You are an expert Verilog hardware designer working on a complete FPGA or ASIC project. You are given the full source code of a Verilog project. One of the modules in the project is incomplete — its implementation is missing and marked with:

// >>> Module Implementation Begin

// <<< Module Implementation End

Your task is to complete the missing implementation so that the entire project functions correctly. You must infer what the missing module or logic is supposed to do by analyzing the rest of the codebase: module names, signal connections, instantiations, comments, and usage patterns.

Follow these rules:

1. Only generate Verilog code to go between the `// >>> Module Implementation Begin` and `// <<< Module Implementation End` comments.
2. Use Verilog-2001 syntax. Ensure the code is synthesizable.
3. Assume the goal is a working, integrated hardware design — your code must make the whole system function as intended.
4. Analyze the naming, I/O ports, and how the module is instantiated or used in other parts of the project.
5. Implement correct behavior based on design context. Do not guess randomly — reason it out.
6. Avoid vendor-specific features unless explicitly required.
7. Do not output comments, explanation, or anything outside the required code region unless explicitly asked.

Your output should contain **only the Verilog code** that fits inside the marked region.

/*
 * Copyright (c) 2024 Adonai Cruz
 * SPDX-License-Identifier: Apache-2.0
 */

`default_nettype none

module tt_um_adonairc_dda (
    input  wire [7:0] ui_in,    // Dedicated inputs
    output wire [7:0] uo_out,   // Dedicated outputs
    input  wire [7:0] uio_in,   // IOs: Input path
    output wire [7:0] uio_out,  // IOs: Output path
    output wire [7:0] uio_oe,   // IOs: Enable path (active high: 0=input, 1=output)
    input  wire       ena,      // always 1 when the design is powered, so you can ignore it
    input  wire       clk,      // clock
    input  wire       rst_n     // reset_n - low to reset
);
	parameter N = 16; // Posit word size
	parameter ES = 1; // Posit ES

	// All output pins must be assigned. If not used, assign to 0.
	assign uo_out  = 0;
	assign uio_out[7:3] = 0;
	assign uio_out[1:0] = 0;
	assign uio_oe[7:4]  = 0;
	
	// SPI pins
	wire sclk, cs, mosi, miso;

	// Output port (MISO)
	assign uio_oe[2] = 1; 

	// Input ports
	assign cs  = uio_in[0];
	assign uio_oe[0] = 0;
	assign mosi  = uio_in[1];
	assign uio_oe[1] = 0;
	assign sclk  = uio_in[3];
	assign uio_oe[3] = 0;
	
	// Dynamical system parameters
	reg [N-1:0] icx, icy; // Initial conditions for variables x and y

	wire [N-1:0] mu; // Van-der-Pol parameter
	reg [N-1:0] r_mu;

	// DDA
	reg clk_dda; // DDA clock

	assign mu = r_mu;

	// SPI
	reg [2:0] r_SCK;
	reg [2:0] r_CS;
	reg [1:0] r_MOSI;
	reg [4:0] bitcnt; // bit counter
	reg is_data_received; // high when a byte has been received
	reg [31:0] data_received;
	reg [31:0] data_sent;
	reg [31:0] cnt;

	// state variables
	wire [N-1:0] x,y;

	// Van-Der-Pol DDA instance
	dda #(.N(N), .ES(ES)) van_der_pol(
		.clk(clk_dda),
		.rst_n(rst_n),
		.x(x),
		.y(y),
		.icx(icx),
		.icy(icy),
		.mu(mu)
	);

	// Reset to enable DDA and set initial conditions
	always @(posedge clk)  begin
		if (!rst_n) begin
			icx <= 16'h3000;
			icy <= 16'h3000;
		end
	end

	// 
	// SPI interface
	//

	// Sync SCK to FPGA clock using a 3-bits shift register;
	always @(posedge clk) r_SCK <= {r_SCK[1:0], sclk};
	wire SCK_risingedge = (r_SCK[2:1]==2'b01); // detect SCK rising edge
	wire SCK_fallingedge = (r_SCK[2:1]==2'b10); // SCK falling edge

	// Sync CS
	always @(posedge clk) r_CS <= {r_CS[1:0], cs};
	wire CS_active = ~r_CS[1]; // CS is active low
	wire CS_startmessage = (r_CS[2:1]==2'b10); // message starts at falling edge
	wire CS_endmessage = (r_CS[2:1]==2'b01); // message stops at rising edge

	// Sync MOSI
	always @(posedge clk) r_MOSI <= {r_MOSI[0], mosi};
	wire MOSI_data = r_MOSI[1];

	// Handle SPI data in 32-bits word format
	always @(posedge clk)
	begin
		if (~CS_active)
			bitcnt <= 0;
		else
		if (SCK_risingedge)
		begin
			bitcnt <= bitcnt + 1;
			// implement a shift-left register (since we receive the data MSB first)
			data_received <= {data_received[30:0], MOSI_data};
		end
	end

	always @(posedge clk) is_data_received <= CS_active &&  SCK_risingedge && (bitcnt == 5'b11111);

	// Updates system parameter mu
	always @(posedge clk) if(is_data_received) r_mu <= data_received[15:0];

	// Data write
	always @(posedge clk) begin
		if(!rst_n) begin
			clk_dda <= 0;
		end
		if (CS_startmessage) begin
			clk_dda <= !clk_dda; // clock the DDA
		end
	end

	always @(posedge clk)
	if (CS_active)
	begin
		if (CS_startmessage) data_sent <= {x,y};
		else
		if (SCK_fallingedge) data_sent <= (bitcnt == 5'b00000)? x : {data_sent[30:0], 1'b0};
	end

	assign uio_out[2] = data_sent[31]; // send MSB first (MISO)

endmodule


/*
 * Copyright (c) 2024 Adonai Cruz
 * SPDX-License-Identifier: Apache-2.0
 */

module dda (
    input clk, // Clock
    input rst_n, // Reset (active low)
    output [N-1:0] x,y, // Dynamical system state variables
    input [N-1:0] icx, icy, // Initial conditions
    input [N-1:0] mu // Van-der-Pol parameter
);

// Posit parameters
parameter N = 16;
parameter ES = 1;

// Multiplications
wire [N-1:0] w_mult1, w_mult2, w_mult3;

posit_mult #(.N(N),.ES(ES)) mult1(.in1(x), .in2(x), .out(w_mult1)); // Multiply x*x
posit_mult #(.N(N),.ES(ES)) mult2(.in1(mu), .in2(w_sub1), .out(w_mult2)); // Multiply mu*(1-x*x)
posit_mult #(.N(N),.ES(ES)) mult3(.in1(w_mult2), .in2(y), .out(w_mult3)); // Multiply mu*(1-x*x)*y

// Subtraction
wire [N-1:0] w_sub1, w_sub2;

wire [N-1:0] w_neg_mult1;
assign w_neg_mult1 = {~w_mult1[N-1], ~w_mult1[N-2:0]+1'b1};

wire [N-1:0] w_neg_x;
assign w_neg_x = {~x[N-1], ~x[N-2:0]+1'b1};

posit_add #(.N(N),.ES(ES)) sub_1_mult_x_x(.in1(16'b0100000000000000), .in2(w_neg_mult1), .out(w_sub1)); // Subtract 1 - x*x
posit_add #(.N(N),.ES(ES)) sub_rho_z(.in1(w_mult3), .in2(w_neg_x), .out(w_sub2)); // Subtract mu*(1-x*x)y - x

// Van der Pol oscillator
// dx/dt = y
// dy/dt = mu*(1 - x*x) - x

euler_integrator  #(.N(N),.ES(ES)) int1(.out(x), .funct(y), .ic(icx), .clk(clk), .rst_n(rst_n));
euler_integrator  #(.N(N),.ES(ES)) int2(.out(y), .funct(w_sub2), .ic(icy), .clk(clk), .rst_n(rst_n));

endmodule


/// Euler integrator
module euler_integrator(out, funct, clk, rst_n, ic);
// >>> Module Implementation Begin

// <<< Module Implementation End
endmodule





/*
 * Copyright (c) 2024 Adonai Cruz
 * SPDX-License-Identifier: Apache-2.0
 */

///////////
// Adder //
///////////
module posit_add (in1, in2, out);

function [31:0] log2;
input reg [31:0] value;
	begin
	value = value-1;
	for (log2=0; value>0; log2=log2+1)
        	value = value>>1;
      	end
endfunction

parameter N = 16;
parameter Bs = log2(N); 
parameter ES = 1;

input [N-1:0] in1, in2;
output [N-1:0] out;
wire inf, zero;

wire s1 = in1[N-1];
wire s2 = in2[N-1];
wire zero_tmp1 = |in1[N-2:0];
wire zero_tmp2 = |in2[N-2:0];
wire inf1 = in1[N-1] & (~zero_tmp1),
	inf2 = in2[N-1] & (~zero_tmp2);
wire zero1 = ~(in1[N-1] | zero_tmp1),
	zero2 = ~(in2[N-1] | zero_tmp2);
assign inf = inf1 | inf2,
	zero = zero1 & zero2;

//Data Extraction
wire rc1, rc2;
wire [Bs-1:0] regime1, regime2;
wire [ES-1:0] e1, e2;
wire [N-ES-1:0] mant1, mant2;
wire [N-1:0] xin1 = s1 ? -in1 : in1;
wire [N-1:0] xin2 = s2 ? -in2 : in2;
data_extract_v1 #(.N(N),.ES(ES)) uut_de1(.in(xin1), .rc(rc1), .regime(regime1), .exp(e1), .mant(mant1));
data_extract_v1 #(.N(N),.ES(ES)) uut_de2(.in(xin2), .rc(rc2), .regime(regime2), .exp(e2), .mant(mant2));

wire [N-ES:0] m1 = {zero_tmp1,mant1}, 
	m2 = {zero_tmp2,mant2};

//Large Checking and Assignment
wire in1_gt_in2 = (xin1[N-2:0] >= xin2[N-2:0]) ? 1'b1 : 1'b0;

wire ls = in1_gt_in2 ? s1 : s2;
wire op = s1 ~^ s2;

wire lrc = in1_gt_in2 ? rc1 : rc2;
wire src = in1_gt_in2 ? rc2 : rc1;

wire [Bs-1:0] lr = in1_gt_in2 ? regime1 : regime2;
wire [Bs-1:0] sr = in1_gt_in2 ? regime2 : regime1;

wire [ES-1:0] le = in1_gt_in2 ? e1 : e2;
wire [ES-1:0] se = in1_gt_in2 ? e2 : e1;

wire [N-ES:0] lm = in1_gt_in2 ? m1 : m2;
wire [N-ES:0] sm = in1_gt_in2 ? m2 : m1;

//Exponent Difference: Lower Mantissa Right Shift Amount
wire [ES+Bs+1:0] diff;
wire [Bs:0] lr_N;
wire [Bs:0] sr_N;
a_abs_regime #(.N(Bs)) uut_abs_regime1 (lrc, lr, lr_N);
a_abs_regime #(.N(Bs)) uut_abs_regime2 (src, sr, sr_N);
a_sub_N #(.N(ES+Bs+1)) uut_ediff ({lr_N,le}, {sr_N, se}, diff);
wire [Bs-1:0] exp_diff = (|diff[ES+Bs:Bs]) ? {Bs{1'b1}} : diff[Bs-1:0];

//DSR Right Shifting
wire [N-1:0] DSR_right_in;
generate
	if (ES >= 2) 
	assign DSR_right_in = {sm,{ES-1{1'b0}}};
	else 
	assign DSR_right_in = sm;
endgenerate

wire [N-1:0] DSR_right_out;
wire [Bs-1:0] DSR_e_diff  = exp_diff;
DSR_right_N_S #(.N(N), .S(Bs))  dsr1(.a(DSR_right_in), .b(DSR_e_diff), .c(DSR_right_out));

//Mantissa Addition
wire [N-1:0] add_m_in1;
generate
	if (ES >= 2) 
	assign add_m_in1 = {lm,{ES-1{1'b0}}};
	else 
	assign add_m_in1 = lm;
endgenerate

wire [N:0] add_m;
a_add_sub_N #(.N(N)) uut_add_sub_N (op, add_m_in1, DSR_right_out, add_m);
wire [1:0] mant_ovf = add_m[N:N-1];

//LOD
wire [N-1:0] LOD_in = {(add_m[N] | add_m[N-1]), add_m[N-2:0]};
wire [Bs-1:0] left_shift;
LOD_N #(.N(N)) l2(.in(LOD_in), .out(left_shift));

//DSR Left Shifting
wire [N-1:0] DSR_left_out_t;
DSR_left_N_S #(.N(N), .S(Bs)) dsl1(.a(add_m[N:1]), .b(left_shift), .c(DSR_left_out_t));
wire [N-1:0] DSR_left_out = DSR_left_out_t[N-1] ? DSR_left_out_t[N-1:0] : {DSR_left_out_t[N-2:0],1'b0}; 


//Exponent and Regime Computation
wire [ES+Bs+1:0] le_o_tmp, le_o;
a_sub_N #(.N(ES+Bs+1)) sub3 ({lr_N,le}, {{ES+1{1'b0}},left_shift}, le_o_tmp);
add_1 #(.N(ES+Bs+1)) uut_add_mantovf (le_o_tmp, mant_ovf[1], le_o);

wire [ES-1:0] e_o;
wire [Bs-1:0] r_o;
a_reg_exp_op #(.ES(ES), .Bs(Bs)) uut_reg_ro (le_o[ES+Bs:0], e_o, r_o);

//Exponent and Mantissa Packing
wire [2*N-1+3:0] tmp_o;
generate
	if(ES > 2)
		assign tmp_o = { {N{~le_o[ES+Bs]}}, le_o[ES+Bs], e_o, DSR_left_out[N-2:ES-2], |DSR_left_out[ES-3:0]};
	else 
		assign tmp_o = { {N{~le_o[ES+Bs]}}, le_o[ES+Bs], e_o, DSR_left_out[N-2:0], {3-ES{1'b0}} };

endgenerate

//Including/Pushing Regime bits in Exponent-Mantissa Packing
wire [3*N-1+3:0] tmp1_o;
DSR_right_N_S #(.N(3*N+3), .S(Bs)) dsr2 (.a({tmp_o,{N{1'b0}}}), .b(r_o), .c(tmp1_o));

//Rounding RNE : ulp_add = G.(R + S) + L.G.(~(R+S))
wire L = tmp1_o[N+4], G = tmp1_o[N+3], R = tmp1_o[N+2], St = |tmp1_o[N+1:0],
     ulp = ((G & (R | St)) | (L & G & ~(R | St)));
wire [N-1:0] rnd_ulp = {{N-1{1'b0}},ulp};

wire [N:0] tmp1_o_rnd_ulp;
a_add_N #(.N(N)) uut_add_ulp (tmp1_o[2*N-1+3:N+3], rnd_ulp, tmp1_o_rnd_ulp);
wire [N-1:0] tmp1_o_rnd = (r_o < N-ES-2) ? tmp1_o_rnd_ulp[N-1:0] : tmp1_o[2*N-1+3:N+3];

//Final Output
wire [N-1:0] tmp1_oN = ls ? -tmp1_o_rnd : tmp1_o_rnd;
assign out = inf|zero|(~DSR_left_out[N-1]) ? {inf,{N-1{1'b0}}} : {ls, tmp1_oN[N-1:1]};

endmodule

////////////////
// Multiplier //
////////////////
module posit_mult(in1, in2, out);

function [31:0] log2;
input reg [31:0] value;
	begin
	value = value-1;
	for (log2=0; value>0; log2=log2+1)
        	value = value>>1;
      	end
endfunction

parameter N = 16;
parameter Bs = log2(N); 
parameter ES = 1;

input [N-1:0] in1, in2;
output [N-1:0] out;
wire inf, zero;

wire s1 = in1[N-1];
wire s2 = in2[N-1];
wire zero_tmp1 = |in1[N-2:0];
wire zero_tmp2 = |in2[N-2:0];
wire inf1 = in1[N-1] & (~zero_tmp1),
	inf2 = in2[N-1] & (~zero_tmp2);
wire zero1 = ~(in1[N-1] | zero_tmp1),
	zero2 = ~(in2[N-1] | zero_tmp2);
assign inf = inf1 | inf2,
	zero = zero1 & zero2;

//Data Extraction
wire rc1, rc2;
wire [Bs-1:0] regime1, regime2;
wire [ES-1:0] e1, e2;
wire [N-ES-1:0] mant1, mant2;
wire [N-1:0] xin1 = s1 ? -in1 : in1;
wire [N-1:0] xin2 = s2 ? -in2 : in2;
data_extract_v1 #(.N(N),.ES(ES)) uut_de1(.in(xin1), .rc(rc1), .regime(regime1), .exp(e1), .mant(mant1));
data_extract_v1 #(.N(N),.ES(ES)) uut_de2(.in(xin2), .rc(rc2), .regime(regime2), .exp(e2), .mant(mant2));

wire [N-ES:0] m1 = {zero_tmp1,mant1}, 
	m2 = {zero_tmp2,mant2};

//Sign, Exponent and Mantissa Computation
wire mult_s = s1 ^ s2;

wire [2*(N-ES)+1:0] mult_m = m1*m2;
wire mult_m_ovf = mult_m[2*(N-ES)+1];
wire [2*(N-ES)+1:0] mult_mN = ~mult_m_ovf ? mult_m << 1'b1 : mult_m;

wire [Bs+1:0] r1 = rc1 ? {2'b0,regime1} : -regime1;
wire [Bs+1:0] r2 = rc2 ? {2'b0,regime2} : -regime2;
wire [Bs+ES+1:0] mult_e;
m_add_N_Cin #(.N(Bs+ES+1)) uut_add_exp ({r1,e1}, {r2,e2}, mult_m_ovf, mult_e);

//Exponent and Regime Computation
wire [ES-1:0] e_o;
wire [Bs:0] r_o;
m_reg_exp_op #(.ES(ES), .Bs(Bs)) uut_reg_ro (mult_e[ES+Bs+1:0], e_o, r_o);

//Exponent, Mantissa and GRS Packing
wire [2*N-1+3:0]tmp_o = {{N{~mult_e[ES+Bs+1]}},mult_e[ES+Bs+1],e_o,mult_mN[2*(N-ES):2*(N-ES)-(N-ES-1)+1], mult_mN[2*(N-ES)-(N-ES-1):2*(N-ES)-(N-ES-1)-1], |mult_mN[2*(N-ES)-(N-ES-1)-2:0] }; 


//Including Regime bits in Exponent-Mantissa Packing
wire [3*N-1+3:0] tmp1_o;
DSR_right_N_S #(.N(3*N+3), .S(Bs+1)) dsr2 (.a({tmp_o,{N{1'b0}}}), .b(r_o[Bs] ? {Bs{1'b1}} : r_o), .c(tmp1_o));

//Rounding RNE : ulp_add = G.(R + S) + L.G.(~(R+S))
wire L = tmp1_o[N+4], G = tmp1_o[N+3], R = tmp1_o[N+2], St = |tmp1_o[N+1:0],
     ulp = ((G & (R | St)) | (L & G & ~(R | St)));
wire [N-1:0] rnd_ulp = {{N-1{1'b0}},ulp};

wire [N:0] tmp1_o_rnd_ulp;
m_add_N #(.N(N)) uut_add_ulp (tmp1_o[2*N-1+3:N+3], rnd_ulp, tmp1_o_rnd_ulp);
wire [N-1:0] tmp1_o_rnd = (r_o < N-ES-2) ? tmp1_o_rnd_ulp[N-1:0] : tmp1_o[2*N-1+3:N+3];


//Final Output
wire [N-1:0] tmp1_oN = mult_s ? -tmp1_o_rnd : tmp1_o_rnd;
assign out = inf|zero|(~mult_mN[2*(N-ES)+1]) ? {inf,{N-1{1'b0}}} : {mult_s, tmp1_oN[N-1:1]};
endmodule


///////////////////////////////////////////////
// Multiply by hard-coded constant in (16,1) //
/// dt = 0x8000 					         //
///////////////////////////////////////////////
module posit_dt_mult(in1, out);

function [31:0] log2;
input reg [31:0] value;
	begin
	value = value-1;
	for (log2=0; value>0; log2=log2+1)
        	value = value>>1;
      	end
endfunction

parameter N = 16;
parameter Bs = log2(N); 
parameter ES = 1;

input [N-1:0] in1;
output [N-1:0] out;

wire s1 = in1[N-1];
wire s2; //

assign s2 = 0;

wire zero_tmp1 = |in1[N-2:0];
wire inf = in1[N-1] & (~zero_tmp1);
wire zero = ~(in1[N-1] | zero_tmp1);

//Data Extraction
wire rc1, rc2;
wire [Bs-1:0] regime1, regime2;
wire [ES-1:0] e1, e2;
wire [N-ES-1:0] mant1, mant2;
wire [N-1:0] xin1 = s1 ? -in1 : in1;
wire [N-1:0] xin2;

data_extract_v1 #(.N(N),.ES(ES)) uut_de1(.in(xin1), .rc(rc1), .regime(regime1), .exp(e1), .mant(mant1));

// data_extract_v1 #(.N(N),.ES(ES)) uut_de2(.in(xin2), .rc(rc2), .regime(regime2), .exp(e2), .mant(mant2));
assign regime2 = 4;
assign xin2 = 16'h0400;
assign rc2 = 0;
assign e2 = 0;
assign mant2 = 0;

wire [N-ES:0] m1 = {zero_tmp1,mant1};
wire [N-ES:0] m2;

assign m2 = 16'h8000;

//Sign, Exponent and Mantissa Computation
wire mult_s = s1;

wire [2*(N-ES)+1:0] mult_m = m1*m2;
wire mult_m_ovf = mult_m[2*(N-ES)+1];
wire [2*(N-ES)+1:0] mult_mN = ~mult_m_ovf ? mult_m << 1'b1 : mult_m;

wire [Bs+1:0] r1 = rc1 ? {2'b0,regime1} : -regime1;
wire [Bs+1:0] r2 = rc2 ? {2'b0,regime2} : -regime2;
wire [Bs+ES+1:0] mult_e;
m_add_N_Cin #(.N(Bs+ES+1)) uut_add_exp ({r1,e1}, {r2,e2}, mult_m_ovf, mult_e);

//Exponent and Regime Computation
wire [ES-1:0] e_o;
wire [Bs:0] r_o;
m_reg_exp_op #(.ES(ES), .Bs(Bs)) uut_reg_ro (mult_e[ES+Bs+1:0], e_o, r_o);

//Exponent, Mantissa and GRS Packing
wire [2*N-1+3:0]tmp_o = {{N{~mult_e[ES+Bs+1]}},mult_e[ES+Bs+1],e_o,mult_mN[2*(N-ES):2*(N-ES)-(N-ES-1)+1], mult_mN[2*(N-ES)-(N-ES-1):2*(N-ES)-(N-ES-1)-1], |mult_mN[2*(N-ES)-(N-ES-1)-2:0] }; 


//Including Regime bits in Exponent-Mantissa Packing
wire [3*N-1+3:0] tmp1_o;
DSR_right_N_S #(.N(3*N+3), .S(Bs+1)) dsr2 (.a({tmp_o,{N{1'b0}}}), .b(r_o[Bs] ? {Bs{1'b1}} : r_o), .c(tmp1_o));

//Rounding RNE : ulp_add = G.(R + S) + L.G.(~(R+S))
wire L = tmp1_o[N+4], G = tmp1_o[N+3], R = tmp1_o[N+2], St = |tmp1_o[N+1:0],
     ulp = ((G & (R | St)) | (L & G & ~(R | St)));
wire [N-1:0] rnd_ulp = {{N-1{1'b0}},ulp};

wire [N:0] tmp1_o_rnd_ulp;
m_add_N #(.N(N)) uut_add_ulp (tmp1_o[2*N-1+3:N+3], rnd_ulp, tmp1_o_rnd_ulp);
wire [N-1:0] tmp1_o_rnd = (r_o < N-ES-2) ? tmp1_o_rnd_ulp[N-1:0] : tmp1_o[2*N-1+3:N+3];


//Final Output
wire [N-1:0] tmp1_oN = mult_s ? -tmp1_o_rnd : tmp1_o_rnd;
assign out = inf|zero|(~mult_mN[2*(N-ES)+1]) ? {inf,{N-1{1'b0}}} : {mult_s, tmp1_oN[N-1:1]};
endmodule


/////////////////////////////
// Adder auxiliary modules //
////////////////////////////


/////////////////
module a_sub_N (a,b,c);
parameter N=10;
input [N-1:0] a,b;
output [N:0] c;
wire [N:0] ain = {1'b0,a};
wire [N:0] bin = {1'b0,b};
a_sub_N_in #(.N(N)) s1 (ain,bin,c);
endmodule

/////////////////////////
module a_add_N (a,b,c);
parameter N=10;
input [N-1:0] a,b;
output [N:0] c;
wire [N:0] ain = {1'b0,a};
wire [N:0] bin = {1'b0,b};
a_add_N_in #(.N(N)) a1 (ain,bin,c);
endmodule

/////////////////////////
module a_sub_N_in (a,b,c);
parameter N=10;
input [N:0] a,b;
output [N:0] c;
assign c = a - b;
endmodule

/////////////////////////
module a_add_N_in (a,b,c);
parameter N=10;
input [N:0] a,b;
output [N:0] c;
assign c = a + b;
endmodule

/////////////////////////
module a_add_sub_N (op,a,b,c);
parameter N=10;
input op;
input [N-1:0] a,b;
output [N:0] c;
wire [N:0] c_add, c_sub;

a_add_N #(.N(N)) a11 (a,b,c_add);
a_sub_N #(.N(N)) s11 (a,b,c_sub);
assign c = op ? c_add : c_sub;
endmodule

/////////////////////////
module a_abs_regime (rc, regime, regime_N);
parameter N = 10;
input rc;
input [N-1:0] regime;
output [N:0] regime_N;

assign regime_N = rc ? {1'b0,regime} : -{1'b0,regime};
endmodule

/////////////////////////
module a_reg_exp_op (exp_o, e_o, r_o);
parameter ES=3;
parameter Bs=5;
input [ES+Bs:0] exp_o;
output [ES-1:0] e_o;
output [Bs-1:0] r_o;

assign e_o = exp_o[ES-1:0];

wire [ES+Bs:0] exp_oN_tmp;
conv_2c #(.N(ES+Bs)) uut_conv_2c1 (~exp_o[ES+Bs:0],exp_oN_tmp);
wire [ES+Bs:0] exp_oN = exp_o[ES+Bs] ? exp_oN_tmp[ES+Bs:0] : exp_o[ES+Bs:0];
assign r_o = (~exp_o[ES+Bs] || |(exp_oN[ES-1:0])) ? exp_oN[ES+Bs-1:ES] + 1 : exp_oN[ES+Bs-1:ES];
endmodule


///////////////////////////////////
// Multiplier auxiliary modules //
/////////////////////////////////
module m_add_N (a,b,c);
parameter N=10;
input [N-1:0] a,b;
output [N:0] c;
assign c = {1'b0,a} + {1'b0,b};
endmodule

/////////////////////////
module m_add_N_Cin (a,b,cin,c);
parameter N=10;
input [N:0] a,b;
input cin;
output [N:0] c;
assign c = a + b + cin;
endmodule

/////////////////////////
module m_reg_exp_op (exp_o, e_o, r_o);
parameter ES=3;
parameter Bs=5;
input [ES+Bs+1:0] exp_o;
output [ES-1:0] e_o;
output [Bs:0] r_o;

assign e_o = exp_o[ES-1:0];

wire [ES+Bs:0] exp_oN_tmp;
conv_2c #(.N(ES+Bs)) uut_conv_2c1 (~exp_o[ES+Bs:0],exp_oN_tmp);
wire [ES+Bs:0] exp_oN = exp_o[ES+Bs+1] ? exp_oN_tmp[ES+Bs:0] : exp_o[ES+Bs:0];

assign r_o = (~exp_o[ES+Bs+1] || |(exp_oN[ES-1:0])) ? exp_oN[ES+Bs:ES] + 1 : exp_oN[ES+Bs:ES];
endmodule

//////////////////////
// General modules //
////////////////////
module data_extract_v1(in, rc, regime, exp, mant);
function [31:0] log2;
input reg [31:0] value;
	begin
	value = value-1;
	for (log2=0; value>0; log2=log2+1)
        	value = value>>1;
      	end
endfunction

parameter N=16;
parameter Bs=log2(N);
parameter ES = 2;

input [N-1:0] in;
output rc;
output [Bs-1:0] regime;
output [ES-1:0] exp;
output [N-ES-1:0] mant;

wire [N-1:0] xin = in;
assign rc = xin[N-2];

wire [N-1:0] xin_r = rc ? ~xin : xin;

wire [Bs-1:0] k;
LOD_N #(.N(N)) xinst_k(.in({xin_r[N-2:0],rc^1'b0}), .out(k));

assign regime = rc ? k-1 : k;

wire [N-1:0] xin_tmp;
DSR_left_N_S #(.N(N), .S(Bs)) ls (.a({xin[N-3:0],2'b0}),.b(k),.c(xin_tmp));

assign exp= xin_tmp[N-1:N-ES];
assign mant= xin_tmp[N-ES-1:0];

endmodule

/////////////////////////
module add_1 (a,mant_ovf,c);
parameter N=10;
input [N:0] a;
input mant_ovf;
output [N:0] c;
assign c = a + mant_ovf;
endmodule

/////////////////////////
module conv_2c (a,c);
parameter N=10;
input [N:0] a;
output [N:0] c;
assign c = a + 1'b1;
endmodule

/////////////////////////
module DSR_left_N_S(a,b,c);
        parameter N=16;
        parameter S=4;
        input [N-1:0] a;
        input [S-1:0] b;
        output [N-1:0] c;

wire [N-1:0] tmp [S-1:0];
assign tmp[0]  = b[0] ? a << 7'd1  : a; 
genvar i;
generate
	for (i=1; i<S; i=i+1)begin:loop_blk
		assign tmp[i] = b[i] ? tmp[i-1] << 2**i : tmp[i-1];
	end
endgenerate
assign c = tmp[S-1];

endmodule

/////////////////////////
module DSR_right_N_S(a,b,c);
        parameter N=16;
        parameter S=4;
        input [N-1:0] a;
        input [S-1:0] b;
        output [N-1:0] c;

wire [N-1:0] tmp [S-1:0];
assign tmp[0]  = b[0] ? a >> 7'd1  : a; 
genvar i;
generate
	for (i=1; i<S; i=i+1)begin:loop_blk
		assign tmp[i] = b[i] ? tmp[i-1] >> 2**i : tmp[i-1];
	end
endgenerate
assign c = tmp[S-1];

endmodule

/////////////////////////
module LOD_N (in, out);

  function [31:0] log2;
    input reg [31:0] value;
    begin
      value = value-1;
      for (log2=0; value>0; log2=log2+1)
	value = value>>1;
    end
  endfunction

parameter N = 64;
parameter S = log2(N); 
input [N-1:0] in;
output [S-1:0] out;

wire vld;
LOD #(.N(N)) l1 (in, out, vld);
endmodule

/////////////////////////
module LOD (in, out, vld);

  function [31:0] log2;
    input reg [31:0] value;
    begin
      value = value-1;
      for (log2=0; value>0; log2=log2+1)
	value = value>>1;
    end
  endfunction


parameter N = 64;
parameter S = log2(N);

   input [N-1:0] in;
   output [S-1:0] out;
   output vld;

  generate
    if (N == 2)
      begin
	assign vld = |in;
	assign out = ~in[1] & in[0];
      end
    else if (N & (N-1))
      //LOD #(1<<S) LOD ({1<<S {1'b0}} | in,out,vld);
      LOD #(1<<S) LOD ({in,{((1<<S) - N) {1'b0}}},out,vld);
    else
      begin
	wire [S-2:0] out_l, out_h;
	wire out_vl, out_vh;
	LOD #(N>>1) l(in[(N>>1)-1:0],out_l,out_vl);
	LOD #(N>>1) h(in[N-1:N>>1],out_h,out_vh);
	assign vld = out_vl | out_vh;
	assign out = out_vh ? {1'b0,out_h} : {out_vl,out_l};
      end
  endgenerate
endmodule



Reference solution:
rtl/generated.v: 	parameter N = 16;
	parameter ES = 1;

	input clk, rst_n;
	output [N-1:0] out; // state variable
	input [N-1:0] funct; // the dV/dt function
	input [N-1:0] ic; // initial condition

	wire [N-1:0] out, v1new;
	reg [N-1:0] v1;

	wire [N-1:0] out_mult;
	
	// compute new state variable with dt 
	// v1(n+1) = v1(n) + dt*funct(n)

    // Multiply funct(n) by constant dt
	posit_dt_mult  #(
		.N(N),
		.ES(ES)
	) mult (
		.in1(funct),
		.out(out_mult)
	);

	posit_add #(
		.N(N),
		.ES(ES)
	)
	add(
		.in1(out_mult),
		.in2(v1),
		.out(v1new)
	);

	always @(posedge clk)
	begin
		if (!rst_n)
			v1 <= ic;
		else
			v1 <= v1new;
	end
	assign out = v1;