diff --git a/tests/ModelValidationTest.cpp b/tests/ModelValidationTest.cpp
index 7b8e40e4a..1697896d9 100644
--- a/tests/ModelValidationTest.cpp
+++ b/tests/ModelValidationTest.cpp
@@ -1076,8 +1076,10 @@ TEST_CASE("ModelValidation_RigidBodySameOnAnyNumberOfThreads", "[ModelValidation
 }
 
 // The rigid body puts each probe placement on the grid with its own parallel copy of gemmi's gridding
-// (density, solvent mask and their symmetrization). It must give gemmi's grids bit for bit, on any
-// number of threads, for isotropic and anisotropic atoms and for groups with and without centring.
+// (density, solvent mask and their symmetrization). The mask must be gemmi's bit for bit; the density
+// gemmi's but for the last bits of its single-precision walk (a point at a radius' edge can also fall on
+// the other side, adding or losing a term at the cutoff). Both must be the same on any number of threads,
+// for isotropic and anisotropic atoms and for groups with and without centring.
 TEST_CASE("ModelValidation_ParallelGriddingMatchesGemmi", "[ModelValidation]") {
     const char *crysts[] = {
         "CRYST1   40.000   50.000   60.000  90.00  90.00  90.00 P 1           1\n",
@@ -1103,6 +1105,15 @@ TEST_CASE("ModelValidation_ParallelGriddingMatchesGemmi", "[ModelValidation]") {
             return a.data.size() == b.data.size() &&
                    std::memcmp(a.data.data(), b.data.data(), a.data.size() * sizeof(float)) == 0;
         };
+        // gemmi's density to rounding: relative 1e-5, and the few cutoff-sized terms at a radius' edge.
+        const auto close = [](const gemmi::Grid<float> &a, const gemmi::Grid<float> &b, float cutoff) {
+            if (a.data.size() != b.data.size())
+                return false;
+            for (size_t i = 0; i < a.data.size(); i++)
+                if (std::fabs(a.data[i] - b.data[i]) > 1e-5f * std::fabs(b.data[i]) + 4 * cutoff)
+                    return false;
+            return true;
+        };
         for (double d_min : {6.0, 3.5}) {
             gemmi::DensityCalculator<gemmi::IT92<float>, float> gemmi_dc;
             gemmi_dc.d_min = d_min;
@@ -1118,6 +1129,7 @@ TEST_CASE("ModelValidation_ParallelGriddingMatchesGemmi", "[ModelValidation]") {
             gemmi_mask.set_size_from_spacing(gemmi_dc.requested_grid_spacing(), gemmi::GridSizeRounding::Up);
             gemmi::SolventMasker(gemmi::AtomicRadiiSet::Refmac).put_mask_on_grid(gemmi_mask, st.models[0]);
 
+            std::vector<float> one_thread;
             for (size_t nthreads : {1, 4}) {
                 const std::vector<size_t> leaders = OrbitLeaders(gemmi_mask, nthreads);
 
@@ -1128,7 +1140,11 @@ TEST_CASE("ModelValidation_ParallelGriddingMatchesGemmi", "[ModelValidation]") {
                 dc.grid.spacegroup = sg;
                 dc.set_refmac_compatible_blur(st.models[0]);
                 PutModelDensityOnGrid(dc, st.models[0], leaders, nthreads);
-                CHECK(same(dc.grid, gemmi_dc.grid));
+                CHECK(close(dc.grid, gemmi_dc.grid, gemmi_dc.cutoff));
+                if (one_thread.empty())
+                    one_thread = dc.grid.data;
+                else
+                    CHECK(dc.grid.data == one_thread);
 
                 gemmi::Grid<float> mask;
                 mask.unit_cell = st.cell;
@@ -1173,7 +1189,9 @@ TEST_CASE("ModelValidation_GriddingAnAtomWhoseRadiusStartsAtTheCutoff", "[ModelV
     gemmi_dc.put_model_density_on_grid(st.models[0]);
     PutModelDensityOnGrid(dc, st.models[0], {}, 4);
     REQUIRE(dc.grid.data.size() == gemmi_dc.grid.data.size());
-    CHECK(std::memcmp(dc.grid.data.data(), gemmi_dc.grid.data.data(), dc.grid.data.size() * sizeof(float)) == 0);
+    // gemmi's density but for the last bits of the single-precision walk (see above).
+    for (size_t i = 0; i < dc.grid.data.size(); i++)
+        CHECK(std::fabs(dc.grid.data[i] - gemmi_dc.grid.data[i]) <= 1e-5f * std::fabs(gemmi_dc.grid.data[i]) + 4 * dc.cutoff);
     CHECK(std::all_of(dc.grid.data.begin(), dc.grid.data.end(), [](float v) { return std::isfinite(v); }));
 }