<!doctype html>
<html>
<head>
<title>Calculator Arithmetic Overview</title>
<meta charset="UTF-8">
<style>
#toc {
  width:300px;
  border:1px solid #ccc;
  background-color:#efefef;
  float:right;
}
.display {
  color:#666666;
  background-color:#f3f3f3;
}
</style>
</head>
<body onload="init();">
<div id="toc"></div>
<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>

</body>
</html>
<script type="text/javascript">
function generateTOC (rootNode, startLevel) {
  var lastLevel = 0;
  startLevel = startLevel || 2;
  var html = "<ul>";

  for (var i = 0; i < rootNode.childNodes.length; ++i) {
    var node = rootNode.childNodes[i];
    if (!node.tagName || !/H[1-6]/.test(node.tagName)) {
      continue;
    }
    var level = +node.tagName.substr(1);
    if (level < startLevel) { continue; }
    var name = node.innerText;
    if (node.children.length) { name = node.childNodes[0].innerText; }
    if (!name) { continue; }
    var hashable = name.replace(/[.\s\']/g, "-");
    node.id = hashable;
    if (level > lastLevel) {
      html += "";
    } else if (level < lastLevel) {
      html += (new Array(lastLevel - level + 2)).join("</ul></li>");
    } else {
      html += "</ul></li>";
    }
    html += "<li><a class='lvl"+level+"' href='#" + hashable + "'>" + name + "</a><ul>";
    lastLevel = level;
  }

  html += "</ul>";
  return html;
}

function init() {
  document.getElementById("toc").innerHTML = generateTOC(document.body);
}
</script>