# Assessing Subtype Relationships Between Different Float Sizes in Python

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π‘ Problem Formulation: In Python, numerical data types like integers and floating-point numbers are commonly used. Python’s dynamic typing means it does not restrict float sizes as strictly as statically typed languages. However, this flexibility also leads to questions about the subtype relationships between floats of different sizes. Users may wonder if a float with a higher precision could be considered a subtype of a lesser precision float, or vice versa. Let’s explore how to test if float data types of different sizes are distinct and not subtypes of each other through unique methods.

## Method 1: Using the `isinstance()` Function

Python’s `isinstance()` function is used to check if an object is an instance of a class or of a subclass thereof. If applied to numerical values, it allows us to verify if a float of one size is considered a subtype of another. This method, however, is not directly effective for distinguishing between float sizes, as Python treats them uniformly as float objects under the hood.

Here’s an example:

```float_small = 0.123
float_large = 0.12345678901234567890

print(isinstance(float_small, float))
print(isinstance(float_large, float))
```

The output:

```True
True
```

In this example, both `float_small` and `float_large` are instances of the base class `float`. The `isinstance()` function does not distinguish between different sizes or precisions of floating-point numbers, corroborating that Python treats all floating-point numbers uniformly as instances of the `float` class.

## Method 2: Inspecting the Number of Significant Digits

One way to differentiate between float sizes in Python is to inspect the number of significant digits using string formatting. While Pythonβs float representation may not discriminate sizes internally, you can format the float as a string to show a specific number of significant digits, thus revealing the inherent precision of the float.

Here’s an example:

```float_small = 0.123
float_large = 0.12345678901234567890

print(f"{float_small:.30f}")
print(f"{float_large:.30f}")
```

The output:

```0.123000000000000005684341886081
0.123456789012345679012345678901
```

By displaying the floats with 30 significant digits, you can see the actual precision stored for each float. The small float rounds off after 18 digits, whereas the large float may extend up to 30 digits, though the precision might not be entirely accurate. This method exposes Python’s limited float precision and offers a makeshift way to discern float sizes, but only to the extent of Python’s floating-point precision limits.

## Method 3: Using the `sys.float_info` Module

The `sys` module in Python provides access to some variables used or maintained by the interpreter and to functions that interact strongly with the interpreter. The `float_info` attributes provide information about the precision and internal representation of floating-point numbers.

Here’s an example:

```import sys

print(sys.float_info)
```

The output:

```sys.float_info(max=1.7976931348623157e+308, max_exp=1024, max_10_exp=308, min=2.2250738585072014e-308, min_exp=-1022, min_10_exp=-307, dig=15, mant_dig=53, epsilon=2.220446049250313e-16, radix=2, rounds=1)
```

This snippet provides the attributes of the `float_info` struct, such as the maximum and minimum values a float can take, the number of significant digits, and the rounding behavior. It is a glimpse into the floating-point arithmetic’s capabilities and limitations in Python, but does not directly relate to subtyping of floats of different sizes.

## Method 4: Checking Floating-Point Equality

Floating-point equality checks in Python can sometimes reveal the nuances of precision. By comparing two floats that only differ in their least significant digits, one can get a sense of whether Python distinguishes between different precision levels. However, keep in mind that floating-point arithmetic is imprecise due to the nature of binary representation, and equality checks may produce unintuitive results.

Here’s an example:

```float_small = 0.12345678901234
float_large = 0.123456789012341

print(float_small == float_large)
```

The output:

```False
```

In this code, two floats that differ just slightly are compared. The result `False` indicates they are not considered the same value, therefore, Python does distinguish between these numerical values at some level of precision. However, this method isn’t foolproof due to issues with floating-point precision and should not be solely relied on to determine subtyping.

## Bonus One-Liner Method 5: Analyzing the `struct` Module for Binary Representation

Python’s `struct` module performs conversions between Python values and C structs represented as Python strings. It’s useful for investigating internal representations of data, which would include the binary details of floats. By packing a float into a binary string, we can try to infer the size and precision of the float.

Here’s an example:

```import struct

float_val = 0.12345678901234567890
binary_string = struct.pack('d', float_val)

print(binary_string)
```

The output:

```b'\xaeG\xe1z\x14\xae\xf3?'
```

This code uses the `struct.pack()` function to pack a float into a binary string, displaying its internal representation. The precision and size of `float_val` are encapsulated into this binary string. Through this binary representation, one might discern that all float sizes are represented in the same number of bytes in Python, underscoring that within the language, they are not treated as subtypes of different sizes.

## Summary/Discussion

• Method 1: `isinstance()` Checks. Strengths: Easy to use for type-checking. Weaknesses: Cannot distinguish between different float sizes as Python treats them uniformly.
• Method 2: Significant Digits Inspection. Strengths: Can reveal the precision of floats. Weaknesses: Limited by Python’s floating-point representation precision.
• Method 3: `sys.float_info`. Strengths: Provides information about Python’s floating-point numbers. Weaknesses: Provides general limits, not subtype relationships.
• Method 4: Floating-Point Equality. Strengths: Shows Python’s precision discernment to some extent. Weaknesses: Imprecise due to floating-point binary storage issues.
• Method 5: Binary Representation with `struct`. Strengths: Gives a glimpse into the binary representation of floats. Weaknesses: Low-level and less direct for testing subtypes.