Do not trust Floating Point - Versionsgeschichte

Cg: /* Example 2 */

2024-06-17T14:42:44Z

Example 2

← Nächstältere Version		Version vom 17. Juni 2024, 14:42 Uhr
Zeile 182:		Zeile 182:
	(However, be reminded that LargeFloat and OctaFloat arithmetic is much slower)		(However, be reminded that LargeFloat and OctaFloat arithmetic is much slower)

	2) see [https://people.eecs.berkeley.edu/~wkahan/JAVAhurt.pdf "How Java’s Floating-Point Hurts Everyone Everywhere"]		2) see [https://people.eecs.berkeley.edu/~wkahan/JAVAhurt.pdf "How Java’s Floating-Point Hurts Everyone Everywhere pg35"]

Cg: /* Example 2 */

2024-06-17T14:41:15Z

Example 2

← Nächstältere Version		Version vom 17. Juni 2024, 14:41 Uhr
Zeile 174:		Zeile 174:

	x := LargeFloat readFrom:'0.02' precision:400. "/ LargeFloat-400		x := LargeFloat readFrom:'0.02' precision:400. "/ LargeFloat-400
			=> 0.0333486812981771543916836365420930826209673623104313000386284441111102503364815348...
	=> 0.0333486812981771543916836365420930826209673623104313000386284441111102503364815348730355721274824080136352822772341652
	0.0333486812981771543916836365420930826209673623104313000386284441...		0.0333486812981771543916836365420930826209673623104313000386284441...

Cg: /* Example 2 */

2024-06-17T14:40:43Z

Example 2

← Nächstältere Version		Version vom 17. Juni 2024, 14:40 Uhr
Zeile 173:		Zeile 173:
	0.033348681298177154391683636542093082620967362310431300038628\|4441...		0.033348681298177154391683636542093082620967362310431300038628\|4441...

	x := LargeFloat readFrom:'0.02' precision:400. "/ LargeFloat-400		x := LargeFloat readFrom:'0.02' precision:400. "/ LargeFloat-400
	=> 0.0333486812981771543916836365420930826209673623104313000386284441111102503364815348730355721274824080136352822772341652		=> 0.0333486812981771543916836365420930826209673623104313000386284441111102503364815348730355721274824080136352822772341652
	0.0333486812981771543916836365420930826209673623104313000386284441...		0.0333486812981771543916836365420930826209673623104313000386284441...

	Notice, that currently QDouble is less accurate than LargeFloat-200, despite its slightly higher number of bits.		Notice, that (currently?) QDouble is less accurate than LargeFloat-200, despite its slightly higher number of bits.
	This seems to result from internal rounding errors which accumulate, whereas LargeFloat first computes the trigonometric function with a higher precision, and rounds at the end.		This seems to result from internal rounding errors which accumulate, whereas LargeFloat first computes the trigonometric function with a higher precision, and rounds at the end. This will likely remain as-is, as the open-source code for QDouble is used as-is.

	(However, be reminded that LargeFloat and OctaFloat arithmetic is much slower)		(However, be reminded that LargeFloat and OctaFloat arithmetic is much slower)

Cg: /* Example 2 */

2024-06-17T14:39:12Z

Example 2

← Nächstältere Version		Version vom 17. Juni 2024, 14:39 Uhr
Zeile 149:		Zeile 149:
	"0.02QO" for float256 and "0.02QL" for LargeFloat (which defaults to 200 bits),		"0.02QO" for float256 and "0.02QL" for LargeFloat (which defaults to 200 bits),
	we get:		we get:
	x := 0.02. "/ 64 bits		x := 0.02. "/ Float: 64 bits
	=> 0.03334\|73483202229		=> 0.03334\|73483202229
	0.03334\|86812981771543916836365420930826209673623104313000386284441		0.03334\|86812981771543916836365420930826209673623104313000386284441

	x := 0.02q. "/ 80 bits		x := 0.02q. "/ LongFloat: 80 bits
	=> 0.033348681\|07362584807		=> 0.033348681\|07362584807
	0.033348681\|2981771543916836365420930826209673623104313000386284441		0.033348681\|2981771543916836365420930826209673623104313000386284441

	x := QuadFloat fromString:'0.02'. "/ 128 bits		x := QuadFloat fromString:'0.02'. "/ QuadFloat: 128 bits
	=> 0.03334868129817715439168\|58584082		=> 0.03334868129817715439168\|58584082
	0.03334868129817715439168\|36365420930826209673623104313000386284441		0.03334868129817715439168\|36365420930826209673623104313000386284441

	x := LargeFloat fromString:'0.02'. "/ defaults to 200 bits		x := LargeFloat fromString:'0.02'. "/ LargeFloat: defaults to 200 bits
	=> 0.0333486812981771543916836365420930826209673623104\|45685286582		=> 0.0333486812981771543916836365420930826209673623104\|45685286582
	0.0333486812981771543916836365420930826209673623104\|313000386284441		0.0333486812981771543916836365420930826209673623104\|313000386284441

	x := 0.02QD. "/ 204 bits		x := 0.02QD. "/ QDouble: 204 bits
	=> 0.03334868129817715439168363654209308262096\|68260470426318400894		=> 0.03334868129817715439168363654209308262096\|68260470426318400894
	0.03334868129817715439168363654209308262096\|73623104313000386284441		0.03334868129817715439168363654209308262096\|73623104313000386284441

	x := OctaFloat fromString:'0.02'. "/ 256 bits		x := OctaFloat fromString:'0.02'. "/ OctaFloat: 256 bits
	=> 0.033348681298177154391683636542093082620967362310431300038628\|8528889811		=> 0.033348681298177154391683636542093082620967362310431300038628\|8528889811
	0.033348681298177154391683636542093082620967362310431300038628\|4441...		0.033348681298177154391683636542093082620967362310431300038628\|4441...

	x := LargeFloat readFrom:'0.02' precision:400.		x := LargeFloat readFrom:'0.02' precision:400. "/ LargeFloat-400
	=> 0.0333486812981771543916836365420930826209673623104313000386284441111102503364815348730355721274824080136352822772341652		=> 0.0333486812981771543916836365420930826209673623104313000386284441111102503364815348730355721274824080136352822772341652
	0.0333486812981771543916836365420930826209673623104313000386284441...		0.0333486812981771543916836365420930826209673623104313000386284441...

Cg: /* Example 2 */

2024-06-17T14:37:09Z

Example 2

← Nächstältere Version		Version vom 17. Juni 2024, 14:37 Uhr
Zeile 172:		Zeile 172:
	=> 0.033348681298177154391683636542093082620967362310431300038628\|8528889811		=> 0.033348681298177154391683636542093082620967362310431300038628\|8528889811
	0.033348681298177154391683636542093082620967362310431300038628\|4441...		0.033348681298177154391683636542093082620967362310431300038628\|4441...


	x := LargeFloat readFrom:'0.02' precision:400.		x := LargeFloat readFrom:'0.02' precision:400.
	=> 0.0333486812981771543916836365420930826209673623104313000386284441111102503364815348730355721274824080136352822772341652		=> 0.0333486812981771543916836365420930826209673623104313000386284441111102503364815348730355721274824080136352822772341652
Zeile 181:		Zeile 181:
	(However, be reminded that LargeFloat and OctaFloat arithmetic is much slower)		(However, be reminded that LargeFloat and OctaFloat arithmetic is much slower)

	2) see [https://people.eecs.berkeley.edu/~wkahan/JAVAhurt.pdf]		2) see [https://people.eecs.berkeley.edu/~wkahan/JAVAhurt.pdf "How Java’s Floating-Point Hurts Everyone Everywhere"]

Cg: /* Example 2 */

2024-06-17T14:35:53Z

Example 2

← Nächstältere Version		Version vom 17. Juni 2024, 14:35 Uhr
Zeile 130:		Zeile 130:
	As an example, take		As an example, take
	( tan(sin(x)) - sin(tan(x)) ) / x<sup>7</sup>		( tan(sin(x)) - sin(tan(x)) ) / x<sup>7</sup>
	with:		with:
	x = 0.02		x = 0.02

Cg: /* Exampe 2 */

2024-06-17T14:35:40Z

Exampe 2

← Nächstältere Version		Version vom 17. Juni 2024, 14:35 Uhr
Zeile 122:		Zeile 122:
	1) see [https://books.google.de/books?id=ZqnuDwAAQBAJ&pg=PT244&lpg=PT244&dq=rumps+royal+pain&source=bl&ots=mQeqSQWHtM&sig=ACfU3U3DLU3vpu4yMlx99xFDeorzlXLGEA&hl=de&sa=X&ved=2ahUKEwiEhIjmlPL0AhVdS_EDHSw8A10Q6AF6BAgPEAM#v=onepage&q=rumps%20royal%20pain&f=false Rump's Pain in books.google]		1) see [https://books.google.de/books?id=ZqnuDwAAQBAJ&pg=PT244&lpg=PT244&dq=rumps+royal+pain&source=bl&ots=mQeqSQWHtM&sig=ACfU3U3DLU3vpu4yMlx99xFDeorzlXLGEA&hl=de&sa=X&ved=2ahUKEwiEhIjmlPL0AhVdS_EDHSw8A10Q6AF6BAgPEAM#v=onepage&q=rumps%20royal%20pain&f=false Rump's Pain in books.google]

	== ~~Exampe~~ 2 ==		== Example 2 ==

	Trigonometric functions are computed by Taylor- or Powerseries,		Trigonometric functions are computed by Taylor- or Powerseries,

Cg: /* An Example */

2024-06-17T14:35:18Z

An Example

@@ Zeile 4: / Zeile 4: @@
 This may sound strange, but you have to be aware all the time, that computations involving floating point numbers are always in danger of being inexact and sometimes completely (absurdly) wrong. It is a good idea (i.e. highly recommended) that you check the expected error, precision and value ranges when doing floating point math.
-== An Example ==
+== Example 1 ==
 === Rump's Royal Pain<sup>1)</sup> ===
@@ Zeile 121: / Zeile 121: @@
 ) see [https://books.google.de/books?id=ZqnuDwAAQBAJ&pg=PT244&lpg=PT244&dq=rumps+royal+pain&source=bl&ots=mQeqSQWHtM&sig=ACfU3U3DLU3vpu4yMlx99xFDeorzlXLGEA&hl=de&sa=X&ved=2ahUKEwiEhIjmlPL0AhVdS_EDHSw8A10Q6AF6BAgPEAM#v=onepage&q=rumps%20royal%20pain&f=false Rump's Pain in books.google]

Cg: /* Rump's Royal Pain1) */

2024-06-15T12:28:15Z

Rump's Royal Pain1)

← Nächstältere Version		Version vom 15. Juni 2024, 12:28 Uhr
Zeile 82:		Zeile 82:
	\|x := 77617f.<br>y := 33096f.		\|x := 77617f.<br>y := 33096f.
	\| -1.18059162071741e+21		\| -1.18059162071741e+21
	\|WRONG		\|WRONG (ABSURDLY)
	\|-		\|-
	\|IEEE Double		\|IEEE Double
	\|x := 77617.<br>y := 33096.		\|x := 77617.<br>y := 33096.
	\| 1.18059162071741e+21		\| 1.18059162071741e+21
	\|WRONG		\|WRONG (ABSURDLY)
	\|-		\|-
	\|IEEE Extended		\|IEEE Extended
	\|x := 77617q.<br>y := 33096q.		\|x := 77617q.<br>y := 33096q.
	\| 576460752303423489.2		\| 576460752303423489.2
	\|WRONG		\|WRONG (still way off)
	\|-		\|-
	\|IEEE Quadruple		\|IEEE Quadruple
	\|x := 77617Q.<br>y := 33096Q.		\|x := 77617Q.<br>y := 33096Q.
	\| 1.17260394005317863185883490452		\| 1.17260394005317863185883490452
	\|WRONG		\|WRONG (coming closer)
	\|-		\|-
	\|IEEE Octuple		\|IEEE Octuple

Cg: /* Rump's Royal Pain1) */

2022-01-29T11:31:09Z

Rump's Royal Pain1)

← Nächstältere Version		Version vom 29. Januar 2022, 11:31 Uhr
Zeile 116:		Zeile 116:
	\|Fractions		\|Fractions
	\|x := 77617.<br>y := 33096.		\|x := 77617.<br>y := 33096.
	\| (-54767/66192)<br>which ~~asFloat~~ is -0.~~827396059946821~~		\| (-54767/66192)<br>which asLargeFloat is -0.8273960599468213681411650954...
	\|EXACT		\|EXACT
	\|}		\|}