From 8f14ef855de42ac3535dd6b9b135162d8716dd14 Mon Sep 17 00:00:00 2001 From: Scott Lahteine Date: Mon, 28 May 2018 19:24:51 -0500 Subject: [PATCH] More concise commentary in planner.cpp --- Marlin/src/module/planner.cpp | 230 +++++++++++++--------------------- 1 file changed, 85 insertions(+), 145 deletions(-) diff --git a/Marlin/src/module/planner.cpp b/Marlin/src/module/planner.cpp index 7d7e36982b..4676d32433 100644 --- a/Marlin/src/module/planner.cpp +++ b/Marlin/src/module/planner.cpp @@ -235,145 +235,86 @@ void Planner::init() { #if ENABLED(S_CURVE_ACCELERATION) #ifdef __AVR__ - // This routine, for AVR, returns 0x1000000 / d, but trying to get the inverse as - // fast as possible. A fast converging iterative Newton-Raphson method is able to - // reach full precision in just 1 iteration, and takes 211 cycles (worst case, mean - // case is less, up to 30 cycles for small divisors), instead of the 500 cycles a - // normal division would take. - // - // Inspired by the following page, - // https://stackoverflow.com/questions/27801397/newton-raphson-division-with-big-integers - // - // Suppose we want to calculate - // floor(2 ^ k / B) where B is a positive integer - // Then - // B must be <= 2^k, otherwise, the quotient is 0. - // - // The Newton - Raphson iteration for x = B / 2 ^ k yields: - // q[n + 1] = q[n] * (2 - q[n] * B / 2 ^ k) - // - // We can rearrange it as: - // q[n + 1] = q[n] * (2 ^ (k + 1) - q[n] * B) >> k - // - // Each iteration of this kind requires only integer multiplications - // and bit shifts. - // Does it converge to floor(2 ^ k / B) ?: Not necessarily, but, in - // the worst case, it eventually alternates between floor(2 ^ k / B) - // and ceiling(2 ^ k / B)). - // So we can use some not-so-clever test to see if we are in this - // case, and extract floor(2 ^ k / B). - // Lastly, a simple but important optimization for this approach is to - // truncate multiplications (i.e.calculate only the higher bits of the - // product) in the early iterations of the Newton - Raphson method.The - // reason to do so, is that the results of the early iterations are far - // from the quotient, and it doesn't matter to perform them inaccurately. - // Finally, we should pick a good starting value for x. Knowing how many - // digits the divisor has, we can estimate it: - // - // 2^k / x = 2 ^ log2(2^k / x) - // 2^k / x = 2 ^(log2(2^k)-log2(x)) - // 2^k / x = 2 ^(k*log2(2)-log2(x)) - // 2^k / x = 2 ^ (k-log2(x)) - // 2^k / x >= 2 ^ (k-floor(log2(x))) - // floor(log2(x)) simply is the index of the most significant bit set. - // - // If we could improve this estimation even further, then the number of - // iterations can be dropped quite a bit, thus saving valuable execution time. - // The paper "Software Integer Division" by Thomas L.Rodeheffer, Microsoft - // Research, Silicon Valley,August 26, 2008, that is available at - // https://www.microsoft.com/en-us/research/wp-content/uploads/2008/08/tr-2008-141.pdf - // suggests , for its integer division algorithm, that using a table to supply the - // first 8 bits of precision, and due to the quadratic convergence nature of the - // Newton-Raphon iteration, then just 2 iterations should be enough to get - // maximum precision of the division. - // If we precompute values of inverses for small denominator values, then - // just one Newton-Raphson iteration is enough to reach full precision - // We will use the top 9 bits of the denominator as index. - // - // The AVR assembly function is implementing the following C code, included - // here as reference: - // - // uint32_t get_period_inverse(uint32_t d) { - // static const uint8_t inv_tab[256] = { - // 255,253,252,250,248,246,244,242,240,238,236,234,233,231,229,227, - // 225,224,222,220,218,217,215,213,212,210,208,207,205,203,202,200, - // 199,197,195,194,192,191,189,188,186,185,183,182,180,179,178,176, - // 175,173,172,170,169,168,166,165,164,162,161,160,158,157,156,154, - // 153,152,151,149,148,147,146,144,143,142,141,139,138,137,136,135, - // 134,132,131,130,129,128,127,126,125,123,122,121,120,119,118,117, - // 116,115,114,113,112,111,110,109,108,107,106,105,104,103,102,101, - // 100,99,98,97,96,95,94,93,92,91,90,89,88,88,87,86, - // 85,84,83,82,81,80,80,79,78,77,76,75,74,74,73,72, - // 71,70,70,69,68,67,66,66,65,64,63,62,62,61,60,59, - // 59,58,57,56,56,55,54,53,53,52,51,50,50,49,48,48, - // 47,46,46,45,44,43,43,42,41,41,40,39,39,38,37,37, - // 36,35,35,34,33,33,32,32,31,30,30,29,28,28,27,27, - // 26,25,25,24,24,23,22,22,21,21,20,19,19,18,18,17, - // 17,16,15,15,14,14,13,13,12,12,11,10,10,9,9,8, - // 8,7,7,6,6,5,5,4,4,3,3,2,2,1,0,0 - // }; - // - // // For small denominators, it is cheaper to directly store the result, - // // because those denominators would require 2 Newton-Raphson iterations - // // to converge to the required result precision. For bigger ones, just - // // ONE Newton-Raphson iteration is enough to get maximum precision! - // static const uint32_t small_inv_tab[111] PROGMEM = { - // 16777216,16777216,8388608,5592405,4194304,3355443,2796202,2396745,2097152,1864135,1677721,1525201,1398101,1290555,1198372,1118481, - // 1048576,986895,932067,883011,838860,798915,762600,729444,699050,671088,645277,621378,599186,578524,559240,541200, - // 524288,508400,493447,479349,466033,453438,441505,430185,419430,409200,399457,390167,381300,372827,364722,356962, - // 349525,342392,335544,328965,322638,316551,310689,305040,299593,294337,289262,284359,279620,275036,270600,266305, - // 262144,258111,254200,250406,246723,243148,239674,236298,233016,229824,226719,223696,220752,217885,215092,212369, - // 209715,207126,204600,202135,199728,197379,195083,192841,190650,188508,186413,184365,182361,180400,178481,176602, - // 174762,172960,171196,169466,167772,166111,164482,162885,161319,159783,158275,156796,155344,153919,152520 - // }; - // - // // For small divisors, it is best to directly retrieve the results - // if (d <= 110) - // return pgm_read_dword(&small_inv_tab[d]); - // - // // Compute initial estimation of 0x1000000/x - - // // Get most significant bit set on divider - // uint8_t idx = 0; - // uint32_t nr = d; - // if (!(nr & 0xFF0000)) { - // nr <<= 8; - // idx += 8; - // if (!(nr & 0xFF0000)) { - // nr <<= 8; - // idx += 8; - // } - // } - // if (!(nr & 0xF00000)) { - // nr <<= 4; - // idx += 4; - // } - // if (!(nr & 0xC00000)) { - // nr <<= 2; - // idx += 2; - // } - // if (!(nr & 0x800000)) { - // nr <<= 1; - // idx += 1; - // } - // - // // Isolate top 9 bits of the denominator, to be used as index into the initial estimation table - // uint32_t tidx = nr >> 15; // top 9 bits. bit8 is always set - // uint32_t ie = inv_tab[tidx & 0xFF] + 256; // Get the table value. bit9 is always set - // uint32_t x = idx <= 8 ? (ie >> (8 - idx)) : (ie << (idx - 8)); // Position the estimation at the proper place - // - // // Now, refine estimation by newton-raphson. 1 iteration is enough - // x = uint32_t((x * uint64_t((1 << 25) - x * d)) >> 24); - // - // // Estimate remainder - // uint32_t r = (1 << 24) - x * d; - // - // // Check if we must adjust result - // if (r >= d) x++; - // - // // x holds the proper estimation - // return uint32_t(x); - // } - // + /** + * This routine returns 0x1000000 / d, getting the inverse as fast as possible. + * A fast-converging iterative Newton-Raphson method can reach full precision in + * just 1 iteration, and takes 211 cycles (worst case; the mean case is less, up + * to 30 cycles for small divisors), instead of the 500 cycles a normal division + * would take. + * + * Inspired by the following page: + * https://stackoverflow.com/questions/27801397/newton-raphson-division-with-big-integers + * + * Suppose we want to calculate floor(2 ^ k / B) where B is a positive integer + * Then, B must be <= 2^k, otherwise, the quotient is 0. + * + * The Newton - Raphson iteration for x = B / 2 ^ k yields: + * q[n + 1] = q[n] * (2 - q[n] * B / 2 ^ k) + * + * This can be rearranged to: + * q[n + 1] = q[n] * (2 ^ (k + 1) - q[n] * B) >> k + * + * Each iteration requires only integer multiplications and bit shifts. + * It doesn't necessarily converge to floor(2 ^ k / B) but in the worst case + * it eventually alternates between floor(2 ^ k / B) and ceil(2 ^ k / B). + * So it checks for this case and extracts floor(2 ^ k / B). + * + * A simple but important optimization for this approach is to truncate + * multiplications (i.e., calculate only the higher bits of the product) in the + * early iterations of the Newton - Raphson method. This is done so the results + * of the early iterations are far from the quotient. Then it doesn't matter if + * they are done inaccurately. + * It's important to pick a good starting value for x. Knowing how many + * digits the divisor has, it can be estimated: + * + * 2^k / x = 2 ^ log2(2^k / x) + * 2^k / x = 2 ^(log2(2^k)-log2(x)) + * 2^k / x = 2 ^(k*log2(2)-log2(x)) + * 2^k / x = 2 ^ (k-log2(x)) + * 2^k / x >= 2 ^ (k-floor(log2(x))) + * floor(log2(x)) is simply the index of the most significant bit set. + * + * If this estimation can be improved even further the number of iterations can be + * reduced a lot, saving valuable execution time. + * The paper "Software Integer Division" by Thomas L.Rodeheffer, Microsoft + * Research, Silicon Valley,August 26, 2008, available at + * https://www.microsoft.com/en-us/research/wp-content/uploads/2008/08/tr-2008-141.pdf + * suggests, for its integer division algorithm, using a table to supply the first + * 8 bits of precision, then, due to the quadratic convergence nature of the + * Newton-Raphon iteration, just 2 iterations should be enough to get maximum + * precision of the division. + * By precomputing values of inverses for small denominator values, just one + * Newton-Raphson iteration is enough to reach full precision. + * This code uses the top 9 bits of the denominator as index. + * + * The AVR assembly function implements this C code using the data below: + * + * // For small divisors, it is best to directly retrieve the results + * if (d <= 110) return pgm_read_dword(&small_inv_tab[d]); + * + * // Compute initial estimation of 0x1000000/x - + * // Get most significant bit set on divider + * uint8_t idx = 0; + * uint32_t nr = d; + * if (!(nr & 0xFF0000)) { + * nr <<= 8; idx += 8; + * if (!(nr & 0xFF0000)) { nr <<= 8; idx += 8; } + * } + * if (!(nr & 0xF00000)) { nr <<= 4; idx += 4; } + * if (!(nr & 0xC00000)) { nr <<= 2; idx += 2; } + * if (!(nr & 0x800000)) { nr <<= 1; idx += 1; } + * + * // Isolate top 9 bits of the denominator, to be used as index into the initial estimation table + * uint32_t tidx = nr >> 15, // top 9 bits. bit8 is always set + * ie = inv_tab[tidx & 0xFF] + 256, // Get the table value. bit9 is always set + * x = idx <= 8 ? (ie >> (8 - idx)) : (ie << (idx - 8)); // Position the estimation at the proper place + * + * x = uint32_t((x * uint64_t(_BV(25) - x * d)) >> 24); // Refine estimation by newton-raphson. 1 iteration is enough + * const uint32_t r = _BV(24) - x * d; // Estimate remainder + * if (r >= d) x++; // Check whether to adjust result + * return uint32_t(x); // x holds the proper estimation + * + */ static uint32_t get_period_inverse(uint32_t d) { static const uint8_t inv_tab[256] PROGMEM = { @@ -409,13 +350,12 @@ void Planner::init() { }; // For small divisors, it is best to directly retrieve the results - if (d <= 110) - return pgm_read_dword(&small_inv_tab[d]); + if (d <= 110) return pgm_read_dword(&small_inv_tab[d]); - register uint8_t r8 = d & 0xFF; - register uint8_t r9 = (d >> 8) & 0xFF; - register uint8_t r10 = (d >> 16) & 0xFF; - register uint8_t r2,r3,r4,r5,r6,r7,r11,r12,r13,r14,r15,r16,r17,r18; + register uint8_t r8 = d & 0xFF, + r9 = (d >> 8) & 0xFF, + r10 = (d >> 16) & 0xFF, + r2,r3,r4,r5,r6,r7,r11,r12,r13,r14,r15,r16,r17,r18; register const uint8_t* ptab = inv_tab; __asm__ __volatile__(