internal static (__m128, __m128) Inverse(__m128 p1, __m128 p2)
{
// s, t computed as in the normalization
__m128 b2 = Detail.dp_bc(p1, p1);
__m128 s = Detail.rsqrt_nr1(b2);
__m128 bc = Detail.dp_bc(_mm_xor_ps(p1, _mm_set_ss(-0f)), p2);
__m128 b2Inv = Detail.rcp_nr1(b2);
__m128 t = _mm_mul_ps(_mm_mul_ps(bc, b2Inv), s);
__m128 neg = _mm_set_ps(-0f, -0f, -0f, 0f);
// p1 * (s + t e0123)^2 = (s * p1 - t P1perp) * (s + t e0123)
// = s^2 p1 - s t P1perp - s t P1perp
// = s^2 p1 - 2 s t P1perp
// (the scalar component above needs to be negated)
// p2 * (s + t e0123)^2 = s^2 p2 NOTE: s^2 = b2_inv
__m128 st = _mm_mul_ps(s, t);
st = _mm_mul_ps(p1, st);
p2 = _mm_sub_ps(_mm_mul_ps(p2, b2Inv),
_mm_xor_ps(_mm_add_ps(st, st), _mm_set_ss(-0f)));
p2 = _mm_xor_ps(p2, neg);
p1 = _mm_xor_ps(_mm_mul_ps(p1, b2Inv), neg);
return(p1, p2);
}
public static Motor operator /(Motor r, float s)
{
__m128 vs = Detail.rcp_nr1(_mm_set1_ps(s));
var p1 = _mm_mul_ps(r.P1, vs);
var p2 = _mm_mul_ps(r.P2, vs);
return(new Motor(p1, p2));
}
public Point Inverse()
{
__m128 p3 = P3;
__m128 invNorm = Detail.rcp_nr1(_mm_swizzle_ps(p3, 0 /* 0, 0, 0, 0 */));
p3 = _mm_mul_ps(invNorm, p3);
p3 = _mm_mul_ps(invNorm, p3);
return(new Point(p3));
}
public static Branch Log(Rotor r)
{
float cos_ang = _mm_store_ss(r.P1);
float ang = MathF.Acos(cos_ang);
float sin_ang = MathF.Sin(ang);
var p1 = _mm_mul_ps(r.P1, Detail.rcp_nr1(_mm_set1_ps(sin_ang)));
p1 = _mm_mul_ps(p1, _mm_set1_ps(ang));
p1 = Sse41.IsSupported
? _mm_blend_ps(p1, _mm_setzero_ps(), 1)
: _mm_and_ps(p1, _mm_castsi128_ps(_mm_set_epi32(-1, -1, -1, 0)));
return(new Branch(p1));
}
internal static (__m128, __m128) Normalized(__m128 p1, __m128 p2)
{
// l = b + c where b is p1 and c is p2
// l * ~l = |b|^2 - 2(b1 c1 + b2 c2 + b3 c3)e0123
//
// sqrt(l*~l) = |b| - (b1 c1 + b2 c2 + b3 c3)/|b| e0123
//
// 1/sqrt(l*~l) = 1/|b| + (b1 c1 + b2 c2 + b3 c3)/|b|^3 e0123
// = s + t e0123
__m128 b2 = Detail.hi_dp_bc(p1, p1);
__m128 s = Detail.rsqrt_nr1(b2);
__m128 bc = Detail.hi_dp_bc(p1, p2);
__m128 t = _mm_mul_ps(_mm_mul_ps(bc, Detail.rcp_nr1(b2)), s);
// p1 * (s + t e0123) = s * p1 - t P1perp
__m128 tmp = _mm_mul_ps(p2, s);
p2 = _mm_sub_ps(tmp, _mm_mul_ps(p1, t));
p1 = _mm_mul_ps(p1, s);
return(p1, p2);
}
internal static (__m128, __m128) Normalized(__m128 p1, __m128 p2)
{
// m = b + c where b is p1 and c is p2
//
// m * ~m = |b|^2 + 2(b0 c0 - b1 c1 - b2 c2 - b3 c3)e0123
//
// The square root is given as:
// |b| + (b0 c0 - b1 c1 - b2 c2 - b3 c3)/|b| e0123
//
// The inverse of this is given by:
// 1/|b| + (-b0 c0 + b1 c1 + b2 c2 + b3 c3)/|b|^3 e0123 = s + t e0123
//
// Multiplying our original Motor by this inverse will give us a
// normalized Motor.
__m128 b2 = Detail.dp_bc(p1, p1);
__m128 s = Detail.rsqrt_nr1(b2);
__m128 bc = Detail.dp_bc(_mm_xor_ps(p1, _mm_set_ss(-0f)), p2);
__m128 t = _mm_mul_ps(_mm_mul_ps(bc, Detail.rcp_nr1(b2)), s);
// (s + t e0123) * Motor =
//
// s b0 +
// s b1 e23 +
// s b2 e31 +
// s b3 e12 +
// (s c0 + t b0) e0123 +
// (s c1 - t b1) e01 +
// (s c2 - t b2) e02 +
// (s c3 - t b3) e03
__m128 tmp = _mm_mul_ps(p2, s);
p2 = _mm_sub_ps(tmp, _mm_xor_ps(_mm_mul_ps(p1, t), _mm_set_ss(-0f)));
p1 = _mm_mul_ps(p1, s);
return(p1, p2);
}
public static Plane operator /(Plane p, float s)
{
return(new Plane(_mm_mul_ps(p.P0, Detail.rcp_nr1(_mm_set1_ps(s)))));
}
public static Direction operator /(Direction a, float s)
{
return(new Direction(_mm_mul_ps(a.P3, Detail.rcp_nr1(_mm_set1_ps(s)))));
}
public static Point operator /(Point p, float s)
{
return(new Point(_mm_mul_ps(p.P3, Detail.rcp_nr1(_mm_set1_ps(s)))));
}
public Point Normalized()
{
__m128 tmp = Detail.rcp_nr1(_mm_swizzle_ps(P3, 0 /* 0, 0, 0, 0 */));
return(new Point(_mm_mul_ps(P3, tmp)));
}
public static IdealLine operator /(IdealLine a, float s)
{
return(new IdealLine(_mm_mul_ps(a.P2, Detail.rcp_nr1(_mm_set1_ps(s)))));
}
public static Branch operator /(Branch b, float s)
{
return(new Branch(_mm_mul_ps(b.P1, Detail.rcp_nr1(_mm_set1_ps(s)))));
}