Add a little hack for visualizing KLEE branching.

 - This consumes the treestream files produced with --write-paths or
   --write-sym-paths, and renders out the tree in a very ad-hoc funky way.
   Your mileage may vary! :)

Example image: http://klee.llvm.org/data/treegraph_example.jpg
Example movie: http://klee.llvm.org/data/treegraph_example.avi

git-svn-id: https://llvm.org/svn/llvm-project/klee/trunk@102869 91177308-0d34-0410-b5e6-96231b3b80d8

9 files changed, 559 insertions, 0 deletions

diff --git a/utils/hacks/TreeGraphs/Graphics/Geometry/Intersect2D.py b/utils/hacks/TreeGraphs/Graphics/Geometry/Intersect2D.py
new file mode 100644
index 00000000..a600a64f
--- /dev/null
+++ b/utils/hacks/TreeGraphs/Graphics/Geometry/Intersect2D.py
@@ -0,0 +1,59 @@
+import vec2, math

+

+def intersectLineCircle((p, no), (C, r)):

+	x = vec2.sub(p,C)

+	b = 2.*vec2.dot(no, x)

+	c = vec2.sqr(x) - r*r

+	dist = b*b - 4*c

+	

+	if dist<0:

+		return None

+	else:

+		d = math.sqrt(dist)

+		t0 = (-b - d)*.5

+		t1 = (-b + d)*.5

+		return (t0,t1)

+	

+#def intersectLineCircle((lineP,lineN),(circleP,circleR)):

+#	dx,dy = vec2.sub(lineP,circleP)

+#	nx,ny = lineN

+#

+#	a =   (nx*nx + ny*ny)

+#	b = 2*(dx*nx + dy*ny)

+#	c =   (dx*dx + dy*dy) - circleR*circleR

+#	

+#	k = b*b - 4*a*c

+#	if k<0:

+#		return None

+#	else:

+#		d = math.sqrt(k)

+#		t1 = (-b - d)/2*a

+#		t0 = (-b + d)/2*a

+#		return t0,t1

+	

+def intersectCircleCircle(c0P, c0R, c1P, c1R):

+	v = vec2.sub(c1P, c0P)

+	d = vec2.length(v)

+	

+	R = c0R

+	r = c1R

+

+	try:	

+		x = (d*d - r*r + R*R)/(2*d)

+	except ZeroDivisionError:

+		if R<r:

+			return 'inside',()

+		elif r>R:

+			return 'outside',()

+		else:

+			return 'coincident',()

+	

+	k = R*R - x*x

+	if k<0:

+		if x<0:

+			return 'inside',()

+		else:

+			return 'outside',()

+	else:

+		y = math.sqrt(k)

+		return 'intersect',(vec2.toangle(v),vec2.toangle((x,y)))

diff --git a/utils/hacks/TreeGraphs/Graphics/Geometry/__init__.py b/utils/hacks/TreeGraphs/Graphics/Geometry/__init__.py
new file mode 100644
index 00000000..e69de29b
--- /dev/null
+++ b/utils/hacks/TreeGraphs/Graphics/Geometry/__init__.py
diff --git a/utils/hacks/TreeGraphs/Graphics/Geometry/mat2.py b/utils/hacks/TreeGraphs/Graphics/Geometry/mat2.py
new file mode 100644
index 00000000..05320d5e
--- /dev/null
+++ b/utils/hacks/TreeGraphs/Graphics/Geometry/mat2.py
@@ -0,0 +1,20 @@
+import vec2

+

+def det(m):

+	((m00,m01),(m10,m11))= m

+	

+	return m00*m11 - m01*m10

+

+def mul(a,b):

+	b_trans= zip(* b)

+	return tuple([transmulvec2(b_trans, a_r) for a_r in a])

+

+	# multiple vector v by a transposed matrix

+def transmulvec2(m_trans,v):

+	return tuple([vec2.dot(v, m_c) for m_c in m_trans])

+

+def mulvec2(m,v):

+	return transmulvec2(zip(* m), v)

+

+def mulN(m,N):

+	return tuple([vec2.mulN(v,N) for v in m])

diff --git a/utils/hacks/TreeGraphs/Graphics/Geometry/mat3.py b/utils/hacks/TreeGraphs/Graphics/Geometry/mat3.py
new file mode 100644
index 00000000..b8705a32
--- /dev/null
+++ b/utils/hacks/TreeGraphs/Graphics/Geometry/mat3.py
@@ -0,0 +1,40 @@
+import vec3,mat2

+

+def identity():

+	return ((1.0, 0.0, 0.0),

+			(0.0, 1.0, 0.0),

+			(0.0, 0.0, 1.0))

+

+def fromscale(scale):

+	x,y,z= scale

+	x,y,z= float(x),float(y),float(z)

+	return ((  x, 0.0, 0.0),

+			(0.0,   y, 0.0),

+			(0.0, 0.0,   z))

+def fromscaleN(n):

+	return fromscale((n,n,n))

+

+def mul(a,b):

+	b_trans= zip(* b)

+	return tuple([transmulvec3(b_trans, a_r) for a_r in a])

+

+	# multiple vector v by a transposed matrix

+def transmulvec3(m_trans,v):

+	return tuple([vec3.dot(v, m_c) for m_c in m_trans])

+

+def mulvec3(m,v):

+	return transmulvec3(zip(* m), v)

+

+def mulN(m,N):

+	return tuple([vec3.mulN(v,N) for v in m])

+

+def det(m):

+	((m00,m01,m02),

+	 (m10,m11,m12),

+	 (m20,m21,m22))= m

+

+	a= m00 * mat2.det( ((m11, m12), (m21, m22)) );

+	b= m10 * mat2.det( ((m01, m02), (m21, m22)) );

+	c= m20 * mat2.det( ((m01, m02), (m11, m12)) );

+	

+	return a-b+c

diff --git a/utils/hacks/TreeGraphs/Graphics/Geometry/mat4.py b/utils/hacks/TreeGraphs/Graphics/Geometry/mat4.py
new file mode 100644
index 00000000..8dc7d35c
--- /dev/null
+++ b/utils/hacks/TreeGraphs/Graphics/Geometry/mat4.py
@@ -0,0 +1,153 @@
+import vec4,mat3

+

+def identity():

+	return ((1.0, 0.0, 0.0, 0.0),

+			(0.0, 1.0, 0.0, 0.0),

+			(0.0, 0.0, 1.0, 0.0),

+			(0.0, 0.0, 0.0, 1.0))

+

+def fromtrans(trans):

+	return ((1.0, 0.0, 0.0, 0.0),

+			(0.0, 1.0, 0.0, 0.0),

+			(0.0, 0.0, 1.0, 0.0),

+			(trans[0], trans[1], trans[2], 1.0))

+

+def fromscale(scale):

+	x,y,z= scale

+	x,y,z= float(x),float(y),float(z)

+	return ((  x, 0.0, 0.0, 0.0),

+			(0.0,   y, 0.0, 0.0),

+			(0.0, 0.0,   z, 0.0),

+			(0.0, 0.0, 0.0, 1.0))

+def fromscaleN(n):

+	return fromscale((n,n,n))

+

+def fromortho(left,right,bottom,top,znear,zfar):

+	m0= ( 2.0/(right-left), 0.0, 0.0, 0.0)

+	m1= (0.0,  2.0/(top-bottom), 0.0, 0.0)

+	m2= (0.0, 0.0, -2.0/(zfar-znear), 0.0)

+	m3= (	-((right+left)/(right-left)),

+			-((top+bottom)/(top-bottom)),

+			-((zfar+znear)/(zfar-znear)),

+			1.0)

+	return (m0,m1,m2,m3)

+

+def mulN(m,N):

+	return tuple([vec4.mulN(v,N) for v in m])

+	

+def mul(a,b):

+	b_trans= zip(* b)

+	return tuple([transmulvec4(b_trans, a_r) for a_r in a])

+

+	# multiple vector v by a transposed matrix

+def transmulvec4(m_trans,v):

+	return tuple([vec4.dot(v, m_c) for m_c in m_trans])

+

+def mulvec4(m,v):

+	return transmulvec4(zip(* m), v)

+	

+def trans(m):

+	((m00,m01,m02,m03),	

+	 (m10,m11,m12,m13),	

+	 (m20,m21,m22,m23),	

+	 (m30,m31,m32,m33))= m

+	 

+	return (	(m00,m10,m20,m30),	

+				(m01,m11,m21,m31),	

+				(m02,m12,m22,m32),	

+				(m03,m13,m23,m33))

+

+def det(m):

+	((m00,m01,m02,m03),	

+	 (m10,m11,m12,m13),	

+	 (m20,m21,m22,m23),	

+	 (m30,m31,m32,m33))= m

+	 

+	a= m00 * mat3.det(	((m11, m12, m13), 

+						 (m21, m22, m23), 

+						 (m31, m32, m33)) );

+	b= m10 * mat3.det(	((m01, m02, m03), 

+						 (m21, m22, m23), 

+						 (m31, m32, m33)) );

+	c= m20 * mat3.det(  ((m01, m02, m03), 

+						 (m11, m12, m13), 

+						 (m31, m32, m33)) );

+	d= m30 * mat3.det(	((m01, m02, m03), 

+						 (m11, m12, m13), 

+						 (m21, m22, m23)) );

+	

+	return a-b+c-d;

+

+def adj(m):

+	((m00,m01,m02,m03),	

+	 (m10,m11,m12,m13),	

+	 (m20,m21,m22,m23),	

+	 (m30,m31,m32,m33))= m

+	 

+	t00=  mat3.det( ((m11, m12, m13), 

+					 (m21, m22, m23),

+					 (m31, m32, m33)) )

+	t01= -mat3.det( ((m10, m12, m13), 

+					 (m20, m22, m23),

+					 (m30, m32, m33)) )

+	t02=  mat3.det( ((m10, m11, m13), 

+					 (m20, m21, m23),

+					 (m30, m31, m33)) )

+	t03= -mat3.det( ((m10, m11, m12), 

+					 (m20, m21, m22),

+					 (m30, m31, m32)) )

+

+

+	t10= -mat3.det( ((m01, m02, m03), 

+					 (m21, m22, m23),

+					 (m31, m32, m33)) )

+	t11=  mat3.det( ((m00, m02, m03), 

+					 (m20, m22, m23),

+					 (m30, m32, m33)) )

+	t12= -mat3.det( ((m00, m01, m03), 

+					 (m20, m21, m23),

+					 (m30, m31, m33)) )

+	t13=  mat3.det( ((m00, m01, m02), 

+					 (m20, m21, m22),

+					 (m30, m31, m32)) )

+

+	t20=  mat3.det( ((m01, m02, m03), 

+					 (m11, m12, m13),

+					 (m31, m32, m33)) )

+	t21= -mat3.det( ((m00, m02, m03), 

+					 (m10, m12, m13),

+					 (m30, m32, m33)) )

+	t22=  mat3.det( ((m00, m01, m03), 

+					 (m10, m11, m13),

+					 (m30, m31, m33)) )

+	t23= -mat3.det( ((m00, m01, m02), 

+					 (m10, m11, m12),

+					 (m30, m31, m32)) )

+

+	t30= -mat3.det( ((m01, m02, m03), 

+					 (m11, m12, m13),

+					 (m21, m22, m23)) )

+	t31=  mat3.det( ((m00, m02, m03), 

+					 (m10, m12, m13),

+					 (m20, m22, m23)) )

+	t32= -mat3.det( ((m00, m01, m03), 

+					 (m10, m11, m13),

+					 (m20, m21, m23)) )

+	t33=  mat3.det( ((m00, m01, m02), 

+					 (m10, m11, m12),

+					 (m20, m21, m22)) )

+					

+	return ((t00,t01,t02,t03),

+			(t10,t11,t12,t13),

+			(t20,t21,t22,t23),

+			(t30,t31,t32,t33))

+	

+def inv(m):

+	d= det(m)

+	t= trans(adj(m))

+	v= 1.0/d

+	return tuple([(a*v,b*v,c*v,d*v) for a,b,c,d in t])

+	

+def toGL(m):

+	m0,m1,m2,m3= m

+	return m0+m1+m2+m3
\ No newline at end of file
diff --git a/utils/hacks/TreeGraphs/Graphics/Geometry/quat.py b/utils/hacks/TreeGraphs/Graphics/Geometry/quat.py
new file mode 100644
index 00000000..f7837891
--- /dev/null
+++ b/utils/hacks/TreeGraphs/Graphics/Geometry/quat.py
@@ -0,0 +1,107 @@
+from __future__ import division

+

+import math

+import vec3, vec4

+

+def identity():

+	return (0.0,0.0,0.0,1.0)

+

+def fromaxisangle(axisangle):

+	axis,angle= axisangle

+	ang_2= angle/2.0

+	s_ang= math.sin(ang_2)

+	c_ang= math.cos(ang_2)

+	

+	q= vec3.mulN(axis, s_ang) + (c_ang,)

+	return normalize(q)

+

+def fromnormals(n1,n2):

+	axis,angle= vec3.normalize(vec3.cross(n1, n2)), math.acos(vec3.dot(n1, n2))

+	return fromaxisangle((axis,angle))

+

+	# avoid trigonmetry

+def fromnormals_faster(n1,n2):

+	axis= vec3.normalize(vec3.cross(n1, n2))

+

+	half_n= vec3.normalize(vec3.add(n1, n2))

+	cos_half_angle= vec3.dot(n1, half_n)

+	sin_half_angle= 1.0 - cos_half_angle**2

+	

+	return vec3.mulN(axis, sin_half_angle) + (cos_half_angle,)

+

+def fromvectors(v1,v2):

+	return fromnormals(vec3.normalize(v1), vec3.normalize(v2))

+

+def magnitude(q):

+	return vec4.length(q)

+

+def normalize(q):

+	return vec4.divN(q, magnitude(q))

+

+def conjugate(q):

+	x,y,z,w= q

+	return (-x, -y, -z, w)

+

+def mulvec3(q, v):

+	t= mul(q, v+(0.0,))

+	t= mul(t, conjugate(q))

+	return t[:3]

+

+def mul(a, b):

+	ax,ay,az,aw= a

+	bx,by,bz,bw= b

+

+	x= aw*bx + ax*bw + ay*bz - az*by

+	y= aw*by + ay*bw + az*bx - ax*bz

+	z= aw*bz + az*bw + ax*by - ay*bx 

+	w= aw*bw - ax*bx - ay*by - az*bz

+

+	return (x,y,z,w)

+

+def toaxisangle(q):

+	tw= math.acos(q[3])

+	scale= math.sin(tw)

+	angle= tw*2.0

+

+	try:

+		axis= vec3.divN(q[:3], scale)

+	except ZeroDivisionError:

+		axis= (1.0,0.0,0.0)

+

+	return axis,angle

+

+def tomat3x3(q):

+	x,y,z,w= q

+

+	m0= (	1.0 -	2.0 * ( y*y + z*z ),

+					2.0 * ( x*y - z*w ),

+					2.0 * ( x*z + y*w ))

+	m1=	(			2.0 * ( x*y + z*w ),

+			1.0 -	2.0 * ( x*x + z*z ),

+					2.0 * ( y*z - x*w )) 

+	m2=	(			2.0 * ( x*z - y*w ),

+					2.0 * ( y*z + x*w ),

+			1.0 -	2.0 * ( x*x + y*y ))

+

+	return m0,m1,m2

+

+def tomat4x4(q):

+	m0,m1,m2= tomat3x3(q)

+	return (m0 + (0.0,),

+			m1 + (0.0,),

+			m2 + (0.0,),

+			(0.0, 0.0, 0.0, 1.0))

+

+def slerp(a, b, t):

+	raise NotImplementedError

+

+	cos_omega= vec4.dot(a, b)

+	

+	if (cos_omega<0.0):

+		cos_omega= -cos_omega

+		b= vec4.neg(b)

+

+	imega= math.acos(cos_omega)

+	t= sin(t*omega)/sin(omega)

+

+	return vec4.lerp(a, b, t)

diff --git a/utils/hacks/TreeGraphs/Graphics/Geometry/vec2.py b/utils/hacks/TreeGraphs/Graphics/Geometry/vec2.py
new file mode 100644
index 00000000..80159043
--- /dev/null
+++ b/utils/hacks/TreeGraphs/Graphics/Geometry/vec2.py
@@ -0,0 +1,79 @@
+from __future__ import division

+from math import ceil,floor,sqrt,atan2,pi,cos,sin

+import random

+_abs,_min,_max= abs,min,max

+

+def random(rng=random):

+	return (rng.random()*2-1,rng.random()*2-1)

+

+def getangle(a):

+	x,y= a

+	if y>=0:

+		return atan2(y,x)

+	else:

+		return pi*2 + atan2(y,x)

+toangle = getangle

+

+def topolar(pt):

+	return getangle(pt),length(pt)

+def fromangle(angle,radius=1.):

+	return (cos(angle)*radius, sin(angle)*radius)

+frompolar = fromangle

+

+def rotate((x,y),angle):

+	c_a,s_a = cos(angle),sin(angle)

+	return (c_a*x - s_a*y, s_a*x + c_a*y)

+

+def rotate90((x,y)):

+	return (-y,x)

+	

+def abs(a): return (_abs(a[0]),_abs(a[1]))	

+def inv(a):	return (-a[0], -a[1])

+

+def add(a,b):	return (a[0]+b[0], a[1]+b[1])

+def sub(a,b):	return (a[0]-b[0], a[1]-b[1])

+def mul(a,b):	return (a[0]*b[0], a[1]*b[1])

+def div(a,b):	return (a[0]/b[0], a[1]/b[1])

+def mod(a,b):	return (a[0]%b[0], a[1]%b[1])

+def dot(a,b):	return (a[0]*b[0]+ a[1]*b[1])

+

+def addN(a,n):	return (a[0]+n, a[1]+n)

+def subN(a,n):	return (a[0]-n, a[1]-n)

+def mulN(a,n):	return (a[0]*n, a[1]*n)

+def modN(a,n):	return (a[0]%n, a[1]%n)

+def divN(a,n):	return (a[0]/n, a[1]/n)

+

+def sqr(a):			return dot(a,a)

+def length(a):		return sqrt(sqr(a))

+def avg(a,b):		return mulN(add(a,b),0.5)

+def distance(a,b):	return length(sub(a,b))

+

+def normalize(a):

+	return mulN(a, 1.0/length(a))

+

+def normalizeOrZero(a):

+	try:

+		return mulN(a, 1.0/length(a))

+	except ZeroDivisionError:

+		return (0.0,0.0)

+	

+def min((a0,a1),(b0,b1)):

+	return (_min(a0,b0),_min(a1,b1))

+def max((a0,a1),(b0,b1)):

+	return (_max(a0,b0),_max(a1,b1))

+

+def lerp(a,b,t):

+	return add(mulN(a,1.0-t), mulN(b, t))

+

+def toint(a):

+	return (int(a[0]), int(a[1]))

+def tofloor(a):

+	return (floor(a[0]), floor(a[1]))

+def toceil(a):

+	return (ceil(a[0]), ceil(a[1]))

+

+def sumlist(l):

+	return reduce(add, l)

+def avglist(l):

+	return mulN(sumlist(l), 1.0/len(l))

+

diff --git a/utils/hacks/TreeGraphs/Graphics/Geometry/vec3.py b/utils/hacks/TreeGraphs/Graphics/Geometry/vec3.py
new file mode 100644
index 00000000..ed4d8114
--- /dev/null
+++ b/utils/hacks/TreeGraphs/Graphics/Geometry/vec3.py
@@ -0,0 +1,55 @@
+from __future__ import division

+from math import ceil,floor,sqrt

+from random import random as _random

+_min,_max= min,max

+

+def random():

+	return (_random()*2-1,_random()*2-1,_random()*2-1)

+

+def inv(a):	return (-a[0],-a[1],-a[2])

+

+def add(a,b):	return (a[0]+b[0], a[1]+b[1], a[2]+b[2])

+def sub(a,b):	return (a[0]-b[0], a[1]-b[1], a[2]-b[2])

+def mul(a,b):	return (a[0]*b[0], a[1]*b[1], a[2]*b[2])

+def div(a,b):	return (a[0]/b[0], a[1]/b[1], a[2]/b[2])

+def mod(a,b):	return (a[0]%b[0], a[1]%b[1], a[2]%b[2])

+def dot(a,b):	return (a[0]*b[0]+ a[1]*b[1]+ a[2]*b[2])

+

+def addN(a,n):	return (a[0]+n, a[1]+n, a[2]+n)

+def subN(a,n):	return (a[0]-n, a[1]-n, a[2]-n)

+def mulN(a,n):	return (a[0]*n, a[1]*n, a[2]*n)

+def modN(a,n):	return (a[0]%n, a[1]%n, a[2]%n)

+def divN(a,n):	return (a[0]/n, a[1]/n, a[2]/n)

+

+def sqr(a):			return dot(a,a)

+def length(a):		return sqrt(sqr(a))

+def normalize(a):	return mulN(a, 1.0/length(a))

+def avg(a,b):		return mulN(add(a,b),0.5)

+def distance(a,b):	return length(sub(a,b))

+

+def cross(a, b):

+	return (a[1]*b[2] - a[2]*b[1],

+			a[2]*b[0] - a[0]*b[2],

+			a[0]*b[1] - a[1]*b[0])

+def reflect(a, b):

+	return sub(mulN(b, dot(a, b)*2), a)

+

+def lerp(a,b,t):

+	return add(mulN(a,1.0-t), mulN(b, t))

+

+def min((a0,a1,a2),(b0,b1,b2)):

+	return (_min(a0,b0),_min(a1,b1),_min(a2,b2))

+def max((a0,a1,a2),(b0,b1,b2)):

+	return (_max(a0,b0),_max(a1,b1),_max(a2,b2))

+

+def toint(a):

+	return (int(a[0]), int(a[1]), int(a[2]))

+def tofloor(a):

+	return (floor(a[0]), floor(a[1]), floor(a[2]))

+def toceil(a):

+	return (ceil(a[0]), ceil(a[1]), ceil(a[2]))

+

+def sumlist(l):

+	return reduce(add, l)

+def avglist(l):

+	return mulN(sumlist(l), 1.0/len(l))

diff --git a/utils/hacks/TreeGraphs/Graphics/Geometry/vec4.py b/utils/hacks/TreeGraphs/Graphics/Geometry/vec4.py
new file mode 100644
index 00000000..3a542272
--- /dev/null
+++ b/utils/hacks/TreeGraphs/Graphics/Geometry/vec4.py
@@ -0,0 +1,46 @@
+from __future__ import division

+from math import ceil,floor,sqrt

+import vec3

+_min,_max= min,max

+

+def inv(a):	return (-a[0], -a[1], -a[2], -a[3])

+

+def add(a,b):	return (a[0]+b[0], a[1]+b[1], a[2]+b[2], a[3]+b[3])

+def sub(a,b):	return (a[0]-b[0], a[1]-b[1], a[2]-b[2], a[3]-b[3])

+def mul(a,b):	return (a[0]*b[0], a[1]*b[1], a[2]*b[2], a[3]*b[3])

+def div(a,b):	return (a[0]/b[0], a[1]/b[1], a[2]/b[2], a[3]/b[3])

+def mod(a,b):	return (a[0]%b[0], a[1]%b[1], a[2]%b[2], a[3]%b[3])

+def dot(a,b):	return (a[0]*b[0]+ a[1]*b[1]+ a[2]*b[2]+ a[3]*b[3])

+

+def addN(a,n):	return (a[0]+n, a[1]+n, a[2]+n, a[3]+n)

+def subN(a,n):	return (a[0]-n, a[1]-n, a[2]-n, a[3]-n)

+def mulN(a,n):	return (a[0]*n, a[1]*n, a[2]*n, a[3]*n)

+def modN(a,n):	return (a[0]%n, a[1]%n, a[2]%n, a[3]%n)

+def divN(a,n):	return (a[0]/n, a[1]/n, a[2]/n, a[3]/n)

+

+def sqr(a):		return dot(a,a)

+def length(a):	return sqrt(sqr(a))

+def avg(a,b):	return mulN(add(a,b),0.5)

+def normalize(a):	return mulN(a, 1.0/length(a))

+

+def lerp(a,b,t):

+	return add(mulN(a,1.0-t), mulN(b, t))

+

+def min((a0,a1,a2,a3),(b0,b1,b2,b3)):

+	return (_min(a0,b0),_min(a1,b1),_min(a2,b2),_min(a3,b3))

+def max((a0,a1,a2,a3),(b0,b1,b2,b3)):

+	return (_max(a0,b0),_max(a1,b1),_max(a2,b2),_max(a3,b3))

+

+def toint(a):

+	return (int(a[0]), int(a[1]), int(a[2]), int(a[3]))

+def tofloor(a):

+	return (floor(a[0]), floor(a[1]), floor(a[2]), floor(a[3]))

+def toceil(a):

+	return (ceil(a[0]), ceil(a[1]), ceil(a[2]), ceil(a[3]))

+def tovec3(a):

+	return vec3.divN(a, a[3])

+

+def sumlist(l):

+	return reduce(add, l)

+def avglist(l):

+	return mulN(sumlist(l), 1.0/len(l))