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One of the first programming languages to provide floating-point data types was Fortran.[citation needed] Before the widespread adoption of IEEE 754-1985, the representation and properties of floating-point data types depended on the computer manufacturer and computer model, and upon decisions made by programming-language implementers. E.g., GW-BASIC's double-precision data type was the 64-bit MBF floating-point format.
Double-precision binary floating-point is a commonly used format on PCs, due to its wider range over single-precision floating point, in spite of its performance and bandwidth cost. It is commonly known simply as double. The IEEE 754 standard specifies a binary64 as having:
The double-precision binary floating-point exponent is encoded using an offset-binary representation, with the zero offset being 1023; also known as exponent bias in the IEEE 754 standard. Examples of such representations would be:
Encodings of qNaN and sNaN are not completely specified in IEEE 754 and depend on the processor. Most processors, such as the x86 family and the ARM family processors, use the most significant bit of the significand field to indicate a quiet NaN; this is what is recommended by IEEE 754. The PA-RISC processors use the bit to indicate a signaling NaN.
Using double-precision floating-point variables is usually slower than working with their single precision counterparts. One area of computing where this is a particular issue is parallel code running on GPUs. For example, when using NVIDIA's CUDA platform, calculations with double precision can take, depending on hardware, from 2 to 32 times as long to complete compared to those done using single precision.[4]
Doubles are implemented in many programming languages in different ways such as the following. On processors with only dynamic precision, such as x86 without SSE2 (or when SSE2 is not used, for compatibility purpose) and with extended precision used by default, software may have difficulties to fulfill some requirements.
C and C++ offer a wide variety of arithmetic types. Double precision is not required by the standards (except by the optional annex F of C99, covering IEEE 754 arithmetic), but on most systems, the double type corresponds to double precision. However, on 32-bit x86 with extended precision by default, some compilers may not conform to the C standard or the arithmetic may suffer from double rounding.[5]
Common Lisp provides the types SHORT-FLOAT, SINGLE-FLOAT, DOUBLE-FLOAT and LONG-FLOAT. Most implementations provide SINGLE-FLOATs and DOUBLE-FLOATs with the other types appropriate synonyms. Common Lisp provides exceptions for catching floating-point underflows and overflows, and the inexact floating-point exception, as per IEEE 754. No infinities and NaNs are described in the ANSI standard, however, several implementations do provide these as extensions.
On Java before version 1.2, every implementation had to be IEEE 754 compliant. Version 1.2 allowed implementations to bring extra precision in intermediate computations for platforms like x87. Thus a modifier strictfp was introduced to enforce strict IEEE 754 computations. Strict floating point has been restored in Java 17.[6]
The JSON data encoding format supports numeric values, and the grammar to which numeric expressions must conform has no limits on the precision or range of the numbers so encoded. However, RFC 8259 advises that, since IEEE 754 binary64 numbers are widely implemented, good interoperability can be achieved by implementations processing JSON if they expect no more precision or range than binary64 offers.[8]
I've read about the difference between double precision and single precision. However, in most cases, float and double seem to be interchangeable, i.e. using one or the other does not seem to affect the results. Is this really the case? When are floats and doubles interchangeable? What are the differences between them?
Also, the maximum value of float is about 3e38, but double is about 1.7e308, so using float can hit "infinity" (i.e. a special floating-point number) much more easily than double for something simple, e.g. computing the factorial of 60.
Of course, sometimes, even double isn't accurate enough, hence we sometimes have long double[1] (the above example gives 9.000000000000000066 on Mac), but all floating point types suffer from round-off errors, so if precision is very important (e.g. money processing) you should use int or a fraction class.
[1]: The C and C++ standards do not specify the representation of float, double and long double. It is possible that all three are implemented as IEEE double-precision. Nevertheless, for most architectures (gcc, MSVC; x86, x64, ARM) float is indeed a IEEE single-precision floating point number (binary32), and double is a IEEE double-precision floating point number (binary64).
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.
I would suggest having a look at the excellent What Every Computer Scientist Should Know About Floating-Point Arithmetic that covers the IEEE floating-point standard in depth. You'll learn about the representation details and you'll realize there is a tradeoff between magnitude and precision. The precision of the floating point representation increases as the magnitude decreases, hence floating point numbers between -1 and 1 are those with the most precision.
(In fact, the above is not the best way of solving quadratic equations using either single- or double-precision floating-point numbers, but the answer remains unchanged even if one uses a more stable method.)
In essence, if you're performing a calculation and the result is an irrational number or recurring decimal, then there will be rounding errors when that number is squashed into the finite size data structure you're using. Since double is twice the size of float then the rounding error will be a lot smaller.
Type long double is nominally 80 bits, though a given compiler/OS pairing may store it as 12-16 bytes for alignment purposes. The long double has an exponent that just ridiculously huge and should have 19 digits precision. Microsoft, in their infinite wisdom, limits long double to 8 bytes, the same as plain double.
Generally speaking, just use type double when you need a floating point value/variable. Literal floating point values used in expressions will be treated as doubles by default, and most of the math functions that return floating point values return doubles. You'll save yourself many headaches and typecastings if you just use double.
When using floating point numbers you cannot trust that your local tests will be exactly the same as the tests that are done on the server side. The environment and the compiler are probably different on you local system and where the final tests are run. I have seen this problem many times before in some TopCoder competitions especially if you try to compare two floating point numbers.
But perhaps even more important is the qualitative difference. Type float has good precision, which will often be good enough for whatever you're doing. Type double, on the other hand, has excellent precision, which will almost always be good enough for whatever you're doing.
The upshot, which is not nearly as well known as it should be, is that you should almost always use type double. Unless you have some particularly special need, you should almost never use type float.
And the thing that's not necessarily different between type float and double is execution speed. On most of today's general-purpose processors, arithmetic operations on type float and double take more or less exactly the same amount of time. Everything's done in parallel, so you don't pay a speed penalty for the greater range and precision of type double. That's why it's safe to make the recommendation that you should almost never use type float: Using double shouldn't cost you anything in speed, and it shouldn't cost you much in space, and it will almost definitely pay off handsomely in freedom from precision and roundoff error woes.
(With that said, though, one of the "special needs" where you may need type float is when you're doing embedded work on a microcontroller, or writing code that's optimized for a GPU. On those processors, type double can be significantly slower, or practically nonexistent, so in those cases programmers do typically choose type float for speed, and maybe pay for it in precision.)
If one works with embedded processing, eventually the underlying hardware (e.g. FPGA or some specific processor / microcontroller model) will have float implemented optimally in hardware whereas double will use software routines. So if the precision of a float is enough to handle the needs, the program will execute some times faster with float then double. As noted on other answers, beware of accumulation errors.
Unlike an int (whole number), a float have a decimal point, and so can a double.But the difference between the two is that a double is twice as detailed as a float, meaning that it can have double the amount of numbers after the decimal point.
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