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Can you simplify this mathematical algorithm

Posted on 2014-11-19
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Last Modified: 2014-11-20
Is it possible to refactor this mathematical algorithm to minimize the use of trigonometric functions?

input:
  z      :=  0 to ½PI
  Ma, Mb :=  0 to ∞
  Pa, Pb :=  0 to 2PI

algorithm:
  r = Ma * sin(Pa)
  i = Mb * sin(Pb)
  Sr = r + i
  Dr = r - i
  r = Ma * cos(Pa)
  i = Mb * cos(Pb)
  Si = r + i
  Di = r - i
  r = -cos(z)
  i = -sin(z)
  Tr = r * Si + i * Dr
  Ti = i * Si - r * Dr

output:
  Ra :=  Sr + Tr
  Ia :=  Ti + Di
  Rb :=  Sr - Tr
  RI :=  Ti - Di

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Question by:JasonMewes
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ozo earned 500 total points
ID: 40451983
What do Ra,Ia,Rb,RI represent?
Unless you can establish some relationship between the inputs, there may not be much more you can do than to reduce half of the trigonometric functions to sqrt functions.
Or if you can transform your inputs so that instead of working with Ma,Mb,Pa,Pb,
you're working with inputs that correspond to  Ma*sin(Pa),Ma*cos(Pa),Mb*sin(Pb),Mb*cos(Pb),
But that may depend on refactoring other parts of the algorithm that use this part.
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Author Comment

by:JasonMewes
ID: 40452008
Errors (for reference, no effect on question/solution):
Line 3 - bounds are actually -∞ to +∞
Line 24 - the name RI is actually Ib

Ra, Ia, Rb, Ib represent the four needed outputs of the algorithm.
There is no relationship between the inputs, only the bounds stated.

Other parts cannot be refactored, as inputs are external and outputs at this stage must be as specified.

SQRT is costlier than SIN and COS in this case.

Essentially you are saying that this is already as optimized as it can get?
Not possible to use identities or relationships (such as sin(A + B) = sin(A)*cos(B) + cos(A)*sin(B)) to reduce?
Given your (ozo) track record I trust your opinion implicitly!
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Author Comment

by:JasonMewes
ID: 40452018
This probably does not help:

Ma, Pa = magnitude, phase pair A
Mb, Pb = magnitude, phase pair B

Ra, Ia = complex number out A
Rb, Ib = complex number out B
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Expert Comment

by:ozo
ID: 40452126
I'm not sure if this comes out simpler, but
Tr = -(Ma * cos(z-Pa) + Mb * cos(z+Pb))
Ti = -(Ma * sin(z-Pa) + Mb * sin(z+Pb))
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Author Comment

by:JasonMewes
ID: 40452169
Since Sr and Di (requiring four sin/cos to calculate) are still required to calculate output - it seems calculating Tr and Ti this way would add additional processing.
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Expert Comment

by:dhsindy
ID: 40452943
This page shows a history of how trig functions have been computed in the past and how complicated it becomes.  If you read this you will probably be convinced to just use the functions defined in program.  That is probably the simplest way.

http://www.clarku.edu/~djoyce/trig/compute.html
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