Floating-point arithmetic

Floating-point numbers

A number representation specifies some way of encoding a number, usually as a string of digits.

There are several mechanisms by which strings of digits can represent numbers. In standard mathematical notation, the digit string can be of any length, and the location of the radix point is indicated by placing an explicit "point" character (dot or comma) there. If the radix point is not specified, then the string implicitly represents an integer and the unstated radix point would be off the right-hand end of the string, next to the least significant digit. In fixed-point systems, a position in the string is specified for the radix point. So a fixed-point scheme might use a string of 8 decimal digits with the decimal point in the middle, whereby "00012345" would represent 0001.2345.

In scientific notation, the given number is scaled by a power of 10, so that it lies within a specific range—typically between 1 and 10, with the radix point appearing immediately after the first digit. As a power of ten, the scaling factor is then indicated separately at the end of the number. For example, the orbital period of Jupiter's moon Io is 152,853.5047 seconds, a value that would be represented in standard-form scientific notation as 1.528535047×10⁵ seconds.

Floating-point representation is similar in concept to scientific notation. Logically, a floating-point number consists of:

• A signed (meaning positive or negative) digit string of a given length in a given base (or radix). This digit string is referred to as the significand, mantissa, or coefficient.^{[nb 1]} The length of the significand determines the precision to which numbers can be represented. The radix point position is assumed always to be somewhere within the significand—often just after or just before the most significant digit, or to the right of the rightmost (least significant) digit. This article generally follows the convention that the radix point is set just after the most significant (leftmost) digit.
• A signed integer exponent (also referred to as the characteristic, or scale),^{[nb 2]} which modifies the magnitude of the number.

To derive the value of the floating-point number, the significand is multiplied by the base raised to the power of the exponent, equivalent to shifting the radix point from its implied position by a number of places equal to the value of the exponent—to the right if the exponent is positive or to the left if the exponent is negative.

Using base-10 (the familiar decimal notation) as an example, the number 152,853.5047, which has ten decimal digits of precision, is represented as the significand 1,528,535,047 together with 5 as the exponent. To determine the actual value, a decimal point is placed after the first digit of the significand and the result is multiplied by 10⁵ to give 1.528535047×10⁵, or 152,853.5047. In storing such a number, the base (10) need not be stored, since it will be the same for the entire range of supported numbers, and can thus be inferred.

Symbolically, this final value is: $${\displaystyle {\frac {s}{b^{\,p-1}}}\times b^{e},}$$

where s is the significand (ignoring any implied decimal point), p is the precision (the number of digits in the significand), b is the base (in our example, this is the number ten), and e is the exponent.

Historically, several number bases have been used for representing floating-point numbers, with base two (binary) being the most common, followed by base ten (decimal floating point), and other less common varieties, such as base sixteen (hexadecimal floating point^{[4][5][nb 3]}), base eight (octal floating point^{[1][5][6][4][nb 4]}), base four (quaternary floating point^{[7][5][nb 5]}), base three (balanced ternary floating point^[1]) and even base 256^{[5][nb 6]} and base 65,536.^{[8][nb 7]}

A floating-point number is a rational number, because it can be represented as one integer divided by another; for example 1.45×10³ is (145/100)×1000 or 145,000/100. The base determines the fractions that can be represented; for instance, 1/5 cannot be represented exactly as a floating-point number using a binary base, but 1/5 can be represented exactly using a decimal base (0.2, or 2×10⁻¹). However, 1/3 cannot be represented exactly by either binary (0.010101...) or decimal (0.333...), but in base 3, it is trivial (0.1 or 1×3⁻¹) . The occasions on which infinite expansions occur depend on the base and its prime factors.

The way in which the significand (including its sign) and exponent are stored in a computer is implementation-dependent. The common IEEE formats are described in detail later and elsewhere, but as an example, in the binary single-precision (32-bit) floating-point representation, ${\displaystyle p=24}$, and so the significand is a string of 24 bits. For instance, the number π's first 33 bits are: $${\displaystyle 11001001\ 00001111\ 1101101{\underline {0}}\ 10100010\ 0.}$$

In this binary expansion, let us denote the positions from 0 (leftmost bit, or most significant bit) to 32 (rightmost bit). The 24-bit significand will stop at position 23, shown as the underlined bit 0 above. The next bit, at position 24, is called the round bit or rounding bit. It is used to round the 33-bit approximation to the nearest 24-bit number (there are specific rules for halfway values, which is not the case here). This bit, which is 1 in this example, is added to the integer formed by the leftmost 24 bits, yielding: $${\displaystyle 11001001\ 00001111\ 1101101{\underline {1}}.}$$

When this is stored in memory using the IEEE 754 encoding, this becomes the significand s. The significand is assumed to have a binary point to the right of the leftmost bit. So, the binary representation of π is calculated from left-to-right as follows: $${\displaystyle {\begin{aligned}&\left(\sum _{n=0}^{p-1}{\text{bit}}_{n}\times 2^{-n}\right)\times 2^{e}\\={}&\left(1\times 2^{-0}+1\times 2^{-1}+0\times 2^{-2}+0\times 2^{-3}+1\times 2^{-4}+\cdots +1\times 2^{-23}\right)\times 2^{1}\\\approx {}&1.57079637\times 2\\\approx {}&3.1415927\end{aligned}}}$$

where p is the precision (24 in this example), n is the position of the bit of the significand from the left (starting at 0 and finishing at 23 here) and e is the exponent (1 in this example).

It can be required that the most significant digit of the significand of a non-zero number be non-zero (except when the corresponding exponent would be smaller than the minimum one). This process is called normalization. For binary formats (which uses only the digits 0 and 1), this non-zero digit is necessarily 1. Therefore, it does not need to be represented in memory, allowing the format to have one more bit of precision. This rule is variously called the leading bit convention, the implicit bit convention, the hidden bit convention,^[1] or the assumed bit convention.

Alternatives to floating-point numbers

The floating-point representation is by far the most common way of representing in computers an approximation to real numbers. However, there are alternatives:

• Fixed-point representation uses integer hardware operations controlled by a software implementation of a specific convention about the location of the binary or decimal point, for example, 6 bits or digits from the right. The hardware to manipulate these representations is less costly than floating point, and it can be used to perform normal integer operations, too. Binary fixed point is usually used in special-purpose applications on embedded processors that can only do integer arithmetic, but decimal fixed point is common in commercial applications.
• Logarithmic number systems (LNSs) represent a real number by the logarithm of its absolute value and a sign bit. The value distribution is similar to floating point, but the value-to-representation curve (i.e., the graph of the logarithm function) is smooth (except at 0). Conversely to floating-point arithmetic, in a logarithmic number system multiplication, division and exponentiation are simple to implement, but addition and subtraction are complex. The (symmetric) level-index arithmetic (LI and SLI) of Charles Clenshaw, Frank Olver and Peter Turner is a scheme based on a generalized logarithm representation.
• Tapered floating-point representation, used in Unum.
• Some simple rational numbers (e.g., 1/3 and 1/10) cannot be represented exactly in binary floating point, no matter what the precision is. Using a different radix allows one to represent some of them (e.g., 1/10 in decimal floating point), but the possibilities remain limited. Software packages that perform rational arithmetic represent numbers as fractions with integral numerator and denominator, and can therefore represent any rational number exactly. Such packages generally need to use "bignum" arithmetic for the individual integers.
• Interval arithmetic allows one to represent numbers as intervals and obtain guaranteed bounds on results. It is generally based on other arithmetics, in particular floating point.
• Computer algebra systems such as Mathematica, Maxima, and Maple can often handle irrational numbers like ${\displaystyle \pi }$ or ${\displaystyle {\sqrt {3}}}$ in a completely "formal" way (symbolic computation), without dealing with a specific encoding of the significand. Such a program can evaluate expressions like "${\displaystyle \sin(3\pi )}$" exactly, because it is programmed to process the underlying mathematics directly, instead of using approximate values for each intermediate calculation.

Internal representation

Floating-point numbers are typically packed into a computer datum as the sign bit, the exponent field, and the significand or mantissa, from left to right. For the IEEE 754 binary formats (basic and extended) which have extant hardware implementations, they are apportioned as follows:

While the exponent can be positive or negative, in binary formats it is stored as an unsigned number that has a fixed "bias" added to it. Values of all 0s in this field are reserved for the zeros and subnormal numbers; values of all 1s are reserved for the infinities and NaNs. The exponent range for normal numbers is [−126, 127] for single precision, [−1022, 1023] for double, or [−16382, 16383] for quad. Normal numbers exclude subnormal values, zeros, infinities, and NaNs.

In the IEEE binary interchange formats the leading 1 bit of a normalized significand is not actually stored in the computer datum. It is called the "hidden" or "implicit" bit. Because of this, the single-precision format actually has a significand with 24 bits of precision, the double-precision format has 53, and quad has 113.

For example, it was shown above that π, rounded to 24 bits of precision, has:

• sign = 0 ; e = 1 ; s = 110010010000111111011011 (including the hidden bit)

The sum of the exponent bias (127) and the exponent (1) is 128, so this is represented in the single-precision format as

• 0 10000000 10010010000111111011011 (excluding the hidden bit) = 40490FDB^[27] as a hexadecimal number.

An example of a layout for 32-bit floating point is

and the 64-bit ("double") layout is similar.

Rounding modes

Rounding is used when the exact result of a floating-point operation (or a conversion to floating-point format) would need more digits than there are digits in the significand. IEEE 754 requires correct rounding: that is, the rounded result is as if infinitely precise arithmetic was used to compute the value and then rounded (although in implementation only three extra bits are needed to ensure this). There are several different rounding schemes (or rounding modes). Historically, truncation was the typical approach. Since the introduction of IEEE 754, the default method (round to nearest, ties to even, sometimes called Banker's Rounding) is more commonly used. This method rounds the ideal (infinitely precise) result of an arithmetic operation to the nearest representable value, and gives that representation as the result.^{[nb 8]} In the case of a tie, the value that would make the significand end in an even digit is chosen. The IEEE 754 standard requires the same rounding to be applied to all fundamental algebraic operations, including square root and conversions, when there is a numeric (non-NaN) result. It means that the results of IEEE 754 operations are completely determined in all bits of the result, except for the representation of NaNs. ("Library" functions such as cosine and log are not mandated.)

Alternative rounding options are also available. IEEE 754 specifies the following rounding modes:

• round to nearest, where ties round to the nearest even digit in the required position (the default and by far the most common mode)
• round to nearest, where ties round away from zero (optional for binary floating-point and commonly used in decimal)
• round up (toward +∞; negative results thus round toward zero)
• round down (toward −∞; negative results thus round away from zero)
• round toward zero (truncation; it is similar to the common behavior of float-to-integer conversions, which convert −3.9 to −3 and 3.9 to 3)

Alternative modes are useful when the amount of error being introduced must be bounded. Applications that require a bounded error are multi-precision floating-point, and interval arithmetic. The alternative rounding modes are also useful in diagnosing numerical instability: if the results of a subroutine vary substantially between rounding to + and − infinity then it is likely numerically unstable and affected by round-off error.^[34]

Binary-to-decimal conversion with minimal number of digits

Converting a double-precision binary floating-point number to a decimal string is a common operation, but an algorithm producing results that are both accurate and minimal did not appear in print until 1990, with Steele and White's Dragon4. Some of the improvements since then include:

• David M. Gay's dtoa.c, a practical open-source implementation of many ideas in Dragon4.^[35]
• Grisu3, with a 4× speedup as it removes the use of bignums. Must be used with a fallback, as it fails for ~0.5% of cases.^[36]
• Errol3, an always-succeeding algorithm similar to, but slower than, Grisu3. Apparently not as good as an early-terminating Grisu with fallback.^[37]
• Ryū, an always-succeeding algorithm that is faster and simpler than Grisu3.^[38]
• Schubfach, an always-succeeding algorithm that is based on a similar idea to Ryū, developed almost simultaneously and independently.^[39] Performs better than Ryū and Grisu3 in certain benchmarks.^[40]

Many modern language runtimes use Grisu3 with a Dragon4 fallback.^[41]

Decimal-to-binary conversion

The problem of parsing a decimal string into a binary FP representation is complex, with an accurate parser not appearing until Clinger's 1990 work (implemented in dtoa.c).^[35] Further work has likewise progressed in the direction of faster parsing.^[42]

Addition and subtraction

A simple method to add floating-point numbers is to first represent them with the same exponent. In the example below, the second number (with the smaller exponent) is shifted right by three digits, and one then proceeds with the usual addition method:

  123456.7 = 1.234567 × 10^5
  101.7654 = 1.017654 × 10^2 = 0.001017654 × 10^5

  Hence:
  123456.7 + 101.7654 = (1.234567 × 10^5) + (1.017654 × 10^2)
                      = (1.234567 × 10^5) + (0.001017654 × 10^5)
                      = (1.234567 + 0.001017654) × 10^5
                      =  1.235584654 × 10^5

In detail:

  e=5;  s=1.234567     (123456.7)
+ e=2;  s=1.017654     (101.7654)

  e=5;  s=1.234567
+ e=5;  s=0.001017654  (after shifting)
--------------------
  e=5;  s=1.235584654  (true sum: 123558.4654)

This is the true result, the exact sum of the operands. It will be rounded to seven digits and then normalized if necessary. The final result is

  e=5;  s=1.235585    (final sum: 123558.5)

The lowest three digits of the second operand (654) are essentially lost. This is round-off error. In extreme cases, the sum of two non-zero numbers may be equal to one of them:

  e=5;  s=1.234567
+ e=−3; s=9.876543

  e=5;  s=1.234567
+ e=5;  s=0.00000009876543 (after shifting)
----------------------
  e=5;  s=1.23456709876543 (true sum)
  e=5;  s=1.234567         (after rounding and normalization)

In the above conceptual examples it would appear that a large number of extra digits would need to be provided by the adder to ensure correct rounding; however, for binary addition or subtraction using careful implementation techniques only a guard bit, a rounding bit and one extra sticky bit need to be carried beyond the precision of the operands.^{[43][44]: 218–220}

Another problem of loss of significance occurs when approximations to two nearly equal numbers are subtracted. In the following example e = 5; s = 1.234571 and e = 5; s = 1.234567 are approximations to the rationals 123457.1467 and 123456.659.

  e=5;  s=1.234571
− e=5;  s=1.234567
----------------
  e=5;  s=0.000004
  e=−1; s=4.000000 (after rounding and normalization)

The floating-point difference is computed exactly because the numbers are close—the Sterbenz lemma guarantees this, even in case of underflow when gradual underflow is supported. Despite this, the difference of the original numbers is e = −1; s = 4.877000, which differs more than 20% from the difference e = −1; s = 4.000000 of the approximations. In extreme cases, all significant digits of precision can be lost.^[43][45] This cancellation illustrates the danger in assuming that all of the digits of a computed result are meaningful. Dealing with the consequences of these errors is a topic in numerical analysis; see also Accuracy problems.

Multiplication and division

To multiply, the significands are multiplied while the exponents are added, and the result is rounded and normalized.

  e=3;  s=4.734612
× e=5;  s=5.417242
-----------------------
  e=8;  s=25.648538980104 (true product)
  e=8;  s=25.64854        (after rounding)
  e=9;  s=2.564854        (after normalization)

Similarly, division is accomplished by subtracting the divisor's exponent from the dividend's exponent, and dividing the dividend's significand by the divisor's significand.

There are no cancellation or absorption problems with multiplication or division, though small errors may accumulate as operations are performed in succession.^[43] In practice, the way these operations are carried out in digital logic can be quite complex (see Booth's multiplication algorithm and Division algorithm).^{[nb 9]}

Literal syntax

Literals for floating-point numbers depend on languages. They typically use e or E to denote scientific notation. The C programming language and the IEEE 754 standard also define a hexadecimal literal syntax with a base-2 exponent instead of 10. In languages like C, when the decimal exponent is omitted, a decimal point is needed to differentiate them from integers. Other languages do not have an integer type (such as JavaScript), or allow overloading of numeric types (such as Haskell). In these cases, digit strings such as 123 may also be floating-point literals.

Examples of floating-point literals are:

• 99.9
• -5000.12
• 6.02e23
• -3e-45
• 0x1.fffffep+127 in C and IEEE 754

Incidents

• On 25 February 1991, a loss of significance in a MIM-104 Patriot missile battery prevented it from intercepting an incoming Scud missile in Dhahran, Saudi Arabia, contributing to the death of 28 soldiers from the U.S. Army's 14th Quartermaster Detachment.^[50] The error was actually introduced by a fixed-point computation,^[51] but the underlying issue would have been the same with floating-point arithmetic.
• Salami slicing is the practice of removing the 'invisible' part of a transaction into a separate account.^{[clarification needed]}

Machine precision and backward error analysis

Machine precision is a quantity that characterizes the accuracy of a floating-point system, and is used in backward error analysis of floating-point algorithms. It is also known as unit roundoff or machine epsilon. Usually denoted Ε_mach, its value depends on the particular rounding being used.

With rounding to zero, $${\displaystyle \mathrm {E} _{\text{mach}}=B^{1-P},\,}$$ whereas rounding to nearest, $${\displaystyle \mathrm {E} _{\text{mach}}={\tfrac {1}{2}}B^{1-P},}$$ where B is the base of the system and P is the precision of the significand (in base B).

This is important since it bounds the relative error in representing any non-zero real number x within the normalized range of a floating-point system: $${\displaystyle \left|{\frac {\operatorname {fl} (x)-x}{x}}\right|\leq \mathrm {E} _{\text{mach}}.}$$

Backward error analysis, the theory of which was developed and popularized by James H. Wilkinson, can be used to establish that an algorithm implementing a numerical function is numerically stable.^[52] The basic approach is to show that although the calculated result, due to roundoff errors, will not be exactly correct, it is the exact solution to a nearby problem with slightly perturbed input data. If the perturbation required is small, on the order of the uncertainty in the input data, then the results are in some sense as accurate as the data "deserves". The algorithm is then defined as backward stable. Stability is a measure of the sensitivity to rounding errors of a given numerical procedure; by contrast, the condition number of a function for a given problem indicates the inherent sensitivity of the function to small perturbations in its input and is independent of the implementation used to solve the problem.^[53]

As a trivial example, consider a simple expression giving the inner product of (length two) vectors ${\displaystyle x}$ and ${\displaystyle y}$, then $${\displaystyle {\begin{aligned}\operatorname {fl} (x\cdot y)&=\operatorname {fl} {\big (}\operatorname {fl} (x_{1}\cdot y_{1})+\operatorname {fl} (x_{2}\cdot y_{2}){\big )},&&{\text{ where }}\operatorname {fl} (){\text{ indicates correctly rounded floating-point arithmetic}}\\&=\operatorname {fl} {\big (}(x_{1}\cdot y_{1})(1+\delta _{1})+(x_{2}\cdot y_{2})(1+\delta _{2}){\big )},&&{\text{ where }}\delta _{n}\leq \mathrm {E} _{\text{mach}},{\text{ from above}}\\&={\big (}(x_{1}\cdot y_{1})(1+\delta _{1})+(x_{2}\cdot y_{2})(1+\delta _{2}){\big )}(1+\delta _{3})\\&=(x_{1}\cdot y_{1})(1+\delta _{1})(1+\delta _{3})+(x_{2}\cdot y_{2})(1+\delta _{2})(1+\delta _{3}),\end{aligned}}}$$ and so $${\displaystyle \operatorname {fl} (x\cdot y)={\hat {x}}\cdot {\hat {y}},}$$

where

$${\displaystyle {\begin{aligned}{\hat {x}}_{1}&=x_{1}(1+\delta _{1});&{\hat {x}}_{2}&=x_{2}(1+\delta _{2});\\{\hat {y}}_{1}&=y_{1}(1+\delta _{3});&{\hat {y}}_{2}&=y_{2}(1+\delta _{3}),\\\end{aligned}}}$$

where

$${\displaystyle \delta _{n}\leq \mathrm {E} _{\text{mach}}}$$

by definition, which is the sum of two slightly perturbed (on the order of Ε_mach) input data, and so is backward stable. For more realistic examples in numerical linear algebra, see Higham 2002^[54] and other references below.

Minimizing the effect of accuracy problems

Although individual arithmetic operations of IEEE 754 are guaranteed accurate to within half a ULP, more complicated formulae can suffer from larger errors for a variety of reasons. The loss of accuracy can be substantial if a problem or its data are ill-conditioned, meaning that the correct result is hypersensitive to tiny perturbations in its data. However, even functions that are well-conditioned can suffer from large loss of accuracy if an algorithm numerically unstable for that data is used: apparently equivalent formulations of expressions in a programming language can differ markedly in their numerical stability. One approach to remove the risk of such loss of accuracy is the design and analysis of numerically stable algorithms, which is an aim of the branch of mathematics known as numerical analysis. Another approach that can protect against the risk of numerical instabilities is the computation of intermediate (scratch) values in an algorithm at a higher precision than the final result requires,^[55] which can remove, or reduce by orders of magnitude,^[56] such risk: IEEE 754 quadruple precision and extended precision are designed for this purpose when computing at double precision.^{[57][nb 11]}

For example, the following algorithm is a direct implementation to compute the function A(x) = (x−1) / (exp(x−1) − 1) which is well-conditioned at 1.0,^{[nb 12]} however it can be shown to be numerically unstable and lose up to half the significant digits carried by the arithmetic when computed near 1.0.^[58]

double A(double X)
{
        double Y, Z;  // [1]
        Y = X - 1.0;
        Z = exp(Y);
        if (Z != 1.0)
                Z = Y / (Z - 1.0); // [2]
        return Z;
}

If, however, intermediate computations are all performed in extended precision (e.g. by setting line [1] to C99 long double), then up to full precision in the final double result can be maintained.^{[nb 13]} Alternatively, a numerical analysis of the algorithm reveals that if the following non-obvious change to line [2] is made:

Z = log(Z) / (Z - 1.0);

then the algorithm becomes numerically stable and can compute to full double precision.

To maintain the properties of such carefully constructed numerically stable programs, careful handling by the compiler is required. Certain "optimizations" that compilers might make (for example, reordering operations) can work against the goals of well-behaved software. There is some controversy about the failings of compilers and language designs in this area: C99 is an example of a language where such optimizations are carefully specified to maintain numerical precision. See the external references at the bottom of this article.

A detailed treatment of the techniques for writing high-quality floating-point software is beyond the scope of this article, and the reader is referred to,^[54][59] and the other references at the bottom of this article. Kahan suggests several rules of thumb that can substantially decrease by orders of magnitude^[59] the risk of numerical anomalies, in addition to, or in lieu of, a more careful numerical analysis. These include: as noted above, computing all expressions and intermediate results in the highest precision supported in hardware (a common rule of thumb is to carry twice the precision of the desired result, i.e. compute in double precision for a final single-precision result, or in double extended or quad precision for up to double-precision results^[60]); and rounding input data and results to only the precision required and supported by the input data (carrying excess precision in the final result beyond that required and supported by the input data can be misleading, increases storage cost and decreases speed, and the excess bits can affect convergence of numerical procedures:^[61] notably, the first form of the iterative example given below converges correctly when using this rule of thumb). Brief descriptions of several additional issues and techniques follow.

As decimal fractions can often not be exactly represented in binary floating-point, such arithmetic is at its best when it is simply being used to measure real-world quantities over a wide range of scales (such as the orbital period of a moon around Saturn or the mass of a proton), and at its worst when it is expected to model the interactions of quantities expressed as decimal strings that are expected to be exact.^[56][59] An example of the latter case is financial calculations. For this reason, financial software tends not to use a binary floating-point number representation.^[62] The "decimal" data type of the C# and Python programming languages, and the decimal formats of the IEEE 754-2008 standard, are designed to avoid the problems of binary floating-point representations when applied to human-entered exact decimal values, and make the arithmetic always behave as expected when numbers are printed in decimal.

Expectations from mathematics may not be realized in the field of floating-point computation. For example, it is known that ${\displaystyle (x+y)(x-y)=x^{2}-y^{2}\,}$, and that ${\displaystyle \sin ^{2}{\theta }+\cos ^{2}{\theta }=1\,}$, however these facts cannot be relied on when the quantities involved are the result of floating-point computation.

The use of the equality test (if (x==y) ...) requires care when dealing with floating-point numbers. Even simple expressions like 0.6/0.2-3==0 will, on most computers, fail to be true^[63] (in IEEE 754 double precision, for example, 0.6/0.2 - 3 is approximately equal to -4.44089209850063e-16). Consequently, such tests are sometimes replaced with "fuzzy" comparisons (if (abs(x-y) < epsilon) ..., where epsilon is sufficiently small and tailored to the application, such as 1.0E−13). The wisdom of doing this varies greatly, and can require numerical analysis to bound epsilon.^[54] Values derived from the primary data representation and their comparisons should be performed in a wider, extended, precision to minimize the risk of such inconsistencies due to round-off errors.^[59] It is often better to organize the code in such a way that such tests are unnecessary. For example, in computational geometry, exact tests of whether a point lies off or on a line or plane defined by other points can be performed using adaptive precision or exact arithmetic methods.^[64]

Small errors in floating-point arithmetic can grow when mathematical algorithms perform operations an enormous number of times. A few examples are matrix inversion, eigenvector computation, and differential equation solving. These algorithms must be very carefully designed, using numerical approaches such as iterative refinement, if they are to work well.^[65]

Summation of a vector of floating-point values is a basic algorithm in scientific computing, and so an awareness of when loss of significance can occur is essential. For example, if one is adding a very large number of numbers, the individual addends are very small compared with the sum. This can lead to loss of significance. A typical addition would then be something like

3253.671
+  3.141276
-----------
3256.812

The low 3 digits of the addends are effectively lost. Suppose, for example, that one needs to add many numbers, all approximately equal to 3. After 1000 of them have been added, the running sum is about 3000; the lost digits are not regained. The Kahan summation algorithm may be used to reduce the errors.^[54]

Round-off error can affect the convergence and accuracy of iterative numerical procedures. As an example, Archimedes approximated π by calculating the perimeters of polygons inscribing and circumscribing a circle, starting with hexagons, and successively doubling the number of sides. As noted above, computations may be rearranged in a way that is mathematically equivalent but less prone to error (numerical analysis). Two forms of the recurrence formula for the circumscribed polygon are:^{[citation needed]}

• ${\textstyle t_{0}={\frac {1}{\sqrt {3}}}}$
• First form: ${\textstyle t_{i+1}={\frac {{\sqrt {t_{i}^{2}+1}}-1}{t_{i}}}}$
• Second form: ${\textstyle t_{i+1}={\frac {t_{i}}{{\sqrt {t_{i}^{2}+1}}+1}}}$
• ${\displaystyle \pi \sim 6\times 2^{i}\times t_{i}}$, converging as ${\displaystyle i\rightarrow \infty }$

Here is a computation using IEEE "double" (a significand with 53 bits of precision) arithmetic:

 i   6 × 2i × ti, first form    6 × 2i × ti, second form
---------------------------------------------------------
 0   3.4641016151377543863      3.4641016151377543863
 1   3.2153903091734710173      3.2153903091734723496
 2   3.1596599420974940120      3.1596599420975006733
 3   3.1460862151314012979      3.1460862151314352708
 4   3.1427145996453136334      3.1427145996453689225
 5   3.1418730499801259536      3.1418730499798241950
 6   3.1416627470548084133      3.1416627470568494473
 7   3.1416101765997805905      3.1416101766046906629
 8   3.1415970343230776862      3.1415970343215275928
 9   3.1415937488171150615      3.1415937487713536668
10   3.1415929278733740748      3.1415929273850979885
11   3.1415927256228504127      3.1415927220386148377
12   3.1415926717412858693      3.1415926707019992125
13   3.1415926189011456060      3.1415926578678454728
14   3.1415926717412858693      3.1415926546593073709
15   3.1415919358822321783      3.1415926538571730119
16   3.1415926717412858693      3.1415926536566394222
17   3.1415810075796233302      3.1415926536065061913
18   3.1415926717412858693      3.1415926535939728836
19   3.1414061547378810956      3.1415926535908393901
20   3.1405434924008406305      3.1415926535900560168
21   3.1400068646912273617      3.1415926535898608396
22   3.1349453756585929919      3.1415926535898122118
23   3.1400068646912273617      3.1415926535897995552
24   3.2245152435345525443      3.1415926535897968907
25                              3.1415926535897962246
26                              3.1415926535897962246
27                              3.1415926535897962246
28                              3.1415926535897962246
              The true value is 3.14159265358979323846264338327...

While the two forms of the recurrence formula are clearly mathematically equivalent,^{[nb 14]} the first subtracts 1 from a number extremely close to 1, leading to an increasingly problematic loss of significant digits. As the recurrence is applied repeatedly, the accuracy improves at first, but then it deteriorates. It never gets better than about 8 digits, even though 53-bit arithmetic should be capable of about 16 digits of precision. When the second form of the recurrence is used, the value converges to 15 digits of precision.

"Fast math" optimization

The aforementioned lack of associativity of floating-point operations in general means that compilers cannot as effectively reorder arithmetic expressions as they could with integer and fixed-point arithmetic, presenting a roadblock in optimizations such as common subexpression elimination and auto-vectorization.^[66] The "fast math" option on many compilers (ICC, GCC, Clang, MSVC...) turns on reassociation along with unsafe assumptions such as a lack of NaN and infinite numbers in IEEE 754. Some compilers also offer more granular options to only turn on reassociation. In either case, the programmer is exposed to many of the precision pitfalls mentioned above for the portion of the program using "fast" math.^[67]

In some compilers (GCC and Clang), turning on "fast" math may cause the program to disable subnormal floats at startup, affecting the floating-point behavior of not only the generated code, but also any program using such code as a library.^[68]

In most Fortran compilers, as allowed by the ISO/IEC 1539-1:2004 Fortran standard, reassociation is the default, with breakage largely prevented by the "protect parens" setting (also on by default). This setting stops the compiler from reassociating beyond the boundaries of parentheses.^[69] Intel Fortran Compiler is a notable outlier.^[70]

A common problem in "fast" math is that subexpressions may not be optimized identically from place to place, leading to unexpected differences. One interpretation of the issue is that "fast" math as implemented currently has a poorly defined semantics. One attempt at formalizing "fast" math optimizations is seen in Icing, a verified compiler.^[71]