package squidpony.squidmath;
/**
* This is Ken Perlin's third revision of his noise function. It is sometimes
* referred to as "Simplex Noise". Results are bound by (-1, 1) inclusive.
*
*
* It is significantly faster than his earlier versions. This particular version
* was originally from Stefan Gustavson. This is much preferred to the earlier
* versions of Perlin Noise due to the reasons noted in the articles:
* <ul>
* <li>http://www.itn.liu.se/~stegu/simplexnoise/simplexnoise.pdf</li>
* <li>http://webstaff.itn.liu.se/~stegu/TNM022-2005/perlinnoiselinks/ch02.pdf</li>
* </ul>
* But, Gustavson's paper is not without its own issues, particularly for 2D noise.
* More detail is noted here,
* http://stackoverflow.com/questions/18885440/why-does-simplex-noise-seem-to-have-more-artifacts-than-classic-perlin-noise#21568753
* and some changes have been made to 2D noise generation to reduce angular artifacts.
* Specifically for the 2D gradient table, code based on Gustavson's paper used 12
* points, with some duplicates, and not all on the unit circle. In this version,
* points are used on the unit circle starting at (1,0) and moving along the circle
* in increments of 1.61803398875 radians, that is, the golden ratio phi, getting the
* sin and cosine of 15 points after the starting point and storing them as constants.
* This definitely doesn't have a noticeable 45 degree angle artifact, though it does
* have, to a lesser extent, some other minor artifacts.
* <br>
* You can also consider {@link WhirlingNoise} as an alternative, which can be faster
* and also reduces the likelihood of angular artifacts. WhirlingNoise does not scale its
* input (it doesn't need to), so it won't produce the same results as PerlinNoise for the
* same inputs, but it will produce similar shape, density, and aesthetic quality of noise.
* @see WhirlingNoise A subclass that has a faster implementation and some different qualities.
*/
public class PerlinNoise {
protected static final double phi = 1.61803398875,
epi = 1.0 / Math.E / Math.PI, unit1_4 = 0.70710678118, unit1_8 = 0.38268343236, unit3_8 = 0.92387953251;
protected static final double[][] grad2 = {
{1, 0}, {-1, 0}, {0, 1}, {0, -1},/*
{1, 0}, {-1, 0}, {0, 1}, {0, -1},
{1, 0}, {-1, 0}, {0, 1}, {0, -1},
{1, 0}, {-1, 0}, {0, 1}, {0, -1},*/
{unit3_8, unit1_8}, {unit3_8, -unit1_8}, {-unit3_8, unit1_8}, {-unit3_8, -unit1_8},
{unit1_4, unit1_4}, {unit1_4, -unit1_4}, {-unit1_4, unit1_4}, {-unit1_4, -unit1_4},
{unit1_8, unit3_8}, {unit1_8, -unit3_8}, {-unit1_8, unit3_8}, {-unit1_8, -unit3_8}};
protected static final double[][] phiGrad2 = {
{1, 0}, {Math.cos(phi), Math.sin(phi)},
{Math.cos(phi*2), Math.sin(phi*2)}, {Math.cos(phi*3), Math.sin(phi*3)},
{Math.cos(phi*4), Math.sin(phi*4)}, {Math.cos(phi*5), Math.sin(phi*5)},
{Math.cos(phi*6), Math.sin(phi*6)}, {Math.cos(phi*7), Math.sin(phi*7)},
{Math.cos(phi*8), Math.sin(phi*8)}, {Math.cos(phi*9), Math.sin(phi*9)},
{Math.cos(phi*10), Math.sin(phi*10)}, {Math.cos(phi*11), Math.sin(phi*11)},
{Math.cos(phi*13), Math.sin(phi*12)}, {Math.cos(phi*13), Math.sin(phi*13)},
{Math.cos(phi*14), Math.sin(phi*14)}, {Math.cos(phi*15), Math.sin(phi*15)},
};
protected static final int[][] grad3 = {{1, 1, 0}, {-1, 1, 0}, {1, -1, 0},
{-1, -1, 0}, {1, 0, 1}, {-1, 0, 1},
{1, 0, -1}, {-1, 0, -1}, {0, 1, 1},
{0, -1, 1}, {0, 1, -1}, {0, -1, -1}};
protected static final int[][] grad4 = {{0, 1, 1, 1}, {0, 1, 1, -1},
{0, 1, -1, 1}, {0, 1, -1, -1},
{0, -1, 1, 1}, {0, -1, 1, -1},
{0, -1, -1, 1}, {0, -1, -1, -1},
{1, 0, 1, 1}, {1, 0, 1, -1},
{1, 0, -1, 1}, {1, 0, -1, -1},
{-1, 0, 1, 1}, {-1, 0, 1, -1},
{-1, 0, -1, 1}, {-1, 0, -1, -1},
{1, 1, 0, 1}, {1, 1, 0, -1},
{1, -1, 0, 1}, {1, -1, 0, -1},
{-1, 1, 0, 1}, {-1, 1, 0, -1},
{-1, -1, 0, 1}, {-1, -1, 0, -1},
{1, 1, 1, 0}, {1, 1, -1, 0},
{1, -1, 1, 0}, {1, -1, -1, 0},
{-1, 1, 1, 0}, {-1, 1, -1, 0},
{-1, -1, 1, 0}, {-1, -1, -1, 0}};
private static final int p[] = {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};
protected static final double F2 = 0.5 * (Math.sqrt(3.0) - 1.0);
protected static final double G2 = (3.0 - Math.sqrt(3.0)) / 6.0;
protected static final double F3 = 1.0 / 3.0;
protected static final double G3 = 1.0 / 6.0;
protected static final double F4 = (Math.sqrt(5.0) - 1.0) / 4.0;
protected static final double G4 = (5.0 - Math.sqrt(5.0)) / 20.0;
// To remove the need for index wrapping, double the permutation table
// length
protected static final int perm[] = new int[512];
static {
for (int i = 0; i < 512; i++) {
perm[i] = p[i & 255];
}
}
protected PerlinNoise()
{
}
// A lookup table to traverse the simplex around a given point in 4D.
// Details can be found where this table is used, in the 4D noise method.
protected static final int simplex[][]
= {{0, 1, 2, 3}, {0, 1, 3, 2}, {0, 0, 0, 0}, {0, 2, 3, 1},
{0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}, {1, 2, 3, 0},
{0, 2, 1, 3}, {0, 0, 0, 0}, {0, 3, 1, 2}, {0, 3, 2, 1},
{0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}, {1, 3, 2, 0},
{0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0},
{0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0},
{1, 2, 0, 3}, {0, 0, 0, 0}, {1, 3, 0, 2}, {0, 0, 0, 0},
{0, 0, 0, 0}, {0, 0, 0, 0}, {2, 3, 0, 1}, {2, 3, 1, 0},
{1, 0, 2, 3}, {1, 0, 3, 2}, {0, 0, 0, 0}, {0, 0, 0, 0},
{0, 0, 0, 0}, {2, 0, 3, 1}, {0, 0, 0, 0}, {2, 1, 3, 0},
{0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0},
{0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0},
{2, 0, 1, 3}, {0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0},
{3, 0, 1, 2}, {3, 0, 2, 1}, {0, 0, 0, 0}, {3, 1, 2, 0},
{2, 1, 0, 3}, {0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0},
{3, 1, 0, 2}, {0, 0, 0, 0}, {3, 2, 0, 1}, {3, 2, 1, 0}};
protected static double dot(double g[], double x, double y) {
return g[0] * x + g[1] * y;
}
protected static double dot(int g[], double x, double y, double z) {
return g[0] * x + g[1] * y + g[2] * z;
}
protected static double dot(int g[], double x, double y, double z, double w) {
return g[0] * x + g[1] * y + g[2] * z + g[3] * w;
}
/**
* 2D simplex noise.
* This doesn't use its parameters verbatim; xin and yin are both effectively divided by
* ({@link Math#E} * {@link Math#PI}), because without a step like that, any integer parameters would return 0 and
* only doubles with a decimal component would produce actual noise. This step allows integers to be passed in a
* arguments, and changes the cycle at which 0 is repeated to multiples of (E*PI).
*
* @param xin x input; works well if between 0.0 and 1.0, but anything is accepted
* @param yin y input; works well if between 0.0 and 1.0, but anything is accepted
* @return noise from -1.0 to 1.0, inclusive
*/
public static double noise(double xin, double yin) {
xin *= epi;
yin *= epi;
double noise0, noise1, noise2; // from the three corners
// Skew the input space to determine which simplex cell we're in
double skew = (xin + yin) * F2; // Hairy factor for 2D
int i = (int) Math.floor(xin + skew);
int j = (int) Math.floor(yin + skew);
double t = (i + j) * G2;
double X0 = i - t; // Unskew the cell origin back to (x,y) space
double Y0 = j - t;
double x0 = xin - X0; // The x,y distances from the cell origin
double y0 = yin - 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
double x1 = x0 - i1 + G2; // Offsets for middle corner in (x,y)
// unskewed coords
double y1 = y0 - j1 + G2;
double x2 = x0 - 1.0 + 2.0 * G2; // Offsets for last corner in (x,y)
// unskewed coords
double y2 = y0 - 1.0 + 2.0 * G2;
// Work out the hashed gradient indices of the three simplex corners
int ii = i & 255;
int jj = j & 255;
int gi0 = perm[ii + perm[jj]] & 15;
int gi1 = perm[ii + i1 + perm[jj + j1]] & 15;
int gi2 = perm[ii + 1 + perm[jj + 1]] & 15;
// Calculate the contribution from the three corners
double t0 = 0.5 - x0 * x0 - y0 * y0;
if (t0 < 0) {
noise0 = 0.0;
} else {
t0 *= t0;
noise0 = t0 * t0 * dot(phiGrad2[gi0], x0, y0); // (x,y) of grad3 used
// for 2D gradient
}
double t1 = 0.5 - x1 * x1 - y1 * y1;
if (t1 < 0) {
noise1 = 0.0;
} else {
t1 *= t1;
noise1 = t1 * t1 * dot(phiGrad2[gi1], x1, y1);
}
double t2 = 0.5 - x2 * x2 - y2 * y2;
if (t2 < 0) {
noise2 = 0.0;
} else {
t2 *= t2;
noise2 = t2 * t2 * dot(phiGrad2[gi2], x2, y2);
}
// Add contributions from each corner to get the final noise value.
// The result is scaled to return values in the interval [-1,1].
return 70.0 * (noise0 + noise1 + noise2);
}
/**
* 3D simplex noise.
*
* @param xin X input
* @param yin Y input
* @param zin Z input
* @return noise from -1.0 to 1.0, inclusive
*/
public static double noise(double xin, double yin, double zin) {
xin *= epi;
yin *= epi;
zin *= epi;
double n0, n1, n2, n3; // Noise contributions from the four corners
// Skew the input space to determine which simplex cell we're in
double s = (xin + yin + zin) * F3; // Very nice and simple skew
// factor for 3D
int i = (int) Math.floor(xin + s);
int j = (int) Math.floor(yin + s);
int k = (int) Math.floor(zin + s);
double t = (i + j + k) * G3;
double X0 = i - t; // Unskew the cell origin back to (x,y,z) space
double Y0 = j - t;
double Z0 = k - t;
double x0 = xin - X0; // The x,y,z distances from the cell origin
double y0 = yin - Y0;
double z0 = zin - 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
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.
double x1 = x0 - i1 + G3; // Offsets for second corner in (x,y,z)
// coords
double y1 = y0 - j1 + G3;
double z1 = z0 - k1 + G3;
double x2 = x0 - i2 + 2.0 * G3; // Offsets for third corner in
// (x,y,z) coords
double y2 = y0 - j2 + 2.0 * G3;
double z2 = z0 - k2 + 2.0 * G3;
double x3 = x0 - 1.0 + 3.0 * G3; // Offsets for last corner in
// (x,y,z) coords
double y3 = y0 - 1.0 + 3.0 * G3;
double z3 = z0 - 1.0 + 3.0 * G3;
// Work out the hashed gradient indices of the four simplex corners
int ii = i & 255;
int jj = j & 255;
int kk = k & 255;
int gi0 = perm[ii + perm[jj + perm[kk]]] % 12;
int gi1 = perm[ii + i1 + perm[jj + j1 + perm[kk + k1]]] % 12;
int gi2 = perm[ii + i2 + perm[jj + j2 + perm[kk + k2]]] % 12;
int gi3 = perm[ii + 1 + perm[jj + 1 + perm[kk + 1]]] % 12;
// Calculate the contribution from the four corners
double t0 = 0.6 - x0 * x0 - y0 * y0 - z0 * z0;
if (t0 < 0) {
n0 = 0.0;
} else {
t0 *= t0;
n0 = t0 * t0 * dot(grad3[gi0], x0, y0, z0);
}
double t1 = 0.6 - x1 * x1 - y1 * y1 - z1 * z1;
if (t1 < 0) {
n1 = 0.0;
} else {
t1 *= t1;
n1 = t1 * t1 * dot(grad3[gi1], x1, y1, z1);
}
double t2 = 0.6 - x2 * x2 - y2 * y2 - z2 * z2;
if (t2 < 0) {
n2 = 0.0;
} else {
t2 *= t2;
n2 = t2 * t2 * dot(grad3[gi2], x2, y2, z2);
}
double t3 = 0.6 - x3 * x3 - y3 * y3 - z3 * z3;
if (t3 < 0) {
n3 = 0.0;
} else {
t3 *= t3;
n3 = t3 * t3 * dot(grad3[gi3], x3, y3, z3);
}
// Add contributions from each corner to get the final noise value.
// The result is scaled to stay just inside [-1,1]
return 32.0 * (n0 + n1 + n2 + n3);
}
/**
* 4D simplex noise.
*
* @param x X position
* @param y Y position
* @param z Z position
* @param w Fourth-dimensional position. It is I, the Fourth-Dimensional Ziltoid the Omniscient!
* @return noise from -1.0 to 1.0, inclusive
*/
public static double noise(double x, double y, double z, double w) {
x *= epi;
y *= epi;
z *= epi;
w *= epi;
// The skewing and unskewing factors are hairy again for the 4D case
double n0, n1, n2, n3, n4; // Noise contributions from the five
// corners
// Skew the (x,y,z,w) space to determine which cell of 24 simplices
// we're in
double s = (x + y + z + w) * F4; // Factor for 4D skewing
int i = (int) Math.floor(x + s);
int j = (int) Math.floor(y + s);
int k = (int) Math.floor(z + s);
int l = (int) Math.floor(w + s);
double t = (i + j + k + l) * G4; // Factor for 4D unskewing
double X0 = i - t; // Unskew the cell origin back to (x,y,z,w) space
double Y0 = j - t;
double Z0 = k - t;
double W0 = l - t;
double x0 = x - X0; // The x,y,z,w distances from the cell origin
double y0 = y - Y0;
double z0 = z - Z0;
double w0 = w - W0;
// For the 4D case, the simplex is a 4D shape I won't even try to
// describe.
// To find out which of the 24 possible simplices we're in, we need
// to
// determine the magnitude ordering of x0, y0, z0 and w0.
// The method below is a good way of finding the ordering of x,y,z,w
// and
// then find the correct traversal order for the simplex we’re in.
// First, six pair-wise comparisons are performed between each
// possible pair
// of the four coordinates, and the results are used to add up binary
// bits
// for an integer index.
int c =
(x0 > y0 ? 32 : 0)
+ (x0 > z0 ? 16 : 0)
+ (y0 > z0 ? 8 : 0 )
+ (x0 > w0 ? 4 : 0 )
+ (y0 > w0 ? 2 : 0 )
+ (z0 > w0 ? 1 : 0 );
int i1, j1, k1, l1; // The integer offsets for the second simplex
// corner
int i2, j2, k2, l2; // The integer offsets for the third simplex
// corner
int i3, j3, k3, l3; // The integer offsets for the fourth simplex
// corner
// simplex[c] is a 4-vector with the numbers 0, 1, 2 and 3 in some
// order.
// Many values of c will never occur, since e.g. x>y>z>w makes x<z,
// y<w and x<w
// impossible. Only the 24 indices which have non-zero entries make
// any sense.
// We use a thresholding to set the coordinates in turn from the
// largest magnitude.
// The number 3 in the "simplex" array is at the position of the
// largest coordinate.
i1 = simplex[c][0] >= 3 ? 1 : 0;
j1 = simplex[c][1] >= 3 ? 1 : 0;
k1 = simplex[c][2] >= 3 ? 1 : 0;
l1 = simplex[c][3] >= 3 ? 1 : 0;
// The number 2 in the "simplex" array is at the second largest
// coordinate.
i2 = simplex[c][0] >= 2 ? 1 : 0;
j2 = simplex[c][1] >= 2 ? 1 : 0;
k2 = simplex[c][2] >= 2 ? 1 : 0;
l2 = simplex[c][3] >= 2 ? 1 : 0;
// The number 1 in the "simplex" array is at the second smallest
// coordinate.
i3 = simplex[c][0] >= 1 ? 1 : 0;
j3 = simplex[c][1] >= 1 ? 1 : 0;
k3 = simplex[c][2] >= 1 ? 1 : 0;
l3 = simplex[c][3] >= 1 ? 1 : 0;
// The fifth corner has all coordinate offsets = 1, so no need to
// look that up.
double x1 = x0 - i1 + G4; // Offsets for second corner in (x,y,z,w)
// coords
double y1 = y0 - j1 + G4;
double z1 = z0 - k1 + G4;
double w1 = w0 - l1 + G4;
double x2 = x0 - i2 + 2.0 * G4; // Offsets for third corner in
// (x,y,z,w) coords
double y2 = y0 - j2 + 2.0 * G4;
double z2 = z0 - k2 + 2.0 * G4;
double w2 = w0 - l2 + 2.0 * G4;
double x3 = x0 - i3 + 3.0 * G4; // Offsets for fourth corner in
// (x,y,z,w) coords
double y3 = y0 - j3 + 3.0 * G4;
double z3 = z0 - k3 + 3.0 * G4;
double w3 = w0 - l3 + 3.0 * G4;
double x4 = x0 - 1.0 + 4.0 * G4; // Offsets for last corner in
// (x,y,z,w) coords
double y4 = y0 - 1.0 + 4.0 * G4;
double z4 = z0 - 1.0 + 4.0 * G4;
double w4 = w0 - 1.0 + 4.0 * G4;
// Work out the hashed gradient indices of the five simplex corners
int ii = i & 255;
int jj = j & 255;
int kk = k & 255;
int ll = l & 255;
int gi0 = perm[ii + perm[jj + perm[kk + perm[ll]]]] & 31;
int gi1
= perm[ii + i1 + perm[jj + j1 + perm[kk + k1 + perm[ll + l1]]]] & 31;
int gi2
= perm[ii + i2 + perm[jj + j2 + perm[kk + k2 + perm[ll + l2]]]] & 31;
int gi3
= perm[ii + i3 + perm[jj + j3 + perm[kk + k3 + perm[ll + l3]]]] & 31;
int gi4
= perm[ii + 1 + perm[jj + 1 + perm[kk + 1 + perm[ll + 1]]]] & 31;
// Calculate the contribution from the five corners
double t0 = 0.6 - x0 * x0 - y0 * y0 - z0 * z0 - w0 * w0;
if (t0 < 0) {
n0 = 0.0;
} else {
t0 *= t0;
n0 = t0 * t0 * dot(grad4[gi0], x0, y0, z0, w0);
}
double t1 = 0.6 - x1 * x1 - y1 * y1 - z1 * z1 - w1 * w1;
if (t1 < 0) {
n1 = 0.0;
} else {
t1 *= t1;
n1 = t1 * t1 * dot(grad4[gi1], x1, y1, z1, w1);
}
double t2 = 0.6 - x2 * x2 - y2 * y2 - z2 * z2 - w2 * w2;
if (t2 < 0) {
n2 = 0.0;
} else {
t2 *= t2;
n2 = t2 * t2 * dot(grad4[gi2], x2, y2, z2, w2);
}
double t3 = 0.6 - x3 * x3 - y3 * y3 - z3 * z3 - w3 * w3;
if (t3 < 0) {
n3 = 0.0;
} else {
t3 *= t3;
n3 = t3 * t3 * dot(grad4[gi3], x3, y3, z3, w3);
}
double t4 = 0.6 - x4 * x4 - y4 * y4 - z4 * z4 - w4 * w4;
if (t4 < 0) {
n4 = 0.0;
} else {
t4 *= t4;
n4 = t4 * t4 * dot(grad4[gi4], x4, y4, z4, w4);
}
// Sum up and scale the result to cover the range [-1,1]
return 27.0 * (n0 + n1 + n2 + n3 + n4);
}
}