Is the most significant decimal digits precision that can be converted to binary and back to decimal without loss of significance 6 or 7.225?
Solution 1:
These are talking about two slightly different things.
The 7.2251 digits is the precision with which a number can be stored internally. For one example, if you did a computation with a double precision number (so you were starting with something like 15 digits of precision), then rounded it to a single precision number, the precision you'd have left at that point would be approximately 7 digits.
The 6 digits is talking about the precision that can be maintained through a round-trip conversion from a string of decimal digits, into a floating point number, then back to another string of decimal digits.
So, let's assume I start with a number like 1.23456789 as a string, then convert that to a float32, then convert the result back to a string. When I've done this, I can expect 6 digits to match exactly. The seventh digit might be rounded though, so I can't necessarily expect it to match (though it probably will be +/- 1 of the original string.
For example, consider the following code:
#include <iostream>
#include <iomanip>
int main() {
double init = 987.23456789;
for (int i = 0; i < 100; i++) {
float f = init + i / 100.0;
std::cout << std::setprecision(10) << std::setw(20) << f;
}
}
This produces a table like the following:
987.2345581 987.2445679 987.2545776 987.2645874
987.2745972 987.2845459 987.2945557 987.3045654
987.3145752 987.324585 987.3345947 987.3445435
987.3545532 987.364563 987.3745728 987.3845825
987.3945923 987.404541 987.4145508 987.4245605
987.4345703 987.4445801 987.4545898 987.4645386
987.4745483 987.4845581 987.4945679 987.5045776
987.5145874 987.5245972 987.5345459 987.5445557
987.5545654 987.5645752 987.574585 987.5845947
987.5945435 987.6045532 987.614563 987.6245728
987.6345825 987.6445923 987.654541 987.6645508
987.6745605 987.6845703 987.6945801 987.7045898
987.7145386 987.7245483 987.7345581 987.7445679
987.7545776 987.7645874 987.7745972 987.7845459
987.7945557 987.8045654 987.8145752 987.824585
987.8345947 987.8445435 987.8545532 987.864563
987.8745728 987.8845825 987.8945923 987.904541
987.9145508 987.9245605 987.9345703 987.9445801
987.9545898 987.9645386 987.9745483 987.9845581
987.9945679 988.0045776 988.0145874 988.0245972
988.0345459 988.0445557 988.0545654 988.0645752
988.074585 988.0845947 988.0945435 988.1045532
988.114563 988.1245728 988.1345825 988.1445923
988.154541 988.1645508 988.1745605 988.1845703
988.1945801 988.2045898 988.2145386 988.2245483
If we look through this, we can see that the first six significant digits always follow the pattern precisely (i.e., each result is exactly 0.01 greater than its predecessor). As we can see in the original double, the value is actually 98x.xx456--but when we convert the single-precision float to decimal, we can see that the 7th digit frequently would not be read back in correctly--since the subsequent digit is greater than 5, it should round up to 98x.xx46, but some of the values won't (e.g,. the second to last item in the first column is 988.154541, which would be round down instead of up, so we'd end up with 98x.xx45 instead of 46. So, even though the value (as stored) is precise to 7 digits (plus a little), by the time we round-trip the value through a conversion to decimal and back, we can't depend on that seventh digit matching precisely any more (even though there's enough precision that it will a lot more often than not).
1. That basically means 7 digits, and the 8th digit will be a little more accurate than nothing, but not a whole lot--for example, if we were converting from a double of 1.2345678, the .225 digits of precision mean that the last digit would be with about +/- .775 of the what started out there (whereas without the .225 digits of precision, it would be basically +/- 1 of what started out there).
Solution 2:
what is the most significant decimal digits precision that can be converted to binary and back to decimal without loss of significance?
The most significant decimal digits precision that can be converted to binary and back to decimal without loss of significance (for single-precision floating-point numbers or 24-bits) is 6 decimal digits.
Why do these two numbers differ...
The numbers 6 and 7.225 differ, because they define two different things. 6 is the most decimal digits that can be round-tripped. 7.225 is the approximate number of decimal digits precision for a 24-bit binary integer because a 24-bit binary integer can have 7 or 8 decimal digits depending on its specific value.
7.225 was found using the specific binary integer formula.
dspec = b·log10(2) (dspec = specific decimal digits, b = bits)
However, what you normally need to know, are the minimum and maximum decimal digits for a b-bit integer. The following formulas are used to find the min and max decimal digits (7 and 8 respectively for 24-bits) of a specific binary integer.
dmin = ⌈(b-1)·log10(2)⌉ (dmin = min decimal digits, b = bits, ⌈x⌉ = smallest integer ≥ x)
dmax = ⌈b·log10(2)⌉ (dmax = max decimal digits, b = bits, ⌈x⌉ = smallest integer ≥ x)
To learn more about how these formulas are derived, read Number of Decimal Digits In a Binary Integer, written by Rick Regan.
This is all well and good, but you may ask, why is 6 the most decimal digits for a round-trip conversion if you say that the span of decimal digits for a 24-bit number is 7 to 8?
The answer is — because the above formulas only work for integers and not floating-point numbers!
Every decimal integer has an exact value in binary. However, the same cannot be said for every decimal floating-point number. Take .1 for example. .1 in binary is the number 0.000110011001100..., which is a repeating or recurring binary. This can produce rounding error.
Moreover, it takes one more bit to represent a decimal floating-point number than it does to represent a decimal integer of equal significance. This is because floating-point numbers are more precise the closer they are to 0, and less precise the further they are from 0. Because of this, many floating-point numbers near the minimum and maximum value ranges (emin = -126 and emax = +127 for single-precision) lose 1 bit of precision due to rounding error. To see this visually, look at What every computer programmer should know about floating point, part 1, written by Josh Haberman.
Furthermore, there are at least 784,757 positive seven-digit decimal numbers that cannot retain their original value after a round-trip conversion. An example of such a number that cannot survive the round-trip is 8.589973e9. This is the smallest positive number that does not retain its original value.
Here's the formula that you should be using for floating-point number precision that will give you 6 decimal digits for round-trip conversion.
dmax = ⌊(b-1)·log10(2)⌋ (dmax = max decimal digits, b = bits, ⌊x⌋ = largest integer ≤ x)
To learn more about how this formula is derived, read Number of Digits Required For Round-Trip Conversions, also written by Rick Regan. Rick does an excellent job showing the formulas derivation with references to rigorous proofs.
As a result, you can utilize the above formulas in a constructive way; if you understand how they work, you can apply them to any programming language that uses floating-point data types. All you have to know is the number of significant bits that your floating-point data type has, and you can find their respective number of decimal digits that you can count on to have no loss of significance after a round-trip conversion.
June 18, 2017 Update: I want to include a link to Rick Regan's new article which goes into more detail and in my opinion better answers this question than any answer provided here. His article is "Decimal Precision of Binary Floating-Point Numbers" and can be found on his website www.exploringbinary.com.
Solution 3:
Do keep in mind that they are the exact same formulas. Remember your high-school math book identity:
Log(x^y) == y * Log(x)
It helps to actually calculate the values for N = 24 with your calculator:
Kahan's: 23 * Log(2) = 6.924
Wikipedia's: Log(2^24) = 7.225
Kahan was forced to truncate 6.924 down to 6 digits because of floor(), bummer. The only actual difference is that Kahan used 1 less bit of precision.
Pretty hard to guess why, the professor might have relied on old notes. Written before IEEE-754 and not taking into account that the 24th bit of precision is for free. The format uses a trick, the most significant bit of a floating point value that isn't 0 is always 1. So it doesn't need to be stored. The processor adds it back before it performs a calculation. Turning 23 bits of stored precision into 24 of effective precision.
Or he took into account that the conversion from a decimal string to a binary floating point value itself generates an error. Many nice round decimal values, like 0.1, cannot be perfectly converted to binary. It has an endless number of digits, just like 1/3 in decimal. That however generates a result that is off by +/- 0.5 bits, achieved by simple rounding. So the result is accurate to 23.5 * Log(2) = 7.074 decimal digits. If he assumed that the conversion routine is clumsy and doesn't properly round then the result can be off by +/-1 bit and N-1 is appropriate. They are not clumsy.
Or he thought like a typical scientist or (heaven forbid) accountant and wants the result of a calculation converted back to decimal as well. Such as you'd get when you trivially look for a 7 digit decimal number whose conversion back-and-forth does not produce the same number. Yes, that adds another +/- 0.5 bit error, summing up to 1 bit error total.
But never, never make that mistake, you always have to include any errors you get from manipulating the number in a calculation. Some of them lose significant digits very quickly, subtraction in particular is very dangerous.