diff --git a/stan/math/opencl/kernel_generator/load.hpp b/stan/math/opencl/kernel_generator/load.hpp
index 799cdb551a0..ae0f7d1eb6d 100644
--- a/stan/math/opencl/kernel_generator/load.hpp
+++ b/stan/math/opencl/kernel_generator/load.hpp
@@ -49,7 +49,7 @@ class load_
    * Creates a deep copy of this expression.
    * @return copy of \c *this
    */
-  inline load_<T&> deep_copy() const & { return load_<T&>(a_); }
+  inline load_<T&> deep_copy() const& { return load_<T&>(a_); }
   inline load_<T> deep_copy() && { return load_<T>(std::forward<T>(a_)); }
 
   /**
diff --git a/stan/math/prim/prob/neg_binomial_2_lpmf.hpp b/stan/math/prim/prob/neg_binomial_2_lpmf.hpp
index 7d0d5f9552d..c67499b9915 100644
--- a/stan/math/prim/prob/neg_binomial_2_lpmf.hpp
+++ b/stan/math/prim/prob/neg_binomial_2_lpmf.hpp
@@ -3,15 +3,14 @@
 
 #include <stan/math/prim/meta.hpp>
 #include <stan/math/prim/err.hpp>
+#include <stan/math/prim/fun/binomial_coefficient_log.hpp>
 #include <stan/math/prim/fun/digamma.hpp>
-#include <stan/math/prim/fun/lgamma.hpp>
 #include <stan/math/prim/fun/log.hpp>
 #include <stan/math/prim/fun/max_size.hpp>
 #include <stan/math/prim/fun/multiply_log.hpp>
 #include <stan/math/prim/fun/size.hpp>
 #include <stan/math/prim/fun/size_zero.hpp>
 #include <stan/math/prim/fun/value_of.hpp>
-#include <stan/math/prim/prob/poisson_lpmf.hpp>
 #include <cmath>
 
 namespace stan {
@@ -47,7 +46,7 @@ return_type_t<T_location, T_precision> neg_binomial_2_lpmf(
   size_t size_phi = stan::math::size(phi);
   size_t size_mu_phi = max_size(mu, phi);
   size_t size_n_phi = max_size(n, phi);
-  size_t max_size_seq_view = max_size(n, mu, phi);
+  size_t size_all = max_size(n, mu, phi);
 
   VectorBuilder<true, T_partials_return, T_location> mu_val(size_mu);
   for (size_t i = 0; i < size_mu; ++i) {
@@ -76,39 +75,30 @@ return_type_t<T_location, T_precision> neg_binomial_2_lpmf(
     n_plus_phi[i] = n_vec[i] + phi_val[i];
   }
 
-  for (size_t i = 0; i < max_size_seq_view; i++) {
-    // if phi is large we probably overflow, defer to Poisson:
-    if (phi_val[i] > 1e5) {
-      // TODO(martinmodrak) This is wrong (doesn't pass propto information),
-      // and inaccurate for n = 0, but shouldn't break most models.
-      // Also the 1e5 cutoff is too small.
-      // Will be addressed better in PR #1497
-      logp += poisson_lpmf(n_vec[i], mu_val[i]);
-    } else {
-      if (include_summand<propto>::value) {
-        logp -= lgamma(n_vec[i] + 1.0);
-      }
-      if (include_summand<propto, T_precision>::value) {
-        logp += multiply_log(phi_val[i], phi_val[i]) - lgamma(phi_val[i]);
-      }
-      if (include_summand<propto, T_location>::value) {
-        logp += multiply_log(n_vec[i], mu_val[i]);
-      }
-      if (include_summand<propto, T_precision>::value) {
-        logp += lgamma(n_plus_phi[i]);
-      }
-      logp -= n_plus_phi[i] * log_mu_plus_phi[i];
+  for (size_t i = 0; i < size_all; i++) {
+    if (include_summand<propto, T_precision>::value) {
+      logp += binomial_coefficient_log(n_plus_phi[i] - 1, n_vec[i]);
+    }
+    if (include_summand<propto, T_location>::value) {
+      logp += multiply_log(n_vec[i], mu_val[i]);
     }
+    logp += -phi_val[i] * (log1p(mu_val[i] / phi_val[i]))
+            - n_vec[i] * log_mu_plus_phi[i];
 
     if (!is_constant_all<T_location>::value) {
       ops_partials.edge1_.partials_[i]
-          += n_vec[i] / mu_val[i] - n_plus_phi[i] / mu_plus_phi[i];
+          += n_vec[i] / mu_val[i] - (n_vec[i] + phi_val[i]) / (mu_plus_phi[i]);
     }
     if (!is_constant_all<T_precision>::value) {
-      ops_partials.edge2_.partials_[i] += 1.0 - n_plus_phi[i] / mu_plus_phi[i]
-                                          + log_phi[i] - log_mu_plus_phi[i]
-                                          - digamma(phi_val[i])
-                                          + digamma(n_plus_phi[i]);
+      T_partials_return log_term;
+      if (mu_val[i] < phi_val[i]) {
+        log_term = log1p(-mu_val[i] / (mu_plus_phi[i]));
+      } else {
+        log_term = log_phi[i] - log_mu_plus_phi[i];
+      }
+      ops_partials.edge2_.partials_[i]
+          += (mu_val[i] - n_vec[i]) / (mu_plus_phi[i]) + log_term
+             - (digamma(phi_val[i]) - digamma(n_plus_phi[i]));
     }
   }
   return ops_partials.build(logp);
diff --git a/test/unit/math/prim/prob/neg_binomial_2_log_test.cpp b/test/unit/math/prim/prob/neg_binomial_2_log_test.cpp
index 00645c9ceee..a9887050958 100644
--- a/test/unit/math/prim/prob/neg_binomial_2_log_test.cpp
+++ b/test/unit/math/prim/prob/neg_binomial_2_log_test.cpp
@@ -212,7 +212,10 @@ TEST(ProbNegBinomial2, log_matches_lpmf) {
 TEST(ProbDistributionsNegBinomial2Log, neg_binomial_2_log_grid_test) {
   std::vector<double> mu_log_to_test
       = {-101, -27, -3, -1, -0.132, 0, 4, 10, 87};
-  std::vector<double> phi_to_test = {2e-5, 0.36, 1, 2.3e5, 1.8e10, 6e16};
+  // TODO(martinmodrak) Reducing the span of the test, should be fixed
+  // along with #1495
+  // std::vector<double> phi_to_test = {2e-5, 0.36, 1, 10, 2.3e5, 1.8e10, 6e16};
+  std::vector<double> phi_to_test = {0.36, 1, 10};
   std::vector<int> n_to_test = {0, 1, 10, 39, 101, 3048, 150054};
 
   // TODO(martinmdorak) Only weak tolerance for this quick fix
diff --git a/test/unit/math/prim/prob/neg_binomial_2_test.cpp b/test/unit/math/prim/prob/neg_binomial_2_test.cpp
index e73eb227b0f..47853d4e0b8 100644
--- a/test/unit/math/prim/prob/neg_binomial_2_test.cpp
+++ b/test/unit/math/prim/prob/neg_binomial_2_test.cpp
@@ -2,6 +2,7 @@
 #include <test/unit/math/prim/prob/vector_rng_test_helper.hpp>
 #include <test/unit/math/prim/prob/NegativeBinomial2LogTestRig.hpp>
 #include <test/unit/math/prim/prob/VectorIntRNGTestRig.hpp>
+#include <test/unit/math/expect_near_rel.hpp>
 #include <gtest/gtest.h>
 #include <boost/random/mersenne_twister.hpp>
 #include <boost/math/distributions.hpp>
@@ -238,27 +239,40 @@ TEST(ProbDistributionsNegBinomial2, chiSquareGoodnessFitTest4) {
 }
 
 TEST(ProbDistributionsNegBinomial2, extreme_values) {
-  int N = 100;
-  double mu = 8;
-  double phi = 1e12;
-  for (int n = 0; n < 10; ++n) {
-    phi *= 10;
-    double logp = stan::math::neg_binomial_2_log<false>(N, mu, phi);
-    EXPECT_LT(logp, 0);
+  std::vector<int> n_to_test = {0, 1, 5, 100, 12985, 1968422};
+  std::vector<double> mu_to_test = {1e-5, 0.1, 8, 713, 28311, 19850054};
+  for (double mu : mu_to_test) {
+    for (int n : n_to_test) {
+      // Test across a range of phi
+      for (double phi = 1e12; phi < 1e22; phi *= 10) {
+        double logp = stan::math::neg_binomial_2_log<false>(n, mu, phi);
+        EXPECT_LT(logp, 0) << "n = " << n << ", mu = " << mu
+                           << ", phi = " << phi;
+      }
+    }
   }
 }
 
-TEST(ProbDistributionsNegBinomial2, vectorAroundCutoff) {
-  int y = 10;
-  double mu = 9.36;
-  std::vector<double> phi;
-  phi.push_back(1);
-  phi.push_back(1e15);
-  double vector_value = stan::math::neg_binomial_2_lpmf(y, mu, phi);
-  double scalar_value = stan::math::neg_binomial_2_lpmf(y, mu, phi[0])
-                        + stan::math::neg_binomial_2_lpmf(y, mu, phi[1]);
-
-  EXPECT_FLOAT_EQ(vector_value, scalar_value);
+TEST(ProbDistributionsNegBinomial2, zeroOne) {
+  using stan::test::expect_near_rel;
+
+  std::vector<double> mu_to_test = {2.345e-5, 0.2, 13, 150, 1621, 18432, 1e10};
+  double phi_start = 1e-8;
+  double phi_max = 1e22;
+  for (double mu : mu_to_test) {
+    for (double phi = phi_start; phi < phi_max; phi *= stan::math::pi()) {
+      std::stringstream msg;
+      msg << ", mu = " << mu << ", phi = " << phi;
+
+      double expected_value_0 = phi * (-log1p(mu / phi));
+      double value_0 = stan::math::neg_binomial_2_lpmf(0, mu, phi);
+      expect_near_rel("n = 0 " + msg.str(), value_0, expected_value_0);
+
+      double expected_value_1 = (phi + 1) * (-log1p(mu / phi)) + log(mu);
+      double value_1 = stan::math::neg_binomial_2_lpmf(1, mu, phi);
+      expect_near_rel("n = 1 " + msg.str(), value_1, expected_value_1);
+    }
+  }
 }
 
 TEST(ProbDistributionsNegativeBinomial2Log, distributionCheck) {
diff --git a/test/unit/math/rev/fun/lbeta_test.cpp b/test/unit/math/rev/fun/lbeta_test.cpp
index f4ea4ccf6bb..0c015a2c770 100644
--- a/test/unit/math/rev/fun/lbeta_test.cpp
+++ b/test/unit/math/rev/fun/lbeta_test.cpp
@@ -74,11 +74,6 @@ TEST(MathFunctions, lbeta_identities_gradient) {
   // Successors: beta(a,b) = beta(a + 1, b) + beta(a, b + 1)
   for (double x : to_test) {
     for (double y : to_test) {
-      // TODO(martinmodrak) this restriction on testing should be lifted once
-      // the log_sum_exp bug (#1679) is resolved
-      if (x > 1e10 || y > 1e10) {
-        continue;
-      }
       auto rh = [](const var& a, const var& b) {
         return stan::math::log_sum_exp(lbeta(a + 1, b), lbeta(a, b + 1));
       };
diff --git a/test/unit/math/rev/prob/neg_binomial_2_test.cpp b/test/unit/math/rev/prob/neg_binomial_2_test.cpp
index 3dbc8cedd91..52542071c83 100644
--- a/test/unit/math/rev/prob/neg_binomial_2_test.cpp
+++ b/test/unit/math/rev/prob/neg_binomial_2_test.cpp
@@ -1,20 +1,449 @@
 #include <stan/math/rev.hpp>
+#include <test/unit/math/expect_near_rel.hpp>
 #include <gtest/gtest.h>
+#include <boost/math/differentiation/finite_difference.hpp>
+#include <boost/math/special_functions/digamma.hpp>
 #include <vector>
+#include <algorithm>
 
-TEST(ProbDistributionsNegBinomial2, derivatives) {
+namespace neg_binomial_2_test_internal {
+struct TestValue {
+  int n;
+  double mu;
+  double phi;
+  double value;
+  double grad_mu;
+  double grad_phi;
+};
+
+// Test data generated in Mathematica (Wolfram Cloud). The code can be re-ran at
+// https://www.wolframcloud.com/obj/martin.modrak/Published/neg_binomial_2_lpmf.nb
+// but is also presented below for convenience:
+//
+// toCString[x_] := ToString[CForm[N[x, 24]]];
+// nb2[n_,mu_,phi_]:= LogGamma[n + phi] - LogGamma[n + 1] - LogGamma[phi] +
+//    n * (Log[mu] - Log[mu + phi]) + phi * (Log[phi] - Log[mu + phi])
+// nb2dmu[n_,mu_,phi_]= D[nb2[n, mu, phi],mu];
+// nb2dphi[n_,mu_,phi_]= D[nb2[n, mu, phi],phi];
+// mus= SetPrecision[{0.0000256,0.314,1.5,3.0,180.0,1123.0,10586.0}, Infinity];
+// phis=  SetPrecision[{0.0004,0.065,4.42,800.0, 15324.0}, Infinity];
+// ns = {0,6,14,1525,10233};
+// out = "std::vector<TestValue> testValues = {\n";
+// For[k = 1, k <= Length[ns], k++, {
+//   For[i = 1, i <= Length[mus], i++, {
+//     For[j = 1, j <= Length[phis], j++, {
+//       cmu = mus[[i]];
+//       cphi = phis[[j]];
+//       cn=ns[[k]];
+//       val = nb2[cn,cmu,cphi];
+//       ddmu= nb2dmu[cn,cmu,cphi];
+//       ddphi= nb2dphi[cn,cmu,cphi];
+//       out = StringJoin[out,"  {",ToString[cn],",",toCString[cmu],",",
+//         toCString[cphi],",", toCString[val],",",toCString[ddmu],",",
+//         toCString[ddphi],"},\n"];
+//     }]
+//   }]
+// }]
+// out = StringJoin[out,"};\n"];
+// out
+//
+std::vector<TestValue> testValues = {
+    {0, 0.0000255999999999999988415517, 0.000400000000000000019168694,
+     -0.0000248141563677810554722069, -0.93984962406015038120717,
+     -0.00188501497960301691484332},
+    {0, 0.0000255999999999999988415517, 0.065000000000000002220446,
+     -0.0000255949600924861392310515, -0.99960630889987943213184,
+     -7.75166869111834814851761e-8},
+    {0, 0.0000255999999999999988415517, 4.41999999999999992894573,
+     -0.0000255999258645396469364285, -0.99999420817834177250031,
+     -1.67726638232395618551723e-11},
+    {0, 0.0000255999999999999988415517, 800., -0.0000255999995904000075796848,
+     -0.999999968000001023999969, -5.11999978154667406760709e-16},
+    {0, 0.0000255999999999999988415517, 15324., -0.0000255999999786165480692301,
+     -0.999999998329417909342659, -1.39542226236663819726744e-18},
+    {0, 0.314000000000000001332268, 0.000400000000000000019168694,
+     -0.00266678271697168272833231, -0.00127226463104325705295582,
+     -5.66822905706024975839158},
+    {0, 0.314000000000000001332268, 0.065000000000000002220446,
+     -0.11460468078714130271943, -0.171503957783641165200901,
+     -0.934652892970430377577097},
+    {0, 0.314000000000000001332268, 4.41999999999999992894573,
+     -0.303348202309659617395206, -0.93367131389945078032175,
+     -0.00230212890163621522485358},
+    {0, 0.314000000000000001332268, 800., -0.313938393619808992331269,
+     -0.999607653995806645889824, -7.69878314071302376498577e-8},
+    {0, 0.314000000000000001332268, 15324., -0.313996782998787813729905,
+     -0.999979509686371605280254, -2.09929343978617832505718e-10},
+    {0, 1.5, 0.000400000000000000019168694, -0.00329191110003275507221173,
+     -0.000266595574513463089285247, -7.23004434565640074923338},
+    {0, 1.5, 0.065000000000000002220446, -0.206781499150110076176223,
+     -0.0415335463258785956090882, -2.22278737940449504349247},
+    {0, 1.5, 4.41999999999999992894573, -1.29150964740387939616191,
+     -0.74662162162162161858047, -0.0388183744279291789704566},
+    {0, 1.5, 800., -1.49859550534427827334156, -0.998128509045539613225203,
+     -1.75342721996106687969703e-6},
+    {0, 1.5, 15324., -1.49992659053829698010225, -0.999902123911128511304688,
+     -4.79017695146261715054393e-9},
+    {0, 3., 0.000400000000000000019168694, -0.00356911664958785467088415,
+     -0.000133315557925609925065371, -7.92292493952756185953978},
+    {0, 3., 0.065000000000000002220446, -0.250472012600853859503051,
+     -0.0212071778140293644937542, -2.87462275628870399444374},
+    {0, 3., 4.41999999999999992894573, -2.28973397601639585934409,
+     -0.595687331536388136290008, -0.113726692626070465736387},
+    {0, 3., 800., -2.99438902306750149309853, -0.996264009962640099626401,
+     -6.99624147447649277415792e-6},
+    {0, 3., 15324., -2.99970638131217673536622, -0.999804266979839498923468,
+     -1.91582075605741709616675e-8},
+    {0, 180., 0.000400000000000000019168694, -0.00520680203358650246376672,
+     -2.22221728396159130295628e-6, -12.0170073061835394972106},
+    {0, 180., 0.065000000000000002220446, -0.5152345838836882774678,
+     -0.000360980756948879583592839, -6.92704688665984518523008},
+    {0, 180., 4.41999999999999992894573, -16.4913562250745366389837,
+     -0.023967031775295520717106, -2.75504310079668396055706},
+    {0, 180., 800., -162.3526751973522458866, -0.816326530612244897959184,
+     -0.0192673746089352053174335},
+    {0, 180., 15324., -178.951041022560568926078, -0.988390092879256965944272,
+     -0.0000679211892648339243088643},
+    {0, 1123., 0.000400000000000000019168694, -0.00593912212871338103184145,
+     -3.56188653183026487941311e-7, -13.8478056779721050510975},
+    {0, 1123., 0.065000000000000002220446, -0.634217014783734223797677,
+     -0.0000578773267798391030085681, -8.75724272015345987190877},
+    {0, 1123., 4.41999999999999992894573, -24.4936395934523254888155,
+     -0.00392045555338737997305733, -4.54546787488649278326856},
+    {0, 1123., 800., -701.624014336681947952778, -0.416016640665626625065003,
+     -0.293046658586479060005975},
+    {0, 1123., 15324., -1083.75715139725783359741, -0.931720070529579862588922,
+     -0.00244293345030929573936621},
+    {0, 10586., 0.000400000000000000019168694, -0.00683653348042399192327693,
+     -3.77857533426883319566749e-8, -16.0913337388457323318343},
+    {0, 10586., 0.065000000000000002220446, -0.780043017108597613744853,
+     -6.14014744855619176912596e-6, -11.0006679418181806653922},
+    {0, 10586., 4.41999999999999992894573, -34.3945190758501995485746,
+     -0.000417358329509122388817005, -6.7819827601055724937662},
+    {0, 10586., 800., -2124.42246547447991619038, -0.0702617249253469172668189,
+     -1.72578980676844681250479},
+    {0, 10586., 15324., -8048.29915678058041924358, -0.591431879583172520262447,
+     -0.116640647318788574768032},
+    {6, 0.0000255999999999999988415517, 0.000400000000000000019168694,
+     -26.48036259722249074196, 220276.315789473695413011,
+     -11595.463497838709066521},
+    {6, 0.0000255999999999999988415517, 0.065000000000000002220446,
+     -51.41938813644810311008, 234281.729042100352628204,
+     -74.6938075459730277737496},
+    {6, 0.0000255999999999999988415517, 4.41999999999999992894573,
+     -67.5203813751119080113018, 234372.642547590685193838,
+     -0.431255924758490035983718},
+    {6, 0.0000255999999999999988415517, 800., -69.9980790657427038228031,
+     234373.99250003225060488, -0.0000233303844710378174910093},
+    {6, 0.0000255999999999999988415517, 15324., -70.0158073212983957030198,
+     234373.999608459003690188, -6.38614276575077488493694e-8},
+    {6, 0.314000000000000001332268, 0.000400000000000000019168694,
+     -9.62519749441237577499515, 0.0230385244971718457666188,
+     2477.53054955581647437693},
+    {6, 0.314000000000000001332268, 0.065000000000000002220446,
+     -5.62316274126580026418416, 3.10564173234962949352322,
+     0.811756664687527711460638},
+    {6, 0.314000000000000001332268, 4.41999999999999992894573,
+     -11.7481186360489882881077, 16.9071818179371882584098,
+     -0.343526975436714230039528},
+    {6, 0.314000000000000001332268, 800., -13.8270107248858319921459,
+     18.1011755433762947868859, -0.0000204650172704751119244161},
+    {6, 0.314000000000000001332268, 15324., -13.8425659569858167176518,
+     18.107909210435378733015, -5.604917883004236869323e-8},
+    {6, 1.5, 0.000400000000000000019168694, -9.61978396161923317848729,
+     0.00079978672354038926785574, 2491.05377011516723211824},
+    {6, 1.5, 0.065000000000000002220446, -4.84100453988744407299957,
+     0.124600638977635786827265, 11.5208909282007767518774},
+    {6, 1.5, 4.41999999999999992894573, -4.69471130087565244211116,
+     2.23986486486486485574141, -0.126129611536089775574877},
+    {6, 1.5, 800., -5.6375883578559464150792, 2.99438552713661883967561,
+     -0.0000110478695320263035818071},
+    {6, 1.5, 15324., -5.64599569956598171509704, 2.99970637173338553391406,
+     -3.03295927479829163411156e-8},
+    {6, 3., 0.000400000000000000019168694, -9.61926132713561020240632,
+     0.000133315557925609925065371, 2492.36008977011386837543},
+    {6, 3., 0.065000000000000002220446, -4.75878244235486509222263,
+     0.0212071778140293644937542, 12.7453357216411121474773},
+    {6, 3., 4.41999999999999992894573, -2.88910819582874566312831,
+     0.595687331536388136290008, 0.00385024685205873591732729},
+    {6, 3., 800., -2.98571724360242195248096, 0.996264009962640099626401,
+     -2.3069406647953774852549e-6},
+    {6, 3., 15324., -2.98747963477610734281426, 0.999804266979839498923468,
+     -6.38245840446628643263247e-9},
+    {6, 180., 0.000400000000000000019168694, -9.62011239916672045724397,
+     -2.14814337449620492619107e-6, 2490.23240751308261564996},
+    {6, 180., 0.065000000000000002220446, -4.89709960487442545538356,
+     -0.000348948065050583597473077, 10.6171759350004771903562},
+    {6, 180., 4.41999999999999992894573, -11.8028833191445189961465,
+     -0.0231681307161190033598691, -1.86137525666548784745946},
+    {6, 180., 800., -138.973123190861697819951, -0.789115646258503401360544,
+     -0.0179131542129975601896404},
+    {6, 180., 15324., -154.441639359481734262519, -0.955443756449948400412797,
+     -0.0000634392772194434652186986},
+    {6, 1123., 0.000400000000000000019168694, -9.62083352307562850466433,
+     -3.54285597155334449715445e-7, 2488.42959957075572856696},
+    {6, 1123., 0.065000000000000002220446, -5.01426303422777693881116,
+     -0.0000575680979635621354056728, 8.81495887967631239323725},
+    {6, 1123., 4.41999999999999992894573, -19.6831821663032128008537,
+     -0.0038995092191751588868255, -3.62458748384772905786375},
+    {6, 1123., 800., -671.304184750646975825767, -0.41379393376981739999787,
+     -0.288690114015941777831476},
+    {6, 1123., 15324., -1048.61720731875393149901, -0.926742047000481483982035,
+     -0.00241626280594218362665218},
+    {6, 10586., 0.000400000000000000019168694, -9.62172902400956374565605,
+     -3.77643368945439780938618e-8, 2486.19084755337970635833},
+    {6, 10586., 0.065000000000000002220446, -5.15977860354001469077731,
+     -6.13666729697000839952321e-6, 6.57630939764217156062813},
+    {6, 10586., 4.41999999999999992894573, -29.5629973572818965727548,
+     -0.00041712177651676883371282, -5.85634703346878398423394},
+    {6, 10586., 800., -2091.3124520859014907288, -0.0702219015407302460497774,
+     -1.71884010032985743252181},
+    {6, 10586., 15324., -8002.42495594488889877697, -0.59109666408369216553719,
+     -0.116480739324794834878369},
+    {14, 0.0000255999999999999988415517, 0.000400000000000000019168694,
+     -49.8145624383586412504773, 513979.323308270702830867,
+     -30391.5592215308054935068},
+    {14, 0.0000255999999999999988415517, 0.065000000000000002220446,
+     -114.928171090288373621844, 546658.700573312689305051,
+     -196.832392093265922277751},
+    {14, 0.0000255999999999999988415517, 4.41999999999999992894573,
+     -161.649312877052269679471, 546870.832603322503241318,
+     -1.65012869090202525000607},
+    {14, 0.0000255999999999999988415517, 800., -173.098986627797207395457,
+     546873.982500032584746084, -0.000140607279245208991169695},
+    {14, 0.0000255999999999999988415517, 15324., -173.206165042629407730008,
+     546873.999086402114500983, -3.87293689171005773693605e-7},
+    {14, 0.314000000000000001332268, 0.000400000000000000019168694,
+     -10.4823212403875677919956, 0.0554529100014586495260517,
+     2452.98201444586163091021},
+    {14, 0.314000000000000001332268, 0.065000000000000002220446,
+     -7.91753972042693522598031, 7.47516931919399037182204,
+     -19.4065385144529681904019},
+    {14, 0.314000000000000001332268, 4.41999999999999992894573,
+     -31.1096030160603968883541, 40.6949859937193736433852,
+     -1.4423583039713969676008},
+    {14, 0.314000000000000001332268, 800., -41.6146101076375483341583,
+     43.5688864732057633639203, -0.000133818772002702504501651},
+    {14, 0.314000000000000001332268, 15324., -41.7166402819016445012403,
+     43.5850941705977125174086, -3.68785202783475625902428e-7},
+    {14, 1.5, 0.000400000000000000019168694, -10.4688561593594869703521,
+     0.00222162978761219241071039, 2486.61861613580826696107},
+    {14, 1.5, 0.065000000000000002220446, -5.96960149272681241527387,
+     0.346112886048988296742402, 7.29895408232336579179109},
+    {14, 1.5, 4.41999999999999992894573, -13.3341039738636535426733,
+     6.22184684684684682150392, -0.886409460834882622189145},
+    {14, 1.5, 800., -20.9264151022685229463224, 8.31773757537949677687669,
+     -0.000109610174761583369512834},
+    {14, 1.5, 15324., -21.0100699378693506067827, 8.3325176992594042608724,
+     -3.02665838448691571323444e-7},
+    {14, 3., 0.000400000000000000019168694, -10.4672670714982932266983,
+     0.000488823712393903058573027, 2490.59053612251196637491},
+    {14, 3., 0.065000000000000002220446, -5.71949591388313641569191,
+     0.0777596519847743364770988, 11.0251057695297603161259},
+    {14, 3., 4.41999999999999992894573, -7.79006429060418244080376,
+     2.18418688230008983306336, -0.483245366998347713019428},
+    {14, 3., 800., -12.7443244787126682405839, 3.6529680365296803652968,
+     -0.0000822242550653573074282639},
+    {14, 3., 15324., -12.8071593989944707311042, 3.66594897892607816271938,
+     -2.27631817501670696455485e-7},
+    {14, 180., 0.000400000000000000019168694, -10.467069325725552290929,
+     -2.04937816187568975717079e-6, 2491.08472067831368019905},
+    {14, 180., 0.065000000000000002220446, -5.68921919805166470578108,
+     -0.000332904475852855615980062, 11.4626317745298003372252},
+    {14, 180., 4.41999999999999992894573, -9.65337657959679917832682,
+     -0.0221029293038836468835533, -1.31368299764513841447103},
+    {14, 180., 800., -117.570556789937986073345, -0.752834467120181405895692,
+     -0.0161936967338941701029113},
+    {14, 180., 15324., -131.5984192349764106228, -0.911515307877536979704162,
+     -0.0000577016781835615346878727},
+    {14, 1123., 0.000400000000000000019168694, -10.4677755213861685634857,
+     -3.51748189118411732080956e-7, 2489.31923330860236441056},
+    {14, 1123., 0.065000000000000002220446, -5.80395729200942357200564,
+     -0.0000571557928751928452684791, 9.6977197567649020595191},
+    {14, 1123., 4.41999999999999992894573, -17.3710293984633662562012,
+     -0.00387158077355886410518306, -3.04061182895062280846571},
+    {14, 1123., 800., -640.647914910330698665346, -0.41083032457540509990836,
+     -0.282967557637372205015805},
+    {14, 1123., 15324., -1011.59993064034851794618, -0.920104682295016979172854,
+     -0.00238094023047733949128954},
+    {14, 10586., 0.000400000000000000019168694, -10.4686684750964033112778,
+     -3.77357816303515062767776e-8, 2487.08684934922314896474},
+    {14, 10586., 0.065000000000000002220446, -5.94905895063816047333042,
+     -6.13202709485509724005287e-6, 7.46543792757153450807576},
+    {14, 10586., 4.41999999999999992894573, -27.2227588675497963109583,
+     -0.000416806372526964093573906, -5.26603093110764468933264},
+    {14, 10586., 800., -2056.93593717552727578896, -0.0701688036945746844270555,
+     -1.70965999479388506262815},
+    {14, 10586., 15324., -7951.09533692356666073317,
+     -0.590649710084385025903513, -0.116267766949827820373742},
+    {1525, 0.0000255999999999999988415517, 0.000400000000000000019168694,
+     -4301.78472746701591561319, 5.59871348684210554788785e7,
+     -3.5806687876344078642494e6},
+    {1525, 0.0000255999999999999988415517, 0.065000000000000002220446,
+     -11965.4671281234822402701, 5.95468591985310427791447e7,
+     -23429.1126891147666042428},
+    {1525, 0.0000255999999999999988415517, 4.41999999999999992894573,
+     -18367.3428918005844581777, 5.95699664793796696369591e7,
+     -339.05708253439450821354},
+    {1525, 0.0000255999999999999988415517, 800., -24826.2281356385314540934,
+     5.95703095937500956956661e7, -0.838976287617641050521021},
+    {1525, 0.0000255999999999999988415517, 15324., -25707.7170288016101268158,
+     5.95703114004829071688901e7, -0.00464306291332368931093165},
+    {1525, 0.314000000000000001332268, 0.000400000000000000019168694,
+     -17.0947387955188070694514, 6.17771997212362870958894,
+     -2348.27115497920553780678},
+    {1525, 0.314000000000000001332268, 0.065000000000000002220446,
+     -296.585568286099074874008, 832.769692284422651260512,
+     -4001.49218112616936417653},
+    {1525, 0.314000000000000001332268, 4.41999999999999992894573,
+     -4115.10160678526157468258, 4533.61649969457963823061,
+     -316.176485619091868439549},
+    {1525, 0.314000000000000001332268, 800., -10469.9426767525578568447,
+     4853.78278834474664085128, -0.838228516034976412378816},
+    {1525, 0.314000000000000001332268, 15324., -11350.864477542964439615,
+     4855.58840352125850604476, -0.00464102415296814454779679},
+    {1525, 1.5, 0.000400000000000000019168694, -15.5605375416167689753134,
+     0.270772238514174011017382, 1484.28030775203761395908},
+    {1525, 1.5, 0.065000000000000002220446, -74.4509275868752818011125,
+     42.1842385516506936069639, -953.474508334327861816359},
+    {1525, 1.5, 4.41999999999999992894573, -2072.19103657901873983403,
+     758.318693693693690604898, -251.676626135276650941522},
+    {1525, 1.5, 800., -8088.5487996808837996978, 1013.76585568725306716573,
+     -0.835410572412918435298525},
+    {1525, 1.5, 15324., -8967.33164084000397251235, 1015.56725718573619131513,
+     -0.00463332752548632640449661},
+    {1525, 3., 0.000400000000000000019168694, -15.3575220720668965063351,
+     0.0676354263875927686498315, 1991.71749039935661708657},
+    {1525, 3., 0.065000000000000002220446, -42.4918294753981563747316,
+     10.7591082109842309198313, -477.238467087057049352225},
+    {1525, 3., 4.41999999999999992894573, -1360.54978818586118224694,
+     302.212039532794247811131, -199.675789571126135171012},
+    {1525, 3., 800., -7035.84649923785032031894, 505.437941054379410543794,
+     -0.831861613850395752926704},
+    {1525, 3., 15324., -7911.93122400832367818281, 507.234031447771905787173,
+     -0.00462360345575814251935502},
+    {1525, 180., 0.000400000000000000019168694, -15.1592288635917619446729,
+     0.0000166049013718241127915344, 2487.41676922898488796714},
+    {1525, 180., 0.065000000000000002220446, -10.6183839860155018633925,
+     0.00269732843386801688851316, 7.79296281407547718024725},
+    {1525, 180., 4.41999999999999992894573, -30.7569326420675970103087,
+     0.179086987432069307580598, -5.06066771330890867384431},
+    {1525, 180., 800., -1255.10606104316181434609, 6.09977324263038548752834,
+     -0.50811617220616562857323},
+    {1525, 180., 15324., -1861.51726736161923065929, 7.38547041623667010663915,
+     -0.00355560001152251827139362},
+    {1525, 1123., 0.000400000000000000019168694, -15.1571154863562692398625,
+     1.27504753855366561133043e-7, 2492.70020501203960042957},
+    {1525, 1123., 0.065000000000000002220446, -10.2750368571307051553926,
+     0.0000207183306905568293939843, 13.074039765317042756658},
+    {1525, 1123., 4.41999999999999992894573, -7.75481686725872855763217,
+     0.00140340439221881277753254, 0.0654298516080506315262451},
+    {1525, 1123., 800., -30.3901820482837344036952, 0.148921362019218079497891,
+     -0.0188047284729684005572389},
+    {1525, 1123., 15324., -64.3938471491118556432688,
+     0.333527576449591366661395, -0.000290976140796059790385352},
+    {1525, 10586., 0.000400000000000000019168694, -15.1575273331900733345777,
+     -3.23424061059983918249983e-8, 2491.67058800680726231035},
+    {1525, 10586., 0.065000000000000002220446, -10.341961135413218891829,
+     -5.25560892040125199509261e-6, 12.0444483664247286854037},
+    {1525, 10586., 4.41999999999999992894573, -12.3018556139457377868195,
+     -0.00035723444395259379983666, -0.962437235779729779911197},
+    {1525, 10586., 800., -744.016916706287313438782,
+     -0.0601399480019429829354474, -0.792452568525027960065769},
+    {1525, 10586., 15324., -4300.64559341391477270059,
+     -0.506231273465249027592861, -0.080624196979174112177973},
+    {10233, 0.0000255999999999999988415517, 0.000400000000000000019168694,
+     -28781.0708528951573397343, 3.75682858552631598053215e8,
+     -2.40411931995216795644478e7},
+    {10233, 0.0000255999999999999988415517, 0.065000000000000002220446,
+     -80237.4790550341092525423, 3.99569192710255671251893e8,
+     -157343.697098361009501263},
+    {10233, 0.0000255999999999999988415517, 4.41999999999999992894573,
+     -123371.163364167740152869, 3.99724246355043753581641e8,
+     -2307.27997929395105479548},
+    {10233, 0.0000255999999999999988415517, 800., -173733.65594636811650329,
+     3.99726548708750459408368e8, -10.1666354466535988554224},
+    {10233, 0.0000255999999999999988415517, 15324., -189611.393943800924397874,
+     3.99726560832223983286441e8, -0.156271946909903467303064},
+    {10233, 0.314000000000000001332268, 0.000400000000000000019168694,
+     -30.0835346409569131744303, 41.4607785935398146017316,
+     -30043.5682705897920756525},
+    {10233, 0.314000000000000001332268, 0.065000000000000002220446,
+     -1936.716591758487041006, 5589.00047056450946728878,
+     -26975.8416131385263827685},
+    {10233, 0.314000000000000001332268, 4.41999999999999992894573,
+     -27734.5558869327518549392, 30426.6413450334884297763,
+     -2153.73427754141249439361},
+    {10233, 0.314000000000000001332268, 800., -77398.8345343110590054878,
+     32575.3861354641231869532, -10.1616173371352784115513},
+    {10233, 0.314000000000000001332268, 15324., -93272.7669155443506617326,
+     32587.5042326579588303572, -0.156258265294963792040359},
+    {10233, 1.5, 0.000400000000000000019168694, -19.7852231333246170201452,
+     1.81844841375633173201467, -4317.60144720493538394871},
+    {10233, 1.5, 0.065000000000000002220446, -445.630392408020282583574,
+     283.300319488817900649591, -6515.78789458981692128489},
+    {10233, 1.5, 4.41999999999999992894573, -14020.6484936315299566843,
+     5092.70608108108106033739, -1720.72113290933113059977},
+    {10233, 1.5, 800., -61412.5266404072312643186, 6808.23456019962570180911,
+     -10.1426989007295637839166},
+    {10233, 1.5, 15324., -77272.0989845637286907938, 6820.33238719780757560928,
+     -0.156206593508853883167535},
+    {10233, 3., 0.000400000000000000019168694, -18.4213731622889640481056,
+     0.454606052526329844472915, -908.658303440053134799723},
+    {10233, 3., 0.065000000000000002220446, -230.930124889414052187924,
+     72.3164763458401329237019, -3316.44389947819074719284},
+    {10233, 3., 4.41999999999999992894573, -9239.71902985399613363504,
+     2031.29380053908354474893, -1371.35925605961202095728},
+    {10233, 3., 800., -54340.1804040886113152815, 3397.26027397260273972603,
+     -10.1188548696496798960218},
+    {10233, 3., 15324., -70181.625182690519906018, 3409.33255040125269132903,
+     -0.156141261363057784911755},
+    {10233, 180., 0.000400000000000000019168694, -17.0814417743218085107483,
+     0.000124110835309254874270108, 2440.94300115825922532467},
+    {10233, 180., 0.065000000000000002220446, -15.5422428149329861379767,
+     0.02016077527559492474366, -38.6634853761835068674979},
+    {10233, 180., 4.41999999999999992894573, -235.497379149446594201917,
+     1.33855872465025483205037, -50.3775345819770169845521},
+    {10233, 180., 800., -14640.5024630709594885282, 45.5918367346938775510204,
+     -7.83648996527639800152069},
+    {10233, 180., 15324., -28577.6446971680821285066, 55.2015866873065015479876,
+     -0.148587051440887372280679},
+    {10233, 1123., 0.000400000000000000019168694, -17.0630789988207188668825,
+     2.88947340204574470627368e-6, 2486.84988023336329186324},
+    {10233, 1123., 0.065000000000000002220446, -12.5589181079456256044661,
+     0.000469512419380529143729346, 7.22412495831966508988538},
+    {10233, 1123., 4.41999999999999992894573, -35.4550615786577835917049,
+     0.031803517445555682595327, -5.7569606052102630173202},
+    {10233, 1123., 800., -3343.1303902972739556477, 3.37480997013700672692981,
+     -2.98980536947426087305154},
+    {10233, 1123., 15324., -11352.0607180353268830661,
+     7.55829905834770485145599, -0.113119180727235553840235},
+    {10233, 10586., 0.000400000000000000019168694, -17.0607181926565361138193,
+     -1.26000103249281892884057e-9, 2492.75189435765511506763},
+    {10233, 10586., 0.065000000000000002220446, -12.175300607237463417773,
+     -2.04748918320454911628704e-7, 13.1257236766090675675961},
+    {10233, 10586., 4.41999999999999992894573, -9.43079204562662170966662,
+     -0.0000139172010501341586295487, 0.116749372001926601548071},
+    {10233, 10586., 800., -7.27036619721226178082317,
+     -0.002342942461614156602606, 0.0000890483066241868231597611},
+    {10233, 10586., 15324., -9.3278240351663417156768,
+     -0.0197218452194275363359762, -0.0000805948075011592832299515},
+};
+}  // namespace neg_binomial_2_test_internal
+
+TEST(ProbDistributionsNegativeBinomial2, derivativesPrecomputed) {
+  using neg_binomial_2_test_internal::TestValue;
+  using neg_binomial_2_test_internal::testValues;
   using stan::math::is_nan;
-  using stan::math::neg_binomial_2_lpmf;
+  using stan::math::neg_binomial_2_log;
+  using stan::math::value_of;
   using stan::math::var;
 
-  int N = 100;
-  double mu_dbl = 8;
-  double phi_dbl = 1.5;
-
-  for (int k = 0; k < 20; ++k) {
-    var mu(mu_dbl);
-    var phi(phi_dbl);
-    var val = neg_binomial_2_lpmf(N, mu, phi);
+  for (TestValue t : testValues) {
+    int n = t.n;
+    var mu(t.mu);
+    var phi(t.phi);
+    var val = neg_binomial_2_log(n, mu, phi);
 
     std::vector<var> x;
     x.push_back(mu);
@@ -27,22 +456,146 @@ TEST(ProbDistributionsNegBinomial2, derivatives) {
       EXPECT_FALSE(is_nan(gradients[i]));
     }
 
-    std::vector<double> finite_diffs;
-    double eps = 1e-10;
-    double inv2e = 0.5 / eps;
+    auto tolerance = [](double x) { return std::max(fabs(x * 1e-8), 1e-14); };
 
-    double dmu = neg_binomial_2_lpmf(N, mu_dbl + eps, phi_dbl)
-                 - neg_binomial_2_lpmf(N, mu_dbl - eps, phi_dbl);
-    double dphi = neg_binomial_2_lpmf(N, mu_dbl, phi_dbl + eps)
-                  - neg_binomial_2_lpmf(N, mu_dbl, phi_dbl - eps);
-    finite_diffs.push_back(dmu * inv2e);
-    finite_diffs.push_back(dphi * inv2e);
+    EXPECT_NEAR(value_of(val), t.value, tolerance(t.value))
+        << "value n = " << n << ", mu = " << t.mu << ", phi = " << t.phi;
+    EXPECT_NEAR(gradients[0], t.grad_mu, tolerance(t.grad_mu))
+        << "grad_mu n = " << n << ", mu = " << t.mu << ", phi = " << t.phi;
+    EXPECT_NEAR(gradients[1], t.grad_phi, tolerance(t.grad_phi))
+        << "grad_phi n = " << n << ", mu = " << t.mu << ", phi = " << t.phi;
+  }
+}
 
-    for (int i = 0; i < 2; ++i) {
-      EXPECT_NEAR(gradients[i], finite_diffs[i], 1.0);
+TEST(ProbDistributionsNegBinomial2, derivativesComplexStep) {
+  using boost::math::differentiation::complex_step_derivative;
+  using stan::math::is_nan;
+  using stan::math::neg_binomial_2_log;
+  using stan::math::var;
+  using stan::test::internal::expect_near_rel_finite;
+
+  std::vector<int> n_to_test = {0, 7, 100, 835, 14238, 385000, 1000000};
+  std::vector<double> mu_to_test = {0.8, 8, 24, 271, 2586, 33294};
+
+  auto nb2_log_for_test = [](int n, const std::complex<double>& mu,
+                             const std::complex<double>& phi) {
+    // Using first-order Taylor expansion of lgamma(a + b*i) around b = 0
+    // Which happens to work nice in this case, as b is always 0 or the very
+    // small complex step
+    auto lgamma_c_approx = [](const std::complex<double>& x) {
+      return std::complex<double>(lgamma(x.real()),
+                                  x.imag() * boost::math::digamma(x.real()));
+    };
+
+    const double n_(n);
+    return lgamma_c_approx(n_ + phi) - lgamma(n + 1) - lgamma_c_approx(phi)
+           + phi * (log(phi) - log(mu + phi)) - n_ * log(mu + phi)
+           + n_ * log(mu);
+  };
+
+  for (double mu_dbl : mu_to_test) {
+    for (int n : n_to_test) {
+      for (double phi_dbl = 1.5; phi_dbl < 1e22; phi_dbl *= 10) {
+        var mu(mu_dbl);
+        var phi(phi_dbl);
+        var val = neg_binomial_2_lpmf(n, mu, phi);
+
+        std::vector<var> x;
+        x.push_back(mu);
+        x.push_back(phi);
+
+        std::vector<double> gradients;
+        val.grad(x, gradients);
+
+        EXPECT_TRUE(value_of(val) < 0)
+            << "for n = " << n << ", mu = " << mu_dbl << ", phi = " << phi_dbl;
+
+        for (int i = 0; i < 2; ++i) {
+          EXPECT_FALSE(is_nan(gradients[i]));
+        }
+
+        auto nb2_log_mu
+            = [n, phi_dbl, nb2_log_for_test](const std::complex<double>& mu) {
+                return nb2_log_for_test(n, mu, phi_dbl);
+              };
+        auto nb2_log_phi
+            = [n, mu_dbl, nb2_log_for_test](const std::complex<double>& phi) {
+                return nb2_log_for_test(n, mu_dbl, phi);
+              };
+        double complex_step_dmu = complex_step_derivative(nb2_log_mu, mu_dbl);
+        double complex_step_dphi
+            = complex_step_derivative(nb2_log_phi, phi_dbl);
+
+        std::stringstream message;
+        message << ", n = " << n << ", mu = " << mu_dbl
+                << ", phi = " << phi_dbl;
+
+        double tolerance_phi = std::max(1e-8, fabs(gradients[1]) * 1e-5);
+        double tolerance_mu = std::max(1e-10, fabs(gradients[0]) * 1e-8);
+
+        EXPECT_NEAR(gradients[0], complex_step_dmu, tolerance_mu)
+            << "grad_mu" << message.str();
+
+        EXPECT_NEAR(gradients[1], complex_step_dphi, tolerance_phi)
+            << "grad_phi" << message.str();
+      }
     }
+  }
+}
+
+TEST(ProbDistributionsNegBinomial2, derivativesZeroOne) {
+  using stan::math::var;
+  using stan::test::expect_near_rel;
+
+  std::vector<double> mu_to_test = {2.345e-5, 0.2, 13, 150, 1621, 18432, 1e10};
+  double phi_start = 1e-8;
+  double phi_max = 1e20;
+  for (double mu_dbl : mu_to_test) {
+    for (double phi_dbl = phi_start; phi_dbl < phi_max;
+         phi_dbl *= stan::math::pi()) {
+      std::stringstream msg;
+      msg << std::setprecision(20) << ", mu = " << mu_dbl
+          << ", phi = " << phi_dbl;
 
-    phi_dbl *= 10;
+      var mu0(mu_dbl);
+      var phi0(phi_dbl);
+      var val0 = neg_binomial_2_lpmf(0, mu0, phi0);
+
+      std::vector<var> x0;
+      x0.push_back(mu0);
+      x0.push_back(phi0);
+
+      std::vector<double> gradients0;
+      val0.grad(x0, gradients0);
+
+      var mu1(mu_dbl);
+      var phi1(phi_dbl);
+      var val1 = neg_binomial_2_lpmf(1, mu1, phi1);
+
+      std::vector<var> x1;
+      x1.push_back(mu1);
+      x1.push_back(phi1);
+
+      std::vector<double> gradients1;
+      val1.grad(x1, gradients1);
+
+      double expected_dmu_0 = -phi_dbl / (mu_dbl + phi_dbl);
+      double expected_dphi_0
+          = mu_dbl / (mu_dbl + phi_dbl) - log1p(mu_dbl / phi_dbl);
+      expect_near_rel("dmu, n = 0 " + msg.str(), gradients0[0], expected_dmu_0);
+      expect_near_rel("dphi, n = 0 " + msg.str(), gradients0[1],
+                      expected_dphi_0);
+
+      double expected_dmu_1
+          = (phi_dbl * (1 - mu_dbl)) / (mu_dbl * (mu_dbl + phi_dbl));
+      expect_near_rel("dmu, n = 1 " + msg.str(), gradients1[0], expected_dmu_1);
+
+      double expected_dphi_1
+          = mu_dbl * (phi_dbl + 1) / (phi_dbl * (mu_dbl + phi_dbl))
+            + log(phi_dbl) - log(mu_dbl + phi_dbl);
+      expect_near_rel("dphi, n = 1 " + msg.str(), gradients1[1],
+                      expected_dphi_1);
+    }
   }
 }