replacement for fastled sqrt16() (#4426)

* added bitwise operation based sqrt16

- replacement for fastled, it is about 10% slower for numbers smaller 128 but faster for larger numbers. speed difference is irrelevant to WLED but it saves some flash.

* updated to 32bit, improved for typical WLED use

- making it 32bits allows for larger numbers
- added another initial condition check for medium sized numbers
- increased the "small number" optimization to larger numbers: the function is currently only used to calculate sqrt(x^2+y^2) which even for small segments is larger than the initially used 64, so optimizing for 1024 makes more sense, although the value is arbitrarily chosen

wled00/FX.cpp

@@ -6179,15 +6179,15 @@ uint16_t mode_2Dmetaballs(void) {   // Metaballs by Stefan Petrick. Cannot have
       // and add them together with weightening
       uint16_t dx = abs(x - x1);
       uint16_t dy = abs(y - y1);
-      uint16_t dist = 2 * sqrt16((dx * dx) + (dy * dy));
+      uint16_t dist = 2 * sqrt32_bw((dx * dx) + (dy * dy));
 
       dx = abs(x - x2);
       dy = abs(y - y2);
-      dist += sqrt16((dx * dx) + (dy * dy));
+      dist += sqrt32_bw((dx * dx) + (dy * dy));
 
       dx = abs(x - x3);
       dy = abs(y - y3);
-      dist += sqrt16((dx * dx) + (dy * dy));
+      dist += sqrt32_bw((dx * dx) + (dy * dy));
 
       // inverse result
       byte color = dist ? 1000 / dist : 255;
@@ -11744,7 +11744,7 @@ uint16_t mode_particle1DsonicStream(void) {
       else PartSys->particles[i].ttl = 0;
     }
     if (SEGMENT.check1) { // modulate colors by mid frequencies
-      int mids = sqrt16((int)fftResult[5] + (int)fftResult[6] + (int)fftResult[7] + (int)fftResult[8] + (int)fftResult[9] + (int)fftResult[10]); // average the mids, bin 5 is ~500Hz, bin 10 is ~2kHz (see audio_reactive.h)
+      int mids = sqrt32_bw((int)fftResult[5] + (int)fftResult[6] + (int)fftResult[7] + (int)fftResult[8] + (int)fftResult[9] + (int)fftResult[10]); // average the mids, bin 5 is ~500Hz, bin 10 is ~2kHz (see audio_reactive.h)
       PartSys->particles[i].hue += (mids * perlin8(PartSys->particles[i].x << 2, SEGMENT.step << 2)) >> 9; // color by perlin noise from mid frequencies
     }
   }
@@ -11832,7 +11832,7 @@ uint16_t mode_particle1DsonicBoom(void) {
   // particle manipulation
   for (uint32_t i = 0; i < PartSys->usedParticles; i++) {
     if (SEGMENT.check1) { // modulate colors by mid frequencies
-      int mids = sqrt16((int)fftResult[5] + (int)fftResult[6] + (int)fftResult[7] + (int)fftResult[8] + (int)fftResult[9] + (int)fftResult[10]); // average the mids, bin 5 is ~500Hz, bin 10 is ~2kHz (see audio_reactive.h)
+      int mids = sqrt32_bw((int)fftResult[5] + (int)fftResult[6] + (int)fftResult[7] + (int)fftResult[8] + (int)fftResult[9] + (int)fftResult[10]); // average the mids, bin 5 is ~500Hz, bin 10 is ~2kHz (see audio_reactive.h)
       PartSys->particles[i].hue += (mids * perlin8(PartSys->particles[i].x << 2, SEGMENT.step << 2)) >> 9; // color by perlin noise from mid frequencies
     }
     if (PartSys->particles[i].ttl > 16) {

wled00/FX_fcn.cpp

@@ -932,7 +932,7 @@ uint16_t Segment::calc_virtualLength() const {
         break;
       case M12_pArc:
         { unsigned vLen2 = vW * vW + vH * vH;            // length ^2
-          if (vLen2 < UINT16_MAX) vLen = sqrt16(vLen2);  // use faster function for 16bit values
+          if (vLen2 < UINT16_MAX) vLen = sqrt32_bw(vLen2);  // use faster function for 16bit values
           else vLen = sqrtf(vLen2);                      // fall-back to float if bigger
           if (vW != vH) vLen++; // round up
         }

wled00/fcn_declare.h

@@ -588,6 +588,7 @@ float fmod_t(float num, float denom);
 #define cos_t cosf
 #define tan_t tanf
 */
+uint32_t sqrt32_bw(uint32_t x);
 
 //wled_serial.cpp
 void handleSerial();

wled00/wled_math.cpp

@@ -234,4 +234,28 @@ float fmod_t(float num, float denom) {
   return res;
 }
 
-#endif  // WLEDMM
+#endif  // WLEDMM
+
+// bit-wise integer square root calculation (exact)
+uint32_t sqrt32_bw(uint32_t x) {
+  uint32_t res = 0;
+  uint32_t bit;
+  uint32_t num = x; // use 32bit for faster calculation
+
+  if(num < 1 << 10)  bit = 1 << 10; // speed optimization for small numbers < 32^2
+  else if (num < 1 << 20) bit = 1 << 20; // speed optimization for medium numbers < 1024^2
+  else bit = 1 << 30; // start with highest power of 4 <= 2^32
+
+  while (bit > num) bit >>= 2; // reduce iterations
+
+  while (bit != 0) {
+    if (num >= res + bit) {
+      num -= res + bit;
+      res = (res >> 1) + bit;
+    } else {
+      res >>= 1;
+    }
+    bit >>= 2;
+  }
+  return res;
+}