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From: Jesús Carrete Montaña on 28 Aug 2006 03:09 GWB wrote: > Oops, I was wrong! Even Delphi extended type slips on this. > > Recalculating sin(3.141592654) as 3 sin(3.141592654/3) - > 4(sin(3.141592654/3))^3 I obtained -4.10206535406132E-10 (in Delphi). > This should be a more accurate result since the sine function would > have no problem in the pi/3 boundary. But I am not sure about this > result either because of rounding errors and other issues I am not > aware of. Could someone compute both sin(3.141592654) and the > trigonometric identity above in Maple to 30 places so we can see how > many digits match? With apcalc: config("epsilon",1e-100); # To be on the safe side. display(30) config("mode","exp") sin(3.141592654) # Gives -4.102067615373566167089928953970e-10. 3*sin(3.141592654/3)-4*sin(3.141592654/3)**3 # Also #gives -4.102067615373566167089928953970e-10. So the result obtained using the hp is correct to its precission. Just for fun, some more digits: config("epsilon",1e-1000) display(500) sin(3.141592654) #Gives -4.10206761537356616708992895396990923551313900371008706300203233699044266235232737732532009660484494153109030260754980676826986152238002680807920492955063221606617329395817871511104599839211170698745034243375467268878274373987950218374947168043141231591672762471453073839737252342896581262524175713904046508929945590534458569511685937555452897102991737765264461447922306865167920320804119829366266902993855256010952987603391962218753530044290662353728783426447411688680181736430818048702693811840353422e-10
From: Veli-Pekka Nousiainen on 28 Aug 2006 08:33 Jes?s Carrete Monta?a wrote: > GWB wrote: > >> Oops, I was wrong! Even Delphi extended type slips on this. >> >> Recalculating sin(3.141592654) as 3 sin(3.141592654/3) - >> 4(sin(3.141592654/3))^3 I obtained -4.10206535406132E-10 (in Delphi). >> This should be a more accurate result since the sine function would >> have no problem in the pi/3 boundary. But I am not sure about this >> result either because of rounding errors and other issues I am not >> aware of. Could someone compute both sin(3.141592654) and the >> trigonometric identity above in Maple to 30 places so we can see how >> many digits match? > > With apcalc: > > config("epsilon",1e-100); # To be on the safe side. > display(30) > config("mode","exp") > sin(3.141592654) # Gives -4.102067615373566167089928953970e-10. > 3*sin(3.141592654/3)-4*sin(3.141592654/3)**3 # Also > #gives -4.102067615373566167089928953970e-10. > > So the result obtained using the hp is correct to its precission. > > Just for fun, some more digits: > > config("epsilon",1e-1000) > display(500) > sin(3.141592654) > #Gives > -4.10206761537356616708992895396990923551313900371008706300203233699044266235232737732532009660484494153109030260754980676826986152238002680807920492955063221606617329395817871511104599839211170698745034243375467268878274373987950218374947168043141231591672762471453073839737252342896581262524175713904046508929945590534458569511685937555452897102991737765264461447922306865167920320804119829366266902993855256010952987603391962218753530044290662353728783426447411688680181736430818048702693811840353422e-10 Jes?s! That's a lot of digits HP-49 series has also DIGITS & LongFloat library What functions are available is a different story but I like to use it mostly for matrices -- Veli-Pekka
From: GWB on 28 Aug 2006 10:12 Zeno wrote: > Sorry, but both your Delphi answer are wrong > > What i stated before is correct I AM sorry. You're pretty right. HP answer is correct as we can see by Paul Schlyter's and Jesús Montaña's results. Looks like I cannot use my Delphi Calculator to track satellite's orbits anymore :-) (will rely on my good old 42S, 48GX or 49G for that task ) Regards, Gerson.
From: GWB on 28 Aug 2006 11:37 Jesús Carrete Montaña wrote: > GWB wrote: > > > Oops, I was wrong! Even Delphi extended type slips on this. > > > > Recalculating sin(3.141592654) as 3 sin(3.141592654/3) - > > 4(sin(3.141592654/3))^3 I obtained -4.10206535406132E-10 (in Delphi). > > This should be a more accurate result since the sine function would > > have no problem in the pi/3 boundary. But I am not sure about this > > result either because of rounding errors and other issues I am not > > aware of. Could someone compute both sin(3.141592654) and the > > trigonometric identity above in Maple to 30 places so we can see how > > many digits match? > > With apcalc: > > config("epsilon",1e-100); # To be on the safe side. > display(30) > config("mode","exp") > sin(3.141592654) # Gives -4.102067615373566167089928953970e-10. > 3*sin(3.141592654/3)-4*sin(3.141592654/3)**3 # Also > #gives -4.102067615373566167089928953970e-10. > > So the result obtained using the hp is correct to its precission. > > Just for fun, some more digits: > > config("epsilon",1e-1000) > display(500) > sin(3.141592654) > #Gives -4.10206761537356616708992895396990923551313900371008706300203233699044266235232737732532009660484494153109030260754980676826986152238002680807920492955063221606617329395817871511104599839211170698745034243375467268878274373987950218374947168043141231591672762471453073839737252342896581262524175713904046508929945590534458569511685937555452897102991737765264461447922306865167920320804119829366266902993855256010952987603391962218753530044290662353728783426447411688680181736430818048702693811840353422e-10 Muchas gracias, Jesús! Gerson
From: Fv on 28 Aug 2006 14:27
Explanation: Since the slope of sin is 1 where it crosses the x axis, sin (Pi + tiny number) = tiny number In this case the "tiny number" is simply = 3.141592654 - Pi = 4.1020676153735661672049711580283060062489417902505540769218359371379100137....*10^-10 So, of course the TI answer is correct too, since it corresponds to 14 sig. dig.. To justify the accuracy given by HP, both Pi and 3.141592654 would have to be given to about 22 digits (that would be something like 3.141592654000000000000 and 3.141592653589793238463). Of course HP does not in general compute to this accuracy. HP algorithm gives an accurate answer that is not justified by the number of digits it can handle in general, and this is OK in a calculator. It would be perfect if HP could give arbitrary accuracy numbers or machine accuracy results depending on a setting. "GWB" <gerson.w.barbosa(a)gmail.com> wrote in message news:1156779438.616210.261840(a)m79g2000cwm.googlegroups.com... Jes?s Carrete Monta?a wrote: > GWB wrote: > > > Oops, I was wrong! Even Delphi extended type slips on this. > > > > Recalculating sin(3.141592654) as 3 sin(3.141592654/3) - > > 4(sin(3.141592654/3))^3 I obtained -4.10206535406132E-10 (in Delphi). > > This should be a more accurate result since the sine function would > > have no problem in the pi/3 boundary. But I am not sure about this > > result either because of rounding errors and other issues I am not > > aware of. Could someone compute both sin(3.141592654) and the > > trigonometric identity above in Maple to 30 places so we can see how > > many digits match? > > With apcalc: > > config("epsilon",1e-100); # To be on the safe side. > display(30) > config("mode","exp") > sin(3.141592654) # Gives -4.102067615373566167089928953970e-10. > 3*sin(3.141592654/3)-4*sin(3.141592654/3)**3 # Also > #gives -4.102067615373566167089928953970e-10. > > So the result obtained using the hp is correct to its precission. > > Just for fun, some more digits: > > config("epsilon",1e-1000) > display(500) > sin(3.141592654) > #Gives -4.10206761537356616708992895396990923551313900371008706300203233699044266235232737732532009660484494153109030260754980676826986152238002680807920492955063221606617329395817871511104599839211170698745034243375467268878274373987950218374947168043141231591672762471453073839737252342896581262524175713904046508929945590534458569511685937555452897102991737765264461447922306865167920320804119829366266902993855256010952987603391962218753530044290662353728783426447411688680181736430818048702693811840353422e-10 Muchas gracias, Jes?s! Gerson |