Add Julia fwd, and batch example.

Author: ludgerpaehler

Diff for: julia_fwd_and_batch/fwd_and_batch.jl

@@ -0,0 +1,37 @@
+using Enzyme
+
+# Defining our test function
+function f(x::Array{Float64}, y::Array{Float64})
+    y[1] = x[1] * x[1] + x[2] * x[1]
+    return nothing
+end;
+
+# To use forward-mode we then have to seed the gradient computation with
+x  = [2.0, 2.0];
+dx = [1.0, 0.0];
+y  = [0.0];
+dy = [0.0];
+
+# Of which the second duplicated element then stores the tangent
+Enzyme.autodiff(Forward, f, Duplicated(x, dx), Duplicated(y, dy))
+
+# If we then seek to propagate multiple tangents at the same time to obtain the Hessian in one autodiff call, we then seed
+# the following way
+y = [0.0];
+x = [2.0, 2.0];
+
+vdy = ([0.0],[0.0]);
+vdx = ([1.0, 0.0], [0.0, 1.0]);
+
+bx = [0.0, 0.0];
+by = [1.0];
+vdbx = ([0.0, 0.0], [0.0, 0.0]);
+vdby = ([0.0], [0.0]);
+
+# The AD-call then takes the following form
+Enzyme.autodiff(
+    Forward,
+    (x,y) -> Enzyme.autodiff_deferred(f, x, y),
+    BatchDuplicated(Duplicated(x, bx), Duplicated.(vdx, vdbx)),
+    BatchDuplicated(Duplicated(y, by), Duplicated.(vdy, vdby)),
+);