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diff --git a/docs/arithmetic-overview.html b/docs/arithmetic-overview.html new file mode 100644 index 0000000..68dd24e --- /dev/null +++ b/docs/arithmetic-overview.html @@ -0,0 +1,271 @@ +<!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> diff --git a/docs/implementation-overview.html b/docs/implementation-overview.html new file mode 100644 index 0000000..a06e73b --- /dev/null +++ b/docs/implementation-overview.html @@ -0,0 +1,51 @@ +<!doctype html> +<html> +<head> +<title>Calculator Implementation Overview</title> +<meta charset="UTF-8"> +</head> +<h1>M Calculator Implementation Overview</h1> +<p>Although the appearance of the calculator has changed little from Lollipop, and some of the UI +code is indeed the same, the rest of the code has changed substantially. Unsurprisingly, +<b>Calculator.java</b> implements the main UI. The other major parts of the implementation +are:</p> + +<p><b>CR.java</b> in <b>external/crcalc</b> provides the underlying demand-driven ("constructive +real") arithmetic implementation. Numbers are represented primarily as objects with a method that +can compute arbitrarily precise approximations. The actual arithmetic performed by these methods +is based on Java's <tt>java.util.BigInteger</tt> arithmetic, with appropriate implicit +scaling.</p> + +<p><b>BoundedRational.java</b> is a rational arithmetic package that is used to provide finite +exact answers in "easy" cases. It is used primarily to determine when an approximation provided +by CR.java is actually exact. This is used in turn both to limit the length of displayed results +and scrolling, as well as to identify errors such as division by zero, that would otherwise result +in timeouts during computations. It is in some sense not needed to produce correct results, but +it significantly improves the usability of the calculator. It is also used for the "display as +fraction" option in the overflow menu.</p> + +<p><b>CalculatorExpr.java</b> implements calculator arithmetic expressions. It supports editing, +saving, restoring, and evaluation of expressions. Evaluation produces a constructive real (CR) +and possibly a BoundedRational result. Unlike the "arity" library used in earlier versions, the +underlying expression is represented as a sequence of "tokens", many of which are represented by +Button ids, not as a character string.</p> + +<p><b>Evaluator.java</b> implements much of the actual calculator logic, particularly background +expression evaluation. Expression evaluation here includes both using CalculatorExpr.java to +evaluate the expression, and then invoking the resulting CR value to actually produce finite +approximations and convert them to decimal. Two types of expression evaluation are supported: +(1) Initial evaluation of the expression and producing an initial decimal approximation, and (2) +reevaluation to higher precision. (1) is invoked directly from the Calculator UI, while (2) is +invoked from the calculator display, commonly in response to scrolling. When the display requests +a result, a "result" is immediately returned, though it may contains blank placeholders. The +display is then notified when the real result becomes available.</p> + +<p><b>CalculatorText.java</b> is the TextView subclass used to display the formula.</p> + +<p><b>CalculatorResult.java</b> is the TextView subclass used to display the result. It handles +result formatting, scrolling, etc. After the user hits "=", the CalculatorResult widget moves +into the top position, replacing the formula display. Currently it remains in that position until +the formula is again modified.</p> +</body> +</html> + |