diff --git a/Modules/clinic/cmathmodule.c.h b/Modules/clinic/cmathmodule.c.h
index 7f9e65baf120ea..774505f2fe0954 100644
--- a/Modules/clinic/cmathmodule.c.h
+++ b/Modules/clinic/cmathmodule.c.h
@@ -744,7 +744,7 @@ PyDoc_STRVAR(cmath_rect__doc__,
 #define CMATH_RECT_METHODDEF    \
     {"rect", _PyCFunction_CAST(cmath_rect), METH_FASTCALL, cmath_rect__doc__},
 
-static PyObject *
+static Py_complex
 cmath_rect_impl(PyObject *module, double r, double phi);
 
 static PyObject *
@@ -753,6 +753,7 @@ cmath_rect(PyObject *module, PyObject *const *args, Py_ssize_t nargs)
     PyObject *return_value = NULL;
     double r;
     double phi;
+    Py_complex _return_value;
 
     if (!_PyArg_CheckPositional("rect", nargs, 2, 2)) {
         goto exit;
@@ -777,7 +778,18 @@ cmath_rect(PyObject *module, PyObject *const *args, Py_ssize_t nargs)
             goto exit;
         }
     }
-    return_value = cmath_rect_impl(module, r, phi);
+    _return_value = cmath_rect_impl(module, r, phi);
+    if (errno == EDOM) {
+        PyErr_SetString(PyExc_ValueError, "math domain error");
+        goto exit;
+    }
+    else if (errno == ERANGE) {
+        PyErr_SetString(PyExc_OverflowError, "math range error");
+        goto exit;
+    }
+    else {
+        return_value = PyComplex_FromCComplex(_return_value);
+    }
 
 exit:
     return return_value;
@@ -985,4 +997,4 @@ cmath_isclose(PyObject *module, PyObject *const *args, Py_ssize_t nargs, PyObjec
 exit:
     return return_value;
 }
-/*[clinic end generated code: output=631db17fb1c79d66 input=a9049054013a1b77]*/
+/*[clinic end generated code: output=7b68f197b24bcb78 input=a9049054013a1b77]*/
diff --git a/Modules/cmathmodule.c b/Modules/cmathmodule.c
index 81cbf0d554de3c..c49ae7dd060f84 100644
--- a/Modules/cmathmodule.c
+++ b/Modules/cmathmodule.c
@@ -438,6 +438,9 @@ cmath_cosh_impl(PyObject *module, Py_complex z)
         x_minus_one = z.real - copysign(1., z.real);
         r.real = cos(z.imag) * cosh(x_minus_one) * Py_MATH_E;
         r.imag = sin(z.imag) * sinh(x_minus_one) * Py_MATH_E;
+        if (isnan(r.imag)) {
+            r.imag = copysign(0., z.real*z.imag);
+        }
     } else {
         r.real = cos(z.imag) * cosh(z.real);
         r.imag = sin(z.imag) * sinh(z.real);
@@ -674,6 +677,9 @@ cmath_sinh_impl(PyObject *module, Py_complex z)
         x_minus_one = z.real - copysign(1., z.real);
         r.real = cos(z.imag) * sinh(x_minus_one) * Py_MATH_E;
         r.imag = sin(z.imag) * cosh(x_minus_one) * Py_MATH_E;
+        if (isnan(r.imag)) {
+            r.imag = copysign(0., z.imag);
+        }
     } else {
         r.real = cos(z.imag) * sinh(z.real);
         r.imag = sin(z.imag) * cosh(z.real);
@@ -972,7 +978,7 @@ cmath_polar_impl(PyObject *module, Py_complex z)
 static Py_complex rect_special_values[7][7];
 
 /*[clinic input]
-cmath.rect
+cmath.rect -> Py_complex_protected
 
     r: double
     phi: double
@@ -981,9 +987,9 @@ cmath.rect
 Convert from polar coordinates to rectangular coordinates.
 [clinic start generated code]*/
 
-static PyObject *
+static Py_complex
 cmath_rect_impl(PyObject *module, double r, double phi)
-/*[clinic end generated code: output=385a0690925df2d5 input=24c5646d147efd69]*/
+/*[clinic end generated code: output=74ff3d17585f3388 input=50e60c5d28c834e6]*/
 {
     Py_complex z;
     errno = 0;
@@ -1027,11 +1033,7 @@ cmath_rect_impl(PyObject *module, double r, double phi)
         z.imag = r * sin(phi);
         errno = 0;
     }
-
-    if (errno != 0)
-        return math_error();
-    else
-        return PyComplex_FromCComplex(z);
+    return z;
 }
 
 /*[clinic input]