Note: Much of this section is motivated by a desire to have a self-contained description of numerics in commodity computer systems, as well as an/a interpretation / explanation / rationale of the standard text that's at least more useful in terms of practical usage than the standard text itself.
IEEE-754 specifies the following rounding modes:
roundTiesToEven: This is MANDATORY and SHALL be the default within a thread when the thread starts. The floating point value closest to the infinitely precise result is returned. If there are two such values, the one with an even digit value at the position corresponding to the least significant of the least significant digits of the two values will be returned.
roundTowardPositive: The least representable floating point value no less than the infinitely precise result is returned.
roundTowardNegative: The greatest representable floating point value no greater than the infinitely precise result is returned.
roundTowardZero: The representable floating point value with greatest magnitude no greater than that of the infinitely precise result is returned.
The standard library provides facility for setting and querying the rounding mode in the current thread. The presence of other rounding modes (e.g. roundTiesToAway, roundToOdd, etc.) are implementation-defined.
Infinity and NaNs are not numbers. It is the interpretation of @dannyniu that they exist in numerical computation strictly to serve as error recovery and reporting mechanism.
IEEE-754 specifies the following 5 exceptions:
invalid: known as "invalid operation" in standard's term. This is when:
pole: known as "division by zero" in standard's term. A pole results when operation by an operand results in an infinite limit. Particular cases of this include 1/0, tan(90°), log(0), etc.
overflow: this is when and only when the result exceeds the magnitude
of the largest representable finite number of the floating point data type
after rounding. The data type is double a.k.a. binary64 in our language.
underflow: this is when a tiny non-zero result having an absolute value below bemin, where b is the radix of the floating point data type - 2 in our case , and emin is, in our case -1022.
Note: emin can be derived as: 2 - 2ebits-1, where ebits is the number of bits in the exponents, which is 11 in our case.
inexact: this is when the result after rounding differs from what would be the actual result if it were calculated to unbounded precision and range.
The standard library provides facility for querying, clearing, and raising exceptions. Alternate exception handling attributes are implemented in the language as error-handling flow-control constructs, such as null-coalescing expression and phrases operators, as well as execution control functions.
Floating points have a fixed significand width as well as limited range(s) of exponents, as such, they're very similar to scientific notations, further as such, they suffer from the same inaccuracy problems as any notation that truncates a large fraction of value digits. However, this do yield a favorable trade-off in terms of implementation (and to some extent, usage) efficiency.
IEEE-754 recommends that language standard provide a mean to derive a sequence (graph actually, if taken dependencies into account) of computation in a way that is deterministic. Many C compilers provide options that make maths work faster using arithmetic associativity, commutativity, distributivity and other laws (e.g. fast-math options), cxing make no provision that prevents this - people favoring efficiency and people favoring accuracy should both be audience of this language.
The root cause of calculation errors stem from the fact that the significand of floating point datum are limited. This error is amplified in calculations. A way to quantify this error is using the "unit(s) in the last place" - ULP. There are various definitions of ULP. Vendors of mathematical libraries may at their discretion document the error amplification behavior of their library routines for users to consult; framework and library standards may at their discretion specify requirements in terms error amplification limits. Developers are reminded again to recognize, and evaluate at their discretion, the trade-off between accuracy and efficiency.
Because of the existence of calculation errors, floating point datum are recommended as instrument of data exchange. In fact, earlier versions of the IEEE-754 standard distinguished between interchange formats and arithmetic formats. Because arithmetics and the format where it's carried out are essentially black-box implementation details, the significance of arithmetic formats is no longer emphasized in IEEE-754.
The recommended methodology of arithmetic, is to first derive procedure of calculation that is a simplified version of the full algorithm, eliminating as much amplification of error as possible, then feed the input datum elements into the algorithm to obtain the output data. The procedure so derived should take into account of any exceptions that might occur.
For example, (a+b)(c+d) = ac+ad + bc+bd have
2 additions and 1 multiplication on the left-hand side and
3 additions and 4 multiplications on the right-hand side.
a program may first attempt to calculate the left hand side, because it has
less chance of error amplification. However, if the addition of c and d
overflows but they're individually small enough such that their multiplication
with either a and b won't overflow, yet the sum of a and b underflows
in a certain way that's catastrophic, the the whole expression may become NaN.
In this case, a fallback expression may then compute the right-hand side of the expression, possibly yielding a finite result, or at least one that arithmetically make sense (i.e. infinity).
The result of computation carried out using such "derived" procedure will certainly deviate from the result from of a "complete" algorithm. Developers should recognize that robustness may be more important in some applications than they may expect. In the limited circumstances where an application in reality is less important, or in fact be prototyping, developer may at their careful discretion, excercise less engineering effort when coding a numerical program.
Finally, it is recognized that large existing body of sophisticated numerical programs are written using 3rd-party libraries, and/or using techniques that're under active research and not specified and beyond the scope of many standards. Developers requiring high numerial sophistication and robustness are encouraged to consult these research, and evaluate (again) the accuracy and efficiency requirements at their careful discretion.
The recommended applications of floating points in computer, are Computer Graphics, Signal Processing, Artificial Intelligence, etc.
Typical characteristics of these applications include: