关于float8等浮点类型数字位数及Numpy dtype文档换算的技术咨询

Hey, great questions about floating-point precision—let’s break this down clearly, since this is a common point of confusion when working with numeric types!

First, a quick background: The number of reliable decimal significant digits a float can represent depends on the total precision bits of its mantissa (including the implicit leading 1 that’s not stored in IEEE 754 normalized floats). We calculate this using log₁₀(2^p) where p is the total mantissa precision bits (stored bits + 1 implicit bit). Here’s the breakdown for each type:

• float8: There are two common IEEE 754 variants:
  • e5m2: 2 stored mantissa bits → total precision 3 bits. This gives ~1-2 reliable decimal significant digits.
  • e4m3: 3 stored mantissa bits → total precision 4 bits. This gives ~2-3 reliable decimal significant digits.
• float16 (half-precision): 10 stored mantissa bits → total precision 11 bits. You get ~3-4 reliable decimal significant digits (most 3-digit decimals are exact, some 4-digit ones are too).
• float32 (single-precision): 23 stored mantissa bits → total precision 24 bits. This gives ~7-8 reliable decimal significant digits (7 is the standard "safe" count—any 8-digit decimal may lose precision when converted to float32).
• float64 (double-precision): 52 stored mantissa bits → total precision 53 bits. You get ~15-16 reliable decimal significant digits (15 is the go-to safe number for most calculations).
• float128 (quadruple-precision): 112 stored mantissa bits → total precision 113 bits. This gives ~33-34 reliable decimal significant digits.

Numpy’s docs only list exponent and mantissa storage bits, but converting that to decimal significant digits is straightforward once you know the trick:

Simple Formula #

For any IEEE 754 float type in Numpy:

1. Take the y bits mantissa value from the docs.
2. Add 1 to get the total precision bits p (accounting for the implicit leading 1).
3. Calculate the decimal significant digits with log₁₀(2^p)—the integer part gives you the safe number of significant digits.

In code terms (if you want to compute it on the fly):

import math

def get_decimal_digits(mantissa_stored_bits):
    total_precision = mantissa_stored_bits + 1
    return math.log10(2 ** total_precision)

# Example for float32: mantissa_stored_bits=23
print(get_decimal_digits(23))  # ~7.22 → safe to use 7 digits

Quick Reference Table #

Here’s a ready-to-use table mapping Numpy’s dtype info to decimal significant digits:

A quick note: This gives you significant digits, not fixed digits before/after the decimal point. Floats use scientific notation under the hood, so a float32 can accurately represent 1234567 (7 significant digits) just as well as 0.0001234567 (also 7 significant digits)—it’s about the total number of meaningful digits, not their position relative to the decimal point.

内容的提问来源于stack exchange，提问作者mathguy