Improve JD acosx implementation (#17817)

Marlin/src/module/planner.cpp

         const float junction_acceleration = limit_value_by_axis_maximum(block->acceleration, junction_unit_vec),
                     sin_theta_d2 = SQRT(0.5f * (1.0f - junction_cos_theta)); // Trig half angle identity. Always positive.
 
-        vmax_junction_sqr = (junction_acceleration * junction_deviation_mm * sin_theta_d2) / (1.0f - sin_theta_d2);
+        vmax_junction_sqr = JUNC_SQ(junction_acceleration, sin_theta_d2);
 
         if (block->millimeters < 1) {
-          // Fast acos approximation (max. error +-0.033 rads)
-          // Based on MinMax polynomial published by W. Randolph Franklin, see
-          // https://wrf.ecse.rpi.edu/Research/Short_Notes/arcsin/onlyelem.html
-          // (acos(x) = pi / 2 - asin(x))
-
           const float neg = junction_cos_theta < 0 ? -1 : 1,
-                      t = neg * junction_cos_theta,
-                      asinx =       0.032843707f
-                            + t * (-1.451838349f
-                            + t * ( 29.66153956f
-                            + t * (-131.1123477f
-                            + t * ( 262.8130562f
-                            + t * (-242.7199627f + t * 84.31466202f) )))),
-                      junction_theta = RADIANS(90) - neg * asinx;
+                      t = neg * junction_cos_theta;
 
           // If angle is greater than 135 degrees (octagon), find speed for approximate arc
-          if (junction_theta > RADIANS(135)) {
-            // NOTE: MinMax acos approximation and thereby also junction_theta top out at pi-0.033, which avoids division by 0
-            const float limit_sqr = block->millimeters / (RADIANS(180) - junction_theta) * junction_acceleration;
+          if (t < -0.7071067812f) {
+
+            #if ENABLED(JD_USE_MATH_ACOS)
+
+              #error "TODO: Inline maths with the MCU / FPU."
+
+            #elif ENABLED(JD_USE_LOOKUP_TABLE)
+
+              // Fast acos approximation (max. error +-0.01 rads)
+              // Based on LUT table and linear interpolation
+
+              /**
+               *  // Generate the JD Lookup Table
+               *  constexpr float c = 1.00751317f; // Correction factor to center error around 0
+               *  for (int i = 0; i < jd_lut_count - 1; ++i) {
+               *    const float x0 = (sq(i) - 1) / sq(i),
+               *                y0 = acos(x0) * (i ? c : 1),
+               *                x1 = 0.5 * x0 + 0.5,
+               *                y1 = acos(x1) * c;
+               *    jd_lut_k[i] = (y0 - y1) / (x0 - x1);
+               *    jd_lut_b[i] = (y1 * x0 - y0 * x1) / (x0 - x1);
+               *  }
+               *  jd_lut_k[jd_lut_count - 1] = jd_lut_b[jd_lut_count - 1] = 0;
+               *
+               *  // Compute correction factor (Set c to 1.0f first!)
+               *  float min = INFINITY, max = -min;
+               *  for (float t = 0; t <= 1; t += 0.0003f) {
+               *    const float e = acos(t) / approx(t);
+               *    if (isfinite(e)) {
+               *      NOMORE(min, e);
+               *      NOLESS(max, e);
+               *    }
+               *  }
+               *  fprintf(stderr, "%.9gf, ", (min + max) / 2);
+               */
+              static constexpr int16_t  jd_lut_count = 15;
+              static constexpr uint16_t jd_lut_tll   = 1 << jd_lut_count;
+              static constexpr int16_t  jd_lut_tll0  = __builtin_clz(jd_lut_tll) + 1; // i.e., 16 - jd_lut_count
+              static constexpr float jd_lut_k[jd_lut_count] PROGMEM = {
+                -1.03146219f, -1.30760407f, -1.75205469f, -2.41705418f, -3.37768555f,
+                -4.74888229f, -6.69648552f, -9.45659828f, -13.3640289f, -18.8927879f,
+                -26.7136307f, -37.7754059f, -53.4200745f, -75.5457306f,   0.0f };
+              static constexpr float jd_lut_b[jd_lut_count] PROGMEM = {
+                1.57079637f, 1.70886743f, 2.04220533f, 2.62408018f, 3.52467203f,
+                4.85301876f, 6.77019119f, 9.50873947f, 13.4009094f, 18.9188652f,
+                26.7320709f, 37.7884521f, 53.4292908f, 75.5522461f,  0.0f };
+
+              const int16_t idx = (t == 0.0f) ? 0 : __builtin_clz(int16_t((1.0f - t) * jd_lut_tll)) - jd_lut_tll0;
+
+              float junction_theta = t * pgm_read_float(&jd_lut_k[idx]) + pgm_read_float(&jd_lut_b[idx]);
+              if (neg > 0) junction_theta = RADIANS(180) - junction_theta;
+
+            #else
+
+              // Fast acos(-t) approximation (max. error +-0.033rad = 1.89°)
+              // Based on MinMax polynomial published by W. Randolph Franklin, see
+              // https://wrf.ecse.rpi.edu/Research/Short_Notes/arcsin/onlyelem.html
+              //  acos( t) = pi / 2 - asin(x)
+              //  acos(-t) = pi - acos(t) ... pi / 2 + asin(x)
+
+              const float asinx =       0.032843707f
+                                + t * (-1.451838349f
+                                + t * ( 29.66153956f
+                                + t * (-131.1123477f
+                                + t * ( 262.8130562f
+                                + t * (-242.7199627f
+                                + t * ( 84.31466202f ) ))))),
+                          junction_theta = RADIANS(90) + neg * asinx; // acos(-t)
+
+              // NOTE: junction_theta bottoms out at 0.033 which avoids divide by 0.
+
+            #endif
+
+            const float limit_sqr = (block->millimeters * junction_acceleration) / junction_theta;
             NOMORE(vmax_junction_sqr, limit_sqr);
           }
         }
     // Start with a safe speed (from which the machine may halt to stop immediately).
     float safe_speed = nominal_speed;
 
-    #ifdef TRAVEL_EXTRA_XYJERK
-      const float extra_xyjerk = (de <= 0) ? TRAVEL_EXTRA_XYJERK : 0;
-    #else
-      constexpr float extra_xyjerk = 0;
+    #ifndef TRAVEL_EXTRA_XYJERK
+      #define TRAVEL_EXTRA_XYJERK 0
     #endif
+    const float extra_xyjerk = (de <= 0) ? TRAVEL_EXTRA_XYJERK : 0;
 
     uint8_t limited = 0;
     TERN(HAS_LINEAR_E_JERK, LOOP_XYZ, LOOP_XYZE)(i) {

Marlin/src/module/planner.h

 
 #include "../MarlinCore.h"
 
+#if HAS_JUNCTION_DEVIATION
+  // Enable this option for perfect accuracy but maximum
+  // computation. Should be fine on ARM processors.
+  //#define JD_USE_MATH_ACOS
+
+  // Disable this option to save 120 bytes of PROGMEM,
+  // but incur increased computation and a reduction
+  // in accuracy.
+  #define JD_USE_LOOKUP_TABLE
+#endif
+
 #include "motion.h"
 #include "../gcode/queue.h"
 
       static void autotemp_update();
     #endif
 
+    #define JUNC_SQ(N,ST) (junction_deviation_mm * (N) * (ST) / (1.0f - (ST)))
+
     #if HAS_LINEAR_E_JERK
       FORCE_INLINE static void recalculate_max_e_jerk() {
-        #define GET_MAX_E_JERK(N) SQRT(SQRT(0.5) * junction_deviation_mm * (N) * RECIPROCAL(1.0 - SQRT(0.5)))
+        #define GET_MAX_E_JERK(N) SQRT(JUNC_SQ(N,SQRT(0.5)))
         #if ENABLED(DISTINCT_E_FACTORS)
           LOOP_L_N(i, EXTRUDERS)
             max_e_jerk[i] = GET_MAX_E_JERK(settings.max_acceleration_mm_per_s2[E_AXIS_N(i)]);