c - Why floating-points number's significant numbers is 7 or 6

Question

I see this in Wikipedia log 2²⁴ = 7.22.

I have no idea why we should calculate 2^24 and why we should take log10......I really really need your help.

Answer 1

why floating-points number's significant numbers is 7 or 6 (?)

Consider some thoughts employing the Pigeonhole principle:

1. binary32 float can encode about 2³² different numbers exactly. The numbers one can write in text like 42.0, 1.0, 3.1415623... are infinite, even if we restrict ourselves to a range like -10³⁸ ... +10³⁸. Any time code has a textual value like 0.1f, it is encoded to a nearby float, which may not be the exact same text value. The question is: how many digits can we code and still maintain distinctive float?
2. For the various powers-of-2 range, 2²³ (8,388,608) values are normally linearly encoded.
3. In the range [1.0 ... 2.0), 2²³ (8,388,608) values are linearly encoded.
4. In the range [2³³ or 8,589,934,592 ... 2³⁴ or 17,179,869,184), again, 2²³ (8,388,608) values are linearly encoded: 1024.0 apart from each other. In the sub range [9,000,000,000 and 10,000,000,000), there are about 976,562 different values.

Put this together ...

5. As text, the range [1.000_000 ... 2.000_000), using 1 lead digit and 6 trailings ones, there are 1,000,000 different values. Per #3, In the same range, with 8,388,608 different float exist, allowing each textual value to map to a different float. In this range we can use 7 digits.

6. As text, the range [9,000,000*10³ and 10,000,000*10³), using 1 lead digit and 6 trailings ones, there are 1,000,000 different values. Per #4, In the same range, there are less than 1,000,000 different float values. Thus some decimal textual values will convert to the same float. In this range we can use 6, not 7, digits for distinctive conversions.

The worse case for typical float is 6 significant digits. To find the limit for your float:

#include <float.h>
printf("FLT_DIG = %d
", FLT_DIG);  // this commonly prints 6

---

... no idea why we should calculate 2^24 and why we should take log10

2^24 is a generalization as with common float and its 24 bits of binary precision, that corresponds to fanciful decimal system with 7.22... digits. We take log10 to compare the binary float to decimal text.

2²⁴ == 10^7.22...

Yet we should not take 2²⁴. Let us look into how FLT_DIG is defined from C11dr §5.2.4.2.2 11:

number of decimal digits, q, such that any floating-point number with q decimal digits can be rounded into a floating-point number with p radix b digits and back again without change to the q decimal digits,

p log₁₀ b ............. if b is a power of 10
?(p ? 1) log₁₀ b?.. otherwise

Notice "log₁₀ 2²⁴" is same as "24 log₁₀ 2".

As a float, the values are distributed linearly between powers of 2 as shown in #2,3,4.

As text, values are distributed linearly between powers of 10 like a 7 significant digit values of [1.000000 ... 9.999999]*10^{some_exponent}.

The transition of these 2 groups happen at different values. 1,2,4,8,16,32... versus 1,10,100, ... In determining the worst case, we subtract 1 from the 24 bits to account for the mis-alignment.

?(p ? 1) log₁₀ b? --> floor((24 ? 1) log10(2)) --> floor(6.923...) --> 6.

Had our float used base 10, 100, or 1000, rather than very common 2, the transition of these 2 groups happen at same values and we would not subtract one.