Shrinking Giants: A Word on Floating-Point Precision in LLM Domain for Faster, Cheaper Models

#floatingpoints #fp4 #fp16 #llm

Ever wondered how floating-point decision can have an impact on LLM’s output?

🔢 What is Floating-Point Precision?

Floating-point is the standard way computers represent real numbers (numbers with a fractional part, like 3.14 or 1.2×10−5).

A floating-point number is generally composed of three parts: a sign bit, an exponent, and a mantissa (or significand).

Sign bit: Determines if the number is positive or negative.
Exponent: Determines the scale or magnitude of the number (how large or small it is).
Mantissa: Determines the precision (the number of significant digits).
The number following “FP” (e.g., 4 or 16) indicates the total number of bits used to store the number. Fewer bits mean less memory and faster computation, but also less precision and a smaller range of representable values.

The terms FP4 and FP16 in the context of Large Language Models (LLMs) refer to the floating-point precision used to represent the model’s weights and perform its calculations. This is a form of quantization, which is the process of reducing the number of bits required to store model data, leading to significant efficiency improvements.

💡 FP16 (16-bit Floating Point)

FP16, also known as half-precision, uses 16 bits to store each number.

Structure: Typically 1 sign bit, 5 exponent bits, and 10 mantissa bits.
Memory/Speed: Reduces the memory required for the model weights by half compared to the traditional FP32 (single-precision), allowing larger LLMs to fit into GPU memory and enabling faster processing on modern hardware (like NVIDIA’s Tensor Cores).
Trade-off: While generally effective, the reduced range and precision can sometimes lead to numerical instability issues during training, such as gradient underflow (gradients becoming too small to matter). This is often mitigated using mixed-precision training, where most calculations use FP16 for speed, but critical steps (like weight updates) use a higher precision like FP32 or BF16.

⚡ FP4 (4-bit Floating Point)

FP4 uses an even more aggressive form of quantization, storing each number in just 4 bits. This is a leading-edge technique focused on ultra-efficiency, particularly for inference (running the model after it’s trained).

Structure: Formats vary (e.g., NVIDIA’s NVFP4 uses E2M1: 1 sign bit, 2 exponent bits, 1 mantissa bit) and often employ sophisticated scaling strategies.
Memory/Speed: It offers a massive reduction in memory footprint — up to 4x less memory than FP16 — and significantly faster throughput. This makes it possible to run extremely large LLMs on less powerful hardware or to run them much faster.
Trade-off: Moving to only 4 bits dramatically limits the possible values a number can take, increasing the risk of quantization error and a noticeable drop in accuracy compared to FP16 or FP32. Advanced techniques like micro-block scaling and specialized quantization-aware training are required to maintain model quality.

| Feature                 | FP32 (Full)              | FP16 (Half)                        | FP4 (Ultra-Low)                             |
| ----------------------- | ------------------------ | ---------------------------------- | ------------------------------------------- |
| **Bit-Width**           | 32 bits                  | 16 bits                            | 4 bits                                      |
| **Model Size/Memory**   | Largest                  | Half of FP32                       | Up to 1/8th of FP32                         |
| **Computational Speed** | Standard                 | Faster (with specialized hardware) | Ultra-Fast                                  |
| **Primary Use in LLMs** | Baseline, critical steps | Common for training & inference    | State-of-the-art for efficient inference    |
| **Accuracy/Stability**  | Highest                  | Good (requires mixed precision)    | Requires advanced techniques (Quantization) |

🐍 Python Example: Quantization Error

Disclaimer: Samples found around the net! Didn’t actually use them.

The main goal is to show how the drastic reduction in the number of bits — especially the mantissa (which governs precision) — causes significant rounding error.

Set up PyTorch and Define Values: First, we set a high-precision value (like FP32, which is Python’s default float) and define a target 16-bit value.

import torch

# 1. Define a high-precision value (FP32/standard Python float)
original_value = 12.3456789
original_tensor = torch.tensor(original_value, dtype=torch.float32)

print(f"Original Value (FP32): {original_tensor.item()}")

# 2. Convert to FP16 (Half Precision)
fp16_tensor = original_tensor.half()
fp16_value = fp16_tensor.item()

print(f"FP16 Value:           {fp16_value}")

Simulate FP4 Quantization: Since there’s no native torch.float4, we simulate a highly constrained 4-bit format (like an E2M1, which has only 1 bit for the mantissa) by aggressively rounding and scaling the FP16 value. In real-world LLMs, FP4 uses complex scaling to preserve model quality. For this simple demo, we demonstrate the effect of limited precision by forcing the number to a very small set of representable values.

# Function to simulate a very simple, lossy FP4-like quantization
def quantize_to_fp4_like(val, scale_factor=4.0):
    # 1. Scale the value up
    scaled_val = val / scale_factor

    # 2. Round aggressively (Simulating very few mantissa bits)
    # This step is the "loss of precision"
    rounded_val = round(scaled_val)

    # 3. Scale the value back down
    fp4_like_val = rounded_val * scale_factor
    return fp4_like_val

# Apply the simulated FP4 quantization to the FP16 value
fp4_like_value = quantize_to_fp4_like(fp16_value)

print(f"Simulated FP4 Value:  {fp4_like_value}")

Compare Quantization Error: Finally, you can calculate and compare the absolute error caused by the two different precision formats:

# Calculate Errors
error_fp16 = abs(original_value - fp16_value)
error_fp4_like = abs(original_value - fp4_like_value)

print("\n--- Error Comparison ---")
print(f"FP16 Absolute Error: {error_fp16:.8f}")
print(f"FP4-like Absolute Error: {error_fp4_like:.8f}")

What we get: The FP4-like Absolute Error will be significantly higher than the FP16 Absolute Error, demonstrating how the much more aggressive 4-bit quantization causes a greater loss of original information (precision).

🚀 Testing in a Real LLM Context

For a practical test of LLM performance:

Use a Quantization Library: Use libraries like Hugging Face’s bitsandbytes (which supports 4-bit and 8-bit quantization) or AutoAWQ/GPTQ for post-training quantization.
Load the Model: Load a model (e.g., a small Llama model) first in torch.float16 and then load a second copy in a 4-bit quantized format (e.g., load_in_4bit=True).
Run Inference: Run the same prompt through both models and compare:

GPU Memory Usage: The 4-bit model will use ≈41 the memory of the FP16 model.
Inference Speed: The 4-bit model will often be faster, especially with optimized kernels.
Output Quality: For simple tasks, the difference may be minimal, but for complex, subtle reasoning, the FP16 model might retain slightly higher accuracy.

Evolution-🧠 BF16 (Brain Floating Point) vs. FP16 (Half Precision) - Preferred 16-bit format for training the largest LLMs due to its superior stability.

Both BF16 and FP16 use 16 bits in total, resulting in the same memory footprint (half of FP32). The critical difference is how they allocate those 16 bits between the exponent (which controls the numerical range) and the mantissa (which controls the precision).

| Feature                       | FP32 (Single Precision)      | FP16 (Half Precision)              | BF16 (Brain Float 16)     |
| ----------------------------- | ---------------------------- | ---------------------------------- | ------------------------- |
| **Total Bits**                | 32                           | 16                                 | 16                        |
| **Exponent Bits (Range)**     | 8 bits                       | **5 bits**                         | **8 bits**                |
| **Mantissa Bits (Precision)** | 23 bits                      | **10 bits**                        | **7 bits**                |
| **Dynamic Range**             | Wide                         | Narrow                             | **Wide (Matches FP32)**   |
| **Primary Advantage**         | Highest Precision            | Better Precision for small numbers | **Stability/Wider Range** |
| **Primary Use in LLMs**       | Master Weights, Accumulators | Inference, older architectures     | **Modern LLM Training**   |

1. The Dynamic Range Advantage (Why BF16 Wins for Training)

FP16’s Problem (Narrow Range): With only 5 exponent bits, FP16 has a narrow dynamic range. During the complex calculations of deep neural network training (especially in the backward pass where gradients are calculated), very large or very small numbers can easily cause:
Overflow: Numbers become too large and round to infinity (inf).
Underflow: Numbers become too small and round to zero. If critical gradients underflow, the model stops learning, a major stability issue.
BF16’s Solution (Wide Range): BF16 uses 8 exponent bits, which is the same number as FP32. This gives it a huge dynamic range, making it highly resistant to underflow and overflow issues. This numerical robustness is crucial for stable training, particularly for models with many layers and complex gradient landscapes like LLMs.

2. The Precision Trade-off

BF16 only retains 7 mantissa bits, giving it lower precision than FP16’s 10 mantissa bits.
However, researchers at Google and others found that deep learning model accuracy is less sensitive to the precision of the fractional part (mantissa) than it is to the overall scale/magnitude of the numbers (exponent).
By matching the FP32 exponent, BF16 offers a “just works” solution that rarely requires the complex and often manual tuning of loss scaling techniques necessary when training with FP16.

🤝 Mixed Precision Training

Because both FP16 and BF16 save memory and boost speed, they are almost always used in Mixed Precision Training.

In this approach:

Low Precision (BF16/FP16) is used for all memory-intensive operations (weights, activations, and most computations) to maximize speed and fit the model on the GPU.
Full Precision (FP32) is used for numerically sensitive steps, such as the master copy of the weights and the optimizer updates, to ensure the model’s accuracy doesn’t degrade.

Key Players

Google: Originally developed and championed BF16 for its TPUs (Tensor Processing Units), which heavily rely on this format for large-scale training.
NVIDIA: Introduced native support for BF16 on its modern GPUs (starting with the A100 and continuing with the H100), effectively putting BF16 on par with FP16 in terms of hardware-accelerated speed.
The result is that BF16 is generally the default choice for LLM training, while FP16 remains a popular format for efficient inference (once the model is already trained), especially on hardware that is optimized for it.

Conclusion

The drive towards FP16 and, increasingly, FP4 precision is an essential engineering response to the exponential growth of Large Language Models. This technology is adopted not for marginal gains, but for fundamental necessity: FP16 halves the memory footprint and significantly speeds up calculations compared to FP32, making the training and deployment of multi-billion parameter models economically feasible. FP4, as an ultra-low precision format, pushes this efficiency to its limit, enabling the highest throughput and lowest memory consumption for model inference — the process of actually running the model — which dramatically lowers the operational cost of serving AI.

This shift is heavily championed by hardware and software leaders, most notably NVIDIA, which has integrated specialized Tensor Cores into its GPUs (like the A100, H100, and Blackwell) specifically optimized for high-speed, low-precision arithmetic like FP16, BF16, and its proprietary NVFP4 format. The adoption is widespread across the AI ecosystem, with major cloud providers and AI research leveraging these precision formats to build and deploy their state-of-the-art models. Ultimately, low-precision computing is the critical innovation that allows LLMs to escape the confines of theoretical research and become fast, accessible, and scalable products powering the modern digital world.