diff --git a/pkgs/racket-test-core/tests/racket/math.rktl b/pkgs/racket-test-core/tests/racket/math.rktl
index 85e1bf459e..b196ac02f3 100644
--- a/pkgs/racket-test-core/tests/racket/math.rktl
+++ b/pkgs/racket-test-core/tests/racket/math.rktl
@@ -279,10 +279,10 @@
 (test -inf.0 sinh -inf.0)
 (test -inf.0 sinh -max.0)
 (test #t double=? sinh-1 (sinh -1.0))
-(test -0.0 sinh -min.0)
+(test -min.0 sinh -min.0)
 (test -0.0 sinh -0.0)
 (test 0.0 sinh 0.0)
-(test 0.0 sinh +min.0)
+(test +min.0 sinh +min.0)
 (test #t double=? sinh+1 (sinh 1.0))
 (test +inf.0 sinh +max.0)
 (test +inf.0 sinh +inf.0)
diff --git a/racket/collects/racket/math.rkt b/racket/collects/racket/math.rkt
index 50a0919ddc..12bfe5b8cd 100644
--- a/racket/collects/racket/math.rkt
+++ b/racket/collects/racket/math.rkt
@@ -58,6 +58,12 @@
           [(real? z)
            (let loop ([z z])
              (cond [(z . < . 0) (- (loop (- z)))]
+                   [(< z 1e-8) (exact->inexact z)]
+                   [(< z .13)
+                    ;; Taylor expansion near 0 to avoid cancellation
+                    ;~ z+z^3/3!+z^5/5!+...
+                    (define z^2 (* z z))
+                    (+ z (* z z^2 (+ #i1/6 (* z^2 (+ #i1/120 (* z^2 (+ #i1/5040 (* z^2 #i1/362880))))))))]
                    [else        (/ (- (exp z) (exp (- z))) 2)]))]
           [else (/ (- (exp z) (exp (- z))) 2)]))