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@Fraktality
Created March 9, 2017 02:33
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local ln = math.log
local abs = math.abs
local sin = math.sin
local cos = math.cos
local exp = math.exp
local acos = math.acos
local sqrt = math.sqrt
local function MulLn(w0, x0, y0, z0, w1, x1, y1, z1)
local w, x, y, z =
w0*w1 - x0*x1 - y0*y1 - z0*z1,
x0*w1 + w0*x1 - z0*y1 + y0*z1,
y0*w1 + z0*x1 + w0*y1 - x0*z1,
z0*w1 - y0*x1 + x0*y1 + w0*z1
local normv = x*x + y*y + z*z
local absq = sqrt(w*w + normv)
local t = normv < 1e-15 and 0 or acos(abs(w)/absq)/sqrt(normv)
return ln(absq), x*t, y*t, z*t
end
local function CMap(w0, x0, y0, z0, w1, x1, y1, z1, w2, x2, y2, z2)
local norm1 = w1*w1 + x1*x1 + y1*y1 + z1*z1
local iqw, iqx, iqy, iqz = w1/norm1, x1/-norm1, y1/-norm1, z1/-norm1
local bw0, bx0, by0, bz0 = MulLn(w2, x2, y2, z2, iqw, iqx, iqy, iqz)
local bw1, bx1, by1, bz1 = MulLn(w0, x0, y0, z0, iqw, iqx, iqy, iqz)
local mx, my, mz =
(bx0 + bx1)/-4,
(by0 + by1)/-4,
(bz0 + bz1)/-4
local normv = sqrt(mx*mx + my*my + mz*mz)
local u = exp((bw0 + bw1)/-4)
local v = normv > 1e-15 and u*sin(normv)/normv or 0
local ew, ex, ey, ez = u*cos(normv), v*mx, v*my, v*mz
return
ew*w1 - ex*x1 - ey*y1 - ez*z1,
ex*w1 + ew*x1 - ez*y1 + ey*z1,
ey*w1 + ez*x1 + ew*y1 - ex*z1,
ez*w1 - ey*x1 + ex*y1 + ew*z1
end
local function NSlerpParams(p)
return
(1.0904 + p*(p*(3.55645 - p*1.43519) - 3.2452)),
0.848013 + p*(p*0.215638 - 1.06021)
end
local function Squad(q0, q1, q2, q3)
local q1w, q1x, q1y, q1z = q1[1], q1[2], q1[3], q1[4]
local q2w, q2x, q2y, q2z = q2[1], q2[2], q2[3], q2[4]
local p0w, p0x, p0y, p0z = CMap(
q0[1], q0[2], q0[3], q0[4],
q1w, q1x, q1y, q1z,
q2w, q2x, q2y, q2z
)
local p1w, p1x, p1y, p1z = CMap(
q1w, q1x, q1y, q1z,
q2w, q2x, q2y, q2z,
q3[1], q3[2], q3[3], q3[4]
)
local d0, d1 =
q1w*q2w + q1x*q2x + q1y*q2y + q1z*q2z,
p0w*p1w + p0x*p1x + p0y*p1y + p0z*p1z
local u0, v0 = NSlerpParams(abs(d0))
local u1, v1 = NSlerpParams(abs(d1))
d0, d1 = d0 < 0, d1 < 0
return function(t)
local awx, awm =
t*(t - 0.5)*(t - 1),
(t - 0.5)*(t - 0.5)
local o = t + awx*(u0*awm + v0)
local t0, t1 = 1 - o, d0 and -o or o
local w, x, y, z =
q1w*t0 + q2w*t1,
q1x*t0 + q2x*t1,
q1y*t0 + q2y*t1,
q1z*t0 + q2z*t1
local n = sqrt(w*w + x*x + y*y + z*z)
local c0w, c0x, c0y, c0z = w/n, x/n, y/n, z/n
o = t + awx*(u1*awm + v1)
t0, t1 = 1 - o, d1 and -o or o
w, x, y, z =
p0w*t0 + p1w*t1,
p0x*t0 + p1x*t1,
p0y*t0 + p1y*t1,
p0z*t0 + p1z*t1
n = sqrt(w*w + x*x + y*y + z*z)
local c1w, c1x, c1y, c1z = w/n, x/n, y/n, z/n
local d2 = c0w*c1w + c0x*c1x + c0y*c1y + c0z*c1z
local d2a = abs(d2)
local h = 2*t*(1 - t)
o = h + h*(h - 0.5)*(h - 1)*(
(1.0904 + d2a*(d2a*(3.55645 - d2a*1.43519) - 3.2452))*
(h - 0.5)*(h - 0.5) + 0.848013 + d2a*(d2a*0.215638 - 1.06021)
)
t0, t1 = 1 - o, d2 < 0 and -o or o
w, x, y, z =
c0w*t0 + c1w*t1,
c0x*t0 + c1x*t1,
c0y*t0 + c1y*t1,
c0z*t0 + c1z*t1
n = sqrt(w*w + x*x + y*y + z*z)
return w/n, x/n, y/n, z/n
end
end
@Validark
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It is faster to do (x * z)^0.5 than sqrt(x * z)

@Fraktality
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localized sqrt call is about the same speed as a^b on Lua 5.1.5 and substantially faster than a^b on LuaJIT

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