 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
35import Base
36import bend-ml-tensor-array@0.1.14.0/main.bend as TA
37import bend-ml-tensor-gpu@0.1.2.1/main.bend as TG
38import bendtorch@1.0.1.0/main.bend as T
39
40# ---- Linear: y = x·Wᵀ + b ----
41
42def lin_fwd(+n: Nat, +i: Nat, +o: Nat, p: T.Lin<i, o>, x: TA.Mat<n, i>) -> T.Fwd<T.Lin<i, o>, TA.Mat<n, o>, TA.Mat<n, i>>:
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
47def lin_b2(+n: Nat, +i: Nat, +o: Nat, b: TA.Mat<1n, o>, x: TA.Mat<n, i>, g: T.Lin<i, o>, r: TA.MMul<n, o, i>) -> T.Bwd<T.Lin<i, o>, TA.Mat<n, i>>:
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
52def lin_bwd(+n: Nat, +i: Nat, +o: Nat, p: T.Lin<i, o>, x: TA.Mat<n, i>, dy: TA.Mat<n, o>, g: T.Lin<i, o>) -> T.Bwd<T.Lin<i, o>, TA.Mat<n, i>>:
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
64type EN is Data:
65 EN{n: U32}
66
67def nn(u: Unit) -> Array<F32>:
68 TG.none(u)
69
70# out[i] = f(x[i]) for the leaf's cells
71def ew1_loop(~f: @+a: F32 -> F32, k: Nat, +i: U32, +l: U32, o: Array<F32>, r: Array<F32> & F32) -> Array<F32>:
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
76def ew1(~f: @+a: F32 -> F32, +t: U32, +l: U32, +e: EN, x: Array<F32>, o: Array<F32>) -> Array<F32>:
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
81def ew2_loop(~f: @+a: F32 -> @+b: F32 -> F32, k: Nat, +i: U32, +l: U32, o: Array<F32>, ra: Array<F32> & F32, rb: Array<F32> & F32) -> Array<F32>:
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
86def ew2(~f: @+a: F32 -> @+b: F32 -> F32, +t: U32, +l: U32, +e: EN, x: Array<F32>, y: Array<F32>, o: Array<F32>) -> Array<F32>:
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)
91def ewi_loop(~f: @+a: F32 -> @+b: F32 -> F32, k: Nat, +i: U32, +l: U32, ro: Array<F32> & F32, ry: Array<F32> & F32) -> Array<F32>:
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
96def ewi(~f: @+a: F32 -> @+b: F32 -> F32, +t: U32, +l: U32, +e: EN, o: Array<F32>, y: Array<F32>) -> Array<F32>:
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
100def f32_id(+x: F32) -> F32:
101 x
102
103def lf_copy(+t: U32, +l: U32, +e: EN, x1: Array<F32>, x2: Array<F32>, x3: Array<F32>, x4: Array<F32>, o: Array<F32>) -> Array<F32>:
104 ew1(~f32_id, t, l, e, x1, o)
105
106def lf_silu(+t: U32, +l: U32, +e: EN, x1: Array<F32>, x2: Array<F32>, x3: Array<F32>, x4: Array<F32>, o: Array<F32>) -> Array<F32>:
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)
110def lf_silud(+t: U32, +l: U32, +e: EN, x1: Array<F32>, x2: Array<F32>, x3: Array<F32>, x4: Array<F32>, o: Array<F32>) -> Array<F32>:
111 ewi(~T.silu_d, t, l, e, o, x1)
112
113# o = a, written in place, x1 = b: a + b
114def lf_add(+t: U32, +l: U32, +e: EN, x1: Array<F32>, x2: Array<F32>, x3: Array<F32>, x4: Array<F32>, o: Array<F32>) -> Array<F32>:
115 ewi(~T.f32_add, t, l, e, o, x1)
116
117def lf_mul(+t: U32, +l: U32, +e: EN, x1: Array<F32>, x2: Array<F32>, x3: Array<F32>, x4: Array<F32>, o: Array<F32>) -> Array<F32>:
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
121def mulb_loop(k: Nat, +i: U32, +l: U32, rd: Array<F32> & F32, ra: Array<F32> & F32, rb: Array<F32> & 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
128def lf_mulb(+t: U32, +l: U32, +e: EN, x1: Array<F32>, x2: Array<F32>, x3: Array<F32>, x4: Array<F32>, 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
132def en(+cnt: Nat) -> EN:
133 EN{U32.from_nat(cnt)}
134
135def r1_a(r: TG.R1) -> Array<F32>:
136 match r:
137 case TG.R1{x1, x2, x3, x4, a}: a
138
139# ---- element-wise layers ----
140
141def dup_fin(-r: Nat, -c: Nat, q: TG.R1) -> T.Two<TA.Mat<r, c>, TA.Mat<r, c>>:
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
146def Mat.dup(+r: Nat, +c: Nat, m: TA.Mat<r, c>) -> T.Two<TA.Mat<r, c>, TA.Mat<r, c>>:
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)
151def Mat.sum2(+r: Nat, +c: Nat, a: TA.Mat<r, c>, b: TA.Mat<r, c>) -> TA.Mat<r, c>:
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
155def silu_f2(-n: Nat, -c: Nat, q: TG.R1) -> T.Fwd<Unit, TA.Mat<n, c>, TA.Mat<n, c>>:
156 match q:
157 case TG.R1{x1, x2, x3, x4, a}: T.Fwd{Unit{}, TA.Mat{a}, TA.Mat{x1}}
158
159def silu_f(+n: Nat, +c: Nat, x: TA.Mat<n, c>) -> T.Fwd<Unit, TA.Mat<n, c>, TA.Mat<n, c>>:
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
163def silu_b(+n: Nat, +c: Nat, s: TA.Mat<n, c>, dy: TA.Mat<n, c>) -> T.Bwd<Unit, TA.Mat<n, c>>:
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
168def SiLU(+n: Nat, +c: Nat) -> T.Layer<Unit, TA.Mat<n, c>, TA.Mat<n, c>, TA.Mat<n, c>>:
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
171def mul2_f2(-n: Nat, -c: Nat, q: TG.R1) -> T.Fwd<Unit, TA.Mat<n, c>, T.Two<TA.Mat<n, c>, TA.Mat<n, c>>>:
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
175def mul2_f(+n: Nat, +c: Nat, x: T.Two<TA.Mat<n, c>, TA.Mat<n, c>>) -> T.Fwd<Unit, TA.Mat<n, c>, T.Two<TA.Mat<n, c>, TA.Mat<n, c>>>:
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
179def mul2_b2(-n: Nat, -c: Nat, q: TG.R2) -> T.Bwd<Unit, T.Two<TA.Mat<n, c>, TA.Mat<n, c>>>:
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
183def mul2_b(+n: Nat, +c: Nat, s: T.Two<TA.Mat<n, c>, TA.Mat<n, c>>, dy: TA.Mat<n, c>) -> T.Bwd<Unit, T.Two<TA.Mat<n, c>, TA.Mat<n, c>>>:
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
188def Mul2(+n: Nat, +c: Nat) -> T.Layer<Unit, T.Two<TA.Mat<n, c>, TA.Mat<n, c>>, TA.Mat<n, c>, T.Two<TA.Mat<n, c>, TA.Mat<n, c>>>:
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
191def add2_f(+n: Nat, +c: Nat, x: T.Two<TA.Mat<n, c>, TA.Mat<n, c>>) -> T.Fwd<Unit, TA.Mat<n, c>, Unit>:
192 match x:
193 case T.Two{a, b}: T.Fwd{Unit{}, Mat.sum2(n, c, a, b), Unit{}}
194
195def Add2(+n: Nat, +c: Nat) -> T.Layer<Unit, T.Two<TA.Mat<n, c>, TA.Mat<n, c>>, TA.Mat<n, c>, 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
199def Residual(~n: Nat, ~c: Nat, ~P: Type, ~S: Type, ~f: T.Layer<P, TA.Mat<n, c>, TA.Mat<n, c>, S>) -> T.Layer<T.Two<T.Two<Unit, P>, Unit>, TA.Mat<n, c>, TA.Mat<n, c>, T.Two<T.Two<Unit, S>, Unit>>:
200 T.Seq(~T.Two<Unit, P>, ~Unit, ~TA.Mat<n, c>, ~T.Two<TA.Mat<n, c>, TA.Mat<n, c>>, ~TA.Mat<n, c>, ~T.Two<Unit, S>, ~Unit, ~T.Fork(~Unit, ~P, ~TA.Mat<n, c>, ~TA.Mat<n, c>, ~TA.Mat<n, c>, ~Unit, ~S, ~(m => Mat.dup(n, c, m)), ~(a => b => Mat.sum2(n, c, a, b)), ~T.Id(~TA.Mat<n, c>), ~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
204type ESGD is Data:
205 ESGD{n: U32, lr: F32}
206
207def sgd_loop(k: Nat, +i: U32, +l: U32, +lr: F32, rw: Array<F32> & F32, rg: Array<F32> & 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
213def lf_sgd(+t: U32, +l: U32, +e: ESGD, x1: Array<F32>, x2: Array<F32>, x3: Array<F32>, x4: Array<F32>, 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
217type EMOM is Data:
218 EMOM{n: U32, lr: F32, mu: F32}
219
220def mom_loop(k: Nat, +i: U32, +l: U32, +lr: F32, +mu: F32, rw: Array<F32> & F32, rg: Array<F32> & F32, rm: Array<F32> & 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
228def lf_mom(+t: U32, +l: U32, +e: EMOM, x1: Array<F32>, x2: Array<F32>, x3: Array<F32>, x4: Array<F32>, 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
232type 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
235def 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> & F32, rg: Array<F32> & F32, rm: Array<F32> & F32, rv: Array<F32> & 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
246def lf_adam(+t: U32, +l: U32, +e: EADAM, x1: Array<F32>, x2: Array<F32>, x3: Array<F32>, x4: Array<F32>, 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
250def sgd_fin(-r: Nat, -c: Nat, q: TG.R2) -> T.MatStep<r, c>:
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
254def mom_fin(-r: Nat, -c: Nat, q: TG.R3) -> T.MatStep<r, c>:
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
258def opt_fin(-r: Nat, -c: Nat, q: TG.R4) -> T.MatStep<r, c>:
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
263def opt_mat(+r: Nat, +c: Nat, +o: T.Opt, w: TA.Mat<r, c>, g: TA.Mat<r, c>, st: T.OptState<TA.Mat<r, c>>) -> T.MatStep<r, c>:
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 case _ w g st: T.MatStep{w, g, st}
272
273def mat_step(+r: Nat, +c: Nat, +op: T.Opt, p: TA.Mat<r, c>, g: TA.Mat<r, c>, st: T.OptState<TA.Mat<r, c>>) -> T.Step<TA.Mat<r, c>>:
274 T.mat_step_fin(r, c, opt_mat(r, c, op, p, g, st))
275
276def lin_step2(+i: Nat, +o: Nat, +op: T.Opt, p: T.Lin<i, o>, g: T.Lin<i, o>, st: T.Two<T.OptState<TA.Mat<o, i>>, T.OptState<TA.Mat<1n, o>>>) -> T.Step<T.Lin<i, o>>:
277 match p g st:
278 case T.Lin{w, b} T.Lin{gw, gb} T.Two{sw, sb}:
279 T.lin_s2(i, o, opt_mat(o, i, op, w, gw, sw), opt_mat(1n, o, op, b, gb, sb))
280
281def lin_step(+i: Nat, +o: Nat, +op: T.Opt, p: T.Lin<i, o>, g: T.Lin<i, o>, st: T.OptState<T.Lin<i, o>>) -> T.Step<T.Lin<i, o>>:
282 lin_step2(i, o, op, p, g, T.OptState.split(TA.Mat<o, i>, TA.Mat<1n, o>, T.OptState.map(T.Lin<i, o>, T.Two<TA.Mat<o, i>, TA.Mat<1n, o>>, st, x => T.lin_pair(i, o, x), x => T.lin_pair(i, o, x))))
283
284# ---- the linear layers ----
285
286def LinearF(+n: Nat, +i: Nat, +o: Nat, +par: Nat) -> T.Layer<T.Lin<i, o>, TA.Mat<n, i>, TA.Mat<n, o>, TA.Mat<n, i>>:
287 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))}
288
289# LinearFirst: no dX (the first layer's input needs no gradient)
290def linf_bwd(+n: Nat, +i: Nat, +o: Nat, p: T.Lin<i, o>, x: TA.Mat<n, i>, dy: TA.Mat<n, o>, g: T.Lin<i, o>) -> T.Bwd<T.Lin<i, o>, TA.Mat<n, i>>:
291 match p g:
292 case T.Lin{w, b} T.Lin{gw, gb}:
293 T.linf_b3(n, i, o, w, b, gb, TG.matmul_tn_acc(o, n, i, 0n, dy, x, gw))
294
295def LinearFirstF(+n: Nat, +i: Nat, +o: Nat, +par: Nat) -> T.Layer<T.Lin<i, o>, TA.Mat<n, i>, TA.Mat<n, o>, TA.Mat<n, i>>:
296 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))}
297
298# LinearNB: y = x·Wᵀ, no bias (Llama's projections)
299def lnb_b2(+n: Nat, +i: Nat, +o: Nat, x: TA.Mat<n, i>, g: TA.Mat<o, i>, r: TA.MMul<n, o, i>) -> T.Bwd<TA.Mat<o, i>, TA.Mat<n, i>>:
300 match r:
301 case TA.MMul{dy, w, dx}:
302 T.lnb_b3(n, i, o, w, dx, TG.matmul_tn_acc(o, n, i, 0n, dy, x, g))
303
304def LinearNBF(+n: Nat, +i: Nat, +o: Nat, +par: Nat) -> T.Layer<TA.Mat<o, i>, TA.Mat<n, i>, TA.Mat<n, o>, TA.Mat<n, i>>:
305 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))}
306
307# ---- RMSNorm: a leaf a row with main.bend's own row code (rmsf_row) ----
308
309type ERF is Data:
310 ERF{n: U32, d: Nat, d32: U32, df: F32, eps: F32}
311
312def r3_y(s: T.R3) -> Array<F32>:
313 match s:
314 case T.R3{x, g, y}: y
315
316def rmsf_if(ok: Bool, +t: U32, +d: Nat, +d32: U32, +df: F32, +eps: F32, x: Array<F32>, g: Array<F32>, y: Array<F32>) -> Array<F32>:
317 match ok:
318 case True{}: r3_y(T.rmsf_row(d, (t * d32 : U32), df, eps, T.R3{x, g, y}))
319 case False{}: y
320
321# x1 = x, x2 = the gain g; y a row a leaf
322def lf_rmsf(+t: U32, +l: U32, +e: ERF, x1: Array<F32>, x2: Array<F32>, x3: Array<F32>, x4: Array<F32>, y: Array<F32>) -> Array<F32>:
323 match e:
324 case ERF{+n, +d, +d32, +df, +eps}: rmsf_if(U32.is_lt(t, n), t, d, d32, df, eps, x1, x2, y)
325
326def erf(+n: Nat, +d: Nat, +eps: F32) -> ERF:
327 ERF{U32.from_nat(n), d, U32.from_nat(d), F32.from_nat(d), eps}
328
329def rms_f2(-n: Nat, -d: Nat, q: TG.R1) -> T.Fwd<TA.Mat<1n, d>, TA.Mat<n, d>, TA.Mat<n, d>>:
330 match q:
331 case TG.R1{x1, x2, x3, x4, y}: T.Fwd{TA.Mat{x2}, TA.Mat{y}, TA.Mat{x1}}
332
333def rms_f(+n: Nat, +d: Nat, +eps: F32, p: TA.Mat<1n, d>, x: TA.Mat<n, d>) -> T.Fwd<TA.Mat<1n, d>, TA.Mat<n, d>, TA.Mat<n, d>>:
334 match p x:
335 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))))
336
337# 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
338# 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
339# row's term into dg
340
341def gdx_w(k: Nat, +j: U32, +off: U32, +r: F32, +m: F32, dx: Array<F32>, rx: Array<F32> & F32, rd: Array<F32> & F32, rg: Array<F32> & F32) -> Array<F32>:
342 match k rx rd rg:
343 case 0n Tuple{x, +a} Tuple{dy, +b} Tuple{g, +c}:
344 +xh = (a * r : F32)
345 Array.set(F32, dx, (off + j : U32), (((b * c : F32) - (xh * m : F32) : F32) * r : F32))
346 case 1n+p Tuple{x, +a} Tuple{dy, +b} Tuple{g, +c}:
347 +xh = (a * r : F32)
348 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)))
349
350def gdx_3(+d: Nat, +off: U32, +df: F32, +r: F32, +t: U32, dx: Array<F32>, rr: Array<F32>, q: T.R3S) -> TG.O2:
351 match q:
352 case T.R3S{x, dy, g, +s}:
353 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)}
354
355def gdx_2(+d: Nat, +off: U32, +df: F32, +eps: F32, +t: U32, dy: Array<F32>, g: Array<F32>, dx: Array<F32>, rr: Array<F32>, q: T.AS) -> TG.O2:
356 match q:
357 case T.AS{x, +s}:
358 +r = T.rms_inv(s, df, eps)
359 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)))
360
361def gdx_if(ok: Bool, +t: U32, +d: Nat, +d32: U32, +df: F32, +eps: F32, x: Array<F32>, dy: Array<F32>, g: Array<F32>, dx: Array<F32>, rr: Array<F32>) -> TG.O2:
362 match ok:
363 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))))
364 case False{}: TG.O2{dx, rr}
365
366# x1 = x, x2 = dy, x3 = g; outputs dx and 1/rms of each row
367def lf_gdx(+t: U32, +l: U32, +e: ERF, x1: Array<F32>, x2: Array<F32>, x3: Array<F32>, x4: Array<F32>, o: TG.O2) -> TG.O2:
368 match e o:
369 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)
370
371# acc + dy[i]·(x[i]·r[row]) down a column, i from j by d
372def gdg_loop(k: Nat, +i: U32, +d: U32, +row: U32, acc: F32, rx: Array<F32> & F32, rd: Array<F32> & F32, rr: Array<F32> & F32) -> F32:
373 match k rx rd rr:
374 case 0n Tuple{x, +a} Tuple{dy, +b} Tuple{r, +c}: (acc + (b * (a * c : F32) : F32) : F32)
375 case 1n+p Tuple{x, +a} Tuple{dy, +b} Tuple{r, +c}:
376 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)))
377
378def gdg_put(+j: U32, +s: F32, r: Array<F32> & F32) -> Array<F32>:
379 match r:
380 case Tuple{ga, +v}: Array.set(F32, ga, j, (v + s : F32))
381
382type EDG is Data:
383 EDG{d: U32, n: Nat}
384
385def gdg_if(ok: Bool, +j: U32, +n: Nat, +d: U32, x: Array<F32>, dy: Array<F32>, rr: Array<F32>, ga: Array<F32>) -> Array<F32>:
386 match ok:
387 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))
388 case False{}: ga
389
390# x1 = x, x2 = dy, x3 = 1/rms; the gradient of the gain, in place: a leaf a column
391def lf_gdg(+t: U32, +l: U32, +e: EDG, x1: Array<F32>, x2: Array<F32>, x3: Array<F32>, x4: Array<F32>, ga: Array<F32>) -> Array<F32>:
392 match e:
393 case EDG{+d, +n}: gdg_if(U32.is_lt(t, d), t, n, d, x1, x2, x3, ga)
394
395def rms_b3(-n: Nat, -d: Nat, g: Array<F32>, dx: Array<F32>, q: TG.R1) -> T.Bwd<TA.Mat<1n, d>, TA.Mat<n, d>>:
396 match q:
397 case TG.R1{x1, x2, x3, x4, ga}: T.Bwd{TA.Mat{g}, TA.Mat{dx}, TA.Mat{ga}}
398
399def rms_b2(+n: Nat, +d: Nat, ga: Array<F32>, q: TG.R2) -> T.Bwd<TA.Mat<1n, d>, TA.Mat<n, d>>:
400 match q:
401 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))
402
403def rms_b(+n: Nat, +d: Nat, +eps: F32, p: TA.Mat<1n, d>, x: TA.Mat<n, d>, dy: TA.Mat<n, d>, gacc: TA.Mat<1n, d>) -> T.Bwd<TA.Mat<1n, d>, TA.Mat<n, d>>:
404 match p x dy gacc:
405 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)))
406
407# RMSNorm over the last dimension, its gain trained (main.bend's RMSNorm)
408def RMSNorm(+n: Nat, +d: Nat, +eps: F32) -> T.Layer<TA.Mat<1n, d>, TA.Mat<n, d>, TA.Mat<n, d>, TA.Mat<n, d>>:
409 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)}
410
411# ---- RoPE: the tables on the host (main.bend's rope_tab, so the host's cos and sin), each pair rotated in place by
412# main.bend's rope_one, the pairs dealt to the leaves round-robin ----
413
414type ERO is Data:
415 ERO{n: U32, fwd: Bool, hh: U32, half: U32, w: U32, t: U32}
416
417def rope_loop(k: Nat, +e: U32, +l: U32, +fwd: Bool, +hh: U32, +half: U32, +w: U32, +t: U32, st: T.RS3) -> T.RS3:
418 match k:
419 case 0n: st
420 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))
421
422def rs3_x(st: T.RS3) -> Array<F32>:
423 match st:
424 case T.RS3{x, c, s}: x
425
426# x1 = cos, x2 = sin; x rotated in place
427def lf_rope(+t: U32, +l: U32, +e: ERO, x1: Array<F32>, x2: Array<F32>, x3: Array<F32>, x4: Array<F32>, x: Array<F32>) -> Array<F32>:
428 match e:
429 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}))
430
431def rope_go2(+n: Nat, +w: Nat, +fwd: Bool, +t: Nat, +h: Nat, +half: Nat, x: Array<F32>, tb: T.CS) -> TA.Mat<n, w>:
432 match tb:
433 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))}
434
435def rope_go(+n: Nat, +w: Nat, +fwd: Bool, +t: Nat, +h: Nat, +half: Nat, +base: F32, m: TA.Mat<n, w>) -> TA.Mat<n, w>:
436 match m:
437 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))}))
438
439# RoPE on n = b·t rows of h heads of 2·half (main.bend's RoPE)
440def 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<Unit, TA.Mat<n, w>, TA.Mat<n, w>, Unit>:
441 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}
442
443type S4 is Type:
444 S4{a: Array<F32>, b: Array<F32>, c: Array<F32>, d: Array<F32>}
445
446# ---- attention in cell bangs and row bangs. A leaf handles cells or rows t, t + l, .. of one bang. A cell is one number
447# 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
448# 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
449# 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
450# main.bend's row code (psmc, g_set2), a tail loop over the row's i + 1 positions. Every number is computed by
451# main.bend's expression in main.bend's order, so the bits are main.bend's (but CUDA's exp in the softmax).
452# forward: 0. cells: kᵀ 1. cells: P's scores (q·k scaled, j <= i) 2. rows: the softmax over the i + 1
453# 3. cells: o = P·V over the query's i + 1 positions
454# backward: 0. cells: kᵀ and vᵀ 1. cells: P's scores again and dP = dO·V into G 2. rows: P's softmax, then
455# G = (P * (dP - s)) * sc with s = dP·P over the row, as gr3 3. cells: dQ = G·K, dK = Σ Gᵀ·Q and
456# dV = Σ Pᵀ·dO over the group's heads, dK written over kᵀ and dV over V (neither read any more)
457# 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
458# 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
459# sizes on the RTX 4050; this one takes 17 ms, about half of it filling the new arrays (one lane, Array.new). ----
460
461# 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
462# of a head's P
463type EAC is Data:
464 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}
465
466# 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
467# 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
468# started at +0.0, which leaves it as it is; so the sums below skip them and give the same bits (for finite numbers:
469# 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)
470def toff(+bh: U32, +i: U32, +tr: U32) -> U32:
471 ((bh * tr : U32) + U32.div((i * (i + 1 : U32) : U32), 2) : U32)
472
473def eac_n(+c: EAC) -> U32:
474 match c:
475 case EAC{+n, +t, +t32, +h, +g, +gn, +hd, +hd32, +w, +kvw, +sc, +nw, +nk, +tr}: n
476
477type S3 is Type:
478 S3{a: Array<F32>, b: Array<F32>, c: Array<F32>}
479
480def s3_c(s: S3) -> Array<F32>:
481 match s:
482 case S3{a, b, c}: c
483
484def sc_fin(y: Array<F32>, s: S3) -> TG.O2:
485 match s:
486 case S3{q, k, p}: TG.O2{p, y}
487
488# 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
489# i + 1 positions, · V's (or K's) column of the head's key/value head
490def pv_val(+e: U32, +c: EAC, a: Array<F32>, b: Array<F32>) -> T.D2:
491 match c:
492 case EAC{+n, +t, +t32, +h, +g, +gn, +hd, +hd32, +w, +kvw, +sc, +nw, +nk, +tr}:
493 +row = U32.div(e, w)
494 +col = U32.mod(e, w)
495 +bi = U32.div(row, t32)
496 +hi = U32.div(col, hd32)
497 +i = U32.mod(row, t32)
498 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)
499
500# 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
501# rows (sb apart), from row i on: dot0's sum, but with the step in a growing by one a row
502def tdot(k: Nat, +ia: U32, +ib: U32, +i: U32, +sb: U32, acc: F32, ra: Array<F32> & F32, rb: Array<F32> & F32) -> T.D2:
503 match k ra rb:
504 case 0n Tuple{a, x} Tuple{b, y}: T.D2{a, b, acc}
505 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)))
506
507# 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
508# apart a head), rows j..t-1
509def 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:
510 match k r:
511 case 0n T.D2{a, b, +s}: T.D2{a, b, (acc + s : F32)}
512 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))))
513
514# 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
515# heads: the head's column j of a (rows j..t-1) · the head's column of b
516def kv_val(+e: U32, +c: EAC, a: Array<F32>, b: Array<F32>) -> T.D2:
517 match c:
518 case EAC{+n, +t, +t32, +h, +g, +gn, +hd, +hd32, +w, +kvw, +sc, +nw, +nk, +tr}:
519 +row = U32.div(e, kvw)
520 +col = U32.mod(e, kvw)
521 +bi = U32.div(row, t32)
522 +jj = U32.div(col, hd32)
523 +j = U32.mod(row, t32)
524 +ia = (toff(((bi * h : U32) + (jj * g : U32) : U32), j, tr) + j : U32)
525 +ib = ((((bi * t32 : U32) + j : U32) * w : U32) + (((jj * g : U32) * hd32 : U32) + U32.mod(col, hd32) : U32) : U32)
526 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)))
527
528# K and V transposed for the score cells (kᵀ: b·kv·hd·t, by sequence, key/value head, column c, position j), so
529# that neighbouring lanes, which take neighbouring keys j, read neighbouring cells. dot0 reads kᵀ with stride t: the same
530# numbers in the same order as reading k's row, so the same bits.
531def tk_src(+e: U32, +t32: U32, +kv: U32, +hd32: U32, +kvw: U32) -> U32:
532 +r = U32.div(e, t32)
533 +bj = U32.div(r, hd32)
534 +row = ((U32.div(bj, kv) * t32 : U32) + U32.mod(e, t32) : U32)
535 ((row * kvw : U32) + ((U32.mod(bj, kv) * hd32 : U32) + U32.mod(r, hd32) : U32) : U32)
536
537def tk_put(+e: U32, o: Array<F32>, r: Array<F32> & F32) -> TG.O2:
538 match r:
539 case Tuple{a, +x}: TG.O2{a, Array.set(F32, o, e, x)}
540
541def tk_loop(k: Nat, +e: U32, +l: U32, +t32: U32, +kv: U32, +hd32: U32, +kvw: U32, s: TG.O2) -> TG.O2:
542 match k s:
543 case 0n r: r
544 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))))
545
546def o2_b(s: TG.O2) -> Array<F32>:
547 match s:
548 case TG.O2{a, b}: b
549
550# x1 = k; kᵀ written (made here)
551def lf_tk(+t: U32, +l: U32, +c: EAC, x1: Array<F32>, x2: Array<F32>, x3: Array<F32>, x4: Array<F32>, o: Array<F32>) -> Array<F32>:
552 match c:
553 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}))
554
555# both: cells [0, nk) are kᵀ's from k, the next nk vᵀ's from v (S4{k, v, kᵀ, vᵀ})
556def tkv_k(v: Array<F32>, vt: Array<F32>, r: TG.O2) -> S4:
557 match r:
558 case TG.O2{k, kt}: S4{k, v, kt, vt}
559
560def tkv_v(k: Array<F32>, kt: Array<F32>, r: TG.O2) -> S4:
561 match r:
562 case TG.O2{v, vt}: S4{k, v, kt, vt}
563
564def tkv_sel(lo: Bool, +e: U32, +nk: U32, +t32: U32, +kv: U32, +hd32: U32, +kvw: U32, s: S4) -> S4:
565 match lo s:
566 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))))
567 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))))
568
569def tkv_loop(k: Nat, +e: U32, +l: U32, +nk: U32, +t32: U32, +kv: U32, +hd32: U32, +kvw: U32, s: S4) -> S4:
570 match k:
571 case 0n: s
572 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))
573
574def tkv_o(dq: Array<F32>, s: S4) -> TG.O3:
575 match s:
576 case S4{k, v, kt, vt}: TG.O3{kt, vt, dq}
577
578# x1 = k, x2 = v; outputs kᵀ and vᵀ (written), dQ (made here, written by backward 3)
579def lf_tkv(+t: U32, +l: U32, +c: EAC, x1: Array<F32>, x2: Array<F32>, x3: Array<F32>, x4: Array<F32>, o: TG.O3) -> TG.O3:
580 match c o:
581 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}))
582
583# 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
584# 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);
585# in the middle row of an odd t the second half repeats the first and is skipped (ok false)
586type IJ is Data:
587 IJ{ok: Bool, bh: U32, i: U32, j: U32}
588
589def ij_of(+e: U32, +t32: U32) -> IJ:
590 +np = U32.div((t32 + 1 : U32), 2)
591 +c = U32.mod(e, (t32 + 1 : U32))
592 +pr = U32.div(e, (t32 + 1 : U32))
593 +p = U32.mod(pr, np)
594 +lo = U32.is_le(c, p)
595 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))}
596
597# 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})
598def sc_put(+e: U32, +sc: F32, p: Array<F32>, r: T.D2) -> S3:
599 match r:
600 case T.D2{q, k, +s}: S3{q, k, Array.set(F32, p, e, (s * sc : F32))}
601
602def sc_if(ok: Bool, +e: U32, +qa: U32, +ka: U32, +hd: Nat, +t32: U32, +sc: F32, s: S3) -> S3:
603 match ok s:
604 case True{} S3{q, k, p}: sc_put(e, sc, p, T.dot0(hd, qa, ka, 1, t32, q, k))
605 case False{} S3{q, k, p}: S3{q, k, p}
606
607def sc_ij(+a: IJ, +t32: U32, +h: U32, +g: U32, +hd: Nat, +hd32: U32, +w: U32, +sc: F32, +tr: U32, s: S3) -> S3:
608 match a:
609 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)
610
611def sc_cell(+e: U32, +c: EAC, s: S3) -> S3:
612 match c:
613 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)
614
615def sc_loop(k: Nat, +e: U32, +l: U32, +c: EAC, s: S3) -> S3:
616 match k:
617 case 0n: s
618 case 1n+q: sc_loop(q, (e + l : U32), l, c, sc_cell(e, c, s))
619
620# x1 = q, x2 = k; outputs P (written) and o (made here, written by forward 3)
621def lf_sc(+t: U32, +l: U32, +c: EAC, x1: Array<F32>, x2: Array<F32>, x3: Array<F32>, x4: Array<F32>, o: TG.O2) -> TG.O2:
622 match o:
623 case TG.O2{p, y}: sc_fin(y, sc_loop(TG.share(t, eac_n(c), l), t, l, c, S3{x1, x2, p}))
624
625# forward 2: P's row r (query i = r % t) by main.bend's softmax over its i + 1 positions
626def sm_loop(k: Nat, +r: U32, +l: U32, +t32: U32, +tr: U32, p: Array<F32>) -> Array<F32>:
627 match k:
628 case 0n: p
629 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)))
630
631def lf_sm(+t: U32, +l: U32, +c: EAC, x1: Array<F32>, x2: Array<F32>, x3: Array<F32>, x4: Array<F32>, p: Array<F32>) -> Array<F32>:
632 match c:
633 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)
634
635# forward 3 (and backward 3's dQ): o[e] from pv_val (S3{a, b, o})
636def put_c(+e: U32, o: Array<F32>, r: T.D2) -> S3:
637 match r:
638 case T.D2{a, b, +s}: S3{a, b, Array.set(F32, o, e, s)}
639
640def pv_loop(k: Nat, +e: U32, +l: U32, +c: EAC, s: S3) -> S3:
641 match k s:
642 case 0n r: r
643 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)))
644
645# x1 = P, x2 = v; o written in place
646def lf_pv(+t: U32, +l: U32, +c: EAC, x1: Array<F32>, x2: Array<F32>, x3: Array<F32>, x4: Array<F32>, o: Array<F32>) -> Array<F32>:
647 s3_c(pv_loop(TG.share(t, eac_n(c), l), t, l, c, S3{x1, x2, o}))
648
649# 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})
650type S6 is Type:
651 S6{q: Array<F32>, k: Array<F32>, v: Array<F32>, d: Array<F32>, p: Array<F32>, g: Array<F32>}
652
653def b1_g(+e: U32, q: Array<F32>, k: Array<F32>, p: Array<F32>, g: Array<F32>, r: T.D2) -> S6:
654 match r:
655 case T.D2{d, v, +s}: S6{q, k, v, d, p, Array.set(F32, g, e, s)}
656
657def b1_p(+e: U32, +sc: F32, +qa: U32, +ka: U32, +hd: Nat, +t32: U32, v: Array<F32>, d: Array<F32>, p: Array<F32>, g: Array<F32>, r: T.D2) -> S6:
658 match r:
659 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))
660
661def b1_if(ok: Bool, +e: U32, +qa: U32, +ka: U32, +hd: Nat, +t32: U32, +sc: F32, s: S6) -> S6:
662 match ok s:
663 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))
664 case False{} S6{q, k, v, d, p, g}: S6{q, k, v, d, p, g}
665
666def b1_ij(+a: IJ, +t32: U32, +h: U32, +g: U32, +hd: Nat, +hd32: U32, +w: U32, +sc: F32, +tr: U32, s: S6) -> S6:
667 match a:
668 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)
669
670def b1_cell(+e: U32, +c: EAC, s: S6) -> S6:
671 match c:
672 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)
673
674def b1_loop(k: Nat, +e: U32, +l: U32, +c: EAC, s: S6) -> S6:
675 match k:
676 case 0n: s
677 case 1n+q: b1_loop(q, (e + l : U32), l, c, b1_cell(e, c, s))
678
679def b1_o(s: S6) -> TG.O2:
680 match s:
681 case S6{q, k, v, d, p, g}: TG.O2{p, g}
682
683# x1 = q, x2 = kᵀ, x3 = vᵀ, x4 = dO; outputs P and G (made here, written)
684def lf_b1(+t: U32, +l: U32, +c: EAC, x1: Array<F32>, x2: Array<F32>, x3: Array<F32>, x4: Array<F32>, o: TG.O2) -> TG.O2:
685 match o:
686 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}))
687
688# 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,
689# main.bend's gr3 and g_set2
690def gr_2(+n: Nat, +off: U32, +sc: F32, g: Array<F32>, r: T.PC) -> T.A2:
691 match r:
692 case T.PC{p, c}: T.g_set2(n, sc, off, T.dot0(n, off, off, 1, 1, g, p))
693
694def gr_loop(k: Nat, +r: U32, +l: U32, +t32: U32, +tr: U32, +sc: F32, s: T.A2) -> T.A2:
695 match k s:
696 case 0n r: r
697 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)))
698
699def a2_o2(s: T.A2) -> TG.O2:
700 match s:
701 case T.A2{p, g}: TG.O2{p, g}
702
703def lf_gr(+t: U32, +l: U32, +c: EAC, x1: Array<F32>, x2: Array<F32>, x3: Array<F32>, x4: Array<F32>, o: TG.O2) -> TG.O2:
704 match c o:
705 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}))
706
707# 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})
708type S8 is Type:
709 S8{q: Array<F32>, k: Array<F32>, d: Array<F32>, g: Array<F32>, p: Array<F32>, dq: Array<F32>, dk: Array<F32>, v: Array<F32>}
710
711def b3_q(+e: U32, q: Array<F32>, d: Array<F32>, p: Array<F32>, dq: Array<F32>, dk: Array<F32>, v: Array<F32>, r: T.D2) -> S8:
712 match r:
713 case T.D2{g, k, +s}: S8{q, k, d, g, p, Array.set(F32, dq, e, s), dk, v}
714
715def b3_k(+e: U32, k: Array<F32>, d: Array<F32>, p: Array<F32>, dq: Array<F32>, dk: Array<F32>, v: Array<F32>, r: T.D2) -> S8:
716 match r:
717 case T.D2{g, q, +s}: S8{q, k, d, g, p, dq, Array.set(F32, dk, e, s), v}
718
719def b3_v(+e: U32, q: Array<F32>, k: Array<F32>, g: Array<F32>, dq: Array<F32>, dk: Array<F32>, v: Array<F32>, r: T.D2) -> S8:
720 match r:
721 case T.D2{p, d, +s}: S8{q, k, d, g, p, dq, dk, Array.set(F32, v, e, s)}
722
723def b3_sel(a: Bool, b: Bool, +e: U32, +nw: U32, +nk: U32, +c: EAC, s: S8) -> S8:
724 match a b s:
725 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))
726 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))
727 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))
728
729def b3_cell(+e: U32, +c: EAC, s: S8) -> S8:
730 match c:
731 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)
732
733def b3_loop(k: Nat, +e: U32, +l: U32, +c: EAC, s: S8) -> S8:
734 match k:
735 case 0n: s
736 case 1n+q: b3_loop(q, (e + l : U32), l, c, b3_cell(e, c, s))
737
738def b3_o(s: S8) -> TG.O4:
739 match s:
740 case S8{q, k, d, g, p, dq, dk, v}: TG.O4{p, dq, dk, v}
741
742# x1 = q, x2 = k, x3 = dO, x4 = G; P read from the first output (unchanged), dQ, dK and dV (over V) written
743def lf_b3(+t: U32, +l: U32, +c: EAC, x1: Array<F32>, x2: Array<F32>, x3: Array<F32>, x4: Array<F32>, o: TG.O4) -> TG.O4:
744 match o:
745 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}))
746
747# the sizes of a bang of cnt cells (or rows)
748def eac(+cnt: Nat, +n: Nat, +w: Nat, +kvw: Nat, +t: Nat, +kv: Nat, +g: Nat, +hd: Nat, +sc: F32) -> EAC:
749 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)}
750
751# 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)
752def pcells(+b: Nat, +t: Nat, +kv: Nat, +g: Nat) -> Nat:
753 Nat.mul(Nat.mul(b, Nat.mul(kv, g)), Nat.div(Nat.mul(t, 1n+t), 2n))
754
755def tcells(+b: Nat, +t: Nat, +kv: Nat, +g: Nat) -> Nat:
756 Nat.mul(Nat.mul(Nat.mul(b, Nat.mul(kv, g)), Nat.div(1n+t, 2n)), 1n+t)
757
758def prows(+b: Nat, +t: Nat, +kv: Nat, +g: Nat) -> Nat:
759 Nat.mul(Nat.mul(b, Nat.mul(kv, g)), t)
760
761def af3(-n: Nat, -w: Nat, -kvw: Nat, q: Array<F32>, k: Array<F32>, r: TG.R1) -> T.Fwd<Unit, TA.Mat<n, w>, T.AttnS>:
762 match r:
763 case TG.R1{p, v, x3, x4, o}: T.Fwd{Unit{}, TA.Mat{o}, T.AttnS{q, k, v}}
764
765def af2(+n: Nat, +w: Nat, +kvw: Nat, +b: Nat, +t: Nat, +kv: Nat, +g: Nat, +hd: Nat, +sc: F32, q: Array<F32>, k: Array<F32>, v: Array<F32>, o: Array<F32>, r: TG.R1) -> T.Fwd<Unit, TA.Mat<n, w>, T.AttnS>:
766 match r:
767 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))
768
769def af1(+n: Nat, +w: Nat, +kvw: Nat, +b: Nat, +t: Nat, +kv: Nat, +g: Nat, +hd: Nat, +sc: F32, k: Array<F32>, v: Array<F32>, r: TG.R2) -> T.Fwd<Unit, TA.Mat<n, w>, T.AttnS>:
770 match r:
771 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))
772
773def af0(+n: Nat, +w: Nat, +kvw: Nat, +b: Nat, +t: Nat, +kv: Nat, +g: Nat, +hd: Nat, +sc: F32, q: Array<F32>, v: Array<F32>, r: TG.R1) -> T.Fwd<Unit, TA.Mat<n, w>, T.AttnS>:
774 match r:
775 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))))
776
777def attn_f(+n: Nat, +w: Nat, +kvw: Nat, +b: Nat, +t: Nat, +kv: Nat, +g: Nat, +hd: Nat, +sc: F32, x: T.Two<TA.Mat<n, w>, T.Two<TA.Mat<n, kvw>, TA.Mat<n, kvw>>>) -> T.Fwd<Unit, TA.Mat<n, w>, T.AttnS>:
778 match x:
779 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))))
780
781def ab4(-n: Nat, -w: Nat, -kvw: Nat, r: TG.R4) -> T.Bwd<Unit, T.Two<TA.Mat<n, w>, T.Two<TA.Mat<n, kvw>, TA.Mat<n, kvw>>>>:
782 match r:
783 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{}}
784
785# backward 3 writes dK over kᵀ and dV over v (neither read any more)
786def ab3(+n: Nat, +w: Nat, +kvw: Nat, +b: Nat, +t: Nat, +kv: Nat, +g: Nat, +hd: Nat, +sc: F32, q: Array<F32>, k: Array<F32>, v: Array<F32>, d: Array<F32>, dq: Array<F32>, kt: Array<F32>, r: TG.R2) -> T.Bwd<Unit, T.Two<TA.Mat<n, w>, T.Two<TA.Mat<n, kvw>, TA.Mat<n, kvw>>>>:
787 match r:
788 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))
789
790def ab2(+n: Nat, +w: Nat, +kvw: Nat, +b: Nat, +t: Nat, +kv: Nat, +g: Nat, +hd: Nat, +sc: F32, k: Array<F32>, v: Array<F32>, dq: Array<F32>, r: TG.R2) -> T.Bwd<Unit, T.Two<TA.Mat<n, w>, T.Two<TA.Mat<n, kvw>, TA.Mat<n, kvw>>>>:
791 match r:
792 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))
793
794def ab1(+n: Nat, +w: Nat, +kvw: Nat, +b: Nat, +t: Nat, +kv: Nat, +g: Nat, +hd: Nat, +sc: F32, q: Array<F32>, d: Array<F32>, r: TG.R3) -> T.Bwd<Unit, T.Two<TA.Mat<n, w>, T.Two<TA.Mat<n, kvw>, TA.Mat<n, kvw>>>>:
795 match r:
796 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))))
797
798def 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<n, w>) -> T.Bwd<Unit, T.Two<TA.Mat<n, w>, T.Two<TA.Mat<n, kvw>, TA.Mat<n, kvw>>>>:
799 match s dy:
800 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))))
801
802# causal attention with grouped heads over b sequences of t positions (main.bend's Attention)
803def 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<Unit, T.Two<TA.Mat<n, w>, T.Two<TA.Mat<n, kvw>, TA.Mat<n, kvw>>>, TA.Mat<n, w>, T.AttnS>:
804 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}
805
806# ---- MSE on the device. main.bend's MSE writes each gradient cell 2 (y - t) / (n·c) over y and sums the squares with
807# tree_sum, whose passes add neighbouring pairs; its first p passes carry no odd cell when 2^p divides the count, so a
808# leaf that sums a block of 2^p squares as a balanced tree of pairs gets exactly the number those passes leave at the
809# block's place. The leaves do that on the device (y written in place), and the host finishes tree_sum over the
810# count / 2^p block sums: the same bits as main.bend's MSE, with only the block sums read back. ----
811
812type EMS is Data:
813 EMS{nb: U32, p: Nat, bs: U32, kk: F32}
814
815type MS is Type:
816 MS{y: Array<F32>, t: Array<F32>, s: F32}
817
818def ms_add(+s: F32, r: MS) -> MS:
819 match r:
820 case MS{y, t, +s2}: MS{y, t, (s + s2 : F32)}
821
822def ms_then(r: MS, k: @+s: F32 -> Array<F32> -> Array<F32> -> MS) -> MS:
823 match r:
824 case MS{y, t, +s}: k(s, y, t)
825
826def ms_cell(+i: U32, +kk: F32, ry: Array<F32> & F32, rt: Array<F32> & F32) -> MS:
827 match ry rt:
828 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)}
829
830# the squares of the block of 2^p cells from i summed as a balanced tree of pairs (h = 2^(p-1)), gradients written
831def ms_blk(p: Nat, +i: U32, +h: U32, +kk: F32, y: Array<F32>, t: Array<F32>) -> MS:
832 match p:
833 case 0n: ms_cell(i, kk, Array.get(F32, y, i), Array.get(F32, t, i))
834 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)))
835
836def ms_put(+t: U32, ps: Array<F32>, r: MS) -> TG.O2:
837 match r:
838 case MS{y, tt, +s}: TG.O2{y, Array.set(F32, ps, t, s)}
839
840def ms_if(ok: Bool, +t: U32, +p: Nat, +bs: U32, +kk: F32, tg: Array<F32>, y: Array<F32>, ps: Array<F32>) -> TG.O2:
841 match ok:
842 case True{}: ms_put(t, ps, ms_blk(p, (t * bs : U32), U32.shr(bs), kk, y, tg))
843 case False{}: TG.O2{y, ps}
844
845# x1 = the target; y written in place with the gradient, ps[t] = leaf t's block sum (one leaf a block)
846def lf_mse(+t: U32, +l: U32, +e: EMS, x1: Array<F32>, x2: Array<F32>, x3: Array<F32>, x4: Array<F32>, o: TG.O2) -> TG.O2:
847 match e o:
848 case EMS{+nb, +p, +bs, +kk} TG.O2{y, ps}: ms_if(U32.is_lt(t, nb), t, p, bs, kk, x1, y, ps)
849
850# the largest p <= k with 2^p dividing m (m > 0)
851def ms_p(k: Nat, +m: U32) -> Nat:
852 match k:
853 case 0n: 0n
854 case 1n+q: Bool.pick(Nat, U32.is_eq(U32.mod(m, 2), 0), 1n+ms_p(q, U32.div(m, 2)), 0n)
855
856def mse_fin(-n: Nat, -c: Nat, +nb: U32, +nc: F32, q: TG.R2) -> T.LossOut<TA.Mat<n, c>>:
857 match q:
858 case TG.R2{x1, x2, x3, x4, TG.O2{y, ps}}: T.LossOut{(T.tree_sum(nb, ps) / nc : F32), TA.Mat{y}}
859
860def mse_go(+n: Nat, +c: Nat, +m: U32, +p: Nat, y: Array<F32>, t: Array<F32>) -> T.LossOut<TA.Mat<n, c>>:
861 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)))))
862
863# mean squared error over every number (main.bend's MSE, the same bits)
864def MSE(+n: Nat, +c: Nat, y: TA.Mat<n, c>, t: TA.Mat<n, c>) -> T.LossOut<TA.Mat<n, c>>:
865 match y t:
866 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)
867
868# ---- 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
869# 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 ----
870
871def u2f(x: U32) -> F32:
872 match x:
873 case U32{w}: F32{w}
874
875def f2u(x: F32) -> U32:
876 match x:
877 case F32{w}: U32{w}
878
879type IA is Type:
880 IA{a: Array<F32>, k: Nat}
881
882def ia_go(left: Nat, ids: List<&2, Nat>, +i: U32, k: Nat, a: Array<F32>) -> IA:
883 match left ids:
884 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))))
885 case _ _: IA{a, k}
886
887def ids_arr(+n: Nat, ids: List<&2, Nat>) -> IA:
888 ia_go(n, ids, 0, 0n, T.zeros_a(n))
889
890# ---- Embedding: a leaf a column j of the table (d of them), each looping over the ids in order: the forward pass
891# 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
892# after the other, so every cell of the gradient gets its rows added in main.bend's order, with no two leaves writing
893# one cell. Neighbouring lanes read neighbouring cells. ----
894
895type EEM is Data:
896 EEM{k: Nat, d: U32}
897
898type ES3 is Type:
899 ES3{ids: Array<F32>, a: Array<F32>, o: Array<F32>}
900
901def ef_3(+i: U32, +j: U32, +d: U32, ids: Array<F32>, o: Array<F32>, r: Array<F32> & F32) -> ES3:
902 match r:
903 case Tuple{e, +x}: ES3{ids, e, Array.set(F32, o, ((i * d : U32) + j : U32), x)}
904
905def ef_2(+i: U32, +j: U32, +d: U32, e: Array<F32>, o: Array<F32>, r: Array<F32> & F32) -> ES3:
906 match r:
907 case Tuple{ids, +v}: ef_3(i, j, d, ids, o, Array.get(F32, e, ((f2u(v) * d : U32) + j : U32)))
908
909# o[i, j] = e[ids[i], j]
910def ef_cell(+i: U32, +j: U32, +d: U32, st: ES3) -> ES3:
911 match st:
912 case ES3{ids, e, o}: ef_2(i, j, d, e, o, Array.get(F32, ids, i))
913
914def ef_loop(k: Nat, +i: U32, +j: U32, +d: U32, st: ES3) -> ES3:
915 match k:
916 case 0n: st
917 case 1n+q: ef_loop(q, (i + 1 : U32), j, d, ef_cell(i, j, d, st))
918
919def es3_o(st: ES3) -> Array<F32>:
920 match st:
921 case ES3{ids, a, o}: o
922
923def ef_if(ok: Bool, +t: U32, +k: Nat, +d: U32, e: Array<F32>, ids: Array<F32>, o: Array<F32>) -> Array<F32>:
924 match ok:
925 case True{}: es3_o(ef_loop(k, 0, t, d, ES3{ids, e, o}))
926 case False{}: o
927
928# x1 = the table, x2 = the ids; o (made on the device, zeros) gets the rows
929def lf_embf(+t: U32, +l: U32, +e: EEM, x1: Array<F32>, x2: Array<F32>, x3: Array<F32>, x4: Array<F32>, o: Array<F32>) -> Array<F32>:
930 match e:
931 case EEM{+k, +d}: ef_if(U32.is_lt(t, d), t, k, d, x1, x2, o)
932
933def eb_4(+gi: U32, ids: Array<F32>, g: Array<F32>, +a: F32, r: Array<F32> & F32) -> ES3:
934 match r:
935 case Tuple{dy, +b}: ES3{ids, dy, Array.set(F32, g, gi, (a + b : F32))}
936
937def eb_3(+i: U32, +j: U32, +d: U32, +gi: U32, ids: Array<F32>, dy: Array<F32>, r: Array<F32> & F32) -> ES3:
938 match r:
939 case Tuple{g, +a}: eb_4(gi, ids, g, a, Array.get(F32, dy, ((i * d : U32) + j : U32)))
940
941def eb_2(+i: U32, +j: U32, +d: U32, dy: Array<F32>, g: Array<F32>, r: Array<F32> & F32) -> ES3:
942 match r:
943 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)))
944
945# g[ids[i], j] = g[ids[i], j] + dy[i, j]
946def eb_cell(+i: U32, +j: U32, +d: U32, st: ES3) -> ES3:
947 match st:
948 case ES3{ids, dy, g}: eb_2(i, j, d, dy, g, Array.get(F32, ids, i))
949
950def eb_loop(k: Nat, +i: U32, +j: U32, +d: U32, st: ES3) -> ES3:
951 match k:
952 case 0n: st
953 case 1n+q: eb_loop(q, (i + 1 : U32), j, d, eb_cell(i, j, d, st))
954
955def eb_if(ok: Bool, +t: U32, +k: Nat, +d: U32, dy: Array<F32>, ids: Array<F32>, g: Array<F32>) -> Array<F32>:
956 match ok:
957 case True{}: es3_o(eb_loop(k, 0, t, d, ES3{ids, dy, g}))
958 case False{}: g
959
960# x1 = dy, x2 = the ids; the gradient g, in place
961def lf_embb(+t: U32, +l: U32, +e: EEM, x1: Array<F32>, x2: Array<F32>, x3: Array<F32>, x4: Array<F32>, g: Array<F32>) -> Array<F32>:
962 match e:
963 case EEM{+k, +d}: eb_if(U32.is_lt(t, d), t, k, d, x1, x2, g)
964
965def emb_f3(-n: Nat, -v: Nat, -d: Nat, +ids: List<&2, Nat>, q: TG.R1) -> T.Fwd<TA.Mat<v, d>, TA.Mat<n, d>, List<&2, Nat>>:
966 match q:
967 case TG.R1{x1, x2, x3, x4, o}: T.Fwd{TA.Mat{x1}, TA.Mat{o}, ids}
968
969def emb_f2(+n: Nat, +v: Nat, +d: Nat, +ids: List<&2, Nat>, ea: Array<F32>, r: IA) -> T.Fwd<TA.Mat<v, d>, TA.Mat<n, d>, List<&2, Nat>>:
970 match r:
971 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))))
972
973def emb_f(+n: Nat, +v: Nat, +d: Nat, +ids: List<&2, Nat>, e: TA.Mat<v, d>) -> T.Fwd<TA.Mat<v, d>, TA.Mat<n, d>, List<&2, Nat>>:
974 match e:
975 case TA.Mat{ea}: emb_f2(n, v, d, ids, ea, ids_arr(n, ids))
976
977def emb_b3(-v: Nat, -d: Nat, e: TA.Mat<v, d>, q: TG.R1) -> T.Bwd<TA.Mat<v, d>, List<&2, Nat>>:
978 match q:
979 case TG.R1{x1, x2, x3, x4, g}: T.Bwd{e, Nil{}, TA.Mat{g}}
980
981def emb_b2(+v: Nat, +d: Nat, e: TA.Mat<v, d>, da: Array<F32>, ga: Array<F32>, r: IA) -> T.Bwd<TA.Mat<v, d>, List<&2, Nat>>:
982 match r:
983 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))
984
985def emb_bwd(+n: Nat, +v: Nat, +d: Nat, e: TA.Mat<v, d>, ids: List<&2, Nat>, dy: TA.Mat<n, d>, g: TA.Mat<v, d>) -> T.Bwd<TA.Mat<v, d>, List<&2, Nat>>:
986 match dy g:
987 case TA.Mat{da} TA.Mat{ga}: emb_b2(v, d, e, da, ga, ids_arr(n, ids))
988
989# 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
990def Embedding(+n: Nat, +v: Nat, +d: Nat) -> T.Layer<TA.Mat<v, d>, List<&2, Nat>, TA.Mat<n, d>, List<&2, Nat>>:
991 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))}
992
993# ---- cross-entropy on the device, the same numbers as main.bend's CrossEntropy (but CUDA's exp on the GPU), in four
994# bangs; only 3 numbers a row come back to the host, and the gradient stays on the device:
995# 1. a leaf a row: its max m by main.bend's pmax0 (the same order of comparisons) and x[y] - m;
996# 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
997# 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
998# these pairs (no odd cell is carried while 2^p divides the count), so a block's sum is the number they leave there;
999# 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;
1000# 4. a leaf for 2^14 cells at a time: ((e / s) - [j == y]) / n, pdiv's and ce_g's two expressions.
1001# 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
1002# they are, as in main.bend; labels past the n rows are not read (main.bend would write past the matrix). ----
1003
1004type TS is Type:
1005 TS{a: Array<F32>, s: F32}
1006
1007def ts_leaf(r: Array<F32> & F32) -> TS:
1008 match r:
1009 case Tuple{a, +x}: TS{a, x}
1010
1011def ts_plus(+s: F32, r: TS) -> TS:
1012 match r:
1013 case TS{a, +s2}: TS{a, (s + s2 : F32)}
1014
1015def ts_then(r: TS, k: @+s: F32 -> Array<F32> -> TS) -> TS:
1016 match r:
1017 case TS{a, +s}: k(s, a)
1018
1019# 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 =
1020# 2^(l-1) cells) and its right child (the rest) when there is a rest (big = h < len), and is its left child otherwise
1021# (tree_sum's odd last cell carried up a pass)
1022def tnode(l: Nat, big: Bool, +lo: U32, +len: U32, +h: U32, a: Array<F32>) -> TS:
1023 match l big:
1024 case 0n _: ts_leaf(Array.get(F32, a, lo))
1025 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)))
1026 case 1n+q False{}: tnode(q, U32.is_lt(U32.shr(h), len), lo, len, U32.shr(h), a)
1027
1028# the passes tree_sum makes over m cells: the smallest l with 2^l >= m (f is fuel)
1029def ts_lv(f: Nat, +m: U32, +p: U32) -> Nat:
1030 match f:
1031 case 0n: 0n
1032 case 1n+q: Bool.pick(Nat, U32.is_lt(p, m), 1n+ts_lv(q, m, (p * 2 : U32)), 0n)
1033
1034# 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
1035# last class
1036type LS is Type:
1037 LS{a: Array<F32>, l: F32}
1038
1039def ce_t2(ok: Bool, +loss: F32, +xm: F32, r: Array<F32> & F32) -> LS:
1040 match ok r:
1041 case True{} Tuple{a, +s}: LS{a, (loss - (xm - F32.log(s) : F32) : F32)}
1042 case False{} Tuple{a, s}: LS{a, (loss - F32.log(F32.max(0.0, 0.000000000001)) : F32)}
1043
1044def ce_g4(+n: Nat, -c: Nat, z: Array<F32>, r: LS) -> T.LossOut<TA.Mat<n, c>>:
1045 match r:
1046 case LS{a, +l}: T.LossOut{(l / F32.from_nat(n) : F32), TA.Mat{z}}
1047
1048# 1. a leaf a row: st[3t] = x[y] - m, st[3t + 1] = m (x[y] read only when y is a class)
1049type EC1 is Data:
1050 EC1{k: U32, c1: Nat, c32: U32}
1051
1052def c1_3(+t: U32, +m: F32, st: Array<F32>, r: Array<F32> & F32) -> Array<F32>:
1053 match r:
1054 case Tuple{z, +xy}: Array.set(F32, Array.set(F32, st, (t * 3 : U32), (xy - m : F32)), ((t * 3 : U32) + 1 : U32), m)
1055
1056def c1_2(ok: Bool, +t: U32, +off: U32, +y: U32, st: Array<F32>, r: T.AS) -> Array<F32>:
1057 match ok r:
1058 case True{} T.AS{z, +m}: c1_3(t, m, st, Array.get(F32, z, (off + y : U32)))
1059 case False{} T.AS{z, +m}: c1_3(t, m, st, Tuple{z, 0.0})
1060
1061def c1_1(+t: U32, +off: U32, +c1: Nat, +c32: U32, z: Array<F32>, st: Array<F32>, r: Array<F32> & F32) -> Array<F32>:
1062 match r:
1063 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)))
1064
1065def c1_if(ok: Bool, +t: U32, +c1: Nat, +c32: U32, lab: Array<F32>, z: Array<F32>, st: Array<F32>) -> Array<F32>:
1066 match ok:
1067 case True{}: c1_1(t, (t * c32 : U32), c1, c32, z, st, Array.get(F32, lab, t))
1068 case False{}: st
1069
1070def lf_c1(+t: U32, +l: U32, +e: EC1, x1: Array<F32>, x2: Array<F32>, x3: Array<F32>, x4: Array<F32>, st: Array<F32>) -> Array<F32>:
1071 match e:
1072 case EC1{+k, +c1, +c32}: c1_if(U32.is_lt(t, k), t, c1, c32, x1, x2, st)
1073
1074# 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))
1075def tsx_leaf(+lo: U32, +m: F32, r: Array<F32> & F32) -> TS:
1076 match r:
1077 case Tuple{a, +x}:
1078 +e = F32.exp((x - m : F32))
1079 TS{Array.set(F32, a, lo, e), e}
1080
1081def tsx(l: Nat, +lo: U32, +h: U32, +m: F32, a: Array<F32>) -> TS:
1082 match l:
1083 case 0n: tsx_leaf(lo, m, Array.get(F32, a, lo))
1084 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)))
1085
1086# the same over 8 cells in straight-line code (no closure a node): e0..e7 written, then ((e0 + e1) + (e2 + e3)) +
1087# ((e4 + e5) + (e6 + e7)), the balanced tree's three levels
1088def x8_7(+lo: U32, +m: F32, +e0: F32, +e1: F32, +e2: F32, +e3: F32, +e4: F32, +e5: F32, +e6: F32, r: Array<F32> & F32) -> TS:
1089 match r:
1090 case Tuple{a, +x}:
1091 +e7 = F32.exp((x - m : F32))
1092 TS{Array.set(F32, a, (lo + 7 : U32), e7), ((((e0 + e1 : F32) + (e2 + e3 : F32) : F32) + ((e4 + e5 : F32) + (e6 + e7 : F32) : F32)) : F32)}
1093
1094def x8_6(+lo: U32, +m: F32, +e0: F32, +e1: F32, +e2: F32, +e3: F32, +e4: F32, +e5: F32, r: Array<F32> & F32) -> TS:
1095 match r:
1096 case Tuple{a, +x}:
1097 +e = F32.exp((x - m : F32))
1098 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)))
1099
1100def x8_5(+lo: U32, +m: F32, +e0: F32, +e1: F32, +e2: F32, +e3: F32, +e4: F32, r: Array<F32> & F32) -> TS:
1101 match r:
1102 case Tuple{a, +x}:
1103 +e = F32.exp((x - m : F32))
1104 x8_6(lo, m, e0, e1, e2, e3, e4, e, Array.get(F32, Array.set(F32, a, (lo + 5 : U32), e), (lo + 6 : U32)))
1105
1106def x8_4(+lo: U32, +m: F32, +e0: F32, +e1: F32, +e2: F32, +e3: F32, r: Array<F32> & F32) -> TS:
1107 match r:
1108 case Tuple{a, +x}:
1109 +e = F32.exp((x - m : F32))
1110 x8_5(lo, m, e0, e1, e2, e3, e, Array.get(F32, Array.set(F32, a, (lo + 4 : U32), e), (lo + 5 : U32)))
1111
1112def x8_3(+lo: U32, +m: F32, +e0: F32, +e1: F32, +e2: F32, r: Array<F32> & F32) -> TS:
1113 match r:
1114 case Tuple{a, +x}:
1115 +e = F32.exp((x - m : F32))
1116 x8_4(lo, m, e0, e1, e2, e, Array.get(F32, Array.set(F32, a, (lo + 3 : U32), e), (lo + 4 : U32)))
1117
1118def x8_2(+lo: U32, +m: F32, +e0: F32, +e1: F32, r: Array<F32> & F32) -> TS:
1119 match r:
1120 case Tuple{a, +x}:
1121 +e = F32.exp((x - m : F32))
1122 x8_3(lo, m, e0, e1, e, Array.get(F32, Array.set(F32, a, (lo + 2 : U32), e), (lo + 3 : U32)))
1123
1124def x8_1(+lo: U32, +m: F32, +e0: F32, r: Array<F32> & F32) -> TS:
1125 match r:
1126 case Tuple{a, +x}:
1127 +e = F32.exp((x - m : F32))
1128 x8_2(lo, m, e0, e, Array.get(F32, Array.set(F32, a, (lo + 1 : U32), e), (lo + 2 : U32)))
1129
1130def x8_0(+lo: U32, +m: F32, r: Array<F32> & F32) -> TS:
1131 match r:
1132 case Tuple{a, +x}:
1133 +e = F32.exp((x - m : F32))
1134 x8_1(lo, m, e, Array.get(F32, Array.set(F32, a, lo, e), (lo + 1 : U32)))
1135
1136# tsx over 2^(l+3) cells, its last three levels in blocks of 8
1137def tsx8(l: Nat, +lo: U32, +h: U32, +m: F32, a: Array<F32>) -> TS:
1138 match l:
1139 case 0n: x8_0(lo, m, Array.get(F32, a, lo))
1140 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)))
1141
1142# 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
1143type EC2 is Data:
1144 EC2{nb: U32, per: U32, p: Nat, bsz: U32, c32: U32}
1145
1146type C3S is Type:
1147 C3S{a: Array<F32>, b: Array<F32>, c: Array<F32>}
1148
1149def c2_put(+b: U32, st: Array<F32>, bs: Array<F32>, r: TS) -> C3S:
1150 match r:
1151 case TS{z, +s}: C3S{st, z, Array.set(F32, bs, b, s)}
1152
1153def c2_m(b8: Bool, +b: U32, +lo: U32, +p: Nat, +h: U32, z: Array<F32>, bs: Array<F32>, r: Array<F32> & F32) -> C3S:
1154 match b8 r:
1155 case True{} Tuple{st, +m}: c2_put(b, st, bs, tsx8(Nat.sub(p, 3n), lo, h, m, z))
1156 case False{} Tuple{st, +m}: c2_put(b, st, bs, tsx(p, lo, h, m, z))
1157
1158def c2_cell(+b: U32, +per: U32, +p: Nat, +bsz: U32, +c32: U32, s: C3S) -> C3S:
1159 match s:
1160 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)))
1161
1162def c2_loop(k: Nat, +b: U32, +l: U32, +per: U32, +p: Nat, +bsz: U32, +c32: U32, s: C3S) -> C3S:
1163 match k:
1164 case 0n: s
1165 case 1n+q: c2_loop(q, (b + l : U32), l, per, p, bsz, c32, c2_cell(b, per, p, bsz, c32, s))
1166
1167def c2_o(s: C3S) -> TG.O2:
1168 match s:
1169 case C3S{st, z, bs}: TG.O2{z, bs}
1170
1171def lf_c2(+t: U32, +l: U32, +e: EC2, x1: Array<F32>, x2: Array<F32>, x3: Array<F32>, x4: Array<F32>, o: TG.O2) -> TG.O2:
1172 match e o:
1173 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}))
1174
1175# 3. a leaf a row: st[3t + 2] = s, tree_sum's remaining passes over the row's per block sums
1176type EC3 is Data:
1177 EC3{k: U32, per: U32, tl: Nat, th: U32}
1178
1179def c3_put(+t: U32, st: Array<F32>, r: TS) -> Array<F32>:
1180 match r:
1181 case TS{bs, +s}: Array.set(F32, st, ((t * 3 : U32) + 2 : U32), s)
1182
1183def c3_if(ok: Bool, +t: U32, +per: U32, +tl: Nat, +th: U32, bs: Array<F32>, st: Array<F32>) -> Array<F32>:
1184 match ok:
1185 case True{}: c3_put(t, st, tnode(tl, U32.is_lt(th, per), (t * per : U32), per, th, bs))
1186 case False{}: st
1187
1188def lf_c3(+t: U32, +l: U32, +e: EC3, x1: Array<F32>, x2: Array<F32>, x3: Array<F32>, x4: Array<F32>, st: Array<F32>) -> Array<F32>:
1189 match e:
1190 case EC3{+k, +per, +tl, +th}: c3_if(U32.is_lt(t, k), t, per, tl, th, x1, st)
1191
1192# 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
1193type EC4 is Data:
1194 EC4{n: U32, c32: U32, nf: F32}
1195
1196def c4_4(+i: U32, +j: U32, +y: U32, +s: F32, +nf: F32, lab: Array<F32>, st: Array<F32>, r: Array<F32> & F32) -> C3S:
1197 match r:
1198 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))}
1199
1200def c4_3(+i: U32, +j: U32, +y: U32, +nf: F32, lab: Array<F32>, z: Array<F32>, r: Array<F32> & F32) -> C3S:
1201 match r:
1202 case Tuple{st, +s}: c4_4(i, j, y, s, nf, lab, st, Array.get(F32, z, i))
1203
1204def c4_2(+i: U32, +row: U32, +j: U32, +nf: F32, st: Array<F32>, z: Array<F32>, r: Array<F32> & F32) -> C3S:
1205 match r:
1206 case Tuple{lab, +yv}: c4_3(i, j, f2u(yv), nf, lab, z, Array.get(F32, st, ((row * 3 : U32) + 2 : U32)))
1207
1208def c4_cell(+i: U32, +c32: U32, +nf: F32, s: C3S) -> C3S:
1209 match s:
1210 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)))
1211
1212def c4_loop(k: Nat, +i: U32, +l: U32, +c32: U32, +nf: F32, s: C3S) -> C3S:
1213 match k:
1214 case 0n: s
1215 case 1n+q: c4_loop(q, (i + l : U32), l, c32, nf, c4_cell(i, c32, nf, s))
1216
1217def c3s_c(s: C3S) -> Array<F32>:
1218 match s:
1219 case C3S{a, b, c}: c
1220
1221def lf_c4(+t: U32, +l: U32, +e: EC4, x1: Array<F32>, x2: Array<F32>, x3: Array<F32>, x4: Array<F32>, z: Array<F32>) -> Array<F32>:
1222 match e:
1223 case EC4{+n, +c32, +nf}: c3s_c(c4_loop(TG.share(t, n, l), t, l, c32, nf, C3S{x1, x2, z}))
1224
1225# the host's part over st, row by row
1226def cx_t1(ok: Bool, +i: U32, +loss: F32, r: Array<F32> & F32) -> LS:
1227 match r:
1228 case Tuple{a, +xm}: ce_t2(ok, loss, xm, Array.get(F32, a, ((i * 3 : U32) + 2 : U32)))
1229
1230def cx_ls(left: Nat, ls: List<&2, Nat>, +i: U32, +c: Nat, r: LS) -> LS:
1231 match left ls r:
1232 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))))
1233 case _ _ r: r
1234
1235def cx_5(+n: Nat, +c: Nat, +labels: List<&2, Nat>, +k: Nat, q: TG.R1) -> T.LossOut<TA.Mat<n, c>>:
1236 match q:
1237 case TG.R1{x1, x2, x3, x4, z}: ce_g4(n, c, z, cx_ls(k, labels, 0, c, LS{x2, 0.0}))
1238
1239def cx_4(+n: Nat, +c: Nat, +labels: List<&2, Nat>, +k: Nat, la: Array<F32>, z: Array<F32>, q: TG.R1) -> T.LossOut<TA.Mat<n, c>>:
1240 match q:
1241 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))
1242
1243def cx_3(+n: Nat, +c: Nat, +labels: List<&2, Nat>, +k: Nat, +per: U32, la: Array<F32>, q: TG.R2) -> T.LossOut<TA.Mat<n, c>>:
1244 match q:
1245 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))
1246
1247def cx_2(+n: Nat, +c: Nat, +labels: List<&2, Nat>, +k: Nat, +p: Nat, q: TG.R1) -> T.LossOut<TA.Mat<n, c>>:
1248 match q:
1249 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))))))
1250
1251def cx_1(+n: Nat, +c: Nat, +labels: List<&2, Nat>, z: Array<F32>, r: IA) -> T.LossOut<TA.Mat<n, c>>:
1252 match r:
1253 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))))
1254
1255# cross-entropy over n rows of c classes against the labels (main.bend's CrossEntropy; its bits but CUDA's exp)
1256def CrossEntropy(+n: Nat, +c: Nat, z: TA.Mat<n, c>, +labels: List<&2, Nat>) -> T.LossOut<TA.Mat<n, c>>:
1257 match z:
1258 case TA.Mat{d}: cx_1(n, c, labels, d, ids_arr(n, labels))