Chirag Ahuja

[EdgeAI - Part 1]: From PyTorch to Silicon - The Alchemist's Secret to Efficient AI

Aug 3, 2025
9 minute read

We live in an age of digital magic. We speak to our phones, generate stunning images from text, and get instant answers to complex questions. At the core of this magic are neural networks, often designed and trained in high-level frameworks like PyTorch or TensorFlow. These models are written in the language of floating-point mathematics—a world of decimal points and high precision.

But here’s the puzzle: the tiny, power-sipping chip inside your smart speaker or camera doesn’t have the luxury of a supercomputer’s floating-point unit (FPU). Running complex float operations is slow, power-hungry, and takes up precious space on the silicon.

So how do we bridge the gap? How do we translate the beautiful, precise math of a Python model into the brutally efficient language of a hardware chip?

The answer lies in a kind of digital alchemy. We’re going to turn expensive floating-point operations into cheap, lightning-fast integer math. This process, broadly known as quantization, is what allows powerful AI to run on the edge. Let’s open the hardware designer’s toolkit and reveal the secrets.

The Toolkit: Four Tricks to Rule Them All

At its core, our goal is to eliminate floats and replace expensive operations with cheaper ones. Here are the four foundational techniques to do it.

1. The Secret Code: Fixed-Point Arithmetic

Instead of storing a number like 3.14159, we store a large integer like 314159 and simply agree on where the decimal point should be. This agreement is managed by a scale factor.

  • The Math: float_value ≈ integer_value × scale
  • The Analogy: It’s like writing down “1250” and telling your friend it represents dollars, so they know it means “$12.50”. The scale factor is our shared secret.
# A floating-point value we want to represent
float_val = 3.14
scale = 0.01

# Quantize to an integer
# The hardware stores this integer value
int_val = round(float_val / scale)  # Result: 314

# Dequantize to get the approximate float value back
reconstructed_float = int_val * scale  # Result: 3.14

This is the bedrock of quantization. Every floating-point number in our model—every weight, bias, and activation—will be converted into an integer and a corresponding scale factor.

2. The Inverse Trick: Replacing Division with Multiplication

On a chip, integer division is a slow, multi-cycle nightmare. Multiplication, on the other hand, is incredibly fast. We avoid division at all costs.

  • The Math: To calculate x / D, we can instead calculate x * (1/D). We just need to find an integer representation for 1/D. Using our fixed-point trick, we find an integer multiplier M and a bit-shift s so that 1/D ≈ M / 2^s.
  • The Hardware Operation: The slow y = x / D becomes the lightning-fast y = (x * M) >> s (where >> s is a right bit-shift, a single-cycle hardware operation equivalent to dividing by 2<sup>s</sup>).
# Goal: Calculate y = x / 1024 without using division
x = 51200

# An offline compiler calculates the best M and s for 1/1024.
# In this simple case, M=1 and s=10 works perfectly.
M = 1
s = 10

# The fast hardware operation:
# A multiplication followed by a right bit-shift
y = (x * M) >> s  # (51200 * 1) >> 10  => 51200 / 1024 => 50

But where do these “magic” integer constants M and s come from? This is the job of special software functions that run “offline”—before the model is deployed. These offline software tools effectively act as specialized compilers for the hardware. They take a high-level goal, like “divide by 1024,” and calculate the perfect integer multiplier M and shift s that the simple hardware needs to achieve that result. This pre-calculation step is what makes the final hardware execution so incredibly fast.

3. The Impostor Strategy: Taming Complex Functions

What about functions like exp(), sqrt(), or tanh()? You can’t do those with simple integer math. So, we cheat. We replace them with impostors that are “good enough.”

A) The Cheat Sheet: Look-Up Tables (LUTs) for exp() in Softmax

If the range of inputs is small (like an 8-bit integer with 256 possible values), we can pre-calculate the function’s result for every single input and store it in a tiny, on-chip memory. This is the perfect strategy for the exp() function in a quantized Softmax.

  • The Goal: Calculate exp(x) where x is an 8-bit integer.
  • The Preparation: Before the chip is even made, we calculate exp(n) for every possible input n from -128 to 127. We don’t store the results as floats. Instead, we convert them into large 32-bit fixed-point integers using a carefully chosen scale factor. This creates a table with 256 entries, where each entry is a 32-bit integer. This table is etched directly into the hardware’s memory (a ROM).
# A tiny, pre-calculated LUT for exp(x)
# In hardware, this is a read-only memory block (ROM)
exp_lookup_table = {
    -2: 14,   # Represents a scaled version of exp(-2)
    -1: 37,   # Represents a scaled version of exp(-1)
     0: 100,  # Represents a scaled version of exp(0)
     1: 272,  # Represents a scaled version of exp(1)
     2: 739,  # Represents a scaled version of exp(2)
}

# Hardware needs to calculate exp(1) for an input value of 1
input_val = 1

# It just reads from the table! This is a single-cycle operation.
output_int = exp_lookup_table[input_val] # Result: 272

This turns a mathematically complex transcendental function into one of the fastest operations possible in hardware.

B) The Body Double: Polynomial Approximation for GELU

If the input range is too large for a LUT (like a 16-bit integer with 65,536 values), we need another trick. We find a much simpler function that looks “close enough” to the real thing. For GELU, the complex tanh-based curve is often replaced with a simple piecewise function: a parabola for small inputs, and a straight line for large inputs. This is a form of hardware-friendly curve fitting, where engineers find the simplest possible polynomial that mimics the shape of the true function over the most important range of inputs.

  • The Goal: Approximate the S-shaped curve inside the GELU function.
  • The Approximation: The hardware implements a simple rule:
    1. If the absolute value of the input |x| is less than some integer threshold T, the result is calculated using a quadratic formula: y = a*x² + c.
    2. If |x| is greater than or equal to T, the result is simply a fixed large value (it saturates).
# Simplified hardware logic for an integer-based GELU approximation
# These integer coefficients are pre-calculated by a "compiler" function.
T = 177 # Threshold
a = 28  # Quadratic coefficient
c = 10  # Offset

def hardware_gelu_approx(x_int):
    # 1. Check if input is in the quadratic region
    # In hardware, this is a fast comparator circuit.
    if abs(x_int) < T:
        # 2. Apply the simple quadratic formula
        # In hardware, this is just a multiplier and an adder.
        y = a * x_int * x_int + c
    else:
        # 3. Saturate if outside the region
        y = 32767 # Max 16-bit integer value

    return y

4. The Safety Net: Managing Overflow

When you add or multiply many large integers, the result can exceed the maximum value your hardware accumulator can hold (e.g., 32 bits). This is called an overflow.

  • The Math: To calculate a sum of squares Σ(x²), we can pre-shift every value to make it smaller before squaring and summing: Σ((x >> k)²).
  • The Hardware Operation: This strategic shift (sum_shift in the RMSNorm code) ensures the intermediate sum never overflows the accumulator. We just have to account for that shift in the final scaling step.
# Goal: Calculate sum of squares for [300, 400, 500]
# The hardware accumulator's max value is 300,000
inputs = [300, 400, 500]
sum_shift = 2 # A pre-calculated safety shift

# Without the shift, 500*500 = 250,000. Adding the next term
# 400*400 = 160,000 would overflow (250k + 160k > 300k).

sum_sq = 0
for x in inputs:
    # Shift before squaring to keep intermediate values small
    x_shifted = x >> sum_shift # 75, 100, 125
    sum_sq += x_shifted * x_shifted

# sum_sq = 75*75 + 100*100 + 125*125 = 31,250
# This is well below the 300,000 limit. No overflow!
# The final result can be scaled back up later if needed.

A Hardware Blueprint for LayerNorm

Let’s see how these tricks come together to build a real-world operation like Layer Normalization.

  1. The Goal: Calculate y = (x - mean(x)) / sqrt(variance(x)) * weight + bias.
  2. The Hardware Pipeline:
    • Mean Block: Calculates mean(x) using the Inverse Trick (multiplying by 1/num_elements).
    • Variance Block: Calculates the sum of squares, using the Safety Net (sum_shift) to prevent overflow.
    • Square Root Block: Uses a fast, iterative integer algorithm (a hardware-friendly version of the Newton-Raphson method) to find the square root.
    • Division Block: Divides (x - mean) by the result from the square root block, again using the Inverse Trick.
    • Final Scaling: Applies the weight and bias (which are also stored in Fixed-Point format) using simple integer multiplication and addition.

The Python reference code we analyzed is a perfect software simulation of this exact hardware pipeline, translating a single line of math into a dozen carefully orchestrated integer operations.

The Magic is in the Method

The breathtaking performance of modern AI hardware isn’t just about making transistors smaller. It’s about this elegant dance between software and silicon. It’s about the alchemists—the compiler engineers and hardware architects—who turn the gold of high-level mathematics into the efficient, powerful silicon that runs our world.

As we continue our journey to build a complete linear algebra and quantization library in Rust, these are the foundational principles we will build upon, crafting code that is not just correct, but truly hardware-aware.