§3.1 Arithmetics and Error Measures

Computer arithmetic is described for the binary based system with base 2; another system that has been used is the hexadecimal system with base 16.

A nonzero normalized binary floating-point machine number $x$ is represented as

where $s$ is equal to $1$ or $0$, each $b_j$, $j\ge 1$, is either $0$ or $1$, $b_1$ is the most significant bit, $p$ ($\in \mathbb{N}$) is the number of significant bits $b_j$, $b_{p-1}$ is the least significant bit, $E$ is an integer called the exponent, $b_0.b_1{b}_2\mathrm{\dots }{b}_{p-1}$ is the significand, and $f=\mathrm{.}{b}_1{b}_2\mathrm{\dots }{b}_{p-1}$ is the fractional part.

The set of machine numbers ${ℝ}_{\mathrm{fl}}$ is the union of $0$ and the set

with $b_0=1$ and all allowable choices of $E$, $p$, $s$, and $b_j$.

Let $E_{\min }\le E\le E_{\max }$ with and $E_{\max }>0$. For given values of $E_{\min }$, $E_{\max }$, and $p$, the format width in bits $N$ of a computer word is the total number of bits: the sign (one bit), the significant bits $b_1,b_2,\mathrm{\dots },b_{p-1}$ ($p-1$ bits), and the bits allocated to the exponent (the remaining $N-p$ bits). The integers $p$, $E_{\min }$, and $E_{\max }$ are characteristics of the machine. The machine epsilon ${ϵ}_M$, that is, the distance between $1$ and the next larger machine number with $E=0$ is given by ${ϵ}_M=2^{-p+1}$. The machine precision is $\frac{1}{2}{ϵ}_M=2^{-p}$. The lower and upper bounds for the absolute values of the nonzero machine numbers are given by

Underflow (overflow) after computing $x\ne 0$ occurs when $|x|$ is smaller (larger) than $N_{\min }$ ($N_{\max }$).

Interval arithmetic is intended for bounding the total effect of rounding errors of calculations with machine numbers. With this arithmetic the computed result can be proved to lie in a certain interval, which leads to validated computing with guaranteed and rigorous inclusion regions for the results.

Let $G$ be the set of closed intervals $\{[a,b]\}$. The elementary arithmetical operations on intervals are defined as follows:

where $\mathrm{\ast }\in \{\mathrm{+},\mathrm{-},\mathrm{\cdot },\mathrm{/}\}$, with appropriate roundings of the end points of $I\ast J$ when machine numbers are being used. Division is possible only if the divisor interval does not contain zero.

A basic text on interval arithmetic and analysis is Alefeld and Herzberger (1983), and for applications and further information see Moore (1979) and Petković and Petković (1998). The last reference includes analogs for arithmetic in the complex plane $ℂ$. For interval arithmetic, one should refer to the IEEE Standards for Interval Arithmetic IEEE (2015, 2018).

Computer algebra systems use exact rational arithmetic with rational numbers $p/q$, where $p$ and $q$ are multi-length integers. During the calculations common divisors are removed from the rational numbers, and the final results can be converted to decimal representations of arbitrary length. For further information see Matula and Kornerup (1980).

To eliminate overflow or underflow in finite-precision arithmetic numbers are represented by using generalized logarithms ${\ln }_{\mathrm{\ell }}(x)$ given by

with $x\ge 0$ and $\mathrm{\ell }$ the unique nonnegative integer such that $a\equiv {\ln }_{\mathrm{\ell }}(x)\in [0,1)$. In level-index arithmetic $x$ is represented by $\mathrm{\ell }+a$ (or $-(\mathrm{\ell }+a)$ for negative numbers). Also in this arithmetic generalized precision can be defined, which includes absolute error and relative precision (§3.1(v)) as special cases.

For further information see Clenshaw and Olver (1984) and Clenshaw et al. (1989). For applications see Lozier (1993).

For further references on level-index arithmetic (and also other arithmetics) see Anuta et al. (1996). See also Hayes (2009).

If $x^{\ast }$ is an approximation to a real or complex number $x$, then the absolute error is

If $x\ne 0$, the relative error is

The relative precision is

where $x{x}^{\ast }>0$ for real variables, and $x{x}^{\ast }\ne 0$ for complex variables (with the principal value of the logarithm).

The mollified error is

For error measures for complex arithmetic see Olver (1983).