Tensor Math: Broadcasting, Operations, and Precision

Tensor math specification for EMEP. Defines broadcasting rules, elementwise operations, reductions, matmul shape rules for transformer weights, and precision ranges for fp32, bf16, fp16, fp8, int8, and int4.

Tensor math is the foundation of all operations in TensorEngine. This page defines broadcasting rules, elementwise operations, reductions, matrix multiplication shape contracts, and precision ranges.

Broadcasting Rules

Two tensors are broadcastable if, for each dimension from trailing to leading, the dimensions are equal or one of them is 1. A missing dimension is treated as 1.

Shape A	Shape B	Broadcasted Shape
(H, I)	(H, 1)	(H, I)
(H, I)	(1, I)	(H, I)
(H, I)	(I,)	(H, I)
(V, H)	(H,)	(V, H)
(H,)	()	(H,)
(H, I)	(H, I, K)	Not broadcastable

Broadcasting is applied before elementwise operations. The TensorEngine validates broadcastability before executing any op.

Elementwise Operations

For elementwise add, sub, mul, div:

C = A op B

shape(C) = broadcast(shape(A), shape(B))
dtype(C) = promote(dtype(A), dtype(B))

dtype A	dtype B	Promoted dtype
fp32	any	fp32
bf16	fp16	fp32
bf16	bf16	bf16
fp16	fp16	fp16
int8	int8	int16
int4	any	fp32 (dequantize first)

Reductions

Operation	Input	Output	Notes
sum	(H, I)	scalar	Accumulate in fp32
mean	(H, I)	scalar	Accumulate in fp32
max	(H, I)	scalar	Direct on input dtype
argmax	(H, I)	scalar	Returns index
L2 norm	flattened	scalar	sqrt(sum(x^2)) in fp32
L1 norm	flattened	scalar	sum(	x	) in fp32

Matmul Shape Rules for Transformer Weights

For a linear layer with weight W and input x:

x: (batch, seq_len, H)
W: (H, I)
output: (batch, seq_len, I)

For attention Q/K/V/O projections:

x: (batch, seq_len, H)
W_q: (H, H)
W_k: (H, K * H / A)
W_v: (H, K * H / A)
W_o: (H, H)

For MLP SwiGLU:

x: (batch, seq_len, H)
W_up: (H, I)
W_gate: (H, I)
W_down: (I, H)

Precision and Numerical Range

Precision	Bits	Exponent	Mantissa	Approx Range	Notes
fp32	32	8	23	1.4e-45 to 3.4e38	Safe for all compute
bf16	16	8	7	1.2e-38 to 3.4e38	Same range as fp32, less precision
fp16	16	5	10	5.96e-8 to 65504	Overflow risk in large norms
fp8	8	varies	varies	Narrow	Experimental, limited support
int8	8	N/A	N/A	-128 to 127	Quantized, requires scale
int4	4	N/A	N/A	0 to 15	Highly quantized, dequantize mandatory

Numerical Stability Rules

1. All elementwise accumulation uses fp32 intermediates.
2. Norms are computed in fp32 regardless of input dtype.
3. SLERP dot products use fp64 for precision.
4. int4 and int8 weights are dequantized to fp32 before merge ops.
5. Results are cast to the target dtype only at the final step.

Cross-Links

• Merge Math for shared notation and weight shapes.
• Tensor Operations for the TensorEngine operation catalog.
• SLERP Math for spherical interpolation formulas.
• TIES Math for trim and disjoint merge.
• DARE Math for sparsification and rescaling.