Cycles: optimize ensure_valid_reflection(), reduces render time by about 1%

This is an implementation that is about 1.5-2.1 times faster. It gives a result
that is on average 6° different from the old implementation. The difference is
because normals (Ng, N, N') are not selected to be coplanar, but instead
reflection R is lifted the least amount and the N' is computed as a bisector.

Differential Revision: https://developer.blender.org/D10084

2 files changed, 36 insertions, 150 deletions

diff --git a/intern/cycles/kernel/kernel_montecarlo.h b/intern/cycles/kernel/kernel_montecarlo.h
index ce37bd0b15e..ba25c0e24e4 100644
--- a/intern/cycles/kernel/kernel_montecarlo.h
+++ b/intern/cycles/kernel/kernel_montecarlo.h
@@ -195,108 +195,31 @@ ccl_device float2 regular_polygon_sample(float corners, float rotation, float u,
 
 ccl_device float3 ensure_valid_reflection(float3 Ng, float3 I, float3 N)
 {
-  float3 R = 2 * dot(N, I) * N - I;
-
-  /* Reflection rays may always be at least as shallow as the incoming ray. */
-  float threshold = min(0.9f * dot(Ng, I), 0.01f);
-  if (dot(Ng, R) >= threshold) {
-    return N;
-  }
-
-  /* Form coordinate system with Ng as the Z axis and N inside the X-Z-plane.
-   * The X axis is found by normalizing the component of N that's orthogonal to Ng.
-   * The Y axis isn't actually needed.
-   */
-  float NdotNg = dot(N, Ng);
-  float3 X = normalize(N - NdotNg * Ng);
-
-  /* Keep math expressions. */
-  /* clang-format off */
-  /* Calculate N.z and N.x in the local coordinate system.
-   *
-   * The goal of this computation is to find a N' that is rotated towards Ng just enough
-   * to lift R' above the threshold (here called t), therefore dot(R', Ng) = t.
-   *
-   * According to the standard reflection equation,
-   * this means that we want dot(2*dot(N', I)*N' - I, Ng) = t.
-   *
-   * Since the Z axis of our local coordinate system is Ng, dot(x, Ng) is just x.z, so we get
-   * 2*dot(N', I)*N'.z - I.z = t.
-   *
-   * The rotation is simple to express in the coordinate system we formed -
-   * since N lies in the X-Z-plane, we know that N' will also lie in the X-Z-plane,
-   * so N'.y = 0 and therefore dot(N', I) = N'.x*I.x + N'.z*I.z .
-   *
-   * Furthermore, we want N' to be normalized, so N'.x = sqrt(1 - N'.z^2).
-   *
-   * With these simplifications,
-   * we get the final equation 2*(sqrt(1 - N'.z^2)*I.x + N'.z*I.z)*N'.z - I.z = t.
-   *
-   * The only unknown here is N'.z, so we can solve for that.
-   *
-   * The equation has four solutions in general:
-   *
-   * N'.z = +-sqrt(0.5*(+-sqrt(I.x^2*(I.x^2 + I.z^2 - t^2)) + t*I.z + I.x^2 + I.z^2)/(I.x^2 + I.z^2))
-   * We can simplify this expression a bit by grouping terms:
-   *
-   * a = I.x^2 + I.z^2
-   * b = sqrt(I.x^2 * (a - t^2))
-   * c = I.z*t + a
-   * N'.z = +-sqrt(0.5*(+-b + c)/a)
-   *
-   * Two solutions can immediately be discarded because they're negative so N' would lie in the
-   * lower hemisphere.
-   */
-  /* clang-format on */
-
-  float Ix = dot(I, X), Iz = dot(I, Ng);
-  float Ix2 = sqr(Ix), Iz2 = sqr(Iz);
-  float a = Ix2 + Iz2;
-
-  float b = safe_sqrtf(Ix2 * (a - sqr(threshold)));
-  float c = Iz * threshold + a;
-
-  /* Evaluate both solutions.
-   * In many cases one can be immediately discarded (if N'.z would be imaginary or larger than
-   * one), so check for that first. If no option is viable (might happen in extreme cases like N
-   * being in the wrong hemisphere), give up and return Ng. */
-  float fac = 0.5f / a;
-  float N1_z2 = fac * (b + c), N2_z2 = fac * (-b + c);
-  bool valid1 = (N1_z2 > 1e-5f) && (N1_z2 <= (1.0f + 1e-5f));
-  bool valid2 = (N2_z2 > 1e-5f) && (N2_z2 <= (1.0f + 1e-5f));
-
-  float2 N_new;
-  if (valid1 && valid2) {
-    /* If both are possible, do the expensive reflection-based check. */
-    float2 N1 = make_float2(safe_sqrtf(1.0f - N1_z2), safe_sqrtf(N1_z2));
-    float2 N2 = make_float2(safe_sqrtf(1.0f - N2_z2), safe_sqrtf(N2_z2));
-
-    float R1 = 2 * (N1.x * Ix + N1.y * Iz) * N1.y - Iz;
-    float R2 = 2 * (N2.x * Ix + N2.y * Iz) * N2.y - Iz;
-
-    valid1 = (R1 >= 1e-5f);
-    valid2 = (R2 >= 1e-5f);
-    if (valid1 && valid2) {
-      /* If both solutions are valid, return the one with the shallower reflection since it will be
-       * closer to the input (if the original reflection wasn't shallow, we would not be in this
-       * part of the function). */
-      N_new = (R1 < R2) ? N1 : N2;
+  float3 R;
+  float NI = dot(N, I);
+  float NgR, threshold;
+
+  /* Check if the incident ray is coming from behind normal N. */
+  if (NI > 0) {
+    /* Normal reflection */
+    R = (2 * NI) * N - I;
+    NgR = dot(Ng, R);
+
+    /* Reflection rays may always be at least as shallow as the incoming ray. */
+    threshold = min(0.9f * dot(Ng, I), 0.01f);
+    if (NgR >= threshold) {
+      return N;
     }
-    else {
-      /* If only one reflection is valid (= positive), pick that one. */
-      N_new = (R1 > R2) ? N1 : N2;
-    }
-  }
-  else if (valid1 || valid2) {
-    /* Only one solution passes the N'.z criterium, so pick that one. */
-    float Nz2 = valid1 ? N1_z2 : N2_z2;
-    N_new = make_float2(safe_sqrtf(1.0f - Nz2), safe_sqrtf(Nz2));
   }
   else {
-    return Ng;
+    /* Bad incident */
+    R = -I;
+    NgR = dot(Ng, R);
+    threshold = 0.01f;
   }
 
-  return N_new.x * X + N_new.y * Ng;
+  R = R + Ng * (threshold - NgR);            /* Lift the reflection above the threshold. */
+  return normalize(I * len(R) + R * len(I)); /* Find a bisector. */
 }
 
 CCL_NAMESPACE_END
diff --git a/intern/cycles/kernel/shaders/stdcycles.h b/intern/cycles/kernel/shaders/stdcycles.h
index dd604da68ce..af7b645d9a2 100644
--- a/intern/cycles/kernel/shaders/stdcycles.h
+++ b/intern/cycles/kernel/shaders/stdcycles.h
@@ -84,67 +84,30 @@ closure color principled_hair(normal N,
 closure color henyey_greenstein(float g) BUILTIN;
 closure color absorption() BUILTIN;
 
-normal ensure_valid_reflection(normal Ng, vector I, normal N)
+normal ensure_valid_reflection(normal Ng, normal I, normal N)
 {
   /* The implementation here mirrors the one in kernel_montecarlo.h,
    * check there for an explanation of the algorithm. */
-
-  float sqr(float x)
-  {
-    return x * x;
-  }
-
-  vector R = 2 * dot(N, I) * N - I;
-
-  float threshold = min(0.9 * dot(Ng, I), 0.01);
-  if (dot(Ng, R) >= threshold) {
-    return N;
-  }
-
-  float NdotNg = dot(N, Ng);
-  vector X = normalize(N - NdotNg * Ng);
-
-  float Ix = dot(I, X), Iz = dot(I, Ng);
-  float Ix2 = sqr(Ix), Iz2 = sqr(Iz);
-  float a = Ix2 + Iz2;
-
-  float b = sqrt(Ix2 * (a - sqr(threshold)));
-  float c = Iz * threshold + a;
-
-  float fac = 0.5 / a;
-  float N1_z2 = fac * (b + c), N2_z2 = fac * (-b + c);
-  int valid1 = (N1_z2 > 1e-5) && (N1_z2 <= (1.0 + 1e-5));
-  int valid2 = (N2_z2 > 1e-5) && (N2_z2 <= (1.0 + 1e-5));
-
-  float N_new_x, N_new_z;
-  if (valid1 && valid2) {
-    float N1_x = sqrt(1.0 - N1_z2), N1_z = sqrt(N1_z2);
-    float N2_x = sqrt(1.0 - N2_z2), N2_z = sqrt(N2_z2);
-
-    float R1 = 2 * (N1_x * Ix + N1_z * Iz) * N1_z - Iz;
-    float R2 = 2 * (N2_x * Ix + N2_z * Iz) * N2_z - Iz;
-
-    valid1 = (R1 >= 1e-5);
-    valid2 = (R2 >= 1e-5);
-    if (valid1 && valid2) {
-      N_new_x = (R1 < R2) ? N1_x : N2_x;
-      N_new_z = (R1 < R2) ? N1_z : N2_z;
-    }
-    else {
-      N_new_x = (R1 > R2) ? N1_x : N2_x;
-      N_new_z = (R1 > R2) ? N1_z : N2_z;
+  vector R;
+  float NI = dot(N, I);
+  float NgR, threshold;
+
+  if (NI > 0) {
+    R = (2 * NI) * N - I;
+    NgR = dot(Ng, R);
+    threshold = min(0.9 * dot(Ng, I), 0.01);
+    if (NgR >= threshold) {
+      return N;
     }
   }
-  else if (valid1 || valid2) {
-    float Nz2 = valid1 ? N1_z2 : N2_z2;
-    N_new_x = sqrt(1.0 - Nz2);
-    N_new_z = sqrt(Nz2);
-  }
   else {
-    return Ng;
+    R = -I;
+    NgR = dot(Ng, R);
+    threshold = 0.01;
   }
 
-  return N_new_x * X + N_new_z * Ng;
+  R = R + Ng * (threshold - NgR);
+  return normalize(I * length(R) + R * length(I));
 }
 
 #endif /* CCL_STDOSL_H */