425 lines
8.1 KiB
HTML
425 lines
8.1 KiB
HTML
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<!DOCTYPE HTML>
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>sqrt, isqrt | Common Lisp Nova Spec</TITLE
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>Common Lisp Nova Spec</A
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>→ <A HREF="f_sqrt.html"
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>sqrt, isqrt</A
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><HR
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><TABLE WIDTH="100%" CELLSPACING="0" CELLPADDING="0"
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><TR
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><TD ALIGN="LEFT" VALIGN="BASELINE" WIDTH="100%" CLASS="name"
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><SPAN CLASS="idx" DATA-KIND="idxref" DATA-TERM="sqrt, isqrt"
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></SPAN
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><B
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>sqrt, isqrt</B
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></TD
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><TD ALIGN="RIGHT" VALIGN="BASELINE" WIDTH="0" NOWRAP="NOWRAP" CLASS="ftype"
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><I
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>Function</I
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></TD
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></TR
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></TABLE
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><HR
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></DIV
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><UL CLASS="subtoc"
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></UL
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><DL
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><DT
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><B
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>Syntax</B
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></DT
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><DD
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><P CLASS="j"
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><B
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>sqrt</B
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> <SPAN CLASS="cmssi"
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>number</SPAN
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> <SPAN CLASS="arrow"
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>→</SPAN
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> <SPAN CLASS="cmssi"
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>root</SPAN
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></P
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><P CLASS="j"
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><B
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>isqrt</B
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> <SPAN CLASS="cmssi"
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>natural</SPAN
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> <SPAN CLASS="arrow"
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>→</SPAN
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> <SPAN CLASS="cmssi"
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>natural-root</SPAN
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></P
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></DD
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><DT
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><B
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>Arguments and Values</B
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></DT
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><DD
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><P CLASS="j"
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><VAR CLASS="param"
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>number</VAR
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>, <VAR CLASS="param"
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>root</VAR
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> — a <A HREF="26_1_Glossary.html#number"
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><EM CLASS="term"
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>number</EM
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></A
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>. </P
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><P CLASS="j"
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><VAR CLASS="param"
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>natural</VAR
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>, <VAR CLASS="param"
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>natural-root</VAR
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> — a non-negative <A HREF="26_1_Glossary.html#integer"
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><EM CLASS="term"
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>integer</EM
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></A
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>.</P
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></DD
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><DT
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><B
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>Description</B
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></DT
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><DD
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><P CLASS="j"
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><A HREF="f_sqrt.html" CLASS="funref"
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><B
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>sqrt</B
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></A
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> and <A HREF="f_sqrt.html" CLASS="funref"
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><B
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>isqrt</B
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></A
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> compute square roots. </P
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><P CLASS="j"
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><A HREF="f_sqrt.html" CLASS="funref"
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><B
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>sqrt</B
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></A
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> returns the <A HREF="26_1_Glossary.html#principal"
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><EM CLASS="term"
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>principal</EM
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></A
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> square root of <VAR CLASS="param"
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>number</VAR
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>. If the <VAR CLASS="param"
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>number</VAR
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> is not a <A HREF="26_1_Glossary.html#complex"
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><EM CLASS="term"
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>complex</EM
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></A
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> but is negative, then the result is a <A HREF="26_1_Glossary.html#complex"
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><EM CLASS="term"
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>complex</EM
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></A
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>. </P
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><P CLASS="j"
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><A HREF="f_sqrt.html" CLASS="funref"
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><B
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>isqrt</B
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></A
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> returns the greatest <A HREF="26_1_Glossary.html#integer"
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><EM CLASS="term"
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>integer</EM
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></A
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> less than or equal to the exact positive square root of <VAR CLASS="param"
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>natural</VAR
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>. </P
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><P CLASS="j"
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>If <VAR CLASS="param"
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>number</VAR
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> is a positive <A HREF="26_1_Glossary.html#rational"
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><EM CLASS="term"
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>rational</EM
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></A
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>, it is <A HREF="26_1_Glossary.html#implementation-dependent"
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><EM CLASS="term"
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>implementation-dependent</EM
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></A
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> whether <VAR CLASS="param"
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>root</VAR
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> is a <A HREF="26_1_Glossary.html#rational"
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><EM CLASS="term"
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>rational</EM
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></A
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> or a <A HREF="26_1_Glossary.html#float"
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><EM CLASS="term"
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>float</EM
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></A
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>. If <VAR CLASS="param"
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>number</VAR
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> is a negative <A HREF="26_1_Glossary.html#rational"
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><EM CLASS="term"
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>rational</EM
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></A
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>, it is <A HREF="26_1_Glossary.html#implementation-dependent"
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><EM CLASS="term"
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>implementation-dependent</EM
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></A
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> whether <VAR CLASS="param"
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>root</VAR
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> is a <A HREF="26_1_Glossary.html#complex_rational"
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><EM CLASS="term"
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>complex rational</EM
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></A
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> or a <A HREF="26_1_Glossary.html#complex_float"
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><EM CLASS="term"
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>complex float</EM
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></A
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>. </P
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><P CLASS="j"
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>The mathematical definition of complex square root (whether or not minus zero is supported) follows: </P
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><P CLASS="j"
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><CODE CLASS="f"
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>(sqrt <I CLASS="i"
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><I
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>x</I
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></I
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>) = (exp (/ (log <I CLASS="i"
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><I
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>x</I
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></I
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>) 2))</CODE
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> </P
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><P CLASS="j"
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>The branch cut for square root lies along the negative real axis, continuous with quadrant II. The range consists of the right half-plane, including the non-negative imaginary axis and excluding the negative imaginary axis.</P
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></DD
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><DT
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><B
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>Examples</B
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></DT
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><DD
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><PRE CLASS="screen"
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> (sqrt 9.0) <SPAN CLASS="cmsy"
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><SPAN CLASS="arrow"
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>→</SPAN
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></SPAN
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> 3.0
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(sqrt -9.0) <SPAN CLASS="cmsy"
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><SPAN CLASS="arrow"
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>→</SPAN
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></SPAN
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> #C(0.0 3.0)
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(isqrt 9) <SPAN CLASS="cmsy"
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><SPAN CLASS="arrow"
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>→</SPAN
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></SPAN
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> 3
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(sqrt 12) <SPAN CLASS="cmsy"
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><SPAN CLASS="arrow"
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>→</SPAN
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></SPAN
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> 3.4641016
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(isqrt 12) <SPAN CLASS="cmsy"
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><SPAN CLASS="arrow"
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>→</SPAN
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></SPAN
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> 3
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(isqrt 300) <SPAN CLASS="cmsy"
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><SPAN CLASS="arrow"
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>→</SPAN
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></SPAN
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> 17
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(isqrt 325) <SPAN CLASS="cmsy"
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><SPAN CLASS="arrow"
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>→</SPAN
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></SPAN
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> 18
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(sqrt 25)
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<SPAN CLASS="cmsy"
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><SPAN CLASS="arrow"
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>→</SPAN
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></SPAN
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> 5
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<SPAN CLASS="sp"
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><SPAN CLASS="arrow withabove"
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><SPAN CLASS="above"
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>or </SPAN
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>→</SPAN
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></SPAN
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> 5.0
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(isqrt 25) <SPAN CLASS="cmsy"
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><SPAN CLASS="arrow"
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>→</SPAN
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></SPAN
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> 5
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(sqrt -1) <SPAN CLASS="cmsy"
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><SPAN CLASS="arrow"
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>→</SPAN
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></SPAN
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> #C(0.0 1.0)
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(sqrt #c(0 2)) <SPAN CLASS="cmsy"
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><SPAN CLASS="arrow"
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>→</SPAN
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></SPAN
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> #C(1.0 1.0)</PRE
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></DD
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><DT
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><B
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>Exceptional Situations</B
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></DT
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><DD
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><P CLASS="j"
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>The <A HREF="26_1_Glossary.html#function"
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><EM CLASS="term"
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>function</EM
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></A
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> <A HREF="f_sqrt.html" CLASS="funref"
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><B
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>sqrt</B
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></A
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> should signal <A HREF="t_type-error.html" CLASS="typeref"
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><B
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>type-error</B
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></A
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> if its argument is not a <A HREF="26_1_Glossary.html#number"
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><EM CLASS="term"
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>number</EM
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></A
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>. </P
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><P CLASS="j"
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>The <A HREF="26_1_Glossary.html#function"
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><EM CLASS="term"
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>function</EM
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></A
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> <A HREF="f_sqrt.html" CLASS="funref"
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><B
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>isqrt</B
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></A
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> should signal <A HREF="t_type-error.html" CLASS="typeref"
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><B
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>type-error</B
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></A
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> if its argument is not a non-negative <A HREF="26_1_Glossary.html#integer"
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><EM CLASS="term"
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>integer</EM
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></A
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>. </P
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><P CLASS="j"
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>The functions <A HREF="f_sqrt.html" CLASS="funref"
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><B
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>sqrt</B
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></A
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> and <A HREF="f_sqrt.html" CLASS="funref"
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><B
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>isqrt</B
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></A
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> might signal <A HREF="t_arithmetic-error.html" CLASS="typeref"
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><B
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>arithmetic-error</B
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></A
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>.</P
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></DD
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><DT
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><B
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>See Also</B
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></DT
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><DD
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><P CLASS="j"
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><A HREF="f_exp.html" CLASS="funref"
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><B
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>exp</B
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></A
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>, <A HREF="f_log.html" CLASS="funref"
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><B
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>log</B
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></A
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>, <A HREF="12_1_Number_Concepts.html#sec_12_1_3_3" CLASS="secref"
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><SPAN CLASS="cmr"
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>Section</SPAN
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> <SPAN CLASS="cmr"
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>12.1.3.3</SPAN
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> <SPAN CLASS="cmr"
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>(Rule</SPAN
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> <SPAN CLASS="cmr"
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|
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>of</SPAN
|
||
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> <SPAN CLASS="cmr"
|
||
|
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>Float</SPAN
|
||
|
|
> <SPAN CLASS="cmr"
|
||
|
|
>Substitutability)</SPAN
|
||
|
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></A
|
||
|
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></P
|
||
|
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></DD
|
||
|
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><DT
|
||
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><B
|
||
|
|
>Notes</B
|
||
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|
></DT
|
||
|
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><DD
|
||
|
|
><PRE CLASS="screen"
|
||
|
|
>(isqrt x) <SPAN CLASS="cmsy"
|
||
|
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><SPAN STYLE="font-size:16px;vertical-align:-2px"
|
||
|
|
>≡</SPAN
|
||
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></SPAN
|
||
|
|
> (values (floor (sqrt x)))</PRE
|
||
|
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><P CLASS="j"
|
||
|
|
>but it is potentially more efficient.</P
|
||
|
|
></DD
|
||
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></DL
|
||
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></DIV
|
||
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></DIV
|
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><DIV CLASS="footer"
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><DIV CLASS="btmnav"
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><A HREF="f_signum.html" CLASS="prev"
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>←</A
|
||
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><A HREF="t_random-state.html" CLASS="next"
|
||
|
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>→</A
|
||
|
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></DIV
|
||
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><DIV CLASS="trail"
|
||
|
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>Conversion to HTML copyright 2023 by Gilbert Baumann</DIV
|
||
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></DIV
|
||
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></DIV
|
||
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