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Timeline: sophisticated helper function to resolve problems with numeric precision

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While the calculation was already basically under control, I just was not content
with the achieved numeric precision -- and the fact that the test case in fact
misses the bar, making it difficult do demonstrate that the calculation
is not derailed. I just had the gut feeling that it must be somehow possible
to achieve an absolute error level, not just a relative error level of 1e-6

Thus I reworked this part into a generic helper function, see #1262

The end result is:
 * partial failure. we can only ''guarantee'' the relative error margin of 1e-6
 * but in most cases not out to the most extreme numbers, the sophisticated
   solution achieves much better results way below the absolute error level of 1µ-Tick

Thus with using rational numbers, we have now a solution that is absolutely precise
in the regular case, and gradually introduces errors at the domain bound
but with an guaranteed relative error margin of 1e-6 (== Time::SCALE)
This commit is contained in:
Fischlurch 2022-12-02 22:23:14 +01:00
parent 2ddbed693b
commit 4d79bdce5f
4 changed files with 317 additions and 66 deletions
Show stats Download patch file Download diff file Expand all files Collapse all files

4
src/stage/model/zoom-window.cpp
View file

@ -24,6 +24,10 @@
/** @file zoom-window.cpp
** Common implementation details related to zoom handling and
** transformation into screen coordinates.
** @todo do we need this dedicated translation unit?
** As of 11/2022, the ZoomWindow component was developed header-only...
** However the amount of code is significant and the details are complex,
** and there is indeed a major implementation-only block with private functions.
**
*/

128
src/stage/model/zoom-window.hpp
View file

@ -96,6 +96,7 @@
#include <limits>
#include <functional>
#include <array>
namespace stage {
@ -182,8 +183,9 @@ namespace model {
* @remark due to the common divisor normalisation, and the typical time computations,
* DENOMINATOR * Time::Scale has to stay below INT_MAX, with some safety margin
*/
const int64_t LIM_HAZARD{int64_t{1} << 40 };
const int64_t LIM_HAZARD {int64_t{1} << 40 };
const int64_t HAZARD_DEGREE{util::ilog2(LIM_HAZARD)};
const int64_t MAXDIM {util::ilog2 (std::numeric_limits<int64_t>::max())};
inline int
toxicDegree (Rat poison, const int64_t THRESHOLD =HAZARD_DEGREE)
@ -477,20 +479,7 @@ namespace model {
}
/* === establish and maintain invariants === */
/*
* - oriented and non-empty windows
* - never alter given pxWidth
* - zoom metric factor < max zoom
* - visibleWindow ⊂ Canvas
*/
static TimeValue
ensureNonEmpty (Time start, TimeValue endPoint)
{
return (start < endPoint)? endPoint
: start + Time{DEFAULT_CANVAS};
}
/* === utility functions to handle dangerous fractional values === */
/**
* Check and possibly sanitise a rational number to avoid internal numeric overflow.
@ -511,6 +500,7 @@ namespace model {
* @note the check is based on the 2-logarithm of numerator and denominator, which is
* pretty much the fastest possibility (even a simple comparison would have
* to do the same). Values below threshold are simply passed-through.
* @todo this utility function could be factored out into a `FSecs` or `RSec` class //////////////////TICKET #1262
*/
static Rat
detox (Rat poison)
@ -522,8 +512,13 @@ namespace model {
/**
* Scale a possibly large time duration by a rational factor, while attempting to avoid
* integer wrap-around. Obviously this is only a heuristic, yet adequate within the
* framework of ZoomWindow, where the end result is pixel aligned anyway.
* integer wrap-around. In the typical use-case, the multiplication can be just computed
* precisely and safe, but at least we check the limits. In the danger and boundary zone,
* a slight error is introduced to allow cancelling out a common factor, so that the result
* can be just constructed without any further dangerous computation. Obviously this is only
* a heuristic, yet adequate within the framework of ZoomWindow, where the end result is
* pixel aligned anyway.
* @todo this utility function could be factored out into a `FSecs` or `RSec` class //////////////////TICKET #1262
*/
static FSecs
scaleSafe (FSecs duration, Rat factor)
@ -538,16 +533,79 @@ namespace model {
return MAX_TIMESPAN * sgn(guess); // exceeds limits of time representation => cap the result
if (0 == guess)
return 0;
// slightly adjust the factor so that the time-base denominator cancels out,
// allowing to calculate the product without dangerous multiplication of large numbers
// Note: we normalise 1/factor, i.e. we normalise the numerator to be equal to duration.denominator()
int64_t inverseFactor = reQuant (factor.denominator(), factor.numerator(), Time::SCALE);
// some common factors might have been cancelled out, but we know these are integral
/**
* Descriptor for a Strategy to reduce the numbers to keep them in domain.
* After cross-wise cancelling out one part in each factor, the result can be
* constructed without any further multiplication. To achieve that, a slight
* error is introduced into one of the four participating numbers
*/
struct ReductionStrategy
{
int64_t f1; ///< factor one is safe and will not be changed
int64_t u; ///< the counterpart of f1 is used as quantiser and cancelled out
int64_t q; ///< the diagonal counterpart of u is scaled to u and cancelled
int64_t f2; ///< the counterpart of #q is re-quantised to u; it acts as limit
bool invert; ///< Strategy will be applied to the inverse 1/x
int64_t
determineLimit()
{
REQUIRE (u != 0);
return isFeasible()? u : 0;
}
Rat
calculateResult()
{
REQUIRE (isFeasible());
f2 = reQuant (f2, q, u);
return invert? Rat{f2, f1}
: Rat{f1, f2};
}
bool
isFeasible()
{ // Note: factors are nonzero,
REQUIRE (u and q and f2);// otherwise exit after pre-check above
int dim_u = util::ilog2 (abs (u));
int dim_q = util::ilog2 (abs (q));
if (dim_q > dim_u) return true; // requantisation will reduce size and thus no danger
int dim_f = util::ilog2 (abs (f2));
int deltaQ = dim_u - dim_q; // how much q must be increased to match u
int headroom = MAXDIM - dim_f; // how much the counter factor f2 can be increased
return headroom > deltaQ;
}
};
using Cases = std::array<ReductionStrategy, 4>;
// There are four possible strategy configurations.
// One case stands out, insofar this factor is guaranteed to be present:
// because one of the numbers is a quantised Time, it has Time::SCALE as denominator,
// maybe after cancelling out some further common integral factors
int64_t reduction = Time::SCALE / duration.denominator();
int64_t durationTicks = duration.numerator()*reduction;
// result can be constructed without calculation, due to shared common factors
return detox (Rat{durationTicks, inverseFactor});
//-f1--------------------+-u-------------------+-q---------------------+-f2--------------------+-invert--
Cases cases{{{durationTicks , Time::SCALE , factor.numerator() , factor.denominator() , false}
,{factor.numerator() , factor.denominator(), duration.numerator() , duration.denominator(), false}
,{duration.denominator(), duration.numerator(), factor.denominator() , factor.numerator() , true}
,{factor.denominator() , factor.numerator() , duration.denominator(), duration.numerator() , true}
}};
// However, some of the other cases may yield a larger denominator to be cancelled out,
// and thus lead to a smaller error margin. Attempt thus to find the best strategy...
ReductionStrategy* solution{nullptr};
int64_t maxLimit = 0;
for (auto& candidate: cases)
{
int64_t limit = candidate.determineLimit();
if (limit > maxLimit)
{
maxLimit = limit;
solution = &candidate;
}
}
ASSERT (solution and maxLimit > 0);
return detox (solution->calculateResult());
}
}
@ -556,6 +614,7 @@ namespace model {
* Again, this is a heuristics, based on re-quantisation to a smaller common denominator.
* @return exact result if representable, otherwise approximation
* @note result is capped to MAX_TIMESPAN when exceeding domain
* @todo this utility function could be factored out into a `FSecs` or `RSec` class //////////////////TICKET #1262
*/
static FSecs
addSafe (FSecs t1, FSecs t2)
@ -586,7 +645,7 @@ namespace model {
u = Time::SCALE;
else //re-quantise to common denominator more fine-grained than µ-grid
if (s1*s2 > 0 // check numerators to detect danger of wrap-around
and (62<util::ilog2(n1) or 62<util::ilog2(n2)))
and (MAXDIM<=util::ilog2(n1) or MAXDIM<=util::ilog2(n2)))
u >>= 1; // danger zone! wrap-around imminent
n1 = d1==u? n1 : reQuant (n1,d1, u);
@ -600,6 +659,23 @@ namespace model {
}
/* === establish and maintain invariants === */
/*
* - oriented and non-empty windows
* - never alter given pxWidth
* - zoom metric factor < max zoom
* - visibleWindow ⊂ Canvas
*/
static TimeValue
ensureNonEmpty (Time start, TimeValue endPoint)
{
return (start < endPoint)? endPoint
: start + Time{DEFAULT_CANVAS};
}
static Rat
establishMetric (uint pxWidth, Time startWin, Time afterWin)
{

38
tests/stage/model/zoom-window-test.cpp
View file

@ -632,26 +632,26 @@ namespace test {
CHECK (win.px_per_sec() == 575000000_r/856350691); // the new metric however is comprised of sanitised fractional numbers
CHECK (win.pxWidth() == 575); // and the existing pixel width was not changed
SHOW_EXPR(win.overallSpan());
SHOW_EXPR(_raw(win.visible().duration()));
SHOW_EXPR(win.px_per_sec());
SHOW_EXPR(win.pxWidth());
CHECK (win.overallSpan().start() == Time::ZERO);
CHECK (win.overallSpan().duration() == TimeValue{307445734561825860});
CHECK (win.visible().duration() == TimeValue{856350691});
win.setVisiblePos (poison); // Yet another way to sneak in our toxic value...
SHOW_EXPR(win.overallSpan());
SHOW_EXPR(_raw(win.overallSpan()));
SHOW_EXPR(_raw(win.overallSpan().duration()));
SHOW_EXPR(_raw(win.visible().duration()));
SHOW_EXPR(win.px_per_sec());
SHOW_EXPR(win.pxWidth());
SHOW_EXPR(_raw(win.overallSpan().duration()) * rational_cast<double> (poison))
TimeValue targetPos{gavl_time_t(_raw(win.overallSpan().duration())
* rational_cast<double> (poison))};
SHOW_EXPR(targetPos);
SHOW_EXPR(_raw(targetPos));
SHOW_EXPR(_raw(win.visible().start()))
SHOW_EXPR(_raw(win.visible().end()))
SHOW_EXPR(bool(win.visible().start() < targetPos))
SHOW_EXPR(bool(win.visible().end() > targetPos))
CHECK (win.overallSpan().start() == Time::ZERO);
CHECK (win.overallSpan().duration() == TimeValue{307445734561825860}); // However, all base values turn out unaffected
CHECK (win.visible().duration() == TimeValue{856350691});
TimeValue targetPos{gavl_time_t(_raw(win.overallSpan().duration()) // based on the overall span...
* rational_cast<double> (poison))}; // the given toxic factor would point at that target position
CHECK (targetPos == TimeValue{206435633551724864});
CHECK (win.visible().start() == TimeValue{206435633106265625}); // the visible window has been moved to enclose this target
CHECK (win.visible().end() == TimeValue{206435633962616316});
CHECK (win.visible().start() < targetPos);
CHECK (win.visible().end() > targetPos);
CHECK (win.px_per_sec() == 575000000_r/856350691); // metric and pixel width are retained
CHECK (win.pxWidth() == 575);
}

213
wiki/thinkPad.ichthyo.mm
View file

@ -39762,17 +39762,20 @@
<node CREATED="1668267238564" ID="ID_1057480192" MODIFIED="1668267248362" TEXT="startWin_ = startAll_ + (canvasOffset - partBeforeAnchor);">
<icon BUILTIN="info"/>
</node>
<node BACKGROUND_COLOR="#eee5c3" COLOR="#990000" CREATED="1668267251650" ID="ID_1280626173" MODIFIED="1668267290312" TEXT="potentieller TimeValue-wrap">
<node COLOR="#435e98" CREATED="1668267251650" ID="ID_1280626173" MODIFIED="1670016135487" TEXT="potentieller TimeValue-wrap">
<icon BUILTIN="broken-line"/>
<node CREATED="1669507976363" ID="ID_1360914257" MODIFIED="1669508005798" TEXT="Duration und Canvas-size k&#xf6;nnen gro&#xdf; sein">
<icon BUILTIN="messagebox_warning"/>
</node>
<node CREATED="1669507940458" ID="ID_355998747" MODIFIED="1669507940458" TEXT="Rat posFactor = canvasOffset / FSecs{afterAll_-startAll_};"/>
<node CREATED="1669507965470" ID="ID_1858701509" MODIFIED="1669507965470" TEXT="FSecs partBeforeAnchor = posFactor * duration;"/>
<node COLOR="#338800" CREATED="1670016112689" ID="ID_1869447563" LINK="#ID_289279043" MODIFIED="1670016131455" TEXT="hierf&#xfc;r eigene Hilfsfunktion geschaffen">
<icon BUILTIN="button_ok"/>
</node>
</node>
</node>
<node CREATED="1668267393360" ID="ID_1700728412" MODIFIED="1668267398176" TEXT="giftige Br&#xfc;che">
<node BACKGROUND_COLOR="#eee5c3" COLOR="#990000" CREATED="1669508070174" ID="ID_171647736" MODIFIED="1669508178509" TEXT="Aua: so ziemlich alles kann hier giftig sein">
<node COLOR="#435e98" CREATED="1669508070174" ID="ID_171647736" MODIFIED="1670016138169" TEXT="Aua: so ziemlich alles kann hier giftig sein">
<linktarget COLOR="#e32c65" DESTINATION="ID_171647736" ENDARROW="Default" ENDINCLINATION="8;76;" ID="Arrow_ID_1341085031" SOURCE="ID_196873456" STARTARROW="None" STARTINCLINATION="577;-36;"/>
<icon BUILTIN="broken-line"/>
<node CREATED="1669508081900" ID="ID_1740115162" MODIFIED="1669508382889" TEXT="zwar kann das Ergebnis nicht grob entgleisen">
@ -39922,11 +39925,15 @@
<node COLOR="#338800" CREATED="1669478154709" ID="ID_499422673" MODIFIED="1669488173372" TEXT="Fall-2 : hinreichend gro&#xdf;er Canvas">
<icon BUILTIN="button_ok"/>
</node>
<node BACKGROUND_COLOR="#eee5c3" COLOR="#990000" CREATED="1669479930775" ID="ID_196873456" MODIFIED="1669508183093" TEXT="Fall-3 : setVisiblePos mit giftigem faktor">
<node COLOR="#338800" CREATED="1669479930775" ID="ID_196873456" MODIFIED="1670016092876" TEXT="Fall-3 : setVisiblePos mit giftigem faktor">
<arrowlink COLOR="#e32c65" DESTINATION="ID_171647736" ENDARROW="Default" ENDINCLINATION="8;76;" ID="Arrow_ID_1341085031" STARTARROW="None" STARTINCLINATION="577;-36;"/>
<icon BUILTIN="flag-yellow"/>
<node BACKGROUND_COLOR="#eef0c5" COLOR="#990000" CREATED="1669682888293" HGAP="67" ID="ID_1431682003" MODIFIED="1669682917272" TEXT="erweist sich als ziemlich schwer unter Kontrolle zu bekommen" VSHIFT="-1">
<icon BUILTIN="button_ok"/>
<node COLOR="#435e98" CREATED="1669682888293" HGAP="67" ID="ID_1431682003" MODIFIED="1670016089041" TEXT="erweist sich als ziemlich schwer unter Kontrolle zu bekommen" VSHIFT="-1">
<edge COLOR="#01a450"/>
<icon BUILTIN="broken-line"/>
<node COLOR="#338800" CREATED="1670016009463" HGAP="38" ID="ID_413813658" LINK="#ID_264195573" MODIFIED="1670016072649" TEXT="daraus sind nun Support-Funktionen entstanden" VSHIFT="7">
<icon BUILTIN="idea"/>
</node>
</node>
</node>
<node BACKGROUND_COLOR="#eee5c3" COLOR="#990000" CREATED="1669480019215" ID="ID_930072077" MODIFIED="1669480034942" TEXT="Fall-4 : setVisiblePos mit extrem entfernter Position">
@ -40377,7 +40384,7 @@
</node>
<node CREATED="1669680416417" ID="ID_264195573" MODIFIED="1669680423082" TEXT="L&#xf6;sungs-Techniken">
<icon BUILTIN="idea"/>
<node CREATED="1669680424602" ID="ID_267834802" MODIFIED="1669682824592" TEXT="K&#xfc;rzungs-Trick">
<node COLOR="#435e98" CREATED="1669680424602" ID="ID_267834802" MODIFIED="1670015986498" TEXT="K&#xfc;rzungs-Trick">
<linktarget COLOR="#fbf8c7" DESTINATION="ID_267834802" ENDARROW="Default" ENDINCLINATION="538;-591;" ID="Arrow_ID_1957682133" SOURCE="ID_1644190283" STARTARROW="None" STARTINCLINATION="668;44;"/>
<node CREATED="1669680431912" ID="ID_180315930" MODIFIED="1669680447650" TEXT="einen Partner re-Quantisieren"/>
<node CREATED="1669680448333" ID="ID_969768863" MODIFIED="1669680473406" TEXT="so da&#xdf; sich dann kreuzweise ein Z&#xe4;hler/Nenner wegk&#xfc;rzen"/>
@ -40386,19 +40393,19 @@
<node COLOR="#338800" CREATED="1669682825470" ID="ID_906501375" MODIFIED="1669682838860" TEXT="verallgemeinert: scaleSafe()">
<icon BUILTIN="button_ok"/>
</node>
<node BACKGROUND_COLOR="#eee5c3" COLOR="#990000" CREATED="1669911724009" ID="ID_289279043" MODIFIED="1669911765633" TEXT="verallgemeinert: addSafe()">
<linktarget COLOR="#a32f55" DESTINATION="ID_289279043" ENDARROW="Default" ENDINCLINATION="167;16;" ID="Arrow_ID_333407901" SOURCE="ID_797370792" STARTARROW="None" STARTINCLINATION="-174;-8;"/>
<icon BUILTIN="flag-yellow"/>
<node COLOR="#338800" CREATED="1669911724009" ID="ID_289279043" MODIFIED="1670015959526" TEXT="verallgemeinert: addSafe()">
<linktarget COLOR="#2f7ca3" DESTINATION="ID_289279043" ENDARROW="Default" ENDINCLINATION="167;16;" ID="Arrow_ID_333407901" SOURCE="ID_797370792" STARTARROW="None" STARTINCLINATION="-174;-8;"/>
<icon BUILTIN="button_ok"/>
</node>
<node BACKGROUND_COLOR="#eee5c3" COLOR="#990000" CREATED="1669932544504" ID="ID_750418137" MODIFIED="1669932569275" TEXT="FSecs &#x27fc; &#xb5;-Tick absichern">
<icon BUILTIN="flag-yellow"/>
<node COLOR="#338800" CREATED="1669932544504" ID="ID_750418137" MODIFIED="1670015851933" TEXT="FSecs &#x27fc; &#xb5;-Tick absichern">
<icon BUILTIN="button_ok"/>
<node CREATED="1669932578600" ID="ID_974803158" MODIFIED="1669932591544" TEXT="das erweist sich als h&#xe4;ufige Quelle von wrap-arounds">
<icon BUILTIN="messagebox_warning"/>
</node>
<node BACKGROUND_COLOR="#e0ceaa" COLOR="#690f14" CREATED="1669932596081" ID="ID_1661291024" MODIFIED="1669932618249" TEXT="dabei ist es einfach blo&#xdf; naiv gecodet">
<icon BUILTIN="smiley-oh"/>
</node>
<node CREATED="1669932620086" ID="ID_673992117" MODIFIED="1669932631571" TEXT="stattdessen die re-Quantisierung nutzen">
<node COLOR="#435e98" CREATED="1669932620086" ID="ID_673992117" MODIFIED="1670015849357" TEXT="stattdessen die re-Quantisierung nutzen">
<icon BUILTIN="idea"/>
<node CREATED="1669932632788" ID="ID_831065868" MODIFIED="1669932761462" TEXT="nur dann hier auf 1/1000000">
<icon BUILTIN="yes"/>
@ -40409,7 +40416,7 @@
<node COLOR="#338800" CREATED="1669932653473" ID="ID_120882361" MODIFIED="1669932684292" TEXT="kann sogar den Rat-Fall hierauf zur&#xfc;ckf&#xfc;hren">
<icon BUILTIN="idea"/>
</node>
<node COLOR="#435e98" CREATED="1669932686823" ID="ID_1630699394" MODIFIED="1669932696052" TEXT="Numerik-Probleme">
<node COLOR="#435e98" CREATED="1669932686823" FOLDED="true" ID="ID_1630699394" MODIFIED="1669932696052" TEXT="Numerik-Probleme">
<icon BUILTIN="broken-line"/>
<node CREATED="1669932700323" ID="ID_1606021601" MODIFIED="1669932716796" TEXT="tja... Float bleibt Float selbst bei 128bit">
<icon BUILTIN="clanbomber"/>
@ -40501,6 +40508,9 @@
</node>
</node>
</node>
<node COLOR="#338800" CREATED="1670015941129" ID="ID_1262505892" MODIFIED="1670015955338" TEXT="kann nun (wohl) nicht mehr entgleisen">
<icon BUILTIN="button_ok"/>
</node>
</node>
</node>
</node>
@ -40511,8 +40521,8 @@
</node>
<node BACKGROUND_COLOR="#eee5c3" COLOR="#990000" CREATED="1668707479892" ID="ID_1964644327" MODIFIED="1668707490200" TEXT="ansonsten durch Ausreizen der Extremwerte testen">
<icon BUILTIN="flag-yellow"/>
<node BACKGROUND_COLOR="#eef0c5" COLOR="#990000" CREATED="1669682928809" ID="ID_499417740" MODIFIED="1669682938471" TEXT="setVisiblePos(gift)">
<icon BUILTIN="pencil"/>
<node COLOR="#338800" CREATED="1669682928809" ID="ID_499417740" MODIFIED="1670015827119" TEXT="setVisiblePos(gift)">
<icon BUILTIN="button_ok"/>
<node CREATED="1669682950712" ID="ID_1730356591" MODIFIED="1669682974366" TEXT="macht was Anderes als nur die Zoom-Metrik anpassen"/>
<node CREATED="1669682975415" ID="ID_1386857509" MODIFIED="1669682993901" TEXT="die Verbindung mit L&#xe4;ngen/Positions-Kalkulationen ist gef&#xe4;hrlich">
<icon BUILTIN="messagebox_warning"/>
@ -40520,7 +40530,7 @@
<node COLOR="#338800" CREATED="1669683004871" ID="ID_1503834170" MODIFIED="1669683025116" TEXT="wrap-around bekomme ich mit detox() und scaleSafe() in den Griff">
<icon BUILTIN="button_ok"/>
</node>
<node BACKGROUND_COLOR="#eee5c3" COLOR="#990000" CREATED="1669683026004" ID="ID_1867241917" MODIFIED="1669943704863" TEXT="Ergebnis ist aber noch nicht plausibel">
<node COLOR="#435e98" CREATED="1669683026004" ID="ID_1867241917" MODIFIED="1670015746454" TEXT="Ergebnis ist aber noch nicht plausibel">
<richcontent TYPE="NOTE"><html>
<head>
@ -40581,7 +40591,7 @@
</node>
</node>
<node COLOR="#435e98" CREATED="1669911687704" ID="ID_797370792" MODIFIED="1669943698789" TEXT="k&#xf6;nnte man im nun etablierten Schema auch ein addSafe() definieren">
<arrowlink COLOR="#a32f55" DESTINATION="ID_289279043" ENDARROW="Default" ENDINCLINATION="167;16;" ID="Arrow_ID_333407901" STARTARROW="None" STARTINCLINATION="-174;-8;"/>
<arrowlink COLOR="#2f7ca3" DESTINATION="ID_289279043" ENDARROW="Default" ENDINCLINATION="167;16;" ID="Arrow_ID_333407901" STARTARROW="None" STARTINCLINATION="-174;-8;"/>
<icon BUILTIN="help"/>
<node COLOR="#338800" CREATED="1669911787964" ID="ID_147783330" MODIFIED="1669911798248" TEXT="Erkennung und Behandlung der Grenzbedingungen analog">
<icon BUILTIN="button_ok"/>
@ -40677,7 +40687,7 @@
</node>
</node>
</node>
<node BACKGROUND_COLOR="#eee5c3" COLOR="#990000" CREATED="1669943663699" ID="ID_693461392" MODIFIED="1669943718965" TEXT="Ergebnis nun im Fehlerband &#x2014; aber leider immer noch daneben">
<node COLOR="#435e98" CREATED="1669943663699" ID="ID_693461392" MODIFIED="1670015348017" TEXT="Ergebnis nun im Fehlerband &#x2014; aber leider immer noch daneben">
<icon BUILTIN="messagebox_warning"/>
<node CREATED="1669943776106" ID="ID_1895721471" MODIFIED="1669943795208" TEXT="bei re-Quantisierung auf &#xb5;-Tick kann ich nicht genauer sein als 1e-6"/>
<node CREATED="1669943801006" ID="ID_1524475083" MODIFIED="1669943916288" TEXT="trotzdem will mir dieser Befund irgendwie logisch nicht in den Kopf">
@ -40693,7 +40703,7 @@
</html></richcontent>
<icon BUILTIN="smiley-angry"/>
</node>
<node BACKGROUND_COLOR="#f0d5c5" COLOR="#990033" CREATED="1669943918742" ID="ID_1902626416" MODIFIED="1669943941550" TEXT="der Fehler wird relativ zur Ergebnishzahl wirksam &#x2014; warum?">
<node COLOR="#435e98" CREATED="1669943918742" ID="ID_1902626416" MODIFIED="1669994406710" TEXT="der Fehler wird relativ zur Ergebnishzahl wirksam &#x2014; warum?">
<icon BUILTIN="help"/>
<node CREATED="1669948612766" ID="ID_773874014" MODIFIED="1669948622948" TEXT="der Fehler tritt nicht in der Subtraktion auf"/>
<node CREATED="1669948623670" ID="ID_549263730" MODIFIED="1669948642617" TEXT="sondern bereits beim Berechnen der Bezugs-Position"/>
@ -40708,6 +40718,41 @@
</p>
</body>
</html></richcontent>
<icon BUILTIN="forward"/>
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<head>
</head>
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<p>
...und damit die rationale Arithmetik als M&#246;glichkeit zu erhalten; sie w&#228;re nicht sinnvoll nutzbar, wenn man st&#228;ndig die Sorge haben m&#252;&#223;te, da&#223; einem die Zahlen &#187;explodieren&#171; k&#246;nnen. So ist es nun schon besser, im Regelfall eine hochpr&#228;zise Berechnung zu haben (pr&#228;ziser als floating-point), die in Grenzf&#228;llen ungenau wird, mit letztlich kontrollierbarem Fehler-Level
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<icon BUILTIN="yes"/>
<node CREATED="1670015358471" ID="ID_1559004584" MODIFIED="1670015734747" TEXT="hatte erst so meine Zweifel">
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<node CREATED="1670015367683" ID="ID_1687855073" MODIFIED="1670015724868" TEXT="jetzt bin ich froh, da&#xdf; ich&apos;s gemacht hab">
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</head>
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<p>
Es f&#252;hlt sich gut an, das Problem nun doch befriedigend bezwungen zu haben; ich hatte immer das Bauchgef&#252;hl, da&#223; da mehr Genauigkeit m&#246;glich sein sollte. Allerdings &#8212; wie sich zeigte &#8212; nur in g&#252;nstigen F&#228;llen
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<node BACKGROUND_COLOR="#e0ceaa" COLOR="#690f14" CREATED="1670015379779" ID="ID_636584956" MODIFIED="1670015481254" TEXT="...nun sind es eigentlich schon generische Library-Funktionen">
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<node CREATED="1669949048959" ID="ID_595183859" MODIFIED="1669949241962" TEXT="im konkreten Einzelfall k&#xf6;nnte man auch einen gr&#xf6;&#xdf;eren Faktor nehmen">
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@ -40720,12 +40765,134 @@
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<node CREATED="1669949243427" ID="ID_1428834110" MODIFIED="1669949269501" TEXT="eine solcherma&#xdf;en erh&#xf6;hte Komplexit&#xe4;t erscheint nicht gerechtfertigt"/>
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<p>
Fazit: es ist das <b>gesicherte Minimum</b>
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<node CREATED="1669994521693" ID="ID_983699676" MODIFIED="1670000518187" TEXT="mu&#xdf; dazu jeweils die Machbarkeit pr&#xfc;fen">
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<node CREATED="1669994530717" ID="ID_246811570" MODIFIED="1670000510847" TEXT="vier symmetrische F&#xe4;lle">
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<node COLOR="#338800" CREATED="1669994676320" ID="ID_277762782" MODIFIED="1669997271457" TEXT="der Fall mit Time::SCALE ist speziell">
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<p>
hier wissen wir, da&#223; wir den Quantiser ggfs hochskalieren k&#246;nnen auf 1e6 (er hat sich u.u mit der &#181;-Tick-Zahl gek&#252;rzt). Da es sich um eine valide Duration handelt, ist dieses Hochskalieren stets garantiert. Mithin ist dieser Faktor das garantierte Minimum (die schlechtest m&#246;gliche Genauigkeit &#8793; 1e-6
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<node COLOR="#338800" CREATED="1669994539603" ID="ID_429120779" MODIFIED="1670000489650" TEXT="als Klasse formulieren">
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<head>
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<p>
aus Gr&#252;nden der Symmetrie kann man das gleiche Argument jeweils auch auf den Kehrwert anwenden, deshalb haben wir ja 4 F&#228;lle (bei zwei Eingangs-Faktoren). Man mu&#223; nur ggfs. dann den Kehrwert vom Ergebnis ausgeben. Beispiel: wir nehmen den Z&#228;hler vom ersten Faktor als Quantisierer. Dann ist f1 der Nenner vom ersten Faktor, und mu&#223; daher auch im Ergebnis im Nenner landen, nicht im Z&#228;hler wie beim regul&#228;ren Schema
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<node COLOR="#338800" CREATED="1670000524698" ID="ID_229993223" MODIFIED="1670000571148" TEXT="...und damit ist das Ergebnis nun hinreichend genau f&#xfc;r diesen Testfall">
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<head>
</head>
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<p>
#--&#9670;--# _raw(targetPos) ?&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;= 206435633551724864
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<p>
#--&#9670;--# _raw(win.visible().start()) ? = 206435633106265625
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<p>
#--&#9670;--# _raw(win.visible().end()) ?&#160;&#160;&#160;= 206435633962616316
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<node BACKGROUND_COLOR="#eee5c3" COLOR="#990000" CREATED="1670016158206" ID="ID_1376502254" MODIFIED="1670016169130" TEXT="setVisiblePos(weit-weg)">
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@ -40748,6 +40915,10 @@
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<icon BUILTIN="hourglass"/>
<node BACKGROUND_COLOR="#eee5c3" COLOR="#990000" CREATED="1670015422301" ID="ID_1681510241" MODIFIED="1670015457619" TEXT="#1262 extract helpers for handling problematic factions">
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