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More concise commentary in planner.cpp

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Scott Lahteine 2018-05-28 19:27:55 -05:00
parent dfe90d552d
commit 2c4d8761ec
1 changed files with 85 additions and 145 deletions
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Marlin/planner.cpp
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@ -223,145 +223,86 @@ void Planner::init() {
#if ENABLED(S_CURVE_ACCELERATION)
// 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 = {
@ -397,13 +338,12 @@ 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__(