What Is the Precision of Long Double in C++

What is the precision of long double in C++?

You can find out with std::numeric_limits:

#include <iostream>     // std::cout
#include <limits>       // std::numeric_limits
int main(){
    std::cout << std::numeric_limits<long double>::digits10 << std::endl;
}

sizeof long double and precision not matching?

The long double format in your C implementation uses an Intel format with a one-bit sign, a 15-bit exponent, and a 64-bit significand (ten bytes total). The compiler allocates 16 bytes for it, which is wasteful but useful for some things such as alignment. However, the 64 bits provide only log₁₀(2⁶⁴) digits of significance, which is about 20 digits.

How to identify the precision of long double datatype

I want to know until how many digits the precision can handle.

LDBL_DIG, in <float.h>, is the number of minimum number of significant decimal digits that will be converted from text to long double distinctively. It is at least 10 and may be 18 or so on your platform. 0.1234567812345678 and 123.4567812345678 have 16 significant decimal digits.

My compiler is telling me that the size of sizeof(long double) is 16 byte, which means it can represent a number up 2^128

It could mean something like that if long double was an integer, yet floating point numbers are distributed logarithmically. Further, some systems do not use all 16 bytes for data. Some are padding. @unwind @Sven Marnach. Use the values in <float.h> to characterize the long double your platform is using.

if x= 0.1234567812345678, can long double identify the exact precision of x?

No, unless you are on a rare platform that uses decimal floating point.

Most platforms use a binary floating point. So unless the number can be represented as an exact sum of powers of 2, like 42.125, the value saved as a long double will be a nearby approximation. Code will convert text 0.1234567812345678 to the nearest long double which may have an exact value of 0.1234567812345677972896140772718354128301143646240234375.

Long Double vs Long Long

long long is integer (possibly with more range than long)

long double is floating point (possibly with more range/precision than double)

long float does not exist.

The integer types sorted by range are

_Bool
char or signed char or unsigned char
short (or short int) or short unsigned
int or unsigned
long (or long int) or long unsigned
long long (or long long int) or long long unsigned

The floating-point types sorted by range/precision are

float
double
long double

Difference between long double and double in C and C++

To quote the C++ standard, §3.9.1 ¶8:

There are three floating point types: float, double, and long double. The type double provides at least as much precision as float, and the type long double provides at least as much precision as double. The set of values of the type float is a subset of the set of values of the type double; the set of values of the type double is a subset of the set of values of the type long double. The value representation of floating-point types is implementation-defined. Integral and floating types are collectively called arithmetic types. Specializations of the standard template std::numeric_limits (18.3) shall specify the maximum and minimum values of each arithmetic type for an implementation.

That is to say that double takes at least as much memory for its representation as float and long double at least as much as double. That extra memory is used for more precise representation of a number.

On x86 systems, float is typically 4 bytes long and can store numbers as large as about 3×10³⁸ and about as small as 1.4×10⁻⁴⁵. It is an IEEE 754 single-precision number that stores about 7 decimal digits of a fractional number.

Also on x86 systems, double is 8 bytes long and can store numbers in the IEEE 754 double-precision format, which has a much larger range and stores numbers with more precision, about 15 decimal digits. On some other platforms, double may not be 8 bytes long and may indeed be the same as a single-precision float.

The standard only requires that long double is at least as precise as double, so some compilers will simply treat long double as if it is the same as double. But, on most x86 chips, the 10-byte extended precision format 80-bit number is available through the CPU's floating-point unit, which provides even more precision than 64-bit double, with about 21 decimal digits of precision.

Some compilers instead support a 16-byte (128-bit) IEEE 754 quadruple precision number format with yet more precise representations and a larger range.

Long double precision error saturation in RK integrator

but for some reason, the precision of my integrator maxes out at 16 significant digits.

At a minimum, use more correct values of long double initialization with long double quotients rather than double quotients.

// long double a2 = 2.0 / 7.0;
long double a2 = 2.0L / 7.0L;

Like-wise for the other 19 initializations.

What Is the Precision of Long Double in C++