diff --git a/library/init/meta/float.lean b/library/init/meta/float.lean
index 772249a9ab..697985e9f4 100644
--- a/library/init/meta/float.lean
+++ b/library/init/meta/float.lean
@@ -116,13 +116,17 @@ meta constant acosh : float → float
 meta constant atanh : float → float
 
 meta constant abs : float → float
-/-- Nearest integer not less than the given value. -/
+/-- Nearest integer not less than the given value.
+Returns 0 if the input is not finite. -/
 meta constant ceil : float → int
-/-- Nearest integer not greater than the given value. -/
+/-- Nearest integer not greater than the given value.
+Returns 0 if the input is not finite. -/
 meta constant floor : float → int
-/-- Nearest integer not greater in magnitude than the given value. -/
+/-- Nearest integer not greater in magnitude than the given value.
+Returns 0 if the input is not finite. -/
 meta constant trunc : float → int
-/-- Round to the nearest integer, rounding away from zero in halfway cases. -/
+/-- Round to the nearest integer, rounding away from zero in halfway cases.
+Returns 0 if the input is not finite. -/
 meta constant round : float → int
 
 meta constant lt : float → float → bool
diff --git a/src/frontends/lean/parser.cpp b/src/frontends/lean/parser.cpp
index 31b05f55bd..bd342343ea 100644
--- a/src/frontends/lean/parser.cpp
+++ b/src/frontends/lean/parser.cpp
@@ -725,7 +725,7 @@ unsigned parser::get_small_nat() {
         maybe_throw_error({"invalid numeral, value does not fit in a machine integer", pos()});
         return 0;
     }
-    return val.get<unsigned>();
+    return static_cast<unsigned>(val);
 }
 
 pair<ast_id, std::string> parser::parse_string_lit() {
diff --git a/src/library/string.cpp b/src/library/string.cpp
index 2bdb51bf7d..554a182f7b 100644
--- a/src/library/string.cpp
+++ b/src/library/string.cpp
@@ -189,13 +189,13 @@ optional<unsigned> to_char_core(expr const & e) {
     expr const & fn = get_app_args(e, args);
     if (fn == *g_char_mk && args.size() == 2) {
         if (auto n = to_num(args[0])) {
-            return optional<unsigned>(n->get<unsigned>());
+            return optional<unsigned>(static_cast<unsigned>(*n));
         } else {
             return optional<unsigned>();
         }
     } else if (fn == *g_char_of_nat && args.size() == 1) {
         if (auto n = to_num(args[0])) {
-            return optional<unsigned>(n->get<unsigned>());
+            return optional<unsigned>(static_cast<unsigned>(*n));
         } else {
             return optional<unsigned>();
         }
diff --git a/src/library/type_context.cpp b/src/library/type_context.cpp
index 403f63162b..651bedc439 100644
--- a/src/library/type_context.cpp
+++ b/src/library/type_context.cpp
@@ -2962,7 +2962,7 @@ static optional<mpz> eval_num(expr const & e) {
 optional<unsigned> type_context_old::to_small_num(expr const & e) {
     if (optional<mpz> r = eval_num(e)) {
         if (r->is<unsigned>()) {
-            unsigned r1 = r->get<unsigned>();
+            unsigned r1 = static_cast<unsigned>(*r);
             if (r1 <= m_cache->get_nat_offset_cnstr_threshold())
                 return optional<unsigned>(r1);
         }
diff --git a/src/library/vm/vm.cpp b/src/library/vm/vm.cpp
index fff1feb1da..3ff7cc7c05 100644
--- a/src/library/vm/vm.cpp
+++ b/src/library/vm/vm.cpp
@@ -712,7 +712,7 @@ vm_instr mk_constructor_instr(unsigned cidx, unsigned nfields) {
 vm_instr mk_num_instr(mpz const & v) {
     if (v < LEAN_MAX_SMALL_NAT) {
         vm_instr r(opcode::SConstructor);
-        r.m_num = v.get<unsigned>();
+        r.m_num = static_cast<unsigned>(v);
         return r;
     } else {
         vm_instr r(opcode::Num);
diff --git a/src/library/vm/vm_float.cpp b/src/library/vm/vm_float.cpp
index 96c4c9b04b..6b21ce3313 100644
--- a/src/library/vm/vm_float.cpp
+++ b/src/library/vm/vm_float.cpp
@@ -41,10 +41,11 @@ float to_float(vm_obj const & o) {
 }
 
 vm_obj float_of_nat(vm_obj const & a) {
-    // [TODO] check that the nat isn't too big to fit in an unsigned
-    return mk_vm_float(static_cast<float>(to_unsigned(a)));
+    return is_simple(a) ? mk_vm_float(cidx(a)) : mk_vm_float(to_mpz(a).get_double());
+}
+vm_obj float_of_int(vm_obj const & i) {
+    return is_simple(i) ? mk_vm_float(to_int(i)) : mk_vm_float(to_mpz(i).get_double());
 }
-vm_obj float_of_int(vm_obj const & i) { return mk_vm_float(static_cast<float>(to_int(i))); }
 
 vm_obj float_repr(vm_obj const & a) {
     std::ostringstream out;
diff --git a/src/library/vm/vm_int.cpp b/src/library/vm/vm_int.cpp
index 720e0e3335..faf664ba69 100644
--- a/src/library/vm/vm_int.cpp
+++ b/src/library/vm/vm_int.cpp
@@ -5,6 +5,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Author: Leonardo de Moura
 */
 #include <iostream>
+#include <limits>
 #include "library/vm/vm.h"
 #include "library/vm/vm_nat.h"
 
@@ -14,33 +15,42 @@ Author: Leonardo de Moura
 namespace lean {
 // =======================================
 // Builtin int operations
-inline bool is_small_int(int n)         { return LEAN_MIN_SMALL_INT <= n && n < LEAN_MAX_SMALL_INT; }
-inline bool is_small_int(unsigned n)    { return n < LEAN_MAX_SMALL_INT; }
-inline bool is_small_int(long long n)   { return LEAN_MIN_SMALL_INT <= n && n < LEAN_MAX_SMALL_INT; }
-inline bool is_small_int(mpz const & n) { return LEAN_MIN_SMALL_INT <= n && n < LEAN_MAX_SMALL_INT; }
+template<typename T>
+inline typename std::enable_if<std::numeric_limits<T>::is_signed, bool>::type is_small_int(const T& n) {
+    return LEAN_MIN_SMALL_INT <= n && n < LEAN_MAX_SMALL_INT;
+}
+template<typename T>
+inline typename std::enable_if<!std::numeric_limits<T>::is_signed, bool>::type is_small_int(const T& n) {
+    return n < LEAN_MAX_SMALL_INT;
+}
 
-inline unsigned to_unsigned(int n) {
+template<typename T>
+inline unsigned to_unsigned(T n) {
     lean_assert(is_small_int(n));
-    unsigned r = static_cast<unsigned>(n) & 0x7FFFFFFF;
+    // small ints are strictly smaller than `signed`, so this is safe for `T = mpz`
+    signed ns = static_cast<signed>(n);
+    unsigned r = static_cast<unsigned>(ns) & 0x7FFFFFFF;
     lean_assert(r < LEAN_MAX_SMALL_NAT);
     return r;
 }
 
-inline  int of_unsigned(unsigned n) {
+inline int of_unsigned(unsigned n) {
     return static_cast<int>(n << 1) / 2;
 }
 
-vm_obj mk_vm_int(int n) {
-    return is_small_int(n) ? mk_vm_simple(to_unsigned(n)) : mk_vm_mpz(mpz(n));
+template<typename T>
+vm_obj mk_vm_int_impl(T && n) {
+    return is_small_int(n) ? mk_vm_simple(to_unsigned(n)) : mk_vm_mpz(mpz(std::forward<T>(n)));
 }
 
-vm_obj mk_vm_int(unsigned n) {
-    return is_small_int(n) ? mk_vm_simple(to_unsigned(n)) : mk_vm_mpz(mpz(n));
-}
-
-vm_obj mk_vm_int(mpz const & n) {
-    return is_small_int(n) ? mk_vm_simple(to_unsigned(n.get<int>())) : mk_vm_mpz(n);
-}
+vm_obj mk_vm_int(int n) { return mk_vm_int_impl(n); }
+vm_obj mk_vm_int(unsigned int n) { return mk_vm_int_impl(n); }
+vm_obj mk_vm_int(long n) { return mk_vm_int_impl(n); }
+vm_obj mk_vm_int(unsigned long n) { return mk_vm_int_impl(n); }
+vm_obj mk_vm_int(long long n) { return mk_vm_int_impl(n); }
+vm_obj mk_vm_int(unsigned long long n) { return mk_vm_int_impl(n); }
+vm_obj mk_vm_int(double n) { return std::isfinite(n) ? mk_vm_int_impl(n) : mk_vm_int(0); }
+vm_obj mk_vm_int(mpz const & n) { return mk_vm_int_impl(n); }
 
 inline int to_small_int(vm_obj const & o) {
     lean_assert(is_simple(o));
@@ -51,7 +61,7 @@ int to_int(vm_obj const & o) {
     if (is_simple(o))
         return to_small_int(o);
     else
-        return to_mpz(o).get<int>();
+        return static_cast<int>(to_mpz(o));
 }
 
 optional<int> try_to_int(vm_obj const & o) {
@@ -60,7 +70,7 @@ optional<int> try_to_int(vm_obj const & o) {
     } else {
         mpz const & v = to_mpz(o);
         if (v.is<int>())
-            return optional<int>(v.get<int>());
+            return optional<int>(static_cast<int>(v));
         else
             return optional<int>();
     }
@@ -235,7 +245,7 @@ vm_obj int_test_bit(vm_obj const & a1, vm_obj const & a2) {
         mpz const & v1 = to_mpz1(a1);
         mpz const & v2 = to_mpz2(a2);
         if (v2.is<unsigned long int>())
-            return mk_vm_bool(v1.test_bit(v2.get<unsigned long int>()));
+            return mk_vm_bool(v1.test_bit(static_cast<unsigned long int>(v2)));
         else
             return mk_vm_bool(false);
     }
diff --git a/src/library/vm/vm_int.h b/src/library/vm/vm_int.h
index bf9638cb5e..3d8566db8a 100644
--- a/src/library/vm/vm_int.h
+++ b/src/library/vm/vm_int.h
@@ -11,7 +11,16 @@ namespace lean {
 int to_int(vm_obj const & o);
 optional<int> try_to_int(vm_obj const & o);
 int force_to_int(vm_obj const & o, int def);
-vm_obj mk_vm_int(int i);
+
+vm_obj mk_vm_int(int n);
+vm_obj mk_vm_int(unsigned int n);
+vm_obj mk_vm_int(long n);
+vm_obj mk_vm_int(unsigned long n);
+vm_obj mk_vm_int(long long n);
+vm_obj mk_vm_int(unsigned long long n);
+vm_obj mk_vm_int(double n);
+vm_obj mk_vm_int(mpz const & n);
+
 void initialize_vm_int();
 void finalize_vm_int();
 }
diff --git a/src/library/vm/vm_nat.cpp b/src/library/vm/vm_nat.cpp
index 6cea424ce6..32f3799ff8 100644
--- a/src/library/vm/vm_nat.cpp
+++ b/src/library/vm/vm_nat.cpp
@@ -21,7 +21,7 @@ vm_obj mk_vm_nat(unsigned n) {
 
 vm_obj mk_vm_nat(mpz const & n) {
     if (LEAN_LIKELY(n < LEAN_MAX_SMALL_NAT))
-        return mk_vm_simple(n.get<unsigned>());
+        return mk_vm_simple(static_cast<unsigned>(n));
     else
         return mk_vm_mpz(n);
 }
@@ -30,7 +30,7 @@ unsigned to_unsigned(vm_obj const & o) {
     if (LEAN_LIKELY(is_simple(o)))
         return cidx(o);
     else
-        return to_mpz(o).get<unsigned>();
+        return static_cast<unsigned>(to_mpz(o));
 }
 
 optional<unsigned> try_to_unsigned(vm_obj const & o) {
@@ -39,7 +39,7 @@ optional<unsigned> try_to_unsigned(vm_obj const & o) {
     } else {
         mpz const & v = to_mpz(o);
         if (v.is<unsigned>())
-            return optional<unsigned>(v.get<unsigned>());
+            return optional<unsigned>(static_cast<unsigned>(v));
         else
             return optional<unsigned>();
     }
@@ -262,7 +262,7 @@ vm_obj nat_test_bit(vm_obj const & a1, vm_obj const & a2) {
         mpz const & v1 = to_mpz1(a1);
         mpz const & v2 = to_mpz2(a2);
         if (v2.is<unsigned long int>())
-            return mk_vm_bool(v1.test_bit(v2.get<unsigned long int>()));
+            return mk_vm_bool(v1.test_bit(static_cast<unsigned long int>(v2)));
         else
             return mk_vm_bool(false);
     }
diff --git a/src/tests/util/numerics/mpz.cpp b/src/tests/util/numerics/mpz.cpp
index 1ee98fef07..0125c5a990 100644
--- a/src/tests/util/numerics/mpz.cpp
+++ b/src/tests/util/numerics/mpz.cpp
@@ -82,13 +82,27 @@ static void tst5() {
     mpz m_max(max);
     lean_assert(m_max.is<T>());
     lean_assert(!(m_max + 1).is<T>());
-    lean_assert(m_max.get<T>() == max);
+    lean_assert(static_cast<T>(m_max) == max);
 
     T min = std::numeric_limits<T>::min();
     mpz m_min(min);
     lean_assert(m_min.is<T>());
     lean_assert(!(m_min - 1).is<T>());
-    lean_assert(m_min.get<T>() == min);
+    lean_assert(static_cast<T>(m_min) == min);
+
+    if (std::numeric_limits<T>::is_signed) {
+        T neg_one = -1;
+        mpz m_neg_one(neg_one);
+        lean_assert(m_neg_one.is<T>());
+        lean_assert(static_cast<T>(m_neg_one) == neg_one);
+    }
+}
+
+static void tst6() {
+    // the largest representable double is integral, so is fine to store in mpz
+    double max = std::numeric_limits<double>::max();
+    mpz n1(max);
+    lean_assert(n1.get_double() == max);
 }
 
 int main() {
@@ -102,5 +116,6 @@ int main() {
     tst5<unsigned long>();
     tst5<long long>();
     tst5<unsigned long long>();
+    tst6();
     return has_violations() ? 1 : 0;
 }
diff --git a/src/util/numerics/mpz.cpp b/src/util/numerics/mpz.cpp
index dd2e3fb199..f2de7f5d46 100644
--- a/src/util/numerics/mpz.cpp
+++ b/src/util/numerics/mpz.cpp
@@ -25,19 +25,19 @@ mpz::mpz(int64 v) : mpz(static_cast<unsigned>(v)) {
     mpz_add(m_val, m_val, tmp.m_val);
 }
 
-template<> long long int mpz::get() const {
+mpz::operator long long int() const {
     lean_assert(is<long long int>());
     mpz high_m, low_m;
     mpz_fdiv_r_2exp(low_m.m_val, m_val, 32);
     mpz_fdiv_q_2exp(high_m.m_val, m_val, 32);
-    return static_cast<long long int>(high_m.get<signed>()) << 32 | low_m.get<unsigned>();
+    return static_cast<long long int>(high_m.operator signed()) << 32 | low_m.operator unsigned();
 }
-template<> unsigned long long int mpz::get() const {
+mpz::operator unsigned long long int() const {
     lean_assert(is<unsigned long long int>());
     mpz high_m, low_m;
     mpz_fdiv_r_2exp(low_m.m_val, m_val, 32);
     mpz_fdiv_q_2exp(high_m.m_val, m_val, 32);
-    return static_cast<unsigned long long int>(high_m.get<unsigned>()) << 32 | low_m.get<unsigned>();
+    return static_cast<unsigned long long int>(high_m.operator unsigned()) << 32 | low_m.operator unsigned();
 }
 
 unsigned mpz::log2() const {
diff --git a/src/util/numerics/mpz.h b/src/util/numerics/mpz.h
index be177330c0..43196af8a8 100644
--- a/src/util/numerics/mpz.h
+++ b/src/util/numerics/mpz.h
@@ -33,6 +33,7 @@ class mpz {
     explicit mpz(int v) { mpz_init_set_si(m_val, v); }
     explicit mpz(uint64 v);
     explicit mpz(int64 v);
+    explicit mpz(double v) { mpz_init_set_d(m_val, v); }
     mpz(mpz const & s) { mpz_init_set(m_val, s.m_val); }
     mpz(mpz && s):mpz() { mpz_swap(m_val, s.m_val); }
     ~mpz() { mpz_clear(m_val); }
@@ -58,9 +59,16 @@ class mpz {
     bool even() const { return mpz_even_p(m_val) != 0; }
     bool odd() const { return !even(); }
 
-    template <typename T> bool is() const;
-    template <typename T> T get() const;
+    template <typename T> bool is() const = delete;
 
+    explicit operator long int() const;
+    explicit operator unsigned long int() const;
+    explicit operator int() const;
+    explicit operator unsigned int() const;
+    explicit operator long long int() const;
+    explicit operator unsigned long long int() const;
+
+    // not a cast operator, to match `mpz`
     double get_double() const { return mpz_get_d(m_val); }
 
     mpz & operator=(mpz const & v) { mpz_set(m_val, v.m_val); return *this; }
@@ -70,6 +78,7 @@ class mpz {
     mpz & operator=(long int v) { mpz_set_si(m_val, v); return *this; }
     mpz & operator=(unsigned int v) { return operator=(static_cast<unsigned long int>(v)); }
     mpz & operator=(int v) { return operator=(static_cast<long int>(v)); }
+    mpz & operator=(double v) { mpz_set_d(m_val, v); return *this; }
 
     friend int cmp(mpz const & a, mpz const & b) { return mpz_cmp(a.m_val, b.m_val); }
     friend int cmp(mpz const & a, unsigned b) { return mpz_cmp_ui(a.m_val, b); }
@@ -234,12 +243,11 @@ template<> inline bool mpz::is<long long>() const {
 template<> inline bool mpz::is<unsigned long long>() const {
     return mpz(std::numeric_limits<unsigned long long>::min()) <= *this && *this <= mpz(std::numeric_limits<unsigned long long>::max()); }
 
-template<> inline long int mpz::get()          const { lean_assert(is<long int>());          return mpz_get_si(m_val); }
-template<> inline unsigned long int mpz::get() const { lean_assert(is<unsigned long int>()); return mpz_get_ui(m_val); }
-template<> inline int mpz::get()               const { lean_assert(is<int>());          return static_cast<int>(get<long int>()); }
-template<> inline unsigned int mpz::get()      const { lean_assert(is<unsigned int>()); return static_cast<unsigned>(get<unsigned long int>()); }
-template<> long long int mpz::get()            const;
-template<> unsigned long long int mpz::get()   const;
+// we can't define these until the `is` specializations are declared
+inline mpz::operator long int()          const { lean_assert(is<long int>());          return mpz_get_si(m_val); }
+inline mpz::operator unsigned long int() const { lean_assert(is<unsigned long int>()); return mpz_get_ui(m_val); }
+inline mpz::operator int()               const { lean_assert(is<int>());          return static_cast<long>(operator long int()); }
+inline mpz::operator unsigned int()      const { lean_assert(is<unsigned int>()); return static_cast<unsigned>(operator unsigned long int()); }
 
 struct mpz_cmp_fn {
     int operator()(mpz const & v1, mpz const & v2) const { return cmp(v1, v2); }
diff --git a/tests/lean/float.lean b/tests/lean/float.lean
index 3199e38ea4..5434a8b874 100644
--- a/tests/lean/float.lean
+++ b/tests/lean/float.lean
@@ -95,6 +95,11 @@ list.foldl band tt $ floats.map (λ ⟨x,y⟩, prop x y)
 #eval cos pi
 #eval tan pi ≃ 0
 
+#eval of_nat 0x100000000  -- 2 ^ 32
+#eval of_int 0x100000000  -- 2 ^ 32
+#eval of_nat 0x10000000000000000   -- 2 ^ 64
+#eval of_int 0x10000000000000000   -- 2 ^ 64
+
 #eval (of_int (-12341234))
 
 #eval to_bool $ float.floor (2.5)  =  2
@@ -106,6 +111,14 @@ list.foldl band tt $ floats.map (λ ⟨x,y⟩, prop x y)
 #eval to_bool $ float.round (2.5)  =  3
 #eval to_bool $ float.round (-2.5) = -3
 
+#eval to_bool $ float.round (of_int 0x100000000) = 0x100000000
+#eval to_bool $ float.round (of_int 0x10000000000000000) = 0x10000000000000000
+
+#eval (native.float.infinity : native.float).floor
+#eval (native.float.infinity : native.float).ceil
+#eval (native.float.infinity : native.float).trunc
+#eval (native.float.infinity : native.float).round
+
 #eval (of_string "hello")
 #eval (of_string "0.123E4")
 #eval (of_string "-123.123")
diff --git a/tests/lean/float.lean.expected.out b/tests/lean/float.lean.expected.out
index 611dfe7f67..52843de39f 100644
--- a/tests/lean/float.lean.expected.out
+++ b/tests/lean/float.lean.expected.out
@@ -70,6 +70,10 @@ tt
 tt
 -1
 tt
+4.29497e+09
+4.29497e+09
+1.84467e+19
+1.84467e+19
 -1.23412e+07
 tt
 tt
@@ -79,6 +83,12 @@ tt
 tt
 tt
 tt
+tt
+tt
+0
+0
+0
+0
 none
 (some 1230)
 (some -123.123)