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Add kernel generator cast operation #2472

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merged 5 commits into from
May 5, 2021
Merged

Add kernel generator cast operation #2472

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aa77ead
add cast operation and use it in distributions
t4c1 Apr 16, 2021
a3cdcd8
[Jenkins] auto-formatting by clang-format version 6.0.0-1ubuntu2~16.0…
stan-buildbot Apr 16, 2021
1722825
fixed cpplint and test names
t4c1 Apr 16, 2021
3962c69
Merge branch 'kg_cast' of https://github.com/bstatcomp/math into kg_cast
t4c1 Apr 16, 2021
fcaff77
bugfix inv_gamma_lpdf
t4c1 Apr 19, 2021
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1 change: 1 addition & 0 deletions stan/math/opencl/kernel_generator.hpp
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Expand Up @@ -131,6 +131,7 @@
#include <stan/math/opencl/kernel_generator/index.hpp>
#include <stan/math/opencl/kernel_generator/indexing.hpp>
#include <stan/math/opencl/kernel_generator/opencl_code.hpp>
#include <stan/math/opencl/kernel_generator/cast.hpp>

#include <stan/math/opencl/kernel_generator/multi_result_kernel.hpp>
#include <stan/math/opencl/kernel_generator/get_kernel_source_for_evaluating_into.hpp>
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101 changes: 101 additions & 0 deletions stan/math/opencl/kernel_generator/cast.hpp
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@@ -0,0 +1,101 @@
#ifndef STAN_MATH_OPENCL_KERNEL_GENERATOR_CAST_HPP
#define STAN_MATH_OPENCL_KERNEL_GENERATOR_CAST_HPP
#ifdef STAN_OPENCL

#include <stan/math/prim/meta.hpp>
#include <stan/math/opencl/matrix_cl_view.hpp>
#include <stan/math/opencl/kernel_generator/common_return_scalar.hpp>
#include <stan/math/opencl/kernel_generator/type_str.hpp>
#include <stan/math/opencl/kernel_generator/name_generator.hpp>
#include <stan/math/opencl/kernel_generator/operation_cl.hpp>
#include <stan/math/opencl/kernel_generator/as_operation_cl.hpp>
#include <array>
#include <string>
#include <type_traits>
#include <set>
#include <utility>

namespace stan {
namespace math {

/** \addtogroup opencl_kernel_generator
* @{
*/

/**
* Represents a typecast os scalar in kernel generator expressions.
* @tparam Derived derived type
* @tparam T type of argument
* @tparam Scal type of the scalar of result
*/
template <typename Scal, typename T>
class cast_ : public operation_cl<cast_<Scal, T>, Scal, T> {
public:
using Scalar = Scal;
using base = operation_cl<cast_<Scal, T>, Scalar, T>;
using base::var_name_;

/**
* Constructor
* @param args argument expression(s)
*/
explicit cast_(T&& arg) : base(std::forward<T>(arg)) {}

/**
* Generates kernel code for this expression.
* @param row_index_name row index variable name
* @param col_index_name column index variable name
* @param view_handled whether whether caller already handled matrix view
* @param var_names_arg variable names of the nested expressions
* @return part of kernel with code for this expression
*/
inline kernel_parts generate(const std::string& row_index_name,
const std::string& col_index_name,
const bool view_handled,
const std::string& var_name_arg) const {
kernel_parts res{};

res.body = type_str<Scalar>() + " " + var_name_ + " = ("
+ type_str<Scalar>() + ")" + var_name_arg + ";\n";
Comment on lines +58 to +59
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So this line here generates Scalar var_name = (Scalar)input_var_name;? If so then idt I'm understanding how this works for char?

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Yes. I don't understand what is the confusion with char. It works exactly the same as for any other supported scalar type.

Side note, not necessary for this PR, but I think this is an example where it would be nice to have something in the tests or an easy scheme available from runTests.py for generating the OpenCL kernel code.

We do have reference kernels in tests, although I am not using them for operations that generate really trivial code, such as typecasting in this PR.

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So I'm looking at

auto any_n_negative = colwise_max(cast<char>(n < 0));

And here this is turned into

char var_name = (char)input_var_name;

Apologies if this is silly but I'm not following how that is used in an if etc. Like is it that if input_var_name is 0 then the char is 0 and the if is false?

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Yes, exactly. input_var_name is in this case the result of the comparison n < 0, which is either 0 or 1. 0 or 1 converted into char is still 0 or 1. That gets written into global memory and than copied to host. Is there still anything you are not understanding?

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Aight then I think that makes sense

return res;
}

inline auto deep_copy() const {
auto&& arg_copy = this->template get_arg<0>().deep_copy();
return cast_<Scalar, std::remove_reference_t<decltype(arg_copy)>>{
std::move(arg_copy)};
}
};

/**
* Typecast a kernel generator expression scalar.
*
* @tparam T type of argument
* @param a input argument
* @return Typecast of given expression
*/
template <typename Scalar, typename T,
require_all_kernel_expressions_and_none_scalar_t<T>* = nullptr>
inline auto cast(T&& a) {
auto&& a_operation = as_operation_cl(std::forward<T>(a)).deep_copy();
return cast_<Scalar, std::remove_reference_t<decltype(a_operation)>>(
std::move(a_operation));
}

/**
* Typecast a scalar.
*
* @tparam T type of argument
* @param a input argument
* @return Typecast of given expression
*/
template <typename Scalar, typename T, require_stan_scalar_t<T>* = nullptr>
inline Scalar cast(T a) {
return a;
}

/** @}*/
} // namespace math
} // namespace stan
#endif
#endif
4 changes: 2 additions & 2 deletions stan/math/opencl/prim/bernoulli_cdf.hpp
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Expand Up @@ -49,13 +49,13 @@ return_type_t<T_prob_cl> bernoulli_cdf(const T_n_cl& n,
theta_val, "in the interval [0, 1]");
auto theta_bounded_expr = 0.0 <= theta_val && theta_val <= 1.0;

auto any_n_negative = colwise_max(constant(0, N, 1) + (n < 0));
auto any_n_negative = colwise_max(cast<char>(n < 0));
auto cond = n >= 1;
auto Pi_uncond = 1.0 - theta_val;
auto Pi = select(cond, INFTY, Pi_uncond);
auto P_expr = colwise_prod(select(cond, 1.0, Pi_uncond));

matrix_cl<double> any_n_negative_cl;
matrix_cl<char> any_n_negative_cl;
matrix_cl<double> Pi_cl;
matrix_cl<double> P_cl;

Expand Down
8 changes: 4 additions & 4 deletions stan/math/opencl/prim/bernoulli_lccdf.hpp
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Expand Up @@ -50,13 +50,13 @@ return_type_t<T_prob_cl> bernoulli_lccdf(const T_n_cl& n,
theta_val, "in the interval [0, 1]");
auto theta_bounded_expr = 0.0 <= theta_val && theta_val <= 1.0;

auto any_n_negative = colwise_max(0 + (n < 0));
auto any_n_over_one = colwise_max(constant(0, N, 1) + (n >= 1));
auto any_n_negative = colwise_max(cast<char>(n < 0));
auto any_n_over_one = colwise_max(cast<char>(n >= 1));
auto P_expr = colwise_sum(log(theta_val));
auto deriv = elt_divide(1.0, theta_val);

matrix_cl<double> any_n_negative_cl;
matrix_cl<double> any_n_over_one_cl;
matrix_cl<char> any_n_negative_cl;
matrix_cl<char> any_n_over_one_cl;
matrix_cl<double> P_cl;
matrix_cl<double> deriv_cl;

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4 changes: 2 additions & 2 deletions stan/math/opencl/prim/bernoulli_lcdf.hpp
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Expand Up @@ -50,13 +50,13 @@ return_type_t<T_prob_cl> bernoulli_lcdf(const T_n_cl& n,
theta_val, "in the interval [0, 1]");
auto theta_bounded_expr = 0.0 <= theta_val && theta_val <= 1.0;

auto any_n_negative = colwise_max(0 + (n < 0));
auto any_n_negative = colwise_max(cast<char>(n < 0));
auto Pi = 1.0 - theta_val;
auto cond = n >= 1;
auto P_expr = colwise_sum(select(cond, 0.0, log(Pi)));
auto deriv = select(cond, 0.0, elt_divide(-1.0, Pi));

matrix_cl<double> any_n_negative_cl;
matrix_cl<char> any_n_negative_cl;
matrix_cl<double> P_cl;
matrix_cl<double> deriv_cl;

Expand Down
5 changes: 2 additions & 3 deletions stan/math/opencl/prim/cauchy_cdf.hpp
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Expand Up @@ -63,8 +63,7 @@ return_type_t<T_y_cl, T_loc_cl, T_scale_cl> cauchy_cdf(
= check_cl(function, "Scale parameter", sigma_val, "positive finite");
auto sigma_positive_finite_expr = 0 < sigma_val && isfinite(sigma_val);

auto any_y_neg_inf
= colwise_max(constant(0, N, 1) + (y_val == NEGATIVE_INFTY));
auto any_y_neg_inf = colwise_max(cast<char>(y_val == NEGATIVE_INFTY));
auto cond = y_val == INFTY;
auto sigma_inv = elt_divide(1.0, sigma_val);
auto z = elt_multiply(y_val - mu_val, sigma_inv);
Expand All @@ -76,7 +75,7 @@ return_type_t<T_y_cl, T_loc_cl, T_scale_cl> cauchy_cdf(
elt_divide(sigma_inv, -pi() * elt_multiply(1.0 + square(z), Pn)));
auto sigma_deriv_tmp = elt_multiply(z, mu_deriv_tmp);

matrix_cl<double> any_y_neg_inf_cl;
matrix_cl<char> any_y_neg_inf_cl;
matrix_cl<double> P_cl;
matrix_cl<double> mu_deriv_cl;
matrix_cl<double> y_deriv_cl;
Expand Down
5 changes: 2 additions & 3 deletions stan/math/opencl/prim/exp_mod_normal_cdf.hpp
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Expand Up @@ -72,8 +72,7 @@ return_type_t<T_y_cl, T_loc_cl, T_scale_cl, T_inv_scale_cl> exp_mod_normal_cdf(
= check_cl(function, "Inv_cale parameter", lambda_val, "positive finite");
auto lambda_positive_finite_expr = 0 < lambda_val && isfinite(lambda_val);

auto any_y_neg_inf
= colwise_max(constant(0, N, 1) + (y_val == NEGATIVE_INFTY));
auto any_y_neg_inf = colwise_max(cast<char>(y_val == NEGATIVE_INFTY));
auto inv_sigma = elt_divide(1.0, sigma_val);
auto diff = y_val - mu_val;
auto v = elt_multiply(lambda_val, sigma_val);
Expand Down Expand Up @@ -102,7 +101,7 @@ return_type_t<T_y_cl, T_loc_cl, T_scale_cl, T_inv_scale_cl> exp_mod_normal_cdf(
- elt_multiply(elt_multiply(v, sigma_val) - diff, erf_calc)),
cdf_n);

matrix_cl<double> any_y_neg_inf_cl;
matrix_cl<char> any_y_neg_inf_cl;
matrix_cl<double> cdf_cl;
matrix_cl<double> y_deriv_cl;
matrix_cl<double> mu_deriv_cl;
Expand Down
9 changes: 4 additions & 5 deletions stan/math/opencl/prim/exp_mod_normal_lccdf.hpp
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Expand Up @@ -73,9 +73,8 @@ exp_mod_normal_lccdf(const T_y_cl& y, const T_loc_cl& mu,
= check_cl(function, "Inv_cale parameter", lambda_val, "positive finite");
auto lambda_positive_finite_expr = 0 < lambda_val && isfinite(lambda_val);

auto any_y_neg_inf
= colwise_max(constant(0, N, 1) + (y_val == NEGATIVE_INFTY));
auto any_y_pos_inf = colwise_max(constant(0, N, 1) + (y_val == INFTY));
auto any_y_neg_inf = colwise_max(cast<char>(y_val == NEGATIVE_INFTY));
auto any_y_pos_inf = colwise_max(cast<char>(y_val == INFTY));
auto inv_sigma = elt_divide(1.0, sigma_val);
auto diff = y_val - mu_val;
auto scaled_diff = elt_multiply(diff, inv_sigma * INV_SQRT_TWO);
Expand Down Expand Up @@ -104,8 +103,8 @@ exp_mod_normal_lccdf(const T_y_cl& y, const T_loc_cl& mu,
- INV_SQRT_TWO_PI * elt_multiply(sigma_val, exp_term_2)),
ccdf_n);

matrix_cl<double> any_y_neg_inf_cl;
matrix_cl<double> any_y_pos_inf_cl;
matrix_cl<char> any_y_neg_inf_cl;
matrix_cl<char> any_y_pos_inf_cl;
matrix_cl<double> ccdf_log_cl;
matrix_cl<double> mu_deriv_cl;
matrix_cl<double> y_deriv_cl;
Expand Down
9 changes: 4 additions & 5 deletions stan/math/opencl/prim/exp_mod_normal_lcdf.hpp
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Expand Up @@ -73,9 +73,8 @@ return_type_t<T_y_cl, T_loc_cl, T_scale_cl, T_inv_scale_cl> exp_mod_normal_lcdf(
= check_cl(function, "Inv_cale parameter", lambda_val, "positive finite");
auto lambda_positive_finite_expr = 0 < lambda_val && isfinite(lambda_val);

auto any_y_neg_inf
= colwise_max(constant(0, N, 1) + (y_val == NEGATIVE_INFTY));
auto any_y_pos_inf = colwise_max(constant(0, N, 1) + (y_val == INFTY));
auto any_y_neg_inf = colwise_max(cast<char>(y_val == NEGATIVE_INFTY));
auto any_y_pos_inf = colwise_max(cast<char>(y_val == INFTY));
auto sigma_inv = elt_divide(1.0, sigma_val);
auto diff = y_val - mu_val;
auto scaled_diff = elt_multiply(diff * INV_SQRT_TWO, sigma_inv);
Expand Down Expand Up @@ -105,8 +104,8 @@ return_type_t<T_y_cl, T_loc_cl, T_scale_cl, T_inv_scale_cl> exp_mod_normal_lcdf(
- elt_multiply(elt_multiply(v, sigma_val) - diff, erf_calc)),
cdf_n);

matrix_cl<double> any_y_neg_inf_cl;
matrix_cl<double> any_y_pos_inf_cl;
matrix_cl<char> any_y_neg_inf_cl;
matrix_cl<char> any_y_pos_inf_cl;
matrix_cl<double> cdf_log_cl;
matrix_cl<double> mu_deriv_cl;
matrix_cl<double> y_deriv_cl;
Expand Down
4 changes: 2 additions & 2 deletions stan/math/opencl/prim/gamma_lpdf.hpp
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Expand Up @@ -80,7 +80,7 @@ return_type_t<T_y_cl, T_shape_cl, T_inv_scale_cl> gamma_lpdf(
beta_val, "positive finite");
auto beta_pos_finite_expr = beta_val > 0 && isfinite(beta_val);

auto any_y_negative_expr = colwise_max(constant(0, N, 1) + (y_val < 0));
auto any_y_negative_expr = colwise_max(cast<char>(y_val < 0));
auto log_y_expr = log(y_val);
auto log_beta_expr = log(beta_val);
auto logp1_expr = static_select<include_summand<propto, T_shape_cl>::value>(
Expand All @@ -99,7 +99,7 @@ return_type_t<T_y_cl, T_shape_cl, T_inv_scale_cl> gamma_lpdf(
auto alpha_deriv_expr = log_beta_expr + log_y_expr - digamma(alpha_val);
auto beta_deriv_expr = elt_divide(alpha_val, beta_val) - y_val;

matrix_cl<int> any_y_negative_cl;
matrix_cl<char> any_y_negative_cl;
matrix_cl<double> logp_cl;
matrix_cl<double> y_deriv_cl;
matrix_cl<double> alpha_deriv_cl;
Expand Down
4 changes: 2 additions & 2 deletions stan/math/opencl/prim/inv_chi_square_lpdf.hpp
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Expand Up @@ -74,7 +74,7 @@ return_type_t<T_y_cl, T_dof_cl> inv_chi_square_lpdf(const T_y_cl& y,
= check_cl(function, "Random variable", y_val, "not NaN");
auto y_not_nan = !isnan(y_val);

auto any_y_nonpositive = colwise_max(constant(0, N, 1) + (y_val <= 0));
auto any_y_nonpositive = colwise_max(cast<char>(y_val <= 0));
auto log_y = log(y_val);
auto half_nu = nu_val * 0.5;
auto two_over_y = elt_divide(0.5, y_val);
Expand All @@ -89,7 +89,7 @@ return_type_t<T_y_cl, T_dof_cl> inv_chi_square_lpdf(const T_y_cl& y,
auto y_deriv = elt_divide(two_over_y - half_nu - 1.0, y_val);
auto nu_deriv = -HALF_LOG_TWO - (digamma(half_nu) + log_y) * 0.5;

matrix_cl<int> any_y_nonpositive_cl;
matrix_cl<char> any_y_nonpositive_cl;
matrix_cl<double> logp_cl;
matrix_cl<double> y_deriv_cl;
matrix_cl<double> nu_deriv_cl;
Expand Down
12 changes: 7 additions & 5 deletions stan/math/opencl/prim/inv_gamma_lpdf.hpp
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Expand Up @@ -76,7 +76,7 @@ return_type_t<T_y_cl, T_shape_cl, T_scale_cl> inv_gamma_lpdf(
= check_cl(function, "Scale parameter", beta_val, "positive finite");
auto beta_pos_finite = beta_val > 0 && isfinite(beta_val);

auto any_y_nonpositive = colwise_max(constant(0, N, 1) + (y_val <= 0));
auto any_y_nonpositive = colwise_max(cast<char>(y_val <= 0));
auto log_y = log(y_val);
auto log_beta = log(beta_val);
auto inv_y = elt_divide(1.0, y_val);
Expand All @@ -98,20 +98,22 @@ return_type_t<T_y_cl, T_shape_cl, T_scale_cl> inv_gamma_lpdf(
auto alpha_deriv = log_beta - digamma(alpha_val) - log_y;
auto beta_deriv = elt_divide(alpha_val, beta_val) - inv_y;

matrix_cl<int> any_y_nonpositive_cl;
matrix_cl<char> any_y_nonpositive_cl;
matrix_cl<double> logp_cl;
matrix_cl<double> y_deriv_cl;
matrix_cl<double> alpha_deriv_cl;
matrix_cl<double> beta_deriv_cl;

results(check_alpha_pos_finite, check_beta_pos_finite, check_y_not_nan,
logp_cl, y_deriv_cl, alpha_deriv_cl, beta_deriv_cl)
= expressions(alpha_pos_finite, beta_pos_finite, y_not_nan, logp_expr,
any_y_nonpositive_cl, logp_cl, y_deriv_cl, alpha_deriv_cl,
beta_deriv_cl)
= expressions(alpha_pos_finite, beta_pos_finite, y_not_nan,
any_y_nonpositive, logp_expr,
calc_if<!is_constant<T_y_cl>::value>(y_deriv),
calc_if<!is_constant<T_shape_cl>::value>(alpha_deriv),
calc_if<!is_constant<T_scale_cl>::value>(beta_deriv));

if (from_matrix_cl(any_y_nonpositive).any()) {
if (from_matrix_cl(any_y_nonpositive_cl).any()) {
return LOG_ZERO;
}

Expand Down
5 changes: 2 additions & 3 deletions stan/math/opencl/prim/logistic_cdf.hpp
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Expand Up @@ -63,8 +63,7 @@ return_type_t<T_y_cl, T_loc_cl, T_scale_cl> logistic_cdf(
= check_cl(function, "Scale parameter", sigma_val, "positive finite");
auto sigma_positive_finite_expr = 0 < sigma_val && isfinite(sigma_val);

auto any_y_neg_inf
= colwise_max(constant(0, N, 1) + (y_val == NEGATIVE_INFTY));
auto any_y_neg_inf = colwise_max(cast<char>(y_val == NEGATIVE_INFTY));
auto cond = y_val == INFTY;
auto inv_sigma = elt_divide(1.0, sigma_val);
auto mu_minus_y_div_sigma = elt_multiply(mu_val - y_val, inv_sigma);
Expand All @@ -78,7 +77,7 @@ return_type_t<T_y_cl, T_loc_cl, T_scale_cl> logistic_cdf(
Pn));
auto sigma_deriv_tmp = elt_multiply(y_deriv_tmp, mu_minus_y_div_sigma);

matrix_cl<double> any_y_neg_inf_cl;
matrix_cl<char> any_y_neg_inf_cl;
matrix_cl<double> P_cl;
matrix_cl<double> mu_deriv_cl;
matrix_cl<double> y_deriv_cl;
Expand Down
9 changes: 4 additions & 5 deletions stan/math/opencl/prim/logistic_lccdf.hpp
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Expand Up @@ -63,9 +63,8 @@ return_type_t<T_y_cl, T_loc_cl, T_scale_cl> logistic_lccdf(
= check_cl(function, "Scale parameter", sigma_val, "positive finite");
auto sigma_positive_finite_expr = 0 < sigma_val && isfinite(sigma_val);

auto any_y_neg_inf
= colwise_max(constant(0, N, 1) + (y_val == NEGATIVE_INFTY));
auto any_y_pos_inf = colwise_max(constant(0, N, 1) + (y_val == INFTY));
auto any_y_neg_inf = colwise_max(cast<char>(y_val == NEGATIVE_INFTY));
auto any_y_pos_inf = colwise_max(cast<char>(y_val == INFTY));
auto inv_sigma = elt_divide(1.0, sigma_val);
auto mu_minus_y_div_sigma = elt_multiply(mu_val - y_val, inv_sigma);
auto exp_scaled_diff = exp(mu_minus_y_div_sigma);
Expand All @@ -78,8 +77,8 @@ return_type_t<T_y_cl, T_loc_cl, T_scale_cl> logistic_lccdf(
auto y_deriv = -mu_deriv;
auto sigma_deriv = elt_multiply(-mu_deriv, mu_minus_y_div_sigma);

matrix_cl<double> any_y_neg_inf_cl;
matrix_cl<double> any_y_pos_inf_cl;
matrix_cl<char> any_y_neg_inf_cl;
matrix_cl<char> any_y_pos_inf_cl;
matrix_cl<double> P_cl;
matrix_cl<double> mu_deriv_cl;
matrix_cl<double> y_deriv_cl;
Expand Down
5 changes: 2 additions & 3 deletions stan/math/opencl/prim/logistic_lcdf.hpp
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Expand Up @@ -63,8 +63,7 @@ return_type_t<T_y_cl, T_loc_cl, T_scale_cl> logistic_lcdf(
= check_cl(function, "Scale parameter", sigma_val, "positive finite");
auto sigma_positive_finite_expr = 0 < sigma_val && isfinite(sigma_val);

auto any_y_neg_inf
= colwise_max(constant(0, N, 1) + (y_val == NEGATIVE_INFTY));
auto any_y_neg_inf = colwise_max(cast<char>(y_val == NEGATIVE_INFTY));
auto cond = y_val == INFTY;
auto inv_sigma = elt_divide(1.0, sigma_val);
auto mu_minus_y_div_sigma = elt_multiply(mu_val - y_val, inv_sigma);
Expand All @@ -78,7 +77,7 @@ return_type_t<T_y_cl, T_loc_cl, T_scale_cl> logistic_lcdf(
auto mu_deriv = -y_deriv;
auto sigma_deriv = elt_multiply(y_deriv, mu_minus_y_div_sigma);

matrix_cl<double> any_y_neg_inf_cl;
matrix_cl<char> any_y_neg_inf_cl;
matrix_cl<double> P_cl;
matrix_cl<double> mu_deriv_cl;
matrix_cl<double> y_deriv_cl;
Expand Down
4 changes: 2 additions & 2 deletions stan/math/opencl/prim/lognormal_cdf.hpp
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Expand Up @@ -63,7 +63,7 @@ return_type_t<T_y_cl, T_loc_cl, T_scale_cl> lognormal_cdf(
= check_cl(function, "Scale parameter", sigma_val, "positive finite");
auto sigma_positive_finite_expr = 0 < sigma_val && isfinite(sigma_val);

auto any_y_zero = colwise_max(constant(0, N, 1) + (y_val == 0.0));
auto any_y_zero = colwise_max(cast<char>(y_val == 0.0));
auto log_y = log(y_val);
auto scaled_diff = elt_divide(log_y - mu_val, sigma_val * SQRT_TWO);
auto erfc_m_diff = erfc(-scaled_diff);
Expand All @@ -75,7 +75,7 @@ return_type_t<T_y_cl, T_loc_cl, T_scale_cl> lognormal_cdf(
auto y_deriv_tmp = elt_divide(-mu_deriv_tmp, y_val);
auto sigma_deriv_tmp = elt_multiply(mu_deriv_tmp, scaled_diff * SQRT_TWO);

matrix_cl<double> any_y_zero_cl;
matrix_cl<char> any_y_zero_cl;
matrix_cl<double> cdf_cl;
matrix_cl<double> mu_deriv_cl;
matrix_cl<double> y_deriv_cl;
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