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

That depends on what you mean by "represent". Do you just want round-tripping, or the decimal representation of the exact value? Does it have to be human readable, or could you just treat the 4/8 bytes as opaque binary data and base64 or hex encode them?

Answer 2

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 3

http://en.wikipedia.org/wiki/IEEE_754-1985

The longest representable double:

-2.2250738585072020E−308

has 24 chars.

Answer 4

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 5

Perhaps you can approach the problem from another angle? Just allocate 1024 bytes for your string and feel secure? :)

Answer 6

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 7

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.

Answer 8

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