
// 
// TODO:
// - implement sin, asin, acos, atan, pow, log2, floor, ceil,
// - implement texture1D, texture2D, texture3D, textureCube,
// - implement shadow1D, shadow2D,
// - implement noise1, noise2, noise3, noise4,
// 

// 
// From Shader Spec, ver. 1.10, rev. 59
// 
// The following built-in constants are provided to vertex and fragment shaders.
// 

// 
// Implementation dependent constants. The example values below
// are the minimum values allowed for these maximums.
// 

const int gl_MaxLights = 8;                             // GL 1.0
const int gl_MaxClipPlanes = 6;                         // GL 1.0
const int gl_MaxTextureUnits = 2;                       // GL 1.3
const int gl_MaxTextureCoords = 2;                      // ARB_fragment_program
const int gl_MaxVertexAttribs = 16;                     // ARB_vertex_shader
const int gl_MaxVertexUniformComponents = 512;          // ARB_vertex_shader
const int gl_MaxVaryingFloats = 32;                     // ARB_vertex_shader
const int gl_MaxVertexTextureImageUnits = 0;            // ARB_vertex_shader
const int gl_MaxCombinedTextureImageUnits = 2;          // ARB_vertex_shader
const int gl_MaxTextureImageUnits = 2;                  // ARB_fragment_shader
const int gl_MaxFragmentUniformComponents = 64;         // ARB_fragment_shader
const int gl_MaxDrawBuffers = 1;                        // proposed ARB_draw_buffers

// 
// As an aid to accessing OpenGL processing state, the following uniform variables are built into
// the OpenGL Shading Language. All page numbers and notations are references to the 1.4
// specification.
// 

// 
// Matrix state. p. 31, 32, 37, 39, 40.
// 

uniform mat4 gl_ModelViewMatrix;
uniform mat4 gl_ProjectionMatrix;
uniform mat4 gl_ModelViewProjectionMatrix;
uniform mat4 gl_TextureMatrix[gl_MaxTextureCoords];

//
// Derived matrix state that provides inverse and transposed versions
// of the matrices above. Poorly conditioned matrices may result
// in unpredictable values in their inverse forms.
//
uniform mat3 gl_NormalMatrix; // transpose of the inverse of the
                              // upper leftmost 3x3 of gl_ModelViewMatrix

uniform mat4 gl_ModelViewMatrixInverse;
uniform mat4 gl_ProjectionMatrixInverse;
uniform mat4 gl_ModelViewProjectionMatrixInverse;
uniform mat4 gl_TextureMatrixInverse[gl_MaxTextureCoords];

uniform mat4 gl_ModelViewMatrixTranspose;
uniform mat4 gl_ProjectionMatrixTranspose;
uniform mat4 gl_ModelViewProjectionMatrixTranspose;
uniform mat4 gl_TextureMatrixTranspose[gl_MaxTextureCoords];

uniform mat4 gl_ModelViewMatrixInverseTranspose;
uniform mat4 gl_ProjectionMatrixInverseTranspose;
uniform mat4 gl_ModelViewProjectionMatrixInverseTranspose;
uniform mat4 gl_TextureMatrixInverseTranspose[gl_MaxTextureCoords];

// 
// Normal scaling p. 39.
// 

uniform float gl_NormalScale;

// 
// Depth range in window coordinates, p. 33
// 

struct gl_DepthRangeParameters {
    float near;                                         // n
    float far;                                          // f
    float diff;                                         // f - n
};

uniform gl_DepthRangeParameters gl_DepthRange;

// 
// Clip planes p. 42.
// 

uniform vec4 gl_ClipPlane[gl_MaxClipPlanes];

// 
// Point Size, p. 66, 67.
// 

struct gl_PointParameters {
    float size;
    float sizeMin;
    float sizeMax;
    float fadeThresholdSize;
    float distanceConstantAttenuation;
    float distanceLinearAttenuation;
    float distanceQuadraticAttenuation;
};

uniform gl_PointParameters gl_Point;

// 
// Material State p. 50, 55.
// 

struct gl_MaterialParameters {
    vec4 emission;                                      // Ecm
    vec4 ambient;                                       // Acm
    vec4 diffuse;                                       // Dcm
    vec4 specular;                                      // Scm
    float shininess;                                    // Srm
};

uniform gl_MaterialParameters gl_FrontMaterial;
uniform gl_MaterialParameters gl_BackMaterial;

// 
// Light State p 50, 53, 55.
// 

struct gl_LightSourceParameters {
    vec4 ambient;                                       // Acli
    vec4 diffuse;                                       // Dcli
    vec4 specular;                                      // Scli
    vec4 position;                                      // Ppli
    vec4 halfVector;                                    // Derived: Hi
    vec3 spotDirection;                                 // Sdli
    float spotExponent;                                 // Srli
    float spotCutoff;                                   // Crli
                                                        // (range: [0.0,90.0], 180.0)
    float spotCosCutoff;                                // Derived: cos(Crli)
                                                        // (range: [1.0,0.0],-1.0)
    float constantAttenuation;                          // K0
    float linearAttenuation;                            // K1
    float quadraticAttenuation;                         // K2
};

uniform gl_LightSourceParameters gl_LightSource[gl_MaxLights];

struct gl_LightModelParameters {
    vec4 ambient;                                       // Acs
};

uniform gl_LightModelParameters gl_LightModel;

// 
// Derived state from products of light and material.
// 

struct gl_LightModelProducts {
    vec4 sceneColor;                                    // Derived. Ecm + Acm * Acs
};

uniform gl_LightModelProducts gl_FrontLightModelProduct;
uniform gl_LightModelProducts gl_BackLightModelProduct;

struct gl_LightProducts {
    vec4 ambient;                                       // Acm * Acli
    vec4 diffuse;                                       // Dcm * Dcli
    vec4 specular;                                      // Scm * Scli
};

uniform gl_LightProducts gl_FrontLightProduct[gl_MaxLights];
uniform gl_LightProducts gl_BackLightProduct[gl_MaxLights];

// 
// Texture Environment and Generation, p. 152, p. 40-42.
// 

uniform vec4 gl_TextureEnvColor[gl_MaxTextureImageUnits];
uniform vec4 gl_EyePlaneS[gl_MaxTextureCoords];
uniform vec4 gl_EyePlaneT[gl_MaxTextureCoords];
uniform vec4 gl_EyePlaneR[gl_MaxTextureCoords];
uniform vec4 gl_EyePlaneQ[gl_MaxTextureCoords];
uniform vec4 gl_ObjectPlaneS[gl_MaxTextureCoords];
uniform vec4 gl_ObjectPlaneT[gl_MaxTextureCoords];
uniform vec4 gl_ObjectPlaneR[gl_MaxTextureCoords];
uniform vec4 gl_ObjectPlaneQ[gl_MaxTextureCoords];

// 
// Fog p. 161
// 

struct gl_FogParameters {
    vec4 color;
    float density;
    float start;
    float end;
    float scale;    // Derived:    1.0 / (end - start)
};

uniform gl_FogParameters gl_Fog;

// 
// The OpenGL Shading Language defines an assortment of built-in convenience functions for scalar
// and vector operations. Many of these built-in functions can be used in more than one type
// of shader, but some are intended to provide a direct mapping to hardware and so are available
// only for a specific type of shader.
// 
// The built-in functions basically fall into three categories:
// 
// * They expose some necessary hardware functionality in a convenient way such as accessing
//   a texture map. There is no way in the language for these functions to be emulated by a shader.
// 
// * They represent a trivial operation (clamp, mix, etc.) that is very simple for the user
//   to write, but they are very common and may have direct hardware support. It is a very hard
//   problem for the compiler to map expressions to complex assembler instructions.
// 
// * They represent an operation graphics hardware is likely to accelerate at some point. The
//   trigonometry functions fall into this category.
// 
// Many of the functions are similar to the same named ones in common C libraries, but they support
// vector input as well as the more traditional scalar input.
// 
// Applications should be encouraged to use the built-in functions rather than do the equivalent
// computations in their own shader code since the built-in functions are assumed to be optimal
// (e.g., perhaps supported directly in hardware).
// 
// User code can replace built-in functions with their own if they choose, by simply re-declaring
// and defining the same name and argument list.
// 

// 
// 8.1 Angle and Trigonometry Functions
// 
// Function parameters specified as angle are assumed to be in units of radians. In no case will
// any of these functions result in a divide by zero error. If the divisor of a ratio is 0, then
// results will be undefined.
// 
// These all operate component-wise. The description is per component.
// 

// 
// Converts degrees to radians and returns the result, i.e., result = PI*deg/180.
// 

float radians (float deg) {
    return 3.141593 * deg / 180.0;
}
vec2 radians (vec2 deg) {
    return vec2 (radians (deg.x), radians (deg.y));
}
vec3 radians (vec3 deg) {
    return vec3 (radians (deg.x), radians (deg.y), radians (deg.z));
}
vec4 radians (vec4 deg) {
    return vec4 (radians (deg.x), radians (deg.y), radians (deg.z), radians (deg.w));
}

// 
// Converts radians to degrees and returns the result, i.e., result = 180*rad/PI.
// 

float degrees (float rad) {
    return 180.0 * rad / 3.141593;
}
vec2 degrees (vec2 rad) {
    return vec2 (degrees (rad.x), degrees (rad.y));
}
vec3 degrees (vec3 rad) {
    return vec3 (degrees (rad.x), degrees (rad.y), degrees (rad.z));
}
vec4 degrees (vec4 rad) {
    return vec4 (degrees (rad.x), degrees (rad.y), degrees (rad.z), degrees (rad.w));
}

// 
// The standard trigonometric sine function.
// 
// XXX
float sin (float angle) {
    return 0.0;
}
vec2 sin (vec2 angle) {
    return vec2 (sin (angle.x), sin (angle.y));
}
vec3 sin (vec3 angle) {
    return vec3 (sin (angle.x), sin (angle.y), sin (angle.z));
}
vec4 sin (vec4 angle) {
    return vec4 (sin (angle.x), sin (angle.y), sin (angle.z), sin (angle.w));
}

// 
// The standard trigonometric cosine function.
// 

float cos (float angle) {
    return sin (angle + 1.5708);
}
vec2 cos (vec2 angle) {
    return vec2 (cos (angle.x), cos (angle.y));
}
vec3 cos (vec3 angle) {
    return vec3 (cos (angle.x), cos (angle.y), cos (angle.z));
}
vec4 cos (vec4 angle) {
    return vec4 (cos (angle.x), cos (angle.y), cos (angle.z), cos (angle.w));
}

// 
// The standard trigonometric tangent.
// 

float tan (float angle) {
    return sin (angle) / cos (angle);
}
vec2 tan (vec2 angle) {
    return vec2 (tan (angle.x), tan (angle.y));
}
vec3 tan (vec3 angle) {
    return vec3 (tan (angle.x), tan (angle.y), tan (angle.z));
}
vec4 tan (vec4 angle) {
    return vec4 (tan (angle.x), tan (angle.y), tan (angle.z), tan (angle.w));
}

// 
// Arc sine. Returns an angle whose sine is x. The range of values returned by this function is
// [–PI/2, PI/2]. Results are undefined if |x| > 1.
// 
// XXX
float asin (float x) {
    return 0.0;
}
vec2 asin (vec2 x) {
    return vec2 (asin (x.x), asin (x.y));
}
vec3 asin (vec3 x) {
    return vec3 (asin (x.x), asin (x.y), asin (x.z));
}
vec4 asin (vec4 x) {
    return vec4 (asin (x.x), asin (x.y), asin (x.z), asin (x.w));
}

// 
// Arc cosine. Returns an angle whose cosine is x. The range of values returned by this function is
// [0, PI]. Results are undefined if |x| > 1.
// 
// XXX
float acos (float x) {
    return 0.0;
}
vec2 acos (vec2 x) {
    return vec2 (acos (x.x), acos (x.y));
}
vec3 acos (vec3 x) {
    return vec3 (acos (x.x), acos (x.y), acos (x.z));
}
vec4 acos (vec4 x) {
    return vec4 (acos (x.x), acos (x.y), acos (x.z), acos (x.w));
}

// 
// Arc tangent. Returns an angle whose tangent is y/x. The signs of x and y are used to determine
// what quadrant the angle is in. The range of values returned by this function is [–PI, PI].
// Results are undefined if x and y are both 0.
// 
// XXX
float atan (float x, float y) {
    return 0.0;
}
vec2 atan (vec2 x, vec2 y) {
    return vec2 (atan (x.x, y.x), atan (x.y, y.y));
}
vec3 atan (vec3 x, vec3 y) {
    return vec3 (atan (x.x, y.x), atan (x.y, y.y), atan (x.z, y.z));
}
vec4 atan (vec4 x, vec4 y) {
    return vec4 (atan (x.x, y.x), atan (x.y, y.y), atan (x.z, y.z), atan (x.w, y.w));
}

// 
// Arc tangent. Returns an angle whose tangent is y_over_x. The range of values returned by this
// function is [–PI/2, PI/2].
// 
// XXX
float atan (float y_over_x) {
    return 0.0;
}
vec2 atan (vec2 y_over_x) {
    return vec2 (atan (y_over_x.x), atan (y_over_x.y));
}
vec3 atan (vec3 y_over_x) {
    return vec3 (atan (y_over_x.x), atan (y_over_x.y), atan (y_over_x.z));
}
vec4 atan (vec4 y_over_x) {
    return vec4 (atan (y_over_x.x), atan (y_over_x.y), atan (y_over_x.z), atan (y_over_x.w));
}

// 
// 8.2 Exponential Functions
// 
// These all operate component-wise. The description is per component.
// 

// 
// Returns x raised to the y power, i.e., x^y.
// Results are undefined if x < 0.
// Results are undefined if x = 0 and y <= 0.
// 
// XXX
float pow (float x, float y) {
    return 0.0;
}
vec2 pow (vec2 x, vec2 y) {
    return vec2 (pow (x.x, y.x), pow (x.y, y.y));
}
vec3 pow (vec3 x, vec3 y) {
    return vec3 (pow (x.x, y.x), pow (x.y, y.y), pow (x.z, y.z));
}
vec4 pow (vec4 x, vec4 y) {
    return vec4 (pow (x.x, y.x), pow (x.y, y.y), pow (x.z, y.z), pow (x.w, y.w));
}

// 
// Returns the natural exponentiation of x, i.e., e^x.
// 

float exp (float x) {
    return pow (2.71828183, x);
}
vec2 exp (vec2 x) {
    return vec2 (exp (x.x), exp (x.y));
}
vec3 exp (vec3 x) {
    return vec3 (exp (x.x), exp (x.y), exp (x.z));
}
vec4 exp (vec4 x) {
    return vec4 (exp (x.x), exp (x.y), exp (x.z), exp (x.w));
}

// 
// Returns the natural logarithm of x, i.e., returns the value y which satisfies the equation
// x = e^y.
// Results are undefined if x <= 0.
// 

float log (float x) {
    return log2 (x) / log2 (2.71828183);
}
vec2 log (vec2 x) {
    return vec2 (log (x.x), log (x.y));
}
vec3 log (vec3 x) {
    return vec3 (log (x.x), log (x.y), log (x.z));
}
vec4 log (vec4 x) {
    return vec4 (log (x.x), log (x.y), log (x.z), log (x.w));
}

// 
// Returns 2 raised to the x power, i.e., 2^x
// 

float exp2 (float x) {
    return pow (2.0, x);
}
vec2 exp2 (vec2 x) {
    return vec2 (exp2 (x.x), exp2 (x.y));
}
vec3 exp2 (vec3 x) {
    return vec3 (exp2 (x.x), exp2 (x.y), exp2 (x.z));
}
vec4 exp2 (vec4 x) {
    return vec4 (exp2 (x.x), exp2 (x.y), exp2 (x.z), exp2 (x.w));
}

// 
// Returns the base 2 logarithm of x, i.e., returns the value y which satisfies the equation
// x = 2^y.
// Results are undefined if x <= 0.
// 
// XXX
float log2 (float x) {
    return 0.0;
}
vec2 log2 (vec2 x) {
    return vec2 (log2 (x.x), log2 (x.y));
}
vec3 log2 (vec3 x) {
    return vec3 (log2 (x.x), log2 (x.y), log2 (x.z));
}
vec4 log2 (vec4 x) {
    return vec4 (log2 (x.x), log2 (x.y), log2 (x.z), log2 (x.w));
}

// 
// Returns the positive square root of x.
// Results are undefined if x < 0.
// 

float sqrt (float x) {
    return pow (x, 0.5);
}
vec2 sqrt (vec2 x) {
    return vec2 (sqrt (x.x), sqrt (x.y));
}
vec3 sqrt (vec3 x) {
    return vec3 (sqrt (x.x), sqrt (x.y), sqrt (x.z));
}
vec4 sqrt (vec4 x) {
    return vec4 (sqrt (x.x), sqrt (x.y), sqrt (x.z), sqrt (x.w));
}

// 
// Returns the reciprocal of the positive square root of x.
// Results are undefined if x <= 0.
// 

float inversesqrt (float x) {
    return 1.0 / sqrt (x);
}
vec2 inversesqrt (vec2 x) {
    return vec2 (inversesqrt (x.x), inversesqrt (x.y));
}
vec3 inversesqrt (vec3 x) {
    return vec3 (inversesqrt (x.x), inversesqrt (x.y), inversesqrt (x.z));
}
vec4 inversesqrt (vec4 x) {
    return vec4 (inversesqrt (x.x), inversesqrt (x.y), inversesqrt (x.z), inversesqrt (x.w));
}

// 
// 8.3 Common Functions
// 
// These all operate component-wise. The description is per component.
// 

// 
// Returns x if x >= 0, otherwise it returns –x
// 

float abs (float x) {
    return x >= 0.0 ? x : -x;
}
vec2 abs (vec2 x) {
    return vec2 (abs (x.x), abs (x.y));
}
vec3 abs (vec3 x) {
    return vec3 (abs (x.x), abs (x.y), abs (x.z));
}
vec4 abs (vec4 x) {
    return vec4 (abs (x.x), abs (x.y), abs (x.z), abs (x.w));
}

// 
// Returns 1.0 if x > 0, 0.0 if x = 0, or –1.0 if x < 0
// 

float sign (float x) {
    return x > 0.0 ? 1.0 : x < 0.0 ? -1.0 : 0.0;
}
vec2 sign (vec2 x) {
    return vec2 (sign (x.x), sign (x.y));
}
vec3 sign (vec3 x) {
    return vec3 (sign (x.x), sign (x.y), sign (x.z));
}
vec4 sign (vec4 x) {
    return vec4 (sign (x.x), sign (x.y), sign (x.z), sign (x.w));
}

// 
// Returns a value equal to the nearest integer that is less than or equal to x
// 
// XXX
float floor (float x) {
    return 0.0;
}
vec2 floor (vec2 x) {
    return vec2 (floor (x.x), floor (x.y));
}
vec3 floor (vec3 x) {
    return vec3 (floor (x.x), floor (x.y), floor (x.z));
}
vec4 floor (vec4 x) {
    return vec4 (floor (x.x), floor (x.y), floor (x.z), floor (x.w));
}

// 
// Returns a value equal to the nearest integer that is greater than or equal to x
// 
// XXX
float ceil (float x) {
    return 0.0;
}
vec2 ceil (vec2 x) {
    return vec2 (ceil (x.x), ceil (x.y));
}
vec3 ceil (vec3 x) {
    return vec3 (ceil (x.x), ceil (x.y), ceil (x.z));
}
vec4 ceil (vec4 x) {
    return vec4 (ceil (x.x), ceil (x.y), ceil (x.z), ceil (x.w));
}

// 
// Returns x – floor (x)
// 

float fract (float x) {
    return x - floor (x);
}
vec2 fract (vec2 x) {
    return vec2 (fract (x.x), fract (x.y));
}
vec3 fract (vec3 x) {
    return vec3 (fract (x.x), fract (x.y), fract (x.z));
}
vec4 fract (vec4 x) {
    return vec4 (fract (x.x), fract (x.y), fract (x.z), fract (x.w));
}

// 
// Modulus. Returns x – y * floor (x/y)
// 

float mod (float x, float y) {
    return x - y * floor (x / y);
}
vec2 mod (vec2 x, float y) {
    return vec2 (mod (x.x, y), mod (x.y, y));
}
vec3 mod (vec3 x, float y) {
    return vec3 (mod (x.x, y), mod (x.y, y), mod (x.z, y));
}
vec4 mod (vec4 x, float y) {
    return vec4 (mod (x.x, y), mod (x.y, y), mod (x.z, y), mod (x.w, y));
}
vec2 mod (vec2 x, vec2 y) {
    return vec2 (mod (x.x, y.x), mod (x.y, y.y));
}
vec3 mod (vec3 x, vec3 y) {
    return vec3 (mod (x.x, y.x), mod (x.y, y.y), mod (x.z, y.z));
}
vec4 mod (vec4 x, vec4 y) {
    return vec4 (mod (x.x, y.x), mod (x.y, y.y), mod (x.z, y.z), mod (x.w, y.w));
}

// 
// Returns y if y < x, otherwise it returns x
// 

float min (float x, float y) {
    return y < x ? y : x;
}
vec2 min (vec2 x, float y) {
    return vec2 (min (x.x, y), min (x.y, y));
}
vec3 min (vec3 x, float y) {
    return vec3 (min (x.x, y), min (x.y, y), min (x.z, y));
}
vec4 min (vec4 x, float y) {
    return vec4 (min (x.x, y), min (x.y, y), min (x.z, y), min (x.w, y));
}
vec2 min (vec2 x, vec2 y) {
    return vec2 (min (x.x, y.x), min (x.y, y.y));
}
vec3 min (vec3 x, vec3 y) {
    return vec3 (min (x.x, y.x), min (x.y, y.y), min (x.z, y.z));
}
vec4 min (vec4 x, vec4 y) {
    return vec4 (min (x.x, y.x), min (x.y, y.y), min (x.z, y.z), min (x.w, y.w));
}

// 
// Returns y if x < y, otherwise it returns x
// 

float max (float x, float y) {
    return min (y, x);
}
vec2 max (vec2 x, float y) {
    return vec2 (max (x.x, y), max (x.y, y));
}
vec3 max (vec3 x, float y) {
    return vec3 (max (x.x, y), max (x.y, y), max (x.z, y));
}
vec4 max (vec4 x, float y) {
    return vec4 (max (x.x, y), max (x.y, y), max (x.z, y), max (x.w, y));
}
vec2 max (vec2 x, vec2 y) {
    return vec2 (max (x.x, y.x), max (x.y, y.y));
}
vec3 max (vec3 x, vec3 y) {
    return vec3 (max (x.x, y.x), max (x.y, y.y), max (x.z, y.z));
}
vec4 max (vec4 x, vec4 y) {
    return vec4 (max (x.x, y.x), max (x.y, y.y), max (x.z, y.z), max (x.w, y.w));
}

// 
// Returns min (max (x, minVal), maxVal)
// 
// Note that colors and depths written by fragment shaders will be clamped by the implementation
// after the fragment shader runs.
// 

float clamp (float x, float minVal, float maxVal) {
    return min (max (x, minVal), maxVal);
}
vec2 clamp (vec2 x, float minVal, float maxVal) {
    return vec2 (clamp (x.x, minVal, maxVal), clamp (x.y, minVal, maxVal));
}
vec3 clamp (vec3 x, float minVal, float maxVal) {
    return vec3 (clamp (x.x, minVal, maxVal), clamp (x.y, minVal, maxVal),
        clamp (x.z, minVal, maxVal));
}
vec4 clamp (vec4 x, float minVal, float maxVal) {
    return vec4 (clamp (x.x, minVal, maxVal), clamp (x.y, minVal, maxVal),
        clamp (x.z, minVal, maxVal), clamp (x.w, minVal, maxVal));
}
vec2 clamp (vec2 x, vec2 minVal, vec2 maxVal) {
    return vec2 (clamp (x.x, minVal.x, maxVal.x), clamp (x.y, minVal.y, maxVal.y));
}
vec3 clamp (vec3 x, vec3 minVal, vec3 maxVal) {
    return vec3 (clamp (x.x, minVal.x, maxVal.x), clamp (x.y, minVal.y, maxVal.y),
        clamp (x.z, minVal.z, maxVal.z));
}
vec4 clamp (vec4 x, vec4 minVal, vec4 maxVal) {
    return vec4 (clamp (x.x, minVal.x, maxVal.y), clamp (x.y, minVal.y, maxVal.y),
        clamp (x.z, minVal.z, maxVal.z), clamp (x.w, minVal.w, maxVal.w));
}

// 
// Returns x * (1 – a) + y * a, i.e., the linear blend of x and y
// 

float mix (float x, float y, float a) {
    return x * (1.0 - a) + y * a;
}
vec2 mix (vec2 x, vec2 y, float a) {
    return vec2 (mix (x.x, y.x, a), mix (x.y, y.y, a));
}
vec3 mix (vec3 x, vec3 y, float a) {
    return vec3 (mix (x.x, y.x, a), mix (x.y, y.y, a), mix (x.z, y.z, a));
}
vec4 mix (vec4 x, vec4 y, float a) {
    return vec4 (mix (x.x, y.x, a), mix (x.y, y.y, a), mix (x.z, y.z, a), mix (x.w, y.w, a));
}
vec2 mix (vec2 x, vec2 y, vec2 a) {
    return vec2 (mix (x.x, y.x, a.x), mix (x.y, y.y, a.y));
}
vec3 mix (vec3 x, vec3 y, vec3 a) {
    return vec3 (mix (x.x, y.x, a.x), mix (x.y, y.y, a.y), mix (x.z, y.z, a.z));
}
vec4 mix (vec4 x, vec4 y, vec4 a) {
    return vec4 (mix (x.x, y.x, a.x), mix (x.y, y.y, a.y), mix (x.z, y.z, a.z),
        mix (x.w, y.w, a.w));
}

// 
// Returns 0.0 if x < edge, otherwise it returns 1.0
// 

float step (float edge, float x) {
    return x < edge ? 0.0 : 1.0;
}
vec2 step (float edge, vec2 x) {
    return vec2 (step (edge, x.x), step (edge, x.y));
}
vec3 step (float edge, vec3 x) {
    return vec3 (step (edge, x.x), step (edge, x.y), step (edge, x.z));
}
vec4 step (float edge, vec4 x) {
    return vec4 (step (edge, x.x), step (edge, x.y), step (edge, x.z), step (edge, x.w));
}
vec2 step (vec2 edge, vec2 x) {
    return vec2 (step (edge.x, x.x), step (edge.y, x.y));
}
vec3 step (vec3 edge, vec3 x) {
    return vec3 (step (edge.x, x.x), step (edge.y, x.y), step (edge.z, x.z));
}
vec4 step (vec4 edge, vec4 x) {
    return vec4 (step (edge.x, x.x), step (edge.y, x.y), step (edge.z, x.z), step (edge.w, x.w));
}

// 
// Returns 0.0 if x <= edge0 and 1.0 if x >= edge1 and performs smooth Hermite interpolation
// between 0 and 1 when edge0 < x < edge1. This is useful in cases where you would want a threshold
// function with a smooth transition. This is equivalent to:
// <type> t;
// t = clamp ((x – edge0) / (edge1 – edge0), 0, 1);
// return t * t * (3 – 2 * t);
// 

float smoothstep (float edge0, float edge1, float x) {
    const float t = clamp ((x - edge0) / (edge1 - edge0), 0.0, 1.0);
    return t * t * (3.0 - 2.0 * t);
}
vec2 smoothstep (float edge0, float edge1, vec2 x) {
    return vec2 (smoothstep (edge0, edge1, x.x), smoothstep (edge0, edge1, x.y));
}
vec3 smoothstep (float edge0, float edge1, vec3 x) {
    return vec3 (smoothstep (edge0, edge1, x.x), smoothstep (edge0, edge1, x.y),
        smoothstep (edge0, edge1, x.z));
}
vec4 smoothstep (float edge0, float edge1, vec4 x) {
    return vec4 (smoothstep (edge0, edge1, x.x), smoothstep (edge0, edge1, x.y),
        smoothstep (edge0, edge1, x.z), smoothstep (edge0, edge1, x.w));
}
vec2 smoothstep (vec2 edge0, vec2 edge1, vec2 x) {
    return vec2 (smoothstep (edge0.x, edge1.x, x.x), smoothstep (edge0.y, edge1.y, x.y));
}
vec3 smoothstep (vec3 edge0, vec3 edge1, vec3 x) {
    return vec3 (smoothstep (edge0.x, edge1.x, x.x), smoothstep (edge0.y, edge1.y, x.y),
        smoothstep (edge0.z, edge1.z, x.z));
}
vec4 smoothstep (vec4 edge0, vec4 edge1, vec4 x) {
    return vec4 (smoothstep (edge0.x, edge1.x, x.x), smoothstep (edge0.y, edge1.y, x.y),
        smoothstep (edge0.z, edge1.z, x.z), smoothstep (edge0.w, edge1.w, x.w));
}

// 
// 8.4 Geometric Functions
// 
// These operate on vectors as vectors, not component-wise.
// 

// 
// Returns the dot product of x and y, i.e., result = x[0] * y[0] + x[1] * y[1] + ...
// 

float dot (float x, float y) {
    return x * y;
}
float dot (vec2 x, vec2 y) {
    return dot (x.x, y.x) + dot (x.y, y.y);
}
float dot (vec3 x, vec3 y) {
    return dot (x.x, y.x) + dot (x.y, y.y) + dot (x.z, y.z);
}
float dot (vec4 x, vec4 y) {
    return dot (x.x, y.x) + dot (x.y, y.y) + dot (x.z, y.z) + dot (x.w, y.w);
}

// 
// Returns the length of vector x, i.e., sqrt (x[0] * x[0] + x[1] * x[1] + ...)
// 

float length (float x) {
    return sqrt (dot (x, x));
}
float length (vec2 x) {
    return sqrt (dot (x, x));
}
float length (vec3 x) {
    return sqrt (dot (x, x));
}
float length (vec4 x) {
    return sqrt (dot (x, x));
}

// 
// Returns the distance between p0 and p1, i.e. length (p0 – p1)
// 

float distance (float x, float y) {
    return length (x - y);
}
float distance (vec2 x, vec2 y) {
    return length (x - y);
}
float distance (vec3 x, vec3 y) {
    return length (x - y);
}
float distance (vec4 x, vec4 y) {
    return length (x - y);
}

// 
// Returns the cross product of x and y, i.e.
// result.0 = x[1] * y[2] - y[1] * x[2]
// result.1 = x[2] * y[0] - y[2] * x[0]
// result.2 = x[0] * y[1] - y[0] * x[1]
// 

vec3 cross (vec3 x, vec3 y) {
    return vec3 (x.y * y.z - y.y * x.z, x.z * y.x - y.z * x.x, x.x * y.y - y.x * x.y);
}

// 
// Returns a vector in the same direction as x but with a length of 1.
// 

float normalize (float x) {
    return 1.0;
}
vec2 normalize (vec2 x) {
    return x / length (x);
}
vec3 normalize (vec3 x) {
    return x / length (x);
}
vec4 normalize (vec4 x) {
    return x / length (x);
}

// 
// If dot (Nref, I) < 0 return N otherwise return –N
// 

float faceforward (float N, float I, float Nref) {
    return dot (Nref, I) < 0.0 ? N : -N;
}
vec2 faceforward (vec2 N, vec2 I, vec2 Nref) {
    return dot (Nref, I) < 0.0 ? N : -N;
}
vec3 faceforward (vec3 N, vec3 I, vec3 Nref) {
    return dot (Nref, I) < 0.0 ? N : -N;
}
vec4 faceforward (vec4 N, vec4 I, vec4 Nref) {
    return dot (Nref, I) < 0.0 ? N : -N;
}

// 
// For the incident vector I and surface orientation N, returns the reflection direction:
// result = I - 2 * dot (N, I) * N
// N must already be normalized in order to achieve the desired result.
//

float reflect (float I, float N) {
    return I - 2.0 * dot (N, I) * N;
}
vec2 reflect (vec2 I, vec2 N) {
    return I - 2.0 * dot (N, I) * N;
}
vec3 reflect (vec3 I, vec3 N) {
    return I - 2.0 * dot (N, I) * N;
}
vec4 reflect (vec4 I, vec4 N) {
    return I - 2.0 * dot (N, I) * N;
}

//
// For the incident vector I and surface normal N, and the ratio of inidices of refraction eta,
// return the refraction vector. The returned result is computed by
//
// k = 1.0 - eta * eta * (1.0 - dot (N, I) * dot (N, I))
// if (k < 0.0)
//   result = genType (0.0)
// else
//   result = eta * I - (eta * dot (N, I) + sqrt (k)) * N
//
// The input parameters for the incident vector I and the surface normal N must already be
// normalized to get the desired results.
//

float refract (float I, float N, float eta) {
    const float k = 1.0 - eta * eta * (1.0 - dot (N, I) * dot (N, I));
    if (k < 0.0)
        return 0.0;
    return eta * I - (eta * dot (N, I) + sqrt (k)) * N;
}
vec2 refract (vec2 I, vec2 N, float eta) {
    const float k = 1.0 - eta * eta * (1.0 - dot (N, I) * dot (N, I));
    if (k < 0.0)
        return vec2 (0.0);
    return eta * I - (eta * dot (N, I) + sqrt (k)) * N;
}
vec3 refract (vec3 I, vec3 N, float eta) {
    const float k = 1.0 - eta * eta * (1.0 - dot (N, I) * dot (N, I));
    if (k < 0.0)
        return vec3 (0.0);
    return eta * I - (eta * dot (N, I) + sqrt (k)) * N;
}
vec4 refract (vec4 I, vec4 N, float eta) {
    const float k = 1.0 - eta * eta * (1.0 - dot (N, I) * dot (N, I));
    if (k < 0.0)
        return vec4 (0.0);
    return eta * I - (eta * dot (N, I) + sqrt (k)) * N;
}

// 
// 8.5 Matrix Functions
// 

// 
// Multiply matrix x by matrix y component-wise, i.e., result[i][j] is the scalar product
// of x[i][j] and y[i][j].
// Note: to get linear algebraic matrix multiplication, use the multiply operator (*).
// 

mat2 matrixCompMult (mat2 x, mat2 y) {
    return mat2 (
        x[0].x * y[0].x, x[0].y * y[0].y,
        x[1].x * y[1].x, x[1].y * y[1].y
    );
}
mat3 matrixCompMult (mat3 x, mat3 y) {
    return mat4 (
        x[0].x * y[0].x, x[0].y * y[0].y, x[0].z * y[0].z,
        x[1].x * y[1].x, x[1].y * y[1].y, x[1].z * y[1].z,
        x[2].x * y[2].x, x[2].y * y[2].y, x[2].z * y[2].z
    );
}
mat4 matrixCompMult (mat4 x, mat4 y) {
    return mat4 (
        x[0].x * y[0].x, x[0].y * y[0].y, x[0].z * y[0].z + x[0].w * y[0].w,
        x[1].x * y[1].x, x[1].y * y[1].y, x[1].z * y[1].z + x[1].w * y[1].w,
        x[2].x * y[2].x, x[2].y * y[2].y, x[2].z * y[2].z + x[2].w * y[2].w,
        x[3].x * y[3].x, x[3].y * y[3].y, x[3].z * y[3].z + x[3].w * y[3].w
    );
}

// 
// 8.6 Vector Relational Functions
// 
// Relational and equality operators (<, <=, >, >=, ==, !=) are defined (or reserved) to produce
// scalar Boolean results.
// 

// 
// Returns the component-wise compare of x < y.
// 

bvec2 lessThan (vec2 x, vec2 y) {
    return bvec2 (x.x < y.x, x.y < y.y);
}
bvec3 lessThan (vec3 x, vec3 y) {
    return bvec3 (x.x < y.x, x.y < y.y, x.z < y.z);
}
bvec4 lessThan (vec4 x, vec4 y) {
    return bvec4 (x.x < y.x, x.y < y.y, x.z < y.z, x.w < y.w);
}
bvec2 lessThan (ivec2 x, ivec2 y) {
    return bvec2 (x.x < y.x, x.y < y.y);
}
bvec3 lessThan (ivec3 x, ivec3 y) {
    return bvec3 (x.x < y.x, x.y < y.y, x.z < y.z);
}
bvec4 lessThan (ivec4 x, ivec4 y) {
    return bvec4 (x.x < y.x, x.y < y.y, x.z < y.z, x.w < y.w);
}

// 
// Returns the component-wise compare of x <= y.
// 

bvec2 lessThanEqual (vec2 x, vec2 y) {
    return bvec2 (x.x <= y.x, x.y <= y.y);
}
bvec3 lessThanEqual (vec3 x, vec3 y) {
    return bvec3 (x.x <= y.x, x.y <= y.y, x.z <= y.z);
}
bvec4 lessThanEqual (vec4 x, vec4 y) {
    return bvec4 (x.x <= y.x, x.y <= y.y, x.z <= y.z, x.w <= y.w);
}
bvec2 lessThanEqual (ivec2 x, ivec2 y) {
    return bvec2 (x.x <= y.x, x.y <= y.y);
}
bvec3 lessThanEqual (ivec3 x, ivec3 y) {
    return bvec3 (x.x <= y.x, x.y <= y.y, x.z <= y.z);
}
bvec4 lessThanEqual (ivec4 x, ivec4 y) {
    return bvec4 (x.x <= y.x, x.y <= y.y, x.z <= y.z, x.w <= y.w);
}

// 
// Returns the component-wise compare of x > y.
// 

bvec2 greaterThan (vec2 x, vec2 y) {
    return bvec2 (x.x > y.x, x.y > y.y);
}
bvec3 greaterThan (vec3 x, vec3 y) {
    return bvec3 (x.x > y.x, x.y > y.y, x.z > y.z);
}
bvec4 greaterThan (vec4 x, vec4 y) {
    return bvec4 (x.x > y.x, x.y > y.y, x.z > y.z, x.w > y.w);
}
bvec2 greaterThan (ivec2 x, ivec2 y) {
    return bvec2 (x.x > y.x, x.y > y.y);
}
bvec3 greaterThan (ivec3 x, ivec3 y) {
    return bvec3 (x.x > y.x, x.y > y.y, x.z > y.z);
}
bvec4 greaterThan (ivec4 x, ivec4 y) {
    return bvec4 (x.x > y.x, x.y > y.y, x.z > y.z, x.w > y.w);
}

// 
// Returns the component-wise compare of x >= y.
// 

bvec2 greaterThanEqual (vec2 x, vec2 y) {
    return bvec2 (x.x >= y.x, x.y >= y.y);
}
bvec3 greaterThanEqual (vec3 x, vec3 y) {
    return bvec3 (x.x >= y.x, x.y >= y.y, x.z >= y.z);
}
bvec4 greaterThanEqual (vec4 x, vec4 y) {
    return bvec4 (x.x >= y.x, x.y >= y.y, x.z >= y.z, x.w >= y.w);
}
bvec2 greaterThanEqual (ivec2 x, ivec2 y) {
    return bvec2 (x.x >= y.x, x.y >= y.y);
}
bvec3 greaterThanEqual (ivec3 x, ivec3 y) {
    return bvec3 (x.x >= y.x, x.y >= y.y, x.z >= y.z);
}
bvec4 greaterThanEqual (ivec4 x, ivec4 y) {
    return bvec4 (x.x >= y.x, x.y >= y.y, x.z >= y.z, x.w >= y.w);
}

// 
// Returns the component-wise compare of x == y.
// 

bvec2 equal (vec2 x, vec2 y) {
    return bvec2 (x.x == y.x, x.y == y.y);
}
bvec3 equal (vec3 x, vec3 y) {
    return bvec3 (x.x == y.x, x.y == y.y, x.z == y.z);
}
bvec4 equal (vec4 x, vec4 y) {
    return bvec4 (x.x == y.x, x.y == y.y, x.z == y.z, x.w == y.w);
}
bvec2 equal (ivec2 x, ivec2 y) {
    return bvec2 (x.x == y.x, x.y == y.y);
}
bvec3 equal (ivec3 x, ivec3 y) {
    return bvec3 (x.x == y.x, x.y == y.y, x.z == y.z);
}
bvec4 equal (ivec4 x, ivec4 y) {
    return bvec4 (x.x == y.x, x.y == y.y, x.z == y.z, x.w == y.w);
}

// 
// Returns the component-wise compare of x != y.
// 

bvec2 notEqual (vec2 x, vec2 y) {
    return bvec2 (x.x != y.x, x.y != y.y);
}
bvec3 notEqual (vec3 x, vec3 y) {
    return bvec3 (x.x != y.x, x.y != y.y, x.z != y.z);
}
bvec4 notEqual (vec4 x, vec4 y) {
    return bvec4 (x.x != y.x, x.y != y.y, x.z != y.z, x.w != y.w);
}
bvec2 notEqual (ivec2 x, ivec2 y) {
    return bvec2 (x.x != y.x, x.y != y.y);
}
bvec3 notEqual (ivec3 x, ivec3 y) {
    return bvec3 (x.x != y.x, x.y != y.y, x.z != y.z);
}
bvec4 notEqual (ivec4 x, ivec4 y) {
    return bvec4 (x.x != y.x, x.y != y.y, x.z != y.z, x.w != y.w);
}

// 
// Returns true if any component of x is true.
// 

bool any (bvec2 x) {
    return x.x || x.y;
}
bool any (bvec3 x) {
    return x.x || x.y || x.z;
}
bool any (bvec4 x) {
    return x.x || x.y || x.z || x.w;
}

// 
// Returns true only if all components of x are true.
// 

bool all (bvec2 x) {
    return x.x && x.y;
}
bool all (bvec3 x) {
    return x.x && x.y && x.z;
}
bool all (bvec4 x) {
    return x.x && x.y && x.z && x.w;
}

// 
// Returns the component-wise logical complement of x.
// 

bvec2 not (bvec2 x) {
    return bvec2 (!x.x, !x.y);
}
bvec3 not (bvec3 x) {
    return bvec3 (!x.x, !x.y, !x.z);
}
bvec4 not (bvec4 x) {
    return bvec4 (!x.x, !x.y, !x.z, !x.w);
}

// 
// 8.7 Texture Lookup Functions
// 
// Texture lookup functions are available to both vertex and fragment shaders. However, level
// of detail is not computed by fixed functionality for vertex shaders, so there are some
// differences in operation between vertex and fragment texture lookups. The functions in the table
// below provide access to textures through samplers, as set up through the OpenGL API. Texture
// properties such as size, pixel format, number of dimensions, filtering method, number of mip-map
// levels, depth comparison, and so on are also defined by OpenGL API calls. Such properties are
// taken into account as the texture is accessed via the built-in functions defined below.
// 
// If a non-shadow texture call is made to a sampler that represents a depth texture with depth
// comparisons turned on, then results are undefined. If a shadow texture call is made to a sampler
// that represents a depth texture with depth comparisions turned off, the results are undefined.
// If a shadow texture call is made to a sampler that does not represent a depth texture, then
// results are undefined.
// 
// In all functions below, the bias parameter is optional for fragment shaders. The bias parameter
// is not accepted in a vertex shader. For a fragment shader, if bias is present, it is added to
// the calculated level of detail prior to performing the texture access operation. If the bias
// parameter is not provided, then the implementation automatically selects level of detail:
// For a texture that is not mip-mapped, the texture is used directly. If it is mip-mapped and
// running in a fragment shader, the LOD computed by the implementation is used to do the texture
// lookup. If it is mip-mapped and running on the vertex shader, then the base texture is used.
// 
// The built-ins suffixed with "Lod" are allowed only in a vertex shader. For the "Lod" functions,
// lod is directly used as the level of detail.
// 

// 
// Use the texture coordinate coord to do a texture lookup in the 1D texture currently bound
// to sampler. For the projective ("Proj") versions, the texture coordinate coord.s is divided by
// the last component of coord.
// 
// XXX
vec4 texture1D (sampler1D sampler, float coord) {
    return vec4 (0.0);
}
vec4 texture1DProj (sampler1D sampler, vec2 coord) {
    return texture1D (sampler, coord.s / coord.t);
}
vec4 texture1DProj (sampler1D sampler, vec4 coord) {
    return texture1D (sampler, coord.s / coord.q);
}

// 
// Use the texture coordinate coord to do a texture lookup in the 2D texture currently bound
// to sampler. For the projective ("Proj") versions, the texture coordinate (coord.s, coord.t) is
// divided by the last component of coord. The third component of coord is ignored for the vec4
// coord variant.
// 
// XXX
vec4 texture2D (sampler2D sampler, vec2 coord) {
    return vec4 (0.0);
}
vec4 texture2DProj (sampler2D sampler, vec3 coord) {
    return texture2D (sampler, vec2 (coord.s / coord.p, coord.t / coord.p));
}
vec4 texture2DProj (sampler2D sampler, vec4 coord) {
    return texture2D (sampler, vec2 (coord.s / coord.q, coord.t / coord.q));
}

// 
// Use the texture coordinate coord to do a texture lookup in the 3D texture currently bound
// to sampler. For the projective ("Proj") versions, the texture coordinate is divided by coord.q.
// 
// XXX
vec4 texture3D (sampler3D sampler, vec3 coord) {
    return vec4 (0.0);
}
vec4 texture3DProj (sampler3D sampler, vec4 coord) {
    return texture3D (sampler, vec3 (coord.s / coord.q, coord.t / coord.q, coord.p / coord.q));
}

// 
// Use the texture coordinate coord to do a texture lookup in the cube map texture currently bound
// to sampler. The direction of coord is used to select which face to do a 2-dimensional texture
// lookup in, as described in section 3.8.6 in version 1.4 of the OpenGL specification.
// 
// XXX
vec4 textureCube (samplerCube sampler, vec3 coord) {
    return vec4 (0.0);
}

// 
// Use texture coordinate coord to do a depth comparison lookup on the depth texture bound
// to sampler, as described in section 3.8.14 of version 1.4 of the OpenGL specification. The 3rd
// component of coord (coord.p) is used as the R value. The texture bound to sampler must be a
// depth texture, or results are undefined. For the projective ("Proj") version of each built-in,
// the texture coordinate is divide by coord.q, giving a depth value R of coord.p/coord.q. The
// second component of coord is ignored for the "1D" variants.
// 
// XXX
vec4 shadow1D (sampler1DShadow sampler, vec3 coord) {
    return vec4 (0.0);
}
// XXX
vec4 shadow2D (sampler2DShadow sampler, vec3 coord) {
    return vec4 (0.0);
}
vec4 shadow1DProj (sampler1DShadow sampler, vec4 coord) {
    return shadow1D (sampler, vec3 (coord.s / coord.q, 0.0, coord.p / coord.q));
}
vec4 shadow2DProj (sampler2DShadow sampler, vec4 coord) {
    return shadow2D (sampler, vec3 (coord.s / coord.q, coord.t / coord.q, coord.p / coord.q));
}

// 
// 8.9 Noise Functions
// 
// Noise functions are available to both fragment and vertex shaders. They are stochastic functions
// that can be used to increase visual complexity. Values returned by the following noise functions
// give the appearance of randomness, but are not truly random. The noise functions below are
// defined to have the following characteristics:
// 
// - The return value(s) are always in the range [-1,1], and cover at least the range [-0.6, 0.6],
//   with a gaussian-like distribution.
// * The return value(s) have an overall average of 0.0
// * They are repeatable, in that a particular input value will always produce the same return value
// * They are statistically invariant under rotation (i.e., no matter how the domain is rotated, it
//   has the same statistical character)
// * They have a statistical invariance under translation (i.e., no matter how the domain is
//   translated, it has the same statistical character)
// * They typically give different results under translation.
// - The spatial frequency is narrowly concentrated, centered somewhere between 0.5 to 1.0.
// 

// 
// Returns a 1D noise value based on the input value x.
// 
// XXX
float noise1 (float x) {
    return 0.0;
}
// XXX
float noise1 (vec2 x) {
    return 0.0;
}
// XXX
float noise1 (vec3 x) {
    return 0.0;
}
// XXX
float noise1 (vec4 x) {
    return 0.0;
}

// 
// Returns a 2D noise value based on the input value x.
// 
// XXX
vec2 noise2 (float x) {
    return vec2 (0.0);
}
// XXX
vec2 noise2 (vec2 x) {
    return vec2 (0.0);
}
// XXX
vec2 noise2 (vec3 x) {
    return vec2 (0.0);
}
// XXX
vec2 noise2 (vec4 x) {
    return vec2 (0.0);
}

// 
// Returns a 3D noise value based on the input value x.
// 
// XXX
vec3 noise3 (float x) {
    return vec3 (0.0);
}
// XXX
vec3 noise3 (vec2 x) {
    return vec3 (0.0);
}
// XXX
vec3 noise3 (vec3 x) {
    return vec3 (0.0);
}
// XXX
vec3 noise3 (vec4 x) {
    return vec3 (0.0);
}

// 
// Returns a 4D noise value based on the input value x.
// 
// XXX
vec4 noise4 (float x) {
    return vec4 (0.0);
}
// XXX
vec4 noise4 (vec2 x) {
    return vec4 (0.0);
}
// XXX
vec4 noise4 (vec3 x) {
    return vec4 (0.0);
}
// XXX
vec4 noise4 (vec4 x) {
    return vec4 (0.0);
}

