Why Does Float Precision in Programming Often Range Between 6 and 9 Digits?

Unraveling the Mystery of Floating-Point Precision

Floating-point precision in programming often causes confusion. This article clarifies misconceptions and explains its importance.

Debunking the Microsoft Documentation

Microsoft documentation's claim of 6-9 decimal digit precision for floats is misleading. Floating-point numbers aren't based on decimal digits; they use a sign, a fixed number of binary bits, and an exponent for a base-two power.

The Limits of Conversion

Converting decimal to floating-point numbers introduces inaccuracies. For example, 999999.97 becomes 1,000,000 in a float, highlighting potential decimal digit loss.

Resolution vs. Accuracy

A float's significand has 24 bits, making the resolution of its least significant bit about 6.9 times finer than its most significant bit. This refers to representation resolution, not conversion accuracy. The relative error in float conversion is limited to 1 part in 2²⁴, approximately 7.2 decimal digits.

The Origin of the 6-9 "Rule of Thumb"

The 6 and 9 figures arise from specific aspects of the float format:

• 6: The maximum number of significant decimal digits guaranteed to be preserved during conversion.
• 9: The minimum number of decimal digits needed to reconstruct any finite float value without precision loss.

A Useful Analogy

Imagine a 7.2-unit block on a line of 1-unit bricks. Placing the block at the start covers 7.2 bricks, but starting mid-brick covers only 6. Eight bricks could contain the block, but 9 are needed for non-arbitrary placement.

This illustrates the 6 and 9 limits. The uneven relationship between powers of two and ten affects how values are represented in the float format.

Conclusion

Understanding floating-point numbers requires moving beyond the idea of decimal precision. By focusing on resolution and conversion characteristics, and consulting the IEEE-754 standard and reliable sources, we can better grasp floating-point arithmetic.

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