What is the maximum length in chars needed to represent any double value?

Question

When I convert an unsigned 8-bit int to string then I know the result will always be at most 3 chars (for 255) and for an signed 8-bit int we need 4 chars for e.g. "-128".

Now what I'm actually wondering is the same thing for floating-point values. What is the maximum number of chars required to represent any "double" or "float" value as a string?

Assume a regular C/C++ double (IEEE 754) and normal decimal expansion (i.e. no %e printf-formatting).

I'm not even sure if the really small number (i.e. 0.234234) will be longer than the really huge numbers (doubles representing integers)?

Answer 1

The standard header <float.h> in C, or <cfloat> in C++, contains several constants to do with the range and other metrics of the floating point types. One of these is DBL_MAX_10_EXP, the largest power-of-10 exponent needed to represent all double values. Since 1eN needs N+1 digits to represent, and there might be a negative sign as well, then the answer is

int max_digits = DBL_MAX_10_EXP + 2;

This assumes that the exponent is larger than the number of digits needed to represent the largest possible mantissa value; otherwise, there will also be a decimal point followed by more digits.

CORRECTION

The longest number is actually the smallest representable negative number: it needs enough digits to cover both the exponent and the mantissa. This value is -pow(2, DBL_MIN_EXP - DBL_MANT_DIG), where DBL_MIN_EXP is negative. It's fairly easy to see (and prove by induction) that -pow(2,-N) needs 3+N characters for a non-scientific decimal representation ("-0.", followed by N digits). So the answer is

int max_digits = 3 + DBL_MANT_DIG - DBL_MIN_EXP

For a 64-bit IEEE double, we have

DBL_MANT_DIG = 53
DBL_MIN_EXP = -1023
max_digits = 3 + 53 - (-1023) = 1079

Answer 2

According to IEEE 754-1985, the longest notation for value represented by double type, i.e.:

-2.2250738585072020E-308

has 24 chars.

Answer 3

A correct source of information that goes into more detail than the IEEE-754 Specification are these lecture notes from UC Berkely on page 4, plus a little bit of DIY calculations. These lecture slides are also good for engineering students.

Recommended Buffer Sizes

Single	Double	Extended	Quad
16	24	30	45

These numbers are based on the following calculations:

Maximum Decimal Count of the Integral Portion

Single	Double	Extended	Quad
9	17	21	36

(Quantities listed in decimals.)

Decimal counts are based on the formula: At most Ceiling(1 + NLog_10(2)) decimals, where N is the number of bits in the integral portion*.

Maximum Exponent Lengths

Single	Double	Extended	Quad
5	5	7	7

(Standard format is e-123.)

Fastest Algorithm

The fastest algorithm for printing floating-point numbers is the Grisu2 algorithm detailed in the research paper Printing Floating-point Numbers Quickly and Accurately. The best benchmark I could find can be found here.

Answer 4

You can use snprintf() to check how many chars you need. snprintf() returns the number of chars needed to print whatever is passed to it.

/* NOT TESTED */
#include <stdio.h>
#include <stdlib.h>
int main(void) {
    char dummy[1];
    double value = 42.000042; /* or anything else */
    int siz;
    char *representation;
    siz = snprintf(dummy, sizeof dummy, "%f", value);
    printf("exact length needed to represent 'value' "
           "(without the '\\0' terminator) is %d.\n", siz);
    representation = malloc(siz + 1);
    if (representation) {
        sprintf(representation, "%f", value);
        /* use `representation` */
        free(representation);
    } else {
        /* no memory */
    }
    return 0;
}

Note: snprintf() is a C99 function. If a C89 compiler provides it as an extension, it may not do what the above program expects.

Edit: Changed the link to snprintf() to one that actually describes the functionality imposed by the C99 Standard; the description in the original link is wrong.
2013: Changed the link back to POSIX site which I prefer over the site of the first edit.

Answer 5

Depends on what you mean by "represent". Decimal fraction don't have exact floating-point representations. When you convert decimal fraction -> binary fraction -> decimal, you do not have exact decimal representations and will have noise bits at the end of the binary representation.

The question didn't involve starting from decimal, but all source code (and must user input) is decimal, and involves the possible truncation issue. What does "exact" mean under these circumstances?

Basically, it depends on your floating point representation.

If you have 48 bits of mantissa, this takes about 16 decimal digits. The exponent might be the remaining 14 bits (about 5 decimal digits).

The rule of thumb is that the number of bits is about 3x the number of decimal digits.

Answer 6

You can control the number of digits in the string representation when you convert the float/double to a string by setting the precision. The maximum number of digits would then be equal to the string representation of std::numeric_limits<double>::max() at the precision you specify.

#include <iostream>
#include <limits>
#include <sstream>
#include <iomanip>

int main()
{
 double x = std::numeric_limits<double>::max();

 std::stringstream ss;
 ss << std::setprecision(10) << std::fixed << x;

 std::string double_as_string = ss.str();
 std::cout << double_as_string.length() << std::endl;
}

So, the largest number of digits in a double with a precision of 10 is 320 digits.

Answer 7

1024 is not enough, the smallest negative double value has 1077 decimal digits. Here is some Java code.

double x = Double.longBitsToDouble(0x8000000000000001L);
BigDecimal bd = new BigDecimal(x);
String s = bd.toPlainString();
System.out.println(s.length());
System.out.println(s);

Here is the output of the program.

1077
-0.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000004940656458412465441765687928682213723650598026143247644255856825006755072702087518652998363616359923797965646954457177309266567103559397963987747960107818781263007131903114045278458171678489821036887186360569987307230500063874091535649843873124733972731696151400317153853980741262385655911710266585566867681870395603106249319452715914924553293054565444011274801297099995419319894090804165633245247571478690147267801593552386115501348035264934720193790268107107491703332226844753335720832431936092382893458368060106011506169809753078342277318329247904982524730776375927247874656084778203734469699533647017972677717585125660551199131504891101451037862738167250955837389733598993664809941164205702637090279242767544565229087538682506419718265533447265625

Answer 8

"What is the maximum length in chars needed to represent any double value?"

The exact answer to this question is: 8 ASCII chars - in a hexadicimal format, excluding the '0x' prefix - 100% accuracy :) (but it's not just a joke)

The usable precision of IEEE-754 double is around 16 decimal digits - so excluding educational purposes, representations longer than that are just a waste of resources and computing power:

• Users are not getting more informed when they see a 700-digit-number on the screeen.

• Configuration variables stored in that "more accurate" form are useless - every single operation on such number will destroy the accuracy. (excluding changing the sign bit)

If someone needs better real precision, then there's 80-bit long double with around 18-digit accuracy or f.e. libquadmath.

Regards.

Answer 9

As an improvement on the accepted answer based on Greg A. Woods accurate comment, a more conservative but still adequate number of characters needed is 3 + DBL_DIG + -DBL_MIN_10_EXP (total 325) with 3 being for the leading "-0." that may be needed. If using C-style strings, add one for null ('\0') termination such that an adequately sized buffer (of size 326) may be created with:

#include <limits.h>

char buffer[4 + DBL_DIG + -DBL_MIN_10_EXP];

For those who prefer the C++ numeric limits interface that would be:

#include <limits>

char buffer[4 + std::numeric_limits<double>::digits10 + -std::numeric_limits<double>::min_exponent10];

Answer 10

The maximum number of characters that will be required to print any decimal double value (i.e. in "%f" format) will be for the value of -DBL_MIN (i.e. -0x1p-1022, assuming binary64 IEEE 754 is your double). For that you'll need exactly 325 characters. That's: DBL_DIG + abs(DBL_MIN_10_EXP) + strlen("-0."). This is of course because log10(fabs(DBL_MIN)) is 308, which is also abs(DBL_MIN_10_EXP)+1 (the +1 is because of the leading digit to the left of the decimal place), and that's the number of leading zeros to the left of the significant digits.

int lz;                 /* aka abs(DBL_MIN_10_EXP)+1 */
int dplaces;
int sigdig;             /* aka DBL_DECIMAL_DIG - 1 */
double dbl = -DBL_MIN;

lz = abs((int) lrint(floor(log10(fabs(dbl)))));
sigdig = lrint(ceil(DBL_MANT_DIG * log10((double) FLT_RADIX)));
dplaces = sigdig + lz - 1;
printf("f = %.*f\n", dplaces, dbl);

Answer 11

"For that you'll need exactly 325 characters"

Apparently (and this is a very common case) You don't understand the how the conversion between different numeric bases works.

No matter how accurate the definition of the DBL_MIN is, it is limited by hardware accuracy, which is usually up to 80 bits or 18 decimal digits (x86 and similar architectures)

For that reason, specialized arbitrary-precision-arithmetic libraries has been invented, like f.e. gmp or mpfr.

Answer 12

You should represent floating point numbers as the shortest of either a decimal or in e notation, with 17 decimal digits of precision (with printf("%.17g\n", -1.2345);, this is also how Python does it), then the answer is 24 characters (+1 for the trailing null):

-2.2250738585072014e-308

If you're printing the value as a decimal with 17 digits of precision then the answer is 343 characters (+1 for null):

-0.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000049406564584124654

If you're printing the exact value as a decimal (using this C code for example), then the answer is 1077 characters (+1 for null):

-0.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000004940656458412465441765687928682213723650598026143247644255856825006755072702087518652998363616359923797965646954457177309266567103559397963987747960107818781263007131903114045278458171678489821036887186360569987307230500063874091535649843873124733972731696151400317153853980741262385655911710266585566867681870395603106249319452715914924553293054565444011274801297099995419319894090804165633245247571478690147267801593552386115501348035264934720193790268107107491703332226844753335720832431936092382893458368060106011506169809753078342277318329247904982524730776375927247874656084778203734469699533647017972677717585125660551199131504891101451037862738167250955837389733598993664809941164205702637090279242767544565229087538682506419718265533447265625