diff --git a/acb/lambertw.c b/acb/lambertw.c
index 8b44bf31..fa45f469 100644
--- a/acb/lambertw.c
+++ b/acb/lambertw.c
@@ -263,8 +263,19 @@ void acb_lambertw_main(acb_t res, const acb_t z,
     else
         ebits = fmpz_bits(MAG_EXPREF(err));
 
-    acb_get_mag(err, ez1);
-    mbits = fmpz_bits(MAG_EXPREF(err));
+    if (fmpz_is_zero(k) || (fmpz_is_one(k) && arb_is_negative(acb_imagref(z)))
+                        || (fmpz_equal_si(k, -1) && arb_is_nonnegative(acb_imagref(z))))
+    {
+        acb_get_mag(err, ez1);
+        mbits = -MAG_EXP(err);
+        mbits = FLINT_MAX(mbits, 0);
+        mbits = FLINT_MIN(mbits, prec);
+    }
+    else
+    {
+        mbits = 0;
+    }
+
     kbits = fmpz_bits(k);
 
     extraprec = FLINT_MAX(ebits, kbits);
diff --git a/acb/lambertw_check_branch.c b/acb/lambertw_check_branch.c
index 7427964f..d6e546cf 100644
--- a/acb/lambertw_check_branch.c
+++ b/acb/lambertw_check_branch.c
@@ -24,9 +24,25 @@ _acb_lambertw_check_branch(const acb_t w, const fmpz_t k, slong prec)
     arb_init(b);
 
     /* t = x sinc(y), v = -cos(y) */
-    arb_sinc(t, acb_imagref(w), prec);
+    if (arb_is_exact(acb_imagref(w)))
+    {
+        if (arb_is_zero(acb_imagref(w)))
+        {
+            arb_one(t);
+            arb_one(v);
+        }
+        else
+        {
+            arb_sin_cos(t, v, acb_imagref(w), prec);
+            arb_div(t, t, acb_imagref(w), prec);
+        }
+    }
+    else
+    {
+        arb_sinc(t, acb_imagref(w), prec);
+        arb_cos(v, acb_imagref(w), prec);
+    }
     arb_mul(t, t, acb_realref(w), prec);
-    arb_cos(v, acb_imagref(w), prec);
     arb_neg(v, v);
 
     /* u = y / pi, with conjugate relation for k */
diff --git a/arb/lambertw.c b/arb/lambertw.c
index 47c2f79e..7d5a0211 100644
--- a/arb/lambertw.c
+++ b/arb/lambertw.c
@@ -586,7 +586,7 @@ arb_lambertw(arb_t res, const arb_t x, int flags, slong prec)
     else
     {
         slong k, padding, nextstep, maxstep, *steps;
-        double rate;
+        double rate, nearm1;
 
         steps = flint_malloc(sizeof(slong) * FLINT_BITS);
 
@@ -597,6 +597,21 @@ arb_lambertw(arb_t res, const arb_t x, int flags, slong prec)
         rate = 2.0 + 1.0 / (1.0 + 0.01 * ebits);
         padding = 6 * ebits2;
 
+        /* extra padding near -1/e */
+        nearm1 = arf_get_d(arb_midref(w), ARF_RND_DOWN);
+        if (fabs(nearm1 + 1.0) < 0.01)
+        {
+            arf_add_ui(arb_midref(t), arb_midref(w), 1, 30, ARF_RND_DOWN);
+
+            if (arf_is_zero(arb_midref(t)))
+                padding += prec;
+            else
+            {
+                slong ee = -ARF_EXP(arb_midref(t));
+                padding += FLINT_MIN(FLINT_MAX(2 * ee, 0), prec);
+            }
+        }
+
         maxstep = 0;
         steps[0] = wp;