More concise commentary in planner.cpp
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Scott Lahteine
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@ -223,145 +223,86 @@ void Planner::init() {
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#if ENABLED(S_CURVE_ACCELERATION)
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#if ENABLED(S_CURVE_ACCELERATION)
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// This routine, for AVR, returns 0x1000000 / d, but trying to get the inverse as
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/**
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// fast as possible. A fast converging iterative Newton-Raphson method is able to
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* This routine returns 0x1000000 / d, getting the inverse as fast as possible.
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// reach full precision in just 1 iteration, and takes 211 cycles (worst case, mean
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* A fast-converging iterative Newton-Raphson method can reach full precision in
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// case is less, up to 30 cycles for small divisors), instead of the 500 cycles a
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* just 1 iteration, and takes 211 cycles (worst case; the mean case is less, up
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// normal division would take.
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* to 30 cycles for small divisors), instead of the 500 cycles a normal division
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//
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* would take.
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// Inspired by the following page,
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*
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// https://stackoverflow.com/questions/27801397/newton-raphson-division-with-big-integers
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* Inspired by the following page:
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//
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* https://stackoverflow.com/questions/27801397/newton-raphson-division-with-big-integers
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// Suppose we want to calculate
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*
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// floor(2 ^ k / B) where B is a positive integer
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* Suppose we want to calculate floor(2 ^ k / B) where B is a positive integer
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// Then
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* Then, B must be <= 2^k, otherwise, the quotient is 0.
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// B must be <= 2^k, otherwise, the quotient is 0.
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*
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//
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* The Newton - Raphson iteration for x = B / 2 ^ k yields:
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// The Newton - Raphson iteration for x = B / 2 ^ k yields:
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* q[n + 1] = q[n] * (2 - q[n] * B / 2 ^ k)
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// q[n + 1] = q[n] * (2 - q[n] * B / 2 ^ k)
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*
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//
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* This can be rearranged to:
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// We can rearrange it as:
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* q[n + 1] = q[n] * (2 ^ (k + 1) - q[n] * B) >> k
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// q[n + 1] = q[n] * (2 ^ (k + 1) - q[n] * B) >> k
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*
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//
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* Each iteration requires only integer multiplications and bit shifts.
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// Each iteration of this kind requires only integer multiplications
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* It doesn't necessarily converge to floor(2 ^ k / B) but in the worst case
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// and bit shifts.
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* it eventually alternates between floor(2 ^ k / B) and ceil(2 ^ k / B).
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// Does it converge to floor(2 ^ k / B) ?: Not necessarily, but, in
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* So it checks for this case and extracts floor(2 ^ k / B).
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// the worst case, it eventually alternates between floor(2 ^ k / B)
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*
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// and ceiling(2 ^ k / B)).
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* A simple but important optimization for this approach is to truncate
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// So we can use some not-so-clever test to see if we are in this
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* multiplications (i.e., calculate only the higher bits of the product) in the
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// case, and extract floor(2 ^ k / B).
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* early iterations of the Newton - Raphson method. This is done so the results
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// Lastly, a simple but important optimization for this approach is to
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* of the early iterations are far from the quotient. Then it doesn't matter if
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// truncate multiplications (i.e.calculate only the higher bits of the
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* they are done inaccurately.
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// product) in the early iterations of the Newton - Raphson method.The
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* It's important to pick a good starting value for x. Knowing how many
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// reason to do so, is that the results of the early iterations are far
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* digits the divisor has, it can be estimated:
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// from the quotient, and it doesn't matter to perform them inaccurately.
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*
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// Finally, we should pick a good starting value for x. Knowing how many
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* 2^k / x = 2 ^ log2(2^k / x)
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// digits the divisor has, we can estimate it:
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* 2^k / x = 2 ^(log2(2^k)-log2(x))
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//
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* 2^k / x = 2 ^(k*log2(2)-log2(x))
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// 2^k / x = 2 ^ log2(2^k / x)
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* 2^k / x = 2 ^ (k-log2(x))
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// 2^k / x = 2 ^(log2(2^k)-log2(x))
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* 2^k / x >= 2 ^ (k-floor(log2(x)))
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// 2^k / x = 2 ^(k*log2(2)-log2(x))
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* floor(log2(x)) is simply the index of the most significant bit set.
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// 2^k / x = 2 ^ (k-log2(x))
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*
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// 2^k / x >= 2 ^ (k-floor(log2(x)))
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* If this estimation can be improved even further the number of iterations can be
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// floor(log2(x)) simply is the index of the most significant bit set.
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* reduced a lot, saving valuable execution time.
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//
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* The paper "Software Integer Division" by Thomas L.Rodeheffer, Microsoft
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// If we could improve this estimation even further, then the number of
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* Research, Silicon Valley,August 26, 2008, available at
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// iterations can be dropped quite a bit, thus saving valuable execution time.
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* https://www.microsoft.com/en-us/research/wp-content/uploads/2008/08/tr-2008-141.pdf
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// The paper "Software Integer Division" by Thomas L.Rodeheffer, Microsoft
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* suggests, for its integer division algorithm, using a table to supply the first
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// Research, Silicon Valley,August 26, 2008, that is available at
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* 8 bits of precision, then, due to the quadratic convergence nature of the
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// https://www.microsoft.com/en-us/research/wp-content/uploads/2008/08/tr-2008-141.pdf
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* Newton-Raphon iteration, just 2 iterations should be enough to get maximum
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// suggests , for its integer division algorithm, that using a table to supply the
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* precision of the division.
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// first 8 bits of precision, and due to the quadratic convergence nature of the
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* By precomputing values of inverses for small denominator values, just one
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// Newton-Raphon iteration, then just 2 iterations should be enough to get
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* Newton-Raphson iteration is enough to reach full precision.
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// maximum precision of the division.
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* This code uses the top 9 bits of the denominator as index.
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// If we precompute values of inverses for small denominator values, then
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*
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// just one Newton-Raphson iteration is enough to reach full precision
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* The AVR assembly function implements this C code using the data below:
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// We will use the top 9 bits of the denominator as index.
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*
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//
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* // For small divisors, it is best to directly retrieve the results
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// The AVR assembly function is implementing the following C code, included
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* if (d <= 110) return pgm_read_dword(&small_inv_tab[d]);
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// here as reference:
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*
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//
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* // Compute initial estimation of 0x1000000/x -
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// uint32_t get_period_inverse(uint32_t d) {
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* // Get most significant bit set on divider
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// static const uint8_t inv_tab[256] = {
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* uint8_t idx = 0;
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// 255,253,252,250,248,246,244,242,240,238,236,234,233,231,229,227,
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* uint32_t nr = d;
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// 225,224,222,220,218,217,215,213,212,210,208,207,205,203,202,200,
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* if (!(nr & 0xFF0000)) {
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// 199,197,195,194,192,191,189,188,186,185,183,182,180,179,178,176,
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* nr <<= 8; idx += 8;
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// 175,173,172,170,169,168,166,165,164,162,161,160,158,157,156,154,
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* if (!(nr & 0xFF0000)) { nr <<= 8; idx += 8; }
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// 153,152,151,149,148,147,146,144,143,142,141,139,138,137,136,135,
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* }
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// 134,132,131,130,129,128,127,126,125,123,122,121,120,119,118,117,
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* if (!(nr & 0xF00000)) { nr <<= 4; idx += 4; }
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// 116,115,114,113,112,111,110,109,108,107,106,105,104,103,102,101,
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* if (!(nr & 0xC00000)) { nr <<= 2; idx += 2; }
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// 100,99,98,97,96,95,94,93,92,91,90,89,88,88,87,86,
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* if (!(nr & 0x800000)) { nr <<= 1; idx += 1; }
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// 85,84,83,82,81,80,80,79,78,77,76,75,74,74,73,72,
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*
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// 71,70,70,69,68,67,66,66,65,64,63,62,62,61,60,59,
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* // Isolate top 9 bits of the denominator, to be used as index into the initial estimation table
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// 59,58,57,56,56,55,54,53,53,52,51,50,50,49,48,48,
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* uint32_t tidx = nr >> 15, // top 9 bits. bit8 is always set
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// 47,46,46,45,44,43,43,42,41,41,40,39,39,38,37,37,
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* ie = inv_tab[tidx & 0xFF] + 256, // Get the table value. bit9 is always set
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// 36,35,35,34,33,33,32,32,31,30,30,29,28,28,27,27,
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* x = idx <= 8 ? (ie >> (8 - idx)) : (ie << (idx - 8)); // Position the estimation at the proper place
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// 26,25,25,24,24,23,22,22,21,21,20,19,19,18,18,17,
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*
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// 17,16,15,15,14,14,13,13,12,12,11,10,10,9,9,8,
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* x = uint32_t((x * uint64_t(_BV(25) - x * d)) >> 24); // Refine estimation by newton-raphson. 1 iteration is enough
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// 8,7,7,6,6,5,5,4,4,3,3,2,2,1,0,0
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* const uint32_t r = _BV(24) - x * d; // Estimate remainder
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// };
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* if (r >= d) x++; // Check whether to adjust result
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//
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* return uint32_t(x); // x holds the proper estimation
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// // For small denominators, it is cheaper to directly store the result,
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*
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// // because those denominators would require 2 Newton-Raphson iterations
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*/
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// // to converge to the required result precision. For bigger ones, just
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// // ONE Newton-Raphson iteration is enough to get maximum precision!
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// static const uint32_t small_inv_tab[111] PROGMEM = {
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// 16777216,16777216,8388608,5592405,4194304,3355443,2796202,2396745,2097152,1864135,1677721,1525201,1398101,1290555,1198372,1118481,
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// 1048576,986895,932067,883011,838860,798915,762600,729444,699050,671088,645277,621378,599186,578524,559240,541200,
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// 524288,508400,493447,479349,466033,453438,441505,430185,419430,409200,399457,390167,381300,372827,364722,356962,
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// 349525,342392,335544,328965,322638,316551,310689,305040,299593,294337,289262,284359,279620,275036,270600,266305,
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// 262144,258111,254200,250406,246723,243148,239674,236298,233016,229824,226719,223696,220752,217885,215092,212369,
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// 209715,207126,204600,202135,199728,197379,195083,192841,190650,188508,186413,184365,182361,180400,178481,176602,
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// 174762,172960,171196,169466,167772,166111,164482,162885,161319,159783,158275,156796,155344,153919,152520
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// };
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//
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// // For small divisors, it is best to directly retrieve the results
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// if (d <= 110)
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// return pgm_read_dword(&small_inv_tab[d]);
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//
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// // Compute initial estimation of 0x1000000/x -
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// // Get most significant bit set on divider
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// uint8_t idx = 0;
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// uint32_t nr = d;
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// if (!(nr & 0xFF0000)) {
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// nr <<= 8;
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// idx += 8;
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// if (!(nr & 0xFF0000)) {
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// nr <<= 8;
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// idx += 8;
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// }
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// }
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// if (!(nr & 0xF00000)) {
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// nr <<= 4;
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// idx += 4;
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// }
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// if (!(nr & 0xC00000)) {
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// nr <<= 2;
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// idx += 2;
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// }
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// if (!(nr & 0x800000)) {
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// nr <<= 1;
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// idx += 1;
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// }
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//
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// // Isolate top 9 bits of the denominator, to be used as index into the initial estimation table
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// uint32_t tidx = nr >> 15; // top 9 bits. bit8 is always set
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// uint32_t ie = inv_tab[tidx & 0xFF] + 256; // Get the table value. bit9 is always set
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// uint32_t x = idx <= 8 ? (ie >> (8 - idx)) : (ie << (idx - 8)); // Position the estimation at the proper place
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//
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// // Now, refine estimation by newton-raphson. 1 iteration is enough
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// x = uint32_t((x * uint64_t((1 << 25) - x * d)) >> 24);
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//
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// // Estimate remainder
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// uint32_t r = (1 << 24) - x * d;
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//
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// // Check if we must adjust result
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// if (r >= d) x++;
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//
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// // x holds the proper estimation
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// return uint32_t(x);
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// }
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//
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static uint32_t get_period_inverse(uint32_t d) {
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static uint32_t get_period_inverse(uint32_t d) {
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static const uint8_t inv_tab[256] PROGMEM = {
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static const uint8_t inv_tab[256] PROGMEM = {
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@ -397,13 +338,12 @@ void Planner::init() {
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};
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};
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// For small divisors, it is best to directly retrieve the results
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// For small divisors, it is best to directly retrieve the results
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if (d <= 110)
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if (d <= 110) return pgm_read_dword(&small_inv_tab[d]);
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return pgm_read_dword(&small_inv_tab[d]);
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register uint8_t r8 = d & 0xFF;
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register uint8_t r8 = d & 0xFF,
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register uint8_t r9 = (d >> 8) & 0xFF;
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r9 = (d >> 8) & 0xFF,
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register uint8_t r10 = (d >> 16) & 0xFF;
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r10 = (d >> 16) & 0xFF,
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register uint8_t r2,r3,r4,r5,r6,r7,r11,r12,r13,r14,r15,r16,r17,r18;
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r2,r3,r4,r5,r6,r7,r11,r12,r13,r14,r15,r16,r17,r18;
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register const uint8_t* ptab = inv_tab;
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register const uint8_t* ptab = inv_tab;
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__asm__ __volatile__(
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__asm__ __volatile__(
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