diff --git a/include/cmath b/include/cmath
index f5f62ad..56d17bf 100644
--- a/include/cmath
+++ b/include/cmath
@@ -304,11 +304,15 @@
 #include <__config>
 #include <math.h>
 #include <version>
+#include <type_traits>
 
 #if !defined(_LIBCPP_HAS_NO_PRAGMA_SYSTEM_HEADER)
 #pragma GCC system_header
 #endif
 
+_LIBCPP_PUSH_MACROS
+#include <__undef_macros>
+
 _LIBCPP_BEGIN_NAMESPACE_STD
 
 using ::signbit;
@@ -607,6 +611,38 @@
     return isfinite(__lcpp_x);
 }
 
+template <class _IntT, class _FloatT,
+    bool _FloatBigger = (numeric_limits<_FloatT>::digits > numeric_limits<_IntT>::digits),
+    int _Bits = (numeric_limits<_IntT>::digits - numeric_limits<_FloatT>::digits)>
+_LIBCPP_INLINE_VISIBILITY
+_LIBCPP_CONSTEXPR _IntT __max_representable_int_for_float() _NOEXCEPT {
+  static_assert(is_floating_point<_FloatT>::value, "must be a floating point type");
+  static_assert(is_integral<_IntT>::value, "must be an integral type");
+  static_assert(numeric_limits<_FloatT>::radix == 2, "FloatT has incorrect radix");
+  static_assert(is_same<_FloatT, float>::value || is_same<_FloatT, double>::value
+               || is_same<_FloatT,long double>::value, "unsupported floating point type");
+  return _FloatBigger ? numeric_limits<_IntT>::max() :  (numeric_limits<_IntT>::max() >> _Bits << _Bits);
+}
+
+// Convert a floating point number to the specified integral type after
+// clamping to the integral types representable range.
+//
+// The behavior is undefined if `__r` is NaN.
+template <class _IntT, class _RealT>
+_LIBCPP_INLINE_VISIBILITY
+_IntT __clamp_to_integral(_RealT __r) _NOEXCEPT {
+  using _Lim = std::numeric_limits<_IntT>;
+  const _IntT _MaxVal = std::__max_representable_int_for_float<_IntT, _RealT>();
+  if (__r >= ::nextafter(static_cast<_RealT>(_MaxVal), INFINITY)) {
+    return _Lim::max();
+  } else if (__r <= _Lim::lowest()) {
+    return _Lim::min();
+  }
+  return static_cast<_IntT>(__r);
+}
+
 _LIBCPP_END_NAMESPACE_STD
 
+_LIBCPP_POP_MACROS
+
 #endif  // _LIBCPP_CMATH
diff --git a/include/random b/include/random
index 724bd0f..9b29e14 100644
--- a/include/random
+++ b/include/random
@@ -4593,7 +4593,10 @@
 
 template<class _IntType>
 poisson_distribution<_IntType>::param_type::param_type(double __mean)
-    : __mean_(__mean)
+    // According to the standard `inf` is a valid input, but it causes the
+    // distribution to hang, so we replace it with the maximum representable
+    // mean.
+    : __mean_(isinf(__mean) ? numeric_limits<double>::max() : __mean)
 {
     if (__mean_ < 10)
     {
@@ -4611,7 +4614,7 @@
     {
         __s_ = _VSTD::sqrt(__mean_);
         __d_ = 6 * __mean_ * __mean_;
-        __l_ = static_cast<result_type>(__mean_ - 1.1484);
+        __l_ = std::trunc(__mean_ - 1.1484);
         __omega_ = .3989423 / __s_;
         double __b1_ = .4166667E-1 / __mean_;
         double __b2_ = .3 * __b1_ * __b1_;
@@ -4628,12 +4631,12 @@
 _IntType
 poisson_distribution<_IntType>::operator()(_URNG& __urng, const param_type& __pr)
 {
-    result_type __x;
+    double __tx;
     uniform_real_distribution<double> __urd;
     if (__pr.__mean_ < 10)
     {
-         __x = 0;
-        for (double __p = __urd(__urng); __p > __pr.__l_; ++__x)
+         __tx = 0;
+        for (double __p = __urd(__urng); __p > __pr.__l_; ++__tx)
             __p *= __urd(__urng);
     }
     else
@@ -4643,19 +4646,19 @@
         double __u;
         if (__g > 0)
         {
-            __x = static_cast<result_type>(__g);
-            if (__x >= __pr.__l_)
-                return __x;
-            __difmuk = __pr.__mean_ - __x;
+            __tx = std::trunc(__g);
+            if (__tx >= __pr.__l_)
+                return std::__clamp_to_integral<result_type>(__tx);
+            __difmuk = __pr.__mean_ - __tx;
             __u = __urd(__urng);
             if (__pr.__d_ * __u >= __difmuk * __difmuk * __difmuk)
-                return __x;
+                return std::__clamp_to_integral<result_type>(__tx);
         }
         exponential_distribution<double> __edist;
         for (bool __using_exp_dist = false; true; __using_exp_dist = true)
         {
             double __e;
-            if (__using_exp_dist || __g < 0)
+            if (__using_exp_dist || __g <= 0)
             {
                 double __t;
                 do
@@ -4665,31 +4668,31 @@
                     __u += __u - 1;
                     __t = 1.8 + (__u < 0 ? -__e : __e);
                 } while (__t <= -.6744);
-                __x = __pr.__mean_ + __pr.__s_ * __t;
-                __difmuk = __pr.__mean_ - __x;
+                __tx = std::trunc(__pr.__mean_ + __pr.__s_ * __t);
+                __difmuk = __pr.__mean_ - __tx;
                 __using_exp_dist = true;
             }
             double __px;
             double __py;
-            if (__x < 10)
+            if (__tx < 10 && __tx >= 0)
             {
                 const double __fac[] = {1, 1, 2, 6, 24, 120, 720, 5040,
                                              40320, 362880};
                 __px = -__pr.__mean_;
-                __py = _VSTD::pow(__pr.__mean_, (double)__x) / __fac[__x];
+                __py = _VSTD::pow(__pr.__mean_, (double)__tx) / __fac[static_cast<int>(__tx)];
             }
             else
             {
-                double __del = .8333333E-1 / __x;
+                double __del = .8333333E-1 / __tx;
                 __del -= 4.8 * __del * __del * __del;
-                double __v = __difmuk / __x;
+                double __v = __difmuk / __tx;
                 if (_VSTD::abs(__v) > 0.25)
-                    __px = __x * _VSTD::log(1 + __v) - __difmuk - __del;
+                    __px = __tx * _VSTD::log(1 + __v) - __difmuk - __del;
                 else
-                    __px = __x * __v * __v * (((((((.1250060 * __v + -.1384794) *
+                    __px = __tx * __v * __v * (((((((.1250060 * __v + -.1384794) *
                            __v + .1421878) * __v + -.1661269) * __v + .2000118) *
                            __v + -.2500068) * __v + .3333333) * __v + -.5) - __del;
-                __py = .3989423 / _VSTD::sqrt(__x);
+                __py = .3989423 / _VSTD::sqrt(__tx);
             }
             double __r = (0.5 - __difmuk) / __pr.__s_;
             double __r2 = __r * __r;
@@ -4709,7 +4712,7 @@
             }
         }
     }
-    return __x;
+    return std::__clamp_to_integral<result_type>(__tx);
 }
 
 template <class _CharT, class _Traits, class _IntType>
diff --git a/test/libcxx/numerics/c.math/undef_min_max.pass.cpp b/test/libcxx/numerics/c.math/undef_min_max.pass.cpp
new file mode 100644
index 0000000..ba1696e
--- /dev/null
+++ b/test/libcxx/numerics/c.math/undef_min_max.pass.cpp
@@ -0,0 +1,19 @@
+//
+// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
+// See https://llvm.org/LICENSE.txt for license information.
+// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
+//
+//===----------------------------------------------------------------------===//
+
+#if defined(__GNUC__) || defined(__clang__)
+#pragma GCC diagnostic ignored "-W#warnings"
+#endif
+
+#define min THIS IS A NASTY MACRO!
+#define max THIS IS A NASTY MACRO!
+
+#include <cmath>
+
+#include "test_macros.h"
+
+int main(int, char**) { return 0; }
diff --git a/test/libcxx/numerics/clamp_to_integral.pass.cpp b/test/libcxx/numerics/clamp_to_integral.pass.cpp
new file mode 100644
index 0000000..cb3336f
--- /dev/null
+++ b/test/libcxx/numerics/clamp_to_integral.pass.cpp
@@ -0,0 +1,90 @@
+//===----------------------------------------------------------------------===//
+//
+// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
+// See https://llvm.org/LICENSE.txt for license information.
+// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
+//
+//===----------------------------------------------------------------------===//
+
+// __clamp_to_integral<IntT>(RealT)
+
+// Test the conversion function that truncates floating point types to the
+// closest representable value for the specified integer type, or
+// numeric_limits<IntT>::max()/min() if the value isn't representable.
+
+#include <limits>
+#include <cassert>
+#include <cmath>
+
+template <class IntT>
+void test() {
+  typedef std::numeric_limits<IntT> Lim;
+  const bool MaxIsRepresentable = sizeof(IntT) < 8;
+  const bool IsSigned = std::is_signed<IntT>::value;
+  struct TestCase {
+    double Input;
+    IntT Expect;
+    bool IsRepresentable;
+  } TestCases[] = {
+      {0, 0, true},
+      {1, 1, true},
+      {IsSigned ? static_cast<IntT>(-1) : 0,
+       IsSigned ? static_cast<IntT>(-1) : 0, true},
+      {Lim::lowest(), Lim::lowest(), true},
+      {static_cast<double>(Lim::max()), Lim::max(), MaxIsRepresentable},
+      {static_cast<double>(Lim::max()) + 1, Lim::max(), false},
+      {static_cast<double>(Lim::max()) + 1024, Lim::max(), false},
+      {nextafter(static_cast<double>(Lim::max()), INFINITY), Lim::max(), false},
+  };
+  for (TestCase TC : TestCases) {
+    auto res = std::__clamp_to_integral<IntT>(TC.Input);
+    assert(res == TC.Expect);
+    if (TC.IsRepresentable) {
+      auto other = static_cast<IntT>(std::trunc(TC.Input));
+      assert(res == other);
+    } else
+      assert(res == Lim::min() || res == Lim::max());
+  }
+}
+
+template <class IntT>
+void test_float() {
+  typedef std::numeric_limits<IntT> Lim;
+  const bool MaxIsRepresentable = sizeof(IntT) < 4;
+  ((void)MaxIsRepresentable);
+  const bool IsSigned = std::is_signed<IntT>::value;
+  struct TestCase {
+    float Input;
+    IntT Expect;
+    bool IsRepresentable;
+  } TestCases[] = {
+      {0, 0, true},
+      {1, 1, true},
+      {IsSigned ? static_cast<IntT>(-1) : 0,
+       IsSigned ? static_cast<IntT>(-1) : 0, true},
+      {Lim::lowest(), Lim::lowest(), true},
+      {static_cast<float>(Lim::max()), Lim::max(), MaxIsRepresentable },
+       {nextafter(static_cast<float>(Lim::max()), INFINITY), Lim::max(), false},
+  };
+  for (TestCase TC : TestCases) {
+    auto res = std::__clamp_to_integral<IntT>(TC.Input);
+    assert(res == TC.Expect);
+    if (TC.IsRepresentable) {
+      auto other = static_cast<IntT>(std::trunc(TC.Input));
+      assert(res == other);
+    } else
+      assert(res == Lim::min() || res == Lim::max());
+  }
+}
+
+int main() {
+  test<short>();
+  test<unsigned short>();
+  test<int>();
+  test<unsigned>();
+  test<long long>();
+  test<unsigned long long>();
+  test_float<short>();
+  test_float<int>();
+  test_float<long long>();
+}
diff --git a/test/std/numerics/rand/rand.dis/rand.dist.bern/rand.dist.bern.geo/eval.pass.cpp b/test/std/numerics/rand/rand.dis/rand.dist.bern/rand.dist.bern.geo/eval.pass.cpp
index 6e6072a..3c726a1 100644
--- a/test/std/numerics/rand/rand.dis/rand.dist.bern/rand.dist.bern.geo/eval.pass.cpp
+++ b/test/std/numerics/rand/rand.dis/rand.dist.bern/rand.dist.bern.geo/eval.pass.cpp
@@ -29,6 +29,20 @@
     return x * x;
 }
 
+struct Eng : std::mt19937 {
+  using Base = std::mt19937;
+  using Base::Base;
+};
+
+void test_small_inputs() {
+  Eng engine;
+  std::geometric_distribution<std::int16_t> distribution(5.45361e-311);
+  for (auto i=0; i < 1000; ++i) {
+		volatile auto res = distribution(engine);
+      ((void)res);
+  }
+}
+
 void
 test1()
 {
@@ -295,4 +309,5 @@
     test4();
     test5();
     test6();
+    test_small_inputs();
 }
diff --git a/test/std/numerics/rand/rand.dis/rand.dist.pois/rand.dist.pois.poisson/eval.pass.cpp b/test/std/numerics/rand/rand.dis/rand.dist.pois/rand.dist.pois.poisson/eval.pass.cpp
index 12fcfa3..e7127d7 100644
--- a/test/std/numerics/rand/rand.dis/rand.dist.pois/rand.dist.pois.poisson/eval.pass.cpp
+++ b/test/std/numerics/rand/rand.dis/rand.dist.pois/rand.dist.pois.poisson/eval.pass.cpp
@@ -29,6 +29,68 @@
     return x * x;
 }
 
+void test_bad_ranges() {
+  // Test cases where the mean is around the largest representable integer for
+  // `result_type`. These cases don't generate valid poisson distributions, but
+  // at least they don't blow up.
+  std::mt19937 eng;
+  
+  {
+    std::poisson_distribution<std::int16_t> distribution(32710.9);
+    for (int i=0; i < 1000; ++i) {
+      volatile std::int16_t res = distribution(eng);
+      ((void)res);
+    }
+  }
+  {
+    std::poisson_distribution<std::int16_t> distribution(std::numeric_limits<std::int16_t>::max());
+    for (int i=0; i < 1000; ++i) {
+      volatile std::int16_t res = distribution(eng);
+      ((void)res);
+    }
+  }
+  {
+    std::poisson_distribution<std::int16_t> distribution(
+    static_cast<double>(std::numeric_limits<std::int16_t>::max()) + 10);
+    for (int i=0; i < 1000; ++i) {
+      volatile std::int16_t res = distribution(eng);
+      ((void)res);
+    }
+  }
+  {
+    std::poisson_distribution<std::int16_t> distribution(
+      static_cast<double>(std::numeric_limits<std::int16_t>::max()) * 2);
+      for (int i=0; i < 1000; ++i) {
+        volatile std::int16_t res = distribution(eng);
+        ((void)res);
+      }
+  }
+  {
+    // We convert `INF` to `DBL_MAX` otherwise the distribution will hang.
+    std::poisson_distribution<std::int16_t> distribution(std::numeric_limits<double>::infinity());
+    for (int i=0; i < 1000; ++i) {
+      volatile std::int16_t res = distribution(eng);
+      ((void)res);
+    }
+  }
+  {
+    std::poisson_distribution<std::int16_t> distribution(0);
+    for (int i=0; i < 1000; ++i) {
+      volatile std::int16_t res = distribution(eng);
+      ((void)res);
+    }
+  }
+  {
+    // We convert `INF` to `DBL_MAX` otherwise the distribution will hang.
+    std::poisson_distribution<std::int16_t> distribution(-100);
+    for (int i=0; i < 1000; ++i) {
+      volatile std::int16_t res = distribution(eng);
+      ((void)res);
+    }
+  }
+}
+
+
 int main()
 {
     {
@@ -148,4 +210,6 @@
         assert(std::abs((skew - x_skew) / x_skew) < 0.01);
         assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.01);
     }
+
+  test_bad_ranges();
 }