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<link href="/image.php?url=https://csdnimg.cn/release/download_crawler_static/css/base.min.css" rel="stylesheet"/><link href="/image.php?url=https://csdnimg.cn/release/download_crawler_static/css/fancy.min.css" rel="stylesheet"/><link href="/image.php?url=https://csdnimg.cn/release/download_crawler_static/89630510/raw.css" rel="stylesheet"/><div id="sidebar" style="display: none"><div id="outline"></div></div><div class="pf w0 h0" data-page-no="1" id="pf1"><div class="pc pc1 w0 h0"><img alt="" class="bi x0 y0 w1 h1" src="/image.php?url=https://csdnimg.cn/release/download_crawler_static/89630510/bg1.jpg"/><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls0 ws0">Implementation on FPGA for CORDIC-based Computation of </div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 ls0 ws0">Arcsine and Arccosine </div><div class="t m0 x3 h3 y3 ff2 fs1 fc0 sc0 ls0 ws0">Xiaoning Liu</div><div class="t m0 x4 h4 y4 ff2 fs2 fc0 sc0 ls0 ws0">1</div><div class="t m0 x5 h3 y3 ff2 fs1 fc0 sc0 ls0 ws0">, Yizhuang Xie</div><div class="t m0 x6 h4 y4 ff2 fs2 fc0 sc0 ls0 ws0">1</div><div class="t m0 x7 h3 y3 ff2 fs1 fc0 sc0 ls0 ws0">, He Chen</div><div class="t m0 x8 h4 y4 ff2 fs2 fc0 sc0 ls0 ws0">1</div><div class="t m0 x9 h3 y3 ff2 fs1 fc0 sc0 ls0 ws0">, Bingyi Li</div><div class="t m0 xa h4 y4 ff2 fs2 fc0 sc0 ls0 ws0">1</div><div class="t m0 xb h3 y3 ff2 fs1 fc0 sc0 ls0 ws0"> </div><div class="t m0 xc h4 y5 ff3 fs2 fc0 sc0 ls0 ws0">1</div><div class="t m0 xd h3 y6 ff3 fs1 fc0 sc0 ls0 ws0">Beijing Key Laboratory of Embedde<span class="_ _0"></span>d Real-time Information Processing Technology, B<span class="_ _1"></span>eijing Institute of </div><div class="t m0 xe h3 y7 ff3 fs1 fc0 sc0 ls0 ws0">Technology, Beijing 100081, China </div><div class="t m0 xf h3 y8 ff3 fs1 fc0 sc0 ls0 ws0">liuxiaoning@bit.edu.cn, xyz551_bit@bit.edu.cn<span class="ff4 fs3"> </span></div><div class="t m0 x10 h5 y9 ff5 fs3 fc0 sc0 ls0 ws0">Keywords:<span class="ff6"> <span class="_"> </span>CORDIC, <span class="_"> </span>Double <span class="_"> </span>Iteratio<span class="_ _1"></span>n <span class="_"> </span>A<span class="_ _0"></span>lgorithm, <span class="_"> </span>arcsine, </span></div><div class="t m0 x10 h6 ya ff6 fs3 fc0 sc0 ls0 ws0">arccosine<span class="ff4"> </span></div><div class="t m0 x10 h7 yb ff1 fs1 fc0 sc0 ls0 ws0">Abstract<span class="ff4 fs3"> </span></div><div class="t m0 x10 h6 yc ff4 fs3 fc0 sc0 ls0 ws0">This <span class="_ _2"> </span>paper <span class="_ _2"> </span>concentrates <span class="_ _2"> </span>on <span class="_ _2"> </span>the <span class="_ _2"> </span>hardware <span class="_ _2"> </span>implementation <span class="_ _2"> </span>of </div><div class="t m0 x10 h6 yd ff4 fs3 fc0 sc0 ls0 ws0">CORDIC <span class="_ _3"> </span>algorithm <span class="_ _3"> </span>for <span class="_ _3"> </span>computing <span class="_ _3"> </span>inverse <span class="_ _3"> </span>trigonometric </div><div class="t m0 x10 h6 ye ff4 fs3 fc0 sc0 ls0 ws0">functions <span class="_ _1"></span>like <span class="_ _1"></span>a<span class="_ _1"></span>rcsine <span class="_ _1"></span>a<span class="_ _1"></span>nd <span class="_ _1"></span>arc<span class="_ _1"></span>cosine. <span class="_ _1"></span><span class="ls1">We</span> <span class="_ _1"></span>i<span class="_ _1"></span>mprove <span class="_ _1"></span>the <span class="_ _1"></span>e<span class="_ _1"></span>xisting </div><div class="t m0 x10 h6 yf ff4 fs3 fc0 sc0 ls0 ws0">algorithm <span class="_ _4"> </span><span class="ls2">by<span class="_ _0"></span><span class="ls0"> <span class="_ _4"> </span>changing <span class="_ _4"> </span>the <span class="_ _4"> </span>initial <span class="_ _4"> </span>rotating <span class="_ _4"> </span>vector <span class="_ _4"> </span>of <span class="_ _4"> </span>the </span></span></div><div class="t m0 x10 h6 y10 ff4 fs3 fc0 sc0 ls0 ws0">iterations <span class="_"> </span>a<span class="_ _1"></span>nd <span class="_ _5"> </span>modifying <span class="_"> </span>t<span class="_ _1"></span>he <span class="_"> </span>j<span class="_ _1"></span>udging <span class="_"> </span>conditio<span class="_ _1"></span>n <span class="_"> </span>of <span class="_ _5"> </span>rotati<span class="_ _1"></span>on </div><div class="t m0 x10 h6 y11 ff4 fs3 fc0 sc0 ls0 ws0">direction. <span class="_ _2"> </span>Due <span class="_ _6"></span>to <span class="_ _6"></span>t<span class="_ _1"></span>he <span class="_ _6"></span>impro<span class="_ _1"></span>vement, <span class="_ _2"> </span>two <span class="_ _6"></span>iteration<span class="_ _1"></span>s <span class="_ _6"></span>are <span class="_ _2"> </span>save<span class="ls3">d </span></div><div class="t m0 x10 h6 y12 ff4 fs3 fc0 sc0 ls0 ws0">and the dra<span class="_ _1"></span>w<span class="_ _0"></span>back of t<span class="_ _1"></span>he previous a<span class="_ _1"></span>lgorithm is correc<span class="_ _1"></span>ted. </div><div class="t m0 x10 h6 y13 ff4 fs3 fc0 sc0 ls0 ws0">In <span class="_ _7"> </span>contrasting <span class="_ _7"> </span>with <span class="_ _7"> </span>t<span class="_ _1"></span>he <span class="_ _7"> </span>previous <span class="_ _7"> </span>imple<span class="_ _1"></span>mentation, <span class="_ _7"> </span>the </div><div class="t m0 x10 h6 y14 ff4 fs3 fc0 sc0 ls0 ws0">improved <span class="_ _1"></span>algorithm <span class="_ _1"></span>consumes <span class="_ _1"></span>less <span class="_ _1"></span>hard<span class="_ _1"></span>w<span class="_ _0"></span>are <span class="_ _1"></span>resources <span class="_ _1"></span>and <span class="_ _8"></span><span class="ls4">its</span> </div><div class="t m0 x10 h6 y15 ff4 fs3 fc0 sc0 ls0 ws0">computing results are<span class="_ _1"></span> more accurac<span class="_ _1"></span>y. </div><div class="t m0 x10 h8 y16 ff1 fs1 fc0 sc0 ls0 ws0">1<span class="ff7"> </span>Introduction </div><div class="t m0 x10 h6 y17 ff4 fs3 fc0 sc0 ls0 ws0">The <span class="_ _8"></span>CORDIC <span class="_ _1"></span>algorit<span class="_ _1"></span>hm <span class="_ _8"></span>was <span class="_ _8"></span>originally <span class="_ _1"></span>develop<span class="_ _1"></span>ed <span class="_ _8"></span>in <span class="_ _1"></span>19<span class="_ _1"></span>59 <span class="_ _1"></span>b<span class="_ _1"></span>y </div><div class="t m0 x10 h6 y18 ff4 fs3 fc0 sc0 ls0 ws0">Jack <span class="_ _5"> </span>Volder<span class="_ _1"></span> <span class="_ _5"> </span>as <span class="_ _5"> </span>a <span class="_ _5"> </span>d<span class="_ _1"></span>igital <span class="_ _5"> </span>solution<span class="_ _1"></span> <span class="_ _5"> </span>for <span class="_ _5"> </span>real<span class="_ _1"></span>-time <span class="_ _5"> </span>navigatio<span class="_ _1"></span>n </div><div class="t m0 x10 h6 y19 ff4 fs3 fc0 sc0 ls0 ws0">problems</div><div class="t m0 x11 h9 y1a ff4 fs4 fc0 sc0 ls5 ws0">[1]</div><div class="t m0 x12 h6 y19 ff4 fs3 fc0 sc0 ls0 ws0">and <span class="_ _4"> </span>then <span class="_ _4"> </span>was <span class="_ _9"> </span>extended <span class="_ _9"> </span>to<span class="_ _1"></span> <span class="_ _9"> </span>a <span class="_ _9"> </span>bro<span class="_ _1"></span>ader <span class="_ _9"> </span>class <span class="_ _9"> </span>of </div><div class="t m0 x10 h6 y1b ff4 fs3 fc0 sc0 ls0 ws0">functions</div><div class="t m0 x11 h9 y1c ff4 fs4 fc0 sc0 ls5 ws0">[2]</div><div class="t m0 x12 h6 y1b ff4 fs3 fc0 sc0 ls0 ws0">. <span class="_ _9"> </span>The <span class="_ _a"> </span>algorithm <span class="_ _9"> </span>has <span class="_ _a"> </span>been <span class="_ _a"> </span>used <span class="_ _9"> </span>for <span class="_ _a"> </span>calculating </div><div class="t m0 x10 h6 y1d ff4 fs3 fc0 sc0 ls0 ws0">various <span class="_ _2"> </span>tri<span class="_ _1"></span>gonometric <span class="_ _b"> </span>functions <span class="_ _b"> </span>such <span class="_ _2"> </span>as <span class="_ _b"> </span>sine, <span class="_ _b"> </span>cosine, <span class="_ _b"> </span>arctan </div><div class="t m0 x10 h6 y1e ff4 fs3 fc0 sc0 ls0 ws0">and <span class="_ _5"> </span>it <span class="_"> </span>also <span class="_ _5"> </span>can <span class="_ _5"> </span>be <span class="_ _5"> </span>used <span class="_ _5"> </span>to <span class="_"> </span>co<span class="_ _1"></span>mpute <span class="_"> </span>hyperbo<span class="_ _1"></span>lic <span class="_ _5"> </span>functions, </div><div class="t m0 x10 h6 y1f ff4 fs3 fc0 sc0 ls0 ws0">exponentials <span class="_ _c"> </span>and <span class="_ _c"> </span>square <span class="_ _c"> </span>ro<span class="_ _1"></span>ots<span class="ls6">. <span class="_ _c"> </span></span>CORDIC <span class="_ _c"> </span>has <span class="_ _c"> </span>re<span class="_ _1"></span>mained </div><div class="t m0 x10 h6 y20 ff4 fs3 fc0 sc0 ls0 ws0">popular <span class="_ _8"></span>for <span class="_ _1"></span>its <span class="_ _8"></span>ability <span class="_ _1"></span>to<span class="_ _1"></span> <span class="_ _1"></span>calcu<span class="_ _1"></span>late <span class="_ _1"></span>the <span class="_ _8"></span>results <span class="_ _8"></span>usi<span class="_ _1"></span>ng <span class="_ _1"></span>the <span class="_ _8"></span>simple<span class="_ _1"></span> </div><div class="t m0 x10 h6 y21 ff4 fs3 fc0 sc0 ls0 ws0">operations <span class="_ _8"></span>of <span class="_ _8"></span>ad<span class="_ _1"></span>d, <span class="_ _8"></span>subtract <span class="_ _8"></span>and <span class="_ _d"></span>shift <span class="_ _8"></span>while <span class="_ _8"></span>using <span class="_ _8"></span>small <span class="_ _8"></span>Loo<span class="_ _1"></span>k-</div><div class="t m0 x10 h6 y22 ff4 fs3 fc0 sc0 ls0 ws0">Up Tab<span class="_ _1"></span>les (LUTs) instead o<span class="_ _1"></span>f hardware multipliers<span class="_ _1"></span> on FP<span class="_ _1"></span>GA</div><div class="t m0 x13 h9 y23 ff4 fs4 fc0 sc0 ls5 ws0">[3<span class="ls0">-</span></div><div class="t m0 x10 h9 y24 ff4 fs4 fc0 sc0 ls0 ws0">6].</div><div class="t m0 xd h6 y25 ff4 fs3 fc0 sc0 ls0 ws0"> <span class="_ _1"></span>CORDIC <span class="_ _1"></span>is <span class="_ _8"></span>w<span class="_ _0"></span>idely <span class="_ _1"></span>used <span class="_ _1"></span>in <span class="_ _8"></span>the <span class="_ _1"></span>implementation <span class="_ _1"></span>of <span class="_ _1"></span>re<span class="_ _1"></span><span class="ls7">al</span>-time </div><div class="t m0 x10 h6 y26 ff4 fs3 fc0 sc0 ls0 ws0">SAR <span class="_ _d"></span>imaging <span class="_ _6"></span>processing <span class="_ _d"></span>sinc<span class="_ _1"></span>e <span class="_ _d"></span>there <span class="_ _d"></span>ar<span class="_ _1"></span>e <span class="_ _6"></span>m<span class="_ _0"></span>any <span class="_ _d"></span>tri<span class="_ _1"></span>gonometric </div><div class="t m0 x10 h6 y27 ff4 fs3 fc0 sc0 ls0 ws0">functions <span class="_ _b"> </span>i<span class="_ _1"></span>n <span class="_ _b"> </span>the <span class="_"> </span>SAR <span class="_ _b"> </span>imaging <span class="_"> </span>algorithm. <span class="_ _b"> </span>B<span class="_ _1"></span><span class="ls3">ut</span> <span class="_ _b"> </span>t<span class="_ _1"></span>here <span class="_ _b"> </span>ar<span class="_ _1"></span>e <span class="_ _b"> </span><span class="ls3">no<span class="_ _1"></span></span> </div><div class="t m0 x10 h6 y28 ff4 fs3 fc0 sc0 ls0 ws0">available <span class="_ _3"> </span>IP <span class="_ _3"> </span>cores <span class="_ _e"> </span>for <span class="_ _e"> </span>computing <span class="_ _3"> </span>arcsine <span class="_ _3"> </span>and <span class="_ _e"> </span>arccosine<span class="_ _1"></span> </div><div class="t m0 x10 h6 y29 ff4 fs3 fc0 sc0 ls0 ws0">functions <span class="_ _d"></span>i<span class="_ _1"></span>n <span class="_ _d"></span>t<span class="_ _1"></span>he <span class="_ _6"></span>IDE <span class="_ _6"></span>(Integrated <span class="_ _6"></span>Develop<span class="_ _1"></span>ment <span class="_ _d"></span>E<span class="_ _1"></span>nvironment<span class="ls8">) </span></div><div class="t m0 x10 h6 y2a ff4 fs3 fc0 sc0 ls0 ws0">of <span class="_"> </span>FPGA, <span class="_"> </span>so <span class="_"> </span>users <span class="_"> </span>need <span class="_"> </span>to <span class="_"> </span>realize <span class="_"> </span>the <span class="_"> </span>functions <span class="_"> </span>by <span class="_"> </span>using </div><div class="t m0 x10 h6 y2b ff4 fs3 fc0 sc0 ls0 ws0">CORDIC algorithm.<span class="_ _1"></span> </div><div class="t m0 x10 h6 y2c ff4 fs3 fc0 sc0 ls0 ws0">In <span class="_"> </span>the <span class="_"> </span>computation <span class="_"> </span>o<span class="_ _1"></span>f <span class="_"> </span>inverse <span class="_"> </span>trigono<span class="_ _1"></span>metric <span class="_"> </span>functions <span class="_"> </span>li<span class="_ _1"></span>ke </div><div class="t m0 x10 h6 y2d ff4 fs3 fc0 sc0 ls0 ws0">arcsine <span class="_ _c"> </span>and <span class="_ _f"> </span>arccosine, <span class="_ _f"> </span>the<span class="_ _1"></span> <span class="_ _f"> </span>original <span class="_ _f"> </span>CORDIC <span class="_ _c"> </span>algorith<span class="_ _1"></span>m </div><div class="t m0 x10 h6 y2e ff4 fs3 fc0 sc0 ls0 ws0">displays <span class="_"> </span>its <span class="_"> </span>drawback. <span class="_"> </span>The <span class="_"> </span>accurac<span class="_ _1"></span>y <span class="_"> </span>of<span class="_ _0"></span> <span class="_"> </span>the <span class="_"> </span>result <span class="_"> </span>will <span class="_"> </span>get </div><div class="t m0 x10 h6 y2f ff4 fs3 fc0 sc0 ls0 ws0">worse <span class="_ _6"></span>due <span class="_ _6"></span>to<span class="_ _1"></span> <span class="_ _6"></span>the <span class="_ _6"></span>gain <span class="_ _6"></span>o<span class="_ _1"></span>f <span class="_ _6"></span>the <span class="_ _6"></span>ro<span class="_ _1"></span>tator</div><div class="t m0 x14 h9 y30 ff4 fs4 fc0 sc0 ls5 ws0">[7]</div><div class="t m0 x15 h6 y2f ff4 fs3 fc0 sc0 ls0 ws0">. <span class="_ _6"></span><span class="ls9">Mazen</span>c, <span class="_ _6"></span>C <span class="_ _6"></span>correct<span class="ls7">ed<span class="_ _1"></span></span> </div><div class="t m0 x10 h6 y31 ff4 fs3 fc0 sc0 ls0 ws0">the <span class="_ _8"></span>gain <span class="_ _1"></span>prob<span class="_ _1"></span>lem <span class="_ _1"></span>b<span class="_ _1"></span>y <span class="_ _1"></span>using<span class="_ _1"></span> <span class="_ _8"></span><span class="ff8">“</span>double <span class="_ _8"></span>iteration <span class="_ _1"></span>a<span class="_ _1"></span>lgorithm<span class="ff8">”</span> <span class="_ _8"></span>at <span class="_ _8"></span>the </div><div class="t m0 x10 h6 y32 ff4 fs3 fc0 sc0 ls0 ws0">cost <span class="_ _5"> </span>o<span class="_ _1"></span>f <span class="_ _5"> </span>an <span class="_ _a"> </span>in<span class="_ _0"></span>crease <span class="_ _a"> </span>in<span class="_ _0"></span> <span class="_ _5"> </span>co<span class="_ _1"></span>mplexity</div><div class="t m0 x16 h9 y33 ff4 fs4 fc0 sc0 ls5 ws0">[8]</div><div class="t m0 x17 h6 y32 ff4 fs3 fc0 sc0 ls0 ws0">. <span class="_ _5"> </span>T<span class="_ _1"></span>. <span class="_ _10"> </span>Lang <span class="_ _5"> </span>prop<span class="_ _1"></span>osed <span class="_ _5"> </span>a </div><div class="t m0 x10 h6 y34 ff4 fs3 fc0 sc0 ls0 ws0">method <span class="_ _b"> </span>which <span class="_ _2"> </span>req<span class="_ _1"></span>uires <span class="_ _b"> </span>a <span class="_ _b"> </span>standard <span class="_ _b"> </span>CORDI<span class="_ _1"></span>C <span class="_ _2"> </span>mod<span class="_ _1"></span>ule <span class="_ _2"> </span>plus<span class="_ _1"></span> <span class="_ _2"> </span>a </div><div class="t m0 x10 h6 y35 ff4 fs3 fc0 sc0 ls0 ws0">module to <span class="_ _1"></span>co<span class="_ _1"></span>mpute the directi<span class="_ _1"></span>on of <span class="_ _1"></span>rotation, b<span class="_ _1"></span>ut this app<span class="_ _1"></span>roach </div><div class="t m0 x10 h6 y36 ff4 fs3 fc0 sc0 ls0 ws0">need a hard <span class="_ _1"></span>ware multiplier</div><div class="t m0 x18 h9 y37 ff4 fs4 fc0 sc0 ls5 ws0">[9]</div><div class="t m0 x19 h6 y36 ff4 fs3 fc0 sc0 ls6 ws0">. <span class="ls0"> </span></div><div class="t m0 x10 h6 y38 ff4 fs3 fc0 sc0 lsa ws0">Th<span class="ls0">e <span class="_ _9"> </span>present <span class="_ _9"> </span>work <span class="_ _9"> </span>conce<span class="_ _1"></span>ntrates <span class="_ _9"> </span>on <span class="_ _9"> </span>t<span class="_ _1"></span>he <span class="_ _9"> </span>improveme<span class="_ _1"></span>nt <span class="_ _9"> </span>and </span></div><div class="t m0 x10 h6 y39 ff4 fs3 fc0 sc0 ls0 ws0">optimization <span class="_ _9"> </span>o<span class="_ _1"></span>f <span class="_ _9"> </span>do<span class="_ _1"></span>uble <span class="_ _9"> </span>iter<span class="_ _1"></span>ation <span class="_ _9"> </span>algor<span class="_ _1"></span>ithm. <span class="_ _4"> </span>For <span class="_ _4"> </span>real-time </div><div class="t m0 x10 h6 y3a ff4 fs3 fc0 sc0 ls0 ws0">calculation, <span class="_ _2"> </span>pipeline <span class="_ _6"></span>struct<span class="_ _1"></span>ure <span class="_ _2"> </span>is <span class="_ _2"> </span>needed <span class="_ _6"></span>instead <span class="_ _2"> </span>of <span class="_ _2"></span>feedback </div><div class="t m0 x1a h6 y9 ff4 fs3 fc0 sc0 ls0 ws0">structure, <span class="_ _3"> </span>so <span class="_ _3"> </span>that <span class="_ _3"> </span>the <span class="_ _3"> </span>op<span class="_ _1"></span>eration <span class="_ _3"> </span>module <span class="_ _3"> </span>can <span class="_ _3"> </span>obtain <span class="_ _3"> </span>the </div><div class="t m0 x1a h6 ya ff4 fs3 fc0 sc0 ls0 ws0">maximum <span class="_ _2"> </span>throughput <span class="_ _2"> </span>rate.<span class="_ _1"></span> <span class="_ _2"> </span>The <span class="_ _2"> </span>pipeline <span class="_ _2"> </span>structure <span class="_ _2"> </span>co<span class="_ _1"></span>nsumes </div><div class="t m0 x1a h6 y3b ff4 fs3 fc0 sc0 ls0 ws0">much <span class="_ _d"></span>more <span class="_ _6"></span>hardware <span class="_ _d"></span>r<span class="_ _1"></span>esource<span class="_ _1"></span>s <span class="_ _d"></span>for <span class="_ _d"></span>it <span class="_ _6"></span>must <span class="_ _6"></span>increase <span class="_ _d"></span>iterati<span class="_ _1"></span>on </div><div class="t m0 x1a h6 y3c ff4 fs3 fc0 sc0 ls0 ws0">numbers <span class="_ _1"></span>i<span class="_ _1"></span>n <span class="_ _1"></span>ord<span class="_ _1"></span>er <span class="_ _1"></span>to<span class="_ _1"></span> <span class="_ _8"></span>improve <span class="_ _1"></span>the <span class="_ _8"></span>accuracy. <span class="_ _8"></span>This <span class="_ _8"></span>paper <span class="_ _8"></span>mainly </div><div class="t m0 x1a h6 y3d ff4 fs3 fc0 sc0 ls0 ws0">works <span class="_ _d"></span>o<span class="_ _1"></span>n <span class="_ _d"></span>t<span class="_ _1"></span>he <span class="_ _d"></span>red<span class="_ _1"></span>uction <span class="_ _d"></span>o<span class="_ _1"></span>f <span class="_ _d"></span>ha<span class="_ _1"></span>rdware <span class="_ _6"></span>reso<span class="_ _1"></span>urces <span class="_ _d"></span>b<span class="_ _1"></span>y <span class="_ _6"></span>saving <span class="_ _d"></span>t<span class="_ _1"></span>he </div><div class="t m0 x1a h6 y3e ff4 fs3 fc0 sc0 ls0 ws0">first <span class="_ _f"> </span>two <span class="_ _11"> </span>iterations <span class="_ _f"> </span>without <span class="_ _f"> </span>affecting <span class="_ _11"> </span>the <span class="_ _f"> </span>result <span class="_ _f"> </span>of <span class="_ _11"> </span>the </div><div class="t m0 x1a h6 y3f ff4 fs3 fc0 sc0 ls0 ws0">algorithm <span class="_ _1"></span>and <span class="_ _8"></span>the <span class="_ _1"></span>cor<span class="_ _1"></span>rection <span class="_ _1"></span>of <span class="_ _8"></span>the <span class="_ _1"></span>algorithm <span class="_ _1"></span>d<span class="_ _1"></span>rawback <span class="_ _8"></span>when </div><div class="t m0 x1a h6 y40 ff4 fs3 fc0 sc0 ls0 ws0">the input argument is app<span class="_ _1"></span>roach to 1.<span class="_ _1"></span> </div><div class="t m0 x1a h8 y41 ff1 fs1 fc0 sc0 ls0 ws0">2<span class="ff7"> </span>Computation of Arcsine and Arccosine </div><div class="t m0 x1a h6 y12 ff4 fs3 fc0 sc0 ls0 ws0">The <span class="_ _10"> </span>standard <span class="_ _10"> </span>CORDIC <span class="_ _10"> </span>algorithm <span class="_ _10"> </span>ca<span class="_ _1"></span>n <span class="_ _10"> </span>hard<span class="lsb">ly</span> <span class="_ _5"> </span><span class="lsc">be</span> <span class="_ _10"> </span>used <span class="_ _10"> </span>for </div><div class="t m0 x1a h6 y13 ff4 fs3 fc0 sc0 ls0 ws0">computing <span class="_ _8"></span>arcsine <span class="_ _8"></span>and <span class="_ _8"></span>arccosine <span class="_ _8"></span>functions <span class="_ _8"></span>since <span class="_ _8"></span>it <span class="_ _1"></span>i<span class="_ _1"></span>nvolves <span class="_ _8"></span>a </div><div class="t m0 x1a h6 y14 ff4 fs3 fc0 sc0 ls0 ws0">square <span class="_ _a"> </span>roo<span class="_ _1"></span>t <span class="_ _a"> </span>operatio<span class="_ _1"></span>n <span class="_ _a"> </span>which <span class="_ _9"> </span>cannot <span class="_ _10"> </span>b<span class="_ _1"></span>e <span class="_ _9"> </span>replaced <span class="_ _9"> </span>by <span class="_ _a"> </span>add<span class="_ _1"></span>, </div><div class="t m0 x1a h6 y15 ff4 fs3 fc0 sc0 ls0 ws0">subtract <span class="_ _9"> </span><span class="lsc">or</span> <span class="_ _a"> </span>shift <span class="_ _9"> </span>operations. <span class="_ _9"> </span>Mazenc. <span class="_ _9"> </span>C <span class="_ _a"> </span>pr<span class="_ _1"></span>oposed <span class="_ _9"> </span>double </div><div class="t m0 x1a h6 y42 ff4 fs3 fc0 sc0 ls0 ws0">iteration <span class="_ _5"> </span>algorithm <span class="_ _5"> </span>by <span class="_"> </span>i<span class="_ _1"></span>ntroducing <span class="_ _5"> </span>double <span class="_ _5"> </span>iteratio<span class="_ _1"></span>ns <span class="_ _5"> </span>to <span class="_ _5"> </span>the </div><div class="t m0 x1a h6 y43 ff4 fs3 fc0 sc0 ls0 ws0">standard <span class="_ _11"> </span>CORDIC <span class="_ _11"> </span>algorit<span class="_ _1"></span>hm</div><div class="t m0 x1b h9 y44 ff4 fs4 fc0 sc0 ls5 ws0">[8]</div><div class="t m0 x1c h6 y43 ff4 fs3 fc0 sc0 ls6 ws0">. <span class="_ _11"> </span><span class="ls0">The <span class="_ _11"> </span>main <span class="_ _11"> </span>advantage<span class="_ _1"></span> <span class="_ _11"> </span>of </span></div><div class="t m0 x1a h6 y45 ff4 fs3 fc0 sc0 ls0 ws0">double <span class="_ _d"></span>iterations <span class="_ _d"></span>is <span class="_ _d"></span>that <span class="_ _d"></span>in <span class="_ _d"></span>step <span class="_ _d"></span>n, <span class="_ _d"></span>the<span class="_ _1"></span> <span class="_ _d"></span>iteration <span class="_ _d"></span>t</div><div class="t m0 x1d h9 y46 ff4 fs4 fc0 sc0 ls0 ws0">i<span class="lsd">+1</span></div><div class="t m0 x1e h6 y45 ff4 fs3 fc0 sc0 lse ws0">=t</div><div class="t m0 x1f h9 y46 ff4 fs4 fc0 sc0 ls0 ws0">i</div><div class="t m0 x20 h6 y45 ff4 fs3 fc0 sc0 ls0 ws0">/cos(tan</div><div class="t m0 x21 h9 y47 ff4 fs4 fc0 sc0 ls0 ws0">-</div><div class="t m0 x1a h9 y48 ff4 fs4 fc0 sc0 ls0 ws0">1</div><div class="t m0 x22 h6 y49 ff4 fs3 fc0 sc0 ls0 ws0">2</div><div class="t m0 x6 h9 y48 ff4 fs4 fc0 sc0 ls0 ws0">-i</div><div class="t m0 x7 h6 y49 ff4 fs3 fc0 sc0 ls0 ws0">) <span class="_ _12"> </span>becomes <span class="_ _12"> </span>t</div><div class="t m0 x23 h9 y4a ff4 fs4 fc0 sc0 ls0 ws0">i<span class="lsd">+1</span></div><div class="t m0 x24 h6 y49 ff4 fs3 fc0 sc0 ls0 ws0">= <span class="_ _12"> </span>t</div><div class="t m0 x25 h9 y4a ff4 fs4 fc0 sc0 ls0 ws0">i</div><div class="t m0 x26 h6 y49 ff4 fs3 fc0 sc0 ls0 ws0">/cos</div><div class="t m0 x27 h9 y48 ff4 fs4 fc0 sc0 ls0 ws0">2</div><div class="t m0 xa h6 y49 ff4 fs3 fc0 sc0 ls0 ws0">(tan</div><div class="t m0 x28 h9 y48 ff4 fs4 fc0 sc0 ls0 ws0">-1</div><div class="t m0 x29 h6 y49 ff4 fs3 fc0 sc0 ls0 ws0">2</div><div class="t m0 x2a h9 y48 ff4 fs4 fc0 sc0 ls0 ws0">-i</div><div class="t m0 x2b h6 y49 ff4 fs3 fc0 sc0 ls0 ws0">) <span class="_ _12"> </span>=t</div><div class="t m0 x2c h9 y4a ff4 fs4 fc0 sc0 ls0 ws0">n</div><div class="t m0 x2d h6 y49 ff4 fs3 fc0 sc0 ls0 ws0">(1+2</div><div class="t m0 x2e h9 y48 ff4 fs4 fc0 sc0 ls0 ws0">-2i</div><div class="t m0 x2f h6 y49 ff4 fs3 fc0 sc0 ls0 ws0">); <span class="_ _12"> </span>now, <span class="_ _12"> </span><span class="lsf">a </span></div><div class="t m0 x1a h6 y4b ff4 fs3 fc0 sc0 ls0 ws0">multiplication b<span class="_ _1"></span>y this term reduces to an add and<span class="_ _1"></span> a sh<span class="_ _1"></span>ift. </div><div class="t m0 x1a ha y4c ff1 fs5 fc0 sc0 ls0 ws0">2.1<span class="ff7"> </span>Double Iterati<span class="_ _0"></span>on Algorith<span class="_ _0"></span>m </div><div class="t m0 x1a h6 y4d ff4 fs3 fc0 sc0 ls0 ws0">The <span class="_ _d"></span>double <span class="_ _d"></span>iteration <span class="_ _8"></span>algorithm<span class="_ _1"></span> <span class="_ _8"></span>i<span class="_ _1"></span>n <span class="_ _8"></span>reference[8<span class="_ _1"></span>] <span class="_ _d"></span>for <span class="_ _8"></span>co<span class="_ _1"></span>mputing </div><div class="t m0 x1a h6 y4e ff4 fs3 fc0 sc0 ls0 ws0">arcsine function is:<span class="_ _1"></span> </div><div class="c x30 y4f w2 hb"><div class="t m1 x31 hc y50 ff4 fs6 fc0 sc0 ls0 ws0">0<span class="_ _13"> </span>0<span class="_ _14"> </span>0<span class="_ _15"> </span>0</div><div class="t m1 x32 hc y51 ff4 fs6 fc0 sc0 ls0 ws0">2</div><div class="t m1 x33 hc y52 ff4 fs6 fc0 sc0 ls0 ws0">1</div><div class="t m1 x34 hc y53 ff4 fs6 fc0 sc0 ls0 ws0">1</div><div class="t m1 x35 hc y54 ff4 fs6 fc0 sc0 ls0 ws0">1</div><div class="t m1 x36 hc y55 ff4 fs6 fc0 sc0 ls0 ws0">1</div><div class="t m1 x37 hc y56 ff4 fs6 fc0 sc0 ls0 ws0">2</div><div class="t m1 x38 hc y57 ff4 fs6 fc0 sc0 ls0 ws0">1</div><div class="t m1 x39 hd y58 ff4 fs7 fc0 sc0 ls0 ws0">1<span class="_ _16"> </span>0<span class="_ _17"> </span>0,</div><div class="t m1 x39 hd y59 ff4 fs7 fc0 sc0 ls0 ws0">sign<span class="_ _8"></span>(<span class="_ _18"> </span>)<span class="_ _19"> </span>sign<span class="_ _8"></span>(<span class="_ _18"> </span>)</div><div class="t m1 x3a hd y5a ff4 fs7 fc0 sc0 ls0 ws0">1</div><div class="t m1 x3b hd y5b ff4 fs7 fc0 sc0 ls0 ws0">2</div><div class="t m1 x3c hd y5c ff4 fs7 fc0 sc0 ls0 ws0">2</div><div class="t m1 x3d hd y5d ff4 fs7 fc0 sc0 ls0 ws0">1</div><div class="t m1 x3e hd y5e ff4 fs7 fc0 sc0 ls0 ws0">2<span class="_ _1a"> </span>tan<span class="_ _1b"> </span>2</div><div class="t m1 x3f hd y5f ff4 fs7 fc0 sc0 ls0 ws0">2</div><div class="t m1 x40 he y60 ff3 fs6 fc0 sc0 ls0 ws0">i<span class="_ _1c"> </span>i<span class="_ _15"> </span>i<span class="_ _1d"> </span>i<span class="_ _1e"> </span>i</div><div class="t m1 x41 he y61 ff3 fs6 fc0 sc0 ls0 ws0">i</div><div class="t m1 x36 he y52 ff3 fs6 fc0 sc0 ls10 ws0">ii</div><div class="t m1 x42 he y62 ff3 fs6 fc0 sc0 ls0 ws0">i</div><div class="t m1 x43 he y63 ff3 fs6 fc0 sc0 ls0 ws0">i</div><div class="t m1 x44 he y64 ff3 fs6 fc0 sc0 ls0 ws0">i</div><div class="t m1 x36 he y53 ff3 fs6 fc0 sc0 ls10 ws0">ii</div><div class="t m1 x45 he y54 ff3 fs6 fc0 sc0 ls0 ws0">i</div><div class="t m1 x31 he y55 ff3 fs6 fc0 sc0 ls0 ws0">i<span class="_ _1f"> </span>i<span class="_ _20"> </span>i</div><div class="t m1 x46 he y56 ff3 fs6 fc0 sc0 ls0 ws0">i</div><div class="t m1 x47 he y57 ff3 fs6 fc0 sc0 ls0 ws0">i<span class="_ _21"> </span>i<span class="_ _22"> </span>i</div><div class="t m1 x48 hf y58 ff3 fs7 fc0 sc0 ls0 ws0">x<span class="_ _23"> </span>y<span class="_ _13"> </span>z<span class="_ _24"> </span>t<span class="_ _25"> </span>t</div><div class="t m1 x48 hf y59 ff3 fs7 fc0 sc0 ls0 ws0">d<span class="_ _26"> </span>x<span class="_ _27"> </span>if<span class="_ _7"> </span>y<span class="_ _28"> </span>t<span class="_ _29"> </span>else<span class="_ _2a"> </span>x</div><div class="t m1 x31 hf y65 ff3 fs7 fc0 sc0 ls11 ws0">xx</div><div class="t m1 x49 hf y5b ff3 fs7 fc0 sc0 ls0 ws0">d</div><div class="t m1 x4a hf y5c ff3 fs7 fc0 sc0 ls0 ws0">d</div><div class="t m1 x40 hf y66 ff3 fs7 fc0 sc0 ls11 ws0">yy</div><div class="t m1 x48 hf y5e ff3 fs7 fc0 sc0 ls0 ws0">z<span class="_ _2b"> </span>z<span class="_ _2c"> </span>d</div><div class="t m1 x4b hf y5f ff3 fs7 fc0 sc0 ls0 ws0">t<span class="_ _2d"> </span>t<span class="_ _2e"> </span>t</div><div class="t m1 x4c h10 y61 ff9 fs6 fc0 sc0 ls0 ws0"></div><div class="t m1 x4d h10 y52 ff9 fs6 fc0 sc0 ls0 ws0"></div><div class="t m1 x4e h10 y63 ff9 fs6 fc0 sc0 ls0 ws0"></div><div class="t m1 x4f h10 y53 ff9 fs6 fc0 sc0 ls0 ws0"></div><div class="t m1 x50 h10 y54 ff9 fs6 fc0 sc0 ls12 ws0"></div><div class="t m1 x51 h10 y55 ff9 fs6 fc0 sc0 ls0 ws0"></div><div class="t m1 x3c h10 y56 ff9 fs6 fc0 sc0 ls0 ws0"></div><div class="t m1 x31 h10 y57 ff9 fs6 fc0 sc0 ls0 ws0"></div><div class="t m1 x4d h11 y58 ff9 fs7 fc0 sc0 ls0 ws0"><span class="_ _20"> </span><span class="_ _23"> </span><span class="_ _2f"> </span></div><div class="t m1 x52 h11 y67 ff9 fs7 fc0 sc0 ls0 ws0"></div><div class="t m1 x52 h11 y68 ff9 fs7 fc0 sc0 ls0 ws0"></div><div class="t m1 x53 h11 y59 ff9 fs7 fc0 sc0 ls0 ws0"><span class="_ _30"> </span><span class="_ _31"> </span></div><div class="t m1 x52 h11 y69 ff9 fs7 fc0 sc0 ls0 ws0"></div><div class="t m1 x52 h11 y6a ff9 fs7 fc0 sc0 ls0 ws0"></div><div class="t m1 x54 h11 y6b ff9 fs7 fc0 sc0 ls13 ws0"></div><div class="t m1 x4b h11 y6c ff9 fs7 fc0 sc0 ls0 ws0"><span class="_ _32"> </span><span class="_ _33"> </span><span class="_ _34"> </span></div><div class="t m1 x55 h11 y5b ff9 fs7 fc0 sc0 ls14 ws0"></div><div class="t m1 x52 h11 y6d ff9 fs7 fc0 sc0 ls0 ws0"></div><div class="t m1 x56 h11 y6e ff9 fs7 fc0 sc0 ls0 ws0"></div><div class="t m1 x52 h11 y6f ff9 fs7 fc0 sc0 ls0 ws0"></div><div class="t m1 x54 h11 y70 ff9 fs7 fc0 sc0 ls13 ws0"></div><div class="t m1 x4b h11 y71 ff9 fs7 fc0 sc0 ls0 ws0"><span class="_ _32"> </span><span class="_ _33"> </span><span class="_ _34"> </span></div><div class="t m1 x3a h11 y5c ff9 fs7 fc0 sc0 ls0 ws0"></div><div class="t m1 x4b h11 y72 ff9 fs7 fc0 sc0 ls0 ws0"><span class="_ _32"> </span><span class="_ _33"> </span><span class="_ _34"> </span></div><div class="t m1 x54 h11 y73 ff9 fs7 fc0 sc0 ls13 ws0"></div><div class="t m1 x52 h11 y74 ff9 fs7 fc0 sc0 ls0 ws0"></div><div class="t m1 x52 h11 y75 ff9 fs7 fc0 sc0 ls0 ws0"></div><div class="t m1 x34 h11 y5e ff9 fs7 fc0 sc0 ls0 ws0"><span class="_ _35"> </span><span class="_ _27"> </span><span class="_ _36"> </span></div><div class="t m1 x52 h11 y76 ff9 fs7 fc0 sc0 ls0 ws0"></div><div class="t m1 x52 h11 y77 ff9 fs7 fc0 sc0 ls0 ws0"></div><div class="t m1 x57 h11 y5f ff9 fs7 fc0 sc0 ls0 ws0"><span class="_ _37"> </span><span class="_ _38"> </span></div><div class="t m1 x52 h11 y78 ff9 fs7 fc0 sc0 ls0 ws0"></div><div class="t m1 x58 h12 y58 ffa fs7 fc0 sc0 ls0 ws0">，<span class="_ _15"> </span>，</div></div><div class="t m0 x59 h6 y79 ff4 fs3 fc0 sc0 ls0 ws0"> <span class="ls6"> </span>(1)</div><div class="t m0 x5a h6 y7a ff4 fs3 fc0 sc0 ls0 ws0"> </div><div class="t m0 x1a h6 y7b ff4 fs3 fc0 sc0 ls0 ws0">The <span class="_"> </span>abo<span class="_ _1"></span>ve <span class="_"> </span>formula <span class="_"> </span>gives </div><div class="c x5b y7c w3 h13"><div class="t m2 x3c h14 y7d ff4 fs8 fc0 sc0 ls0 ws0">1</div><div class="t m2 x5c h15 y7e ff4 fs9 fc0 sc0 ls0 ws0">1</div><div class="t m2 x36 h16 y7f ff4 fsa fc0 sc0 ls0 ws0">sin</div><div class="t m2 x3f h17 y7d ff3 fs8 fc0 sc0 ls0 ws0">i</div><div class="t m2 x10 h18 y7f ff3 fsa fc0 sc0 ls15 ws0">tz</div><div class="t m3 x5d h19 y7f ff9 fsb fc0 sc0 ls0 ws0"></div><div class="t m2 x3a h1a y7d ff9 fs8 fc0 sc0 ls0 ws0"></div><div class="t m2 x58 h1b y7e ff9 fs9 fc0 sc0 ls0 ws0"></div><div class="t m2 x47 h1c y7f ff9 fsa fc0 sc0 ls16 ws0"></div></div><div class="t m0 x5e h6 y7b ff4 fs3 fc0 sc0 ls0 ws0">and </div><div class="c x5f y80 w4 h1d"><div class="t m4 x53 h1e y81 ff4 fsc fc0 sc0 ls0 ws0">1</div><div class="t m4 x52 h1f y82 ff4 fsd fc0 sc0 ls0 ws0">co<span class="_ _0"></span>s<span class="_ _18"> </span><span class="ff3 fse">t</span></div><div class="t m4 x38 h20 y81 ff9 fsc fc0 sc0 ls0 ws0"></div></div><div class="t m0 x60 h6 y7b ff4 fs3 fc0 sc0 ls0 ws0">can </div><div class="t m0 x1a h6 y83 ff4 fs3 fc0 sc0 ls0 ws0">be obtained <span class="_ _1"></span>by the relationship</div><div class="c xb y84 w5 h21"><div class="t m5 x53 h22 y85 ff4 fsf fc0 sc0 ls17 ws0">11</div><div class="t m5 x52 h23 y86 ff4 fs10 fc0 sc0 ls0 ws0">c<span class="_ _0"></span>os<span class="_ _39"> </span>si<span class="_ _0"></span>n<span class="_ _3a"></span><span class="fs11">=<span class="_ _3b"> </span>/<span class="_ _b"> </span>2<span class="_ _3c"></span><span class="ff3 ls18">tt</span></span></div><div class="t m6 x61 h24 y86 ff9 fs12 fc0 sc0 ls0 ws0"></div><div class="t m5 x38 h25 y85 ff9 fsf fc0 sc0 ls19 ws0"></div><div class="t m5 x62 h26 y86 ff9 fs11 fc0 sc0 ls0 ws0"></div></div><div class="t m0 x63 h6 y83 ff4 fs3 fc0 sc0 ls0 ws0">. </div><div class="t m0 x1a h6 y87 ff4 fs3 fc0 sc0 ls0 ws0">For <span class="_ _c"> </span>double <span class="_ _c"> </span>iterations, <span class="_ _c"> </span>the <span class="_ _c"> </span>convergence <span class="_ _c"> </span>do<span class="_ _1"></span>main <span class="_ _f"> </span>of <span class="_ _c"> </span>the<span class="_ _1"></span> </div><div class="t m0 x1a h6 y88 ff4 fs3 fc0 sc0 ls0 ws0">algorithm beco<span class="_ _1"></span>mes: </div><div class="c x1a y89 w6 h27"><div class="t m7 x37 h28 y8a ff4 fs13 fc0 sc0 ls1a ws0">11</div><div class="t m7 x64 h28 y8b ff4 fs13 fc0 sc0 ls1a ws0">00</div><div class="t m7 x65 h29 y8c ff4 fs14 fc0 sc0 ls1b ws0">22<span class="_ _3d"></span><span class="fs15 ls0">ta<span class="_ _0"></span>n<span class="_ _3e"> </span>2<span class="_ _7"> </span>,<span class="_ _3f"> </span>ta<span class="_ _0"></span>n<span class="_ _3e"> </span>2<span class="_ _2b"> </span><span class="fs16">[<span class="_ _40"> </span>3.<span class="_ _0"></span>4<span class="_ _0"></span>86<span class="_ _41"></span>,<span class="_ _d"></span>3.<span class="_ _0"></span>4<span class="_ _0"></span>8<span class="_ _0"></span>6]</span></span></div><div class="t m7 x66 h2a y8a ff3 fs13 fc0 sc0 ls1c ws0">ii</div><div class="t m7 xc h2a y8b ff3 fs13 fc0 sc0 ls1c ws0">ii</div><div class="t m8 x5d h2b y8c ff9 fs17 fc0 sc0 ls0 ws0"></div><div class="t m7 xc h2c y8d ff9 fs13 fc0 sc0 ls1d ws0"></div><div class="t m7 x67 h2c y8a ff9 fs13 fc0 sc0 ls0 ws0"><span class="_ _34"> </span><span class="_ _42"> </span><span class="_ _34"> </span></div><div class="t m7 x54 h2c y8b ff9 fs13 fc0 sc0 ls1e ws0"></div><div class="t m7 x48 h2d y8c ff9 fs14 fc0 sc0 ls1f ws0"></div><div class="t m7 x38 h2e y8e ff9 fs16 fc0 sc0 ls20 ws0"></div><div class="t m7 x68 h2e y8c ff9 fs16 fc0 sc0 ls21 ws0"></div><div class="t m7 x38 h2e y8f ff9 fs16 fc0 sc0 ls20 ws0"></div><div class="t m7 x69 h2f y90 ff9 fs18 fc0 sc0 ls22 ws0"></div></div><div class="t m0 x6a h6 y91 ff4 fs3 fc0 sc0 ls0 ws0">. </div><div class="t m0 x1a h6 y92 ff4 fs3 fc0 sc0 ls0 ws0">Therefore <span class="_ _f"> </span>we <span class="_ _11"> </span>can <span class="_ _11"> </span>co<span class="_ _1"></span>mpute </div><div class="c x6b y93 w7 h1d"><div class="t m9 x38 h30 y81 ff4 fs19 fc0 sc0 ls0 ws0">1</div><div class="t m9 x52 h31 y82 ff4 fs1a fc0 sc0 ls0 ws0">sin<span class="_ _3e"> </span><span class="ff3 fsd">t</span></div><div class="t m9 x31 h32 y81 ff9 fs19 fc0 sc0 ls0 ws0"></div></div><div class="t m0 x6c h6 y92 ff4 fs3 fc0 sc0 ls0 ws0"> and </div><div class="c x6d y93 w8 h1d"><div class="t ma x36 h30 y81 ff4 fs19 fc0 sc0 ls0 ws0">1</div><div class="t ma x52 h33 y94 ff4 fs1b fc0 sc0 ls0 ws0">co<span class="_ _0"></span>s<span class="_ _43"> </span><span class="ff3 fs1c">t</span></div><div class="t ma x51 h32 y81 ff9 fs19 fc0 sc0 ls0 ws0"></div></div><div class="t m0 x20 h6 y92 ff4 fs3 fc0 sc0 ls0 ws0">for <span class="_ _11"> </span>an<span class="_ _1"></span>y </div><div class="c x1a y95 w9 h21"><div class="t mb x40 h34 y96 ff9 fs1d fc0 sc0 ls0 ws0"><span class="_ _44"> </span></div><div class="t mc x33 h35 y97 ff4 fs1e fc0 sc0 ls0 ws0">1<span class="_ _45"></span>,<span class="_ _1"></span>1<span class="_ _46"></span><span class="ff3">t<span class="_ _b"> </span><span class="ff9 ls23"></span></span></div></div><div class="t m0 x6e h6 y98 ff4 fs3 fc0 sc0 ls0 ws0">, since </div><div class="c x6f y95 w7 h21"><div class="t md x5d h34 y96 ff9 fs1d fc0 sc0 ls0 ws0"><span class="_ _47"> </span></div><div class="t me x47 h35 y97 ff4 fs1e fc0 sc0 ls0 ws0">1<span class="_ _45"></span>,<span class="_ _1"></span>1<span class="_ _48"></span><span class="ff9"></span></div></div><div class="t m0 x25 h6 y98 ff4 fs3 fc0 sc0 ls0 ws0"> is the domain of defi<span class="_ _1"></span>nition. </div><a class="l"><div class="d mf"></div></a><a class="l"><div class="d mf"></div></a><a class="l"><div class="d mf"></div></a><a class="l"><div class="d mf"></div></a><a class="l"><div class="d mf"></div></a><a class="l"><div class="d mf"></div></a><a class="l"><div class="d mf"></div></a></div><div class="pi" data-data='{"ctm":[1.611562,0.000000,0.000000,1.611562,0.000000,0.000000]}'></div></div>

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