# Converting Floating-Point Numbers to Binary Strings in C

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http://www.exploringbinary.com/converting-floating-point-numbers-to-binary-strings-in-c/

If you want to print a floating-point number in binary using C code, you can’t use printf() — it has no format specifier for it. That’s why I wrote a program to do it, a program I describe in this article.

(If you’re wondering *why* you’d want to print a floating-point number in binary, I’ll tell you that too.)

## Binary to Binary Conversion

You could print a floating-point number in binary by parsing and interpreting its IEEE representation, or you could do it more elegantly by casting it as a base conversion problem — a ** binary to binary conversion**; specifically, a conversion from a binary number to a binary string.

To illustrate the process of converting a number to a string, let’s “convert” the decimal integer 352 to decimal using the classic base conversion algorithm for integers:

- 352/10 = 35 remainder
**2** - 35/10 = 3 remainder
**5** - 3/10 = 0 remainder
**3**

Because we use a divisor of 10, this process simply isolates the digits of the number. If we string them together, we get the original number back: 352.

We can illustrate a similar process for fractional values, for example 0.5943, using the classic base conversion algorithm for fractionals:

- 0.5943 * 10 =
**5**.943 - 0.943 * 10 =
**9**.43 - 0.43 * 10 =
**4**.3 - 0.3 * 10 =
**3**.0

Because we use a multiplier of 10, this process isolates the digits of the number. If we string them together, we get the original number back: 0.5943.

The same two-part algorithm works for binary to binary conversion, if instead you divide and multiply by 2 ** and use binary arithmetic**.

On paper, this is not too exciting. But in a computer, it allows us to convert binary numbers to binary strings. A floating-point binary value is a number, whereas a printed binary value is a string. We can use the binary to binary conversion algorithm to isolate the digits of the number and convert them to ASCII numerals in a string. That’s what I do in the C code below.

## The Code

The function **fp2bin()** converts a number from IEEE double format to an equivalent character string made up of 0s and 1s. It breaks the double into integer and fractional parts and then converts each separately using routines **fp2bin_i()** and **fp2bin_f()**, respectively.

fp2bin_i() and fp2bin_f() use the algorithms described above, which are the same algorithms used in the dec2bin_i() and dec2bin_f() routines in my article Base Conversion in PHP Using BCMath. The algorithms are the same because in each case, the base of the number being converted is the same as the base of the arithmetic used to convert. For the dec2bin* routines, the base is decimal; for the fp2bin* routines, the base is binary.

### fp2bin.h

/***********************************************************/ /* fp2bin.h: Convert IEEE double to binary string */ /* */ /* Rick Regan, http://www.exploringbinary.com */ /* */ /***********************************************************/ /* FP2BIN_STRING_MAX covers the longest binary string (2^-1074 plus "0." and string terminator) */ #define FP2BIN_STRING_MAX 1077 void fp2bin(double fp, char* binString);

### fp2bin.c

/***********************************************************/ /* fp2bin.c: Convert IEEE double to binary string */ /* */ /* Rick Regan, http://www.exploringbinary.com */ /* */ /***********************************************************/ #include <string.h> #include <math.h> #include "fp2bin.h" voidfp2bin_i(double fp_int, char* binString) { int bitCount = 0; int i; char binString_temp[FP2BIN_STRING_MAX]; do { binString_temp[bitCount++] = '0' + (int)fmod(fp_int,2); fp_int = floor(fp_int/2); } while (fp_int > 0); /* Reverse the binary string */ for (i=0; i<bitCount; i++) binString[i] = binString_temp[bitCount-i-1]; binString[bitCount] = 0; //Null terminator } voidfp2bin_f(double fp_frac, char* binString) { int bitCount = 0; double fp_int; while (fp_frac > 0) { fp_frac*=2; fp_frac = modf(fp_frac,&fp_int); binString[bitCount++] = '0' + (int)fp_int; } binString[bitCount] = 0; //Null terminator } voidfp2bin(double fp, char* binString) { double fp_int, fp_frac; /* Separate integer and fractional parts */ fp_frac = modf(fp,&fp_int); /* Convert integer part, if any */ if (fp_int != 0) fp2bin_i(fp_int,binString); else strcpy(binString,"0"); strcat(binString,"."); // Radix point /* Convert fractional part, if any */ if (fp_frac != 0) fp2bin_f(fp_frac,binString+strlen(binString)); //Append else strcpy(binString+strlen(binString),"0"); }

### Notes

- fp2bin() prints binary numbers in their entirety, with no scientific notation.
- fp2bin() only works with positive numbers.
- fp2bin() doesn’t handle the special IEEE values for not-a-number (NaN) and infinity.
- FP2BIN_STRING_MAX can be reduced if you know a priori that you will be converting numbers within a limited range.
- fp2bin_f() terminates, and gives the exact binary fraction, because multiplication by 2 is essentially bit shifting; bits are shifted left out of the number, one at a time, until the number is 0.
- fp2bin_i() can be called independently of fp2bin(), but fp2bin_f() would need modification to run standalone (it doesn’t add the radix point or handle 0 properly).

### Compiling and Running

I compiled and ran this code on both Windows and Linux:

- On Windows, I built a project in Visual C++ and compiled and ran it in there.
- On Linux, I compiled with “gcc fp2binTest.c fp2bin.c -lm -o fp2bin” and then ran with “./fp2bin”.

## Examples

The following program uses fp2bin() to convert five floating-point numbers to binary strings:

/***********************************************************/ /* fp2binTest.c: Test double to binary string conversion */ /* */ /* Rick Regan, http://www.exploringbinary.com */ /* */ /***********************************************************/ #include "fp2bin.h" #include <stdio.h> int main(int argc, char *argv[]) { char binString[FP2BIN_STRING_MAX]; fp2bin(16,binString); printf("2^4 is %s\n",binString); fp2bin(0.00390625,binString); printf("2^-8 is %s\n",binString); fp2bin(25,binString); printf("25 is %s\n",binString); fp2bin(0.1,binString); printf("0.1 is %s\n",binString); fp2bin(0.6,binString); printf("0.6 is %s\n",binString); return (0); }

Here is the output from the program:

2^4 is 10000.0 2^-8 is 0.00000001 25 is 11001.0 0.1 is 0.0001100110011001100110011001100110011001100110011001101 0.6 is 0.10011001100110011001100110011001100110011001100110011

0.1 and 0.6 are interesting because they have terminating expansions in decimal but infinite expansions in binary: 0.1_{10} is 0.00011_{2}, and 0.6_{10} is 0.1001_{2}. Both are rounded to the nearest 53 significant bits, as shown in the values printed above (trailing 0s are not printed).

## Using fp2bin() to Study Binary Numbers

You can use fp2bin() in conjunction with my decimal/binary converter to study how decimal values are approximated with IEEE floating-point. There are two aspects of the approximation in particular you can look at:

- How the exact binary equivalent of a decimal number compares to its IEEE double representation; that is, how it is rounded to fit into 53 significant bits.
- What the exact decimal value of the approximation is.

For example, let’s look at the IEEE approximation of 0.1:

- The decimal/binary converter tells you that 0.1 in binary is 0.00011001100110011001100110011001100110011001100110011001
**1001**…fp2bin(0.1) tells you that 0.1 in double-precision floating-point is 0.0001100110011001100110011001100110011001100110011001101 .

Comparing the two values, you’ll see the IEEE number is the binary number rounded to 53 significant bits. The value of the binary number beyond its 53rd significant bit — its 56th bit overall — is greater than 2

^{-57}; therefore, it is rounded up (the rounding results in a carry to the 52nd significant bit). - The decimal/binary converter tells you that the IEEE approximation to 0.1, 0.0001100110011001100110011001100110011001100110011001101,
is exactly

0.1000000000000000055511151231257827021181583404541015625

in decimal.(If you’re using GCC C, the %f format specifier of printf() can be used to print this value instead of using the converter).

This shows that the IEEE double approximation of 0.1 is accurate only to 17 significant decimal digits.

## Printing floats

fp2bin() will print single-precision floating-point values (floats) as well. Your C compiler will “promote” the float to a double before the call. The resulting double will have the same value, only with extra trailing zeros — which fp2bin() will not print.

June 30th, 2010 at 5:07 am

i want to generate randomly n bits number that divide into W word, where W=m/n

and then i want operate it by adding/xor, multiplying, the process that i mean look below:

input m, n1, n2; m=4, n1=12, n2=9;

assumed: n1>=n2;

n1=12

A[]=[[1001],[0011],[1010]]

n2=9

B[]=[[0001],[0011],[1010]]

three ZERO digit MSB in B[] automatic generate..

so the calculate follow the rule:

A[]=[[1001],[0011],[1010]]

B[]=[[0001],[1010],[0010]]

_______________________xor

C[]=[[1000],[1001],[1000]]

xor per block, how can i implementing it in C, where n1 and n2 up to 100 bit or more?

June 30th, 2010 at 4:16 pm

ikilobo,

This doesn’t seem to have anything to do with converting floating-point numbers to binary strings (and I can’t say I understand the question anyhow). Sorry.

March 24th, 2011 at 12:36 pm

Great article! Thanks!

March 24th, 2011 at 1:45 pm

@Bangon Kali,

I’m glad you liked it. Thanks for the feedback.

July 7th, 2011 at 12:58 am

nicely programmed – it simplified the concept with a good coding style. Thanks a lot!

August 25th, 2011 at 4:10 pm

NOTE: My code depends on the fmod() function, and one reader reports that the MINIX version of fmod() produces incorrect results.August 23rd, 2012 at 1:10 pm

It does not work for negative numbers

Still, great contribution. I appreciate it. Thanks!

H.

August 23rd, 2012 at 7:19 pm

@Herman,

Yes, that’s a known limitation (it is stated in the “Notes” section). Thanks for the feedback.

March 6th, 2013 at 12:18 pm

thanks for the article! helped me a lot!! Could you please include IEEE 754 numbers as well?

March 6th, 2013 at 12:43 pm

@Eric,

I’m not sure what you’re asking for — these

areIEEE 754 numbers I am talking about.October 9th, 2013 at 10:10 pm

If I want the results to consider only 32 bits? There is a easy way to do it? Thanks

October 9th, 2013 at 11:10 pm

@Mauricio,

I’m sorry, I do not understand your question.

October 14th, 2013 at 1:31 pm

The results are presented this way

0.1 is 0.0001100110011001100110011001100110011001100110011001101

it’s double precision,

i need single precision results, that means, the converted vector with a 32-bit precision

October 14th, 2013 at 5:40 pm

@Mauricio,

Do you mean you want to print floats? Calling fp2bin() with a float should work (see section “Printing floats”).

October 14th, 2013 at 10:49 pm

Hello Rick,

thanks, I haven’t seen the printing floats section before. Sorry. I needed the single precision because I am generating values to be used in a hardware implementation of a LUT. I will develop a “rounding” function to generate aways a 32-bit string with fixed size integer and decimal parts. Your functions helped a lot. Thank you.