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21<h1>Arithmetic in the Android M Calculator</h1>
22<p>Most conventional calculators, both the specialized hardware and software varieties, represent
23numbers using fairly conventional machine floating point arithmetic. Each number is stored as an
24exponent, identifying the position of the decimal point, together with the first 10 to 20
25significant digits of the number. For example, 1/300 might be stored as
260.333333333333x10<sup>-2</sup>, i.e. as an exponent of -2, together with the 12 most significant
27digits. This is similar, and sometimes identical to, computer arithmetic used to solve large
28scale scientific problems.</p> <p>This kind of arithmetic works well most of the time, but can
29sometimes produce completely incorrect results. For example, the trigonometric tangent (tan) and
30arctangent (tan<sup>-1</sup>) functions are defined so that tan(tan<sup>-1</sup>(<i>x</i>)) should
31always be <i>x</i>. But on most calculators we have tried, tan(tan<sup>-1</sup>(10<sup>20</sup>))
32is off by at least a factor of 1000. A value around 10<sup>16</sup> or 10<sup>17</sup> is quite
33popular, which unfortunately doesn't make it correct. The underlying problem is that
34tan<sup>-1</sup>(10<sup>17</sup>) and tan<sup>-1</sup>(10<sup>20</sup>) are so close that
35conventional representations don't distinguish them. (They're both 89.9999… degrees with at least
36fifteen 9s beyond the decimal point.) But the tiny difference between them results in a huge
37difference when the tangent function is applied to the result.</p>
38
39<p>Similarly, it may be puzzling to a high school student that while the textbook claims that for
40any <i>x</i>, sin(<i>x</i>) + sin(<i>x</i>+π) = 0, their calculator says that sin(10<sup>15</sup>)
41+ sin(10<sup>15</sup>+π) = <span class="display">-0.00839670971</span>. (Thanks to floating point
42standardization, multiple on-line calculators agree on that entirely bogus value!)</p>
43
44<p>We know that the instantaneous rate of change of a function f, its derivative, can be
45approximated at a point <i>x</i> by computing (<i>f</i>(<i>x</i> + <i>h</i>) - <i>f</i>(<i>x</i>))
46/ <i>h</i>, for very small <i>h</i>. Yet, if you try this in a conventional calculator with
47<i>h</i> = 10<sup>-20</sup> or smaller, you are unlikely to get a useful answer.</p>
48
49<p>In general these problems occur when computations amplify tiny errors, a problem referred to as
50numerical instability. This doesn't happen very often, but as in the above examples, it may
51require some insight to understand when it can and can't happen.</p>
52
53<p>In large scale scientific computations, hardware floating point computations are essential
54since they are the only reasonable way modern computer hardware can produce answers with
55sufficient speed. Experts must be careful to structure computations to avoid such problems. But
56for "computing in the small" problems, like those solved on desk calculators, we can do much
57better!</p>
58
59<h2>Producing accurate answers</h2>
60<p>The Android M Calculator uses a different kind of computer arithmetic. Rather than computing a
61fixed number of digits for each intermediate result, the computation is much more goal directed.
62The user would like to see only correct digits on the display, which we take to mean that the
63displayed answer should always be off by less than one in the last displayed digit. The
64computation is thus performed to whatever precision is required to achieve that.</p>
65
66<p>Let's say we want to compute π+⅓, and the calculator display has 10 digits. We'd compute both π
67and ⅓ to 11 digits each, add them, and round the result to 10 digits. Since π and ⅓ were accurate
68to within 1 in the 11<sup>th</sup> digit, and rounding adds an error of at most 5 in the
6911<sup>th</sup> digit, the result is guaranteed accurate to less than 1 in the 10<sup>th</sup>
70digit, which was our goal.</p>
71
72<p>This is of course an oversimplification of the real implementation. Operations other than
73addition do get appreciably more complicated. Multiplication, for example, requires that we
74approximate one argument in order to determine how much precision we need for the other argument.
75The tangent function requires very high precision for arguments near 90 degrees to produce
76meaningful answers. And so on. And we really use binary rather than decimal arithmetic.
77Nonetheless the above addition method is a good illustration of the approach.</p>
78
79<p>Since we have to be able to produce answers to arbitrary precision, we can also let the user
80specify how much precision she wants, and use that as our goal. In the Android M Calculator, the
81user specifies the requested precision by scrolling the result. As the result is being scrolled,
82the calculator reevaluates it to the newly requested precision. In some cases, the algorithm for
83computing the new higher precision result takes advantage of the old, less accurate result. In
84other cases, it basically starts from scratch. Fortunately modern devices and the Android runtime
85are fast enough that the recomputation delay rarely becomes visible.</p>
86
87<h2>Design Decisions and challenges</h2>
88<p>This form of evaluate-on-demand arithmetic has occasionally been used before, and we use a
89refinement of a previously developed open source package in our implementation. However, the
90scrolling interface, together with the practicailities of a usable general purpose calculator,
91presented some new challenges. These drove a number of not-always-obvious design decisions which
92briefly describe here.</p>
93
94<h3>Indicating position</h3>
95<p>We would like the user to be able to see at a glance which part of the result is currently
96being displayed.</p>
97
98<p>Conventional calculators solve the vaguely similar problem of displaying very large or very
99small numbers by using scientific notation: They display an exponent in addition to the most
100significant digits, analogously to the internal representation they use. We solve that problem in
101exactly the same way, in spite of our different internal representation. If the user enters
102"1÷3⨉10^20", computing ⅓ times 10 to the 20th power, the result may be displayed as <span
103class="display">3.3333333333E19</span>, indicating that the result is approximately 3.3333333333
104times 10<sup>19</sup>. In this version of scientific notation, the decimal point is always
105displayed immediately to the right of the most significant digit, and the exponent indicates where
106it really belongs.</p>
107
108<p>Once the decimal point is scrolled off the display, this style of scientific notation is not
109helpful; it essentially tells us where the decimal point is relative to the most significant
110digit, but the most significant digit is no longer visible. We address this by switching to a
111different variant of scientific notation, in which we interpret the displayed digits as a whole
112number, with an implied decimal point on the right. Instead of displaying <span
113class="display">3.3333333333E19</span>, we hypothetically could display <span
114class="display">33333333333E9</span> or 33333333333 times 10<sup>9</sup>. In fact, we use this
115format only when the normal scientific notation decimal point would not be visible. If we had
116scrolled the above result 2 digits to the left, we would in fact be seeing <span
117ass="display">...33333333333E7</span>. This tells us that the displayed result is very close to a
118whole number ending in 33333333333 times 10<sup>7</sup>. Effectively the <span
119class="display">E7</span> is telling us that the last displayed digit corresponds to the ten
120millions position. In this form, the exponent does tell us the current position in the result.
121The two forms are easily distinguishable by the presence or absence of a decimal point, and the
122ellipsis character at the beginning.</p>
123
124<h3>Rounding vs. scrolling</h3>
125<p>Normally we expect calculators to try to round to the nearest displayable result. If the
126actual computed result were 0.66666666666667, and we could only display 10 digits, we would expect
127a result display of, for example <span class="display">0.666666667</span>, rather than <span
128class="display">0.666666666</span>. For us, this would have the disadvantage that when we
129scrolled the result left to see more digits, the "7" on the right would change to a "6". That
130would be mildly unfortunate. It would be somewhat worse that if the actual result were exactly
1310.99999999999, and we could only display 10 characters at a time, we would see an initial display
132of <span class="display">1.00000000</span>. As we scroll to see more digits, we would
133successively see <span class="display">...000000E-6</span>, then <span
134class="display">...000000E-7</span>, and so on until we get to <span
135class="display">...00000E-10</span>, but then suddenly <span class="display">...99999E-11</span>.
136If we scroll back, the screen would again show zeroes. We decided this would be excessively
137confusing, and thus do not round.</p>
138
139<p>It is still possible for previously displayed digits to change as we're scrolling. But we
140always compute a number of digits more than we actually need, so this is exceedingly unlikely.</p>
141
142<p>Since our goal is an error of strictly less than one in the last displayed digit, we will
143never, for example, display an answer of exactly 2 as <span class="display">1.9999999999</span>.
144That would involve an error of exactly one in the last place, which is too much for us.</p> <p>It
145turns out that there is exactly one case in which the display switches between 9s and 0s: A long
146but finite sequence of 9s (more than 20) in the true result can initially be displayed as a larger
147number ending in 0s. As we scroll, the 0s turn into 9s. When we immediately scroll back, the
148number remains displayed as 9s, since the calculator caches the best known result (though not
149currently across restarts or screen rotations).</p>
150
151<p>We prevent 9s from turning into 0s during scrolling. If we generate a result ending in 9s, our
152error bound implies that the true result is strictly less (in absolute value) than the value
153(ending in 0s) we would get by incrementing the last displayed digit. Thus we can never be forced
154back to generating zeros and will always continue to generate 9s.</p>
155
156<h3>Coping with mathematical limits</h3>
157<p>Internally the calculator essentially represents a number as a program for computing however
158many digits we happen to need. This representation has many nice properties, like never resulting
159in the display of incorrect results. It has one inherent weakness: We provably cannot compute
160precisely whether two numbers are equal. We can compute more and more digits of both numbers, and
161if they ever differ by more than one in the last computed digit, we know they are <i>not</i>
162equal. But if the two numbers were in fact the same, this process will go on forever.</p>
163
164<p>This is still better than machine floating point arithmetic, though machine floating point
165better obscures the problem. With machine floating point arithmetic, two computations that should
166mathematically have given the same answer, may give us substantially different answers, and two
167computations that should have given us different answers may easily produce the same one. We
168can indeed determine whether the floating representations are equal, but this tells us little
169about equality of the true mathematical answers.</p>
170
171<p>The undecidability of equality creates some interesting issues. If we divide a number by
172<i>x</i>, the calculator will compute more and more digits of <i>x</i> until it finds some nonzero
173ones. If <i>x</i> was in fact exactly zero, this process will continue forever.</p> <p>We deal
174with this problem using two complementary techniques:</p>
175
176<ol>
177<li>We always run numeric computations in the background, where they won't interfere with user
178interactions, just in case they take a long time. If they do take a really long time, we time
179them out and inform the user that the computation has been aborted. This is unlikely to happen by
180accident, unless the user entered an ill-defined mathematical expression, like a division by
181zero.</li>
182<li>As we will see below, in many cases we use an additional number representation that does allow
183us to determine that a number is exactly zero. Although this easily handles most cases, it is not
184foolproof. If the user enters "1÷0" we immediately detect the division by zero. If the user
185enters "1÷(π−π)" we time out. (We might choose to explicitly recognize such simple cases in the
186future. But this would always remain a heuristic.)</li>
187</ol>
188
189<h3>Zeros further than the eye can see</h3>
190<p>Prototypes of the M calculator, like mathematicians, treated all real numbers as infinite
191objects, with infinitely many digits to scroll through. If the actual computation happened to be
1922−1, the result was initially displayed as <span class="display">1.00000000</span>, and the user
193could keep scrolling through as many thousands of zeroes to the right of that as he desired.
194Although mathematically sound, this proved unpopular for several good reasons, the first one
195probably more serious than the others:</p>
196
197<ol>
198<li>If we computed $1.23 + $7.89, the result would show up as <span
199class="display">9.1200000000</span> or the like, which is unexpected and harder to read quickly
200than <span class="display">9.12</span>.</li>
201<li>Many users consider the result of 2-1 to be a finite number, and find it confusing to be able
202to scroll through lots of zeros on the right.</li>
203<li>Since the calculator couldn't ever tell that a number wasn't going to be scrolled, it couldn't
204treat any result as short enough to allow the use of a larger font.</li>
205</ol>
206
207<p>As a result, the calculator now also tries to compute the result as an exact fraction whenever
208that is easily possible. It is then easy to tell from the fraction whether a number has a finite
209decimal expansion. If it does, we prevent scrolling past that point, and may use the fact that
210the result has a short representation to increase the font size. Results displayed in a larger
211font are not scrollable. We no longer display any zeros for non-zero results unless there is
212either a nonzero or a displayed decimal point to the right. The fact that a result is not
213scrollable tells the user that the result, as displayed, is exact. This is fallible in the other
214direction. For example, we do not compute a rational representation for π−π, and hence it is
215still possible to scroll through as many zeros of that result as you like.</p>
216
217<p>This underlying fractional representation of the result is also used to detect, for example,
218division by zero without a timeout.</p>
219
220<p>Since we calculate the fractional result when we can in any case, it is also now available to
221the user through the overflow menu.</p>
222
223<h2>More details</h2>
224<p>The underlying evaluate-on-demand arithmetic package is described in H. Boehm, "The
225Constructive Reals as a Java Library'', Special issue on practical development of exact real
226number computation, <i>Journal of Logic and Algebraic Programming 64</i>, 1, July 2005, pp. 3-11.
227(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>
228
229<p>Our version has been slightly refined. Notably it calculates inverse trigonometric functions
230directly instead of using a generic "inverse" function. This is less elegant, but significantly
231improves performance.</p>
232
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