code: def _denominator(t_slice_shape, precision, unroll=1):
      def fwd(qs, ks):
          def body(p, qk):
              q, k = qk
              p += k
              x = jnp.einsum(
                  "...m,...m->...", q, p, precision=precision
              )
              return p, x
  
          p = jnp.zeros(t_slice_shape)
          p, R = lax.scan(body, p, (qs, ks), unroll=unroll)
          return R, (qs, ks, p)
  
      def bwd(qkp, R_ct):
          def body(carry, qkx):
              p, p_ct = carry
              q, k, x_ct = qkx
              q_ct = jnp.einsum(
                  "...,...m->...m", x_ct, p, precision=precision
              )
              p_ct += jnp.einsum(
                  "...,...m->...m", x_ct, q, precision=precision
              )
              k_ct = p_ct
              p -= k
              return (p, p_ct), (q_ct, k_ct)
  
          qs, ks, p = qkp
          _, (qs_ct, ks_ct) = lax.scan(
              body,
              (p, jnp.zeros_like(p)),
              (qs, ks, R_ct),
              reverse=True,
              unroll=unroll,
          )
          return (qs_ct, ks_ct)
  
      @jax.custom_vjp
      def _denominator_impl(qs, ks):
          R, _ = fwd(qs, ks)
          return R
  
      _denominator_impl.defvjp(fwd, bwd)
  
      return _denominator_impl
  
imports: import abc
  import functools
  from collections.abc import Iterable
  import jax
  import jax.numpy
  import numpy
  from absl import logging
  from jax import lax, random
label_desc: Use image node_id 7 for calling a global function with example usage: _denominator(t_slice_shape, precision, unroll) and returns: _denominator_impl, R,  qs,  ks,  p, qs_ct,  ks_ct, R, p,  x, p,  p_ct,  q_ct,  k_ct