Recently Facebook people (G. Lample and F. Charton) wrote about
training neural network to integrate based on derivatives of
random functions. And they showed not so favourable figures
for Mathematica and Maple.
I implemented a little generator which should give similar
distribution as used by Lample and Charton. I write
"similar" as they skip some concrete details and overall
description seem unnecesarily complicated. Anyway,
when you save the attached script 'tree2.input' and do:
)read tree2.input
gen_file("foo_ii.input")
it will generate file with 1000 random integration problems.
ATM there is trouble that some problem contain division by 0.
OTOH it was enough to uncover several bugs in FriCAS
integrator.
It seems that currently FriCAS is able to do about 78% of
random problem. Few relatively simple changes to handle
easy special cases should move this up quite a bit. But
there is also trouble with long running time on some
examples...
--
Waldek Hebisch
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-- s_tab(n) counts of all unary-binary trees with n internal nodes
s_tab := new(31, 0)$IndexedOneDimensionalArray(Integer, 0)
-- b_tab(n, j) like above, but having binary top node with j
-- internal nodes to the left
b_tab := new(31, 30, 0)$IndexedTwoDimensionalArray(Integer, 0, 0)
s_tab(0) := 1
s_tab(1) := 2
for i in 2..30 repeat
bs : Integer := 0
for j in 0..(i-1) repeat
b_tab(i, j) := s_tab(j)*s_tab(i - j - 1)
bs := bs + b_tab(i, j)
s_tab(i) := s_tab(i - 1) + bs
-- Probabities of havinig top unary node
u_prob := new(30, 0)$OneDimensionalArray(DoubleFloat)
-- Probabities for top binary node
b_prob := new(30, 30, 0)$IndexedTwoDimensionalArray(DoubleFloat, 1, 0)
for i in 1..30 repeat
den : DoubleFloat := 1/convert(s_tab(i))@DoubleFloat
u_prob(i) := den*convert(s_tab(i - 1))@DoubleFloat
bsf := u_prob(i)
for j in 0..(i-1) repeat
bsf := bsf + den*convert(b_tab(i, j))@DoubleFloat
b_prob(i, j) := bsf
get_j(p, i) ==
for j in 0..(i - 1) repeat
if p < b_prob(i, j) then return j
i - 1
LARGE := 10^16
I_LARGE := 1/convert(LARGE)@DoubleFloat
get_tree(i) == tree(0, get_tl(i))$Tree(Integer)
-- main workhorse: generate list of childeren
-- with uniform probability
get_tl(i : Integer) : List(Tree(Integer)) ==
i = 0 => []
p := I_LARGE*convert(random(LARGE))@DoubleFloat
p < u_prob(i) => [get_tree(i - 1)]
j := get_j(p, i)
[get_tree(j), get_tree(i - j - 1)]
get_fun(i) == get_fun1(get_tree(i))
u_ops := construct(['log, 'exp, 'sqrt, _
'sin, 'cos, 'tan, 'asin, 'acos, 'atan, _
'sinh, 'cosh, 'tanh, 'atanh, 'acosh, 'asinh _
])$OneDimensionalArray(Symbol)
nu_ops := #u_ops
b_ops := construct(['+, '-, '*, '/])$OneDimensionalArray(Symbol)
nb_ops := #b_ops
-- convert tree to InputForm
get_fun1(t) ==
ch_l := children(t)
empty?(ch_l) =>
k := random(10)
k < 5 => convert(k + 1)@InputForm
convert('x)@InputForm
#ch_l = 1 =>
convert([convert(u_ops(random(nu_ops) + 1))@InputForm,
get_fun1(first(ch_l))])@InputForm
convert([convert(b_ops(random(nb_ops) + 1))@InputForm,
get_fun1(first(ch_l)), get_fun1(second(ch_l))])@InputForm
rline(i) == unparse(get_fun(i))
gen_file(s) ==
f := open("rint0.input", "output")$TextFile
for j in 1..1000 repeat
write!(f, "f := ")
writeLine!(f, rline(15))
writeLine!(f, "integrate(D(f, x), x)")
close!(f)
