Why are Subnormal Numbers a Tradeoff Between Precision and Representation Length?

IEEE 754 Basics

The IEEE 754 floating-point standard organizes numbers into the following format:

• 1 bit: Sign
• 8 bits: Exponent
• 23 bits: Fraction

Subnormal Numbers

Engineers implemented subnormal numbers to address a concern. All numbers except 0.0 have a leading 1 in binary representation. To avoid wasting precision on this leading bit, they created the "leading bit convention."

However, this would make the smallest non-zero number larger than the previous standard. Engineers introduced subnormal numbers as an exception where:

• The exponent is 0
• The leading bit is set to 0
• The exponent is fixed to -126

This allows for the representation of numbers smaller than the smallest non-subnormal value.

Subnormal Numbers Tradeoff

Subnormal numbers represent a tradeoff between precision and representation length. Smaller numbers have reduced precision, but the number of represented values is doubled.

Visualization

Geometrically, subnormals extend the range of exponent 0, doubling the available space for numbers and reducing the spacing within that range.

Implementing Subnormals

C float represents 32-bit IEEE 754 numbers on most desktop machines. An example C program demonstrates the properties of subnormal numbers:

#include <assert.h>
#include <inttypes.h>
#include <math.h>
#include <stdlib.h>
#include <stdio.h>

typedef struct {...} Float32; // Represents the 32-bit floating point

float float_from_bytes(..., uint32_t fraction); // Reconstructs float from individual parts
bool float32_equal(float f, uint32_t sign, uint32_t exponent, uint32_t fraction); // Compares float to individual parts

int main() {
    assert(float32_equal(0.5f, 0, 126, 0));
    assert(isnormal(0.5f));
    ... // More assertions

    return EXIT_SUCCESS;
}

Subnormals in Implementations

Subnormals may be implemented less efficiently on some platforms. ARMv8 provides a "Flush-to-Zero" mode where subnormals are rounded to zero for performance optimization.

Denormal vs. Subnormal

Subnormal and denormal are synonyms, referring to the same type of numbers.

Other Special Cases

• Infinity: Represents numbers too large to be represented with normal numbers.
• NaN (Not a Number): Represents computations that did not produce a valid result, often due to invalid inputs.

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