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+<html lang="en"><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
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+ <title>Jones on Binary to Decimal Conversion</title>
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+
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+
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+ <meta name="Author" content="Douglas W. Jones">
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+ <meta name="Language" content="English">
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+ <meta name="editor" content="/usr/bin/vi">
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+
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+ <meta name="Description" content="A tutorial on binary to decimal conversion using limited precision">
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+ </style>
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+ <meta name="viewport" content="width=device-width, initial-scale=1"></head>
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+ <body bgcolor="#FFFFFF" text="#000000" link="#0000CC" vlink="#880088" alink="#880088">
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+ <div class="HEADBOX" data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="197.5">
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+
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+ <table border="0" cellspacing="0" cellpadding="0" data-pf_style_display="table" data-pf_style_visibility="visible" data-pf_rect_width="1716" data-pf_rect_height="159"><tbody data-pf_style_display="table-row-group" data-pf_style_visibility="visible" data-pf_rect_width="1716" data-pf_rect_height="159"><tr data-pf_style_display="table-row" data-pf_style_visibility="visible" data-pf_rect_width="1716" data-pf_rect_height="159"><td data-pf_style_display="table-cell" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="159"> </td><td data-pf_style_display="table-cell" data-pf_style_visibility="visible" data-pf_rect_width="1708" data-pf_rect_height="158.78125">
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+
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+ <h1 data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1708" data-pf_rect_height="35">Binary to Decimal Conversion in Limited Precision</h1>
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+
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+
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+ <p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1708" data-pf_rect_height="51">
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+
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+ Part of <a href="http://www.cs.uiowa.edu/~jones/bcd/" data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="217.328125" data-pf_rect_height="17">
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+
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+ the Arithmetic Tutorial Collection
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+
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+ </a>
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+ <br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17">
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+
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+
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+
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+ by
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+ <a href="http://www.cs.uiowa.edu/~jones/" data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="114.234375" data-pf_rect_height="17">Douglas W. Jones</a>
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+
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+ <br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17">
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+
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+
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+ <a href="http://www.uiowa.edu/" data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="174.21875" data-pf_rect_height="17">
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+ T<small data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="17.78125" data-pf_rect_height="15">HE</small> U<small data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="70.359375" data-pf_rect_height="15">NIVERSITY</small>
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+ <small data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="17.046875" data-pf_rect_height="15">OF</small> I<small data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="30.359375" data-pf_rect_height="15">OWA</small></a>
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+ <a href="http://www.cs.uiowa.edu/" data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="214.59375" data-pf_rect_height="17">Department of Computer Science</a>
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+
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+
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+
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+ </p><p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1708" data-pf_rect_height="28">
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+ <small data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="1674.484375" data-pf_rect_height="29">
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+ Copyright ©
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+ 1999, Douglas. W. Jones, with major revisions made in 2002 and 2010.
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+ This work may be transmitted or stored in electronic form on any
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+ computer attached to the Internet or World Wide Web so long as this
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+ notice is included in the copy. Individuals may make single copies
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+ for their own use. All other rights are reserved.
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+ </small>
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+
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+ </p></td></tr></tbody></table>
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+
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+ </div>
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+
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+
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+
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+
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+<h2 data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="26">Index</h2>
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+
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+<ul data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="136">
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+<li data-pf_style_display="list-item" data-pf_style_visibility="visible" data-pf_rect_width="1685.84375" data-pf_rect_height="17"> <a href="http://homepage.cs.uiowa.edu/~jones/bcd/decimal.html#intro" data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="79.09375" data-pf_rect_height="17">Introduction</a>
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+</li><li data-pf_style_display="list-item" data-pf_style_visibility="visible" data-pf_rect_width="1685.84375" data-pf_rect_height="17"> <a href="http://homepage.cs.uiowa.edu/~jones/bcd/decimal.html#basic" data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="95.953125" data-pf_rect_height="17">The Basic Idea</a>
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+</li><li data-pf_style_display="list-item" data-pf_style_visibility="visible" data-pf_rect_width="1685.84375" data-pf_rect_height="17"> <a href="http://homepage.cs.uiowa.edu/~jones/bcd/decimal.html#reduce" data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="119.984375" data-pf_rect_height="17">Reduction to Code</a>
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+</li><li data-pf_style_display="list-item" data-pf_style_visibility="visible" data-pf_rect_width="1685.84375" data-pf_rect_height="17"> <a href="http://homepage.cs.uiowa.edu/~jones/bcd/decimal.html#signed" data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="194.203125" data-pf_rect_height="17">Dealing with Signed Numbers</a>
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+</li><li data-pf_style_display="list-item" data-pf_style_visibility="visible" data-pf_rect_width="1685.84375" data-pf_rect_height="17"> <a href="http://homepage.cs.uiowa.edu/~jones/bcd/decimal.html#precision" data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="131.96875" data-pf_rect_height="17">Living Within 8 Bits</a>
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+</li><li data-pf_style_display="list-item" data-pf_style_visibility="visible" data-pf_rect_width="1685.84375" data-pf_rect_height="17"> <a href="http://homepage.cs.uiowa.edu/~jones/bcd/decimal.html#division" data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="297.234375" data-pf_rect_height="17">Doing Without Hardware Multiply and Divide</a>
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+</li><li data-pf_style_display="list-item" data-pf_style_visibility="visible" data-pf_rect_width="1685.84375" data-pf_rect_height="17"> <a href="http://homepage.cs.uiowa.edu/~jones/bcd/decimal.html#radices" data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="126.625" data-pf_rect_height="17">Alternative Radices</a>
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+</li><li data-pf_style_display="list-item" data-pf_style_visibility="visible" data-pf_rect_width="1685.84375" data-pf_rect_height="17"> <a href="http://homepage.cs.uiowa.edu/~jones/bcd/decimal.html#sixtyfour" data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="249.296875" data-pf_rect_height="17">64-bit Conversion on a 32-bit Machine</a>
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+</li></ul>
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+
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+<table border="1" width="100%" data-pf_style_display="table" data-pf_style_visibility="visible" data-pf_rect_width="1725" data-pf_rect_height="24">
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+ <tbody data-pf_style_display="table-row-group" data-pf_style_visibility="visible" data-pf_rect_width="1723" data-pf_rect_height="22"><tr data-pf_style_display="table-row" data-pf_style_visibility="visible" data-pf_rect_width="1723" data-pf_rect_height="18">
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+ <td align="CENTER" data-pf_style_display="table-cell" data-pf_style_visibility="visible" data-pf_rect_width="1719" data-pf_rect_height="18">
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+ <small data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="337.453125" data-pf_rect_height="15">
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+ Rated as a top 50 site by
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+ <a href="http://www.programmersheaven.com/" data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="118.265625" data-pf_rect_height="15">Programmer's Heaven</a>
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+ as of Feb 2002.
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+ </small>
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+ </td>
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+ </tr>
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+</tbody></table>
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+
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+<h2 data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="26"> <a name="intro" data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="130.265625" data-pf_rect_height="26"> Introduction </a> </h2>
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+
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+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="34">
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+Programmers on binary computers have always faced computational problems
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+when dealing with textual input-output because of our convention that printed
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+numeric data should be represented in base 10. These problems have never
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+been insurmountable, but they are particularly annoying when writing programs
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+for machines with limited capacity in their arithmetic units. Suppose, for
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+example, you have an unsigned binary number that you wish to print out in
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+decimal form. The following C code will do this:
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+</p><pre data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="84"> void putdec( unsigned int n ) {
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+ unsigned int rem = n % 10;
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+ unsigned int quot = n / 10;
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+ if (quot > 0) putdec( quot );
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+ putchar( rem + '0' );
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+ }
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+</pre>
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+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="34">
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+The problem with this C code is that it assumes that the computer's arithmetic
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+unit is sufficiently large to handle the entire number n, and it assumes that
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+the arithmetic unit can conveniently do integer division yielding both a
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+quotient and a remainder. These assumptions are not always reasonable, both
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+because many commercially successful computers do not include division
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+hardware and because, even when such hardware is available, it is of little
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+direct use when dividing numbers larger than the ALU can handle.
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+</p><p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="51">
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+The focus of this tutorial is on writing fast code to convert unsigned 16 bit
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+binary numbers to decimal on a machine with an 8-bit ALU and no divide
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+instruction. This situation is common on small microcontrollers, and the
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+techniques generalize in useful ways, for example, to the problem of doing
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+64 bit binary to decimal conversion using a 32 bit ALU. A final section
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+of this tutorial examines how these methods can be extended to larger
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+numbers, for example, converting a 64-bit binary number to decimal on
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+a machine with a 32-bit ALU.
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+</p><p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="17">
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+On machines with no divide instruction where speed is no issue, it may
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+be better to simply use repeated subtraction instead of the fast but
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+algebraically complex code given in the body of this tutorial.
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+Consider the following 16-bit conversion code as an example.
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+</p><pre data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="224"> void putdec( int n ) {
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+ char a, b, c, d;
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+ if (n < 0) {
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+ putchar( '-' );
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+ n = -n;
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+ }
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+ for ( a = '0' - 1; n >= 0; n -= 10000 ) ++a;
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+ for ( b = '9' + 1; n < 0; n += 1000 ) --b;
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+ for ( c = '0' - 1; n >= 0; n -= 100 ) ++c;
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+ for ( d = '9' + 1; n < 0; n += 10 ) --d;
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+ putchar( a );
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+ putchar( b );
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+ putchar( c );
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+ putchar( d );
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+ putchar( n + '0' );
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+ }
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+</pre>
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+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="34">
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+Variants on the clever algorithm given above have been very popular on
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+8-bit microprocessors and microcontrollers, despite the fact that conversions
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+can require over 30 iterations of the for loops in this code. As given here,
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+this code does not work for the extreme negative value -32768; fixing this
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+bug without introducing a special case is an interesting exercise.
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+</p><p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="0">
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+
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+</p><hr data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="2">
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+
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+<h2 data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="26"> <a name="basic" data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="152.015625" data-pf_rect_height="26"> The Basic Idea </a> </h2>
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+
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+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="34">
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+To do the conversion faster, what we'll do is look at our binary number
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+as a base 16 number and then do the conversion in terms of the base 16
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+digits.
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+Consider a 16 bit number <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i>. This can be broken into 4 fields of
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+4 bits each, the base 16 digits; call them <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub>,
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+<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">2</sub>, <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">1</sub> and <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">0</sub>.
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+</p><pre width="70" data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="70"> n
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+ |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
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+ |_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|
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+ | n | n | n | n |
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+ 3 2 1 0
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+</pre>
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+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="17">
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+The value of the number can be expressed in terms of these 4 fields as
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+follows:
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+</p><blockquote data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1645.84375" data-pf_rect_height="17">
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+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1645.84375" data-pf_rect_height="17">
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+<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i> = 4096 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub> + 256 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">2</sub> +
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+ 16 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">1</sub> + 1 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">0</sub>
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+</p></blockquote>
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+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="17">
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+In the same way, if the value of <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i>, expressed in decimal, is
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+<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">d</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">4</sub><i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">d</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub><i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">d</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">2</sub><i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">d</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">1</sub><i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">d</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">0</sub>, where each of
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+<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">d</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">4</sub>, <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">d</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub>, <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">d</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">2</sub>,
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+<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">d</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">1</sub> and <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">d</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">0</sub> are decimal digits, the
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+value of n can be expressed as:
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+</p><blockquote data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1645.84375" data-pf_rect_height="17">
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+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1645.84375" data-pf_rect_height="17">
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+<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i> = 10000 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">d</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">4</sub> + 1000 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">d</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub> +
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+ 100 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">d</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">2</sub> + 10 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">d</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">1</sub> +
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+ 1 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">d</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">0</sub>
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+</p></blockquote>
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+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="17">
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+Our problem then, is to compute the <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">d</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="3.71875" data-pf_rect_height="15">i</sub> from the
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+<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="3.71875" data-pf_rect_height="15">i</sub> without dealing directly with the larger
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+number <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i>.
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+</p><p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="17">
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+To do this, we first note that the factors used in combining the values of
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+<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="3.71875" data-pf_rect_height="15">i</sub> can themselves be expressed as sums of multiples of
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+powers of 10. Therefore, we can rewrite the original expression as:
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+</p><blockquote data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1645.84375" data-pf_rect_height="85">
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+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1645.84375" data-pf_rect_height="85">
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+ <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i> =
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+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17"> <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub>( 4×1000 + 0×100 + 9×10 + 6×1 ) +
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+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17"> <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">2</sub>( 2×100 + 5×10 + 6×1 ) +
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+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17"> <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">1</sub>( 1×10 + 6×1 ) +
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+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17"> <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">0</sub>( 1×1 )
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+</p></blockquote>
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+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="17">
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+If distribute the <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="3.71875" data-pf_rect_height="15">i</sub> over the expressions for each factor,
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+then factor out the multiples of 100, we get the following:
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+</p><blockquote data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1645.84375" data-pf_rect_height="85">
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+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1645.84375" data-pf_rect_height="85">
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+ <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i> =
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+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17"> 1000 (4<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub>) +
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+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17"> 100 (0<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub> + 2<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">2</sub>) +
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+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17"> 10 (9<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub> + 5<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">2</sub> +
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+ 1<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">1</sub>) +
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+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17"> 1 (6<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub> + 6<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">2</sub> +
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+ 6<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">1</sub> +
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+ 1<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">0</sub>)
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+</p></blockquote>
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+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="17">
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+We can use this to arrive at first approximations <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">a</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="3.71875" data-pf_rect_height="15">i</sub>
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+for <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">d</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub> through <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">d</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">0</sub>:
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+</p><blockquote data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1645.84375" data-pf_rect_height="68">
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+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1645.84375" data-pf_rect_height="68">
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+ <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">a</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub> = 4 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub>
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+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17"><i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">a</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">2</sub> = 0 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub> + 2 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">2</sub>
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+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17"><i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">a</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">1</sub> = 9 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub> + 5 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">2</sub> +
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+ 1 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">1</sub>
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+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17"><i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">a</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">0</sub> = 6 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub> + 6 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">2</sub> +
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+ 6 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">1</sub> + 1 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">0</sub>
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+</p></blockquote>
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+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="17">
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+The values of <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">a</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="3.71875" data-pf_rect_height="15">i</sub> are not proper decimal digits because
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+they are not properly bounded in the range
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+0 ≤ <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">a</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="3.71875" data-pf_rect_height="15">i</sub> ≤ 9.
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+Instead, given that each
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+of <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="3.71875" data-pf_rect_height="15">i</sub> is bounded in the range
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+0 ≤ <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="3.71875" data-pf_rect_height="15">i</sub> ≤ 15,
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+the <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">a</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="3.71875" data-pf_rect_height="15">i</sub>
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+are bounded as follows:
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+</p><blockquote data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1645.84375" data-pf_rect_height="68">
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|
+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1645.84375" data-pf_rect_height="68">
|
|
|
+ 0 ≤ <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">a</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub> ≤ 60
|
|
|
+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17">0 ≤ <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">a</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">2</sub> ≤ 30
|
|
|
+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17">0 ≤ <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">a</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">1</sub> ≤ 225
|
|
|
+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17">0 ≤ <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">a</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">0</sub> ≤ 285
|
|
|
+</p></blockquote>
|
|
|
+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="34">
|
|
|
+This is actually quite promising because <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">a</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub> through
|
|
|
+<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">a</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">1</sub> are less than 256 and may therefore be computed using
|
|
|
+an 8 bit ALU, and even <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">a</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">0</sub> may be computed in 8 bits if we
|
|
|
+can use the carry-out bit of the ALU to hold the high bit. Furthermore,
|
|
|
+if our interest is 2's complement arithmetic where the high bit is the
|
|
|
+sign bit, the first step in the output process is to print a minus sign
|
|
|
+and then negate, so the constraint on <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub> becomes
|
|
|
+0 ≤ <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub> ≤ 8
|
|
|
+and we can conclude naively that
|
|
|
+</p><blockquote data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1645.84375" data-pf_rect_height="17">
|
|
|
+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1645.84375" data-pf_rect_height="17">
|
|
|
+0 ≤ <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">a</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">0</sub> ≤ 243
|
|
|
+</p></blockquote>
|
|
|
+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="34">
|
|
|
+Note that we said
|
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|
+<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub> ≤ 8
|
|
|
+and not
|
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|
+<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub> < 8;
|
|
|
+this is because of the one negative number in the 2's complement system
|
|
|
+that cannot be properly negated, -32768. If we exclude this number from
|
|
|
+the range of legal values, the upper bound
|
|
|
+is reduced to 237; in fact, this is the overall upper bound, because
|
|
|
+in the one case where
|
|
|
+<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub> is 8, all of the other
|
|
|
+<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="3.71875" data-pf_rect_height="15">i</sub> are zero, so we can assert globally that
|
|
|
+</p><blockquote data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1645.84375" data-pf_rect_height="17">
|
|
|
+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1645.84375" data-pf_rect_height="17">
|
|
|
+0 ≤ <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">a</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">0</sub> ≤ 237
|
|
|
+</p></blockquote>
|
|
|
+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="17">
|
|
|
+Even with our naive bound, however, it is easy to see that we can
|
|
|
+safely use 8 bit arithmetic to accumulate all of the
|
|
|
+<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">a</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="3.71875" data-pf_rect_height="15">i</sub>.
|
|
|
+</p><p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="17">
|
|
|
+Given that we have computed the <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">a</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="3.71875" data-pf_rect_height="15">i</sub>, we can push them
|
|
|
+into the correct ranges by a series of 8-bit divisions and one 9-bit division,
|
|
|
+as follows:
|
|
|
+</p><blockquote data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1645.84375" data-pf_rect_height="221">
|
|
|
+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1645.84375" data-pf_rect_height="221">
|
|
|
+ <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="7.109375" data-pf_rect_height="17">c</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">1</sub> = <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">a</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">0</sub> / 10
|
|
|
+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17"><i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">d</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">0</sub> = <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">a</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">0</sub> mod 10
|
|
|
+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17">
|
|
|
+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17"><i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="7.109375" data-pf_rect_height="17">c</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">2</sub> = (<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">a</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">1</sub> + <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="7.109375" data-pf_rect_height="17">c</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">1</sub>) / 10
|
|
|
+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17"><i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">d</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">1</sub> = (<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">a</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">1</sub> + <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="7.109375" data-pf_rect_height="17">c</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">1</sub>) mod 10
|
|
|
+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17">
|
|
|
+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17"><i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="7.109375" data-pf_rect_height="17">c</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub> = (<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">a</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">2</sub> + <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="7.109375" data-pf_rect_height="17">c</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">2</sub>) / 10
|
|
|
+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17"><i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">d</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">2</sub> = (<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">a</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">2</sub> + <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="7.109375" data-pf_rect_height="17">c</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">2</sub>) mod 10
|
|
|
+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17">
|
|
|
+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17"><i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="7.109375" data-pf_rect_height="17">c</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">4</sub> = (<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">a</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub> + <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="7.109375" data-pf_rect_height="17">c</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub>) / 10
|
|
|
+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17"><i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">d</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub> = (<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">a</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub> + <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="7.109375" data-pf_rect_height="17">c</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub>) mod 10
|
|
|
+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17">
|
|
|
+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17"><i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">d</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">4</sub> = <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="7.109375" data-pf_rect_height="17">c</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">4</sub>
|
|
|
+</p></blockquote>
|
|
|
+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="17">
|
|
|
+In the above, the <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="7.109375" data-pf_rect_height="17">c</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="3.71875" data-pf_rect_height="15">i</sub> terms represent carries propagated
|
|
|
+from one digit position to the next. The upper bounds on each of the
|
|
|
+<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="7.109375" data-pf_rect_height="17">c</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="3.71875" data-pf_rect_height="15">i</sub> can be computed from the bounds given above for the
|
|
|
+<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">a</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="3.71875" data-pf_rect_height="15">i</sub>:
|
|
|
+</p><blockquote data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1645.84375" data-pf_rect_height="68">
|
|
|
+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1645.84375" data-pf_rect_height="68">
|
|
|
+ <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="7.109375" data-pf_rect_height="17">c</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">1</sub> ≤ 28 [ = 285 / 10 ]
|
|
|
+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17"><i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="7.109375" data-pf_rect_height="17">c</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">2</sub> ≤ 25 [ = (28 + 225) / 10
|
|
|
+ = 253 / 10 ]
|
|
|
+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17"><i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="7.109375" data-pf_rect_height="17">c</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub> ≤ 5 [ = (25 + 30) / 10 = 55 / 10 ]
|
|
|
+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17"><i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="7.109375" data-pf_rect_height="17">c</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">4</sub> ≤ 6 [ = (5 + 60) / 10 = 65 / 10 ]
|
|
|
+</p></blockquote>
|
|
|
+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="34">
|
|
|
+In computing the bound on <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="7.109375" data-pf_rect_height="17">c</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">2</sub>, we came within 2 of
|
|
|
+to 255, the maximum that can be represented in 8 bits, so an 8 bit
|
|
|
+ALU is just barely sufficient for this computation. Again, if we
|
|
|
+negate any negative numbers prior to output, so that the maximum
|
|
|
+value of <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub> is 8, the bound on <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="7.109375" data-pf_rect_height="17">c</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">1</sub>
|
|
|
+becomes 23 instead of 28.
|
|
|
+</p><p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="0">
|
|
|
+
|
|
|
+</p><hr data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="2">
|
|
|
+
|
|
|
+<h2 data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="26"> <a name="reduce" data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="190.671875" data-pf_rect_height="26">Reduction to Code</a> </h2>
|
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+
|
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|
+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="34">
|
|
|
+A first hack at reducing the above idea to code might include named temporaries
|
|
|
+for all of the terms used above, but we can avoid this if we reorganize
|
|
|
+things slightly and note that, after computing each digit of the number, we
|
|
|
+no-longer need one of the <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="3.71875" data-pf_rect_height="15">i</sub>. This leads to the
|
|
|
+following C code:
|
|
|
+</p><pre data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="462"> void putdec( uint16_t n ) {
|
|
|
+ uint8_t d4, d3, d2, d1, q;
|
|
|
+ uint16_t d0;
|
|
|
+
|
|
|
+ d0 = n & 0xF;
|
|
|
+ d1 = (n>>4) & 0xF;
|
|
|
+ d2 = (n>>8) & 0xF;
|
|
|
+ d3 = (n>>12) & 0xF;
|
|
|
+
|
|
|
+ d0 = 6*(d3 + d2 + d1) + d0;
|
|
|
+ q = d0 / 10;
|
|
|
+ d0 = d0 % 10;
|
|
|
+
|
|
|
+ d1 = q + 9*d3 + 5*d2 + d1;
|
|
|
+ q = d1 / 10;
|
|
|
+ d1 = d1 % 10;
|
|
|
+
|
|
|
+ d2 = q + 2*d2;
|
|
|
+ q = d2 / 10;
|
|
|
+ d2 = d2 % 10;
|
|
|
+
|
|
|
+ d3 = q + 4*d3;
|
|
|
+ q = d3 / 10;
|
|
|
+ d3 = d3 % 10;
|
|
|
+
|
|
|
+ d4 = q;
|
|
|
+
|
|
|
+ putchar( d4 + '0' );
|
|
|
+ putchar( d3 + '0' );
|
|
|
+ putchar( d2 + '0' );
|
|
|
+ putchar( d1 + '0' );
|
|
|
+ putchar( d0 + '0' );
|
|
|
+ }
|
|
|
+</pre>
|
|
|
+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="34">
|
|
|
+In the above,
|
|
|
+we use the standard C types <tt data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="50.046875" data-pf_rect_height="15">uint8_t</tt> and <tt data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="57.1875" data-pf_rect_height="15">uint16_t</tt>
|
|
|
+to represent unsigned 8 and 16-bit values. By convention, these should
|
|
|
+be defined in <tt data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="71.484375" data-pf_rect_height="15"><stdint.h></tt>.
|
|
|
+Of course, this code prints all 5 digits of <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i> instead of only
|
|
|
+printing the significant digits, but this can be fixed in many ways. The
|
|
|
+following fix is perhaps the cleanest, making effective use of partial
|
|
|
+results as soon as those results can be used to predict that digits will
|
|
|
+be zero. Even so, adding these features to the code obscures the
|
|
|
+computation being performed.
|
|
|
+</p><pre data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="574"> void putdec( uint16_t n ) {
|
|
|
+ uint8_t d4, d3, d2, d1, q;
|
|
|
+ uint16_t d0;
|
|
|
+
|
|
|
+ d0 = n & 0xF;
|
|
|
+ d1 = (n>>4) & 0xF;
|
|
|
+ d2 = (n>>8) & 0xF;
|
|
|
+ d3 = (n>>12);
|
|
|
+
|
|
|
+ d0 = 6*(d3 + d2 + d1) + d0;
|
|
|
+ q = d0 / 10;
|
|
|
+ d0 = d0 % 10;
|
|
|
+
|
|
|
+ d1 = q + 9*d3 + 5*d2 + d1;
|
|
|
+ if (d1 != 0) {
|
|
|
+ q = d1 / 10;
|
|
|
+ d1 = d1 % 10;
|
|
|
+
|
|
|
+ d2 = q + 2*d2;
|
|
|
+ if ((d2 != 0) || (d3 != 0)) {
|
|
|
+ q = d2 / 10;
|
|
|
+ d2 = d2 % 10;
|
|
|
+
|
|
|
+ d3 = q + 4*d3;
|
|
|
+ if (d3 != 0) {
|
|
|
+ q = d3 / 10;
|
|
|
+ d3 = d3 % 10;
|
|
|
+
|
|
|
+ d4 = q;
|
|
|
+
|
|
|
+ if (d4 != 0) {
|
|
|
+ putchar( d4 + '0' );
|
|
|
+ }
|
|
|
+ putchar( d3 + '0' );
|
|
|
+ }
|
|
|
+ putchar( d2 + '0' );
|
|
|
+ }
|
|
|
+ putchar( d1 + '0' );
|
|
|
+ }
|
|
|
+ putchar( d0 + '0' );
|
|
|
+ }
|
|
|
+</pre>
|
|
|
+
|
|
|
+<hr data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="2">
|
|
|
+
|
|
|
+<h2 data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="26"> <a name="signed" data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="26"></a>Dealing with Signed Numbers</h2>
|
|
|
+
|
|
|
+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="17">
|
|
|
+To print signed values, we can replace the first few lines of the above
|
|
|
+code with the following:
|
|
|
+</p><pre data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="196"> void putdec( int16_t n ) {
|
|
|
+ uint8_t d4, d3, d2, d1, d0, q;
|
|
|
+
|
|
|
+ if (n < 0) {
|
|
|
+ putchar( '-' );
|
|
|
+ n = -n;
|
|
|
+ }
|
|
|
+
|
|
|
+ d0 = n & 0xF;
|
|
|
+ d1 = (n>>4) & 0xF;
|
|
|
+ d2 = (n>>8) & 0xF;
|
|
|
+ d3 = (n>>12) & 0xF;
|
|
|
+
|
|
|
+ /* the remainder of the code is unchanged */
|
|
|
+</pre>
|
|
|
+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="34">
|
|
|
+In the above, we use the standard C type <tt data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="50.046875" data-pf_rect_height="15">int16_t</tt>
|
|
|
+to represent signed 16-bit values and <tt data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="50.046875" data-pf_rect_height="15">uint8_t</tt>
|
|
|
+to represent unsigned 8-bit values. By convention, these should
|
|
|
+be defined in <tt data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="71.484375" data-pf_rect_height="15"><stdint.h></tt>.
|
|
|
+As mentioned earlier, forcing <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i> positive prior to breaking it down
|
|
|
+into nibbles allows us to use an 8-bit type for <tt data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="14.296875" data-pf_rect_height="15">d0</tt> instead
|
|
|
+of the 16-bit type used previously (only 9 bits were actually needed).
|
|
|
+</p><p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="34">
|
|
|
+Note that the computation of <tt data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="14.296875" data-pf_rect_height="15">d3</tt> has also changed as a result of
|
|
|
+the change from unsigned to signed arithmetic. This is because the
|
|
|
+right-shift operator shifts zeros into the high bit when applied to
|
|
|
+unsigned numbers, but it replicates the sign bit when applied to signed
|
|
|
+numbers. There is one case in 16 bit 2's complement arithmetic where
|
|
|
+both <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i> and -<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i> are negative, that is, <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i> = -32768.
|
|
|
+The change we have made gives us the correct output in this case.
|
|
|
+</p><p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="0">
|
|
|
+
|
|
|
+</p><hr data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="2">
|
|
|
+
|
|
|
+<h2 data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="26"> <a name="precision" data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="207.859375" data-pf_rect_height="26">Living Within 8 Bits</a> </h2>
|
|
|
+
|
|
|
+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="34">
|
|
|
+In the code given above for unsigned printing, the fact that
|
|
|
+<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">a</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">0</sub> could be as large as 285 forced us to declare the
|
|
|
+variable <tt data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="14.296875" data-pf_rect_height="15">d0</tt> as a 16-bit integer instead of just 8 bits.
|
|
|
+We can avoid this by observing that the final addition in the sum
|
|
|
+used to compute <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">a</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">0</sub> is the only one that will push
|
|
|
+this sum over 255, and therefore, we need only check the carry bit once
|
|
|
+to detect overflow. Having detected overflow, we can account for it in
|
|
|
+the computation that follows:
|
|
|
+</p><pre data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="308"> void putdec( uint16_t n ) {
|
|
|
+ uint8_t d4, d3, d2, d1, d0, q;
|
|
|
+
|
|
|
+ d0 = n & 0xF;
|
|
|
+ d1 = (n>>4) & 0xF;
|
|
|
+ d2 = (n>>8) & 0xF;
|
|
|
+ d3 = (n>>12) & 0xF;
|
|
|
+
|
|
|
+ if ( (6*(d3 + d2 + d1) + d0) > 255 ) { /* overflow */
|
|
|
+ d0 = (6*(d3 + d2 + d1) + d0) & 0xFF;
|
|
|
+ q = d0/10 + 25;
|
|
|
+ d0 = d0 % 10 + 6;
|
|
|
+ if (d0 >= 10) {
|
|
|
+ d0 = d0 - 10;
|
|
|
+ q = q + 1;
|
|
|
+ }
|
|
|
+ } else { /* no overflow */
|
|
|
+ d0 = 6*(d3 + d2 + d1) + d0;
|
|
|
+ q = d0 / 10;
|
|
|
+ d0 = d0 % 10;
|
|
|
+ }
|
|
|
+
|
|
|
+</pre>
|
|
|
+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="34">
|
|
|
+In the above code, after having noted an overflow in the computation of
|
|
|
+<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">a</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">0</sub>, we retain the least significant bits of the sum
|
|
|
+in <tt data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="14.296875" data-pf_rect_height="15">d0</tt> and then approximately correct the quotient by
|
|
|
+adding 256/10, or 25; correcting this and computing the correct
|
|
|
+remainder is a bit more interesting.
|
|
|
+</p><p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="17">
|
|
|
+Given some integer <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="4.453125" data-pf_rect_height="17">i</i> ≥ 256, if we want to compute
|
|
|
+<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="4.453125" data-pf_rect_height="17">i</i> mod 10 from (<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="4.453125" data-pf_rect_height="17">i</i> - 256) mod 10, we need to note that
|
|
|
+the modulus operator is distributive in an interesting sense:
|
|
|
+</p><blockquote data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1645.84375" data-pf_rect_height="17">
|
|
|
+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1645.84375" data-pf_rect_height="17">
|
|
|
+(<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="4.453125" data-pf_rect_height="17">i</i> + <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="4.453125" data-pf_rect_height="17">j</i>) mod <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="11.5625" data-pf_rect_height="17">m</i> =
|
|
|
+((<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="4.453125" data-pf_rect_height="17">i</i> mod <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="11.5625" data-pf_rect_height="17">m</i>) + (<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="4.453125" data-pf_rect_height="17">j</i> mod <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="11.5625" data-pf_rect_height="17">m</i>)) mod <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="11.5625" data-pf_rect_height="17">m</i>
|
|
|
+</p></blockquote>
|
|
|
+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="17">
|
|
|
+Using this, we can conclude that:
|
|
|
+</p><blockquote data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1645.84375" data-pf_rect_height="34">
|
|
|
+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1645.84375" data-pf_rect_height="34">
|
|
|
+<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="4.453125" data-pf_rect_height="17">i</i> mod 10 = (((<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="4.453125" data-pf_rect_height="17">i</i> - 256) mod 10) + (256 mod 10)) mod 10
|
|
|
+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17"> = (((<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="4.453125" data-pf_rect_height="17">i</i> - 256) mod 10) + 6) mod 10
|
|
|
+</p></blockquote>
|
|
|
+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="34">
|
|
|
+In the case where ((<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="4.453125" data-pf_rect_height="17">i</i> - 256) mod 10) + 6 is greater than 10,
|
|
|
+the truncation of the quotient was also incorrect, so we must add 1
|
|
|
+to the approximate quotient as well. This justifies the code given above,
|
|
|
+and this code has also been tested exhaustively for all integers from
|
|
|
+0 to 65535.
|
|
|
+</p><p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="0">
|
|
|
+
|
|
|
+</p><hr data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="2">
|
|
|
+
|
|
|
+<h2 data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="26"> <a name="division" data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="475.40625" data-pf_rect_height="26">Doing Without Hardware Multiply and Divide</a> </h2>
|
|
|
+
|
|
|
+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="17">
|
|
|
+If flat-out speed is not of the essence, the easiest way to do the
|
|
|
+required divisions is to use the trivial repeated subtraction algorithm.
|
|
|
+The largest dividend we must consider is only 285, so it will take no
|
|
|
+more than 28 iterations for this division.
|
|
|
+</p><pre data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="154"> /* quotient and remainder returned by div10 */
|
|
|
+ unsigned char q, r;
|
|
|
+
|
|
|
+ void div10( uint8_t i ) {
|
|
|
+ q = 0;
|
|
|
+ r = i;
|
|
|
+ while (r > 9) {
|
|
|
+ q = q + 1;
|
|
|
+ r = r - 10;
|
|
|
+ }
|
|
|
+ }
|
|
|
+</pre>
|
|
|
+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="17">
|
|
|
+This can be optimized in several ways. Most notably, termination conditions
|
|
|
+that compare with zero are almost always significantly faster than those
|
|
|
+that compare with other constants, so we might be tempted to write something
|
|
|
+like this:
|
|
|
+</p><pre data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="168"> /* quotient and remainder returned by div10 */
|
|
|
+ uint8_t q, r;
|
|
|
+
|
|
|
+ void div10( uint8_t i ) {
|
|
|
+ q = -1;
|
|
|
+ r = i;
|
|
|
+ do {
|
|
|
+ q = q + 1;
|
|
|
+ r = r - 10;
|
|
|
+ } while (r >= 0); /* see note below */
|
|
|
+ r = r + 10;
|
|
|
+ }
|
|
|
+</pre>
|
|
|
+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="34">
|
|
|
+The above C code is wrong because <tt data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="7.15625" data-pf_rect_height="15">r</tt> is declared to be unsigned leading
|
|
|
+to the fact that the comparison <tt data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="28.59375" data-pf_rect_height="15">r>=0</tt> will always return true.
|
|
|
+Note that in assembly language, this code can be fixed by terminating the loop
|
|
|
+when the condition codes indicate a borrow after the <tt data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="28.59375" data-pf_rect_height="15">r-10</tt> subtraction.
|
|
|
+</p><p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="34">
|
|
|
+<a href="http://homepage.cs.uiowa.edu/~jones/bcd/divide.html" data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="196.40625" data-pf_rect_height="17">Another section of this tutorial</a>
|
|
|
+covers the problem of doing multiplicaton and division using only shift
|
|
|
+and add instructions. We use the results presented there, taking specific
|
|
|
+advantage of the limited ranges of values occupied by our dividends, to
|
|
|
+completely eliminate iteration from our code.
|
|
|
+</p><p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="34">
|
|
|
+There are two ways to get an exact quotient and remainder from
|
|
|
+<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="4.453125" data-pf_rect_height="17">i</i>/10 using reciprocal multiplication in 8 bits.
|
|
|
+Either we multiply by 0.00011001 to get an approximate quotient and then
|
|
|
+compute the remainder and correct the quotient, or we multiply by
|
|
|
+0.00011001101 to get an exact quotient from which we can compute the
|
|
|
+remainder.
|
|
|
+</p><p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="17">
|
|
|
+The former method, yielding an approximation, can be reduced to the
|
|
|
+following C code:
|
|
|
+</p><pre data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="252"> /* quotient and remainder returned by div10 */
|
|
|
+ uint8_t q, r;
|
|
|
+
|
|
|
+ void div10( uint8_t i ) {
|
|
|
+ /* approximate the quotient as q = i*0.000110011 (binary) */
|
|
|
+ q = ((i>>1) + i) >> 1; /* times 0.11 */
|
|
|
+ q = ((q>>4) + q) >> 3; /* times 0.000110011 */
|
|
|
+
|
|
|
+ /* compute the reimainder as r = i - 10*q */
|
|
|
+ r = ((q<<2) + q) << 1; /* times 1010. */
|
|
|
+ r = i - r;
|
|
|
+
|
|
|
+ /* fixup if approximate remainder out of bounds */
|
|
|
+ if (r >= 10) {
|
|
|
+ r = r - 10;
|
|
|
+ q = q + 1;
|
|
|
+ }
|
|
|
+ }
|
|
|
+</pre>
|
|
|
+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="34">
|
|
|
+In first two right-shift-and-add statements used above, we have
|
|
|
+assumed that the carry out of the high bit of the add operation
|
|
|
+is shifted into the high bit of the second of the two right-shift
|
|
|
+operations used in each statement. This can be done efficiently
|
|
|
+in many machine instruction sets, but even good optimizing compilers
|
|
|
+may insist on using a 16 bit accumulator for all intermediate
|
|
|
+results of arithmetic operations.
|
|
|
+</p><p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="17">
|
|
|
+The second approach, using a more elaborate multiplier to give an
|
|
|
+exact quotient, may be coded as follows:
|
|
|
+</p><pre data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="196"> /* quotient and remainder returned by div10 */
|
|
|
+ uint8_t q, r;
|
|
|
+
|
|
|
+ void div10( uint8_t i ) {
|
|
|
+ /* approximate the quotient as q = i*0.00011001101 (binary) */
|
|
|
+ q = ((i>>2) + i) >> 1; /* times 0.101 */
|
|
|
+ q = ((q ) + i) >> 1; /* times 0.1101 */
|
|
|
+ q = ((q>>2) + i) >> 1; /* times 0.1001101 */
|
|
|
+ q = ((q ) + i) >> 4; /* times 0.00011001101 */
|
|
|
+
|
|
|
+ /* compute the reimainder as r = i - 10*q */
|
|
|
+ r = ((q<<2) + q) << 1; /* times 1010. */
|
|
|
+ r = i - r;
|
|
|
+ }
|
|
|
+</pre>
|
|
|
+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="34">
|
|
|
+The particular choice of one or the other of these schemes can only be
|
|
|
+made after compiling them or hand translation to machine code. Depending
|
|
|
+on the specific features offered by the target architecture, either of the
|
|
|
+above may be the fastest or most compact, and careful use of small
|
|
|
+optimizations may tilt the balance one way or the other.
|
|
|
+</p><p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="34">
|
|
|
+Depending on the space-time tradeoffs of the application, the above
|
|
|
+code may be expanded in-line as an open macro, or it may be called as a
|
|
|
+closed function. If in-line expansion is appropriate, it should be noted
|
|
|
+that the computations of <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="7.109375" data-pf_rect_height="17">c</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub> and <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="7.109375" data-pf_rect_height="17">c</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">4</sub>
|
|
|
+involve dividends no greater than 65. For dividends up to 68, the
|
|
|
+multiplier 0.0001101 gives exact results, suggesting the following version
|
|
|
+of the print routine:
|
|
|
+</p><pre data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="476"> void putdec( int16_t n ) {
|
|
|
+ uint8_t d4, d3, d2, d1, d0, q;
|
|
|
+
|
|
|
+ if (n < 0) {
|
|
|
+ putchar( '-' );
|
|
|
+ n = -n;
|
|
|
+ }
|
|
|
+
|
|
|
+ d1 = (n>>4) & 0xF;
|
|
|
+ d2 = (n>>8) & 0xF;
|
|
|
+ d3 = (n>>12) & 0xF;
|
|
|
+
|
|
|
+ d0 = 6*(d3 + d2 + d1) + (n & 0xF);
|
|
|
+ q = (d0 * 0xCD) >> 11;
|
|
|
+ d0 = d0 - 10*q;
|
|
|
+
|
|
|
+ d1 = q + 9*d3 + 5*d2 + d1;
|
|
|
+ q = (d1 * 0xCD) >> 11;
|
|
|
+ d1 = d1 - 10*q;
|
|
|
+
|
|
|
+ d2 = q + 2*d2;
|
|
|
+ q = (d2 * 0x1A) >> 8;
|
|
|
+ d2 = d2 - 10*q;
|
|
|
+
|
|
|
+ d3 = q + 4*d3;
|
|
|
+ d4 = (d3 * 0x1A) >> 8;
|
|
|
+ d3 = d3 - 10*d4;
|
|
|
+
|
|
|
+ putchar( d4 + '0' );
|
|
|
+ putchar( d3 + '0' );
|
|
|
+ putchar( d2 + '0' );
|
|
|
+ putchar( d1 + '0' );
|
|
|
+ putchar( d0 + '0' );
|
|
|
+ }
|
|
|
+</pre>
|
|
|
+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="17">
|
|
|
+All multiplications in the above code can be reduced to short sequences
|
|
|
+of shift and add instructions, and it is straightforward to add the code
|
|
|
+discussed earlier to suppress leading zeros from this code.
|
|
|
+</p><p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="17">
|
|
|
+All of these details are incorporated into the attached
|
|
|
+<a href="http://homepage.cs.uiowa.edu/~jones/bcd/decimalpic.html" data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="428.34375" data-pf_rect_height="17">
|
|
|
+decimal print routine for the 14-bit family of PIC microcontrollers.</a>
|
|
|
+</p><p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="0">
|
|
|
+
|
|
|
+</p><hr data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="2">
|
|
|
+
|
|
|
+<h2 data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="26"> <a name="radices" data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="201.953125" data-pf_rect_height="26">Alternative Radices</a> </h2>
|
|
|
+
|
|
|
+<blockquote data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1645.84375" data-pf_rect_height="14">
|
|
|
+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1645.84375" data-pf_rect_height="14"><small data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="378.484375" data-pf_rect_height="15">
|
|
|
+<b data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="31.09375" data-pf_rect_height="15">Note:</b> The following section was added between
|
|
|
+August 6 and 8, 2012.
|
|
|
+</small>
|
|
|
+</p></blockquote>
|
|
|
+
|
|
|
+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="17">
|
|
|
+All of the above work was based on dividing the 16-bit number to be printed
|
|
|
+into four 4-bit chunks. That is, we worked in radix 16. In fact, there is
|
|
|
+no reason to work in a fixed radix. Consider, for example, breaking a
|
|
|
+16-bit number into 4 chunks as follows:
|
|
|
+</p><pre width="70" data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="70"> n
|
|
|
+ |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
|
|
|
+ |_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|
|
|
|
+ | n | n | n | n |
|
|
|
+ 3 2 1 0
|
|
|
+</pre>
|
|
|
+<blockquote data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1645.84375" data-pf_rect_height="17">
|
|
|
+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1645.84375" data-pf_rect_height="17">
|
|
|
+<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i> = 8192 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub> + 512 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">2</sub> +
|
|
|
+ 32 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">1</sub> + 1 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">0</sub>
|
|
|
+</p></blockquote>
|
|
|
+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="17">
|
|
|
+This leads to the following initial approximations for the decimal digits
|
|
|
+of the number:
|
|
|
+</p><blockquote data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1645.84375" data-pf_rect_height="68">
|
|
|
+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1645.84375" data-pf_rect_height="68">
|
|
|
+ <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">a</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub> = 8 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub>
|
|
|
+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17"><i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">a</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">2</sub> = 1 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub> + 5 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">2</sub>
|
|
|
+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17"><i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">a</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">1</sub> = 9 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub> + 1 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">2</sub> +
|
|
|
+ 3 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">1</sub>
|
|
|
+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17"><i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">a</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">0</sub> = 2 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub> + 2 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">2</sub> +
|
|
|
+ 2 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">1</sub> + 1 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">0</sub>
|
|
|
+</p></blockquote>
|
|
|
+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="17">
|
|
|
+Shifting to a mixed radix system instead of pure base 16 has the net effect
|
|
|
+of changing the range of values in each of initial approximations for the
|
|
|
+decimal digits:
|
|
|
+</p><blockquote data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1645.84375" data-pf_rect_height="68">
|
|
|
+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1645.84375" data-pf_rect_height="68">
|
|
|
+ 0 ≤ <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">a</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub> ≤ 65
|
|
|
+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17">0 ≤ <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">a</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">2</sub> ≤ 95
|
|
|
+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17">0 ≤ <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">a</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">1</sub> ≤ 133
|
|
|
+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17">0 ≤ <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">a</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">0</sub> ≤ 105
|
|
|
+</p></blockquote>
|
|
|
+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="34">
|
|
|
+Enlarging <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">0</sub> to 5 bits had no major effect because the
|
|
|
+multiplier on that term is 1. Shifting the other terms over one place
|
|
|
+changed 3 multiplers from 6 to 12, resulting in smaller terms.
|
|
|
+As a result, 8-bit arithmetic now suffices for all of the computations.
|
|
|
+Reducing this to C code, we get:
|
|
|
+</p><pre data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="420"> void putdec( uint16_t n ) {
|
|
|
+ uint8_t d4, d3, d2, d1, d0, q;
|
|
|
+
|
|
|
+ d0 = n & 0x1F;
|
|
|
+ d1 = (n>>5) & 0xF;
|
|
|
+ d2 = (n>>9) & 0xF;
|
|
|
+ d3 = (n>>13) & 0x7;
|
|
|
+
|
|
|
+ d0 = 2*(d3 + d2 + d1) + d0;
|
|
|
+ q = (d0 * (uint16_t)0x67) >> 10;
|
|
|
+ d0 = d0 - 10*q;
|
|
|
+
|
|
|
+ d1 = 9*d3 + d2 + 3*d1 + q;
|
|
|
+ q = (d1 * (uint16_t)0x67) >> 10;
|
|
|
+ d1 = d1 - 10*q;
|
|
|
+
|
|
|
+ d2 = d3 + 5*d2 + q;
|
|
|
+ q = (d2 * (uint16_t)0x67) >> 10;
|
|
|
+ d2 = d2 - 10*q;
|
|
|
+
|
|
|
+ d3 = 8*d3 + q;
|
|
|
+ q = (d3 * (uint16_t)0x1A) >> 8;
|
|
|
+ d3 = d3 - 10*q;
|
|
|
+
|
|
|
+ putchar(q + '0');
|
|
|
+ putchar(d3 + '0');
|
|
|
+ putchar(d2 + '0');
|
|
|
+ putchar(d1 + '0');
|
|
|
+ putchar(d0 + '0');
|
|
|
+ }
|
|
|
+</pre>
|
|
|
+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="34">
|
|
|
+Enlarging <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">0</sub> to 5 bits had a second consequence. Because
|
|
|
+the upper bounds on the larger terms have been reduced, shortened
|
|
|
+approximations for one tenth can be used. On small
|
|
|
+microcontrollers without hardware support for multiplication, this can
|
|
|
+result in savings of a few instructions per multiply.
|
|
|
+The multiplier 67<sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="13.34375" data-pf_rect_height="15">16</sub> works because 0.0001100111 approximates one
|
|
|
+tenth to sufficient precision to divide any number less than 179 by ten.
|
|
|
+The final digit, <tt data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="14.296875" data-pf_rect_height="15">d3</tt>, can still be computed
|
|
|
+using the even shorter approximation we used previously.
|
|
|
+
|
|
|
+</p><hr data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="2">
|
|
|
+
|
|
|
+<h2 data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="26"> <a name="sixtyfour" data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="394.015625" data-pf_rect_height="26">64-bit Conversion on a 32-bit Machine</a> </h2>
|
|
|
+
|
|
|
+<blockquote data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1645.84375" data-pf_rect_height="14">
|
|
|
+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1645.84375" data-pf_rect_height="14"><small data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="1248.890625" data-pf_rect_height="15">
|
|
|
+<b data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="31.09375" data-pf_rect_height="15">Note:</b> The following section was added on
|
|
|
+June 8, 2010. As of June 10, the constants given in this section
|
|
|
+and the code at the end have been checked
|
|
|
+and appear to be correct. As of August 13, this code has
|
|
|
+undergone extensive testing.
|
|
|
+</small>
|
|
|
+</p></blockquote>
|
|
|
+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="51">
|
|
|
+The techniques explored above are applicable wherever a large integer
|
|
|
+must be converted to decimal on a machine with a smaller ALU. For
|
|
|
+example, consider the problem of converting 64-bit integers to decimal
|
|
|
+on a machine with a 32-bit ALU. This practical problem comes up, for
|
|
|
+example, in the BIOS code for Intel-architecture machines, where the
|
|
|
+code must not assume a 64-bit CPU but must still be able to report
|
|
|
+64-bit values when it reports on such things as the sizes of the
|
|
|
+attached disk drives. When 64-bit arithmetic is supported on a 32-bit
|
|
|
+machine, it is slow enough that significant speedups can be achieved
|
|
|
+by avoiding it.
|
|
|
+</p><p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="17">
|
|
|
+In general, 64-bit values may have up to 20 decimal digits:
|
|
|
+</p><blockquote data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1645.84375" data-pf_rect_height="17">
|
|
|
+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1645.84375" data-pf_rect_height="17">
|
|
|
+2<sup data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="13.34375" data-pf_rect_height="15">64</sup> = 18,446,744,073,709,551,616
|
|
|
+</p></blockquote>
|
|
|
+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="34">
|
|
|
+Normally, we group decimal digits in groups of 3 using commas, as
|
|
|
+illustrated above. This is equivalent to writing the number in base 1000,
|
|
|
+where each base 1000 digit is represented with 3 consecutive decimal digits.
|
|
|
+Because there are 20 digits, grouping them as 5 blocks of 4 makes sense,
|
|
|
+suggesting that we should work in base 10,000. In addition, 10,000
|
|
|
+is the largest power of 10 that can be represented in 16 bits
|
|
|
+(half the precision of our ALU). Therefore, we will write:
|
|
|
+</p><blockquote data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1645.84375" data-pf_rect_height="17">
|
|
|
+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1645.84375" data-pf_rect_height="17">
|
|
|
+2<sup data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="13.34375" data-pf_rect_height="15">64</sup> = 1844,6744,0737,0955,1616
|
|
|
+</p></blockquote>
|
|
|
+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="17">
|
|
|
+We begin by breaking our 64-bit number into 16-bit chunks, just as we
|
|
|
+began the 16-bit conversion by breaking the number into 4-bit chunks.
|
|
|
+Again, we will the number <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i> and we will call the 16-bit chunks
|
|
|
+<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub>, <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">2</sub>, <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">1</sub>
|
|
|
+and <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">0</sub>.
|
|
|
+</p><pre width="70" data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="70"> n
|
|
|
+ |___ ___ ___ ___ ___ ___ ___ ___| 64-bits
|
|
|
+ |___|___|___|___|___|___|___|___| 8 bytes
|
|
|
+ | n | n | n | n | 4 16-bit chunks
|
|
|
+ 3 2 1 0
|
|
|
+</pre>
|
|
|
+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="17">
|
|
|
+The value of the number can be expressed in terms of these 4 fields as
|
|
|
+follows:
|
|
|
+</p><blockquote data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1645.84375" data-pf_rect_height="118">
|
|
|
+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1645.84375" data-pf_rect_height="17">
|
|
|
+<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i> = 2<sup data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="13.34375" data-pf_rect_height="15">48</sup><i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub> +
|
|
|
+ 2<sup data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="13.34375" data-pf_rect_height="15">32</sup><i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">2</sub> +
|
|
|
+ 2<sup data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="13.34375" data-pf_rect_height="15">16</sup><i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">1</sub> +
|
|
|
+ 2<sup data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">0</sup><i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">0</sub>
|
|
|
+</p><p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1645.84375" data-pf_rect_height="85">
|
|
|
+<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i> =
|
|
|
+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17"> 281,474,976,710,656 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub> +
|
|
|
+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17"> 4,294,967,296 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">2</sub> +
|
|
|
+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17"> 65,536 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">1</sub> +
|
|
|
+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17"> 1 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">0</sub>
|
|
|
+</p></blockquote>
|
|
|
+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="17">
|
|
|
+Because we are interested in working in base 10,000, we will regroup the
|
|
|
+digits above and continue with the following version:
|
|
|
+</p><blockquote data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1645.84375" data-pf_rect_height="85">
|
|
|
+<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i> =
|
|
|
+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17"> 281,4749,7671,0656 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub> +
|
|
|
+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17"> 42,9496,7296 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">2</sub> +
|
|
|
+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17"> 6,5536 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">1</sub> +
|
|
|
+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17"> 1 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">0</sub>
|
|
|
+</blockquote>
|
|
|
+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="17">
|
|
|
+Our goal is to express <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i> in base 10,000 as
|
|
|
+<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">d</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">4</sub><i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">d</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub><i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">d</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">2</sub><i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">d</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">1</sub><i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">d</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">0</sub>.
|
|
|
+If each of <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">d</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">4</sub>, <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">d</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub>, <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">d</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">2</sub>,
|
|
|
+<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">d</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">1</sub> and <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">d</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">0</sub> is a single base 10,000
|
|
|
+digit, the value of n can be expressed as:
|
|
|
+</p><blockquote data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1645.84375" data-pf_rect_height="102">
|
|
|
+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1645.84375" data-pf_rect_height="102">
|
|
|
+<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i> =
|
|
|
+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17"> 1,0000,0000,0000,0000 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">d</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">4</sub> +
|
|
|
+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17"> 1,0000,0000,0000 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">d</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub> +
|
|
|
+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17"> 1,0000,0000 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">d</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">2</sub> +
|
|
|
+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17"> 1,0000 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">d</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">1</sub> +
|
|
|
+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17"> 1 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">d</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">0</sub>
|
|
|
+</p></blockquote>
|
|
|
+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="17">
|
|
|
+Our initial approximations of the decimal digits comes from rearranging
|
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+the base 10,000 digits of the multipliers for the 16-bit chunks to conform
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+with the above decimal presentation:
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+</p><p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="0">
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+</p><blockquote data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1645.84375" data-pf_rect_height="85">
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+<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i> =
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+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17"> 1,0000,0000,0000 ( 281 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub> ) +
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+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17"> 1,0000,0000 ( 4749 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub> +
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+ 42 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">2</sub> ) +
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+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17"> 1,0000 ( 7671 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub> + 9496 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">2</sub> +
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+ 6 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">1</sub> ) +
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+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17"> 1 ( 0656 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub> + 7296 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">2</sub> +
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+ 5536 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">1</sub> + 1 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">0</sub> )
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+</blockquote>
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+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="17">
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+The approximations for the four least significant base 10,000 digits fall out
|
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+of the above:
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+</p><blockquote data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1645.84375" data-pf_rect_height="68">
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+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1645.84375" data-pf_rect_height="68">
|
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+<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">a</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub> = 281 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub>
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|
+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17"><i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">a</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">2</sub> = 4749 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub> + 42 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">2</sub>
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|
+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17"><i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">a</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">1</sub> = 7671 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub> +
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+ 9496 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">2</sub> + 6 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">1</sub>
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|
+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17"><i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">a</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">0</sub> = 0656 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub> +
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+ 7296 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">2</sub> +
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+ 5536 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">1</sub> + 1 <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">0</sub>
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|
|
+</p></blockquote>
|
|
|
+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="34">
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+At this point, just as we did with the 4-bit example, we should check the
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|
+range of values we might encounter in each approximation. Here, our
|
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+concern is that some of the approximations might exceed the capacity of a
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+32-bit ALU. The maximum value of each term will occur when all of the
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+<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="3.71875" data-pf_rect_height="15">i</sub> are at their maximum value, which is 65,535.
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|
+</p><blockquote data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1645.84375" data-pf_rect_height="68">
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+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1645.84375" data-pf_rect_height="68">
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|
+ 0 ≤ <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">a</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub>
|
|
|
+ ≤ 18,415,335
|
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|
+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17">0 ≤ <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">a</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">2</sub>
|
|
|
+ ≤ 313,978,185
|
|
|
+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17">0 ≤ <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">a</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">1</sub>
|
|
|
+ ≤ 1,125,432,555
|
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|
+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17">0 ≤ <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">a</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">0</sub>
|
|
|
+ ≤ 884,001,615
|
|
|
+</p></blockquote>
|
|
|
+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="17">
|
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|
+All of our approximations are comfortably less than 2 billion, which means
|
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|
+that we can safely use 32-bit signed division, if that is all that is
|
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|
+available to us.
|
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|
+</p><p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="17">
|
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|
+We are now ready to compute the final values of the digits by a
|
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|
+series of 32-bit divisions, as follows:
|
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|
+</p><blockquote data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1645.84375" data-pf_rect_height="221">
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|
+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1645.84375" data-pf_rect_height="221">
|
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+ <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="7.109375" data-pf_rect_height="17">c</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">1</sub>
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|
+ = <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">a</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">0</sub> / 10,000
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|
+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17"><i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">d</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">0</sub>
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+ = <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">a</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">0</sub> mod 10,000
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+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17">
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+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17"><i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="7.109375" data-pf_rect_height="17">c</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">2</sub>
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+ = (<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">a</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">1</sub> + <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="7.109375" data-pf_rect_height="17">c</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">1</sub>) / 10,000
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+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17"><i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">d</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">1</sub>
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+ = (<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">a</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">1</sub> + <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="7.109375" data-pf_rect_height="17">c</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">1</sub>) mod 10,000
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+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17">
|
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|
+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17"><i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="7.109375" data-pf_rect_height="17">c</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub>
|
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+ = (<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">a</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">2</sub> + <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="7.109375" data-pf_rect_height="17">c</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">2</sub>) / 10,000
|
|
|
+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17"><i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">d</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">2</sub>
|
|
|
+ = (<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">a</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">2</sub> + <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="7.109375" data-pf_rect_height="17">c</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">2</sub>) mod 10,000
|
|
|
+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17">
|
|
|
+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17"><i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="7.109375" data-pf_rect_height="17">c</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">4</sub>
|
|
|
+ = (<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">a</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub> + <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="7.109375" data-pf_rect_height="17">c</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub>) / 10,000
|
|
|
+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17"><i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">d</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub>
|
|
|
+ = (<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">a</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub> + <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="7.109375" data-pf_rect_height="17">c</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub>) mod 10,000
|
|
|
+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17">
|
|
|
+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17"><i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">d</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">4</sub> = <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="7.109375" data-pf_rect_height="17">c</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">4</sub>
|
|
|
+</p></blockquote>
|
|
|
+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="17">
|
|
|
+In the above, the <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="7.109375" data-pf_rect_height="17">c</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="3.71875" data-pf_rect_height="15">i</sub> terms represent carries propagated
|
|
|
+from one digit position to the next. The upper bounds on each of the
|
|
|
+<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="7.109375" data-pf_rect_height="17">c</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="3.71875" data-pf_rect_height="15">i</sub> can be computed from the bounds given above for the
|
|
|
+<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">a</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="3.71875" data-pf_rect_height="15">i</sub>:
|
|
|
+</p><blockquote data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1645.84375" data-pf_rect_height="68">
|
|
|
+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1645.84375" data-pf_rect_height="68">
|
|
|
+ <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="7.109375" data-pf_rect_height="17">c</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">1</sub> ≤ 88,400
|
|
|
+ [ = <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">a</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">0</sub> / 10,000 = 884,001,615 / 10,000 ]
|
|
|
+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17"><i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="7.109375" data-pf_rect_height="17">c</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">2</sub> ≤ 112,552
|
|
|
+ [ = (<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">a</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">1</sub> + <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="7.109375" data-pf_rect_height="17">c</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">1</sub>) / 10,000
|
|
|
+ = 1,125,520,955 / 10,000 ]
|
|
|
+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17"><i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="7.109375" data-pf_rect_height="17">c</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub> ≤ 31,409
|
|
|
+ [ = (<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">a</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">2</sub> + <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="7.109375" data-pf_rect_height="17">c</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">2</sub>) / 10,000
|
|
|
+ = 314,090,737 / 10,000 ]
|
|
|
+<br data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="0" data-pf_rect_height="17"><i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="7.109375" data-pf_rect_height="17">c</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">4</sub> ≤ 1,844
|
|
|
+ [ = (<i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">a</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub> + <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="7.109375" data-pf_rect_height="17">c</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">3</sub>) / 10,000
|
|
|
+ = 18,446,744 / 10,000 ]
|
|
|
+</p></blockquote>
|
|
|
+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="17">
|
|
|
+Note something encouraging: The largest intermediate value encountered
|
|
|
+above is 1,125,520,955, which is safely less than 2 billion and therefore
|
|
|
+safe to represent as either a signed or unsigned integer on a 32-bit machine.
|
|
|
+</p><p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="34">
|
|
|
+Note also that the bound on <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="7.109375" data-pf_rect_height="17">c</i><sub data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="6.671875" data-pf_rect_height="15">4</sub> is a small but useful
|
|
|
+check on our arithmetic. The value, 1,844, is identical to the first 4 decimal
|
|
|
+digits of 2**64, which is exactly what we expect. This does not eliminate
|
|
|
+the possibility that the above might contain many small clerical errors.
|
|
|
+</p><p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="34">
|
|
|
+The code to print the 64-bit number ends up essentially identical to the
|
|
|
+code given previouly to print a 16-bit number, except for the sizes of the
|
|
|
+variables, the values of the various constants, and the code to print each
|
|
|
+digit of the result.
|
|
|
+If we have not made any errors in computing the various constants,
|
|
|
+the following C code should work:
|
|
|
+</p><pre data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="448"> void putdec( uint64_t n ) {
|
|
|
+ uint32_t d4, d3, d2, d1, d0, q;
|
|
|
+
|
|
|
+ d0 = n & 0xFFFF;
|
|
|
+ d1 = (n>>16) & 0xFFFF;
|
|
|
+ d2 = (n>>32) & 0xFFFF;
|
|
|
+ d3 = (n>>48) & 0xFFFF;
|
|
|
+
|
|
|
+ d0 = 656 * d3 + 7296 * d2 + 5536 * d1 + d0;
|
|
|
+ q = d0 / 10000;
|
|
|
+ d0 = d0 % 10000;
|
|
|
+
|
|
|
+ d1 = q + 7671 * d3 + 9496 * d2 + 6 * d1;
|
|
|
+ q = d1 / 10000;
|
|
|
+ d1 = d1 % 10000;
|
|
|
+
|
|
|
+ d2 = q + 4749 * d3 + 42 * d2;
|
|
|
+ q = d2 / 10000;
|
|
|
+ d2 = d2 % 10000;
|
|
|
+
|
|
|
+ d3 = q + 281 * d3;
|
|
|
+ q = d3 / 10000;
|
|
|
+ d3 = d3 % 10000;
|
|
|
+
|
|
|
+ d4 = q;
|
|
|
+
|
|
|
+ printf( "%4.4u", d4 );
|
|
|
+ printf( "%4.4u", d3 );
|
|
|
+ printf( "%4.4u", d2 );
|
|
|
+ printf( "%4.4u", d1 );
|
|
|
+ printf( "%4.4u", d0 );
|
|
|
+ }
|
|
|
+</pre>
|
|
|
+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="34">
|
|
|
+In the above code, we used the standard types
|
|
|
+<tt data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="57.1875" data-pf_rect_height="15">uint32_t</tt> and <tt data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="57.1875" data-pf_rect_height="15">uint64_t</tt> to declare
|
|
|
+32 and 64-bit unsigned integer variables. By convention, these should
|
|
|
+be defined in <tt data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="71.484375" data-pf_rect_height="15"><stdint.h></tt>. Note that
|
|
|
+if you are working on a 32-bit machine with no support for 64-bit arithemtic,
|
|
|
+there is no guarantee that you will be able to declare the 64-bit type
|
|
|
+and you may have to fake it, for example, by representing each 64-bit
|
|
|
+value as a pair of 32-bit components.
|
|
|
+</p><p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="34">
|
|
|
+This code uses <tt data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="57.1875" data-pf_rect_height="15">printf()</tt> to print each 4-digit base 10,000 digit
|
|
|
+of the result. Of course, this code prints all 20 digits of <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="8" data-pf_rect_height="17">n</i>.
|
|
|
+Suppressing leading zeros takes
|
|
|
+a bit more work than it did with the 4-bit nibble code because of the
|
|
|
+complexity of suppressing leading zeros within the printed representation
|
|
|
+of the first nonzero base 10,000 digit.
|
|
|
+</p><p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="0">
|
|
|
+
|
|
|
+</p><hr data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1725.84375" data-pf_rect_height="2">
|
|
|
+
|
|
|
+<blockquote data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1645.84375" data-pf_rect_height="254">
|
|
|
+<p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1645.84375" data-pf_rect_height="14">
|
|
|
+<small data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="1286.265625" data-pf_rect_height="15">
|
|
|
+I don't know where I first encountered the basic idea for the 8-bit
|
|
|
+code presented here. This tutorial began as an attempt to reinvent
|
|
|
+that code after I found myself working on a machine with an 8-bit ALU
|
|
|
+and needing to do 16-bit conversions.
|
|
|
+</small>
|
|
|
+</p><p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1645.84375" data-pf_rect_height="14">
|
|
|
+<small data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="754.4375" data-pf_rect_height="15">
|
|
|
+Thanks to Mark Ramsey, who, on January 22, 2007 pointed out two small typos
|
|
|
+with big consequences in the section on <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="106.28125" data-pf_rect_height="15">Living Within 8 Bits</i>.
|
|
|
+</small>
|
|
|
+</p><p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1645.84375" data-pf_rect_height="14">
|
|
|
+<small data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="1136.6875" data-pf_rect_height="15">
|
|
|
+Thanks to Dennis Vlasenko, who, on June 15, 2007 pointed out more typos in
|
|
|
+the section on <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="188.828125" data-pf_rect_height="15">Doing Without Multiply and Divide</i>, and for his work
|
|
|
+integrate the 16-bit code into the Linux distribution of <tt data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="59.546875" data-pf_rect_height="13">vsprintf()</tt>.
|
|
|
+</small>
|
|
|
+</p><p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1645.84375" data-pf_rect_height="14">
|
|
|
+<small data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="484.625" data-pf_rect_height="15">
|
|
|
+Thanks to Joe Zbiciak, who, on August 23, 2007 pointed out a small error
|
|
|
+the introduction.
|
|
|
+</small>
|
|
|
+</p><p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1645.84375" data-pf_rect_height="14">
|
|
|
+<small data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="1216.828125" data-pf_rect_height="15">
|
|
|
+Thanks to Derick Moore, who, on June 10, 2010, reported that he had
|
|
|
+checked the constants and tested the code in the section
|
|
|
+on <i data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="204" data-pf_rect_height="15">64-bit conversion on a 32-bit machine</i> and verified that it
|
|
|
+worked in the BIOS he was maintaining.
|
|
|
+</small>
|
|
|
+</p><p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1645.84375" data-pf_rect_height="14">
|
|
|
+<small data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="892.5" data-pf_rect_height="15">
|
|
|
+Thanks to Michal Nazarewicz for additional testing reported on August 13, 2010
|
|
|
+and for his work to integrate the 64-bit code into the Linux distribution
|
|
|
+of <tt data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="59.546875" data-pf_rect_height="13">vsprintf()</tt>.
|
|
|
+</small>
|
|
|
+</p><p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1645.84375" data-pf_rect_height="14">
|
|
|
+<small data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="987.078125" data-pf_rect_height="15">
|
|
|
+Thanks to George Spelvin, who on August 3, 2012, suggested that there is no
|
|
|
+reason that all of the chunks should be the same size, leading to the
|
|
|
+addition of the section on other radices.
|
|
|
+</small>
|
|
|
+</p><p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1645.84375" data-pf_rect_height="14">
|
|
|
+<small data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="811.6875" data-pf_rect_height="15">
|
|
|
+Thanks to Ralph Corderoy, who on Septeber 10, 2014, pointed out the weakness
|
|
|
+of the repeated subtraction code when presented with the value -32768.
|
|
|
+</small>
|
|
|
+</p><p data-pf_style_display="block" data-pf_style_visibility="visible" data-pf_rect_width="1645.84375" data-pf_rect_height="14">
|
|
|
+<small data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="915.734375" data-pf_rect_height="15">
|
|
|
+Thanks to Ben Boldt, who on April 9, 2018,
|
|
|
+pointed out an error the first take on optimising <tt data-pf_style_display="inline" data-pf_style_visibility="visible" data-pf_rect_width="29.78125" data-pf_rect_height="13">div10</tt> at the start
|
|
|
+of the section on doing without hardware multiply and divide.
|
|
|
+</small>
|
|
|
+</p></blockquote>
|
|
|
+
|
|
|
+
|
|
|
+
|
|
|
+</body></html>
|
Vladimir Shilov