Generalised floating point addition: Difference between revisions
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===bigatom===
Note this is decimal-only, and that mpfr.e does not really cope because it cannot hold decimal fractions exactly (though it could probably be fudged with a smidgen of rounding)
<!--<lang Phix>
<span style="color: #008080;">include</span> <span style="color: #000000;">bigatom</span><span style="color: #0000FF;">.</span><span style="color: #000000;">e</span>
<span style="color: #0000FF;">{}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">ba_scale</span><span style="color: #0000FF;">(</span><span style="color: #000000;">200</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">string</span> <span style="color: #000000;">s</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">"12345679"</span><span style="color: #0000FF;">,</span>
<span style="color: #000000;">t</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">"123456790"</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">e</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">63</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">n</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">7</span> <span style="color: #008080;">to</span> <span style="color: #000000;">21</span> <span style="color: #008080;">do</span>
<span style="color: #000000;">bigatom</span> <span style="color: #000000;">bd</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">ba_new</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">sprintf</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"%se%d"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">s</span><span style="color: #0000FF;">,</span><span style="color: #000000;">e</span><span style="color: #0000FF;">})),</span>
<span style="color: #000000;">e1</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">ba_new</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">sprintf</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"1e%d"</span><span style="color: #0000FF;">,</span><span style="color: #000000;">e</span><span style="color: #0000FF;">)),</span>
<span style="color: #000000;">res</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">ba_add</span><span style="color: #0000FF;">(</span><span style="color: #000000;">ba_mul</span><span style="color: #0000FF;">(</span><span style="color: #000000;">bd</span><span style="color: #0000FF;">,</span><span style="color: #000000;">81</span><span style="color: #0000FF;">),</span><span style="color: #000000;">e1</span><span style="color: #0000FF;">)</span>
<span style="color: #000080;font-style:italic;">-- res = ba_mul(bd,81)</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"%2d : %s\n"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">n</span><span style="color: #0000FF;">,</span><span style="color: #000000;">ba_sprintf</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"%.eB"</span><span style="color: #0000FF;">,</span><span style="color: #000000;">res</span><span style="color: #0000FF;">)})</span>
<span style="color: #000080;font-style:italic;">-- printf(1,"%2d : %s\n",{n,ba_sprintf("%B",res)})</span>
<span style="color: #000000;">s</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">t</span> <span style="color: #0000FF;">&</span> <span style="color: #000000;">s</span>
<span style="color: #000000;">e</span> <span style="color: #0000FF;">-=</span> <span style="color: #000000;">9</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<!--</lang>-->
{{out|Output (trimmed)}}
(Use the %B format to better see the difference between adding and not adding e1.)
Line 838 ⟶ 840:
=== any base ===
Uses b_add() and b_mul() from [[Generalised_floating_point_multiplication#Phix]]
<!--<lang Phix>(phixonline)--
<span style="color: #000080;font-style:italic;">--
-- demo\rosetta\Generic_addition.exw
-- =================================
--
-- https://rosettacode.org/wiki/Generalised_floating_point_addition#Phix
--
-- Uses b_add() and b_mul() from Generic_multiplication</span>
<span style="color: #008080;">with</span> <span style="color: #008080;">trace</span>
<span style="color: #000080;font-style:italic;">--<next 210 lines copied from Generic_multiplication.exw></span>
<span style="color: #008080;">constant</span> <span style="color: #000000;">decimal</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">"0123456789"</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">negate</span><span style="color: #0000FF;">(</span><span style="color: #004080;">string</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">alphabet</span><span style="color: #0000FF;">)</span>
<span style="color: #000080;font-style:italic;">--
-- negate b (can be balanced or unbalanced)
--</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">alphabet</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]=</span><span style="color: #008000;">'0'</span> <span style="color: #008080;">then</span>
<span style="color: #000080;font-style:italic;">-- traditional: add/remove a leading '-'
-- eg "-123" <==> "123"</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">!=</span><span style="color: #008000;">"0"</span> <span style="color: #008080;">then</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]=</span><span style="color: #008000;">'-'</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">b</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">[</span><span style="color: #000000;">2</span><span style="color: #0000FF;">..$]</span>
<span style="color: #000000;">b</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">"-"</span><span style="color: #0000FF;">&</span><span style="color: #000000;">b</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">else</span>
<span style="color: #000080;font-style:italic;">-- balanced: mirror [non-0] digits
-- eg "-0+" (ie -8) <==> "+0-" (ie +8)</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">do</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]!=</span><span style="color: #008000;">'.'</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">b</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">alphabet</span><span style="color: #0000FF;">[-</span><span style="color: #7060A8;">find</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">],</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">)]</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">b</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">b_trim</span><span style="color: #0000FF;">(</span><span style="color: #004080;">string</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">)</span>
<span style="color: #000080;font-style:italic;">-- (common code)
-- trim trailing ".000"</span>
<span style="color: #008080;">if</span> <span style="color: #7060A8;">find</span><span style="color: #0000FF;">(</span><span style="color: #008000;">'.'</span><span style="color: #0000FF;">,</span><span style="color: #000000;">b</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">b</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">trim_tail</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">trim_tail</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">,</span><span style="color: #008000;">'0'</span><span style="color: #0000FF;">),</span><span style="color: #008000;">'.'</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #000080;font-style:italic;">-- trim leading zeroes, but not "0.nnn" -> ".nnn"
-- [hence we cannot use the standard trim_head()]</span>
<span style="color: #008080;">while</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">)></span><span style="color: #000000;">1</span> <span style="color: #008080;">and</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]=</span><span style="color: #008000;">'0'</span> <span style="color: #008080;">and</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">[</span><span style="color: #000000;">2</span><span style="color: #0000FF;">]!=</span><span style="color: #008000;">'.'</span> <span style="color: #008080;">do</span>
<span style="color: #000000;">b</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">[</span><span style="color: #000000;">2</span><span style="color: #0000FF;">..$]</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">while</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">=</span><span style="color: #008000;">""</span> <span style="color: #008080;">then</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">=</span><span style="color: #008000;">"0"</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">b</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">forward</span> <span style="color: #008080;">function</span> <span style="color: #000000;">b_sub</span><span style="color: #0000FF;">(</span><span style="color: #004080;">string</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">alphabet</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">b_add</span><span style="color: #0000FF;">(</span><span style="color: #004080;">string</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">alphabet</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">base</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">zdx</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">find</span><span style="color: #0000FF;">(</span><span style="color: #008000;">'0'</span><span style="color: #0000FF;">,</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">carry</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">da</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">db</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">digit</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">zdx</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">then</span>
<span style="color: #000080;font-style:italic;">-- (let me know if you can fix this for me!)
-- if a[1]='-' or b[1]='-' then ?9/0 end if -- +ve only</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]=</span><span style="color: #008000;">'-'</span> <span style="color: #008080;">then</span> <span style="color: #000080;font-style:italic;">-- (-a)+b == b-a</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">b_sub</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">,</span><span style="color: #000000;">negate</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">),</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]=</span><span style="color: #008000;">'-'</span> <span style="color: #008080;">then</span> <span style="color: #000080;font-style:italic;">-- a+(-b) == a-b</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">b_sub</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span><span style="color: #000000;">negate</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">,</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">),</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">adt</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">find</span><span style="color: #0000FF;">(</span><span style="color: #008000;">'.'</span><span style="color: #0000FF;">,</span><span style="color: #000000;">a</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">bdt</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">find</span><span style="color: #0000FF;">(</span><span style="color: #008000;">'.'</span><span style="color: #0000FF;">,</span><span style="color: #000000;">b</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">adt</span> <span style="color: #008080;">or</span> <span style="color: #000000;">bdt</span> <span style="color: #008080;">then</span>
<span style="color: #000080;font-style:italic;">-- remove the '.'s and zero-pad the shorter as needed
-- (thereafter treat them as two whole integers)
-- eg "1.23"+"4.5" -> "123"+"450" (leaving adt==2)</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">adt</span> <span style="color: #008080;">then</span> <span style="color: #000000;">adt</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">)-</span><span style="color: #000000;">adt</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">;</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">[-</span><span style="color: #000000;">adt</span><span style="color: #0000FF;">..-</span><span style="color: #000000;">adt</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">""</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">bdt</span> <span style="color: #008080;">then</span> <span style="color: #000000;">bdt</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">)-</span><span style="color: #000000;">bdt</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">;</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">[-</span><span style="color: #000000;">bdt</span><span style="color: #0000FF;">..-</span><span style="color: #000000;">bdt</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">""</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">bdt</span><span style="color: #0000FF;">></span><span style="color: #000000;">adt</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">a</span> <span style="color: #0000FF;">&=</span> <span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(</span><span style="color: #008000;">'0'</span><span style="color: #0000FF;">,</span><span style="color: #000000;">bdt</span><span style="color: #0000FF;">-</span><span style="color: #000000;">adt</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">adt</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">bdt</span>
<span style="color: #008080;">elsif</span> <span style="color: #000000;">adt</span><span style="color: #0000FF;">></span><span style="color: #000000;">bdt</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">b</span> <span style="color: #0000FF;">&=</span> <span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(</span><span style="color: #008000;">'0'</span><span style="color: #0000FF;">,</span><span style="color: #000000;">adt</span><span style="color: #0000FF;">-</span><span style="color: #000000;">bdt</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">if</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">)<</span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">then</span>
<span style="color: #0000FF;">{</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span><span style="color: #000000;">b</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">b</span><span style="color: #0000FF;">,</span><span style="color: #000000;">a</span><span style="color: #0000FF;">}</span> <span style="color: #000080;font-style:italic;">-- ensure b is the shorter</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=-</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #0000FF;">-</span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">by</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">1</span> <span style="color: #008080;">do</span>
<span style="color: #000000;">da</span> <span style="color: #0000FF;">=</span> <span style="color: #008080;">iff</span><span style="color: #0000FF;">(</span><span style="color: #000000;">i</span><span style="color: #0000FF;"><-</span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">)?</span><span style="color: #000000;">0</span><span style="color: #0000FF;">:</span><span style="color: #7060A8;">find</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">],</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">)-</span><span style="color: #000000;">zdx</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">db</span> <span style="color: #0000FF;">=</span> <span style="color: #008080;">iff</span><span style="color: #0000FF;">(</span><span style="color: #000000;">i</span><span style="color: #0000FF;"><-</span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">)?</span><span style="color: #000000;">0</span><span style="color: #0000FF;">:</span><span style="color: #7060A8;">find</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">],</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">)-</span><span style="color: #000000;">zdx</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">digit</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">da</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">db</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">carry</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">zdx</span>
<span style="color: #000000;">carry</span> <span style="color: #0000FF;">=</span> <span style="color: #008080;">iff</span><span style="color: #0000FF;">(</span><span style="color: #000000;">digit</span><span style="color: #0000FF;">></span><span style="color: #000000;">base</span><span style="color: #0000FF;">?+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">:</span><span style="color: #008080;">iff</span><span style="color: #0000FF;">(</span><span style="color: #000000;">digit</span><span style="color: #0000FF;"><</span><span style="color: #000000;">1</span><span style="color: #0000FF;">?-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">:</span><span style="color: #000000;">0</span><span style="color: #0000FF;">))</span>
<span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">alphabet</span><span style="color: #0000FF;">[</span><span style="color: #000000;">digit</span><span style="color: #0000FF;">-</span><span style="color: #000000;">carry</span><span style="color: #0000FF;">*</span><span style="color: #000000;">base</span><span style="color: #0000FF;">]</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">i</span><span style="color: #0000FF;"><-</span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">and</span> <span style="color: #000000;">carry</span><span style="color: #0000FF;">=</span><span style="color: #000000;">0</span> <span style="color: #008080;">then</span> <span style="color: #008080;">exit</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">carry</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">a</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">alphabet</span><span style="color: #0000FF;">[</span><span style="color: #000000;">carry</span><span style="color: #0000FF;">+</span><span style="color: #000000;">zdx</span><span style="color: #0000FF;">]&</span><span style="color: #000000;">a</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">adt</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">a</span><span style="color: #0000FF;">[-</span><span style="color: #000000;">adt</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">..-</span><span style="color: #000000;">adt</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">"."</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #000000;">a</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">b_trim</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">a</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">a_smaller</span><span style="color: #0000FF;">(</span><span style="color: #004080;">string</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">alphabet</span><span style="color: #0000FF;">)</span>
<span style="color: #000080;font-style:italic;">-- return true if a is smaller than b
-- if not balanced then both are +ve</span>
<span style="color: #008080;">if</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">)!=</span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">then</span> <span style="color: #0000FF;">?</span><span style="color: #000000;">9</span><span style="color: #0000FF;">/</span><span style="color: #000000;">0</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span> <span style="color: #000080;font-style:italic;">-- sanity check</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">do</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">da</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">find</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">],</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">db</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">find</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">],</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">c</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">compare</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span><span style="color: #000000;">b</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">c</span><span style="color: #0000FF;">!=</span><span style="color: #000000;">0</span> <span style="color: #008080;">then</span> <span style="color: #008080;">return</span> <span style="color: #000000;">c</span><span style="color: #0000FF;"><</span><span style="color: #000000;">0</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #008080;">return</span> <span style="color: #004600;">false</span> <span style="color: #000080;font-style:italic;">-- (=, which is not <)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">b_sub</span><span style="color: #0000FF;">(</span><span style="color: #004080;">string</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">alphabet</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">base</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">zdx</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">find</span><span style="color: #0000FF;">(</span><span style="color: #008000;">'0'</span><span style="color: #0000FF;">,</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">carry</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">da</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">db</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">digit</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">zdx</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">then</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]=</span><span style="color: #008000;">'-'</span> <span style="color: #008080;">then</span> <span style="color: #000080;font-style:italic;">-- (-a)-b == -(a+b)</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">negate</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b_add</span><span style="color: #0000FF;">(</span><span style="color: #000000;">negate</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">),</span><span style="color: #000000;">b</span><span style="color: #0000FF;">,</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">),</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]=</span><span style="color: #008000;">'-'</span> <span style="color: #008080;">then</span> <span style="color: #000080;font-style:italic;">-- a-(-b) == a+b</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">b_add</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span><span style="color: #000000;">negate</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">,</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">),</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">adt</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">find</span><span style="color: #0000FF;">(</span><span style="color: #008000;">'.'</span><span style="color: #0000FF;">,</span><span style="color: #000000;">a</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">bdt</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">find</span><span style="color: #0000FF;">(</span><span style="color: #008000;">'.'</span><span style="color: #0000FF;">,</span><span style="color: #000000;">b</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">adt</span> <span style="color: #008080;">or</span> <span style="color: #000000;">bdt</span> <span style="color: #008080;">then</span>
<span style="color: #000080;font-style:italic;">-- remove the '.'s and zero-pad the shorter as needed
-- (thereafter treat them as two whole integers)
-- eg "1.23"+"4.5" -> "123"+"450" (leaving adt==2)</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">adt</span> <span style="color: #008080;">then</span> <span style="color: #000000;">adt</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">)-</span><span style="color: #000000;">adt</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">;</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">[-</span><span style="color: #000000;">adt</span><span style="color: #0000FF;">..-</span><span style="color: #000000;">adt</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">""</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">bdt</span> <span style="color: #008080;">then</span> <span style="color: #000000;">bdt</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">)-</span><span style="color: #000000;">bdt</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">;</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">[-</span><span style="color: #000000;">bdt</span><span style="color: #0000FF;">..-</span><span style="color: #000000;">bdt</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">""</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">bdt</span><span style="color: #0000FF;">></span><span style="color: #000000;">adt</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">a</span> <span style="color: #0000FF;">&=</span> <span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(</span><span style="color: #008000;">'0'</span><span style="color: #0000FF;">,</span><span style="color: #000000;">bdt</span><span style="color: #0000FF;">-</span><span style="color: #000000;">adt</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">adt</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">bdt</span>
<span style="color: #008080;">elsif</span> <span style="color: #000000;">adt</span><span style="color: #0000FF;">></span><span style="color: #000000;">bdt</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">b</span> <span style="color: #0000FF;">&=</span> <span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(</span><span style="color: #008000;">'0'</span><span style="color: #0000FF;">,</span><span style="color: #000000;">adt</span><span style="color: #0000FF;">-</span><span style="color: #000000;">bdt</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #004080;">bool</span> <span style="color: #000000;">bNegate</span> <span style="color: #0000FF;">=</span> <span style="color: #004600;">false</span>
<span style="color: #008080;">if</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">)<</span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">or</span> <span style="color: #0000FF;">(</span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">)=</span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">and</span> <span style="color: #000000;">a_smaller</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span><span style="color: #000000;">b</span><span style="color: #0000FF;">,</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">))</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">bNegate</span> <span style="color: #0000FF;">=</span> <span style="color: #004600;">true</span>
<span style="color: #0000FF;">{</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span><span style="color: #000000;">b</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">b</span><span style="color: #0000FF;">,</span><span style="color: #000000;">a</span><span style="color: #0000FF;">}</span> <span style="color: #000080;font-style:italic;">-- ensure b is the shorter/smaller</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=-</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #0000FF;">-</span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">by</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">1</span> <span style="color: #008080;">do</span>
<span style="color: #000000;">da</span> <span style="color: #0000FF;">=</span> <span style="color: #008080;">iff</span><span style="color: #0000FF;">(</span><span style="color: #000000;">i</span><span style="color: #0000FF;"><-</span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">)?</span><span style="color: #000000;">0</span><span style="color: #0000FF;">:</span><span style="color: #7060A8;">find</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">],</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">)-</span><span style="color: #000000;">zdx</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">db</span> <span style="color: #0000FF;">=</span> <span style="color: #008080;">iff</span><span style="color: #0000FF;">(</span><span style="color: #000000;">i</span><span style="color: #0000FF;"><-</span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">)?</span><span style="color: #000000;">0</span><span style="color: #0000FF;">:</span><span style="color: #7060A8;">find</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">],</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">)-</span><span style="color: #000000;">zdx</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">digit</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">da</span> <span style="color: #0000FF;">-</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">db</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">carry</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">zdx</span>
<span style="color: #000000;">carry</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">digit</span><span style="color: #0000FF;"><=</span><span style="color: #000000;">0</span>
<span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">alphabet</span><span style="color: #0000FF;">[</span><span style="color: #000000;">digit</span><span style="color: #0000FF;">+</span><span style="color: #000000;">carry</span><span style="color: #0000FF;">*</span><span style="color: #000000;">base</span><span style="color: #0000FF;">]</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">i</span><span style="color: #0000FF;"><-</span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">and</span> <span style="color: #000000;">carry</span><span style="color: #0000FF;">=</span><span style="color: #000000;">0</span> <span style="color: #008080;">then</span> <span style="color: #008080;">exit</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">carry</span> <span style="color: #008080;">then</span>
<span style="color: #0000FF;">?</span><span style="color: #000000;">9</span><span style="color: #0000FF;">/</span><span style="color: #000000;">0</span> <span style="color: #000080;font-style:italic;">-- should have set bNegate above...</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">adt</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">a</span><span style="color: #0000FF;">[-</span><span style="color: #000000;">adt</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">..-</span><span style="color: #000000;">adt</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">"."</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #000000;">a</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">b_trim</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">bNegate</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">a</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">negate</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">a</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">b_mul</span><span style="color: #0000FF;">(</span><span style="color: #004080;">string</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">alphabet</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">base</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">zdx</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">find</span><span style="color: #0000FF;">(</span><span style="color: #008000;">'0'</span><span style="color: #0000FF;">,</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">dpa</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">find</span><span style="color: #0000FF;">(</span><span style="color: #008000;">'.'</span><span style="color: #0000FF;">,</span><span style="color: #000000;">a</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">dpb</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">find</span><span style="color: #0000FF;">(</span><span style="color: #008000;">'.'</span><span style="color: #0000FF;">,</span><span style="color: #000000;">b</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">ndp</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">dpa</span> <span style="color: #008080;">then</span> <span style="color: #000000;">ndp</span> <span style="color: #0000FF;">+=</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">)-</span><span style="color: #000000;">dpa</span><span style="color: #0000FF;">;</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">dpa</span><span style="color: #0000FF;">..</span><span style="color: #000000;">dpa</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">""</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">dpb</span> <span style="color: #008080;">then</span> <span style="color: #000000;">ndp</span> <span style="color: #0000FF;">+=</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">)-</span><span style="color: #000000;">dpb</span><span style="color: #0000FF;">;</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">[</span><span style="color: #000000;">dpb</span><span style="color: #0000FF;">..</span><span style="color: #000000;">dpb</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">""</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #004080;">string</span> <span style="color: #000000;">pos</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">res</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">"0"</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">zdx</span><span style="color: #0000FF;">!=</span><span style="color: #000000;">1</span> <span style="color: #008080;">then</span>
<span style="color: #000080;font-style:italic;">-- balanced number systems</span>
<span style="color: #004080;">string</span> <span style="color: #000000;">neg</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">negate</span><span style="color: #0000FF;">(</span><span style="color: #000000;">pos</span><span style="color: #0000FF;">,</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">to</span> <span style="color: #000000;">1</span> <span style="color: #008080;">by</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">1</span> <span style="color: #008080;">do</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">m</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">find</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">],</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">)-</span><span style="color: #000000;">zdx</span>
<span style="color: #008080;">while</span> <span style="color: #000000;">m</span> <span style="color: #008080;">do</span>
<span style="color: #000000;">res</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">b_add</span><span style="color: #0000FF;">(</span><span style="color: #000000;">res</span><span style="color: #0000FF;">,</span><span style="color: #008080;">iff</span><span style="color: #0000FF;">(</span><span style="color: #000000;">m</span><span style="color: #0000FF;"><</span><span style="color: #000000;">0</span><span style="color: #0000FF;">?</span><span style="color: #000000;">neg</span><span style="color: #0000FF;">:</span><span style="color: #000000;">pos</span><span style="color: #0000FF;">),</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">m</span> <span style="color: #0000FF;">+=</span> <span style="color: #008080;">iff</span><span style="color: #0000FF;">(</span><span style="color: #000000;">m</span><span style="color: #0000FF;"><</span><span style="color: #000000;">0</span><span style="color: #0000FF;">?+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">:-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">while</span>
<span style="color: #000000;">pos</span> <span style="color: #0000FF;">&=</span> <span style="color: #008000;">'0'</span>
<span style="color: #000000;">neg</span> <span style="color: #0000FF;">&=</span> <span style="color: #008000;">'0'</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #008080;">else</span>
<span style="color: #000080;font-style:italic;">-- non-balanced (normal) number systems</span>
<span style="color: #004080;">bool</span> <span style="color: #000000;">negative</span> <span style="color: #0000FF;">=</span> <span style="color: #004600;">false</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]=</span><span style="color: #008000;">'-'</span> <span style="color: #008080;">then</span> <span style="color: #000000;">a</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">2</span><span style="color: #0000FF;">..$];</span> <span style="color: #000000;">negative</span> <span style="color: #0000FF;">=</span> <span style="color: #004600;">true</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]=</span><span style="color: #008000;">'-'</span> <span style="color: #008080;">then</span> <span style="color: #000000;">b</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">[</span><span style="color: #000000;">2</span><span style="color: #0000FF;">..$];</span> <span style="color: #000000;">negative</span> <span style="color: #0000FF;">=</span> <span style="color: #008080;">not</span> <span style="color: #000000;">negative</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">to</span> <span style="color: #000000;">1</span> <span style="color: #008080;">by</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">1</span> <span style="color: #008080;">do</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">m</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">find</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">],</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">)-</span><span style="color: #000000;">zdx</span>
<span style="color: #008080;">while</span> <span style="color: #000000;">m</span><span style="color: #0000FF;">></span><span style="color: #000000;">0</span> <span style="color: #008080;">do</span>
<span style="color: #000000;">res</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">b_add</span><span style="color: #0000FF;">(</span><span style="color: #000000;">res</span><span style="color: #0000FF;">,</span><span style="color: #000000;">pos</span><span style="color: #0000FF;">,</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">m</span> <span style="color: #0000FF;">-=</span> <span style="color: #000000;">1</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">while</span>
<span style="color: #000000;">pos</span> <span style="color: #0000FF;">&=</span> <span style="color: #008000;">'0'</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">negative</span> <span style="color: #008080;">then</span> <span style="color: #000000;">res</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">negate</span><span style="color: #0000FF;">(</span><span style="color: #000000;">res</span><span style="color: #0000FF;">,</span><span style="color: #000000;">alphabet</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">ndp</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">res</span><span style="color: #0000FF;">[-</span><span style="color: #000000;">ndp</span><span style="color: #0000FF;">..-</span><span style="color: #000000;">ndp</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">"."</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #000000;">res</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">b_trim</span><span style="color: #0000FF;">(</span><span style="color: #000000;">res</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">res</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #000080;font-style:italic;">--</copied from Generic_multiplication.exw></span>
<span style="color: #008080;">function</span> <span style="color: #000000;">normalise_decimal</span><span style="color: #0000FF;">(</span><span style="color: #004080;">string</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">)</span>
<span style="color: #000080;font-style:italic;">--
-- eg "12.34e-3" ==> "0.01234"
-- and "1e2" ==> "100"
--</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">d</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">find</span><span style="color: #0000FF;">(</span><span style="color: #008000;">'.'</span><span style="color: #0000FF;">,</span><span style="color: #000000;">a</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">e</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">find</span><span style="color: #0000FF;">(</span><span style="color: #008000;">'e'</span><span style="color: #0000FF;">,</span><span style="color: #000000;">a</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">e</span> <span style="color: #008080;">then</span>
<span style="color: #004080;">integer</span> <span style="color: #0000FF;">{{</span><span style="color: #000000;">z</span><span style="color: #0000FF;">}}</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">scanf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">e</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">..$],</span><span style="color: #008000;">"%d"</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">a</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">..</span><span style="color: #000000;">e</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">d</span> <span style="color: #008080;">then</span>
<span style="color: #000080;font-style:italic;">-- eg 12.34e-3 == 1234e-5</span>
<span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">d</span><span style="color: #0000FF;">..</span><span style="color: #000000;">d</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">""</span>
<span style="color: #000000;">z</span> <span style="color: #0000FF;">-=</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">)-</span><span style="color: #000000;">d</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">z</span><span style="color: #0000FF;">>=</span><span style="color: #000000;">0</span> <span style="color: #008080;">then</span>
<span style="color: #000080;font-style:italic;">-- eg 1e2 ==> 100</span>
<span style="color: #000000;">a</span> <span style="color: #0000FF;">&=</span> <span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(</span><span style="color: #008000;">'0'</span><span style="color: #0000FF;">,</span><span style="color: #000000;">z</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">elsif</span> <span style="color: #000000;">z</span><span style="color: #0000FF;"><-</span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">then</span>
<span style="color: #000080;font-style:italic;">-- eg 1234e-5 ==> 0.01234</span>
<span style="color: #000000;">a</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">"0."</span><span style="color: #0000FF;">&</span><span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(</span><span style="color: #008000;">'0'</span><span style="color: #0000FF;">,-</span><span style="color: #000000;">z</span><span style="color: #0000FF;">-</span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">))&</span><span style="color: #000000;">a</span>
<span style="color: #008080;">else</span>
<span style="color: #000080;font-style:italic;">-- eg 1234e-3 ==> 1.234</span>
<span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">z</span><span style="color: #0000FF;">..</span><span style="color: #000000;">z</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">"."</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">a</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">e_notation</span><span style="color: #0000FF;">(</span><span style="color: #004080;">string</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">)</span>
<span style="color: #000080;font-style:italic;">-- eg "1000000" ==> "1e6"</span>
<span style="color: #008080;">if</span> <span style="color: #7060A8;">find</span><span style="color: #0000FF;">(</span><span style="color: #008000;">'.'</span><span style="color: #0000FF;">,</span><span style="color: #000000;">a</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">then</span> <span style="color: #0000FF;">?</span><span style="color: #000000;">9</span><span style="color: #0000FF;">/</span><span style="color: #000000;">0</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span> <span style="color: #000080;font-style:italic;">-- (to do)</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">l</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">a</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">trim_tail</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span><span style="color: #008000;">".0"</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">l</span> <span style="color: #0000FF;">-=</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">l</span><span style="color: #0000FF;">!=</span><span style="color: #000000;">0</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">a</span> <span style="color: #0000FF;">&=</span><span style="color: #7060A8;">sprintf</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"e%d"</span><span style="color: #0000FF;">,</span><span style="color: #000000;">l</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">a</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #004080;">string</span> <span style="color: #000000;">s</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">"12345679"</span><span style="color: #0000FF;">,</span>
<span style="color: #000000;">t</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">"123456790"</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">e</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">63</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">n</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">7</span> <span style="color: #008080;">to</span> <span style="color: #000000;">21</span> <span style="color: #008080;">do</span>
<span style="color: #004080;">string</span> <span style="color: #000000;">bd</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">normalise_decimal</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">sprintf</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"%se%d"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">s</span><span style="color: #0000FF;">,</span><span style="color: #000000;">e</span><span style="color: #0000FF;">})),</span>
<span style="color: #000000;">e1</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">normalise_decimal</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">sprintf</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"1e%d"</span><span style="color: #0000FF;">,</span><span style="color: #000000;">e</span><span style="color: #0000FF;">)),</span>
<span style="color: #000000;">res</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">b_add</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b_mul</span><span style="color: #0000FF;">(</span><span style="color: #000000;">bd</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"81"</span><span style="color: #0000FF;">,</span><span style="color: #000000;">decimal</span><span style="color: #0000FF;">),</span><span style="color: #000000;">e1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">decimal</span><span style="color: #0000FF;">)</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"%2d res : %s\n"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">n</span><span style="color: #0000FF;">,</span><span style="color: #000000;">e_notation</span><span style="color: #0000FF;">(</span><span style="color: #000000;">res</span><span style="color: #0000FF;">)})</span>
<span style="color: #000000;">s</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">t</span> <span style="color: #0000FF;">&</span> <span style="color: #000000;">s</span>
<span style="color: #000000;">e</span> <span style="color: #0000FF;">-=</span> <span style="color: #000000;">9</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
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same ouput
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