What is the best approximation for sine?A 1,400 years old approximation to the sine function by...

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What is the best approximation for sine?


A 1,400 years old approximation to the sine function by Mahabhaskariya of Bhaskara IWhat is the purpose of Stirling's approximation to a factorial?How to justify small angle approximation for cosineFinding good approximation for $x^{1/2.4}$A 1,400 years old approximation to the sine function by Mahabhaskariya of Bhaskara Ileast squres vs. lagrange interpolationHow is the derivative truly, literally the “best linear approximation” near a point?Approximation of $e^x$Looking for a better (geometric) approximation of $pi$…single equation that can approximate sine, triangle, and square waves by changing a single variableInverse of Bhāskara I's sine approximation













0












$begingroup$


Can you tell me which is the best approximation for cosine/sine functions. It should also reduce the computational complexity. I've already tried the Bhaskara I's approximation.



Can you suggest me anything better?



Thanks in advance.






approximation transcendental-equations







share|cite|improve this question











$endgroup$












  • $begingroup$
    How accurate do you want it to be, how simple (or complex), and what operations do you allow? Also, do you want it in fixed or floating point?
    $endgroup$
    – marty cohen
    May 11 '15 at 6:13










  • $begingroup$
    I can afford one multiplication, and an addition/subtraction. And I want it for fixed point implementaion
    $endgroup$
    – phanitej
    May 11 '15 at 6:16










  • $begingroup$
    What range of values?
    $endgroup$
    – marty cohen
    May 11 '15 at 6:20










  • $begingroup$
    Actually i want it for simultaneous sine and cosine functions, with the input real/ imag values in between -1 and +1. I want to use these functions for discrete fourier transform
    $endgroup$
    – phanitej
    May 11 '15 at 6:23










  • $begingroup$
    Looks like you are doing this on a quite limited processor. What are the inputs to your computation? From all your restrictions, it's starting to look like the only possibility would be table lookup, perhaps combined with linear interpolation.
    $endgroup$
    – marty cohen
    May 11 '15 at 6:27
















0












$begingroup$


Can you tell me which is the best approximation for cosine/sine functions. It should also reduce the computational complexity. I've already tried the Bhaskara I's approximation.



Can you suggest me anything better?



Thanks in advance.






approximation transcendental-equations







share|cite|improve this question











$endgroup$












  • $begingroup$
    How accurate do you want it to be, how simple (or complex), and what operations do you allow? Also, do you want it in fixed or floating point?
    $endgroup$
    – marty cohen
    May 11 '15 at 6:13










  • $begingroup$
    I can afford one multiplication, and an addition/subtraction. And I want it for fixed point implementaion
    $endgroup$
    – phanitej
    May 11 '15 at 6:16










  • $begingroup$
    What range of values?
    $endgroup$
    – marty cohen
    May 11 '15 at 6:20










  • $begingroup$
    Actually i want it for simultaneous sine and cosine functions, with the input real/ imag values in between -1 and +1. I want to use these functions for discrete fourier transform
    $endgroup$
    – phanitej
    May 11 '15 at 6:23










  • $begingroup$
    Looks like you are doing this on a quite limited processor. What are the inputs to your computation? From all your restrictions, it's starting to look like the only possibility would be table lookup, perhaps combined with linear interpolation.
    $endgroup$
    – marty cohen
    May 11 '15 at 6:27














0












0








0





$begingroup$


Can you tell me which is the best approximation for cosine/sine functions. It should also reduce the computational complexity. I've already tried the Bhaskara I's approximation.



Can you suggest me anything better?



Thanks in advance.






approximation transcendental-equations







share|cite|improve this question











$endgroup$




Can you tell me which is the best approximation for cosine/sine functions. It should also reduce the computational complexity. I've already tried the Bhaskara I's approximation.



Can you suggest me anything better?



Thanks in advance.






approximation transcendental-equations




approximation transcendental-equations






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 22 hours ago









Rócherz

2,8562821




2,8562821










asked May 11 '15 at 6:08









phanitejphanitej

567




567












  • $begingroup$
    How accurate do you want it to be, how simple (or complex), and what operations do you allow? Also, do you want it in fixed or floating point?
    $endgroup$
    – marty cohen
    May 11 '15 at 6:13










  • $begingroup$
    I can afford one multiplication, and an addition/subtraction. And I want it for fixed point implementaion
    $endgroup$
    – phanitej
    May 11 '15 at 6:16










  • $begingroup$
    What range of values?
    $endgroup$
    – marty cohen
    May 11 '15 at 6:20










  • $begingroup$
    Actually i want it for simultaneous sine and cosine functions, with the input real/ imag values in between -1 and +1. I want to use these functions for discrete fourier transform
    $endgroup$
    – phanitej
    May 11 '15 at 6:23










  • $begingroup$
    Looks like you are doing this on a quite limited processor. What are the inputs to your computation? From all your restrictions, it's starting to look like the only possibility would be table lookup, perhaps combined with linear interpolation.
    $endgroup$
    – marty cohen
    May 11 '15 at 6:27


















  • $begingroup$
    How accurate do you want it to be, how simple (or complex), and what operations do you allow? Also, do you want it in fixed or floating point?
    $endgroup$
    – marty cohen
    May 11 '15 at 6:13










  • $begingroup$
    I can afford one multiplication, and an addition/subtraction. And I want it for fixed point implementaion
    $endgroup$
    – phanitej
    May 11 '15 at 6:16










  • $begingroup$
    What range of values?
    $endgroup$
    – marty cohen
    May 11 '15 at 6:20










  • $begingroup$
    Actually i want it for simultaneous sine and cosine functions, with the input real/ imag values in between -1 and +1. I want to use these functions for discrete fourier transform
    $endgroup$
    – phanitej
    May 11 '15 at 6:23










  • $begingroup$
    Looks like you are doing this on a quite limited processor. What are the inputs to your computation? From all your restrictions, it's starting to look like the only possibility would be table lookup, perhaps combined with linear interpolation.
    $endgroup$
    – marty cohen
    May 11 '15 at 6:27
















$begingroup$
How accurate do you want it to be, how simple (or complex), and what operations do you allow? Also, do you want it in fixed or floating point?
$endgroup$
– marty cohen
May 11 '15 at 6:13




$begingroup$
How accurate do you want it to be, how simple (or complex), and what operations do you allow? Also, do you want it in fixed or floating point?
$endgroup$
– marty cohen
May 11 '15 at 6:13












$begingroup$
I can afford one multiplication, and an addition/subtraction. And I want it for fixed point implementaion
$endgroup$
– phanitej
May 11 '15 at 6:16




$begingroup$
I can afford one multiplication, and an addition/subtraction. And I want it for fixed point implementaion
$endgroup$
– phanitej
May 11 '15 at 6:16












$begingroup$
What range of values?
$endgroup$
– marty cohen
May 11 '15 at 6:20




$begingroup$
What range of values?
$endgroup$
– marty cohen
May 11 '15 at 6:20












$begingroup$
Actually i want it for simultaneous sine and cosine functions, with the input real/ imag values in between -1 and +1. I want to use these functions for discrete fourier transform
$endgroup$
– phanitej
May 11 '15 at 6:23




$begingroup$
Actually i want it for simultaneous sine and cosine functions, with the input real/ imag values in between -1 and +1. I want to use these functions for discrete fourier transform
$endgroup$
– phanitej
May 11 '15 at 6:23












$begingroup$
Looks like you are doing this on a quite limited processor. What are the inputs to your computation? From all your restrictions, it's starting to look like the only possibility would be table lookup, perhaps combined with linear interpolation.
$endgroup$
– marty cohen
May 11 '15 at 6:27




$begingroup$
Looks like you are doing this on a quite limited processor. What are the inputs to your computation? From all your restrictions, it's starting to look like the only possibility would be table lookup, perhaps combined with linear interpolation.
$endgroup$
– marty cohen
May 11 '15 at 6:27










2 Answers
2






active

oldest

votes


















1












$begingroup$

For $-pile x le pi $ I found $$left(frac{315}{2}pi^2 - frac{15}{2pi^2} right)x + frac{175}{2pi^6}left( frac{pi^2}{5}-3right)x^3,$$ is it of any help?






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Ya it is of help , but i need to do a modulo operation, to wrap the x^3 back into 0<x<pi.
    $endgroup$
    – phanitej
    May 11 '15 at 6:33










  • $begingroup$
    @phanitej But does it really work? I had just found it on the net, but I'm afraid it's incorrect. Though, I can't check now.
    $endgroup$
    – Vincenzo Oliva
    May 11 '15 at 6:46












  • $begingroup$
    I suppose typo's in this formula. Try using $x=frac pi 2$ or $x=pi$. Could you tell where you did find it ?
    $endgroup$
    – Claude Leibovici
    May 11 '15 at 9:02










  • $begingroup$
    @ClaudeLeibovici Yeah, I checked that once but couldn't doublecheck as I'm at school, so I thought I had typed something wrong in the calculator. It was the result of an Italian graduand, confirmed by who looked like his professor. I guess you're right about the typo.
    $endgroup$
    – Vincenzo Oliva
    May 11 '15 at 10:22












  • $begingroup$
    @phanitej I'll leave this answer here in case someone finds the typo. In the meanwhile, I'll post another answer with a correct inequality.
    $endgroup$
    – Vincenzo Oliva
    May 11 '15 at 10:58



















0












$begingroup$

Hopefully you're interested in the following double inequality, valid for $0le xlepi$: $$xleft(1-frac{x}{pi}right)lesin xle frac{4x}{pi}left(1-frac{x}{pi}right) $$






share|cite|improve this answer









$endgroup$













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    2 Answers
    2






    active

    oldest

    votes








    2 Answers
    2






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    1












    $begingroup$

    For $-pile x le pi $ I found $$left(frac{315}{2}pi^2 - frac{15}{2pi^2} right)x + frac{175}{2pi^6}left( frac{pi^2}{5}-3right)x^3,$$ is it of any help?






    share|cite|improve this answer









    $endgroup$













    • $begingroup$
      Ya it is of help , but i need to do a modulo operation, to wrap the x^3 back into 0<x<pi.
      $endgroup$
      – phanitej
      May 11 '15 at 6:33










    • $begingroup$
      @phanitej But does it really work? I had just found it on the net, but I'm afraid it's incorrect. Though, I can't check now.
      $endgroup$
      – Vincenzo Oliva
      May 11 '15 at 6:46












    • $begingroup$
      I suppose typo's in this formula. Try using $x=frac pi 2$ or $x=pi$. Could you tell where you did find it ?
      $endgroup$
      – Claude Leibovici
      May 11 '15 at 9:02










    • $begingroup$
      @ClaudeLeibovici Yeah, I checked that once but couldn't doublecheck as I'm at school, so I thought I had typed something wrong in the calculator. It was the result of an Italian graduand, confirmed by who looked like his professor. I guess you're right about the typo.
      $endgroup$
      – Vincenzo Oliva
      May 11 '15 at 10:22












    • $begingroup$
      @phanitej I'll leave this answer here in case someone finds the typo. In the meanwhile, I'll post another answer with a correct inequality.
      $endgroup$
      – Vincenzo Oliva
      May 11 '15 at 10:58
















    1












    $begingroup$

    For $-pile x le pi $ I found $$left(frac{315}{2}pi^2 - frac{15}{2pi^2} right)x + frac{175}{2pi^6}left( frac{pi^2}{5}-3right)x^3,$$ is it of any help?






    share|cite|improve this answer









    $endgroup$













    • $begingroup$
      Ya it is of help , but i need to do a modulo operation, to wrap the x^3 back into 0<x<pi.
      $endgroup$
      – phanitej
      May 11 '15 at 6:33










    • $begingroup$
      @phanitej But does it really work? I had just found it on the net, but I'm afraid it's incorrect. Though, I can't check now.
      $endgroup$
      – Vincenzo Oliva
      May 11 '15 at 6:46












    • $begingroup$
      I suppose typo's in this formula. Try using $x=frac pi 2$ or $x=pi$. Could you tell where you did find it ?
      $endgroup$
      – Claude Leibovici
      May 11 '15 at 9:02










    • $begingroup$
      @ClaudeLeibovici Yeah, I checked that once but couldn't doublecheck as I'm at school, so I thought I had typed something wrong in the calculator. It was the result of an Italian graduand, confirmed by who looked like his professor. I guess you're right about the typo.
      $endgroup$
      – Vincenzo Oliva
      May 11 '15 at 10:22












    • $begingroup$
      @phanitej I'll leave this answer here in case someone finds the typo. In the meanwhile, I'll post another answer with a correct inequality.
      $endgroup$
      – Vincenzo Oliva
      May 11 '15 at 10:58














    1












    1








    1





    $begingroup$

    For $-pile x le pi $ I found $$left(frac{315}{2}pi^2 - frac{15}{2pi^2} right)x + frac{175}{2pi^6}left( frac{pi^2}{5}-3right)x^3,$$ is it of any help?






    share|cite|improve this answer









    $endgroup$



    For $-pile x le pi $ I found $$left(frac{315}{2}pi^2 - frac{15}{2pi^2} right)x + frac{175}{2pi^6}left( frac{pi^2}{5}-3right)x^3,$$ is it of any help?







    share|cite|improve this answer












    share|cite|improve this answer



    share|cite|improve this answer










    answered May 11 '15 at 6:31









    Vincenzo OlivaVincenzo Oliva

    5,32611234




    5,32611234












    • $begingroup$
      Ya it is of help , but i need to do a modulo operation, to wrap the x^3 back into 0<x<pi.
      $endgroup$
      – phanitej
      May 11 '15 at 6:33










    • $begingroup$
      @phanitej But does it really work? I had just found it on the net, but I'm afraid it's incorrect. Though, I can't check now.
      $endgroup$
      – Vincenzo Oliva
      May 11 '15 at 6:46












    • $begingroup$
      I suppose typo's in this formula. Try using $x=frac pi 2$ or $x=pi$. Could you tell where you did find it ?
      $endgroup$
      – Claude Leibovici
      May 11 '15 at 9:02










    • $begingroup$
      @ClaudeLeibovici Yeah, I checked that once but couldn't doublecheck as I'm at school, so I thought I had typed something wrong in the calculator. It was the result of an Italian graduand, confirmed by who looked like his professor. I guess you're right about the typo.
      $endgroup$
      – Vincenzo Oliva
      May 11 '15 at 10:22












    • $begingroup$
      @phanitej I'll leave this answer here in case someone finds the typo. In the meanwhile, I'll post another answer with a correct inequality.
      $endgroup$
      – Vincenzo Oliva
      May 11 '15 at 10:58


















    • $begingroup$
      Ya it is of help , but i need to do a modulo operation, to wrap the x^3 back into 0<x<pi.
      $endgroup$
      – phanitej
      May 11 '15 at 6:33










    • $begingroup$
      @phanitej But does it really work? I had just found it on the net, but I'm afraid it's incorrect. Though, I can't check now.
      $endgroup$
      – Vincenzo Oliva
      May 11 '15 at 6:46












    • $begingroup$
      I suppose typo's in this formula. Try using $x=frac pi 2$ or $x=pi$. Could you tell where you did find it ?
      $endgroup$
      – Claude Leibovici
      May 11 '15 at 9:02










    • $begingroup$
      @ClaudeLeibovici Yeah, I checked that once but couldn't doublecheck as I'm at school, so I thought I had typed something wrong in the calculator. It was the result of an Italian graduand, confirmed by who looked like his professor. I guess you're right about the typo.
      $endgroup$
      – Vincenzo Oliva
      May 11 '15 at 10:22












    • $begingroup$
      @phanitej I'll leave this answer here in case someone finds the typo. In the meanwhile, I'll post another answer with a correct inequality.
      $endgroup$
      – Vincenzo Oliva
      May 11 '15 at 10:58
















    $begingroup$
    Ya it is of help , but i need to do a modulo operation, to wrap the x^3 back into 0<x<pi.
    $endgroup$
    – phanitej
    May 11 '15 at 6:33




    $begingroup$
    Ya it is of help , but i need to do a modulo operation, to wrap the x^3 back into 0<x<pi.
    $endgroup$
    – phanitej
    May 11 '15 at 6:33












    $begingroup$
    @phanitej But does it really work? I had just found it on the net, but I'm afraid it's incorrect. Though, I can't check now.
    $endgroup$
    – Vincenzo Oliva
    May 11 '15 at 6:46






    $begingroup$
    @phanitej But does it really work? I had just found it on the net, but I'm afraid it's incorrect. Though, I can't check now.
    $endgroup$
    – Vincenzo Oliva
    May 11 '15 at 6:46














    $begingroup$
    I suppose typo's in this formula. Try using $x=frac pi 2$ or $x=pi$. Could you tell where you did find it ?
    $endgroup$
    – Claude Leibovici
    May 11 '15 at 9:02




    $begingroup$
    I suppose typo's in this formula. Try using $x=frac pi 2$ or $x=pi$. Could you tell where you did find it ?
    $endgroup$
    – Claude Leibovici
    May 11 '15 at 9:02












    $begingroup$
    @ClaudeLeibovici Yeah, I checked that once but couldn't doublecheck as I'm at school, so I thought I had typed something wrong in the calculator. It was the result of an Italian graduand, confirmed by who looked like his professor. I guess you're right about the typo.
    $endgroup$
    – Vincenzo Oliva
    May 11 '15 at 10:22






    $begingroup$
    @ClaudeLeibovici Yeah, I checked that once but couldn't doublecheck as I'm at school, so I thought I had typed something wrong in the calculator. It was the result of an Italian graduand, confirmed by who looked like his professor. I guess you're right about the typo.
    $endgroup$
    – Vincenzo Oliva
    May 11 '15 at 10:22














    $begingroup$
    @phanitej I'll leave this answer here in case someone finds the typo. In the meanwhile, I'll post another answer with a correct inequality.
    $endgroup$
    – Vincenzo Oliva
    May 11 '15 at 10:58




    $begingroup$
    @phanitej I'll leave this answer here in case someone finds the typo. In the meanwhile, I'll post another answer with a correct inequality.
    $endgroup$
    – Vincenzo Oliva
    May 11 '15 at 10:58











    0












    $begingroup$

    Hopefully you're interested in the following double inequality, valid for $0le xlepi$: $$xleft(1-frac{x}{pi}right)lesin xle frac{4x}{pi}left(1-frac{x}{pi}right) $$






    share|cite|improve this answer









    $endgroup$


















      0












      $begingroup$

      Hopefully you're interested in the following double inequality, valid for $0le xlepi$: $$xleft(1-frac{x}{pi}right)lesin xle frac{4x}{pi}left(1-frac{x}{pi}right) $$






      share|cite|improve this answer









      $endgroup$
















        0












        0








        0





        $begingroup$

        Hopefully you're interested in the following double inequality, valid for $0le xlepi$: $$xleft(1-frac{x}{pi}right)lesin xle frac{4x}{pi}left(1-frac{x}{pi}right) $$






        share|cite|improve this answer









        $endgroup$



        Hopefully you're interested in the following double inequality, valid for $0le xlepi$: $$xleft(1-frac{x}{pi}right)lesin xle frac{4x}{pi}left(1-frac{x}{pi}right) $$







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered May 11 '15 at 11:07









        Vincenzo OlivaVincenzo Oliva

        5,32611234




        5,32611234






























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            Nidaros erkebispedøme

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            For det lutherske bispedømet, sjå Nidaros bispedøme. Nidaros erkebispedøme var eit katolsk erkebispedøme med sete i Nidaros. Det vart oppretta av kong Olav Kyrre mellom 1070 og 1080 som bispedøme. Bispedømet vart opphøgd til erkebispedøme i 1153. Erkebispedømet vart innstifta av pave Anastasius IV i 1154, og var i funksjon fram til reformasjonen i 1537. Erkebiskopen sitt hovudsete var Erkebispegarden, rett ved domkyrkja Nidarosdomen. Frå 1152 til 1154 var Nicholas Breakspear i Skandinavia som paveleg utsending, og organiserte opprettinga av erkesetet. Innhaldsliste 1 Utbreiing og jurisdiksjon 2 Overhyrdar 2.1 Lydbiskopar 3 Biskopar 3.1 Biskopar 3.2 Erkebiskopar Utbreiing og jurisdiksjon | Kart over den nordiske kyrkjeprovinsen (1153-1387) Erkebispedømet omfatta i tillegg til Noreg øyane i vest i Noregsveldet. Nidaros var eit metropolitanerkebispedøme med fylgjande underliggjande bispedøme suffraganar: Bjørgvin bispedøme Oslo bispedøme Hamar bispedøme Stavanger bispedøme Kirkjub
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            Birsay

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            Tettstaden Birsay på Orknøyane. Ruinane av jarleplasset om lag midt i biletet. (Fotografi frå 2012) Utsikt mot Birsay og Brough of Brisay (fotografi frå 2009) Birsay er ein tettstad i eit sokn (parish) med same namn, nordvest på øya Mainland [Note 1] på Orknøyane, om lag 31 km frå Kirkwall. Staden er omtala fleire stader i Orknøyingasoga [1] , der han er kalla «Byrgesherad» (norrønt: Birgisherað ). Namnet skriv seg truleg frå elvenamnet Birgisá [2] . Innhaldsliste 1 Den gamle hovudstaden 2 Bispesetet 3 Jarlepalasset 4 Kjelder og utfyllande opplysningar 4.1 Notar 4.2 Fotnotar 4.3 Bakgrunnsstoff Den gamle hovudstaden | Hovudartikkelen for dette emnet er Torfinn Sigurdsson . Soknet er i dag i hovudsak ei jordbrukskommune men er mest kjend for dei mange kulturminna i området. Tettstaden Birsay blir gjerne omtala som «den gamle hovudstaden på Orknøyane» [3] , og ein viser då gjerne til Orknøyingasoga (kap.31) som fortel at orknøyjarlen Torfinn Sigurdsson den mektige «hadde jamleg
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