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1: 3.1 Arithmetics and Error Measures

… ►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. … ►

3.1.2 $${(-1)}^s{2}^E\sum _{j=0}^{p-1}{b}_j{2}^{-j},$$

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Symbols:

$b_p$: bit, $s$: sign, $p$: number of significant bits and $E$: exponent

Permalink:

http://dlmf.nist.gov/3.1.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §3.1(i), §3.1 and Ch.3

… ►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). … ►

3.1.4 $$x=(1.b_1{b}_2\mathrm{\dots }{b}_{p-1}{b}_p{b}_{p+1}\mathrm{\dots })\cdot 2^E,$$

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Symbols:

$b_p$: bit, $p$: number of significant bits and $E$: exponent

Permalink:

http://dlmf.nist.gov/3.1.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §3.1(i), §3.1(i), §3.1 and Ch.3

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2: 19.38 Approximations

… ►The accuracy is controlled by the number of terms retained in the approximation; for real variables the number of significant figures appears to be roughly twice the number of terms retained, perhaps even for $\varphi$ near $\pi /2$ with the improvements made in the 1970 reference. …

3: 6.16 Mathematical Applications

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§6.16(ii) Number-Theoretic Significance of $\mathrm{li}\left(x\right)$

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4: Richard A. Askey

… ►Another significant contribution was the Askey-Gasper inequality for Jacobi polynomials which was published in Positive Jacobi polynomial sums. II (with G. …

5: About Color Map

… ►In doing this, however, we would like to place the mathematically significant phase values, specifically the multiples of $\pi /2$ correponding to the real and imaginary axes, at more immediately recognizable colors. …

6: 19.36 Methods of Computation

… ►If (19.25.9) is used when $0\le k^2\le 1$, cancellations may lead to loss of significant figures when $k^2$ is close to 1 and $\varphi >\pi /4$, as shown by Reinsch and Raab (2000). … ► $E(\varphi ,k)$ can be evaluated by using (19.25.7), and $R_D$ by using (19.21.10), but cancellations may become significant. … ►Near these points there will be loss of significant figures in the computation of $R_J$ or $R_D$. … ►This method loses significant figures in $\rho$ if ${\alpha }^2$ and $k^2$ are nearly equal unless they are given exact values—as they can be for tables. …

7: Mathematical Introduction

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$(a,b]$ or $[a,b)$	half-closed intervals.
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S	significant figures.
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… ►05, and the corresponding function values are tabulated to 8 decimal places or 8 significant figures. …

8: DLMF Project News

error generating summary

9: 2.11 Remainder Terms; Stokes Phenomenon

… ►If the results agree within $S$ significant figures, then it is likely—but not certain—that the truncated asymptotic series will yield at least $S$ correct significant figures for larger values of $x$. … ►In effect, (2.11.7) “corrects” (2.11.6) by introducing a term that is relatively exponentially small in the neighborhood of $\mathrm{ph}z=\pi$, is increasingly significant as $\mathrm{ph}z$ passes from $\pi$ to $\frac{3}{2}\pi$, and becomes the dominant contribution after $\mathrm{ph}z$ passes $\frac{3}{2}\pi$. …

10: Software Index

… ►The following are web-based software repositories with significant holdings in the area of special functions. …