Simplex noise for OSL part 3: flow noise

In this article I present an OSL implementation of flow noise, both in 2D and in 4D

Flow Noise

in two previous posts (I, II) I showed a port to OSL of Stefan Gustavsons implementations of Simplex Noise and a fBM variant. Four dimensional Simplex Noise may be used to simulate the time evolution of 3D noise and in some situations like for example boiling liquids the result might be perfectly adequate. However if we want to capture the effects of swirls as seen in flowing liquids and gasses a better option is to use flow noise.

Flow noise was pioneered by Perlin and Neyret and the concept of adding vorticity (swirls) by rotating the lattice gradient vectors used to create the noise and adding higher octaves of noise at positions displaced by the lower octaves (advection) isn't limited to Perlin noise but can be applied to any lattice gradient noise, including Simplex Noise.

The code presented below is a straightforward and unoptimised port of Stefan Gustavsons original implementation in C. 2D and 3D versions of this noise are provided in a single shader,the type can be selected with the Dim parameter. By keyframing the Angle parameter you can create a swirling motion. The character of this motion may be influenced by altering the Octaves, lacunarity, H and advection parameters. The advection parameter basically determines how much small details in the flow will be swept along by the larger features.

Example node setup

The video of the swirling sphere was created with the following node setup:

The `angle` parameter was varied from 0 to 6.28 in 150 frames.

An additional trick to restrict the advection to some specific point is illustrated in another post.

The shader is an unwieldy large piece of code but nevertheless presented here in its entirety. Just click view plain to download it.

`/* Simplex flow noise in 2 and 3D * * Based on Stefan Gustavsons orignal implementation in srdnoise23.c * (http://webstaff.itn.liu.se/~stegu/simplexnoise/DSOnoises.html) * * OSL port Michel Anders (varkenvarken) 2013-02-04 * original comment is is left mostly in place, OSL specific comments * are preceded by MJA * * This code was placed in the public domain by its original author, * Stefan Gustavson. You may use it as you see fit, but * attribution is appreciated. **/int FASTFLOOR(float x) { int xi = (int)x; return x < xi ? xi-1 : xi;}shader flownoise( point Pos = P, float Scale = 1, int Octaves = 1, float H = 1, float lacunarity=2, float Angle=0, float advection=0, int Dim = 2, output float fac = 0){ int perm[512] = {151,160,137,91,90,15, 131,13,201,95,96,53,194,233,7,225,140,36,103,30,69,142,8,99,37,240,21,10,23, 190, 6,148,247,120,234,75,0,26,197,62,94,252,219,203,117,35,11,32,57,177,33, 88,237,149,56,87,174,20,125,136,171,168, 68,175,74,165,71,134,139,48,27,166, 77,146,158,231,83,111,229,122,60,211,133,230,220,105,92,41,55,46,245,40,244, 102,143,54, 65,25,63,161, 1,216,80,73,209,76,132,187,208, 89,18,169,200,196, 135,130,116,188,159,86,164,100,109,198,173,186, 3,64,52,217,226,250,124,123, 5,202,38,147,118,126,255,82,85,212,207,206,59,227,47,16,58,17,182,189,28,42, 223,183,170,213,119,248,152, 2,44,154,163, 70,221,153,101,155,167, 43,172,9, 129,22,39,253, 19,98,108,110,79,113,224,232,178,185, 112,104,218,246,97,228, 251,34,242,193,238,210,144,12,191,179,162,241, 81,51,145,235,249,14,239,107, 49,192,214, 31,181,199,106,157,184, 84,204,176,115,121,50,45,127, 4,150,254, 138,236,205,93,222,114,67,29,24,72,243,141,128,195,78,66,215,61,156,180, 151,160,137,91,90,15, 131,13,201,95,96,53,194,233,7,225,140,36,103,30,69,142,8,99,37,240,21,10,23, 190, 6,148,247,120,234,75,0,26,197,62,94,252,219,203,117,35,11,32,57,177,33, 88,237,149,56,87,174,20,125,136,171,168, 68,175,74,165,71,134,139,48,27,166, 77,146,158,231,83,111,229,122,60,211,133,230,220,105,92,41,55,46,245,40,244, 102,143,54, 65,25,63,161, 1,216,80,73,209,76,132,187,208, 89,18,169,200,196, 135,130,116,188,159,86,164,100,109,198,173,186, 3,64,52,217,226,250,124,123, 5,202,38,147,118,126,255,82,85,212,207,206,59,227,47,16,58,17,182,189,28,42, 223,183,170,213,119,248,152, 2,44,154,163, 70,221,153,101,155,167, 43,172,9, 129,22,39,253, 19,98,108,110,79,113,224,232,178,185, 112,104,218,246,97,228, 251,34,242,193,238,210,144,12,191,179,162,241, 81,51,145,235,249,14,239,107, 49,192,214, 31,181,199,106,157,184, 84,204,176,115,121,50,45,127, 4,150,254, 138,236,205,93,222,114,67,29,24,72,243,141,128,195,78,66,215,61,156,180}; // MJA precomputing this table instead of calculating it as done in the  // original code saves 30% running time. int permMod12[512] = { 7, 4, 5, 7, 6, 3, 11, 1, 9, 11, 0, 5, 2, 5, 7, 9, 8,  0, 7, 6, 9, 10, 8, 3, 1, 0, 9, 10, 11, 10, 6, 4, 7, 0, 6, 3, 0, 2, 5, 2, 10,  0, 3, 11, 9, 11, 11, 8, 9, 9, 9, 4, 9, 5, 8, 3, 6, 8, 5, 4, 3, 0, 8, 7, 2, 9,  11, 2, 7, 0, 3, 10, 5, 2, 2, 3, 11, 3, 1, 2, 0, 7, 1, 2, 4, 9, 8, 5, 7, 10,  5, 4, 4, 6, 11, 6, 5, 1, 3, 5, 1, 0, 8, 1, 5, 4, 0, 7, 4, 5, 6, 1, 8, 4, 3,  10, 8, 8, 3, 2, 8, 4, 1, 6, 5, 6, 3, 4, 4, 1, 10, 10, 4, 3, 5, 10, 2, 3, 10,  6, 3, 10, 1, 8, 3, 2, 11, 11, 11, 4, 10, 5, 2, 9, 4, 6, 7, 3, 2, 9, 11, 8, 8,  2, 8, 10, 7, 10, 5, 9, 5, 11, 11, 7, 4, 9, 9, 10, 3, 1, 7, 2, 0, 2, 7, 5, 8,  4, 10, 5, 4, 8, 2, 6, 1, 0, 11, 10, 2, 1, 10, 6, 0, 0, 11, 11, 6, 1, 9, 3, 1,  7, 9, 2, 11, 11, 1, 0, 10, 7, 1, 7, 10, 1, 4, 0, 0, 8, 7, 1, 2, 9, 7, 4, 6, 2,  6, 8, 1, 9, 6, 6, 7, 5, 0, 0, 3, 9, 8, 3, 6, 6, 11, 1, 0, 0, 7, 4, 5, 7, 6, 3,  11, 1, 9, 11, 0, 5, 2, 5, 7, 9, 8, 0, 7, 6, 9, 10, 8, 3, 1, 0, 9, 10, 11, 10,  6, 4, 7, 0, 6, 3, 0, 2, 5, 2, 10, 0, 3, 11, 9, 11, 11, 8, 9, 9, 9, 4, 9, 5, 8,  3, 6, 8, 5, 4, 3, 0, 8, 7, 2, 9, 11, 2, 7, 0, 3, 10, 5, 2, 2, 3, 11, 3, 1, 2,  0, 7, 1, 2, 4, 9, 8, 5, 7, 10, 5, 4, 4, 6, 11, 6, 5, 1, 3, 5, 1, 0, 8, 1, 5, 4,  0, 7, 4, 5, 6, 1, 8, 4, 3, 10, 8, 8, 3, 2, 8, 4, 1, 6, 5, 6, 3, 4, 4, 1, 10, 10,  4, 3, 5, 10, 2, 3, 10, 6, 3, 10, 1, 8, 3, 2, 11, 11, 11, 4, 10, 5, 2, 9, 4, 6, 7,  3, 2, 9, 11, 8, 8, 2, 8, 10, 7, 10, 5, 9, 5, 11, 11, 7, 4, 9, 9, 10, 3, 1, 7, 2,  0, 2, 7, 5, 8, 4, 10, 5, 4, 8, 2, 6, 1, 0, 11, 10, 2, 1, 10, 6, 0, 0, 11, 11, 6,  1, 9, 3, 1, 7, 9, 2, 11, 11, 1, 0, 10, 7, 1, 7, 10, 1, 4, 0, 0, 8, 7, 1, 2, 9, 7,  4, 6, 2, 6, 8, 1, 9, 6, 6, 7, 5, 0, 0, 3, 9, 8, 3, 6, 6, 11, 1, 0, 0};  /*  * Gradient tables. These could be programmed the Ken Perlin way with  * some clever bit-twiddling, but this is more clear, and not really slower.  */ vector grad2[8] = {   vector( -1.0, -1.0 , 0.0), vector( 1.0, 0.0 , 0.0) , vector( -1.0, 0.0 , 0.0) , vector( 1.0, 1.0 , 0.0) ,   vector( -1.0, 1.0 , 0.0) , vector( 0.0, -1.0 , 0.0) , vector( 0.0, 1.0 , 0.0) , vector( 1.0, -1.0 , 0.0) }; /*  * For 3D, we define two orthogonal vectors in the desired rotation plane.  * These vectors are based on the midpoints of the 12 edges of a cube,  * they all rotate in their own plane and are never coincident or collinear.  * A larger array of random vectors would also do the job, but these 12  * (including 4 repeats to make the array length a power of two) work better.  * They are not random, they are carefully chosen to represent a small  * isotropic set of directions for any rotation angle.  */ /* a = sqrt(2)/sqrt(3) = 0.816496580 */#define a 0.81649658 vector grad3u[16] = {   vector( 1.0, 0.0, 1.0 ), vector( 0.0, 1.0, 1.0 ), // 12 cube edges   vector( -1.0, 0.0, 1.0 ), vector( 0.0, -1.0, 1.0 ),   vector( 1.0, 0.0, -1.0 ), vector( 0.0, 1.0, -1.0 ),   vector( -1.0, 0.0, -1.0 ), vector( 0.0, -1.0, -1.0 ),   vector( a, a, a ), vector( -a, a, -a ),   vector( -a, -a, a ), vector( a, -a, -a ),   vector( -a, a, a ), vector( a, -a, a ),   vector( a, -a, -a ), vector( -a, a, -a ) }; vector grad3v[16] = {   vector( -a, a, a ), vector( -a, -a, a ),   vector( a, -a, a ), vector( a, a, a ),   vector( -a, -a, -a ), vector( a, -a, -a ),   vector( a, a, -a ), vector( -a, a, -a ),   vector( 1.0, -1.0, 0.0 ), vector( 1.0, 1.0, 0.0 ),   vector( -1.0, 1.0, 0.0 ), vector( -1.0, -1.0, 0.0 ),   vector( 1.0, 0.0, 1.0 ), vector( -1.0, 0.0, 1.0 ), // 4 repeats to make 16   vector( 0.0, 1.0, -1.0 ), vector( 0.0, -1.0, -1.0 ) };#undef a /*  * Helper functions to compute rotated gradients and  * gradients-dot-residualvectors in 2D and 3D.  */ void gradrot2( int hash, float sin_t, float cos_t, float gx, float gy ) {  int h = hash & 7;  float gx0 = grad2[h][0];  float gy0 = grad2[h][1];  gx = cos_t * gx0 - sin_t * gy0;  gy = sin_t * gx0 + cos_t * gy0;  return; } void gradrot3( int hash, float sin_t, float cos_t, float gx, float gy, float gz ) {  int h = hash & 15;  float gux = grad3u[h][0];  float guy = grad3u[h][1];  float guz = grad3u[h][2];  float gvx = grad3v[h][0];  float gvy = grad3v[h][1];  float gvz = grad3v[h][2];  gx = cos_t * gux + sin_t * gvx;  gy = cos_t * guy + sin_t * gvy;  gz = cos_t * guz + sin_t * gvz;  return; } float graddotp3( float gx, float gy, float gz, float x, float y, float z ) {  return gx * x + gy * y + gz * z; } float angle = Angle; // MJA copy input parameter so that we can write it float pwr = 1.0; float pwHL = pow(lacunarity,-H);  /* Skewing factors for 2D simplex grid:  * F2 = 0.5*(sqrt(3.0)-1.0)  * G2 = (3.0-Math.sqrt(3.0))/6.0  */#define F2 0.366025403#define G2 0.211324865 /** 2D simplex noise with derivatives.  * If the last two arguments are not null, the analytic derivative  * (the 2D gradient of the scalar noise field) is also calculated.  */ if( Dim == 2 ){  float x = Pos[0]*Scale;  float y = Pos[1]*Scale;  float dnoise_dx;  float dnoise_dy;  float sin_t, cos_t; /* Sine and cosine for the gradient rotation angle */  sin_t = sin( angle );  cos_t = cos( angle );  for(int p=0; p < Octaves; p++){   float n0, n1, n2; /* Noise contributions from the three simplex corners */   float gx0, gy0, gx1, gy1, gx2, gy2; /* Gradients at simplex corners */   /* Skew the input space to determine which simplex cell we're in */   float s = ( x + y ) * F2; /* Hairy factor for 2D */   float xs = x + s;   float ys = y + s;   int i = FASTFLOOR( xs );   int j = FASTFLOOR( ys );   float t = ( i + j ) * G2;   float X0 = i - t; /* Unskew the cell origin back to (x,y) space */   float Y0 = j - t;   float x0 = x - X0; /* The x,y distances from the cell origin */   float y0 = y - Y0;   /* For the 2D case, the simplex shape is an equilateral triangle.    * Determine which simplex we are in. */   int i1, j1; /* Offsets for second (middle) corner of simplex in (i,j) coords */   if( x0 > y0 ) { i1 = 1; j1 = 0; } /* lower triangle, XY order: (0,0)->(1,0)->(1,1) */   else { i1 = 0; j1 = 1; }      /* upper triangle, YX order: (0,0)->(0,1)->(1,1) */   /* A step of (1,0) in (i,j) means a step of (1-c,-c) in (x,y), and    * a step of (0,1) in (i,j) means a step of (-c,1-c) in (x,y), where    * c = (3-sqrt(3))/6   */   float x1 = x0 - i1 + G2; /* Offsets for middle corner in (x,y) unskewed coords */   float y1 = y0 - j1 + G2;   float x2 = x0 - 1.0 + 2.0 * G2; /* Offsets for last corner in (x,y) unskewed coords */   float y2 = y0 - 1.0 + 2.0 * G2;   /* Wrap the integer indices at 256, to avoid indexing perm[] out of bounds */   int ii = i & 255; // MJA was % 256 but OSL mod is not the same as C %   int jj = j & 255;   /* Calculate the contribution from the three corners */   float t0 = 0.5 - x0 * x0 - y0 * y0;   float t20, t40;   if( t0 < 0.0 ) t40 = t20 = t0 = n0 = gx0 = gy0 = 0.0; /* No influence */   else {     gradrot2( perm[ii + perm[jj]], sin_t, cos_t, gx0, gy0 );     t20 = t0 * t0;     t40 = t20 * t20;     n0 = t40 * ( gx0 * x0 + gy0 * y0 );   }   float t1 = 0.5 - x1 * x1 - y1 * y1;   float t21, t41;   if( t1 < 0.0 ) t21 = t41 = t1 = n1 = gx1 = gy1 = 0.0; /* No influence */   else {     gradrot2( perm[ii + i1 + perm[jj + j1]], sin_t, cos_t, gx1, gy1 );     t21 = t1 * t1;     t41 = t21 * t21;     n1 = t41 * ( gx1 * x1 + gy1 * y1 );   }   float t2 = 0.5 - x2 * x2 - y2 * y2;   float t22, t42;   if( t2 < 0.0 ) t42 = t22 = t2 = n2 = gx2 = gy2 = 0.0; /* No influence */   else {     gradrot2( perm[ii + 1 + perm[jj + 1]], sin_t, cos_t, gx2, gy2 );     t22 = t2 * t2;     t42 = t22 * t22;     n2 = t42 * ( gx2 * x2 + gy2 * y2 );   }   /* Add contributions from each corner to get the final noise value.    * The result is scaled to return values in the interval [-1,1]. */   float noise = 70.0 * ( n0 + n1 + n2 ); // MJA scale factor was 40   /* Compute derivative, if requested by supplying non-null pointers    * for the last two arguments */   if( advection != 0 ){   /*  A straight, unoptimised calculation would be like:    *    *dnoise_dx = -8.0 * t20 * t0 * x0 * ( gx0 * x0 + gy0 * y0 ) + t40 * gx0;    *    *dnoise_dy = -8.0 * t20 * t0 * y0 * ( gx0 * x0 + gy0 * y0 ) + t40 * gy0;    *    *dnoise_dx += -8.0 * t21 * t1 * x1 * ( gx1 * x1 + gy1 * y1 ) + t41 * gx1;    *    *dnoise_dy += -8.0 * t21 * t1 * y1 * ( gx1 * x1 + gy1 * y1 ) + t41 * gy1;    *    *dnoise_dx += -8.0 * t22 * t2 * x2 * ( gx2 * x2 + gy2 * y2 ) + t42 * gx2;    *    *dnoise_dy += -8.0 * t22 * t2 * y2 * ( gx2 * x2 + gy2 * y2 ) + t42 * gy2;    */    float temp0 = t20 * t0 * ( gx0* x0 + gy0 * y0 );    dnoise_dx = temp0 * x0;    dnoise_dy = temp0 * y0;    float temp1 = t21 * t1 * ( gx1 * x1 + gy1 * y1 );    dnoise_dx += temp1 * x1;    dnoise_dy += temp1 * y1;    float temp2 = t22 * t2 * ( gx2* x2 + gy2 * y2 );    dnoise_dx += temp2 * x2;    dnoise_dy += temp2 * y2;    dnoise_dx *= -8.0;    dnoise_dy *= -8.0;    dnoise_dx += t40 * gx0 + t41 * gx1 + t42 * gx2;    dnoise_dy += t40 * gy0 + t41 * gy1 + t42 * gy2;    dnoise_dx *= 70.0; /* Scale derivative to match the noise scaling */ // MJA scale factor was 40    dnoise_dy *= 70.0;        float advect_x = -advection * dnoise_dx;    float advect_y = -advection * dnoise_dy;    fac += noise * pwr;    pwr *= pwHL;    x *= lacunarity;    y *= lacunarity;    x += advect_x*pow(lacunarity,p);    y += advect_y*pow(lacunarity,p);    angle *= lacunarity;   }else{    fac += noise * pwr;    pwr *= pwHL;    x *= lacunarity;    y *= lacunarity;   }  } }else if(Dim==3){ /* Skewing factors for 3D simplex grid:  * F3 = 1/3  * G3 = 1/6 */#define F3 0.333333333#define G3 0.166666667  float x=Pos[0]*Scale;  float y=Pos[1]*Scale;  float z=Pos[2]*Scale;        float dnoise_dx;  float dnoise_dy;  float dnoise_dz;  float n0, n1, n2, n3; /* Noise contributions from the four simplex corners */  float noise;          /* Return value */  float gx0, gy0, gz0, gx1, gy1, gz1; /* Gradients at simplex corners */  float gx2, gy2, gz2, gx3, gy3, gz3;  float sin_t, cos_t; /* Sine and cosine for the gradient rotation angle */  sin_t = sin( angle );  cos_t = cos( angle );  for(int p=0; p < Octaves; p++){   /* Skew the input space to determine which simplex cell we're in */   float s = (x+y+z)*F3; /* Very nice and simple skew factor for 3D */   float xs = x+s;   float ys = y+s;   float zs = z+s;   int i = FASTFLOOR(xs);   int j = FASTFLOOR(ys);   int k = FASTFLOOR(zs);   float t = (float)(i+j+k)*G3;    float X0 = i-t; /* Unskew the cell origin back to (x,y,z) space */   float Y0 = j-t;   float Z0 = k-t;   float x0 = x-X0; /* The x,y,z distances from the cell origin */   float y0 = y-Y0;   float z0 = z-Z0;   /* For the 3D case, the simplex shape is a slightly irregular tetrahedron.    * Determine which simplex we are in. */   int i1, j1, k1; /* Offsets for second corner of simplex in (i,j,k) coords */   int i2, j2, k2; /* Offsets for third corner of simplex in (i,j,k) coords */   /* TODO: This code would benefit from a backport from the GLSL version! */   if(x0>=y0) {     if(y0>=z0)    { i1=1; j1=0; k1=0; i2=1; j2=1; k2=0; } /* X Y Z order */    else if(x0>=z0) { i1=1; j1=0; k1=0; i2=1; j2=0; k2=1; } /* X Z Y order */    else { i1=0; j1=0; k1=1; i2=1; j2=0; k2=1; } /* Z X Y order */     }   else { // x0 < y0     if(y0 < z0) { i1=0; j1=0; k1=1; i2=0; j2=1; k2=1; } /* Z Y X order */     else if(x0 < z0) { i1=0; j1=1; k1=0; i2=0; j2=1; k2=1; } /* Y Z X order */     else { i1=0; j1=1; k1=0; i2=1; j2=1; k2=0; } /* Y X Z order */   }   /* A step of (1,0,0) in (i,j,k) means a step of (1-c,-c,-c) in (x,y,z),    * a step of (0,1,0) in (i,j,k) means a step of (-c,1-c,-c) in (x,y,z), and    * a step of (0,0,1) in (i,j,k) means a step of (-c,-c,1-c) in (x,y,z), where    * c = 1/6.   */   float x1 = x0 - i1 + G3; /* Offsets for second corner in (x,y,z) coords */   float y1 = y0 - j1 + G3;   float z1 = z0 - k1 + G3;   float x2 = x0 - i2 + 2.0 * G3; /* Offsets for third corner in (x,y,z) coords */   float y2 = y0 - j2 + 2.0 * G3;   float z2 = z0 - k2 + 2.0 * G3;   float x3 = x0 - 1.0 + 3.0 * G3; /* Offsets for last corner in (x,y,z) coords */   float y3 = y0 - 1.0 + 3.0 * G3;   float z3 = z0 - 1.0 + 3.0 * G3;   /* Wrap the integer indices at 256, to avoid indexing perm[] out of bounds */   int ii = i & 255; // MJA was % 256 but OSL mod is not the same as C %   int jj = j & 255;   int kk = k & 255;   /* Calculate the contribution from the four corners */   float t0 = 0.6 - x0*x0 - y0*y0 - z0*z0;   float t20, t40;   if(t0 < 0.0) n0 = t0 = t20 = t40 = gx0 = gy0 = gz0 = 0.0;   else {     gradrot3( perm[ii + perm[jj + perm[kk]]], sin_t, cos_t, gx0, gy0, gz0 );     t20 = t0 * t0;     t40 = t20 * t20;     n0 = t40 * graddotp3( gx0, gy0, gz0, x0, y0, z0 );   }   float t1 = 0.6 - x1*x1 - y1*y1 - z1*z1;   float t21, t41;   if(t1 < 0.0) n1 = t1 = t21 = t41 = gx1 = gy1 = gz1 = 0.0;   else {     gradrot3( perm[ii + i1 + perm[jj + j1 + perm[kk + k1]]], sin_t, cos_t, gx1, gy1, gz1 );     t21 = t1 * t1;     t41 = t21 * t21;     n1 = t41 * graddotp3( gx1, gy1, gz1, x1, y1, z1 );   }   float t2 = 0.6 - x2*x2 - y2*y2 - z2*z2;   float t22, t42;   if(t2 < 0.0) n2 = t2 = t22 = t42 = gx2 = gy2 = gz2 = 0.0;   else {     gradrot3( perm[ii + i2 + perm[jj + j2 + perm[kk + k2]]], sin_t, cos_t, gx2, gy2, gz2 );     t22 = t2 * t2;     t42 = t22 * t22;     n2 = t42 * graddotp3( gx2, gy2, gz2, x2, y2, z2 );   }   float t3 = 0.6 - x3*x3 - y3*y3 - z3*z3;   float t23, t43;   if(t3 < 0.0) n3 = t3 = t23 = t43 = gx3 = gy3 = gz3 = 0.0;   else {     gradrot3( perm[ii + 1 + perm[jj + 1 + perm[kk + 1]]], sin_t, cos_t, gx3, gy3, gz3 );     t23 = t3 * t3;     t43 = t23 * t23;     n3 = t43 * graddotp3( gx3, gy3, gz3, x3, y3, z3 );   }   /*  Add contributions from each corner to get the final noise value.    * The result is scaled to return values in the range [-1,1] */   noise = 28.0 * (n0 + n1 + n2 + n3);   /* Compute derivative, if requested by supplying non-null pointers    * for the last three arguments */   if(advection != 0){   /*  A straight, unoptimised calculation would be like:    *     *dnoise_dx = -8.0f * t20 * t0 * x0 * graddotp3(gx0, gy0, gz0, x0, y0, z0) + t40 * gx0;    *    *dnoise_dy = -8.0f * t20 * t0 * y0 * graddotp3(gx0, gy0, gz0, x0, y0, z0) + t40 * gy0;    *    *dnoise_dz = -8.0f * t20 * t0 * z0 * graddotp3(gx0, gy0, gz0, x0, y0, z0) + t40 * gz0;    *    *dnoise_dx += -8.0f * t21 * t1 * x1 * graddotp3(gx1, gy1, gz1, x1, y1, z1) + t41 * gx1;    *    *dnoise_dy += -8.0f * t21 * t1 * y1 * graddotp3(gx1, gy1, gz1, x1, y1, z1) + t41 * gy1;    *    *dnoise_dz += -8.0f * t21 * t1 * z1 * graddotp3(gx1, gy1, gz1, x1, y1, z1) + t41 * gz1;    *    *dnoise_dx += -8.0f * t22 * t2 * x2 * graddotp3(gx2, gy2, gz2, x2, y2, z2) + t42 * gx2;    *    *dnoise_dy += -8.0f * t22 * t2 * y2 * graddotp3(gx2, gy2, gz2, x2, y2, z2) + t42 * gy2;    *    *dnoise_dz += -8.0f * t22 * t2 * z2 * graddotp3(gx2, gy2, gz2, x2, y2, z2) + t42 * gz2;    *    *dnoise_dx += -8.0f * t23 * t3 * x3 * graddotp3(gx3, gy3, gz3, x3, y3, z3) + t43 * gx3;    *    *dnoise_dy += -8.0f * t23 * t3 * y3 * graddotp3(gx3, gy3, gz3, x3, y3, z3) + t43 * gy3;    *    *dnoise_dz += -8.0f * t23 * t3 * z3 * graddotp3(gx3, gy3, gz3, x3, y3, z3) + t43 * gz3;    */    float temp0 = t20 * t0 * graddotp3( gx0, gy0, gz0, x0, y0, z0 );    dnoise_dx = temp0 * x0;    dnoise_dy = temp0 * y0;    dnoise_dz = temp0 * z0;    float temp1 = t21 * t1 * graddotp3( gx1, gy1, gz1, x1, y1, z1 );    dnoise_dx += temp1 * x1;    dnoise_dy += temp1 * y1;    dnoise_dz += temp1 * z1;    float temp2 = t22 * t2 * graddotp3( gx2, gy2, gz2, x2, y2, z2 );    dnoise_dx += temp2 * x2;    dnoise_dy += temp2 * y2;    dnoise_dz += temp2 * z2;    float temp3 = t23 * t3 * graddotp3( gx3, gy3, gz3, x3, y3, z3 );    dnoise_dx += temp3 * x3;    dnoise_dy += temp3 * y3;    dnoise_dz += temp3 * z3;    dnoise_dx *= -8.0;    dnoise_dy *= -8.0;    dnoise_dz *= -8.0;    /* This corrects a bug in the original implementation */    dnoise_dx += t40 * gx0 + t41 * gx1 + t42 * gx2 + t43 * gx3;    dnoise_dy += t40 * gy0 + t41 * gy1 + t42 * gy2 + t43 * gy3;    dnoise_dz += t40 * gz0 + t41 * gz1 + t42 * gz2 + t43 * gz3;    dnoise_dx *= 28.0; /* Scale derivative to match the noise scaling */    dnoise_dy *= 28.0;    dnoise_dz *= 28.0;    float advect_x = -advection * dnoise_dx;    float advect_y = -advection * dnoise_dy;    float advect_z = -advection * dnoise_dz;    fac += noise * pwr;    pwr *= pwHL;    x *= lacunarity;    y *= lacunarity;    x += advect_x*pow(lacunarity,p);    y += advect_y*pow(lacunarity,p);    y += advect_z*pow(lacunarity,p);    angle *= lacunarity;   }else{    fac += noise * pwr;    pwr *= pwHL;    x *= lacunarity;    y *= lacunarity;    z *= lacunarity;   }  } }}`