apply `sumcheck_layer_eval` to compute initial sum in tower verifier

[kunxian-xia]

We should use sumcheck_layer_eval to reduce the cycles in the computation of initial_sum.

The old codes do three things:

1. get initial product evaluations;
   

   ceno/ceno_recursion/src/tower_verifier/program.rs

   Lines 224 to 243 in 1e7dbea

   	let prod_spec_point_n_eval: Array<C, PointAndEvalVariable<C>> =
   	builder.dyn_array(num_prod_spec.clone());
   	iter_zip!(builder, prod_out_evals, prod_spec_point_n_eval).for_each(|ptr_vec, builder| {
   	let ptr = ptr_vec[0];
   	let evals = builder.iter_ptr_get(&prod_out_evals, ptr);
   	let e = evaluate_at_point_degree_1(builder, &evals, &initial_rt);
   	let p_ptr = ptr_vec[1];
   	builder.iter_ptr_set(
   	&prod_spec_point_n_eval,
   	p_ptr,
   	PointAndEvalVariable {
   	point: PointVariable {
   	fs: initial_rt.clone(),
   	},
   	eval: e,
   	},
   	);
   	});

2. get initial logup evaluations and store them in two separate arrays.
   

   ceno/ceno_recursion/src/tower_verifier/program.rs

   Lines 243 to 289 in 1e7dbea

   	let logup_spec_p_point_n_eval: Array<C, PointAndEvalVariable<C>> =
   	builder.dyn_array(num_logup_spec.clone());
   	let logup_spec_q_point_n_eval: Array<C, PointAndEvalVariable<C>> =
   	builder.dyn_array(num_logup_spec.clone());
   	iter_zip!(
   	builder,
   	logup_out_evals,
   	logup_spec_p_point_n_eval,
   	logup_spec_q_point_n_eval
   	)
   	.for_each(|ptr_vec, builder| {
   	let ptr = ptr_vec[0];
   	let evals = builder.iter_ptr_get(&prod_out_evals, ptr);
   	let p_slice = evals.slice(builder, 0, 2);
   	let q_slice = evals.slice(builder, 2, 4);
   	let e1 = evaluate_at_point_degree_1(builder, &p_slice, &initial_rt);
   	let e2 = evaluate_at_point_degree_1(builder, &q_slice, &initial_rt);
   	let p_ptr = ptr_vec[1];
   	let q_ptr = ptr_vec[2];
   	builder.iter_ptr_set(
   	&logup_spec_p_point_n_eval,
   	p_ptr,
   	PointAndEvalVariable {
   	point: PointVariable {
   	fs: initial_rt.clone(),
   	},
   	eval: e1,
   	},
   	);
   	builder.iter_ptr_set(
   	&logup_spec_q_point_n_eval,
   	q_ptr,
   	PointAndEvalVariable {
   	point: PointVariable {
   	fs: initial_rt.clone(),
   	},
   	eval: e2,
   	},
   	);
   	});

3. get initial sum by accumulating the above 3 arrays.
   

   ceno/ceno_recursion/src/tower_verifier/program.rs

   Lines 289 to 308 in 1e7dbea

   	let initial_claim: Ext<C::F, C::EF> = builder.eval(zero + zero);
   	iter_zip!(builder, prod_spec_point_n_eval).for_each(|ptr_vec, builder| {
   	let ptr = ptr_vec[0];
   	let prod_eval = builder.iter_ptr_get(&prod_spec_point_n_eval, ptr);
   	builder.assign(&initial_claim, initial_claim + prod_eval.eval * alpha_acc);
   	builder.assign(&alpha_acc, alpha_acc * alpha);
   	});
   	builder
   	.range(0, num_logup_spec.clone())
   	.for_each(|i_vec, builder| {
   	let p = builder.get(&logup_spec_p_point_n_eval, i_vec[0]);
   	builder.assign(&initial_claim, initial_claim + p.eval * alpha_acc);
   	builder.assign(&alpha_acc, alpha_acc * alpha);
   	let q = builder.get(&logup_spec_q_point_n_eval, i_vec[0]);
   	builder.assign(&initial_claim, initial_claim + q.eval * alpha_acc);
   	builder.assign(&alpha_acc, alpha_acc * alpha);
   	});

[darth-cy]

Completed in: #1249