diff --git a/src/pkg/math/rand/example_test.go b/src/pkg/math/rand/example_test.go
index f429914531..b93a371a04 100644
--- a/src/pkg/math/rand/example_test.go
+++ b/src/pkg/math/rand/example_test.go
@@ -83,8 +83,8 @@ func Example_rand() {
 	// Perm generates a random permutation of the numbers [0, n).
 	show("Perm", r.Perm(5), r.Perm(5), r.Perm(5))
 	// Output:
-	// Float32     0.2635776           0.6358173           0.6718283
-	// Float64     0.628605430454327   0.4504798828572669  0.9562755949377957
+	// Float32     0.73793465          0.38461488          0.9940225
+	// Float64     0.6919607852308565  0.29140004584133117 0.2262092163027547
 	// ExpFloat64  0.3362240648200941  1.4256072328483647  0.24354758816173044
 	// NormFloat64 0.17233959114940064 1.577014951434847   0.04259129641113857
 	// Int31       1501292890          1486668269          182840835
diff --git a/src/pkg/math/rand/rand.go b/src/pkg/math/rand/rand.go
index 04fb67d19d..d3ea840178 100644
--- a/src/pkg/math/rand/rand.go
+++ b/src/pkg/math/rand/rand.go
@@ -95,10 +95,10 @@ func (r *Rand) Intn(n int) int {
 }
 
 // Float64 returns, as a float64, a pseudo-random number in [0.0,1.0).
-func (r *Rand) Float64() float64 { return float64(r.Int63()) / (1 << 63) }
+func (r *Rand) Float64() float64 { return float64(r.Int63n(1<<53)) / (1 << 53) }
 
 // Float32 returns, as a float32, a pseudo-random number in [0.0,1.0).
-func (r *Rand) Float32() float32 { return float32(r.Float64()) }
+func (r *Rand) Float32() float32 { return float32(r.Int31n(1<<24)) / (1 << 24) }
 
 // Perm returns, as a slice of n ints, a pseudo-random permutation of the integers [0,n).
 func (r *Rand) Perm(n int) []int {
diff --git a/src/pkg/math/rand/rand_test.go b/src/pkg/math/rand/rand_test.go
index 4f0a8d0ee9..c174c613f4 100644
--- a/src/pkg/math/rand/rand_test.go
+++ b/src/pkg/math/rand/rand_test.go
@@ -322,6 +322,17 @@ func TestExpTables(t *testing.T) {
 	}
 }
 
+// For issue 6721, the problem came after 7533753 calls, so check 10e6.
+func TestFloat32(t *testing.T) {
+	r := New(NewSource(1))
+	for ct := 0; ct < 10e6; ct++ {
+		f := r.Float32()
+		if f >= 1 {
+			t.Fatal("Float32() should be in range [0,1). ct:", ct, "f:", f)
+		}
+	}
+}
+
 // Benchmarks
 
 func BenchmarkInt63Threadsafe(b *testing.B) {
@@ -358,6 +369,13 @@ func BenchmarkInt31n1000(b *testing.B) {
 	}
 }
 
+func BenchmarkFloat32(b *testing.B) {
+	r := New(NewSource(1))
+	for n := b.N; n > 0; n-- {
+		r.Float32()
+	}
+}
+
 func BenchmarkPerm3(b *testing.B) {
 	r := New(NewSource(1))
 	for n := b.N; n > 0; n-- {