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How I refused to calculate the square root



Very often, when digitally processing signals, it is necessary to calculate the length of the vector, usually this is done by the formula A = SQRT (X ^ 2 + Y ^ 2). Here squaring the value is not difficult, but the operation of calculating the square root is not a simple operation, especially for microcontrollers. In addition, root calculation algorithms are not stable in time, and for algorithms in which there are many such calculations, it becomes difficult to predict the time required for calculations.

I also faced such a problem. About how I refused the procedure for calculating the root, read below.

First, consider four vectors, they lie in different quarters.



We will reduce everything to one quarter - the first. To do this, take the Y coordinate and if it is negative, then turn it into positive. If it is already positive, then we will leave everything unchanged.

In fact, since I have absolute coordinates and values ​​indicated in unsigned integers, we calculate the module.

If Y> 2047, then Y = Y-2047
If Y <= 2047, then Y = 2047-Y.

Here 2047 is a virtual zero, because the project I worked with when I invented this bike had 12 bit ADCs.

Let's see what happened.



And we will do the same for the coordinate X.

If X> 2047, then X = X-2047
If X <= 2047, then X = 2047-X

By the way, it is worth noting that from two 12-bit values, there are two 11-bit values.



Now all the vectors are in the first quarter, but their length, which we are looking for, has not changed.

The next stage of the transformation: “twist around the axis of 45 degrees.”

Vectors can have an angle from zero to 90 degrees, while if in the coordinates of the vector Y> X, then the angle of the vector lies in the range of 45-90 degrees, and if Y <X, then the angle of the vector is less than 45 degrees. Well, if X = Y, then the angle of the vector is 45 degrees.

Reduce vectors to 45 degrees.

If Y> X, then TEMP = Y, Y = X, X = TEMP (swap X and Y in places).
If Y <= X, then we do nothing.

We look what happened. All vectors lie in the range of angles from 0 to 45 degrees.



I want to note that at this stage, I had the idea to check the angle by 22.5 degrees and deploy again. And then do a check and twist by 12.25 degrees. And so on, until the vector “lies” on the X axis, and then its length will be determined by the coordinate X, and Y will become zero.

But I did not go this way, I made a bet on another method, and as it turned out later, I did not lose.

If we divide Y by X, and then take the arctangent from this quotient, we get the angle. And if we divide X by the cosine of this angle, we get the length of the vector.

So, the length of the vector is A = X / COS (ATAN (Y / X))

It seemed difficult when I thought that I needed to get the angle from the table of arc tangents, and then get the coefficient from the table of cosines.

But nothing happened. I decided that you can immediately make a table where the Y / X ratio addresses the cell, and in it lies the coefficient by which X must be divided to get the length of the vector.

A = X / K; where K = COS (ATAN (Y / X))

But multiplying is easier than dividing, therefore:

A = X * K; where K = 1 / COS (ATAN (Y / X))

It is clear that all K, for all (Y / X), must be calculated in advance.

The calculated table contains 2048 values, each of which is 16 bits. That is 4kB of data.

Table
Magnitude_Table
DCW 0x0800,0x0800,0x0800,0x0800,0x0800,0x0800,0x0800,0x0800
DCW 0x0800,0x0800,0x0800,0x0800,0x0800,0x0800,0x0800,0x0800
DCW 0x0800,0x0800,0x0800,0x0800,0x0800,0x0800,0x0800,0x0800
DCW 0x0800,0x0800,0x0800,0x0800,0x0800,0x0800,0x0800,0x0800
DCW 0x0800,0x0800,0x0800,0x0800,0x0800,0x0800,0x0800,0x0800
DCW 0x0800,0x0800,0x0800,0x0800,0x0800,0x0800,0x0800,0x0800
DCW 0x0800,0x0800,0x0800,0x0800,0x0800,0x0800,0x0800,0x0800
DCW 0x0800,0x0800,0x0800,0x0800,0x0800,0x0800,0x0800,0x0800
DCW 0x0801,0x0801,0x0801,0x0801,0x0801,0x0801,0x0801,0x0801
DCW 0x0801,0x0801,0x0801,0x0801,0x0801,0x0801,0x0801,0x0801
DCW 0x0801,0x0801,0x0801,0x0801,0x0801,0x0801,0x0801,0x0801
DCW 0x0801,0x0801,0x0802,0x0802,0x0802,0x0802,0x0802,0x0802
DCW 0x0802,0x0802,0x0802,0x0802,0x0802,0x0802,0x0802,0x0802
DCW 0x0802,0x0802,0x0802,0x0802,0x0802,0x0802,0x0803,0x0803
DCW 0x0803,0x0803,0x0803,0x0803,0x0803,0x0803,0x0803,0x0803
DCW 0x0803,0x0803,0x0803,0x0803,0x0803,0x0803,0x0803,0x0803
DCW 0x0804,0x0804,0x0804,0x0804,0x0804,0x0804,0x0804,0x0804
DCW 0x0804,0x0804,0x0804,0x0804,0x0804,0x0804,0x0804,0x0805
DCW 0x0805,0x0805,0x0805,0x0805,0x0805,0x0805,0x0805,0x0805
DCW 0x0805,0x0805,0x0805,0x0805,0x0806,0x0806,0x0806,0x0806
DCW 0x0806,0x0806,0x0806,0x0806,0x0806,0x0806,0x0806,0x0806
DCW 0x0806,0x0807,0x0807,0x0807,0x0807,0x0807,0x0807,0x0807
DCW 0x0807,0x0807,0x0807,0x0807,0x0807,0x0808,0x0808,0x0808
DCW 0x0808,0x0808,0x0808,0x0808,0x0808,0x0808,0x0808,0x0808
DCW 0x0809,0x0809,0x0809,0x0809,0x0809,0x0809,0x0809,0x0809
DCW 0x0809,0x0809,0x080A,0x080A,0x080A,0x080A,0x080A,0x080A
DCW 0x080A,0x080A,0x080A,0x080A,0x080B,0x080B,0x080B,0x080B
DCW 0x080B,0x080B,0x080B,0x080B,0x080B,0x080B,0x080C,0x080C
DCW 0x080C,0x080C,0x080C,0x080C,0x080C,0x080C,0x080C,0x080D
DCW 0x080D,0x080D,0x080D,0x080D,0x080D,0x080D,0x080D,0x080E
DCW 0x080E,0x080E,0x080E,0x080E,0x080E,0x080E,0x080E,0x080E
DCW 0x080F,0x080F,0x080F,0x080F,0x080F,0x080F,0x080F,0x080F
DCW 0x0810,0x0810,0x0810,0x0810,0x0810,0x0810,0x0810,0x0810
DCW 0x0811,0x0811,0x0811,0x0811,0x0811,0x0811,0x0811,0x0811
DCW 0x0812,0x0812,0x0812,0x0812,0x0812,0x0812,0x0812,0x0813
DCW 0x0813,0x0813,0x0813,0x0813,0x0813,0x0813,0x0814,0x0814
DCW 0x0814,0x0814,0x0814,0x0814,0x0814,0x0814,0x0815,0x0815
DCW 0x0815,0x0815,0x0815,0x0815,0x0816,0x0816,0x0816,0x0816
DCW 0x0816,0x0816,0x0816,0x0817,0x0817,0x0817,0x0817,0x0817
DCW 0x0817,0x0817,0x0818,0x0818,0x0818,0x0818,0x0818,0x0818
DCW 0x0819,0x0819,0x0819,0x0819,0x0819,0x0819,0x0819,0x081A
DCW 0x081A,0x081A,0x081A,0x081A,0x081A,0x081B,0x081B,0x081B
DCW 0x081B,0x081B,0x081B,0x081C,0x081C,0x081C,0x081C,0x081C
DCW 0x081C,0x081D,0x081D,0x081D,0x081D,0x081D,0x081D,0x081E
DCW 0x081E,0x081E,0x081E,0x081E,0x081E,0x081F,0x081F,0x081F
DCW 0x081F,0x081F,0x081F,0x0820,0x0820,0x0820,0x0820,0x0820
DCW 0x0820,0x0821,0x0821,0x0821,0x0821,0x0821,0x0822,0x0822
DCW 0x0822,0x0822,0x0822,0x0822,0x0823,0x0823,0x0823,0x0823
DCW 0x0823,0x0824,0x0824,0x0824,0x0824,0x0824,0x0824,0x0825
DCW 0x0825,0x0825,0x0825,0x0825,0x0826,0x0826,0x0826,0x0826
DCW 0x0826,0x0827,0x0827,0x0827,0x0827,0x0827,0x0828,0x0828
DCW 0x0828,0x0828,0x0828,0x0829,0x0829,0x0829,0x0829,0x0829
DCW 0x082A,0x082A,0x082A,0x082A,0x082A,0x082B,0x082B,0x082B
DCW 0x082B,0x082B,0x082C,0x082C,0x082C,0x082C,0x082C,0x082D
DCW 0x082D,0x082D,0x082D,0x082D,0x082E,0x082E,0x082E,0x082E
DCW 0x082E,0x082F,0x082F,0x082F,0x082F,0x0830,0x0830,0x0830
DCW 0x0830,0x0830,0x0831,0x0831,0x0831,0x0831,0x0831,0x0832
DCW 0x0832,0x0832,0x0832,0x0833,0x0833,0x0833,0x0833,0x0833
DCW 0x0834,0x0834,0x0834,0x0834,0x0835,0x0835,0x0835,0x0835
DCW 0x0835,0x0836,0x0836,0x0836,0x0836,0x0837,0x0837,0x0837
DCW 0x0837,0x0837,0x0838,0x0838,0x0838,0x0838,0x0839,0x0839
DCW 0x0839,0x0839,0x083A,0x083A,0x083A,0x083A,0x083A,0x083B
DCW 0x083B,0x083B,0x083B,0x083C,0x083C,0x083C,0x083C,0x083D
DCW 0x083D,0x083D,0x083D,0x083E,0x083E,0x083E,0x083E,0x083F
DCW 0x083F,0x083F,0x083F,0x0840,0x0840,0x0840,0x0840,0x0840
DCW 0x0841,0x0841,0x0841,0x0841,0x0842,0x0842,0x0842,0x0842
DCW 0x0843,0x0843,0x0843,0x0843,0x0844,0x0844,0x0844,0x0844
DCW 0x0845,0x0845,0x0845,0x0845,0x0846,0x0846,0x0846,0x0847
DCW 0x0847,0x0847,0x0847,0x0848,0x0848,0x0848,0x0848,0x0849
DCW 0x0849,0x0849,0x0849,0x084A,0x084A,0x084A,0x084A,0x084B
DCW 0x084B,0x084B,0x084B,0x084C,0x084C,0x084C,0x084D,0x084D
DCW 0x084D,0x084D,0x084E,0x084E,0x084E,0x084E,0x084F,0x084F
DCW 0x084F,0x0850,0x0850,0x0850,0x0850,0x0851,0x0851,0x0851
DCW 0x0851,0x0852,0x0852,0x0852,0x0853,0x0853,0x0853,0x0853
DCW 0x0854,0x0854,0x0854,0x0854,0x0855,0x0855,0x0855,0x0856
DCW 0x0856,0x0856,0x0856,0x0857,0x0857,0x0857,0x0858,0x0858
DCW 0x0858,0x0858,0x0859,0x0859,0x0859,0x085A,0x085A,0x085A
DCW 0x085A,0x085B,0x085B,0x085B,0x085C,0x085C,0x085C,0x085C
DCW 0x085D,0x085D,0x085D,0x085E,0x085E,0x085E,0x085F,0x085F
DCW 0x085F,0x085F,0x0860,0x0860,0x0860,0x0861,0x0861,0x0861
DCW 0x0861,0x0862,0x0862,0x0862,0x0863,0x0863,0x0863,0x0864
DCW 0x0864,0x0864,0x0864,0x0865,0x0865,0x0865,0x0866,0x0866
DCW 0x0866,0x0867,0x0867,0x0867,0x0868,0x0868,0x0868,0x0868
DCW 0x0869,0x0869,0x0869,0x086A,0x086A,0x086A,0x086B,0x086B
DCW 0x086B,0x086C,0x086C,0x086C,0x086C,0x086D,0x086D,0x086D
DCW 0x086E,0x086E,0x086E,0x086F,0x086F,0x086F,0x0870,0x0870
DCW 0x0870,0x0871,0x0871,0x0871,0x0872,0x0872,0x0872,0x0873
DCW 0x0873,0x0873,0x0874,0x0874,0x0874,0x0874,0x0875,0x0875
DCW 0x0875,0x0876,0x0876,0x0876,0x0877,0x0877,0x0877,0x0878
DCW 0x0878,0x0878,0x0879,0x0879,0x0879,0x087A,0x087A,0x087A
DCW 0x087B,0x087B,0x087B,0x087C,0x087C,0x087C,0x087D,0x087D
DCW 0x087D,0x087E,0x087E,0x087E,0x087F,0x087F,0x087F,0x0880
DCW 0x0880,0x0880,0x0881,0x0881,0x0881,0x0882,0x0882,0x0882
DCW 0x0883,0x0883,0x0883,0x0884,0x0884,0x0885,0x0885,0x0885
DCW 0x0886,0x0886,0x0886,0x0887,0x0887,0x0887,0x0888,0x0888
DCW 0x0888,0x0889,0x0889,0x0889,0x088A,0x088A,0x088A,0x088B
DCW 0x088B,0x088B,0x088C,0x088C,0x088D,0x088D,0x088D,0x088E
DCW 0x088E,0x088E,0x088F,0x088F,0x088F,0x0890,0x0890,0x0890
DCW 0x0891,0x0891,0x0892,0x0892,0x0892,0x0893,0x0893,0x0893
DCW 0x0894,0x0894,0x0894,0x0895,0x0895,0x0895,0x0896,0x0896
DCW 0x0897,0x0897,0x0897,0x0898,0x0898,0x0898,0x0899,0x0899
DCW 0x0899,0x089A,0x089A,0x089B,0x089B,0x089B,0x089C,0x089C
DCW 0x089C,0x089D,0x089D,0x089E,0x089E,0x089E,0x089F,0x089F
DCW 0x089F,0x08A0,0x08A0,0x08A1,0x08A1,0x08A1,0x08A2,0x08A2
DCW 0x08A2,0x08A3,0x08A3,0x08A4,0x08A4,0x08A4,0x08A5,0x08A5
DCW 0x08A5,0x08A6,0x08A6,0x08A7,0x08A7,0x08A7,0x08A8,0x08A8
DCW 0x08A9,0x08A9,0x08A9,0x08AA,0x08AA,0x08AA,0x08AB,0x08AB
DCW 0x08AC,0x08AC,0x08AC,0x08AD,0x08AD,0x08AE,0x08AE,0x08AE
DCW 0x08AF,0x08AF,0x08AF,0x08B0,0x08B0,0x08B1,0x08B1,0x08B1
DCW 0x08B2,0x08B2,0x08B3,0x08B3,0x08B3,0x08B4,0x08B4,0x08B5
DCW 0x08B5,0x08B5,0x08B6,0x08B6,0x08B7,0x08B7,0x08B7,0x08B8
DCW 0x08B8,0x08B9,0x08B9,0x08B9,0x08BA,0x08BA,0x08BB,0x08BB
DCW 0x08BB,0x08BC,0x08BC,0x08BD,0x08BD,0x08BD,0x08BE,0x08BE
DCW 0x08BF,0x08BF,0x08BF,0x08C0,0x08C0,0x08C1,0x08C1,0x08C1
DCW 0x08C2,0x08C2,0x08C3,0x08C3,0x08C3,0x08C4,0x08C4,0x08C5
DCW 0x08C5,0x08C5,0x08C6,0x08C6,0x08C7,0x08C7,0x08C8,0x08C8
DCW 0x08C8,0x08C9,0x08C9,0x08CA,0x08CA,0x08CA,0x08CB,0x08CB
DCW 0x08CC,0x08CC,0x08CD,0x08CD,0x08CD,0x08CE,0x08CE,0x08CF
DCW 0x08CF,0x08CF,0x08D0,0x08D0,0x08D1,0x08D1,0x08D2,0x08D2
DCW 0x08D2,0x08D3,0x08D3,0x08D4,0x08D4,0x08D4,0x08D5,0x08D5
DCW 0x08D6,0x08D6,0x08D7,0x08D7,0x08D7,0x08D8,0x08D8,0x08D9
DCW 0x08D9,0x08DA,0x08DA,0x08DA,0x08DB,0x08DB,0x08DC,0x08DC
DCW 0x08DD,0x08DD,0x08DD,0x08DE,0x08DE,0x08DF,0x08DF,0x08E0
DCW 0x08E0,0x08E0,0x08E1,0x08E1,0x08E2,0x08E2,0x08E3,0x08E3
DCW 0x08E4,0x08E4,0x08E4,0x08E5,0x08E5,0x08E6,0x08E6,0x08E7
DCW 0x08E7,0x08E7,0x08E8,0x08E8,0x08E9,0x08E9,0x08EA,0x08EA
DCW 0x08EB,0x08EB,0x08EB,0x08EC,0x08EC,0x08ED,0x08ED,0x08EE
DCW 0x08EE,0x08EF,0x08EF,0x08EF,0x08F0,0x08F0,0x08F1,0x08F1
DCW 0x08F2,0x08F2,0x08F3,0x08F3,0x08F3,0x08F4,0x08F4,0x08F5
DCW 0x08F5,0x08F6,0x08F6,0x08F7,0x08F7,0x08F8,0x08F8,0x08F8
DCW 0x08F9,0x08F9,0x08FA,0x08FA,0x08FB,0x08FB,0x08FC,0x08FC
DCW 0x08FD,0x08FD,0x08FD,0x08FE,0x08FE,0x08FF,0x08FF,0x0900
DCW 0x0900,0x0901,0x0901,0x0902,0x0902,0x0902,0x0903,0x0903
DCW 0x0904,0x0904,0x0905,0x0905,0x0906,0x0906,0x0907,0x0907
DCW 0x0908,0x0908,0x0908,0x0909,0x0909,0x090A,0x090A,0x090B
DCW 0x090B,0x090C,0x090C,0x090D,0x090D,0x090E,0x090E,0x090F
DCW 0x090F,0x0910,0x0910,0x0910,0x0911,0x0911,0x0912,0x0912
DCW 0x0913,0x0913,0x0914,0x0914,0x0915,0x0915,0x0916,0x0916
DCW 0x0917,0x0917,0x0918,0x0918,0x0918,0x0919,0x0919,0x091A
DCW 0x091A,0x091B,0x091B,0x091C,0x091C,0x091D,0x091D,0x091E
DCW 0x091E,0x091F,0x091F,0x0920,0x0920,0x0921,0x0921,0x0922
DCW 0x0922,0x0923,0x0923,0x0924,0x0924,0x0924,0x0925,0x0925
DCW 0x0926,0x0926,0x0927,0x0927,0x0928,0x0928,0x0929,0x0929
DCW 0x092A,0x092A,0x092B,0x092B,0x092C,0x092C,0x092D,0x092D
DCW 0x092E,0x092E,0x092F,0x092F,0x0930,0x0930,0x0931,0x0931
DCW 0x0932,0x0932,0x0933,0x0933,0x0934,0x0934,0x0935,0x0935
DCW 0x0936,0x0936,0x0937,0x0937,0x0938,0x0938,0x0939,0x0939
DCW 0x093A,0x093A,0x093B,0x093B,0x093C,0x093C,0x093D,0x093D
DCW 0x093E,0x093E,0x093F,0x093F,0x0940,0x0940,0x0941,0x0941
DCW 0x0942,0x0942,0x0943,0x0943,0x0944,0x0944,0x0945,0x0945
DCW 0x0946,0x0946,0x0947,0x0947,0x0948,0x0948,0x0949,0x0949
DCW 0x094A,0x094A,0x094B,0x094B,0x094C,0x094C,0x094D,0x094D
DCW 0x094E,0x094E,0x094F,0x094F,0x0950,0x0950,0x0951,0x0951
DCW 0x0952,0x0952,0x0953,0x0953,0x0954,0x0954,0x0955,0x0956
DCW 0x0956,0x0957,0x0957,0x0958,0x0958,0x0959,0x0959,0x095A
DCW 0x095A,0x095B,0x095B,0x095C,0x095C,0x095D,0x095D,0x095E
DCW 0x095E,0x095F,0x095F,0x0960,0x0960,0x0961,0x0961,0x0962
DCW 0x0962,0x0963,0x0964,0x0964,0x0965,0x0965,0x0966,0x0966
DCW 0x0967,0x0967,0x0968,0x0968,0x0969,0x0969,0x096A,0x096A
DCW 0x096B,0x096B,0x096C,0x096C,0x096D,0x096E,0x096E,0x096F
DCW 0x096F,0x0970,0x0970,0x0971,0x0971,0x0972,0x0972,0x0973
DCW 0x0973,0x0974,0x0974,0x0975,0x0976,0x0976,0x0977,0x0977
DCW 0x0978,0x0978,0x0979,0x0979,0x097A,0x097A,0x097B,0x097B
DCW 0x097C,0x097C,0x097D,0x097E,0x097E,0x097F,0x097F,0x0980
DCW 0x0980,0x0981,0x0981,0x0982,0x0982,0x0983,0x0983,0x0984
DCW 0x0985,0x0985,0x0986,0x0986,0x0987,0x0987,0x0988,0x0988
DCW 0x0989,0x0989,0x098A,0x098B,0x098B,0x098C,0x098C,0x098D
DCW 0x098D,0x098E,0x098E,0x098F,0x098F,0x0990,0x0991,0x0991
DCW 0x0992,0x0992,0x0993,0x0993,0x0994,0x0994,0x0995,0x0996
DCW 0x0996,0x0997,0x0997,0x0998,0x0998,0x0999,0x0999,0x099A
DCW 0x099A,0x099B,0x099C,0x099C,0x099D,0x099D,0x099E,0x099E
DCW 0x099F,0x099F,0x09A0,0x09A1,0x09A1,0x09A2,0x09A2,0x09A3
DCW 0x09A3,0x09A4,0x09A4,0x09A5,0x09A6,0x09A6,0x09A7,0x09A7
DCW 0x09A8,0x09A8,0x09A9,0x09AA,0x09AA,0x09AB,0x09AB,0x09AC
DCW 0x09AC,0x09AD,0x09AD,0x09AE,0x09AF,0x09AF,0x09B0,0x09B0
DCW 0x09B1,0x09B1,0x09B2,0x09B3,0x09B3,0x09B4,0x09B4,0x09B5
DCW 0x09B5,0x09B6,0x09B7,0x09B7,0x09B8,0x09B8,0x09B9,0x09B9
DCW 0x09BA,0x09BA,0x09BB,0x09BC,0x09BC,0x09BD,0x09BD,0x09BE
DCW 0x09BE,0x09BF,0x09C0,0x09C0,0x09C1,0x09C1,0x09C2,0x09C2
DCW 0x09C3,0x09C4,0x09C4,0x09C5,0x09C5,0x09C6,0x09C7,0x09C7
DCW 0x09C8,0x09C8,0x09C9,0x09C9,0x09CA,0x09CB,0x09CB,0x09CC
DCW 0x09CC,0x09CD,0x09CD,0x09CE,0x09CF,0x09CF,0x09D0,0x09D0
DCW 0x09D1,0x09D1,0x09D2,0x09D3,0x09D3,0x09D4,0x09D4,0x09D5
DCW 0x09D6,0x09D6,0x09D7,0x09D7,0x09D8,0x09D8,0x09D9,0x09DA
DCW 0x09DA,0x09DB,0x09DB,0x09DC,0x09DD,0x09DD,0x09DE,0x09DE
DCW 0x09DF,0x09DF,0x09E0,0x09E1,0x09E1,0x09E2,0x09E2,0x09E3
DCW 0x09E4,0x09E4,0x09E5,0x09E5,0x09E6,0x09E7,0x09E7,0x09E8
DCW 0x09E8,0x09E9,0x09E9,0x09EA,0x09EB,0x09EB,0x09EC,0x09EC
DCW 0x09ED,0x09EE,0x09EE,0x09EF,0x09EF,0x09F0,0x09F1,0x09F1
DCW 0x09F2,0x09F2,0x09F3,0x09F4,0x09F4,0x09F5,0x09F5,0x09F6
DCW 0x09F7,0x09F7,0x09F8,0x09F8,0x09F9,0x09FA,0x09FA,0x09FB
DCW 0x09FB,0x09FC,0x09FD,0x09FD,0x09FE,0x09FE,0x09FF,0x09FF
DCW 0x0A00,0x0A01,0x0A01,0x0A02,0x0A03,0x0A03,0x0A04,0x0A04
DCW 0x0A05,0x0A06,0x0A06,0x0A07,0x0A07,0x0A08,0x0A09,0x0A09
DCW 0x0A0A,0x0A0A,0x0A0B,0x0A0C,0x0A0C,0x0A0D,0x0A0D,0x0A0E
DCW 0x0A0F,0x0A0F,0x0A10,0x0A10,0x0A11,0x0A12,0x0A12,0x0A13
DCW 0x0A13,0x0A14,0x0A15,0x0A15,0x0A16,0x0A16,0x0A17,0x0A18
DCW 0x0A18,0x0A19,0x0A1A,0x0A1A,0x0A1B,0x0A1B,0x0A1C,0x0A1D
DCW 0x0A1D,0x0A1E,0x0A1E,0x0A1F,0x0A20,0x0A20,0x0A21,0x0A21
DCW 0x0A22,0x0A23,0x0A23,0x0A24,0x0A25,0x0A25,0x0A26,0x0A26
DCW 0x0A27,0x0A28,0x0A28,0x0A29,0x0A29,0x0A2A,0x0A2B,0x0A2B
DCW 0x0A2C,0x0A2D,0x0A2D,0x0A2E,0x0A2E,0x0A2F,0x0A30,0x0A30
DCW 0x0A31,0x0A32,0x0A32,0x0A33,0x0A33,0x0A34,0x0A35,0x0A35
DCW 0x0A36,0x0A36,0x0A37,0x0A38,0x0A38,0x0A39,0x0A3A,0x0A3A
DCW 0x0A3B,0x0A3B,0x0A3C,0x0A3D,0x0A3D,0x0A3E,0x0A3F,0x0A3F
DCW 0x0A40,0x0A40,0x0A41,0x0A42,0x0A42,0x0A43,0x0A44,0x0A44
DCW 0x0A45,0x0A45,0x0A46,0x0A47,0x0A47,0x0A48,0x0A49,0x0A49
DCW 0x0A4A,0x0A4B,0x0A4B,0x0A4C,0x0A4C,0x0A4D,0x0A4E,0x0A4E
DCW 0x0A4F,0x0A50,0x0A50,0x0A51,0x0A51,0x0A52,0x0A53,0x0A53
DCW 0x0A54,0x0A55,0x0A55,0x0A56,0x0A57,0x0A57,0x0A58,0x0A58
DCW 0x0A59,0x0A5A,0x0A5A,0x0A5B,0x0A5C,0x0A5C,0x0A5D,0x0A5D
DCW 0x0A5E,0x0A5F,0x0A5F,0x0A60,0x0A61,0x0A61,0x0A62,0x0A63
DCW 0x0A63,0x0A64,0x0A64,0x0A65,0x0A66,0x0A66,0x0A67,0x0A68
DCW 0x0A68,0x0A69,0x0A6A,0x0A6A,0x0A6B,0x0A6C,0x0A6C,0x0A6D
DCW 0x0A6D,0x0A6E,0x0A6F,0x0A6F,0x0A70,0x0A71,0x0A71,0x0A72
DCW 0x0A73,0x0A73,0x0A74,0x0A75,0x0A75,0x0A76,0x0A76,0x0A77
DCW 0x0A78,0x0A78,0x0A79,0x0A7A,0x0A7A,0x0A7B,0x0A7C,0x0A7C
DCW 0x0A7D,0x0A7E,0x0A7E,0x0A7F,0x0A80,0x0A80,0x0A81,0x0A81
DCW 0x0A82,0x0A83,0x0A83,0x0A84,0x0A85,0x0A85,0x0A86,0x0A87
DCW 0x0A87,0x0A88,0x0A89,0x0A89,0x0A8A,0x0A8B,0x0A8B,0x0A8C
DCW 0x0A8D,0x0A8D,0x0A8E,0x0A8E,0x0A8F,0x0A90,0x0A90,0x0A91
DCW 0x0A92,0x0A92,0x0A93,0x0A94,0x0A94,0x0A95,0x0A96,0x0A96
DCW 0x0A97,0x0A98,0x0A98,0x0A99,0x0A9A,0x0A9A,0x0A9B,0x0A9C
DCW 0x0A9C,0x0A9D,0x0A9E,0x0A9E,0x0A9F,0x0AA0,0x0AA0,0x0AA1
DCW 0x0AA1,0x0AA2,0x0AA3,0x0AA3,0x0AA4,0x0AA5,0x0AA5,0x0AA6
DCW 0x0AA7,0x0AA7,0x0AA8,0x0AA9,0x0AA9,0x0AAA,0x0AAB,0x0AAB
DCW 0x0AAC,0x0AAD,0x0AAD,0x0AAE,0x0AAF,0x0AAF,0x0AB0,0x0AB1
DCW 0x0AB1,0x0AB2,0x0AB3,0x0AB3,0x0AB4,0x0AB5,0x0AB5,0x0AB6
DCW 0x0AB7,0x0AB7,0x0AB8,0x0AB9,0x0AB9,0x0ABA,0x0ABB,0x0ABB
DCW 0x0ABC,0x0ABD,0x0ABD,0x0ABE,0x0ABF,0x0ABF,0x0AC0,0x0AC1
DCW 0x0AC1,0x0AC2,0x0AC3,0x0AC3,0x0AC4,0x0AC5,0x0AC5,0x0AC6
DCW 0x0AC7,0x0AC7,0x0AC8,0x0AC9,0x0AC9,0x0ACA,0x0ACB,0x0ACB
DCW 0x0ACC,0x0ACD,0x0ACD,0x0ACE,0x0ACF,0x0ACF,0x0AD0,0x0AD1
DCW 0x0AD1,0x0AD2,0x0AD3,0x0AD3,0x0AD4,0x0AD5,0x0AD5,0x0AD6
DCW 0x0AD7,0x0AD8,0x0AD8,0x0AD9,0x0ADA,0x0ADA,0x0ADB,0x0ADC
DCW 0x0ADC,0x0ADD,0x0ADE,0x0ADE,0x0ADF,0x0AE0,0x0AE0,0x0AE1
DCW 0x0AE2,0x0AE2,0x0AE3,0x0AE4,0x0AE4,0x0AE5,0x0AE6,0x0AE6
DCW 0x0AE7,0x0AE8,0x0AE8,0x0AE9,0x0AEA,0x0AEA,0x0AEB,0x0AEC
DCW 0x0AED,0x0AED,0x0AEE,0x0AEF,0x0AEF,0x0AF0,0x0AF1,0x0AF1
DCW 0x0AF2,0x0AF3,0x0AF3,0x0AF4,0x0AF5,0x0AF5,0x0AF6,0x0AF7
DCW 0x0AF7,0x0AF8,0x0AF9,0x0AF9,0x0AFA,0x0AFB,0x0AFC,0x0AFC
DCW 0x0AFD,0x0AFE,0x0AFE,0x0AFF,0x0B00,0x0B00,0x0B01,0x0B02
DCW 0x0B02,0x0B03,0x0B04,0x0B04,0x0B05,0x0B06,0x0B07,0x0B07
DCW 0x0B08,0x0B09,0x0B09,0x0B0A,0x0B0B,0x0B0B,0x0B0C,0x0B0D
DCW 0x0B0D,0x0B0E,0x0B0F,0x0B10,0x0B10,0x0B11,0x0B12,0x0B12
DCW 0x0B13,0x0B14,0x0B14,0x0B15,0x0B16,0x0B16,0x0B17,0x0B18
DCW 0x0B18,0x0B19,0x0B1A,0x0B1B,0x0B1B,0x0B1C,0x0B1D,0x0B1D
DCW 0x0B1E,0x0B1F,0x0B1F,0x0B20,0x0B21,0x0B22,0x0B22,0x0B23
DCW 0x0B24,0x0B24,0x0B25,0x0B26,0x0B26,0x0B27,0x0B28,0x0B28
DCW 0x0B29,0x0B2A,0x0B2B,0x0B2B,0x0B2C,0x0B2D,0x0B2D,0x0B2E
DCW 0x0B2F,0x0B2F,0x0B30,0x0B31,0x0B32,0x0B32,0x0B33,0x0B34
DCW 0x0B34,0x0B35,0x0B36,0x0B36,0x0B37,0x0B38,0x0B39,0x0B39
DCW 0x0B3A,0x0B3B,0x0B3B,0x0B3C,0x0B3D,0x0B3D,0x0B3E,0x0B3F
DCW 0x0B40,0x0B40,0x0B41,0x0B42,0x0B42,0x0B43,0x0B44,0x0B45
DCW 0x0B45,0x0B46,0x0B47,0x0B47,0x0B48,0x0B49,0x0B49,0x0B4A
DCW 0x0B4B,0x0B4C,0x0B4C,0x0B4D,0x0B4E,0x0B4E,0x0B4F,0x0B50

The lowest value is 0x800h, which means 2048. The oldest is 0xB50, and it means 2896.
2896/2048 = 1.414, which corresponds to 1 / cos (45).

To get the index of the array, you must multiply Y by 2048 and divide by X.

The resulting index will point to the cell of the array, from this cell we extract the value, multiply X by it and divide by 2048. The result is the length of the vector.

I want to note that division and multiplication by 2048 is performed by a shift, and another division and multiplication is performed by special commands or algorithms, if the platform used does not have multiplication and / or division commands.

In my case, the STM32F103 microcontroller was used, which had division and multiplication commands.

To summarize the algorithm.

  1. Check the sign by Y and turn if necessary.
  2. Check sign on X and turn if necessary.
  3. Compare X and Y and swap places, if necessary.
  4. Check X for 0.
  5. Multiply Y by 2048 by a shift.
  6. Division Y by X.
  7. Add the resulting value to the address of the array and read the value K.
  8. Multiply X by K.
  9. Divide X by 2048 by a shift.

The resulting assembly language program for STM32F103.

Program
	;r5 - X
	;r6 - Y

Magnitude
	mov32			r2,0x7FF
	mov32			r3,0x7FF
	cmp			r5,r2
	blo			Magnitude_xdn
Magnitude_xup
	sub			r5,r2
	b			Magnitude_X
Magnitude_xdn
	sub			r2,r5
	mov			r5,r2
Magnitude_X		;X in r5
	cmp			r6,r3
	blo			Magnitude_ydn
Magnitude_yup
	sub			r6,r3
	b			Magnitude_Y
Magnitude_ydn
	sub			r3,r6
	mov			r6,r3
Magnitude_Y		;Y in r6
	cmp			r5,r6
	blo			Magnitude_neg
Magnitude_pos
	mov			r1,r5
	mov			r5,r6
	mov			r6,r1
Magnitude_neg		;Rotating 45 degree
	cbz			r6,Magnitude_Zero
	lsl			r5,#0xC
	udiv			r5,r6
	mov32			r2,#0xFFFFFFFE
	and			r5,r2
	mov32			r2,Magnitude_Table
	add			r2,r5
	ldrh			r0,[r2]
	mul			r6,r0
	lsr			r6,#0xB
Magnitude_Zero
	pop			{lr}
	bx			lr
;In r6 length of vector


I will give an example of the algorithm in comparison with the classical method for two of the four vectors that are shown in the images above.

Take the blue vector (from the third area).

Its coordinates are X = -45, Y = -126 (can be checked for pixels)

Classic method:

(-45) ^ 2 + (- 126) ^ 2 = 2025 + 15876 = 17901
SQRT (17901) = 133 (truncated to integer)

Algorithm:

X = 2047-45 = 2002, Y = 2047-126 = 1921

clause 1: 2002> 2047? -> no. X = 2047-2002 = 45
point 2: 1921> 2047? -> no. Y = 2047-1921 = 126
point 3: 126> 45 -> yes. swap X = 126, Y = 45
point 4: X = 0 -> no. continue
paragraph 5: Y = Y * 2048, Y = 45 * 2048 = 92160
point 6: I = Y / X, I = 92160/126 = 731
Point 7: in the table there are 255 rows of 8 values, we are looking for the 731th value.
This is the 91st line and the 3rd value, which is 0x087e, or 2174 in decimal
.
Item 8: X = X * 2174, X = 126 * 2174 = 273924
Item 9: A = X / 2048, A = 273924/2048 = 133 (truncated to an integer)

QED

Now take the green vector (from the fourth area).

Its coordinates are X = -170, Y = 95 (can be checked by pixels).

Classic method:

(-170) ^ 2 + (95) ^ 2 = 28900 + 9025 = 37925
SQRT (37925) = 194 (truncated to the integer)

Algorithm:

X = 2047-170 = 1877, Y = 2047 + 95 = 2142

points 1: 1877> 2047? -> no. X = 2047-1877 = 170
point 2: 2142> 2047? -> yes. Y = 2142-2047 = 95
point 3: 95> 170 -> no. leave as is.
point 4: X = 0 -> no. continue
paragraph 5: Y = Y * 2048, Y = 95 * 2048 = 194560
point 6: I = Y / X, I = 194560/170 = 1144
Point 7: in the table we are looking for the 1144th value.
This is the 143rd line and the 0th value, which is 0x092A, or 2346 in decimal.
Item 8: X = X * 2346, X = 2346 * 170 = 398,820
Item 9: A = X / 2048, A = 398820/2048 = 194 (truncated to an integer)

QED

Total


A similar algorithm can be used in DSPs (like me), for example, you can significantly accelerate the Fast Fourier Transform. It can also be used in computer graphics where ray tracing is used. Perhaps and still there is an application.

That's all. Thank you for the attention.

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