docs/arithmetic-overview.html - platform/packages/apps/ExactCalculator - Gitiles

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 <h1>Arithmetic in the Android M Calculator</h1>
 <p>Most conventional calculators, both the specialized hardware and software varieties, represent
 numbers using fairly conventional machine floating point arithmetic. Each number is stored as an
 exponent, identifying the position of the decimal point, together with the first 10 to 20
 significant digits of the number. For example, 1/300 might be stored as
 0.333333333333x10<sup>-2</sup>, i.e. as an exponent of -2, together with the 12 most significant
 digits. This is similar, and sometimes identical to, computer arithmetic used to solve large
 scale scientific problems.</p> <p>This kind of arithmetic works well most of the time, but can
 sometimes produce completely incorrect results. For example, the trigonometric tangent (tan) and
 arctangent (tan<sup>-1</sup>) functions are defined so that tan(tan<sup>-1</sup>(<i>x</i>)) should
 always be <i>x</i>. But on most calculators we have tried, tan(tan<sup>-1</sup>(10<sup>20</sup>))
 is off by at least a factor of 1000. A value around 10<sup>16</sup> or 10<sup>17</sup> is quite
 popular, which unfortunately doesn't make it correct. The underlying problem is that
 tan<sup>-1</sup>(10<sup>17</sup>) and tan<sup>-1</sup>(10<sup>20</sup>) are so close that
 conventional representations don't distinguish them. (They're both 89.9999… degrees with at least
 fifteen 9s beyond the decimal point.) But the tiny difference between them results in a huge
 difference when the tangent function is applied to the result.</p>

 <p>Similarly, it may be puzzling to a high school student that while the textbook claims that for
 any <i>x</i>, sin(<i>x</i>) + sin(<i>x</i>+π) = 0, their calculator says that sin(10<sup>15</sup>)
 + sin(10<sup>15</sup>+π) = <span class="display">-0.00839670971</span>. (Thanks to floating point
 standardization, multiple on-line calculators agree on that entirely bogus value!)</p>

 <p>We know that the instantaneous rate of change of a function f, its derivative, can be
 approximated at a point <i>x</i> by computing (<i>f</i>(<i>x</i> + <i>h</i>) - <i>f</i>(<i>x</i>))
 / <i>h</i>, for very small <i>h</i>. Yet, if you try this in a conventional calculator with
 <i>h</i> = 10<sup>-20</sup> or smaller, you are unlikely to get a useful answer.</p>

 <p>In general these problems occur when computations amplify tiny errors, a problem referred to as
 numerical instability. This doesn't happen very often, but as in the above examples, it may
 require some insight to understand when it can and can't happen.</p>

 <p>In large scale scientific computations, hardware floating point computations are essential
 since they are the only reasonable way modern computer hardware can produce answers with
 sufficient speed. Experts must be careful to structure computations to avoid such problems. But
 for "computing in the small" problems, like those solved on desk calculators, we can do much
 better!</p>

 <h2>Producing accurate answers</h2>
 <p>The Android M Calculator uses a different kind of computer arithmetic. Rather than computing a
 fixed number of digits for each intermediate result, the computation is much more goal directed.
 The user would like to see only correct digits on the display, which we take to mean that the
 displayed answer should always be off by less than one in the last displayed digit. The
 computation is thus performed to whatever precision is required to achieve that.</p>

 <p>Let's say we want to compute π+⅓, and the calculator display has 10 digits. We'd compute both π
 and ⅓ to 11 digits each, add them, and round the result to 10 digits. Since π and ⅓ were accurate
 to within 1 in the 11<sup>th</sup> digit, and rounding adds an error of at most 5 in the
 11<sup>th</sup> digit, the result is guaranteed accurate to less than 1 in the 10<sup>th</sup>
 digit, which was our goal.</p>

 <p>This is of course an oversimplification of the real implementation. Operations other than
 addition do get appreciably more complicated. Multiplication, for example, requires that we
 approximate one argument in order to determine how much precision we need for the other argument.
 The tangent function requires very high precision for arguments near 90 degrees to produce
 meaningful answers. And so on. And we really use binary rather than decimal arithmetic.
 Nonetheless the above addition method is a good illustration of the approach.</p>

 <p>Since we have to be able to produce answers to arbitrary precision, we can also let the user
 specify how much precision she wants, and use that as our goal. In the Android M Calculator, the
 user specifies the requested precision by scrolling the result. As the result is being scrolled,
 the calculator reevaluates it to the newly requested precision. In some cases, the algorithm for
 computing the new higher precision result takes advantage of the old, less accurate result. In
 other cases, it basically starts from scratch. Fortunately modern devices and the Android runtime
 are fast enough that the recomputation delay rarely becomes visible.</p>

 <h2>Design Decisions and challenges</h2>
 <p>This form of evaluate-on-demand arithmetic has occasionally been used before, and we use a
 refinement of a previously developed open source package in our implementation. However, the
 scrolling interface, together with the practicailities of a usable general purpose calculator,
 presented some new challenges. These drove a number of not-always-obvious design decisions which
 briefly describe here.</p>

 <h3>Indicating position</h3>
 <p>We would like the user to be able to see at a glance which part of the result is currently
 being displayed.</p>

 <p>Conventional calculators solve the vaguely similar problem of displaying very large or very
 small numbers by using scientific notation: They display an exponent in addition to the most
 significant digits, analogously to the internal representation they use. We solve that problem in
 exactly the same way, in spite of our different internal representation. If the user enters
 "1÷3⨉10^20", computing ⅓ times 10 to the 20th power, the result may be displayed as <span
 class="display">3.3333333333E19</span>, indicating that the result is approximately 3.3333333333
 times 10<sup>19</sup>. In this version of scientific notation, the decimal point is always
 displayed immediately to the right of the most significant digit, and the exponent indicates where
 it really belongs.</p>

 <p>Once the decimal point is scrolled off the display, this style of scientific notation is not
 helpful; it essentially tells us where the decimal point is relative to the most significant
 digit, but the most significant digit is no longer visible. We address this by switching to a
 different variant of scientific notation, in which we interpret the displayed digits as a whole
 number, with an implied decimal point on the right. Instead of displaying <span
 class="display">3.3333333333E19</span>, we hypothetically could display <span
 class="display">33333333333E9</span> or 33333333333 times 10<sup>9</sup>. In fact, we use this
 format only when the normal scientific notation decimal point would not be visible. If we had
 scrolled the above result 2 digits to the left, we would in fact be seeing <span
 ass="display">...33333333333E7</span>. This tells us that the displayed result is very close to a
 whole number ending in 33333333333 times 10<sup>7</sup>. Effectively the <span
 class="display">E7</span> is telling us that the last displayed digit corresponds to the ten
 millions position. In this form, the exponent does tell us the current position in the result.
 The two forms are easily distinguishable by the presence or absence of a decimal point, and the
 ellipsis character at the beginning.</p>

 <h3>Rounding vs. scrolling</h3>
 <p>Normally we expect calculators to try to round to the nearest displayable result. If the
 actual computed result were 0.66666666666667, and we could only display 10 digits, we would expect
 a result display of, for example <span class="display">0.666666667</span>, rather than <span
 class="display">0.666666666</span>. For us, this would have the disadvantage that when we
 scrolled the result left to see more digits, the "7" on the right would change to a "6". That
 would be mildly unfortunate. It would be somewhat worse that if the actual result were exactly
 0.99999999999, and we could only display 10 characters at a time, we would see an initial display
 of <span class="display">1.00000000</span>. As we scroll to see more digits, we would
 successively see <span class="display">...000000E-6</span>, then <span
 class="display">...000000E-7</span>, and so on until we get to <span
 class="display">...00000E-10</span>, but then suddenly <span class="display">...99999E-11</span>.
 If we scroll back, the screen would again show zeroes. We decided this would be excessively
 confusing, and thus do not round.</p>

 <p>It is still possible for previously displayed digits to change as we're scrolling. But we
 always compute a number of digits more than we actually need, so this is exceedingly unlikely.</p>

 <p>Since our goal is an error of strictly less than one in the last displayed digit, we will
 never, for example, display an answer of exactly 2 as <span class="display">1.9999999999</span>.
 That would involve an error of exactly one in the last place, which is too much for us.</p> <p>It
 turns out that there is exactly one case in which the display switches between 9s and 0s: A long
 but finite sequence of 9s (more than 20) in the true result can initially be displayed as a larger
 number ending in 0s. As we scroll, the 0s turn into 9s. When we immediately scroll back, the
 number remains displayed as 9s, since the calculator caches the best known result (though not
 currently across restarts or screen rotations).</p>

 <p>We prevent 9s from turning into 0s during scrolling. If we generate a result ending in 9s, our
 error bound implies that the true result is strictly less (in absolute value) than the value
 (ending in 0s) we would get by incrementing the last displayed digit. Thus we can never be forced
 back to generating zeros and will always continue to generate 9s.</p>

 <h3>Coping with mathematical limits</h3>
 <p>Internally the calculator essentially represents a number as a program for computing however
 many digits we happen to need. This representation has many nice properties, like never resulting
 in the display of incorrect results. It has one inherent weakness: We provably cannot compute
 precisely whether two numbers are equal. We can compute more and more digits of both numbers, and
 if they ever differ by more than one in the last computed digit, we know they are <i>not</i>
 equal. But if the two numbers were in fact the same, this process will go on forever.</p>

 <p>This is still better than machine floating point arithmetic, though machine floating point
 better obscures the problem. With machine floating point arithmetic, two computations that should
 mathematically have given the same answer, may give us substantially different answers, and two
 computations that should have given us different answers may easily produce the same one. We
 can indeed determine whether the floating representations are equal, but this tells us little
 about equality of the true mathematical answers.</p>

 <p>The undecidability of equality creates some interesting issues. If we divide a number by
 <i>x</i>, the calculator will compute more and more digits of <i>x</i> until it finds some nonzero
 ones. If <i>x</i> was in fact exactly zero, this process will continue forever.</p> <p>We deal
 with this problem using two complementary techniques:</p>

 <ol>
 <li>We always run numeric computations in the background, where they won't interfere with user
 interactions, just in case they take a long time. If they do take a really long time, we time
 them out and inform the user that the computation has been aborted. This is unlikely to happen by
 accident, unless the user entered an ill-defined mathematical expression, like a division by
 zero.</li>
 <li>As we will see below, in many cases we use an additional number representation that does allow
 us to determine that a number is exactly zero. Although this easily handles most cases, it is not
 foolproof. If the user enters "1÷0" we immediately detect the division by zero. If the user
 enters "1÷(π−π)" we time out. (We might choose to explicitly recognize such simple cases in the
 future. But this would always remain a heuristic.)</li>
 </ol>

 <h3>Zeros further than the eye can see</h3>
 <p>Prototypes of the M calculator, like mathematicians, treated all real numbers as infinite
 objects, with infinitely many digits to scroll through. If the actual computation happened to be
 2−1, the result was initially displayed as <span class="display">1.00000000</span>, and the user
 could keep scrolling through as many thousands of zeroes to the right of that as he desired.
 Although mathematically sound, this proved unpopular for several good reasons, the first one
 probably more serious than the others:</p>

 <ol>
 <li>If we computed $1.23 + $7.89, the result would show up as <span
 class="display">9.1200000000</span> or the like, which is unexpected and harder to read quickly
 than <span class="display">9.12</span>.</li>
 <li>Many users consider the result of 2-1 to be a finite number, and find it confusing to be able
 to scroll through lots of zeros on the right.</li>
 <li>Since the calculator couldn't ever tell that a number wasn't going to be scrolled, it couldn't
 treat any result as short enough to allow the use of a larger font.</li>
 </ol>

 <p>As a result, the calculator now also tries to compute the result as an exact fraction whenever
 that is easily possible. It is then easy to tell from the fraction whether a number has a finite
 decimal expansion. If it does, we prevent scrolling past that point, and may use the fact that
 the result has a short representation to increase the font size. Results displayed in a larger
 font are not scrollable. We no longer display any zeros for non-zero results unless there is
 either a nonzero or a displayed decimal point to the right. The fact that a result is not
 scrollable tells the user that the result, as displayed, is exact. This is fallible in the other
 direction. For example, we do not compute a rational representation for π−π, and hence it is
 still possible to scroll through as many zeros of that result as you like.</p>

 <p>This underlying fractional representation of the result is also used to detect, for example,
 division by zero without a timeout.</p>

 <p>Since we calculate the fractional result when we can in any case, it is also now available to
 the user through the overflow menu.</p>

 <h2>More details</h2>
 <p>The underlying evaluate-on-demand arithmetic package is described in H. Boehm, "The
 Constructive Reals as a Java Library'', Special issue on practical development of exact real
 number computation, <i>Journal of Logic and Algebraic Programming 64</i>, 1, July 2005, pp. 3-11.
 (Also at <a href="http://www.hpl.hp.com/techreports/2004/HPL-2004-70.html">http://www.hpl.hp.com/techreports/2004/HPL-2004-70.html</a>)</p>

 <p>Our version has been slightly refined. Notably it calculates inverse trigonometric functions
 directly instead of using a generic "inverse" function. This is less elegant, but significantly
 improves performance.</p>

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	<h1>Arithmetic in the Android M Calculator</h1>
	<p>Most conventional calculators, both the specialized hardware and software varieties, represent
	numbers using fairly conventional machine floating point arithmetic. Each number is stored as an
	exponent, identifying the position of the decimal point, together with the first 10 to 20
	significant digits of the number. For example, 1/300 might be stored as
	0.333333333333x10<sup>-2</sup>, i.e. as an exponent of -2, together with the 12 most significant
	digits. This is similar, and sometimes identical to, computer arithmetic used to solve large
	scale scientific problems.</p> <p>This kind of arithmetic works well most of the time, but can
	sometimes produce completely incorrect results. For example, the trigonometric tangent (tan) and
	arctangent (tan<sup>-1</sup>) functions are defined so that tan(tan<sup>-1</sup>(<i>x</i>)) should
	always be <i>x</i>. But on most calculators we have tried, tan(tan<sup>-1</sup>(10<sup>20</sup>))
	is off by at least a factor of 1000. A value around 10<sup>16</sup> or 10<sup>17</sup> is quite
	popular, which unfortunately doesn't make it correct. The underlying problem is that
	tan<sup>-1</sup>(10<sup>17</sup>) and tan<sup>-1</sup>(10<sup>20</sup>) are so close that
	conventional representations don't distinguish them. (They're both 89.9999… degrees with at least
	fifteen 9s beyond the decimal point.) But the tiny difference between them results in a huge
	difference when the tangent function is applied to the result.</p>

	<p>Similarly, it may be puzzling to a high school student that while the textbook claims that for
	any <i>x</i>, sin(<i>x</i>) + sin(<i>x</i>+π) = 0, their calculator says that sin(10<sup>15</sup>)
	+ sin(10<sup>15</sup>+π) = <span class="display">-0.00839670971</span>. (Thanks to floating point
	standardization, multiple on-line calculators agree on that entirely bogus value!)</p>

	<p>We know that the instantaneous rate of change of a function f, its derivative, can be
	approximated at a point <i>x</i> by computing (<i>f</i>(<i>x</i> + <i>h</i>) - <i>f</i>(<i>x</i>))
	/ <i>h</i>, for very small <i>h</i>. Yet, if you try this in a conventional calculator with
	<i>h</i> = 10<sup>-20</sup> or smaller, you are unlikely to get a useful answer.</p>

	<p>In general these problems occur when computations amplify tiny errors, a problem referred to as
	numerical instability. This doesn't happen very often, but as in the above examples, it may
	require some insight to understand when it can and can't happen.</p>

	<p>In large scale scientific computations, hardware floating point computations are essential
	since they are the only reasonable way modern computer hardware can produce answers with
	sufficient speed. Experts must be careful to structure computations to avoid such problems. But
	for "computing in the small" problems, like those solved on desk calculators, we can do much
	better!</p>

	<h2>Producing accurate answers</h2>
	<p>The Android M Calculator uses a different kind of computer arithmetic. Rather than computing a
	fixed number of digits for each intermediate result, the computation is much more goal directed.
	The user would like to see only correct digits on the display, which we take to mean that the
	displayed answer should always be off by less than one in the last displayed digit. The
	computation is thus performed to whatever precision is required to achieve that.</p>

	<p>Let's say we want to compute π+⅓, and the calculator display has 10 digits. We'd compute both π
	and ⅓ to 11 digits each, add them, and round the result to 10 digits. Since π and ⅓ were accurate
	to within 1 in the 11<sup>th</sup> digit, and rounding adds an error of at most 5 in the
	11<sup>th</sup> digit, the result is guaranteed accurate to less than 1 in the 10<sup>th</sup>
	digit, which was our goal.</p>

	<p>This is of course an oversimplification of the real implementation. Operations other than
	addition do get appreciably more complicated. Multiplication, for example, requires that we
	approximate one argument in order to determine how much precision we need for the other argument.
	The tangent function requires very high precision for arguments near 90 degrees to produce
	meaningful answers. And so on. And we really use binary rather than decimal arithmetic.
	Nonetheless the above addition method is a good illustration of the approach.</p>

	<p>Since we have to be able to produce answers to arbitrary precision, we can also let the user
	specify how much precision she wants, and use that as our goal. In the Android M Calculator, the
	user specifies the requested precision by scrolling the result. As the result is being scrolled,
	the calculator reevaluates it to the newly requested precision. In some cases, the algorithm for
	computing the new higher precision result takes advantage of the old, less accurate result. In
	other cases, it basically starts from scratch. Fortunately modern devices and the Android runtime
	are fast enough that the recomputation delay rarely becomes visible.</p>

	<h2>Design Decisions and challenges</h2>
	<p>This form of evaluate-on-demand arithmetic has occasionally been used before, and we use a
	refinement of a previously developed open source package in our implementation. However, the
	scrolling interface, together with the practicailities of a usable general purpose calculator,
	presented some new challenges. These drove a number of not-always-obvious design decisions which
	briefly describe here.</p>

	<h3>Indicating position</h3>
	<p>We would like the user to be able to see at a glance which part of the result is currently
	being displayed.</p>

	<p>Conventional calculators solve the vaguely similar problem of displaying very large or very
	small numbers by using scientific notation: They display an exponent in addition to the most
	significant digits, analogously to the internal representation they use. We solve that problem in
	exactly the same way, in spite of our different internal representation. If the user enters
	"1÷3⨉10^20", computing ⅓ times 10 to the 20th power, the result may be displayed as <span
	class="display">3.3333333333E19</span>, indicating that the result is approximately 3.3333333333
	times 10<sup>19</sup>. In this version of scientific notation, the decimal point is always
	displayed immediately to the right of the most significant digit, and the exponent indicates where
	it really belongs.</p>

	<p>Once the decimal point is scrolled off the display, this style of scientific notation is not
	helpful; it essentially tells us where the decimal point is relative to the most significant
	digit, but the most significant digit is no longer visible. We address this by switching to a
	different variant of scientific notation, in which we interpret the displayed digits as a whole
	number, with an implied decimal point on the right. Instead of displaying <span
	class="display">3.3333333333E19</span>, we hypothetically could display <span
	class="display">33333333333E9</span> or 33333333333 times 10<sup>9</sup>. In fact, we use this
	format only when the normal scientific notation decimal point would not be visible. If we had
	scrolled the above result 2 digits to the left, we would in fact be seeing <span
	ass="display">...33333333333E7</span>. This tells us that the displayed result is very close to a
	whole number ending in 33333333333 times 10<sup>7</sup>. Effectively the <span
	class="display">E7</span> is telling us that the last displayed digit corresponds to the ten
	millions position. In this form, the exponent does tell us the current position in the result.
	The two forms are easily distinguishable by the presence or absence of a decimal point, and the
	ellipsis character at the beginning.</p>

	<h3>Rounding vs. scrolling</h3>
	<p>Normally we expect calculators to try to round to the nearest displayable result. If the
	actual computed result were 0.66666666666667, and we could only display 10 digits, we would expect
	a result display of, for example <span class="display">0.666666667</span>, rather than <span
	class="display">0.666666666</span>. For us, this would have the disadvantage that when we
	scrolled the result left to see more digits, the "7" on the right would change to a "6". That
	would be mildly unfortunate. It would be somewhat worse that if the actual result were exactly
	0.99999999999, and we could only display 10 characters at a time, we would see an initial display
	of <span class="display">1.00000000</span>. As we scroll to see more digits, we would
	successively see <span class="display">...000000E-6</span>, then <span
	class="display">...000000E-7</span>, and so on until we get to <span
	class="display">...00000E-10</span>, but then suddenly <span class="display">...99999E-11</span>.
	If we scroll back, the screen would again show zeroes. We decided this would be excessively
	confusing, and thus do not round.</p>

	<p>It is still possible for previously displayed digits to change as we're scrolling. But we
	always compute a number of digits more than we actually need, so this is exceedingly unlikely.</p>

	<p>Since our goal is an error of strictly less than one in the last displayed digit, we will
	never, for example, display an answer of exactly 2 as <span class="display">1.9999999999</span>.
	That would involve an error of exactly one in the last place, which is too much for us.</p> <p>It
	turns out that there is exactly one case in which the display switches between 9s and 0s: A long
	but finite sequence of 9s (more than 20) in the true result can initially be displayed as a larger
	number ending in 0s. As we scroll, the 0s turn into 9s. When we immediately scroll back, the
	number remains displayed as 9s, since the calculator caches the best known result (though not
	currently across restarts or screen rotations).</p>

	<p>We prevent 9s from turning into 0s during scrolling. If we generate a result ending in 9s, our
	error bound implies that the true result is strictly less (in absolute value) than the value
	(ending in 0s) we would get by incrementing the last displayed digit. Thus we can never be forced
	back to generating zeros and will always continue to generate 9s.</p>

	<h3>Coping with mathematical limits</h3>
	<p>Internally the calculator essentially represents a number as a program for computing however
	many digits we happen to need. This representation has many nice properties, like never resulting
	in the display of incorrect results. It has one inherent weakness: We provably cannot compute
	precisely whether two numbers are equal. We can compute more and more digits of both numbers, and
	if they ever differ by more than one in the last computed digit, we know they are <i>not</i>
	equal. But if the two numbers were in fact the same, this process will go on forever.</p>

	<p>This is still better than machine floating point arithmetic, though machine floating point
	better obscures the problem. With machine floating point arithmetic, two computations that should
	mathematically have given the same answer, may give us substantially different answers, and two
	computations that should have given us different answers may easily produce the same one. We
	can indeed determine whether the floating representations are equal, but this tells us little
	about equality of the true mathematical answers.</p>

	<p>The undecidability of equality creates some interesting issues. If we divide a number by
	<i>x</i>, the calculator will compute more and more digits of <i>x</i> until it finds some nonzero
	ones. If <i>x</i> was in fact exactly zero, this process will continue forever.</p> <p>We deal
	with this problem using two complementary techniques:</p>

	<ol>
	<li>We always run numeric computations in the background, where they won't interfere with user
	interactions, just in case they take a long time. If they do take a really long time, we time
	them out and inform the user that the computation has been aborted. This is unlikely to happen by
	accident, unless the user entered an ill-defined mathematical expression, like a division by
	zero.</li>
	<li>As we will see below, in many cases we use an additional number representation that does allow
	us to determine that a number is exactly zero. Although this easily handles most cases, it is not
	foolproof. If the user enters "1÷0" we immediately detect the division by zero. If the user
	enters "1÷(π−π)" we time out. (We might choose to explicitly recognize such simple cases in the
	future. But this would always remain a heuristic.)</li>
	</ol>

	<h3>Zeros further than the eye can see</h3>
	<p>Prototypes of the M calculator, like mathematicians, treated all real numbers as infinite
	objects, with infinitely many digits to scroll through. If the actual computation happened to be
	2−1, the result was initially displayed as <span class="display">1.00000000</span>, and the user
	could keep scrolling through as many thousands of zeroes to the right of that as he desired.
	Although mathematically sound, this proved unpopular for several good reasons, the first one
	probably more serious than the others:</p>

	<ol>
	<li>If we computed $1.23 + $7.89, the result would show up as <span
	class="display">9.1200000000</span> or the like, which is unexpected and harder to read quickly
	than <span class="display">9.12</span>.</li>
	<li>Many users consider the result of 2-1 to be a finite number, and find it confusing to be able
	to scroll through lots of zeros on the right.</li>
	<li>Since the calculator couldn't ever tell that a number wasn't going to be scrolled, it couldn't
	treat any result as short enough to allow the use of a larger font.</li>
	</ol>

	<p>As a result, the calculator now also tries to compute the result as an exact fraction whenever
	that is easily possible. It is then easy to tell from the fraction whether a number has a finite
	decimal expansion. If it does, we prevent scrolling past that point, and may use the fact that
	the result has a short representation to increase the font size. Results displayed in a larger
	font are not scrollable. We no longer display any zeros for non-zero results unless there is
	either a nonzero or a displayed decimal point to the right. The fact that a result is not
	scrollable tells the user that the result, as displayed, is exact. This is fallible in the other
	direction. For example, we do not compute a rational representation for π−π, and hence it is
	still possible to scroll through as many zeros of that result as you like.</p>

	<p>This underlying fractional representation of the result is also used to detect, for example,
	division by zero without a timeout.</p>

	<p>Since we calculate the fractional result when we can in any case, it is also now available to
	the user through the overflow menu.</p>

	<h2>More details</h2>
	<p>The underlying evaluate-on-demand arithmetic package is described in H. Boehm, "The
	Constructive Reals as a Java Library'', Special issue on practical development of exact real
	number computation, <i>Journal of Logic and Algebraic Programming 64</i>, 1, July 2005, pp. 3-11.
	(Also at <a href="http://www.hpl.hp.com/techreports/2004/HPL-2004-70.html">http://www.hpl.hp.com/techreports/2004/HPL-2004-70.html</a>)</p>

	<p>Our version has been slightly refined. Notably it calculates inverse trigonometric functions
	directly instead of using a generic "inverse" function. This is less elegant, but significantly
	improves performance.</p>

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