We asked Llama 3.2 3B Instruct a short question. "What is the capital of France? Answer in one word."
It returned Paris, a full stop and an end-of-turn token. Three token IDs for a one-word answer. We will follow the operations that produced them, from the first embedding lookup to the last token choice.
The Rust extracts come from Foundry, our inference engine. We captured the reference values with Transformers and checked Foundry against the same input IDs. The weights use BF16, a format that stores each number in 16 bits. The run data and comparison report are included at the end of the article.
The previous article covers how the weights are stored on disk. Here we look at how the runtime uses them.
From a prompt to a token
The model gives each possible next token a score based on the tokens already in the context. The runtime picks one, adds it to the sequence and calls the model again. Softmax can turn the scores into a probability for each token.
Here means the model's weights, and are the tokens already in the context. Generating tokens takes forward passes in the loop shown here. The runtime saves earlier work in a cache, so it does not need to process the whole prompt on every pass.
Three kinds of data move through this loop. The parameters stay fixed for the whole run. The activations are the tensors each layer produces for this prompt. The KV cache keeps part of those activations, the keys and values of attention, so the next call doesn't have to compute the past again.
We use greedy decoding throughout this run. Each pass selects the token with the highest score.
Turning the question into vectors
For this chat run, the question is wrapped in the markers Llama saw during training. A turn opens with <|start_header_id|>, a role name and <|end_header_id|>. The user turn closes with <|eot_id|>, and the prompt ends by opening the assistant turn.
The model's chat template also adds a default system turn containing a knowledge cutoff and a date. We fixed the date to 27 Oct 2026 for this run. Keeping that date matters when reproducing the exact token IDs and probabilities.
The tokenizer then cuts this text into 47 pieces. Special tokens like <|eot_id|> have their own IDs above 128,000. Ordinary words are split into common pieces, most of them carrying their leading space. In Rust, the tokenizers crate reads the same tokenizer.json as every other runtime.
use tokenizers::Tokenizer;
let tokenizer = Tokenizer::from_file("tokenizer.json")?;
let question = "What is the capital of France? Answer in one word.";
let prompt = format!(
"<|begin_of_text|><|start_header_id|>system<|end_header_id|>\n\n\
Cutting Knowledge Date: December 2023\nToday Date: 27 Oct 2026\n\n<|eot_id|>\
<|start_header_id|>user<|end_header_id|>\n\n{question}<|eot_id|>\
<|start_header_id|>assistant<|end_header_id|>\n\n"
);
let encoding = tokenizer.encode(prompt, false)?;
// 47 IDs, starting with [128000, 128006, 9125, 128007, ...]
Only 12 of the 47 tokens come from the question. The other 35 come from the template. All 47 pass through every layer, including the role markers and dates.
Each ID now picks one row of the embedding table. The table has 128,256 rows, one per vocabulary entry, and each row holds 3072 numbers. Foundry uses a gather operation to select these rows.
let flat_ids = input_ids.reshape(Shape::try_from([input_ids.shape().numel()])?)?;
let embeds = self.weight.index_select(device, 0, &flat_ids)?;
The result is a tensor of shape [1, 47, 3072]. That means one prompt, 47 positions and 3072 numbers per position. This width is called the hidden size. It stays at 3072 between blocks, though some operations inside a block use wider or narrower vectors.
Inside a Transformer block
A block changes the vectors in two steps. First, attention lets each position read from itself and earlier positions. Then a feed-forward network, the MLP, works on each position alone. Both steps start with a normalization and end by adding their result back to their input. This running sum is called the residual stream.
Foundry's decoder layer follows these four lines almost word for word.
let normed = self.input_layernorm.forward(device, hidden_states)?;
let attn_output = self.self_attn.forward(device, &normed, position_embeddings, attention_mask, past_key_values, past_seen_tokens)?;
let hidden_states = hidden_states.add(device, &attn_output)?;
let normed = self.post_attention_layernorm.forward(device, &hidden_states)?;
let mlp_output = self.mlp.forward(device, &normed)?;
hidden_states.add(device, &mlp_output)
RMSNorm adjusts the scale of each vector. It squares the values, takes their mean, adds a small constant and takes the square root. It divides the original values by that result and applies a learned scale. Foundry does the sums in F32, a 32-bit format, to reduce rounding errors. It converts the normalized result back to BF16 before applying the learned scale.
let x_f32 = x.to_dtype(device, DType::F32)?;
let scale = x_f32
.mul(device, &x_f32)?
.sum_keepdim(device, dim)?
.mul(device, &self.dim_inv)?
.add(device, &self.eps)?
.sqrt(device)?
.reciprocal(device)?;
x_f32.mul(device, &scale)?.to_dtype(device, x.dtype())?.mul(device, &self.weight)
Queries, keys and values
Attention starts with three matrix multiplications. The normalized vectors are multiplied by three weight matrices from the file.
Each query is compared with the keys to give an attention score for every position. The scores determine how much of each value to use. Queries, keys and values are split into groups called heads. Each head has 128 numbers and its own attention scores.
Llama 3.2 3B has 24 query heads but only 8 key heads and 8 value heads. So Q has shape [1, 47, 3072] and K and V have shape [1, 47, 1024]. Each key and value head is shared by a group of 3 query heads. This is grouped-query attention. It makes keys and values three times smaller than they would be with one pair per query head.
Foundry applies each projection with a Linear operation, then reshapes the result into heads.
let query_states = self.q_proj.forward(device, hidden_states)?.reshape(query_shape)?;
let key_states = self.k_proj.forward(device, hidden_states)?.reshape(kv_shape)?;
let value_states = self.v_proj.forward(device, hidden_states)?.reshape(kv_shape)?.permute(&HEAD_MAJOR)?;
Attention on its own doesn't know the order of the tokens. RoPE adds it. Each query and each key is cut into pairs of numbers, and each pair is rotated by an angle that grows with the position of the token. After the rotation, the match between a query and a key depends on how far apart the two tokens are. Values are not rotated.
Different pairs rotate at different rates. The model configuration sets these rates, using a base of 500,000 and Llama 3 position scaling. The code below computes a starting frequency. The scaling rule adjusts some of these frequencies for longer contexts.
let inv_frequency = theta
.powf(exact_f32(dim_index).map_err(too_large_for_f32)? / head_dim_f32)
.recip();
A closer look at softmax
The heart of attention is three operations in a row. A matrix multiplication compares each query with each key. Softmax turns these comparisons into weights. A second matrix multiplication uses the weights to mix the values.
We will follow the recorded scores from layer 15 and query head 1, with both indices starting at zero. These values come from the Transformers reference run.
The first matrix multiplication takes the 47 queries and 47 keys of the head and computes one dot product per pair. That gives a grid of 47 by 47 scores. Each score is divided by the square root of the head size, √128, about 11.3, so the scores don't grow with the size of the head. In Foundry this is one chain of operations. The queries are reshaped so that the 3 heads sharing a key head sit together, and then they are multiplied by the transposed keys in one go.
let scores = query
.contiguous(device)?
.reshape(Shape::try_from([batch, kv_heads, rows, head_dim])?)?
.matmul(device, &key_t)?
.reshape(Shape::try_from([batch.saturating_mul(kv_heads), group, seq_len, kv_length])?)?
.to_dtype(device, DType::F32)?
.mul(device, scale)?
.add(device, attn_mask)?;
The last line adds the causal mask M. A position may only look at itself and at earlier positions. Every score that points to a later position gets minus infinity. Foundry builds the mask as a grid of zeros and minus infinities.
(0..kv_length).map(move |key_position| {
if key_position <= query_position { 0.0_f32 } else { f32::NEG_INFINITY }
})
Softmax turns each row of scores into weights between 0 and 1 that add up to 1. It raises to the power of each score and divides by the sum. A higher score gets a larger share of the weight.
We first subtract the largest score from every score. The final weights stay the same, but the exponentials stay at 1 or below, which helps prevent overflow. A masked score becomes zero after the exponential because to the power of negative infinity is zero. The Rust function follows these steps directly.
fn softmax(scores: &[f32]) -> Vec<f32> {
let max = scores.iter().copied().fold(f32::NEG_INFINITY, f32::max);
let exps: Vec<f32> = scores.iter().map(|&s| (s - max).exp()).collect();
let sum: f32 = exps.iter().sum();
exps.iter().map(|&e| e / sum).collect()
}
We ran this function on all 47 rows of the captured scores, with the causal mask applied. The largest difference from the reference F32 softmax was about 0.00000042.
The last row belongs to prompt position 46. Its scaled scores range from about -6.4 to 2.8. In this head, <|begin_of_text|> receives 77.2 percent of the attention weight. " word" gets 4.4 percent, " France" 3.1 percent and " capital" 2.3 percent.
These numbers describe how this head mixes the value vectors. They do not tell us which word caused the answer. We are looking at one head in one layer, with more operations still to come.
Foundry builds softmax from the same steps, written as tensor operations that run on the GPU.
let x_f32 = x.to_dtype(device, DType::F32)?;
let row_max = x_f32.amax_keepdim(device, dim)?;
let exps = x_f32.sub(device, &row_max)?.exp(device)?;
let row_sum = exps.sum_keepdim(device, dim)?;
exps.div(device, &row_sum)?.to_dtype(device, x.dtype())
In this implementation, the tensor operations run separately on the GPU and write their results to memory. A fused implementation combines steps in one GPU kernel, reducing the trips to memory.
The second matrix multiplication uses the weights to mix the values. The last position gets a vector of 128 numbers. The value vector of <|begin_of_text|> is multiplied by about 0.772, and each other value vector is multiplied by its own weight. These scaled vectors are then added together.
softmax(device, &scores, scores.shape().last_dim()?)?
.to_dtype(device, query.dtype())?
.reshape(Shape::try_from([batch, kv_heads, rows, kv_length])?)?
.matmul(device, value)?
The 24 head outputs are joined back into 3072 numbers, and the output projection mixes them. That result is added to the residual stream, which closes the attention half of the block.
The MLP and the rest of the stack
The MLP works on one position at a time. Two projections each turn its 3072 numbers into 8192 numbers. We apply SiLU, an activation function, to one result and multiply it by the other. A final projection brings the result back to 3072 numbers. This is called a gated MLP, or SwiGLU.
let gate = self.gate_proj.forward(device, x)?;
let up = self.up_proj.forward(device, x)?;
let hidden = self.act_fn.forward(device, &gate)?.mul(device, &up)?;
self.down_proj.forward(device, &hidden)
Llama 3.2 3B stacks 28 of these blocks. They all have the same structure, each with its own weights, and the output of one block is the input of the next. Foundry passes the output through the list of layers, then applies a final RMSNorm.
let hidden_states = self.layers.iter().try_fold(inputs_embeds, |hidden_states, layer| {
layer.forward(device, &hidden_states, [&cos, &sin], &attention_mask, past_key_values, past_seen_tokens)
})?;
self.norm.forward(device, &hidden_states)
Picking the next token
After the last block and the final RMSNorm, each position still holds 3072 numbers. Only the last position matters for the next token, so Foundry drops the others. The LM head multiplies that last vector by a matrix of 128,256 rows and returns one score per vocabulary entry, the logits. Llama 3.2 reuses the embedding table for this, so Foundry wires the two together when it loads the model. The clone creates another handle to the same weight tensor.
let lm_head = Linear::new(model.embed_tokens.weight().clone());
To see the probabilities, we apply softmax again, this time over 128,256 entries instead of 47. The probabilities below come from the Transformers logits. We also checked those logits with the standalone Rust function. Its largest probability difference was about 0.0000065.
let probs = softmax(&logits);
let (id, p) = probs.iter().enumerate().fold((0, 0f32), |best, (i, &p)| if p > best.1 { (i, p) } else { best });
// id 60704, p about 0.99967, decodes to "Paris"
Token 60704, "Paris" with no leading space, gets 99.966 percent in the reference run. The next candidates are "Par" at about 0.023 percent and " Paris" with a leading space at about 0.0024 percent. These are three different vocabulary entries. The probability describes this particular prompt and model run. It does not measure whether an answer is true.
Greedy decoding takes the highest logit directly. It does not need softmax because softmax keeps the scores in the same order. Sampling draws from the probabilities instead, often after applying a temperature to make the distribution sharper or flatter. Foundry uses greedy decoding and finds the top score on the GPU, so only one integer per sequence goes back to the CPU.
let bytes = logits
.argmax_keepdim(self.device, VOCAB_DIM)
.and_then(|token| token.to_bytes(self.device))
.map_err(|error| {
error!(message = "Selecting the next token failed.", %error);
GenerateError::NextToken
})?;
decode_token_ids(&bytes, self.batch_size)
Prefill, then decode
The first forward pass reads the 47 prompt tokens at once, at positions 0 to 46. This is the prefill. The causal mask lets all positions be computed in parallel without seeing the future. Each of the 28 layers writes the keys and values of all 47 positions into its KV cache, with a used region of shape [1, 8, 47, 128] for keys and the same for values. Foundry reserves a larger buffer for the tokens still to come. Only the logits of the last position are used, and they gave "Paris".
Every pass after that is a decode step. The runtime feeds only the newest token, and the model runs all 28 layers for that one position. Attention still needs the keys and values of all earlier tokens. It reads them from the cache and appends one new entry per layer. Positions keep counting. "Paris" enters at position 47, not 0, and RoPE rotates it for position 47.
Foundry's generation loop shows both phases. The first step is the prefill with the whole prompt. Each later step passes only the last token and the number of positions already seen.
let mut tokens = decoder.step(&prompts.concat(), 0)?;
let mut past_seen_tokens = prompt_len;
for _ in 1..max_new_tokens {
if generation.token_ids.iter().all(|row| is_stopped(row, eos_token_id)) {
break;
}
tokens = decoder.step(&tokens, past_seen_tokens)?;
generation.token_ids = extend_rows(generation.token_ids, &tokens, eos_token_id);
past_seen_tokens = past_seen_tokens.checked_add(1).ok_or_else(|| {
error!(message = "Token position overflows usize.", past_seen_tokens);
GenerateError::PositionOverflow
})?;
}
For our question, the model wrote three tokens. "Paris" came from the prefill. The first decode step, at position 47, returned "." as token 13. The second, at position 48, returned <|eot_id|>, token 128009, and generation stopped. This token marks the end of the assistant turn.
The next article looks at how much memory the KV cache needs.
The whole path in one view
The schematic follows the prefill from token IDs to the first answer token. It shows where each matrix multiplication happens and how the tensor shapes change. A reduction combines several numbers into one, such as a sum or a maximum. Pointwise operations work on individual values or pairs, as in the residual adds, SiLU and RoPE.
The residual stream stays 3072 numbers wide between blocks. Inside the MLP, the two projections each produce 8192 values. The LM head produces 128,256 scores, one per vocabulary entry.
We kept the model revision, input IDs and template date fixed for the comparison. The reference run used the stop IDs in the model's generation configuration. All eight module checks passed, and both engines generated the same three token IDs. Their logits differed slightly, so the probabilities in this article come from the Transformers reference run.
The records below include the model revision, software versions, captured scores and comparison results. The module report gives the tolerances and measured differences for each check.
Run data
{
"model": "meta-llama/Llama-3.2-3B-Instruct",
"revision": "0cb88a4f764b7a12671c53f0838cd831a0843b95",
"measured_at": "2026-10-27T10:59:11Z",
"engine": "transformers",
"dtype": "bfloat16",
"device": "mps",
"versions": {
"torch": "2.13.0",
"transformers": "5.12.1",
"numpy": "2.3.5",
"safetensors": "0.8.0",
"huggingface-hub": "1.30.0"
},
"template_date": "27 Oct 2026",
"question": "What is the capital of France? Answer in one word.",
"config": {
"num_hidden_layers": 28,
"hidden_size": 3072,
"intermediate_size": 8192,
"num_attention_heads": 24,
"num_key_value_heads": 8,
"head_dim": 128,
"vocab_size": 128256,
"tie_word_embeddings": true
},
"prompt_ids": [
128000,
128006,
9125,
128007,
271,
38766,
1303,
33025,
2696,
25,
6790,
220,
2366,
18,
198,
15724,
2696,
25,
220,
1544,
5020,
220,
2366,
21,
271,
128009,
128006,
882,
128007,
271,
3923,
374,
279,
6864,
315,
9822,
30,
22559,
304,
832,
3492,
13,
128009,
128006,
78191,
128007,
271
],
"generation": {
"do_sample": false,
"max_new_tokens": 32,
"output_ids": [
60704,
13,
128009
],
"output_tokens": [
"Paris",
".",
"<|eot_id|>"
],
"answer": "Paris."
},
"attention": {
"layer": 15,
"head": 1,
"indices_start_at": 0,
"scaled_scores": [
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