From: Steve Underwood on 5 Oct 2006 03:10 Phil Frisbie, Jr. wrote: > Robert Scott wrote: >> I have an approximation for the Pythagorean distance formula >> (magnitude of >> vector [x,y]) using integer arithmetic that I would like to improve. >> Currently >> I am using this: > <snip> >> >> This is for an ARM processor in an application where I cannot afford >> the time >> for any floating point operations. The integer values of x and y come >> from an >> array of signed 16bit numbers, but x and y themselves are 32bit >> numbers in the >> above formula. The final result (z) will be shifted right one bit >> before being >> stored back into an array of 16bit integers, since the distance >> formula can >> potentially extend the range of x and y by one bit. In case you are >> wondering, >> this is part of calculating the power spectrum at the end of an FFT. >> >> The improvement I am looking for is in accuracy. I would like to try >> for a >> 4fold improvement in accuracy (.5%) without substantially increasing the >> running time of what I have now. Does anyone know of a better >> approximation >> that is almost as fast? > > I like this algorithm. It is snipped from my OpenLPC fixed point codec. > It uses only simple operations. It is likely more accurate than you > need, but I think you can simply truncate the algorithm sooner for less > precision. At the end PRECISION is the fractional bits you are using in > your fixed point code. > > static fixed32 fixsqrt32(fixed32 x) > { > > unsigned long r = 0, s, v = (unsigned long)x; > > #define STEP(k) s = r + (1 << k * 2); r >>= 1; \ > if (s <= v) { v = s; r = (1 << k * 2); } > > STEP(15); > STEP(14); > STEP(13); > STEP(12); > STEP(11); > STEP(10); > STEP(9); > STEP(8); > STEP(7); > STEP(6); > STEP(5); > STEP(4); > STEP(3); > STEP(2); > STEP(1); > STEP(0); > > return (fixed32)(r << (PRECISION / 2)); > } > That only seems to calculate a part of the answer, leaving lots of zeros at the end. Why not use something that calculates as many bits as it can, like: int32_t isqrt32(int32_t h) { int32_t x; int32_t y; int i; /* The answer is calculated as a 32 bit value, where the last 16 bits are fractional. */ /* Calling with negative numbers is not a good idea :) */ x = y = 0; for (i = 0; i < 32; i++) { x = (x << 1)  1; if (y < x) x = 2; else y = x; x++; y <<= 1; if ((h & 0x80000000)) y = 1; h <<= 1; y <<= 1; if ((h & 0x80000000)) y = 1; h <<= 1; } return x; } Of course, both these routines seem like overkill for what the original poster needs. :) Steve
From: Mark Borgerding on 5 Oct 2006 08:07 Robert Scott wrote: > I have an approximation for the Pythagorean distance formula (magnitude of > vector [x,y]) using integer arithmetic that I would like to improve. Currently Look for a paper entitled Filip, A. E. , "A baker's dozen magnitude approximations and their detection statistics" IEEE Trans on Aerospace and Elcectronic Systems, Jan 1976 It is a classic work with many tricks similar to the one you are using. With a little trickery, several of them could be implemented with little or no multiplication.  Mark Borgerding 3dB Labs, Inc Innovate. Develop. Deliver.
From: Everett M. Greene on 4 Oct 2006 22:39 "Phil Frisbie, Jr." <phil(a)hawksoft.com> writes: [snip] > I like this algorithm. It is snipped from my OpenLPC fixed point codec. > It uses only simple operations. It is likely more accurate than you > need, but I think you can simply truncate the algorithm sooner for less > precision. At the end PRECISION is the fractional bits you are using in > your fixed point code. [snip] Isn't computing the exact square root of an integer faster than the algorithm you show?
From: Phil Frisbie, Jr. on 5 Oct 2006 21:26 Everett M. Greene wrote: > "Phil Frisbie, Jr." <phil(a)hawksoft.com> writes: > [snip] > >>I like this algorithm. It is snipped from my OpenLPC fixed point codec. >>It uses only simple operations. It is likely more accurate than you >>need, but I think you can simply truncate the algorithm sooner for less >>precision. At the end PRECISION is the fractional bits you are using in >>your fixed point code. > > [snip] > > Isn't computing the exact square root of an integer > faster than the algorithm you show? That depends on your target CPU/DSP. I am now targeting ARM most of the time so I will make a note to profile that code and see.  Phil Frisbie, Jr. Hawk Software http://www.hawksoft.com
From: Everett M. Greene on 6 Oct 2006 00:34 "Phil Frisbie, Jr." <phil(a)hawksoft.com> writes: > Everett M. Greene wrote: > > > "Phil Frisbie, Jr." <phil(a)hawksoft.com> writes: > > [snip] > > > >>I like this algorithm. It is snipped from my OpenLPC fixed point codec. > >>It uses only simple operations. It is likely more accurate than you > >>need, but I think you can simply truncate the algorithm sooner for less > >>precision. At the end PRECISION is the fractional bits you are using in > >>your fixed point code. > > > > [snip] > > > > Isn't computing the exact square root of an integer > > faster than the algorithm you show? > > That depends on your target CPU/DSP. I am now targeting ARM most > of the time so I will make a note to profile that code and see. It would be interesting to know the results of the comparison.
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