Unlocking the Secrets of Data Formats: Single Precision (FP32) vs Half Precision (FP16)
Dive deep into the world of floating-point numbers with a comparison between FP32 and FP16 formats. These two IEEE formats use either 32-bit or 16-bit binary storage to represent numbers, utilizing a combination of sign bits, exponent bits, and mantissa bits. The allocation of bits to exponent and mantissa in FP32 and FP16 results in distinct value ranges and precisions that impact data processing and computation.
https://en.wikipedia.org/wiki/Bfloat16_floating-point_format
Wondering how to interpret FP16 and FP32 values? Follow the IEEE-754 standards to decode the magic! For FP32, the decimal value can be expressed as (-1)^(sign) × 2^(decimal exponent —127) × (implicit leading 1 + decimal mantissa), where 127 represents the biased exponent value. In contrast, FP16 utilizes (-1)^(sign) × 2^(decimal exponent — 15) × (implicit leading 1 + decimal mantissa), with 15 as the corresponding biased exponent value. Explore more about biased exponent values
here.
In the realm of FP32, values typically range from [-2¹²⁷, 2¹²⁷] ~[-1.7*1e38, 1.7*1e38], while FP16 values fall within the approximate range of [-2¹⁵, 2¹⁵]=[-32768, 32768]. The decimal exponent for FP32 spans 0 to 255, excluding 0xFF which signifies NAN, resulting in a maximum exponent of 127. A similar principle applies to FP16 values, ensuring consistency and logic in numerical representation.
Delving into precision intricacies, it’s vital to acknowledge that both exponent and mantissa play crucial roles in determining precision limits, a concept known as denormalization. For FP32, precision can extend up to 2^(-23)*2^(-126)=2^(-149), whereas FP16 offers a precision threshold of 2^(10)*2^(-14)=2^(-24). Understanding these nuances sheds light on the capabilities and limitations of each format in numerical computations.
Transitioning from FP32 to FP16 representations underscores the significance of mixed precision training, especially in deep learning scenarios. Different layers and operations within neural networks exhibit varying sensitivities to value ranges and precision levels, necessitating a tailored approach to optimize model performance and accuracy.