diff --git a/stan/math/opencl/kernel_generator/elt_function_cl.hpp b/stan/math/opencl/kernel_generator/elt_function_cl.hpp
index 8c2c66ac226..e7dedc1c315 100644
--- a/stan/math/opencl/kernel_generator/elt_function_cl.hpp
+++ b/stan/math/opencl/kernel_generator/elt_function_cl.hpp
@@ -307,7 +307,8 @@ ADD_UNARY_FUNCTION_WITH_INCLUDES(inv_square,
                                  opencl_kernels::inv_square_device_function)
 ADD_UNARY_FUNCTION_WITH_INCLUDES(inv_logit,
                                  opencl_kernels::inv_logit_device_function)
-ADD_UNARY_FUNCTION_WITH_INCLUDES(logit, opencl_kernels::logit_device_function)
+ADD_UNARY_FUNCTION_WITH_INCLUDES(logit, opencl_kernels::log1m_device_function,
+                                 opencl_kernels::logit_device_function)
 ADD_UNARY_FUNCTION_WITH_INCLUDES(Phi, opencl_kernels::phi_device_function)
 ADD_UNARY_FUNCTION_WITH_INCLUDES(Phi_approx,
                                  opencl_kernels::inv_logit_device_function,
diff --git a/stan/math/opencl/kernels/device_functions/Phi.hpp b/stan/math/opencl/kernels/device_functions/Phi.hpp
index ba7ecb6a51f..6946fc6fd28 100644
--- a/stan/math/opencl/kernels/device_functions/Phi.hpp
+++ b/stan/math/opencl/kernels/device_functions/Phi.hpp
@@ -24,11 +24,11 @@ static const char* phi_device_function
             if (x < -37.5) {
               return 0;
             } else if (x < -5.0) {
-              return 0.5 * erfc(-1.0 / sqrt(2.0) * x);
+              return 0.5 * erfc(-M_SQRT1_2 * x);
             } else if (x > 8.25) {
               return 1;
             } else {
-              return 0.5 * (1.0 + erf(1.0 / sqrt(2.0) * x));
+              return 0.5 * (1.0 + erf(M_SQRT1_2 * x));
             }
           }
           // \cond
diff --git a/stan/math/opencl/kernels/device_functions/inv_logit.hpp b/stan/math/opencl/kernels/device_functions/inv_logit.hpp
index 9cb9d56cc27..34c526e4fae 100644
--- a/stan/math/opencl/kernels/device_functions/inv_logit.hpp
+++ b/stan/math/opencl/kernels/device_functions/inv_logit.hpp
@@ -56,7 +56,7 @@ static const char* inv_logit_device_function
            */
           double inv_logit(double x) {
             if (x < 0) {
-              if (x < log(2.2204460492503131E-16)) {
+              if (x < log(DBL_EPSILON)) {
                 return exp(x);
               }
               return exp(x) / (1 + exp(x));
diff --git a/stan/math/opencl/kernels/device_functions/lbeta.hpp b/stan/math/opencl/kernels/device_functions/lbeta.hpp
index f0aeb8061fb..0e8e4d139d3 100644
--- a/stan/math/opencl/kernels/device_functions/lbeta.hpp
+++ b/stan/math/opencl/kernels/device_functions/lbeta.hpp
@@ -95,12 +95,13 @@ static const char* lbeta_device_function
               return lgamma(x) + lgamma(y) - lgamma(x + y);
             }
             double x_over_xy = x / (x + y);
+            double log_xpy = log(x + y);
             if (x < LGAMMA_STIRLING_DIFF_USEFUL) {
               // y large, x small
               double stirling_diff
                   = lgamma_stirling_diff(y) - lgamma_stirling_diff(x + y);
               double stirling
-                  = (y - 0.5) * log1p(-x_over_xy) + x * (1 - log(x + y));
+                  = (y - 0.5) * log1p(-x_over_xy) + x * (1 - log_xpy);
               return stirling + lgamma(x) + stirling_diff;
             }
 
@@ -108,8 +109,9 @@ static const char* lbeta_device_function
             double stirling_diff = lgamma_stirling_diff(x)
                                    + lgamma_stirling_diff(y)
                                    - lgamma_stirling_diff(x + y);
-            double stirling = (x - 0.5) * log(x_over_xy) + y * log1p(-x_over_xy)
-                              + 0.5 * log(2.0 * M_PI) - 0.5 * log(y);
+            double stirling = (x - 0.5) * (log(x) - log_xpy)
+                              + y * log1p(-x_over_xy)
+                              + 0.5 * (M_LN2 + log(M_PI)) - 0.5 * log(y);
             return stirling + stirling_diff;
           }
           // \cond
diff --git a/stan/math/opencl/kernels/device_functions/lgamma_stirling.hpp b/stan/math/opencl/kernels/device_functions/lgamma_stirling.hpp
index 19fc87b968f..3934fe11600 100644
--- a/stan/math/opencl/kernels/device_functions/lgamma_stirling.hpp
+++ b/stan/math/opencl/kernels/device_functions/lgamma_stirling.hpp
@@ -28,7 +28,7 @@ static const char* lgamma_stirling_device_function
            * @return Stirling's approximation to lgamma(x).
            */
           double lgamma_stirling(double x) {
-            return 0.5 * log(2.0 * M_PI) + (x - 0.5) * log(x) - x;
+            return 0.5 * (M_LN2 + log(M_PI)) + (x - 0.5) * log(x) - x;
           }
           // \cond
           ) "\n#endif\n";  // NOLINT
diff --git a/stan/math/opencl/kernels/device_functions/logit.hpp b/stan/math/opencl/kernels/device_functions/logit.hpp
index 8ec1bf8850b..b0b4524214e 100644
--- a/stan/math/opencl/kernels/device_functions/logit.hpp
+++ b/stan/math/opencl/kernels/device_functions/logit.hpp
@@ -49,7 +49,7 @@ static const char* logit_device_function
           * @param x argument
           * @return log odds of argument
           */
-          double logit(double x) { return log(x / (1 - x)); }
+          double logit(double x) { return log(x) - log1m(x); }
           // \cond
           ) "\n#endif\n";  // NOLINT
 // \endcond
diff --git a/stan/math/opencl/kernels/neg_binomial_2_log_glm_lpmf.hpp b/stan/math/opencl/kernels/neg_binomial_2_log_glm_lpmf.hpp
index f79b4449f69..0585df47654 100644
--- a/stan/math/opencl/kernels/neg_binomial_2_log_glm_lpmf.hpp
+++ b/stan/math/opencl/kernels/neg_binomial_2_log_glm_lpmf.hpp
@@ -4,6 +4,7 @@
 
 #include <stan/math/opencl/kernel_cl.hpp>
 #include <stan/math/opencl/kernels/device_functions/digamma.hpp>
+#include <stan/math/opencl/kernels/device_functions/log1p_exp.hpp>
 
 namespace stan {
 namespace math {
@@ -92,9 +93,9 @@ static const char* neg_binomial_2_log_glm_kernel_code = STRINGIFY(
         double log_phi = log(phi);
         double logsumexp_theta_logphi;
         if (theta > log_phi) {
-          logsumexp_theta_logphi = theta + log1p(exp(log_phi - theta));
+          logsumexp_theta_logphi = theta + log1p_exp(log_phi - theta);
         } else {
-          logsumexp_theta_logphi = log_phi + log1p(exp(theta - log_phi));
+          logsumexp_theta_logphi = log_phi + log1p_exp(theta - log_phi);
         }
         double y_plus_phi = y + phi;
         if (need_logp1) {
@@ -196,7 +197,7 @@ const kernel_cl<out_buffer, out_buffer, out_buffer, out_buffer, in_buffer,
                 in_buffer, in_buffer, in_buffer, in_buffer, int, int, int, int,
                 int, int, int, int, int, int, int, int, int>
     neg_binomial_2_log_glm("neg_binomial_2_log_glm",
-                           {digamma_device_function,
+                           {digamma_device_function, log1p_exp_device_function,
                             neg_binomial_2_log_glm_kernel_code},
                            {{"REDUCTION_STEP_SIZE", 4}, {"LOCAL_SIZE_", 64}});
 
diff --git a/stan/math/opencl/kernels/ordered_logistic_glm_lpmf.hpp b/stan/math/opencl/kernels/ordered_logistic_glm_lpmf.hpp
index 74341ff76cc..3b1727b1aa9 100644
--- a/stan/math/opencl/kernels/ordered_logistic_glm_lpmf.hpp
+++ b/stan/math/opencl/kernels/ordered_logistic_glm_lpmf.hpp
@@ -5,6 +5,7 @@
 #include <stan/math/opencl/kernel_cl.hpp>
 #include <stan/math/opencl/kernels/device_functions/log1m_exp.hpp>
 #include <stan/math/opencl/kernels/device_functions/log1p_exp.hpp>
+#include <stan/math/opencl/kernels/device_functions/inv_logit.hpp>
 
 namespace stan {
 namespace math {
@@ -87,20 +88,10 @@ static const char* ordered_logistic_glm_kernel_code = STRINGIFY(
 
           if (need_location_derivative || need_cuts_derivative) {
             double exp_cuts_diff = exp(cut_y2 - cut_y1);
-            if (cut2 > 0) {
-              double exp_m_cut2 = exp(-cut2);
-              d1 = exp_m_cut2 / (1 + exp_m_cut2);
-            } else {
-              d1 = 1 / (1 + exp(cut2));
-            }
+            d1 = inv_logit(-cut2);
             d1 -= exp_cuts_diff / (exp_cuts_diff - 1);
             d2 = 1 / (1 - exp_cuts_diff);
-            if (cut1 > 0) {
-              double exp_m_cut1 = exp(-cut1);
-              d2 -= exp_m_cut1 / (1 + exp_m_cut1);
-            } else {
-              d2 -= 1 / (1 + exp(cut1));
-            }
+            d2 -= inv_logit(-cut1);
 
             if (need_location_derivative) {
               location_derivative[gid] = d1 - d2;
@@ -181,6 +172,7 @@ const kernel_cl<out_buffer, out_buffer, out_buffer, out_buffer, in_buffer,
                 in_buffer, in_buffer, in_buffer, int, int, int, int, int, int>
     ordered_logistic_glm("ordered_logistic_glm",
                          {log1p_exp_device_function, log1m_exp_device_function,
+                          inv_logit_device_function,
                           ordered_logistic_glm_kernel_code},
                          {{"REDUCTION_STEP_SIZE", 4}, {"LOCAL_SIZE_", 64}});
 
diff --git a/stan/math/opencl/kernels/ordered_logistic_lpmf.hpp b/stan/math/opencl/kernels/ordered_logistic_lpmf.hpp
index b06a6d47a46..7b5dbddc169 100644
--- a/stan/math/opencl/kernels/ordered_logistic_lpmf.hpp
+++ b/stan/math/opencl/kernels/ordered_logistic_lpmf.hpp
@@ -5,6 +5,7 @@
 #include <stan/math/opencl/kernel_cl.hpp>
 #include <stan/math/opencl/kernels/device_functions/log1m_exp.hpp>
 #include <stan/math/opencl/kernels/device_functions/log1p_exp.hpp>
+#include <stan/math/opencl/kernels/device_functions/inv_logit.hpp>
 
 namespace stan {
 namespace math {
@@ -83,20 +84,10 @@ static const char* ordered_logistic_kernel_code = STRINGIFY(
 
           if (need_lambda_derivative || need_cuts_derivative) {
             double exp_cuts_diff = exp(cut_y2 - cut_y1);
-            if (cut2 > 0) {
-              double exp_m_cut2 = exp(-cut2);
-              d1 = exp_m_cut2 / (1 + exp_m_cut2);
-            } else {
-              d1 = 1 / (1 + exp(cut2));
-            }
+            d1 = inv_logit(-cut2);
             d1 -= exp_cuts_diff / (exp_cuts_diff - 1);
             d2 = 1 / (1 - exp_cuts_diff);
-            if (cut1 > 0) {
-              double exp_m_cut1 = exp(-cut1);
-              d2 -= exp_m_cut1 / (1 + exp_m_cut1);
-            } else {
-              d2 -= 1 / (1 + exp(cut1));
-            }
+            d2 -= inv_logit(-cut1);
 
             if (need_lambda_derivative) {
               lambda_derivative[gid] = d1 - d2;
@@ -175,7 +166,7 @@ const kernel_cl<out_buffer, out_buffer, out_buffer, in_buffer, in_buffer,
                 in_buffer, int, int, int, int, int, int>
     ordered_logistic("ordered_logistic",
                      {log1p_exp_device_function, log1m_exp_device_function,
-                      ordered_logistic_kernel_code},
+                      inv_logit_device_function, ordered_logistic_kernel_code},
                      {{"REDUCTION_STEP_SIZE", 4}, {"LOCAL_SIZE_", 64}});
 
 }  // namespace opencl_kernels
diff --git a/stan/math/opencl/kernels/tridiagonalization.hpp b/stan/math/opencl/kernels/tridiagonalization.hpp
index b0bf3d3288a..0e0c11ca5ed 100644
--- a/stan/math/opencl/kernels/tridiagonalization.hpp
+++ b/stan/math/opencl/kernels/tridiagonalization.hpp
@@ -84,7 +84,7 @@ static const char* tridiagonalization_householder_kernel_code = STRINGIFY(
       q = q_local[0];
       alpha = q_local[1];
       if (q != 0) {
-        double multi = sqrt(2.) / q;
+        double multi = M_SQRT2 / q;
         // normalize the Householder vector
         for (int i = lid + 1; i < P_span; i += lsize) {
           P[P_start + i] *= multi;
@@ -92,7 +92,7 @@ static const char* tridiagonalization_householder_kernel_code = STRINGIFY(
       }
       if (gid == 0) {
         P[P_rows * (k + j + 1) + k + j]
-            = P[P_rows * (k + j) + k + j + 1] * q / sqrt(2.) + alpha;
+            = P[P_rows * (k + j) + k + j + 1] * q / M_SQRT2 + alpha;
       }
     }
     // \cond
@@ -291,7 +291,7 @@ static const char* tridiagonalization_v_step_3_kernel_code = STRINGIFY(
         v[i] -= acc * u[i];
       }
       if (gid == 0) {
-        P[P_rows * (k + j + 1) + k + j] -= *q / sqrt(2.) * u[0];
+        P[P_rows * (k + j + 1) + k + j] -= *q / M_SQRT2 * u[0];
       }
     }
     // \cond