minor optimization (save a few square roots here and there)

include/mitsuba/core/frame.h

@@ -84,12 +84,24 @@ struct Frame {
 		return s * v.x + t * v.y + n * v.z;
 	}
 
+	/** \brief Assuming that the given direction is in the local coordinate
+	 * system, return the squared cosine of the angle between the normal and v */
+	inline static Float cosTheta2(const Vector &v) {
+		return v.z * v.z;
+	}
+
 	/** \brief Assuming that the given direction is in the local coordinate
 	 * system, return the cosine of the angle between the normal and v */
 	inline static Float cosTheta(const Vector &v) {
 		return v.z;
 	}
 
+	/** \brief Assuming that the given direction is in the local coordinate
+	 * system, return the squared sine of the angle between the normal and v */
+	inline static Float sinTheta2(const Vector &v) {
+		return 1.0f - v.z * v.z;
+	}
+
 	/** \brief Assuming that the given direction is in the local coordinate
 	 * system, return the sine of the angle between the normal and v */
 	inline static Float sinTheta(const Vector &v) {
@@ -109,9 +121,12 @@ struct Frame {
 	}
 
 	/** \brief Assuming that the given direction is in the local coordinate
-	 * system, return the squared sine of the angle between the normal and v */
-	inline static Float sinTheta2(const Vector &v) {
-		return 1.0f - v.z * v.z;
+	 * system, return the squared tangent of the angle between the normal and v */
+	inline static Float tanTheta2(const Vector &v) {
+		Float temp = 1 - v.z*v.z;
+		if (temp <= 0.0f)
+			return 0.0f;
+		return temp / (v.z * v.z);
 	}
 
 	/** \brief Assuming that the given direction is in the local coordinate

src/bsdfs/microfacet.h

@@ -115,19 +115,21 @@ public:
 		switch (m_type) {
 			case EBeckmann: {
 					/* Beckmann distribution function for Gaussian random surfaces */
-					const Float ex = Frame::tanTheta(m) / alphaU;
-					result = math::fastexp(-(ex*ex)) / (M_PI * alphaU*alphaU *
-							std::pow(Frame::cosTheta(m), (Float) 4.0f));
+					const Float exponent  = Frame::tanTheta2(m) / (alphaU*alphaU);
+					const Float cosTheta2 = Frame::cosTheta2(m);
+
+					result = math::fastexp(-exponent) /
+						(M_PI * alphaU*alphaU*cosTheta2*cosTheta2);
 				}
 				break;
 
 			case EGGX: {
 					/* Empirical GGX distribution function for rough surfaces */
-					const Float tanTheta = Frame::tanTheta(m),
-						        cosTheta = Frame::cosTheta(m);
+					const Float tanTheta2 = Frame::tanTheta2(m),
+						        cosTheta2 = Frame::cosTheta2(m);
 
-					const Float root = alphaU / (cosTheta*cosTheta *
-								(alphaU*alphaU + tanTheta*tanTheta));
+					const Float root = alphaU / (cosTheta2 *
+								(alphaU*alphaU + tanTheta2));
 
 					result = INV_PI * (root * root);
 				}

src/libpython/core.cpp

@@ -1201,17 +1201,21 @@ void export_core() {
 		.def("toWorld", &Frame::toWorld, BP_RETURN_VALUE)
 		.def("__repr__", &Frame::toString)
 		.def("cosTheta", &Frame::cosTheta)
+		.def("cosTheta2", &Frame::cosTheta2)
 		.def("sinTheta", &Frame::sinTheta)
 		.def("sinTheta2", &Frame::sinTheta2)
 		.def("tanTheta", &Frame::tanTheta)
+		.def("tanTheta2", &Frame::tanTheta2)
 		.def("sinPhi", &Frame::sinPhi)
 		.def("cosPhi", &Frame::cosPhi)
 		.def("sinPhi2", &Frame::sinPhi2)
 		.def("cosPhi2", &Frame::cosPhi2)
 		.staticmethod("cosTheta")
+		.staticmethod("cosTheta2")
 		.staticmethod("sinTheta")
 		.staticmethod("sinTheta2")
 		.staticmethod("tanTheta")
+		.staticmethod("tanTheta2")
 		.staticmethod("sinPhi")
 		.staticmethod("cosPhi")
 		.staticmethod("sinPhi2")