Quat MUST_USE_RESULT Quat::RotateFromTo(const float4 &sourceDirection, const float4 &targetDirection)
{
#if defined(MATH_AUTOMATIC_SSE) && defined(MATH_SSE)
// Best: 12.289 nsecs / 33.144 ticks, Avg: 12.489 nsecs, Worst: 14.210 nsecs
simd4f cosAngle = dot4_ps(sourceDirection.v, targetDirection.v);
cosAngle = negate3_ps(cosAngle); // [+ - - -]
// XYZ channels use the trigonometric formula sin(x/2) = +/-sqrt(0.5-0.5*cosx))
// The W channel uses the trigonometric formula cos(x/2) = +/-sqrt(0.5+0.5*cosx))
simd4f half = set1_ps(0.5f);
simd4f cosSinHalfAngle = sqrt_ps(add_ps(half, mul_ps(half, cosAngle))); // [cos(x/2), sin(x/2), sin(x/2), sin(x/2)]
simd4f axis = cross_ps(sourceDirection.v, targetDirection.v);
simd4f recipLen = rsqrt_ps(dot4_ps(axis, axis));
axis = mul_ps(axis, recipLen); // [0 z y x]
// Set the w component to one.
simd4f one = add_ps(half, half); // [1 1 1 1]
simd4f highPart = _mm_unpackhi_ps(axis, one); // [_ _ 1 z]
axis = _mm_movelh_ps(axis, highPart); // [1 z y x]
Quat q;
q.q = mul_ps(axis, cosSinHalfAngle);
return q;
#else
// Best: 19.970 nsecs / 53.632 ticks, Avg: 20.197 nsecs, Worst: 21.122 nsecs
assume(EqualAbs(sourceDirection.w, 0.f));
assume(EqualAbs(targetDirection.w, 0.f));
return Quat::RotateFromTo(sourceDirection.xyz(), targetDirection.xyz());
#endif
}
float3x4 float3x4::operator /(float scalar) const
{
assume(!EqualAbs(scalar, 0));
#ifdef MATH_SIMD
float3x4 r;
simd4f s = set1_ps(scalar);
simd4f one = set1_ps(1.f);
s = div_ps(one, s);
r.row[0] = mul_ps(row[0], s);
r.row[1] = mul_ps(row[1], s);
r.row[2] = mul_ps(row[2], s);
#else
float3x4 r = *this;
r /= scalar;
#endif
return r;
}
Quat Quat::operator /(float scalar) const
{
assume(!EqualAbs(scalar, 0.f));
#ifdef MATH_AUTOMATIC_SSE
return div_ps(q, set1_ps(scalar));
#else
return *this * (1.f / scalar);
#endif
}
float3x4 &float3x4::operator /=(float scalar)
{
assume(!EqualAbs(scalar, 0));
#ifdef MATH_SIMD
simd4f s = set1_ps(scalar);
simd4f one = set1_ps(1.f);
s = div_ps(one, s);
row[0] = mul_ps(row[0], s);
row[1] = mul_ps(row[1], s);
row[2] = mul_ps(row[2], s);
#else
float invScalar = 1.f / scalar;
for(int y = 0; y < Rows; ++y)
for(int x = 0; x < Cols; ++x)
v[y][x] *= invScalar;
#endif
return *this;
}
void Quat::SetFromAxisAngle(const float4 &axis, float angle)
{
assume1(EqualAbs(axis.w, 0.f), axis);
assume2(axis.IsNormalized(1e-4f), axis, axis.Length4());
assume1(MATH_NS::IsFinite(angle), angle);
#if defined(MATH_AUTOMATIC_SSE) && defined(MATH_SSE)
// Best: 26.499 nsecs / 71.024 ticks, Avg: 26.856 nsecs, Worst: 27.651 nsecs
simd4f half = set1_ps(0.5f);
simd4f halfAngle = mul_ps(set1_ps(angle), half);
simd4f sinAngle, cosAngle;
sincos_ps(halfAngle, &sinAngle, &cosAngle);
simd4f quat = mul_ps(axis, sinAngle);
// Set the w component to cosAngle.
simd4f highPart = _mm_unpackhi_ps(quat, cosAngle); // [_ _ 1 z]
q = _mm_movelh_ps(quat, highPart); // [1 z y x]
#else
// Best: 36.868 nsecs / 98.312 ticks, Avg: 36.980 nsecs, Worst: 41.477 nsecs
SetFromAxisAngle(axis.xyz(), angle);
#endif
}
float3x4 &float3x4::operator *=(float scalar)
{
#ifdef MATH_SIMD
simd4f s = set1_ps(scalar);
row[0] = mul_ps(row[0], s);
row[1] = mul_ps(row[1], s);
row[2] = mul_ps(row[2], s);
#else
for(int y = 0; y < Rows; ++y)
for(int x = 0; x < Cols; ++x)
v[y][x] *= scalar;
#endif
return *this;
}
float3x4 float3x4::operator *(float scalar) const
{
#ifdef MATH_SIMD
float3x4 r;
simd4f s = set1_ps(scalar);
r.row[0] = mul_ps(row[0], s);
r.row[1] = mul_ps(row[1], s);
r.row[2] = mul_ps(row[2], s);
#else
float3x4 r = *this;
r *= scalar;
#endif
return r;
}
void Quat::ToAxisAngle(float4 &axis, float &angle) const
{
#if defined(MATH_AUTOMATIC_SSE) && defined(MATH_SSE)
// Best: 35.332 nsecs / 94.328 ticks, Avg: 35.870 nsecs, Worst: 57.607 nsecs
assume2(this->IsNormalized(), *this, this->Length());
simd4f cosAngle = _mm_shuffle_ps(q, q, _MM_SHUFFLE(3, 3, 3, 3));
simd4f rcpSinAngle = rsqrt_ps(sub_ps(set1_ps(1.f), mul_ps(cosAngle, cosAngle)));
angle = Acos(s4f_x(cosAngle)) * 2.f;
simd4f a = mul_ps(q, rcpSinAngle);
// Set the w component to zero.
simd4f highPart = _mm_unpackhi_ps(a, zero_ps()); // [_ _ 0 z]
axis.v = _mm_movelh_ps(a, highPart); // [0 z y x]
#else
// Best: 85.258 nsecs / 227.656 ticks, Avg: 85.492 nsecs, Worst: 86.410 nsecs
ToAxisAngle(reinterpret_cast<float3&>(axis), angle);
axis.w = 0.f;
#endif
}
vec Quat::Axis() const
{
assume2(this->IsNormalized(), *this, this->Length());
#if defined(MATH_AUTOMATIC_SSE) && defined(MATH_SSE)
// Best: 6.145 nsecs / 16.88 ticks, Avg: 6.367 nsecs, Worst: 6.529 nsecs
assume2(this->IsNormalized(), *this, this->Length());
simd4f cosAngle = _mm_shuffle_ps(q, q, _MM_SHUFFLE(3, 3, 3, 3));
simd4f rcpSinAngle = rsqrt_ps(sub_ps(set1_ps(1.f), mul_ps(cosAngle, cosAngle)));
simd4f a = mul_ps(q, rcpSinAngle);
// Set the w component to zero.
simd4f highPart = _mm_unpackhi_ps(a, zero_ps()); // [_ _ 0 z]
a = _mm_movelh_ps(a, highPart); // [0 z y x]
return FLOAT4_TO_DIR(a);
#else
// Best: 6.529 nsecs / 18.152 ticks, Avg: 6.851 nsecs, Worst: 8.065 nsecs
// Convert cos to sin via the identity sin^2 + cos^2 = 1, and fuse reciprocal and square root to the same instruction,
// since we are about to divide by it.
float rcpSinAngle = RSqrt(1.f - w*w);
return DIR_VEC(x, y, z) * rcpSinAngle;
#endif
}
bool AABB::IntersectLineAABB_SSE(const float4 &rayPos, const float4 &rayDir, float tNear, float tFar) const
{
assume(rayDir.IsNormalized4());
assume(tNear <= tFar && "AABB::IntersectLineAABB: User gave a degenerate line as input for the intersection test!");
/* For reference, this is the C++ form of the vectorized SSE code below.
float4 recipDir = rayDir.RecipFast4();
float4 t1 = (aabbMinPoint - rayPos).Mul(recipDir);
float4 t2 = (aabbMaxPoint - rayPos).Mul(recipDir);
float4 near = t1.Min(t2);
float4 far = t1.Max(t2);
float4 rayDirAbs = rayDir.Abs();
if (rayDirAbs.x > 1e-4f) // ray is parallel to plane in question
{
tNear = Max(near.x, tNear); // tNear tracks distance to intersect (enter) the AABB.
tFar = Min(far.x, tFar); // tFar tracks the distance to exit the AABB.
}
else if (rayPos.x < aabbMinPoint.x || rayPos.x > aabbMaxPoint.x) // early-out if the ray can't possibly enter the box.
return false;
if (rayDirAbs.y > 1e-4f) // ray is parallel to plane in question
{
tNear = Max(near.y, tNear); // tNear tracks distance to intersect (enter) the AABB.
tFar = Min(far.y, tFar); // tFar tracks the distance to exit the AABB.
}
else if (rayPos.y < aabbMinPoint.y || rayPos.y > aabbMaxPoint.y) // early-out if the ray can't possibly enter the box.
return false;
if (rayDirAbs.z > 1e-4f) // ray is parallel to plane in question
{
tNear = Max(near.z, tNear); // tNear tracks distance to intersect (enter) the AABB.
tFar = Min(far.z, tFar); // tFar tracks the distance to exit the AABB.
}
else if (rayPos.z < aabbMinPoint.z || rayPos.z > aabbMaxPoint.z) // early-out if the ray can't possibly enter the box.
return false;
return tNear < tFar;
*/
simd4f recipDir = rcp_ps(rayDir.v);
// Note: The above performs an approximate reciprocal (11 bits of precision).
// For a full precision reciprocal, perform a div:
// simd4f recipDir = div_ps(set1_ps(1.f), rayDir.v);
simd4f t1 = mul_ps(sub_ps(minPoint, rayPos.v), recipDir);
simd4f t2 = mul_ps(sub_ps(maxPoint, rayPos.v), recipDir);
simd4f nearD = min_ps(t1, t2); // [0 n3 n2 n1]
simd4f farD = max_ps(t1, t2); // [0 f3 f2 f1]
// Check if the ray direction is parallel to any of the cardinal axes, and if so,
// mask those [near, far] ranges away from the hit test computations.
simd4f rayDirAbs = abs_ps(rayDir.v);
const simd4f epsilon = set1_ps(1e-4f);
// zeroDirections[i] will be nonzero for each axis i the ray is parallel to.
simd4f zeroDirections = cmple_ps(rayDirAbs, epsilon);
const simd4f floatInf = set1_ps(FLOAT_INF);
const simd4f floatNegInf = set1_ps(-FLOAT_INF);
// If the ray is parallel to one of the axes, replace the slab range for that axis
// with [-inf, inf] range instead. (which is a no-op in the comparisons below)
nearD = cmov_ps(nearD, floatNegInf, zeroDirections);
farD = cmov_ps(farD, floatInf, zeroDirections);
// Next, we need to compute horizontally max(nearD[0], nearD[1], nearD[2]) and min(farD[0], farD[1], farD[2])
// to see if there is an overlap in the hit ranges.
simd4f v1 = axx_bxx_ps(nearD, farD); // [f1 f1 n1 n1]
simd4f v2 = ayy_byy_ps(nearD, farD); // [f2 f2 n2 n2]
simd4f v3 = azz_bzz_ps(nearD, farD); // [f3 f3 n3 n3]
nearD = max_ps(v1, max_ps(v2, v3));
farD = min_ps(v1, min_ps(v2, v3));
farD = wwww_ps(farD); // Unpack the result from high offset in the register.
nearD = max_ps(nearD, setx_ps(tNear));
farD = min_ps(farD, setx_ps(tFar));
// Finally, test if the ranges overlap.
simd4f rangeIntersects = cmple_ps(nearD, farD); // Only x channel used, higher ones ignored.
// To store out out the interval of intersection, uncomment the following:
// These are disabled, since without these, the whole function runs without a single memory store,
// which has been profiled to be very fast! Uncommenting these causes an order-of-magnitude slowdown.
// For now, using the SSE version only where the tNear and tFar ranges are not interesting.
// _mm_store_ss(&tNear, nearD);
// _mm_store_ss(&tFar, farD);
// To avoid false positives, need to have an additional rejection test for each cardinal axis the ray direction
// is parallel to.
simd4f out2 = cmplt_ps(rayPos.v, minPoint);
simd4f out3 = cmpgt_ps(rayPos.v, maxPoint);
out2 = or_ps(out2, out3);
zeroDirections = and_ps(zeroDirections, out2);
simd4f yOut = yyyy_ps(zeroDirections);
simd4f zOut = zzzz_ps(zeroDirections);
zeroDirections = or_ps(or_ps(zeroDirections, yOut), zOut);
// Intersection occurs if the slab ranges had positive overlap and if the test was not rejected by the ray being
// parallel to some cardinal axis.
simd4f intersects = andnot_ps(zeroDirections, rangeIntersects);
simd4f epsilonMasked = and_ps(epsilon, intersects);
return comieq_ss(epsilon, epsilonMasked) != 0;
}
Quat MUST_USE_RESULT Quat::Slerp(const Quat &q2, float t) const
{
assume(0.f <= t && t <= 1.f);
assume(IsNormalized());
assume(q2.IsNormalized());
#if defined(MATH_AUTOMATIC_SSE) && defined(MATH_SSE)
simd4f angle = dot4_ps(q, q2.q); // <q, q2.q>
simd4f neg = cmplt_ps(angle, zero_ps()); // angle < 0?
neg = and_ps(neg, set1_ps_hex(0x80000000)); // Convert 0/0xFFFFFFFF mask to a 0x/0x80000000 mask.
// neg = s4i_to_s4f(_mm_slli_epi32(s4f_to_s4i(neg), 31)); // A SSE2-esque way to achieve the above would be this, but this seems to clock slower (12.04 clocks vs 11.97 clocks)
angle = xor_ps(angle, neg); // if angle was negative, make it positive.
simd4f one = set1_ps(1.f);
angle = min_ps(angle, one); // If user passed t > 1 or t < -1, clamp the range.
// Compute a fast polynomial approximation to arccos(angle).
// arccos(x): (-0.69813170079773212f * x * x - 0.87266462599716477f) * x + 1.5707963267948966f;
angle = madd_ps(msub_ps(mul_ps(set1_ps(-0.69813170079773212f), angle), angle, set1_ps(0.87266462599716477f)), angle, set1_ps(1.5707963267948966f));
// Shuffle an appropriate vector from 't' and 'angle' for computing two sines in one go.
simd4f T = _mm_set_ss(t); // (.., t)
simd4f oneSubT = sub_ps(one, T); // (.., 1-t)
T = _mm_movelh_ps(T, oneSubT); // (.., 1-t, .., t)
angle = mul_ps(angle, T); // (.., (1-t)*angle, .., t*angle)
// Compute a fast polynomial approximation to sin(t*angle) and sin((1-t)*angle).
// Here could use "angle = sin_ps(angle);" for precision, but favor speed instead with the following polynomial expansion:
// sin(x): ((5.64311797634681035370e-03 * x * x - 1.55271410633428644799e-01) * x * x + 9.87862135574673806965e-01) * x
simd4f angle2 = mul_ps(angle, angle);
angle = mul_ps(angle, madd_ps(madd_ps(angle2, set1_ps(5.64311797634681035370e-03f), set1_ps(-1.55271410633428644799e-01f)), angle2, set1_ps(9.87862135574673806965e-01f)));
// Compute the final lerp factors a and b to scale q and q2.
simd4f a = zzzz_ps(angle);
simd4f b = xxxx_ps(angle);
a = xor_ps(a, neg);
a = mul_ps(q, a);
a = madd_ps(q2, b, a);
// The lerp above generates an unnormalized quaternion which needs to be renormalized.
return mul_ps(a, rsqrt_ps(dot4_ps(a, a)));
#else
float angle = this->Dot(q2);
float sign = 1.f; // Multiply by a sign of +/-1 to guarantee we rotate the shorter arc.
if (angle < 0.f)
{
angle = -angle;
sign = -1.f;
}
float a;
float b;
if (angle < 0.999) // perform spherical linear interpolation.
{
// angle = Acos(angle); // After this, angle is in the range pi/2 -> 0 as the original angle variable ranged from 0 -> 1.
angle = (-0.69813170079773212f * angle * angle - 0.87266462599716477f) * angle + 1.5707963267948966f;
float ta = t*angle;
#ifdef MATH_USE_SINCOS_LOOKUPTABLE
// If Sin() is based on a lookup table, prefer that over polynomial approximation.
a = Sin(angle - ta);
b = Sin(ta);
#else
// Not using a lookup table, manually compute the two sines by using a very rough approximation.
float ta2 = ta*ta;
b = ((5.64311797634681035370e-03f * ta2 - 1.55271410633428644799e-01f) * ta2 + 9.87862135574673806965e-01f) * ta;
a = angle - ta;
float a2 = a*a;
a = ((5.64311797634681035370e-03f * a2 - 1.55271410633428644799e-01f) * a2 + 9.87862135574673806965e-01f) * a;
#endif
}
else // If angle is close to taking the denominator to zero, resort to linear interpolation (and normalization).
{
a = 1.f - t;
b = t;
}
// Lerp and renormalize.
return (*this * (a * sign) + q2 * b).Normalized();
#endif
}