Does 64-bit floating point numbers behave identically on all modern PCs?

Question

I would like to know whether i can assume that same operations on same 64-bit floating point numbers gives exactly the same results on any modern PC and in most common programming languages? (C++, Java, C#, etc.). We can assume, that we are operating on numbers and result is also a number (no NaNs, INFs and so on).

I know there are two very simmilar standards of computation using floating point numbers (IEEE 854-1987 and IEEE 754-2008). However I don't know how it is in practice.

Answer 1

Modern processors that implement 64-bit floating-point typically implement something that is close to the IEEE 754-1985 standard, recently superseded by the 754-2008 standard.

The 754 standard specifies what result you should get from certain basic operations, notably addition, subtraction, multiplication, division, square root, and negation. In most cases, the numeric result is specified precisely: The result must be the representable number that is closest to the exact mathematical result in the direction specified by the rounding mode (to nearest, toward infinity, toward zero, or toward negative infinity). In "to nearest" mode, the standard also specifies how ties are broken.

Because of this, operations that do not involve exception conditions such as overflow will get the same results on different processors that conform to the standard.

However, there are several issues that interfere with getting identical results on different processors. One of them is that the compiler is often free to implement sequences of floating-point operations in a variety of ways. For example, if you write "a = bc + d" in C, where all variables are declared double, the compiler is free to compute "bc" in either double-precision arithmetic or something with more range or precision. If, for example, the processor has registers capable of holding extended-precision floating-point numbers and doing arithmetic with extended-precision does not take any more CPU time than doing arithmetic with double-precision, a compiler is likely to generate code using extended-precision. On such a processor, you might not get the same results as you would on another processor. Even if the compiler does this regularly, it might not in some circumstances because the registers are full during a complicated sequence, so it stores the intermediate results in memory temporarily. When it does that, it might write just the 64-bit double rather than the extended-precision number. So a routine containing floating-point arithmetic might give different results just because it was compiled with different code, perhaps inlined in one place, and the compiler needed registers for something else.

Some processors have instructions to compute a multiply and an add in one instruction, so "bc + d" might be computed with no intermediate rounding and get a more accurate result than on a processor that first computes bc and then adds d.

Your compiler might have switches to control behavior like this.

There are some places where the 754-1985 standard does not require a unique result. For example, when determining whether underflow has occurred (a result is too small to be represented accurately), the standard allows an implementation to make the determination either before or after it rounds the significand (the fraction bits) to the target precision. So some implementations will tell you underflow has occurred when other implementations will not.

A common feature in processors is to have an "almost IEEE 754" mode that eliminates the difficulty of dealing with underflow by substituting zero instead of returning the very small number that the standard requires. Naturally, you will get different numbers when executing in such a mode than when executing in the more compliant mode. The non-compliant mode may be the default set by your compiler and/or operating system, for reasons of performance.

Note that an IEEE 754 implementation is typically not provided just by hardware but by a combination of hardware and software. The processor may do the bulk of the work but rely on the software to handle certain exceptions, set certain modes, and so on.

When you move beyond the basic arithmetic operations to things like sine and cosine, you are very dependent on the library you use. Transcendental functions are generally calculated with carefully engineered approximations. The implementations are developed independently by various engineers and get different results from each other. On one system, the sin function may give results accurate within an ULP (unit of least precision) for small arguments (less than pi or so) but larger errors for large arguments. On another system, the sin function might give results accurate within several ULP for all arguments. No current math library is known to produce correctly rounded results for all inputs. There is a project, crlibm (Correctly Rounded Libm), that has done some good work toward this goal, and they have developed implementations for significant parts of the math library that are correctly rounded and have good performance, but not all of the math library yet.

In summary, if you have a manageable set of calculations, understand your compiler implementation, and are very careful, you can rely on identical results on different processors. Otherwise, getting completely identical results is not something you can rely on.

Answer 2

If you mean getting exactly the same result, then the answer is no.

You might even get different results for debug (non-optimized) builds vs. release builds (optimized) on the same machine in some cases, so don't even assume that the results might be always identical on different machines.

(This can happen e.g. on a computer with an Intel processor, if the optimizer keeps a variable for an intermediate result in a register, that is stored in memory in the unoptimized build. Since Intel FPU registers are 80 bit, and double variables are 64 bit, the intermediate result will be stored with greater precision in the optimized build, causing different values in later results.).

In practice, however, you may often get the same results, but you shouldn't rely on it.

Answer 3

Modern FPUs all implement IEEE754 floats in single and double formats, and some in extended format. A certain set of operations are supported (pretty much anything in math.h), with some special instructions floating around out there.

Answer 4

assuming you are talking about applying multiple operations, I do not think you will get exact numbers. CPU architecture, compiler use, optimization settings will change the results of your computations.

if you mean the exact order of operations (at the assembly level), I think you will still get variations.for example Intel chips use extended precision (80 bits) internally, which may not be the case for other CPUs. (I do not think extended precision is mandated)

Answer 5

The same C# program can bring out different numerical results on the same PC, once compiled in debug mode without optimization, second time compiled in release mode with optimization enabled. That's my personal experience. We did not regard this when we set up an automatic regression test suite for one of our programs for the first time, and were completely surprised that a lot of our tests failed without any apparent reason.

Answer 6

For C# on x86, 80-bit FP registers are used.

The C# standard says that the processor must operate at the same precision as, or greater than, the type itself (i.e. 64-bit in the case of a 'double'). Promotions are allowed, except for storage. That means that locals and parameters could be at greater than 64-bit precision.

In other words, assigning a member variable to a local variable could (and in fact will under certain circumstances) be enough to give an inequality.

See also: Float/double precision in debug/release modes

Answer 7

For the 64-bit data type, I only know of "double precision" / "binary64" from the IEEE 754 (1985 and 2008 don't differ much here for common cases) being used.

Note: The radix types defined in IEEE 854-1987 are superseded by IEEE 754-2008 anyways.