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);
}
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);
}
internal static __m128 Normalized(__m128 p1)
{
__m128 invNorm = Detail.rsqrt_nr1(Detail.dp_bc(p1, p1));
return(_mm_mul_ps(p1, invNorm));
}
