diff --git a/mpmath/functions/bessel.py b/mpmath/functions/bessel.py
index 836bed61..7c69afaa 100644
--- a/mpmath/functions/bessel.py
+++ b/mpmath/functions/bessel.py
@@ -180,11 +180,17 @@ def besselk(ctx, n, z, derivative=0, **kwargs):
 
 @defun_wrapped
 def hankel1(ctx,n,x,**kwargs):
-    return ctx.besselj(n,x,**kwargs) + ctx.j*ctx.bessely(n,x,**kwargs)
+    def terms():
+        return [ctx.besselj(n,x,**kwargs),
+                ctx.j*ctx.bessely(n,x,**kwargs)]
+    return ctx.sum_accurately(terms)
 
 @defun_wrapped
 def hankel2(ctx,n,x,**kwargs):
-    return ctx.besselj(n,x,**kwargs) - ctx.j*ctx.bessely(n,x,**kwargs)
+    def terms():
+        return [ctx.besselj(n,x,**kwargs),
+                -ctx.j*ctx.bessely(n,x,**kwargs)]
+    return ctx.sum_accurately(terms)
 
 @defun_wrapped
 def whitm(ctx,k,m,z,**kwargs):
diff --git a/mpmath/tests/test_functions2.py b/mpmath/tests/test_functions2.py
index 188327fc..610f7ca9 100644
--- a/mpmath/tests/test_functions2.py
+++ b/mpmath/tests/test_functions2.py
@@ -2433,3 +2433,7 @@ def test_issue_634():
 
 def test_issue_908():
     assert mp.besselj(-10+0j, 0+0j) == 0
+
+def test_issue_637():
+    assert hankel1(1, 1 + 30j).ae(-7.25495e-15 - 1.17346e-14j)
+    assert hankel2(1, 1 - 30j).ae(-7.25495e-15 + 1.17346e-14j)