text: r"""
  `Op`\s that have their python implementations taken from SciPy.
  
  As SciPy is not always available, we treat them separately.
  """
  
  import os
  import warnings
  
  import numpy as np
  import scipy.special
  import scipy.stats
  
  from aesara.configdefaults import config
  from aesara.gradient import grad_not_implemented
  from aesara.scalar.basic import (
      BinaryScalarOp,
      ScalarOp,
      UnaryScalarOp,
      complex_types,
      discrete_types,
      exp,
      float64,
      float_types,
      log,
      log1p,
      true_div,
      upcast,
      upgrade_to_float,
      upgrade_to_float64,
      upgrade_to_float_no_complex,
  )
  
  
  class Erf(UnaryScalarOp):
      nfunc_spec = ("scipy.special.erf", 1, 1)
  
      def impl(self, x):
          return scipy.special.erf(x)
  
      def L_op(self, inputs, outputs, grads):
          (x,) = inputs
          (gz,) = grads
          if x.type in complex_types:
              raise NotImplementedError()
          if outputs[0].type in discrete_types:
              if x.type in discrete_types:
                  return [x.zeros_like(dtype=config.floatX)]
              else:
                  return [x.zeros_like()]
  
          cst = np.asarray(
              2.0 / np.sqrt(np.pi), dtype=upcast(x.type.dtype, gz.type.dtype)
          )
          return (gz * cst * exp(-x * x),)
  
      def c_code(self, node, name, inp, out, sub):
          (x,) = inp
          (z,) = out
          if node.inputs[0].type in complex_types:
              raise NotImplementedError("type not supported", type)
          cast = node.outputs[0].type.dtype_specs()[1]
          return f"{z} = erf(({cast}){x});"
  
  
  erf = Erf(upgrade_to_float, name="erf")
  
  
  class Erfc(UnaryScalarOp):
      nfunc_spec = ("scipy.special.erfc", 1, 1)
  
      def impl(self, x):
          return scipy.special.erfc(x)
  
      def L_op(self, inputs, outputs, grads):
          (x,) = inputs
          (gz,) = grads
          if x.type in complex_types:
              raise NotImplementedError()
          if outputs[0].type in discrete_types:
              if x.type in discrete_types:
                  return [x.zeros_like(dtype=config.floatX)]
              else:
                  return [x.zeros_like()]
  
          cst = np.asarray(
              2.0 / np.sqrt(np.pi), dtype=upcast(x.type.dtype, gz.type.dtype)
          )
          return (-gz * cst * exp(-x * x),)
  
      def c_code(self, node, name, inp, out, sub):
          (x,) = inp
          (z,) = out
          if node.inputs[0].type in complex_types:
              raise NotImplementedError("type not supported", type)
          cast = node.outputs[0].type.dtype_specs()[1]
          return f"{z} = erfc(({cast}){x});"
  
  
  # scipy.special.erfc don't support complex. Why?
  erfc = Erfc(upgrade_to_float_no_complex, name="erfc")
  
  
  class Erfcx(UnaryScalarOp):
      """
      Implements the scaled complementary error function exp(x**2)*erfc(x) in a
      numerically stable way for large x. This is useful for calculating things
      like log(erfc(x)) = log(erfcx(x)) - x ** 2 without causing underflow.
      Should only be used if x is known to be large and positive, as using
      erfcx(x) for large negative x may instead introduce overflow problems.
  
      Notes
      -----
      This op can still be executed on GPU, despite not having c_code. When
      running on GPU an optimization will replace it with a gpu version.
  
      """
  
      nfunc_spec = ("scipy.special.erfcx", 1, 1)
  
      def impl(self, x):
          return scipy.special.erfcx(x)
  
      def L_op(self, inputs, outputs, grads):
          (x,) = inputs
          (gz,) = grads
          if x.type in complex_types:
              raise NotImplementedError()
          if outputs[0].type in discrete_types:
              if x.type in discrete_types:
                  return [x.zeros_like(dtype=config.floatX)]
              else:
                  return [x.zeros_like()]
  
          cst = np.asarray(
              2.0 / np.sqrt(np.pi), dtype=upcast(x.type.dtype, gz.type.dtype)
          )
          return (gz * (-cst + (2.0 * x) * erfcx(x)),)
  
      def c_header_dirs(self, **kwargs):
          # Using the Faddeeva.hh (c++) header for Faddeevva.cc
          res = super().c_header_dirs(**kwargs) + [
              os.path.join(os.path.dirname(__file__), "c_code")
          ]
          return res
  
      def c_support_code(self, **kwargs):
          # Using Faddeeva.cc source file from: http://ab-initio.mit.edu/wiki/index.php/Faddeeva_Package
          with open(
              os.path.join(os.path.dirname(__file__), "c_code", "Faddeeva.cc")
          ) as f:
              raw = f.read()
              return raw
  
      def c_code(self, node, name, inp, out, sub):
          (x,) = inp
          (z,) = out
  
          if node.inputs[0].type in float_types:
              dtype = "npy_" + node.outputs[0].dtype
              return f"{z} = ({dtype}) Faddeeva::erfcx({x});"
  
          raise NotImplementedError("type not supported", type)
  
  
  erfcx = Erfcx(upgrade_to_float_no_complex, name="erfcx")
  
  
  class Erfinv(UnaryScalarOp):
      """
      Implements the inverse error function.
  
      Notes
      -----
      This op can still be executed on GPU, despite not having c_code. When
      running on GPU, an optimization will replace it with a GPU version.
  
      (TODO) Find a C implementation of erfinv for CPU.
      """
  
      nfunc_spec = ("scipy.special.erfinv", 1, 1)
  
      def impl(self, x):
          return scipy.special.erfinv(x)
  
      def L_op(self, inputs, outputs, grads):
          (x,) = inputs
          (gz,) = grads
          if x.type in complex_types:
              raise NotImplementedError()
          if outputs[0].type in discrete_types:
              if x.type in discrete_types:
                  return [x.zeros_like(dtype=config.floatX)]
              else:
                  return [x.zeros_like()]
  
          cst = np.asarray(
              np.sqrt(np.pi) / 2.0, dtype=upcast(x.type.dtype, gz.type.dtype)
          )
          return (gz * cst * exp(erfinv(x) ** 2),)
  
      # TODO: erfinv() is not provided by the C standard library
      # def c_code(self, node, name, inp, out, sub):
      #    x, = inp
      #    z, = out
      #    if node.inputs[0].type in complex_types:
      #        raise NotImplementedError('type not supported', type)
      #    return "%(z)s = erfinv(%(x)s);" % locals()
  
  
  erfinv = Erfinv(upgrade_to_float_no_complex, name="erfinv")
  
  
  class Erfcinv(UnaryScalarOp):
      nfunc_spec = ("scipy.special.erfcinv", 1, 1)
  
      def impl(self, x):
          return scipy.special.erfcinv(x)
  
      def L_op(self, inputs, outputs, grads):
          (x,) = inputs
          (gz,) = grads
          if x.type in complex_types:
              raise NotImplementedError()
          if outputs[0].type in discrete_types:
              if x.type in discrete_types:
                  return [x.zeros_like(dtype=config.floatX)]
              else:
                  return [x.zeros_like()]
  
          cst = np.asarray(
              np.sqrt(np.pi) / 2.0, dtype=upcast(x.type.dtype, gz.type.dtype)
          )
          return (-gz * cst * exp(erfcinv(x) ** 2),)
  
      # TODO: erfcinv() is not provided by the C standard library
      # def c_code(self, node, name, inp, out, sub):
      #    x, = inp
      #    z, = out
      #    if node.inputs[0].type in complex_types:
      #        raise NotImplementedError('type not supported', type)
      #    return "%(z)s = erfcinv(%(x)s);" % locals()
  
  
  erfcinv = Erfcinv(upgrade_to_float_no_complex, name="erfcinv")
  
  
  class Gamma(UnaryScalarOp):
      nfunc_spec = ("scipy.special.gamma", 1, 1)
  
      @staticmethod
      def st_impl(x):
          return scipy.special.gamma(x)
  
      def impl(self, x):
          return Gamma.st_impl(x)
  
      def L_op(self, inputs, outputs, gout):
          (x,) = inputs
          (gz,) = gout
          if x.type in complex_types:
              raise NotImplementedError()
          if outputs[0].type in discrete_types:
              if x.type in discrete_types:
                  return [x.zeros_like(dtype=config.floatX)]
              else:
                  return [x.zeros_like()]
  
          return (gz * gamma(x) * psi(x),)
  
      def c_code(self, node, name, inputs, outputs, sub):
          (x,) = inputs
          (z,) = outputs
          if node.inputs[0].type in float_types:
              return f"""{z} = tgamma({x});"""
          raise NotImplementedError("only floating point is implemented")
  
  
  gamma = Gamma(upgrade_to_float, name="gamma")
  
  
  class GammaLn(UnaryScalarOp):
      """
      Log gamma function.
  
      """
  
      nfunc_spec = ("scipy.special.gammaln", 1, 1)
  
      @staticmethod
      def st_impl(x):
          return scipy.special.gammaln(x)
  
      def impl(self, x):
          return GammaLn.st_impl(x)
  
      def L_op(self, inputs, outputs, grads):
          (x,) = inputs
          (gz,) = grads
          if x.type in complex_types:
              raise NotImplementedError()
          if outputs[0].type in discrete_types:
              if x.type in discrete_types:
                  return [x.zeros_like(dtype=config.floatX)]
              else:
                  return [x.zeros_like()]
  
          return [gz * psi(x)]
  
      def c_code(self, node, name, inp, out, sub):
          (x,) = inp
          (z,) = out
          # no c code for complex
          # [u]int* will be casted to float64 before computation
          if node.inputs[0].type in complex_types:
              raise NotImplementedError("gammaln complex c code is not implemented")
          # For some reason, on the GPU, uint64 inputs don't get casted
          # automatically to float64. This make the compilation crash
          cast = node.outputs[0].type.dtype_specs()[1]
          return f"""{z} = lgamma(({cast}){x});"""
  
  
  gammaln = GammaLn(upgrade_to_float, name="gammaln")
  
  
  class Psi(UnaryScalarOp):
      """
      Derivative of log gamma function.
  
      """
  
      nfunc_spec = ("scipy.special.psi", 1, 1)
  
      @staticmethod
      def st_impl(x):
          return scipy.special.psi(x)
  
      def impl(self, x):
          return Psi.st_impl(x)
  
      def L_op(self, inputs, outputs, grads):
          (x,) = inputs
          (gz,) = grads
          if x.type in complex_types:
              raise NotImplementedError()
          if outputs[0].type in discrete_types:
              if x.type in discrete_types:
                  return [x.zeros_like(dtype=config.floatX)]
              else:
                  return [x.zeros_like()]
  
          return [gz * tri_gamma(x)]
  
      def c_support_code(self, **kwargs):
          return """
              // For GPU support
              #ifdef WITHIN_KERNEL
              #define DEVICE WITHIN_KERNEL
              #else
              #define DEVICE
              #endif
  
              #ifndef ga_double
              #define ga_double double
              #endif
  
              #ifndef _PSIFUNCDEFINED
              #define _PSIFUNCDEFINED
              DEVICE double _psi(ga_double x) {
  
              /*taken from
              Bernardo, J. M. (1976). Algorithm AS 103:
              Psi (Digamma) Function. Applied Statistics. 25 (3), 315-317.
              http://www.uv.es/~bernardo/1976AppStatist.pdf */
  
              ga_double y, R, psi_ = 0;
              ga_double S  = 1.0e-5;
              ga_double C = 8.5;
              ga_double S3 = 8.333333333e-2;
              ga_double S4 = 8.333333333e-3;
              ga_double S5 = 3.968253968e-3;
              ga_double D1 = -0.5772156649;
  
              y = x;
  
              if (y <= 0.0)
                 return psi_;
  
              if (y <= S)
                  return D1 - 1.0/y;
  
              while (y < C) {
                  psi_ = psi_ - 1.0 / y;
                  y = y + 1;
              }
  
              R = 1.0 / y;
              psi_ = psi_ + log(y) - .5 * R ;
              R= R*R;
              psi_ = psi_ - R * (S3 - R * (S4 - R * S5));
  
              return psi_;
              }
              #endif
              """
  
      def c_code(self, node, name, inp, out, sub):
          (x,) = inp
          (z,) = out
          if node.inputs[0].type in float_types:
              return f"""{z} =
                  _psi({x});"""
          raise NotImplementedError("only floating point is implemented")
  
  
  psi = Psi(upgrade_to_float, name="psi")
  
  
  class TriGamma(UnaryScalarOp):
      """
      Second derivative of log gamma function.
  
      """
  
      @staticmethod
      def st_impl(x):
          return scipy.special.polygamma(1, x)
  
      def impl(self, x):
          return TriGamma.st_impl(x)
  
      def grad(self, inputs, outputs_gradients):
          raise NotImplementedError()
  
      def c_support_code(self, **kwargs):
          # The implementation has been copied from
          # http://people.sc.fsu.edu/~jburkardt/cpp_src/asa121/asa121.html
          return """
              // For GPU support
              #ifdef WITHIN_KERNEL
              #define DEVICE WITHIN_KERNEL
              #else
              #define DEVICE
              #endif
  
              #ifndef ga_double
              #define ga_double double
              #endif
  
              #ifndef _TRIGAMMAFUNCDEFINED
              #define _TRIGAMMAFUNCDEFINED
  
              DEVICE double _tri_gamma(ga_double x) {
  
                  double a = 0.0001;
                  double b = 5.0;
                  double b2 =  0.1666666667;
                  double b4 = -0.03333333333;
                  double b6 =  0.02380952381;
                  double b8 = -0.03333333333;
                  double value;
                  double y;
                  double z;
  
                  if (x <= 0) {
                      return 0.0;
                  }
  
                  if ( x <= a ) {
                      value = 1.0 / x / x;
                      return value;
                  }
  
                  value = 0.0;
                  z = x;
  
                  while ( z < b ) {
                      value += 1.0 / z / z;
                      z += 1.0;
                  }
  
                  y = 1.0 / z / z;
  
                  value +=  0.5 * y + (1.0 + y * (b2 + y * (b4 + y * (b6 + y * b8 )))) / z;
  
                  return value;
              }
              #endif
              """
  
      def c_code(self, node, name, inp, out, sub):
          (x,) = inp
          (z,) = out
          if node.inputs[0].type in float_types:
              return f"""{z} =
                  _tri_gamma({x});"""
          raise NotImplementedError("only floating point is implemented")
  
  
  tri_gamma = TriGamma(upgrade_to_float, name="tri_gamma")
  
  
  class Chi2SF(BinaryScalarOp):
      """
      Compute (1 - chi2_cdf(x))
          ie. chi2 pvalue (chi2 'survival function')
      """
  
      nfunc_spec = ("scipy.stats.chi2.sf", 2, 1)
  
      @staticmethod
      def st_impl(x, k):
          return scipy.stats.chi2.sf(x, k)
  
      def impl(self, x, k):
          return Chi2SF.st_impl(x, k)
  
      def c_support_code(self, **kwargs):
          with open(os.path.join(os.path.dirname(__file__), "c_code", "gamma.c")) as f:
              raw = f.read()
              return raw
  
      def c_code(self, node, name, inp, out, sub):
          x, k = inp
          (z,) = out
          if node.inputs[0].type in float_types:
              dtype = "npy_" + node.outputs[0].dtype
              return (
                  """%(z)s =
                  (%(dtype)s) 1 - GammaP(%(k)s/2., %(x)s/2.);"""
                  % locals()
              )
          raise NotImplementedError("only floatingpoint is implemented")
  
      def __eq__(self, other):
          return type(self) == type(other)
  
      def __hash__(self):
          return hash(type(self))
  
  
  chi2sf = Chi2SF(upgrade_to_float64, name="chi2sf")
  
  
  class GammaInc(BinaryScalarOp):
      """
      Compute the regularized lower gamma function (P).
      """
  
      nfunc_spec = ("scipy.special.gammainc", 2, 1)
  
      @staticmethod
      def st_impl(k, x):
          return scipy.special.gammainc(k, x)
  
      def impl(self, k, x):
          return GammaInc.st_impl(k, x)
  
      def grad(self, inputs, grads):
          (k, x) = inputs
          (gz,) = grads
          return [
              gz * gammainc_der(k, x),
              gz * exp(-x + (k - 1) * log(x) - gammaln(k)),
          ]
  
      def c_support_code(self, **kwargs):
          with open(os.path.join(os.path.dirname(__file__), "c_code", "gamma.c")) as f:
              raw = f.read()
              return raw
  
      def c_code(self, node, name, inp, out, sub):
          k, x = inp
          (z,) = out
          if node.inputs[0].type in float_types:
              dtype = "npy_" + node.outputs[0].dtype
              return (
                  """%(z)s =
                  (%(dtype)s) GammaP(%(k)s, %(x)s);"""
                  % locals()
              )
          raise NotImplementedError("only floatingpoint is implemented")
  
      def __eq__(self, other):
          return type(self) == type(other)
  
      def __hash__(self):
          return hash(type(self))
  
  
  gammainc = GammaInc(upgrade_to_float, name="gammainc")
  
  
  class GammaIncC(BinaryScalarOp):
      """
      Compute the regularized upper gamma function (Q).
      """
  
      nfunc_spec = ("scipy.special.gammaincc", 2, 1)
  
      @staticmethod
      def st_impl(k, x):
          return scipy.special.gammaincc(k, x)
  
      def impl(self, k, x):
          return GammaIncC.st_impl(k, x)
  
      def grad(self, inputs, grads):
          (k, x) = inputs
          (gz,) = grads
          return [
              gz * gammaincc_der(k, x),
              gz * -exp(-x + (k - 1) * log(x) - gammaln(k)),
          ]
  
      def c_support_code(self, **kwargs):
          with open(os.path.join(os.path.dirname(__file__), "c_code", "gamma.c")) as f:
              raw = f.read()
              return raw
  
      def c_code(self, node, name, inp, out, sub):
          k, x = inp
          (z,) = out
          if node.inputs[0].type in float_types:
              dtype = "npy_" + node.outputs[0].dtype
              return (
                  """%(z)s =
                  (%(dtype)s) GammaQ(%(k)s, %(x)s);"""
                  % locals()
              )
          raise NotImplementedError("only floatingpoint is implemented")
  
      def __eq__(self, other):
          return type(self) == type(other)
  
      def __hash__(self):
          return hash(type(self))
  
  
  gammaincc = GammaIncC(upgrade_to_float, name="gammaincc")
  
  
  class GammaIncDer(BinaryScalarOp):
      """
      Gradient of the the regularized lower gamma function (P) wrt to the first
      argument (k, a.k.a. alpha). Adapted from STAN `grad_reg_lower_inc_gamma.hpp`
  
      Reference: Gautschi, W. (1979). A computational procedure for incomplete gamma functions.
      ACM Transactions on Mathematical Software (TOMS), 5(4), 466-481.
      """
  
      def impl(self, k, x):
  
          if x == 0:
              return 0
  
          sqrt_exp = -756 - x ** 2 + 60 * x
          if (
              (k < 0.8 and x > 15)
              or (k < 12 and x > 30)
              or (sqrt_exp > 0 and k < np.sqrt(sqrt_exp))
          ):
              return -GammaIncCDer.st_impl(k, x)
  
          precision = 1e-10
          max_iters = int(1e5)
  
          log_x = np.log(x)
          log_gamma_k_plus_1 = scipy.special.gammaln(k + 1)
  
          k_plus_n = k
          log_gamma_k_plus_n_plus_1 = log_gamma_k_plus_1
          sum_a = 0.0
          for n in range(0, max_iters + 1):
              term = np.exp(k_plus_n * log_x - log_gamma_k_plus_n_plus_1)
              sum_a += term
  
              if term <= precision:
                  break
  
              log_gamma_k_plus_n_plus_1 += np.log1p(k_plus_n)
              k_plus_n += 1
  
          if n >= max_iters:
              warnings.warn(
                  f"gammainc_der did not converge after {n} iterations",
                  RuntimeWarning,
              )
              return np.nan
  
          k_plus_n = k
          log_gamma_k_plus_n_plus_1 = log_gamma_k_plus_1
          sum_b = 0.0
          for n in range(0, max_iters + 1):
              term = np.exp(
                  k_plus_n * log_x - log_gamma_k_plus_n_plus_1
              ) * scipy.special.digamma(k_plus_n + 1)
              sum_b += term
  
              if term <= precision and n >= 1:  # Require at least two iterations
                  return np.exp(-x) * (log_x * sum_a - sum_b)
  
              log_gamma_k_plus_n_plus_1 += np.log1p(k_plus_n)
              k_plus_n += 1
  
          warnings.warn(
              f"gammainc_der did not converge after {n} iterations",
              RuntimeWarning,
          )
          return np.nan
  
  
  gammainc_der = GammaIncDer(upgrade_to_float, name="gammainc_der")
  
  
  class GammaIncCDer(BinaryScalarOp):
      """
      Gradient of the the regularized upper gamma function (Q) wrt to the first
      argument (k, a.k.a. alpha). Adapted from STAN `grad_reg_inc_gamma.hpp`
      """
  
      @staticmethod
      def st_impl(k, x):
          gamma_k = scipy.special.gamma(k)
          digamma_k = scipy.special.digamma(k)
          log_x = np.log(x)
  
          # asymptotic expansion http://dlmf.nist.gov/8.11#E2
          if (x >= k) and (x >= 8):
              S = 0
              k_minus_one_minus_n = k - 1
              fac = k_minus_one_minus_n
              dfac = 1
              xpow = x
              delta = dfac / xpow
  
              for n in range(1, 10):
                  k_minus_one_minus_n -= 1
                  S += delta
                  xpow *= x
                  dfac = k_minus_one_minus_n * dfac + fac
                  fac *= k_minus_one_minus_n
                  delta = dfac / xpow
                  if np.isinf(delta):
                      warnings.warn(
                          "gammaincc_der did not converge",
                          RuntimeWarning,
                      )
                      return np.nan
  
              return (
                  scipy.special.gammaincc(k, x) * (log_x - digamma_k)
                  + np.exp(-x + (k - 1) * log_x) * S / gamma_k
              )
  
          # gradient of series expansion http://dlmf.nist.gov/8.7#E3
          else:
              log_precision = np.log(1e-6)
              max_iters = int(1e5)
              S = 0
              log_s = 0.0
              s_sign = 1
              log_delta = log_s - 2 * np.log(k)
              for n in range(1, max_iters + 1):
                  S += np.exp(log_delta) if s_sign > 0 else -np.exp(log_delta)
                  s_sign = -s_sign
                  log_s += log_x - np.log(n)
                  log_delta = log_s - 2 * np.log(n + k)
  
                  if np.isinf(log_delta):
                      warnings.warn(
                          "gammaincc_der did not converge",
                          RuntimeWarning,
                      )
                      return np.nan
  
                  if log_delta <= log_precision:
                      return (
                          scipy.special.gammainc(k, x) * (digamma_k - log_x)
                          + np.exp(k * log_x) * S / gamma_k
                      )
  
              warnings.warn(
                  f"gammaincc_der did not converge after {n} iterations",
                  RuntimeWarning,
              )
              return np.nan
  
      def impl(self, k, x):
          return self.st_impl(k, x)
  
  
  gammaincc_der = GammaIncCDer(upgrade_to_float, name="gammaincc_der")
  
  
  class GammaU(BinaryScalarOp):
      """
      compute the upper incomplete gamma function.
      """
  
      # Note there is no basic SciPy version so no nfunc_spec.
  
      @staticmethod
      def st_impl(k, x):
          return scipy.special.gammaincc(k, x) * scipy.special.gamma(k)
  
      def impl(self, k, x):
          return GammaU.st_impl(k, x)
  
      def c_support_code(self, **kwargs):
          with open(os.path.join(os.path.dirname(__file__), "c_code", "gamma.c")) as f:
              raw = f.read()
              return raw
  
      def c_code(self, node, name, inp, out, sub):
          k, x = inp
          (z,) = out
          if node.inputs[0].type in float_types:
              dtype = "npy_" + node.outputs[0].dtype
              return (
                  """%(z)s =
                  (%(dtype)s) upperGamma(%(k)s, %(x)s);"""
                  % locals()
              )
          raise NotImplementedError("only floatingpoint is implemented")
  
      def __eq__(self, other):
          return type(self) == type(other)
  
      def __hash__(self):
          return hash(type(self))
  
  
  gammau = GammaU(upgrade_to_float, name="gammau")
  
  
  class GammaL(BinaryScalarOp):
      """
      Compute the lower incomplete gamma function.
      """
  
      # Note there is no basic SciPy version so no nfunc_spec.
  
      @staticmethod
      def st_impl(k, x):
          return scipy.special.gammainc(k, x) * scipy.special.gamma(k)
  
      def impl(self, k, x):
          return GammaL.st_impl(k, x)
  
      def c_support_code(self, **kwargs):
          with open(os.path.join(os.path.dirname(__file__), "c_code", "gamma.c")) as f:
              raw = f.read()
              return raw
  
      def c_code(self, node, name, inp, out, sub):
          k, x = inp
          (z,) = out
          if node.inputs[0].type in float_types:
              dtype = "npy_" + node.outputs[0].dtype
              return (
                  """%(z)s =
                  (%(dtype)s) lowerGamma(%(k)s, %(x)s);"""
                  % locals()
              )
          raise NotImplementedError("only floatingpoint is implemented")
  
      def __eq__(self, other):
          return type(self) == type(other)
  
      def __hash__(self):
          return hash(type(self))
  
  
  gammal = GammaL(upgrade_to_float, name="gammal")
  
  
  class Jv(BinaryScalarOp):
      """
      Bessel function of the first kind of order v (real).
      """
  
      nfunc_spec = ("scipy.special.jv", 2, 1)
  
      @staticmethod
      def st_impl(v, x):
          return scipy.special.jv(v, x)
  
      def impl(self, v, x):
          return self.st_impl(v, x)
  
      def grad(self, inputs, grads):
          v, x = inputs
          (gz,) = grads
          return [
              grad_not_implemented(self, 0, v),
              gz * (jv(v - 1, x) - jv(v + 1, x)) / 2.0,
          ]
  
  
  jv = Jv(upgrade_to_float, name="jv")
  
  
  class J1(UnaryScalarOp):
      """
      Bessel function of the first kind of order 1.
      """
  
      nfunc_spec = ("scipy.special.j1", 1, 1)
  
      @staticmethod
      def st_impl(x):
          return scipy.special.j1(x)
  
      def impl(self, x):
          return self.st_impl(x)
  
      def grad(self, inputs, grads):
          (x,) = inputs
          (gz,) = grads
          return [gz * (j0(x) - jv(2, x)) / 2.0]
  
      def c_code(self, node, name, inp, out, sub):
          (x,) = inp
          (z,) = out
          if node.inputs[0].type in float_types:
              return f"""{z} =
                  j1({x});"""
          raise NotImplementedError("only floating point is implemented")
  
  
  j1 = J1(upgrade_to_float, name="j1")
  
  
  class J0(UnaryScalarOp):
      """
      Bessel function of the first kind of order 0.
      """
  
      nfunc_spec = ("scipy.special.j0", 1, 1)
  
      @staticmethod
      def st_impl(x):
          return scipy.special.j0(x)
  
      def impl(self, x):
          return self.st_impl(x)
  
      def grad(self, inp, grads):
          (x,) = inp
          (gz,) = grads
          return [gz * -1 * j1(x)]
  
      def c_code(self, node, name, inp, out, sub):
          (x,) = inp
          (z,) = out
          if node.inputs[0].type in float_types:
              return f"""{z} =
                  j0({x});"""
          raise NotImplementedError("only floating point is implemented")
  
  
  j0 = J0(upgrade_to_float, name="j0")
  
  
  class Iv(BinaryScalarOp):
      """
      Modified Bessel function of the first kind of order v (real).
      """
  
      nfunc_spec = ("scipy.special.iv", 2, 1)
  
      @staticmethod
      def st_impl(v, x):
          return scipy.special.iv(v, x)
  
      def impl(self, v, x):
          return self.st_impl(v, x)
  
      def grad(self, inputs, grads):
          v, x = inputs
          (gz,) = grads
          return [
              grad_not_implemented(self, 0, v),
              gz * (iv(v - 1, x) + iv(v + 1, x)) / 2.0,
          ]
  
  
  iv = Iv(upgrade_to_float, name="iv")
  
  
  class I1(UnaryScalarOp):
      """
      Modified Bessel function of the first kind of order 1.
      """
  
      nfunc_spec = ("scipy.special.i1", 1, 1)
  
      @staticmethod
      def st_impl(x):
          return scipy.special.i1(x)
  
      def impl(self, x):
          return self.st_impl(x)
  
      def grad(self, inputs, grads):
          (x,) = inputs
          (gz,) = grads
          return [gz * (i0(x) + iv(2, x)) / 2.0]
  
  
  i1 = I1(upgrade_to_float, name="i1")
  
  
  class I0(UnaryScalarOp):
      """
      Modified Bessel function of the first kind of order 0.
      """
  
      nfunc_spec = ("scipy.special.i0", 1, 1)
  
      @staticmethod
      def st_impl(x):
          return scipy.special.i0(x)
  
      def impl(self, x):
          return self.st_impl(x)
  
      def grad(self, inp, grads):
          (x,) = inp
          (gz,) = grads
          return [gz * i1(x)]
  
  
  i0 = I0(upgrade_to_float, name="i0")
  
  
  class Sigmoid(UnaryScalarOp):
      """
      Logistic sigmoid function (1 / (1 + exp(x)), also known as expit or inverse logit
      """
  
      nfunc_spec = ("scipy.special.expit", 1, 1)
  
      def impl(self, x):
          return scipy.special.expit(x)
  
      def grad(self, inp, grads):
          (x,) = inp
          (gz,) = grads
          y = sigmoid(x)
          rval = gz * y * (1.0 - y)
  
          assert rval.type.dtype.find("float") != -1
  
          return [rval]
  
      def c_code(self, node, name, inp, out, sub):
          (x,) = inp
          (z,) = out
  
          if node.inputs[0].type in float_types:
              if node.inputs[0].type == float64:
                  return f"""{z} = 1.0 / (1.0 + exp(-{x}));"""
              else:
                  return f"""{z} = 1.0f / (1.0f + exp(-{x}));"""
          else:
              raise NotImplementedError("only floatingpoint is implemented")
  
      def c_code_cache_version(self):
          v = super().c_code_cache_version()
          if v:
              return (2,) + v
          else:
              return v
  
  
  sigmoid = Sigmoid(upgrade_to_float, name="sigmoid")
  
  
  class Softplus(UnaryScalarOp):
      r"""
      Compute log(1 + exp(x)), also known as softplus or log1pexp
  
      This function is numerically more stable than the naive approach.
  
      For details, see
      https://cran.r-project.org/web/packages/Rmpfr/vignettes/log1mexp-note.pdf
  
      References
      ----------
      .. [Machler2012] Martin Mächler (2012).
          "Accurately computing `\log(1-\exp(- \mid a \mid))` Assessed by the Rmpfr package"
      """
  
      @staticmethod
      def static_impl(x):
          # If x is an int8 or uint8, numpy.exp will compute the result in
          # half-precision (float16), where we want float32.
          not_int8 = str(getattr(x, "dtype", "")) not in ("int8", "uint8")
          if x < -37.0:
              return np.exp(x) if not_int8 else np.exp(x, signature="f")
          elif x < 18.0:
              return (
                  np.log1p(np.exp(x)) if not_int8 else np.log1p(np.exp(x, signature="f"))
              )
          elif x < 33.3:
              return x + np.exp(-x) if not_int8 else x + np.exp(-x, signature="f")
          else:
              return x
  
      def impl(self, x):
          return Softplus.static_impl(x)
  
      def grad(self, inp, grads):
          (x,) = inp
          (gz,) = grads
          return [gz * sigmoid(x)]
  
      def c_code(self, node, name, inp, out, sub):
          (x,) = inp
          (z,) = out
          # The boundary constants were obtained by looking at the output of
          # python commands like:
          # import numpy, aesara
          # dt='float32'  # or float64
          #  for i in range(750):
          #      print i, repr(numpy.log1p(numpy.exp(_asarray([i,-i], dtype=dt))))
          # the upper boundary check prevents us from generating inf, whereas the
          # the lower boundary check prevents using exp when the result will be 0 anyway.
          # The intermediate constants are taken from Machler (2012).
  
          # We use the float32 limits for float16 for now as the
          # computation will happen in float32 anyway.
          if node.inputs[0].type in float_types:
              if node.inputs[0].type == float64:
                  return (
                      """
                      %(z)s = (
                          %(x)s < -745.0 ? 0.0 :
                          %(x)s < -37.0 ? exp(%(x)s) :
                          %(x)s < 18.0 ? log1p(exp(%(x)s)) :
                          %(x)s < 33.3 ? %(x)s + exp(-%(x)s) :
                          %(x)s
                      );
                      """
                      % locals()
                  )
              else:
                  return (
                      """
                      %(z)s = (
                          %(x)s < -103.0f ? 0.0 :
                          %(x)s < -37.0f ? exp(%(x)s) :
                          %(x)s < 18.0f ? log1p(exp(%(x)s)) :
                          %(x)s < 33.3f ? %(x)s + exp(-%(x)s) :
                          %(x)s
                      );
                      """
                      % locals()
                  )
          else:
              raise NotImplementedError("only floatingpoint is implemented")
  
      def c_code_cache_version(self):
          v = super().c_code_cache_version()
          if v:
              return (2,) + v
          else:
              return v
  
  
  softplus = Softplus(upgrade_to_float, name="scalar_softplus")
  
  
  class Log1mexp(UnaryScalarOp):
      r"""
      Compute log(1 - exp(x)), also known as log1mexp
  
      This function is numerically more stable than the naive approach.
  
      For details, see
      https://cran.r-project.org/web/packages/Rmpfr/vignettes/log1mexp-note.pdf
  
      References
      ----------
      .. [Machler2012] Martin Mächler (2012).
          "Accurately computing `\log(1-\exp(- \mid a \mid))` Assessed by the Rmpfr package"
      """
  
      @staticmethod
      def static_impl(x):
          if x < np.log(0.5):
              return np.log1p(-np.exp(x))
          else:
              return np.log(-np.expm1(x))
  
      def impl(self, x):
          return Log1mexp.static_impl(x)
  
      def grad(self, inp, grads):
          (x,) = inp
          (gz,) = grads
          return [gz * true_div(1.0, 1.0 - exp(-x))]
  
      def c_code(self, node, name, inp, out, sub):
          (x,) = inp
          (z,) = out
  
          if node.inputs[0].type in float_types:
              if node.inputs[0].type == float64:
                  return f"{z} = {x} < -0.6931471805599453 ? log1p(-exp({x})) : log(-expm1({x}));"
              else:
                  return f"{z} = {x} < -0.6931471805599453f ? log1p(-exp({x})) : log(-expm1({x}));"
          else:
              raise NotImplementedError("only floating point is implemented")
  
  
  log1mexp = Log1mexp(upgrade_to_float, name="scalar_log1mexp")
  
  
  class BetaInc(ScalarOp):
      """
      Regularized incomplete beta function
      """
  
      nin = 3
      nfunc_spec = ("scipy.special.betainc", 3, 1)
  
      def impl(self, a, b, x):
          return scipy.special.betainc(a, b, x)
  
      def grad(self, inp, grads):
          a, b, x = inp
          (gz,) = grads
  
          return [
              gz * betainc_der(a, b, x, True),
              gz * betainc_der(a, b, x, False),
              gz
              * exp(
                  log1p(-x) * (b - 1)
                  + log(x) * (a - 1)
                  - (gammaln(a) + gammaln(b) - gammaln(a + b))
              ),
          ]
  
  
  betainc = BetaInc(upgrade_to_float_no_complex, name="betainc")
  
  
  class BetaIncDer(ScalarOp):
      """
      Gradient of the regularized incomplete beta function wrt to the first
      argument (alpha) or the second argument (beta), depending on whether the
      fourth argument to betainc_der is `True` or `False`, respectively.
  
      Reference: Boik, R. J., & Robison-Cox, J. F. (1998). Derivatives of the incomplete beta function.
      Journal of Statistical Software, 3(1), 1-20.
      """
  
      nin = 4
  
      def impl(self, p, q, x, wrtp):
          def _betainc_a_n(f, p, q, n):
              """
              Numerator (a_n) of the nth approximant of the continued fraction
              representation of the regularized incomplete beta function
              """
  
              if n == 1:
                  return p * f * (q - 1) / (q * (p + 1))
  
              p2n = p + 2 * n
              F1 = p ** 2 * f ** 2 * (n - 1) / (q ** 2)
              F2 = (
                  (p + q + n - 2)
                  * (p + n - 1)
                  * (q - n)
                  / ((p2n - 3) * (p2n - 2) ** 2 * (p2n - 1))
              )
  
              return F1 * F2
  
          def _betainc_b_n(f, p, q, n):
              """
              Offset (b_n) of the nth approximant of the continued fraction
              representation of the regularized incomplete beta function
              """
              pf = p * f
              p2n = p + 2 * n
  
              N1 = 2 * (pf + 2 * q) * n * (n + p - 1) + p * q * (p - 2 - pf)
              D1 = q * (p2n - 2) * p2n
  
              return N1 / D1
  
          def _betainc_da_n_dp(f, p, q, n):
              """
              Derivative of a_n wrt p
              """
  
              if n == 1:
                  return -p * f * (q - 1) / (q * (p + 1) ** 2)
  
              pp = p ** 2
              ppp = pp * p
              p2n = p + 2 * n
  
              N1 = -(n - 1) * f ** 2 * pp * (q - n)
              N2a = (-8 + 8 * p + 8 * q) * n ** 3
              N2b = (16 * pp + (-44 + 20 * q) * p + 26 - 24 * q) * n ** 2
              N2c = (10 * ppp + (14 * q - 46) * pp + (-40 * q + 66) * p - 28 + 24 * q) * n
              N2d = 2 * pp ** 2 + (-13 + 3 * q) * ppp + (-14 * q + 30) * pp
              N2e = (-29 + 19 * q) * p + 10 - 8 * q
  
              D1 = q ** 2 * (p2n - 3) ** 2
              D2 = (p2n - 2) ** 3 * (p2n - 1) ** 2
  
              return (N1 / D1) * (N2a + N2b + N2c + N2d + N2e) / D2
  
          def _betainc_da_n_dq(f, p, q, n):
              """
              Derivative of a_n wrt q
              """
              if n == 1:
                  return p * f / (q * (p + 1))
  
              p2n = p + 2 * n
              F1 = (p ** 2 * f ** 2 / (q ** 2)) * (n - 1) * (p + n - 1) * (2 * q + p - 2)
              D1 = (p2n - 3) * (p2n - 2) ** 2 * (p2n - 1)
  
              return F1 / D1
  
          def _betainc_db_n_dp(f, p, q, n):
              """
              Derivative of b_n wrt p
              """
              p2n = p + 2 * n
              pp = p ** 2
              q4 = 4 * q
              p4 = 4 * p
  
              F1 = (p * f / q) * (
                  (-p4 - q4 + 4) * n ** 2 + (p4 - 4 + q4 - 2 * pp) * n + pp * q
              )
              D1 = (p2n - 2) ** 2 * p2n ** 2
  
              return F1 / D1
  
          def _betainc_db_n_dq(f, p, q, n):
              """
              Derivative of b_n wrt to q
              """
              p2n = p + 2 * n
              return -(p ** 2 * f) / (q * (p2n - 2) * p2n)
  
          # Input validation
          if not (0 <= x <= 1) or p < 0 or q < 0:
              return np.nan
  
          if x > (p / (p + q)):
              return -self.impl(q, p, 1 - x, not wrtp)
  
          min_iters = 3
          max_iters = 200
          err_threshold = 1e-12
  
          derivative_old = 0
  
          Am2, Am1 = 1, 1
          Bm2, Bm1 = 0, 1
          dAm2, dAm1 = 0, 0
          dBm2, dBm1 = 0, 0
  
          f = (q * x) / (p * (1 - x))
          K = np.exp(
              p * np.log(x)
              + (q - 1) * np.log1p(-x)
              - np.log(p)
              - scipy.special.betaln(p, q)
          )
          if wrtp:
              dK = (
                  np.log(x)
                  - 1 / p
                  + scipy.special.digamma(p + q)
                  - scipy.special.digamma(p)
              )
          else:
              dK = np.log1p(-x) + scipy.special.digamma(p + q) - scipy.special.digamma(q)
  
          for n in range(1, max_iters + 1):
              a_n_ = _betainc_a_n(f, p, q, n)
              b_n_ = _betainc_b_n(f, p, q, n)
              if wrtp:
                  da_n = _betainc_da_n_dp(f, p, q, n)
                  db_n = _betainc_db_n_dp(f, p, q, n)
              else:
                  da_n = _betainc_da_n_dq(f, p, q, n)
                  db_n = _betainc_db_n_dq(f, p, q, n)
  
              A = a_n_ * Am2 + b_n_ * Am1
              B = a_n_ * Bm2 + b_n_ * Bm1
              dA = da_n * Am2 + a_n_ * dAm2 + db_n * Am1 + b_n_ * dAm1
              dB = da_n * Bm2 + a_n_ * dBm2 + db_n * Bm1 + b_n_ * dBm1
  
              Am2, Am1 = Am1, A
              Bm2, Bm1 = Bm1, B
              dAm2, dAm1 = dAm1, dA
              dBm2, dBm1 = dBm1, dB
  
              if n < min_iters - 1:
                  continue
  
              F1 = A / B
              F2 = (dA - F1 * dB) / B
              derivative = K * (F1 * dK + F2)
  
              errapx = abs(derivative_old - derivative)
              d_errapx = errapx / max(err_threshold, abs(derivative))
              derivative_old = derivative
  
              if d_errapx <= err_threshold:
                  return derivative
  
          warnings.warn(
              f"betainc_der did not converge after {n} iterations",
              RuntimeWarning,
          )
          return np.nan
  
  
  betainc_der = BetaIncDer(upgrade_to_float_no_complex, name="betainc_der")
  