Rational and VFP Numbers - Revision history

WikiSysop: /* Formatted Display */

2019-10-17T15:57:14Z

Formatted Display

← Older revision		Revision as of 11:57, 17 October 2019
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	<apll><pre>		<apll><pre>
	'R4.2' ⎕FMT ∘.÷⍨⍳6<~~pn>x<~~/pn>		'R4.2' ⎕FMT ∘.÷⍨⍳6<_x/>
	1 1<~~pn>r<~~/pn>2 1<~~pn>r<~~/pn>3 1<~~pn>r<~~/pn>4 1<~~pn>r<~~/pn>5 1<~~pn>r<~~/pn>6		1 1<_r/>2 1<_r/>3 1<_r/>4 1<_r/>5 1<_r/>6
	2 1 2<~~pn>r<~~/pn>3 1<~~pn>r<~~/pn>2 2<~~pn>r<~~/pn>5 1<~~pn>r<~~/pn>3		2 1 2<_r/>3 1<_r/>2 2<_r/>5 1<_r/>3
	3 3<~~pn>r<~~/pn>2 1 3<~~pn>r<~~/pn>4 3<~~pn>r<~~/pn>5 1<~~pn>r<~~/pn>2		3 3<_r/>2 1 3<_r/>4 3<_r/>5 1<_r/>2
	4 2 4<~~pn>r<~~/pn>3 1 4<~~pn>r<~~/pn>5 2<~~pn>r<~~/pn>3		4 2 4<_r/>3 1 4<_r/>5 2<_r/>3
	5 5<~~pn>r<~~/pn>2 5<~~pn>r<~~/pn>3 5<~~pn>r<~~/pn>4 1 5<~~pn>r<~~/pn>6		5 5<_r/>2 5<_r/>3 5<_r/>4 1 5<_r/>6
	6 3 2 3<~~pn>r<~~/pn>2 6<~~pn>r<~~/pn>5 1 </pre></apll>		6 3 2 3<_r/>2 6<_r/>5 1 </pre></apll>

	<p>Moreover, the Symbol Substitution (<apll>S<…></apll>) feature of <apll>⎕FMT</apll> allows you to substitute a different symbol for the default <apll><~~pn>r<~~/pn></apll> used to separate the numerator and denominator of a rational number, as in</p>		<p>Moreover, the Symbol Substitution (<apll>S<…></apll>) feature of <apll>⎕FMT</apll> allows you to substitute a different symbol for the default <apll><_r/></apll> used to separate the numerator and denominator of a rational number, as in</p>

	<apll><pre>		<apll><pre>
	'S<r/>R4.2' ⎕FMT ∘.÷⍨⍳6<~~pn>x<~~/pn>		'S<r/>R4.2' ⎕FMT ∘.÷⍨⍳6<_x/>
	1 1/2 1/3 1/4 1/5 1/6 ~~<br />~~		1 1/2 1/3 1/4 1/5 1/6
	2 1 2/3 1/2 2/5 1/3 ~~<br />~~		2 1 2/3 1/2 2/5 1/3
	3 3/2 1 3/4 3/5 1/2 ~~<br />~~		3 3/2 1 3/4 3/5 1/2
	4 2 4/3 1 4/5 2/3 ~~<br />~~		4 2 4/3 1 4/5 2/3
	5 5/2 5/3 5/4 1 5/6 ~~<br />~~		5 5/2 5/3 5/4 1 5/6
	6 3 2 3/2 6/5 1 </pre></apll>		6 3 2 3/2 6/5 1 </pre></apll>

Sudleyplace at 00:57, 16 April 2018

2018-04-16T00:57:05Z

Show changes

Drab: /* Conversions */

2016-12-01T12:20:30Z

Conversions

← Older revision		Revision as of 08:20, 1 December 2016
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	* Integer, floating point, or rational number to VFP, use <apll>0v+</apll>		* Integer, floating point, or rational number to VFP, use <apll>0v+</apll>
	* Rational integer to a 64-bit integer (possibly losing some precision), use <apll>⍎⍕</apll>		* Rational integer to a 64-bit integer (possibly losing some precision), use <apll>⍎⍕</apll>
	* Rational non-integer to a floating point number (possibly losing some precision), use <apll>~~⍎⍕0v~~+</apll>		* Rational non-integer to a floating point number (possibly losing some precision), use <apll>0.0+</apll>
	* VFP number to a floating point number (possibly losing some precision), use <apll>⍎⍕</apll>		* VFP number to a floating point number (possibly losing some precision), use <apll>⍎⍕</apll>
	* VFP number to rational — at the moment, no APL method exists although the underlying libraries do have such a function		* VFP number to rational — at the moment, no APL method exists although the underlying libraries do have such a function

WikiSysop: /* Rationale */

2013-05-24T20:48:29Z

Rationale

← Older revision		Revision as of 16:48, 24 May 2013
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	===Rationale===		===Rationale===

	<p>Simply put: precision. If the 53 bits of precision in the floating point result of, say, <apll>2÷3</apll> is not enough, you now have two more choices: one as an exact number, and one as a ~~decimal~~ number with as much precision as you care to specify.</p>		<p>Simply put: precision. If the 53 bits of precision in the floating point result of, say, <apll>2÷3</apll> is not enough, you now have two more choices: one as an exact number, and one as a binary floating point number with as much precision as you care to specify.</p>

	<p>For example,</p>		<p>For example,</p>

WikiSysop: /* Constants */

2013-04-03T13:27:11Z

Constants

← Older revision		Revision as of 09:27, 3 April 2013
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	* <apll>123v</apll> for the constant <apll>123</apll>		* <apll>123v</apll> for the constant <apll>123</apll>
	* <apll>~~123v4567~~</apll> for the constant <apll>123.4567</apll>		* <apll>123.4567v</apll> for the constant <apll>123.4567</apll>
	* <apll>~~123v4567e3~~</apll> for the constant <apll>123.4567e3</apll>		* <apll>123.4567e3v</apll> for the constant <apll>123.4567e3</apll>

	<p>The above formats for constants (except for the suffix <apll>x</apll>) may be used in other constants such as <apll>1r4p2</apll> to generate a shorter and more accurate value for <apll>π<sup>2</sup>/4</apll> than, say, <apll>((○1)*2)÷4</apll>.</p>		<p>The above formats for constants (except for the suffix <apll>x</apll>) may be used in other constants such as <apll>1r4p2</apll> to generate a shorter and more accurate value for <apll>π<sup>2</sup>/4</apll> than, say, <apll>((○1)*2)÷4</apll>.</p>

WikiSysop: /* Acknowledgments */

2013-03-31T18:25:43Z

Acknowledgments

← Older revision		Revision as of 14:25, 31 March 2013
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	* MPIR (Multiple Precision Integers and Rationals) at [http://www.mpir.org mpir.org].		* MPIR (Multiple Precision Integers and Rationals) at [http://www.mpir.org mpir.org].
	* MPFR (Multiple Precision Floating-Point Reliable Library ) at [http://www.mpfr.org mpfr.org].		* MPFR (Multiple Precision Floating-Point Reliable Library ) at [http://www.mpfr.org mpfr.org].

			===References===

			<p>For a PDF version of this page, view it [http://www.sudleyplace.com/APL/Rational%20&%20Variable-Precision%20FP.pdf here].</p>

WikiSysop: /* Acknowledgments */

2013-03-29T14:13:46Z

Acknowledgments

← Older revision		Revision as of 10:13, 29 March 2013
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	<p>The designers of J are thanked for having the foresight to include [http://www.jsoftware.com/help/dictionary/dictg.htm rational numbers] as a separate datatype.</p>		<p>The designers of J are thanked for having the foresight to include [http://www.jsoftware.com/help/dictionary/dictg.htm rational numbers] as a separate datatype.</p>

	<p>The following ~~GPL~~ libraries have been used to provide support for these datatypes:</p>		<p>The following LGPL libraries have been used to provide support for these datatypes:</p>

	* MPIR (Multiple Precision Integers and Rationals) at [http://www.mpir.org mpir.org].		* MPIR (Multiple Precision Integers and Rationals) at [http://www.mpir.org mpir.org].
	* MPFR (Multiple Precision Floating-Point Reliable Library ) at [http://www.mpfr.org mpfr.org].		* MPFR (Multiple Precision Floating-Point Reliable Library ) at [http://www.mpfr.org mpfr.org].

WikiSysop: /* Rationale */

2013-03-29T13:56:32Z

Rationale

← Older revision		Revision as of 09:56, 29 March 2013
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	<apll>      ⎕PP←60 ⋄ ⎕FPC←512<br />		<apll>      ⎕PP←60 ⋄ ⎕FPC←512<br />
			2÷3<br />
			0.6666666666666666<br />
	2÷3x<br />		2÷3x<br />
	2r3<br />		2r3<br />

WikiSysop: /* Introduction */

2013-03-29T13:52:33Z

Introduction

← Older revision		Revision as of 09:52, 29 March 2013
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	===Introduction===		===Introduction===

	<p><b>Rational numbers</b> complement the 64-bit integer datatype to provide infinite precision (or <apll>WS FULL</apll>) at the cost of some performance. Similarly, <b>Variable-precision Floating Point</b> (VFP) numbers complement the 64-bit floating point datatype to provide more precision (as much as the user cares to specify) again with some cost in performance.</p>		<p><b>Rational numbers</b> complement the existing 64-bit integer datatype to provide infinite precision (or <apll>WS FULL</apll>) at the cost of some performance. Similarly, <b>Variable-precision Floating Point</b> (VFP) numbers complement the existing 64-bit floating point (IEEE-754) datatype to provide more precision (as much as the user cares to specify) again with some cost in performance.</p>

	<p>There is no separate datatype for infinite precision Integers. Instead, they are represented as a special case of rational numbers. In the discussion below, the phrase <b>rational integer</b> means a rational number whose denominator is one.</p>		<p>There is no separate datatype for infinite precision Integers. Instead, they are represented as a special case of rational numbers. In the discussion below, the phrase <b>rational integer</b> means a rational number whose denominator is one.</p>

WikiSysop at 22:20, 27 March 2013

2013-03-27T22:20:34Z

← Older revision		Revision as of 18:20, 27 March 2013
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	===Constants===		===Constants===

	<p>Rational constants may be entered as follows:</p>		<p>Rational constants may be entered by suffixing an integer constant with an <apll>x</apll> (for a rational integer) or separating the numerator and denominator with an <apll>r</apll> (for a rational number) as follows:</p>

	* <apll>123x</apll> for the constant <apll>123</apll> (the suffix <apll>x</apll> is but a shorthand for <apll>r1</apll>)		* <apll>123x</apll> for the constant <apll>123</apll> (the suffix <apll>x</apll> is but a shorthand for <apll>r1</apll>)
	* <apll>123r4567</apll> for the constant <apll>123÷4567</apll>		* <apll>123r4567</apll> for the constant <apll>123÷4567</apll>

	<p>VFP constants may be entered as follows:</p>		<p>VFP constants may be entered by suffixing the integer or floating point constant with a <apll>v</apll> as follows:</p>

	* <apll>123v</apll> for the constant <apll>123</apll>		* <apll>123v</apll> for the constant <apll>123</apll>
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	<p>VFP numbers have <b>user-controlled variable precision</b>, and each number may have a different precision. The default precision at startup is controlled by the value of the system variable <apll>⎕FPC</apll>. The system default of this value is <apll>128</apll> in units of bits of precision of the mantissa (the digits) of the number, not counting the exponent (which is of fixed size). The current precision may be changed as needed by assigning a new value to the system variable. All newly created VFP numbers will have the new precision – the precision of VFP numbers already present in the workspace does not change.</p>		<p>VFP numbers have <b>user-controlled variable precision</b>, and each number may have a different precision. The default precision at startup is controlled by the value of the system variable <apll>⎕FPC</apll>. The system default of this value is <apll>128</apll> in units of bits of precision of the mantissa (the digits) of the number, not counting the exponent (which is of fixed size). The current precision may be changed as needed by assigning a new value to the system variable. All newly created VFP numbers will have the new precision – the precision of VFP numbers already present in the workspace does not change.</p>

	<p>Generally, precision is set once for a particular application and unchanged thereafter. Although not recommended, it is possible to mix VFP numbers of different precisions in a single array – presumably you <b>really</b> know what you are doing. The system function <apll>⎕DR</apll> may be used to display an array's precision(s).</p>		<p>Generally, precision is set once for a particular application and unchanged thereafter. Although not recommended, it is possible to mix VFP numbers of different precisions in a single array – presumably you <b>really</b> know what you are doing. The system function <apll>0 ⎕DR</apll> may be used to display an array's precision(s).</p>

	===Datatype Propagation===		===Datatype Propagation===
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	<p>The obvious expression <apll>¯10↑⍕+/*⍨⍳1000</apll> at first sight seems to solve the problem until you realize that it quickly runs afoul of the limited precision of 64-bit integer and floating point numbers. Clearly, this is a problem for the infinite precision of rational integers.</p>		<p>The obvious expression <apll>¯10↑⍕+/*⍨⍳1000</apll> at first sight seems to solve the problem until you realize that it quickly runs afoul of the limited precision of 64-bit integer and floating point numbers. Clearly, this is a problem for the infinite precision of rational integers.</p>

	<p>As <apll>⍳1000</apll> generates the first thousand integers as an integer datatype (actually an Arithmetic Progression Array), <apll>⍳1000x</apll> generates the same values as rational integers. Next, <apll>⍨⍳1000x</apll> generates the first thousand instances of N<sup>N</sup> as exact rational integers, and unlike its integer counterpart, there is no overflow to floating point, just an increase in precision. Then, <apll>+/⍨⍳1000x</apll> sums them into a single 3001-digit rational integer, and finally <apll>¯10↑⍕+/*⍨⍳1000x</apll> converts the large integer to characters and extracts the low-order ten digits — <apll>9110846700</apll> — all in a small number of milliseconds.</p>		<p>As <apll>⍳1000</apll> generates the first thousand integers as an integer datatype (actually an Arithmetic Progression Array), <apll>⍳1000x</apll> generates the same values as rational integers. Next, <apll>⍨⍳1000x</apll> generates the first thousand instances of N<sup>N</sup> as exact rational integers, and unlike its integer counterpart, there is no overflow to floating point, just an increase in precision (as well as space used in the workspace). Then, <apll>+/⍨⍳1000x</apll> sums them into a single 3001-digit rational integer, and finally <apll>¯10↑⍕+/*⍨⍳1000x</apll> converts the large integer to characters and extracts the low-order ten digits — <apll>9110846700</apll> — all in a small number of milliseconds.</p>

	<p>Note how we started with an obvious expression that failed because of its limited precision, and made a single change to suffix the constant <apll>1000</apll> with an <apll>x</apll> to convert it to a rational integer which then propagates through the calculation with infinite precision to yield the correct result.</p>		<p>Note how we started with an obvious expression that failed because of its limited precision, and made a single change to suffix the constant <apll>1000</apll> with an <apll>x</apll> to convert it to a rational integer which then propagates through the calculation with infinite precision to yield the correct result.</p>
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	===Display===		===Display===

	<p>Rational integers are displayed as an integer with no special adornment; rational non-integers are displayed as a numerator and denominator separated by an <apll>r</apll> as in <apll>34r9</apll>. As with the integer datatype, the numerator and denominator of a rational number are displayed exactly, unaffected by <apll>⎕PP</apll>.</p>		<p>Rational integers are displayed as an integer with no special adornment; rational non-integers are displayed as a numerator and denominator separated by an <apll>r</apll> as in <apll>34r9</apll>. As with the integer datatype, the numerator and denominator of a rational number are displayed exactly, unaffected by the current setting for Printing Precision (<apll>⎕PP</apll>).</p>
	<br />		<br />
	<apll>      !40<br />		<apll>      !40<br />
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	<p>where both of the above displays were limited by the precision of the number, not <apll>⎕PP</apll>.</p>		<p>where both of the above displays were limited by the precision of the number, not <apll>⎕PP</apll>.</p>

	<p>However, the first of the following displays <b>is</b> limited by <apll>⎕PP</apll>~~, and the second one by the explicit <apll>20</apll> in the format function~~:</p>		<p>However, the first of the following displays <b>is</b> limited by <apll>⎕PP</apll>:</p>

	<apll>      ⎕FPC←128<br />		<apll>      ⎕FPC←128<br />
	⎕PP←20<br />		⎕PP←20<br />
	!40v<br />		!40v<br />
	~~81591528324789773435E28<br />~~		81591528324789773435____________________________<br />
	~~      20⍕!40v<br />~~
	~~ 81591528324789773435E28~~<br />
	⎕PP←80<br />		⎕PP←80<br />
	!40v<br />		!40v<br />
	~~815915283247897734345611269596115894272000000000<br />~~		8159152832478977343456112695961158942720________</apll><br />
	~~      39⍕!40v<br />~~
	~~ 815915283247897734345611269596115894272000000000~~</apll><br />

	<p ~~class="note"~~>In ~~other words~~, ~~if a VFP integer appears in E-notation~~, ~~it means that either~~ <apll>~~⎕PP~~</apll> ~~or an explicit display precision to the format function~~ is too small ~~to show all of its significant digits~~.</p>		<p>In the last display, the current setting of Printing Precision is large enough, but the current setting of the Floating Point Control (<apll>⎕FPC</apll>) whose value is in bits is too small, so the display is truncated.</p>

	===Formatted Display===		===Formatted Display===
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	2.718281828459045<br />		2.718281828459045<br />
	*1x<br />		*1x<br />
	2.~~718281828459045235360287471352662497757~~</apll><br />		2.718281828459045235360287471352662497759</apll><br />

	<p>where the datatype of the two results are floating point and VFP, respectively. That is, in a manner similar to how some primitive functions with integer arguments may return floating point results, when a rational number is used as an argument to a primitive function that can't return a result with infinite precision, it returns a VFP number.</p>		<p>where the datatype of the two results are floating point and VFP, respectively. That is, in a manner similar to how some primitive functions with integer arguments may return floating point results, when a rational number is used as an argument to a primitive function that can't return a result with infinite precision, it returns a VFP number.</p>

	<p class="note">The reason irrational, transcendental, and certain other functions on rational numbers ~~don't~~ return rational numbers is that, by definition, the result of such a function is, in general, not representable as a rational number; instead, VFP numbers are better suited to represent irrational results where the end user may control exactly how much precision is desired in an obviously inexact number.</p>		<p class="note">The reason irrational, transcendental, and certain other functions on rational numbers do not return rational numbers is that, by definition, the result of such a function is, in general, not representable as a rational number; instead, VFP numbers are better suited to represent irrational results where the end user may control exactly how much precision is desired in an obviously inexact number.</p>

	<p>Two special functions are the prime decomposition (<apll>πR</apll>)/number theoretic (<apll>LπR</apll>) functions. In these cases, fractional or VFP right arguments are converted to integers or rational integers, respectively, which is the datatype of the result except for <apll>0πR</apll> (Primality Test) which always returns a Boolean result regardless of the type of <apll>R</apll>.</p>		<p>Two special functions are the prime decomposition (<apll>πR</apll>)/number theoretic (<apll>LπR</apll>) functions. In these cases, fractional or VFP right arguments are converted to integers or rational integers, respectively, which is the datatype of the result except for <apll>0πR</apll> (Primality Test) which always returns a Boolean result regardless of the type of <apll>R</apll>.</p>
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	<li><p>Comparisons between a rational number and a floating point number convert both arguments to VFP numbers and compare the two as below.</p></li>		<li><p>Comparisons between a rational number and a floating point number convert both arguments to VFP numbers and compare the two as below.</p></li>

	<li><p>Comparisons between a VFP number and any other number is sensitive to <apll>⎕CT</apll> — just as they are between ~~floats~~.</p></li>		<li><p>Comparisons between a VFP number and any other number is sensitive to the current setting of Comparison Tolerance (<apll>⎕CT</apll>) — just as they are between floating point numbers.</p></li>

	</ul>		</ul>
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	<ul>		<ul>
	<li><p>Both roll (<apll>?R</apll>) and deal (<apll>L?R</apll>) on rational integers use a built-in random number generator so as to use the entire range of rational integers – this algorithm uses its own internal seeds that are much more complicated than the simple integer seed that is <apll>⎕RL</apll>. Thus <apll>⎕RL</apll> is unchanged by these functions on rationals.</p>		<li><p>Both roll (<apll>?R</apll>) and deal (<apll>L?R</apll>) on rational integers use a built-in random number generator so as to use the entire range of rational integers – this algorithm uses its own internal seeds that are much more complicated than the simple integer seed that is <apll>⎕RL</apll> (Random Link). Thus <apll>⎕RL</apll> is unchanged by these functions on rationals.</p>

	<p>For example, if you need <b>really</b> large random numbers</p>		<p>For example, if you need <b>really</b> large random numbers</p>