For a videogame I’m implementing in my spare time, I’ve tried implementing my own versions of sinf(), cosf(), and atan2f(), using lookup tables. The intent is to have implementations that are faster, although with less accuracy.
My initial implementation is below. The functions work, and return good approximate values. The only problem is that they are slower than calling the standard sinf(), cosf(), and atan2f() functions.
So, what am I doing wrong?
// Geometry.h includes definitions of PI, TWO_PI, etc., as // well as the prototypes for the public functions #include 'Geometry.h' namespace { // Number of entries in the sin/cos lookup table const int SinTableCount = 512; // Angle covered by each table entry const float SinTableDelta = TWO_PI / (float)SinTableCount; // Lookup table for Sin() results float SinTable[SinTableCount]; // This object initializes the contents of the SinTable array exactly once class SinTableInitializer { public: SinTableInitializer() { for (int i = 0; i < SinTableCount; ++i) { SinTable[i] = sinf((float)i * SinTableDelta); } } }; static SinTableInitializer sinTableInitializer; // Number of entries in the atan lookup table const int AtanTableCount = 512; // Interval covered by each Atan table entry const float AtanTableDelta = 1.0f / (float)AtanTableCount; // Lookup table for Atan() results float AtanTable[AtanTableCount]; // This object initializes the contents of the AtanTable array exactly once class AtanTableInitializer { public: AtanTableInitializer() { for (int i = 0; i < AtanTableCount; ++i) { AtanTable[i] = atanf((float)i * AtanTableDelta); } } }; static AtanTableInitializer atanTableInitializer; // Lookup result in table. // Preconditions: y > 0, x > 0, y < x static float AtanLookup2(float y, float x) { assert(y > 0.0f); assert(x > 0.0f); assert(y < x); const float ratio = y / x; const int index = (int)(ratio / AtanTableDelta); return AtanTable[index]; } } float Sin(float angle) { // If angle is negative, reflect around X-axis and negate result bool mustNegateResult = false; if (angle < 0.0f) { mustNegateResult = true; angle = -angle; } // Normalize angle so that it is in the interval (0.0, PI) while (angle >= TWO_PI) { angle -= TWO_PI; } const int index = (int)(angle / SinTableDelta); const float result = SinTable[index]; return mustNegateResult? (-result) : result; } float Cos(float angle) { return Sin(angle + PI_2); } float Atan2(float y, float x) { // Handle x == 0 or x == -0 // (See atan2(3) for specification of sign-bit handling.) if (x == 0.0f) { if (y > 0.0f) { return PI_2; } else if (y < 0.0f) { return -PI_2; } else if (signbit(x)) { return signbit(y)? -PI : PI; } else { return signbit(y)? -0.0f : 0.0f; } } // Handle y == 0, x != 0 if (y == 0.0f) { return (x > 0.0f)? 0.0f : PI; } // Handle y == x if (y == x) { return (x > 0.0f)? PI_4 : -(3.0f * PI_4); } // Handle y == -x if (y == -x) { return (x > 0.0f)? -PI_4 : (3.0f * PI_4); } // For other cases, determine quadrant and do appropriate lookup and calculation bool right = (x > 0.0f); bool top = (y > 0.0f); if (right && top) { // First quadrant if (y < x) { return AtanLookup2(y, x); } else { return PI_2 - AtanLookup2(x, y); } } else if (!right && top) { // Second quadrant const float posx = fabsf(x); if (y < posx) { return PI - AtanLookup2(y, posx); } else { return PI_2 + AtanLookup2(posx, y); } } else if (!right && !top) { // Third quadrant const float posx = fabsf(x); const float posy = fabsf(y); if (posy < posx) { return -PI + AtanLookup2(posy, posx); } else { return -PI_2 - AtanLookup2(posx, posy); } } else { // right && !top // Fourth quadrant const float posy = fabsf(y); if (posy < x) { return -AtanLookup2(posy, x); } else { return -PI_2 + AtanLookup2(x, posy); } } return 0.0f; }
‘Premature optimization is the root of all evil’ – Donald Knuth
Nowadays compilers provide very efficient intrinsics for trigonometric functions that get the best from modern processors (SSE etc.), which explains why you can hardly beat the built-in functions. Don’t lose too much time on these parts and instead concentrate on the real bottlenecks that you can spot with a profiler.