5-Dimensional Monte Carlo Integration

This is a simple aplication of the monte-carlo simulation the idea is to just generate a lot of random points and see which ones fit your criteria. Here is my implementation of the solution, I also added some a nice plots to see how the ratio converges. I'm doing 10 simulations with 10000000 points each which gives me 100000000 points to get a good aproximation and allows me to estimate the error by calculating the standard deviation of the outputs, from that I got this as a final result:

Mean: 0.041853749999999995 Standard deviation: 6.970503927263917e-05

Here is my code:

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from tqdm import tqdm
import matplotlib.pyplot as plt
import numpy as np


def monte_carlo_simulation(runs=1001):
    plt.ylabel('Ratio')
    plt.xlabel('Runs')

    positive = 0.
    pbar = tqdm(range(1, runs + 1))
    # Start from 1 so it's easier to graph since i doesn't really metter
    for i in pbar:
        point = np.random.rand(5)
        # Testing for conditions posed
        if (
                point[0] >= point[1] ** 2 and
                point[1] <= np.sin(point[2]) and
                np.exp(point[3]) >= 3 / 2 and
                point[0] * point[4] >= 1 / 2):
            # Adds point to positive test
            positive += 1

        if i % 1000 == 0:
            # Plot calculated ratio every 1000 runs
            plt.plot(i, positive / i, 'ro')

    plt.plot(runs, positive / runs, 'ro')
    plt.show()

    return positive / runs


if __name__ == "__main__":
    data = []
    for i in range(1, 11):
        x = monte_carlo_simulation(10000000)
        print(f"Test {i}: {x}")
        data.append(x)

    print(f"Mean: {np.mean(data)} \t Standard deviation: {np.std(data)}")

And here are 2 of my plots:

The problem has a closed form solution.

$\text{FullSimplify}\left[ \\ \ \ \text{Reduce}\left[a\geq b^2\land b\leq \sin (c)\land e^d\geq \frac{3}{2}\land a\ e\geq \frac{1}{2}\land 0\leq a\leq 1\land \\ \ \ \ \ 0\leq b\leq 1\land 0\leq c\leq 1\land 0\leq d\leq 1\land 0\leq e\leq 1\right]\right] \Longrightarrow$ $\log \left(\frac{3}{2}\right)\leq d\leq 1\land \\ \left(\left(e\leq 1\land \left(\left(a=1\land 2 e\geq 1\land \left(\left(c\leq 1\land \left((b=0\land c\geq 0)\lor \left(b<\sin (1)\land b>0\land \\ c\geq \sin ^{-1}(b)\right)\right)\right)\lor (b=\sin (1)\land c=1)\right)\right)\lor \\ \left(\frac{1}{a}\leq 2 e\land \left(\left(a<1\land \left(\left(c\leq 1\land \left((b=0\land 2 a>1\land c\geq 0)\lor \left(b>0\land a\geq \sin ^2(1)\land c\geq \sin ^{-1}(b)\land b<\sin (1)\right)\right)\right)\lor \\ \left(a\geq \sin ^2(1)\land b=\sin (1)\land c=1\right)\right)\right)\lor \left(a<\sin ^2(1)\land \\ 2 a>1\land \sqrt{a}\geq b\land b>0\land c\leq 1\land c\geq \sin ^{-1}(b)\right)\right)\right)\right)\right)\lor \\ \left(2 a=1\land c\leq 1\land e=1\land \left((b=0\land c\geq 0)\lor \left(b\leq \frac{1}{\sqrt{2}}\land b>0\land c\geq \sin ^{-1}(b)\right)\right)\right)\right)$

Eliminate clauses where a variable is only equal to a value as $\int_z^z f(x) \, dx$ is $0$ .

$\log \left(\frac{3}{2}\right)\leq d\leq 1\land \sin ^{-1}(b)\leq c\leq 1\land \frac{1}{2 a}\leq e\leq 1\land \left(\left(\frac{1}{2}<a<\sin ^2(1)\land 0<b\leq \sqrt{a}\right)\lor \left(\sin ^2(1)<a<1\land 0<b<\sin (1)\right)\right) \Longrightarrow \\ \frac{1}{36} \left(\log \left(\frac{3}{2}\right)-1\right) \left(3 \pi \sqrt{2}+20 \sqrt{2}-18+\cos (3)+36 \log \left(-\left(\sqrt{2}-2\right) \cot \left(\frac{1}{2}\right)\right)+9 \cos (1) (4 \log (\sin (1))-5)\right) \approx \\ 0.041878431746397186653.$