diff --git a/gcc/match.pd b/gcc/match.pd
index de54b17abba..9b4819e5be7 100644
--- a/gcc/match.pd
+++ b/gcc/match.pd
@@ -4902,6 +4902,26 @@ DEFINE_INT_AND_FLOAT_ROUND_FN (RINT)
  )
 )
 
+/* (a > 1) ? 0 : (cast)a is the same as (cast)(a == 1)
+   for unsigned types. */
+(simplify
+ (cond (gt @0 integer_onep@1) integer_zerop (convert? @2))
+ (if (TYPE_UNSIGNED (TREE_TYPE (@0))
+      && bitwise_equal_p (@0, @2))
+  (convert (eq @0 @1))
+ )
+)
+
+/* (a <= 1) & (cast)a is the same as (cast)(a == 1)
+   for unsigned types. */
+(simplify
+ (bit_and:c (convert1? (le @0 integer_onep@1)) (convert2? @2))
+ (if (TYPE_UNSIGNED (TREE_TYPE (@0))
+      && bitwise_equal_p (@0, @2))
+  (convert (eq @0 @1))
+ )
+)
+
 (simplify
  (cond @0 zero_one_valued_p@1 zero_one_valued_p@2)
  (switch
diff --git a/gcc/testsuite/c-c++-common/Wrestrict.c b/gcc/testsuite/c-c++-common/Wrestrict.c
index 9eb02bdbfcb..4d005a618b3 100644
--- a/gcc/testsuite/c-c++-common/Wrestrict.c
+++ b/gcc/testsuite/c-c++-common/Wrestrict.c
@@ -681,7 +681,7 @@ void test_strcpy_range (void)
   ptrdiff_t r;
 
   r = SR (0, 1);
-  T (8, "0", a + r, a);   /* { dg-warning "accessing between 1 and 2 bytes at offsets \\\[0, 1] and 0 overlaps up to 2 bytes at offset \\\[0, 1]" "strcpy" { xfail *-*-*} } */
+  T (8, "0", a + r, a);   /* { dg-warning "accessing 2 bytes at offsets \\\[0, 1] and 0 overlaps between 1 and 2 bytes at offset \\\[0, 1]" "strcpy" } */
 
   r = SR (2, 5);
   T (8, "01",  a + r, a);            /* { dg-warning "accessing 3 bytes at offsets \\\[2, 5] and 0 may overlap 1 byte at offset 2" } */
@@ -860,10 +860,11 @@ void test_strncpy_range (char *d, size_t n)
 
   i = SR (0, 1);
   T ("0123", a, a + i, 0);
-  T ("0123", a, a + i, 1);
-  /* Offset in the range [0, i] is represented as a PHI (&a, &a + i)
-     that the implementation isn't equipped to handle yet.  */
-  T ("0123", a, a + i, 2);   /* { dg-warning "accessing 2 bytes at offsets 0 and \\\[0, 1] may overlap 1 byte at offset 1" "strncpy" { xfail *-*-* } } */
+  T ("0123", a, a + i, 1); /* { dg-warning "accessing 1 byte at offsets 0 and \\\[0, 1] may overlap 1 byte at offset 0" } */
+  /* When i == 1 the following overlaps at least 1 byte: the nul at a[1]
+     (if a + 1 is the empty string).  If a + 1 is not empty then it overlaps
+     it plus as many non-nul characters after it, up to the total of 2.  */
+  T ("0123", a, a + i, 2);   /* { dg-warning "accessing 2 bytes at offsets 0 and \\\[0, 1] overlaps between 1 and 2 bytes at offset \\\[0, 1]" "strncpy" } */
 
   i = SR (1, 5);
   T ("0123", a, a + i, 0);
diff --git a/gcc/testsuite/gcc.dg/tree-ssa/builtin-sprintf-warn-23.c b/gcc/testsuite/gcc.dg/tree-ssa/builtin-sprintf-warn-23.c
index 112b08afc44..051c58892e6 100644
--- a/gcc/testsuite/gcc.dg/tree-ssa/builtin-sprintf-warn-23.c
+++ b/gcc/testsuite/gcc.dg/tree-ssa/builtin-sprintf-warn-23.c
@@ -719,5 +719,5 @@ void test_overlap_with_precision (char *d, int i, int j)
   T (d, "%.*s", i, d + 0);    /* { dg-warning "may overlap" } */
   T (d, "%.*s", i, d + 1);    /* { dg-warning "may overlap" } */
   T (d, "%.*s", i, d + 2);
-  T (d, "%.*s", i, d + i);    /* { dg-warning "may overlap" "" { xfail *-*-* } } */
+  T (d, "%.*s", i, d + i);    /* { dg-warning "may overlap" } */
 }
diff --git a/gcc/testsuite/gcc.dg/tree-ssa/cmpeq-1.c b/gcc/testsuite/gcc.dg/tree-ssa/cmpeq-1.c
new file mode 100644
index 00000000000..1d5b960b663
--- /dev/null
+++ b/gcc/testsuite/gcc.dg/tree-ssa/cmpeq-1.c
@@ -0,0 +1,36 @@
+/* { dg-do compile } */
+/* { dg-options "-O2 -fdump-tree-optimized-raw" } */
+/* PR tree-optimization/109959 */
+
+unsigned fu(unsigned a)
+{
+        _Bool t = a <= 1;
+        return t & a;
+}
+
+_Bool fb(unsigned a)
+{
+        _Bool t = a <= 1;
+        return t & a;
+}
+
+_Bool fb1(unsigned a)
+{
+        _Bool t = a <= 1;
+        _Bool t1 = a;
+        return t & t1;
+}
+
+signed fui(unsigned a)
+{
+        _Bool t = a <= 1;
+        int ai = a;
+        return t & ai;
+}
+
+/* These all should be optimized to `a == 1` */
+/* { dg-final { scan-tree-dump-times "eq_expr," 4 "optimized"} } */
+/* { dg-final { scan-tree-dump-not "le_expr," "optimized"} } */
+/* { dg-final { scan-tree-dump-not "bit_and_expr," "optimized"} } */
+
+
diff --git a/gcc/testsuite/gcc.dg/tree-ssa/cmpeq-2.c b/gcc/testsuite/gcc.dg/tree-ssa/cmpeq-2.c
new file mode 100644
index 00000000000..c727b9e7143
--- /dev/null
+++ b/gcc/testsuite/gcc.dg/tree-ssa/cmpeq-2.c
@@ -0,0 +1,32 @@
+/* { dg-do compile } */
+/* { dg-options "-O2 -fdump-tree-optimized-raw" } */
+/* PR tree-optimization/109959 */
+
+unsigned f(unsigned a)
+{
+  if (a <= 1)
+    return a;
+  return 0;
+}
+
+unsigned f0(unsigned a)
+{
+  if (a > 1)
+    return 0;
+  return a;
+}
+
+_Bool fb(unsigned a)
+{
+  if (a > 1)
+    return 0;
+  return a == 1;
+}
+
+/* These all should be optimized to `a == 1` */
+/* { dg-final { scan-tree-dump-times "eq_expr," 3 "optimized"} } */
+/* { dg-final { scan-tree-dump-not "le_expr," "optimized"} } */
+/* { dg-final { scan-tree-dump-not "bit_and," "optimized"} } */
+/* { dg-final { scan-tree-dump-not "gimple_phi " "optimized"} } */
+
+
diff --git a/gcc/testsuite/gcc.dg/tree-ssa/cmpeq-3.c b/gcc/testsuite/gcc.dg/tree-ssa/cmpeq-3.c
new file mode 100644
index 00000000000..f2546b8813f
--- /dev/null
+++ b/gcc/testsuite/gcc.dg/tree-ssa/cmpeq-3.c
@@ -0,0 +1,22 @@
+/* { dg-do compile } */
+/* { dg-options "-O2 -fdump-tree-optimized-raw" } */
+/* PR tree-optimization/109959 */
+
+
+_Bool f2(unsigned a, int t)
+{
+  void g(void);
+  if (t)
+    return 0;
+  g();
+  if (a > 1)
+    return 0;
+  return a == 1;
+}
+
+/* These all should be optimized to `a == 1` */
+/* { dg-final { scan-tree-dump-times "eq_expr," 1 "optimized"} } */
+/* { dg-final { scan-tree-dump-not "le_expr," "optimized"} } */
+/* { dg-final { scan-tree-dump-not "bit_and_expr," "optimized"} } */
+
+
