code: def _gammatone_iir_kernel(fs, freq, b, a):
      tid = jit.blockIdx.x * jit.blockDim.x + jit.threadIdx.x
  
      EarQ = 9.26449
      minBW = 24.7
      erb = freq / EarQ + minBW
  
      T = 1.0 / fs
      bw = 2 * cupy.pi * 1.019 * erb
      fr = 2 * freq * cupy.pi * T
      bwT = bw * T
  
      # Calculate the gain to normalize the volume at the center frequency
      g1 = -2 * cupy.exp(2j * fr) * T
      g2 = 2 * cupy.exp(-(bwT) + 1j * fr) * T
      g3 = cupy.sqrt(3 + 2 ** (3 / 2)) * cupy.sin(fr)
      g4 = cupy.sqrt(3 - 2 ** (3 / 2)) * cupy.sin(fr)
      g5 = cupy.exp(2j * fr)
  
      g = g1 + g2 * (cupy.cos(fr) - g4)
      g *= g1 + g2 * (cupy.cos(fr) + g4)
      g *= g1 + g2 * (cupy.cos(fr) - g3)
      g *= g1 + g2 * (cupy.cos(fr) + g3)
      g /= (
          -2 / cupy.exp(2 * bwT) - 2 * g5 + 2 * (1 + g5) / cupy.exp(bwT)
      ) ** 4
      g_act = cupy.abs(g)
  
      # Calculate the numerator coefficients
      if tid == 0:
          b[tid] = (T**4) / g_act
          a[tid] = 1
      elif tid == 1:
          b[tid] = -4 * T**4 * cupy.cos(fr) / cupy.exp(bw * T) / g_act
          a[tid] = -8 * cupy.cos(fr) / cupy.exp(bw * T)
      elif tid == 2:
          b[tid] = (
              6
              * T**4
              * cupy.cos(2 * fr)
              / cupy.exp(2 * bw * T)
              / g_act
          )
          a[tid] = 4 * (4 + 3 * cupy.cos(2 * fr)) / cupy.exp(2 * bw * T)
      elif tid == 3:
          b[tid] = (
              -4
              * T**4
              * cupy.cos(3 * fr)
              / cupy.exp(3 * bw * T)
              / g_act
          )
          a[tid] = -8 * (6 * cupy.cos(fr) + cupy.cos(3 * fr))
          a[tid] /= cupy.exp(3 * bw * T)
      elif tid == 4:
          b[tid] = (
              T**4 * cupy.cos(4 * fr) / cupy.exp(4 * bw * T) / g_act
          )
          a[tid] = 2 * (18 + 16 * cupy.cos(2 * fr) + cupy.cos(4 * fr))
          a[tid] /= cupy.exp(4 * bw * T)
      elif tid == 5:
          a[tid] = -8 * (6 * cupy.cos(fr) + cupy.cos(3 * fr))
          a[tid] /= cupy.exp(5 * bw * T)
      elif tid == 6:
          a[tid] = 4 * (4 + 3 * cupy.cos(2 * fr)) / cupy.exp(6 * bw * T)
      elif tid == 7:
          a[tid] = -8 * cupy.cos(fr) / cupy.exp(7 * bw * T)
      elif tid == 8:
          a[tid] = cupy.exp(-8 * bw * T)
  
imports: import operator
  from math import pi
  import warnings
  import cupy
  from cupy.polynomial.polynomial import polyval, polyvalfromroots
  import cupyx.scipy.fft
  from cupyx import jit
  from cupyx.scipy._lib._util import float_factorial
  from cupyx.scipy.signal._polyutils import roots
label_desc: Use image node_id 12 for calling a global function with example usage: _gammatone_iir_kernel(fs, freq, b, a) without return types