printf() and floating point numbers

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Help! My printf is producing digits out of thin air!
Problem setup
One day we had a certain mismatch between two floating point numbers. One number when inspected in an IDE looked much longer than the other, having lots of extra digits. Then a colleague of mine said that it's fine, they might still be the same number, and produced some code similar to this:
#include <stdio.h>
​
int main(void)
{
    double d = 0.1234567890123456;  // 16 digits 
    printf("%.16f\n", d);
    printf("%.55f\n", d);
    return 0;
}
What do you think it will print? Most programmers know that double precision has about 16 significant decimal digits when numbers are in that range (i.e between 0 and 1). So I am printing here 16 digits first and then some more just to see what comes next. It prints this:
➜ gcc 1.c && ./a.out
0.1234567890123456
0.1234567890123455941031593852130754385143518447875976562
This looked like quite a lot of extra digits and it did not even stop there!
I know that we represent real decimal numbers with bits stored according to IEEE 754 floating point standard, so our decimal literal is somewhat imprecisely stored in binary representation. Now it looks like printf is printing that decimal representation. However, what was surprising to me is that this imprecise representation can be expanded to so many new decimal digits.
TL;DR printf prints decimal expansion of the binary number corresponding to our original literal. That binary number approximates the literal. There can be maximum 53 - binary exponent digits in the decimal expansion (at least for "normal" numbers less than 1). Read below for the details.
Plan of attack
Let's figure out step by step what's happening in this little program. The interesting steps with accompanying questions I had are as follows:
1.C compiler has to convert that string representing a decimal C-literal into a double. How that should be done? Are there any restrictions on the literal length?
2.That double is represented with some bits. What bits exactly? How they are laid out in memory? Can I build them by hand?
3.Those bits can be converted back to decimal and printed with printf. How many digits can I expect in this decimal expansion?
Let's first address the representation question, then we'll have terminology to discuss parsing of the literals and finally converting back to decimal with printf.
Representing doubles with IEEE 754
I assume that you more or less know what a floating number is. As a quick reminder, it's a number which is represented as follows: sign * significand * (base ^ exponent)
Sign here can be either -1 or 1. 
Significand (also called mantissa) always starts with '1.' (note the dot at the end!). Since it always starts with 1, there is no need to store that 1 in actual bit representation when we get to it.
Base is 2 in our case. 
Exponent is a scaling factor, it is the number of positions we have to move the point "." to the right or to the left to get back to our number.
For example binary 101.1 can be represented as 1 * 1.011 * (base ^ 2), sign is 1, significand is 1.011 and we need to scale it two positions to the right, so exponent is +2. 
In order to convert a real decimal number into bits of double we can do the following steps.
1. Get exact binary representation of the decimal number we are trying to convert. 
It will most likely contain the repeating fractional part unless it can be represented as P / Q where Q is an exact power of 2 and P is an integer. 
In our case 0.1234567890123456 corresponds to this binary (I calculated it with online converter): 
0.0001111110011010110111010011011101000110111101100101100101101100110...
We haven't even got to the repeating part, but that's alright since we only need the beginning.
2. Take 53 bits starting from the first digit 1 (and including it). 
This is because IEEE 754 double has 53 bits of precision. You can look up those numbers in corresponding Wikipedia table.
IEEE 754 binary formats
So we take this 53-bits long part in the middle:
0.000 11111100110101101110100110111010001101111011001011001 011011...
If the next bit after those 53 is 1 we should also add 1 to that large part for rounding purposes. In our case it's 0 so we are fine.
3. Figure out what the exponent should be. 
Our first 1 is 4 positions to the right from the binary point. So in our case the exponent will be -4. However, in IEEE 754 exponents are stored with a particular bias which has to be added before we store it in bits. For double precision this bias is 1023, so we have to add that to -4 getting 1019 which we need to store as unsigned integer in 11 bits of exponent (those numbers can also be taken from the table above). Why to store the exponent with a bias and not as "sign + absolute value" or "two complement"? The main reason that with this way we can use integer comparator to compare floating-point numbers. Also, it leads to a nice zero representation with all 0 bits. See here for the details.
4. Combine the parts
The memory layout for doubles is as follows (assumes Big Endian order):
Double precision memory layout for IEEE 754
Our number is positive, so we use 0 for sign. Exponent is 1019 which is 01111111011 if represented with 11 bits. And we've got our 53 precision bits of the significand which we need to pack into 52 bits. This is easily done, since the first bit is always 1, so we never store it (it's called "hidden" or "implicit" bit). In the end we get this:
0 01111111011 1111100110101101110100110111010001101111011001011001
which should be 64 bits representing our double 0.1234567890123456 in memory! Whoa, that was a lot of work. Let's check if we did it right with some Rust code (it's easier to print bits in Rust than in C):
fn main() {
    let d: f64 = 0.1234567890123456;
    d.to_be_bytes().iter().for_each(|b| print!("{:08b}", b));
    println!();
}
When we run this we get exactly the bits we calculated before, success!
➜ rustc a.rs && ./a
0011111110111111100110101101110100110111010001101111011001011001
Literals conversion
Now we have the terminology to tackle the next question: how C literals from the program are parsed into doubles? Are there any limitations on length? What if I write a very long literal:
 double d = 1.111111111111111111111111111111111111111111111111;
Will it fail? How many ones will be preserved?
Apparently, according to this StackOverflow answer and C99 standard, there are no limitations on length of double literals (at least I don't see it in the grammar and I can use literals with thousands digits and it compiles just fine). The double representation we'll get should be the closest representable number or one before or after it, depending on the implementation. So yes, you can use literals like 0.123456789012345678901234567890 with 30 digits, but most of those digits would be wasted since it's too precise to be represented in double precision format. 
To quote from C99 standard:
For decimal floating constants [...] the result is either the nearest representable value, or the larger or smaller representable value immediately adjacent to the nearest representable value, chosen in an implementation-defined manner. 
Conversion back to decimal with printf
Now let's see what happens with that printf. It takes all those bits we used for binary representation, converts it back to exact decimal and prints it with specified precision. 
How long we can expect the decimal representation to be, i.e. how many digits does it have before starting the string of zeroes at the end?
Originally I thought that since we have 53 binary digits in significand (mantissa) for numbers close in scale to 1 (with exponent = 0), the smallest number we can represent is about 2−53≈10−16 2^{-53} \approx 10^{-16}2−53≈10−16
 And I thought that it should mean that we should have approximately 16 digits or slightly more in the decimal representation of those bits and if we ask for more we should get zeros. But that is not true. To get an idea why, we can just look at 2−3 2^{-3} 2−3
 which is 0.125. It has 3 digits after the point in its decimal expansion, even though it's very close to 10−1 10^{-1} 10−1
 and by reasoning above should have about 1 or "slightly more" digits. 
Let's look at the table of decimal expansions for 1/2i 1 / 2^i 1/2i
 values for consecutive is. This corresponds to the value of i-th bit in the significand.
Bit number
Fraction
Decimal expansion
Bit 1
1 / 2
0.5
Bit 2
1 / 4
0.25
Bit 3
1 / 8
0.125
Bit 4
1 / 16
0.0625
Bit 5
1 / 32
0.03125
Bit 6
1 / 64
0.015625
It looks like every new bit adds a new digit to the decimal! So, if we have 53 bits in the significand, we can have up to 53 digits in the decimal expansion! (For now we disregard the exponent, which can be just zero for those examples).
Let's prove that 1/2n1 / 2^n1/2n
has n digits after the point in its decimal representation. My friend Igor suggested a nice and easy proof by induction! Basis is always easy: 1/2 = 0.5 and it has 1 digit. Then let's assume by induction hypothesis that 1/2n1/2^n1/2n
 has nnn
 digits and let's see what happens with 1/2n+11/2^{n+1}1/2n+1
. This is just 1/2n1/2^n1/2n
 divided by 222
, and so for every of n digits of the original number we'll have a corresponding digit in the new number (divided by two with truncation and carry-over to the right) plus an extra digit for the last "5" (since it is odd and no carry-over will fix that). Illustration:
.125
-----  / 2
.0625
 Proving that it always ends with 5 can also be done by induction: every 5 at the end leads to another 5 when divided by 2 and we start with 5 for the basis.
And since 1/2n1/2^n1/2n
is the finest unit of precision for n-bit number we can be sure that other higher bits are "coarser", i.e. that they will have less decimal digits in their individual representation.
So now we can see that it's completely fine to have so many digits from printf and we also have an upper bound on them. For example in our case we should expect no more than 53 + 4 digits after the point. "+4" because we use -4 exponent, which can add more digits. Indeed, if we add some more precision to our original program, we can see that decimal representation of our double has 56 digits (and zeros after that):
➜ printf "%.60f\n" 0.1234567890123456
0.123456789012345594103159385213075438514351844787597656250000
References and further reading
A tool to explore floating point numbers, where you can toggle bits and see how it affects the value, inspect decimal representations and do many other things
A very readable and well written article on Wikipedia: "Floating-point arithmetic"​
​"Floating point guide" if you don't want to overload yourself with details and want easy answers
A famous long article "What Every Computer Scientist Should Know About Floating Point Arithmetic." if you want to overload yourself with details and want rigorous answers
Cheatsheet: Option (in Rust) vs Maybe (in Haskell)
More advanced aspects of pattern matching in Rust
Last modified 2yr ago
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Outline
Problem setup
Plan of attack
Representing doubles with IEEE 754
Literals conversion
Conversion back to decimal with printf
References and further reading