code: def _denominator(t_slice_shape, precision, unroll=1):
      """Computes the denominator."""
  
      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 enum
  import functools
  from typing import Any, Callable, Dict, Optional, Sequence, Tuple, Iterable
  from absl import logging
  from flax import linen
  import jax
  from jax import lax
  from jax import random
  import jax.numpy
  import numpy
  from scenic.model_lib.layers import attention_layers
label_desc: Use image node_id 12 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