Support midpoint Jet

Provide autodiff support for C++20 std::midpoint if the standard is
active.

Change-Id: I1308a1e514bef4c74f08f655cbf803fa43503ce5

include/ceres/jet.h

diff --git a/include/ceres/jet.h b/include/ceres/jet.h
index f6789d8..6ebcf45 100644
--- a/include/ceres/jet.h
+++ b/include/ceres/jet.h
@@ -418,6 +418,7 @@
 
 #ifdef CERES_HAS_CPP20
 using std::lerp;
+using std::midpoint;
 #endif  // defined(CERES_HAS_CPP20)
 
 // Legacy names from pre-C++11 days.
@@ -867,6 +868,21 @@
   return Jet<T, N>{lerp(a.a, b.a, t.a),
                    (T(1) - t.a) * a.v + t.a * b.v + (b.a - a.a) * t.v};
 }
+
+// Computes the midpoint a + (b - a) / 2.
+//
+// Differentiating midpoint(a, b) with respect to a and b gives:
+//
+//   d/da midpoint(a, b) = 1/2
+//   d/db midpoint(a, b) = 1/2
+//
+// with the dual representation given by
+//
+//   midpoint(a + da, b + db) ~= midpoint(a, b) + (da + db) / 2 .
+template <typename T, int N>
+inline Jet<T, N> midpoint(const Jet<T, N>& a, const Jet<T, N>& b) {
+  return Jet<T, N>{midpoint(a.a, b.a), T(0.5) * (a.v + b.v)};
+}
 #endif  // defined(CERES_HAS_CPP20)
 
 // atan2(b + db, a + da) ~= atan2(b, a) + (- b da + a db) / (a^2 + b^2)

internal/ceres/jet_test.cc

diff --git a/internal/ceres/jet_test.cc b/internal/ceres/jet_test.cc
index 743405b..0a8f3e5 100644
--- a/internal/ceres/jet_test.cc
+++ b/internal/ceres/jet_test.cc
@@ -943,6 +943,30 @@
     VL << "v = " << v;
     ExpectJetsClose(z, v);
   }
+
+  {  // Check that midpoint(x, y) = (x + y) / 2
+    J z = midpoint(x, y);
+    J v = (x + y) / J{2};
+    VL << "z = " << z;
+    VL << "v = " << v;
+    ExpectJetsClose(z, v);
+  }
+
+  {  // Check that midpoint(x, x) = x
+    J z = midpoint(x, x);
+    VL << "z = " << z;
+    ExpectJetsClose(z, x);
+  }
+
+  {  // Check that midpoint(x, x) = x while avoiding overflow
+    J x = MakeJet(std::numeric_limits<double>::min(), 1, 2);
+    J y = MakeJet(std::numeric_limits<double>::max(), 3, 4);
+    J z = midpoint(x, y);
+    J v = x + (y - x) / J{2};
+    VL << "z = " << z;
+    VL << "v = " << v;
+    ExpectJetsClose(z, v);
+  }
 #endif  // defined(CERES_HAS_CPP20)
 
   {