How Accurate is Float Precision in Decimal Digits?

Floating-Point Precision: A Misunderstood Concept

The notion of float precision in terms of decimal digits is misleading when applied to binary floating-point formats like the IEEE-754 standard. Claims of 6-9 digit precision, as sometimes suggested, are inaccurate and require clarification.

Binary, Not Decimal

Floating-point numbers are inherently binary; they use bits, not decimal digits. A float consists of a sign, a significand (mantissa) with a fixed number of bits, and an exponent. This structure enables efficient representation of a wide range of values, both large and small.

Illustrative Examples

The following examples highlight the limitations of equating float precision to decimal digits:

• 1.0000001f is representable exactly, demonstrating that the precision isn't strictly limited to six digits.
• 100000000f loses precision beyond the leading digit, showing the limitations of the format for large numbers.

Resolution vs. Accuracy

A single-precision float has a resolution of 2²³. This means the smallest distinguishable change in value is approximately 10^-6.9 (since log₁₀2²³ ≈ 6.9). However, resolution is not equivalent to accuracy. Converting a decimal number to a float can introduce an error of up to approximately 10^-7.2.

Origin of the 6-9 Digit Claim

The 6 and 9 digit figures likely stem from the inherent limitations of converting between decimal and binary representations:

• Decimal numbers with up to 6 significant digits are guaranteed to be converted to a float and back to the original decimal without loss.
• Any float can be converted to a 9-digit decimal and then back to the original float value.

These guarantees, however, don't imply that floats possess 6-9 digits of decimal precision.

Conclusion: Understanding the Limitations

The concept of float precision in decimal digits is fundamentally flawed. Accurate understanding of floating-point arithmetic necessitates recognizing its binary nature, its strengths in representing a wide range of values, and its inherent limitations in representing decimal numbers precisely. For numerical computations, the implications of these limitations must be carefully considered.

The above is the detailed content of How Accurate is Float Precision in Decimal Digits?. For more information, please follow other related articles on the PHP Chinese website!