Interest rate in installment payment

Recently I bought a commodity with a loan , Before the loan, the merchant said that the annual interest rate was only 3.11%, It looks really low , But it was later found that due to the staging , The comprehensive annual interest rate is not low .
This product is divided into 36 period , Pay an equal amount in each installment , namely ： Suppose the loan amount is $A$ element , Each installment shall be repaid ：
$$\frac{A+A\ast 0.0311\ast 3}{36}$$ What is the comprehensive annual interest rate ？ Now try to calculate .
hypothesis , The loan amount is $A$ element , The comprehensive annual interest rate is $x$, The first $i$ The amount of repayment is ：$y$（ Equal amount of each period ）, The principal is ：$w(i)$, The interest is ：$v(i)$, Then there are the following equations ：
$$y=\frac{A+A\ast 0.0311\ast 3}{36}\text{\hspace{1em}(1)}$$
$$y=w(i)+v(i)\text{\hspace{1em}(2)}$$
$$v(i)=(A-\sum _{j=1}^{i-1}w(j))\ast \frac{x}{12}\text{\hspace{1em}(3)}$$
$$\sum _{i=1}^{36}v(i)=A\ast 0.0311\ast 3\text{\hspace{1em}(4)}$$

The above is known 4 Equation , seek ：$x$ ？

Poor math skills , Although the formula is listed , But I don't know how to calculate . You can only write the following program manually $(3)$ Type in the $x$, The basic principle is ： Suppose the comprehensive annual interest rate is 0.0311（ The annual interest rate mentioned by the merchant ） To 0.1（ It is estimated that it will not exceed 10%） Between , Then you can go from [0.0311, 0.1] Select evenly 1000 A little bit , From here 1000 Choose one of the points to make $(4)$ The error of the left and right ends of the formula is the smallest $x$ value . Carefully observe the above formulas , It can be found in $(4)$ After expansion , Both sides of the equation can simultaneously reduce the loan principal , in other words , The loan principal has nothing to do with the comprehensive annual interest rate , The following code assumes that the loan principal is 100000. The code is as follows ：

#include <stdio.h>

#define A 100000
#define CANDIDATE_MAX_NUM 1000

struct candidate {
    
	double rate;
	double result;
	double diff;
};

double abs_double(double num)
{
    
	return num >= 0 ? num : -num;
}

double v_sum(double rate)
{
    
	double y = (A + A * 0.0311 * 3) / 36;
	double vi, wi, w_sum = 0, _v_sum = 0;
	int i = 0;

	for (i = 1; i <= 36; ++i) {
    
		vi = (A - w_sum) * rate * 1 / 12;
		wi = y - vi;
		w_sum += wi;
		_v_sum += vi;
	}

	return _v_sum;
}

void calc(void)
{
    
	double min = 0.0311;
	double max = 0.1000;
	double step = (max - min) / CANDIDATE_MAX_NUM;
	double r = min;
	struct candidate cand = {
    0, 0, 0};
	double target = A * 0.0311 * 3;
	double result, diff;
	int i = 0;

	for (i = 0; i < CANDIDATE_MAX_NUM; ++i) {
    
		result = v_sum(r);
		diff = abs_double(result - target);
		if (i == 0 || diff < cand.diff) {
    
			cand.rate = r;
			cand.result = result;
			cand.diff = diff;
		}
		r += step;
	}

	printf("rate = %lf, result = %lf, target = %lf, diff = %lf\n",
		cand.rate, cand.result, target, cand.diff);
}

int main(void)
{
    
	calc();

	return 0;
}

The program runs as follows ：

[[email protected]]$ ./main 
rate = 0.058867, result = 9334.592459, target = 9330.000000, diff = 4.592459

It can be seen from the above running results , The calculated comprehensive annual interest rate $x$ by 5.88%, The annual interest rate is much higher than the business said , Like to see , The comprehensive annual interest rate of installment payment is not low .