From 8f14ef855de42ac3535dd6b9b135162d8716dd14 Mon Sep 17 00:00:00 2001
From: Scott Lahteine <github@thinkyhead.com>
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__(
--
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