How to speed up a script for maximum speed in python?

Without analyzing your code in depth, just by tapping the eye quickly, I've come to realize a critical point that can be easily improved: the sqrt call to calculate "distance."

Calculating the square root is quite costly , and you do it very often. If you avoid it, it will greatly increase the performance of your algorithm. In your case it is quite simple because its comparison is with the value 1 , whose power of 2 is 1 ! So, just take the call of sqrt (if you compared it with another value, you would use its power in the comparison).

With this unique change in code, the runtime already falls by half (I also used the time measurement method suggested by colleague @Miguel in his reply ).

. . .
def pis (scale):
    pi = 0
    points = []
    for i in range(scale):
        points.append([random(),random()])
    for p in points:
        if (p[0] * p[0] + p[1] * p[1]) <= 1: # <=== Alteração aqui
            pi += 1
    pi /= scale
    pi *= 4
    return pi
. . .

Here in my tests, running your original code took about 14.5 seconds and produced:

. . .
99%  encontrado: 3.188
99%  encontrado: 3.108
99%  encontrado: 3.128
99%  encontrado: 3.008
3.1419439999999845

While running with the above change took about 8.6 seconds and produced:

. . .
99%  encontrado: 3.032
99%  encontrado: 3.208
99%  encontrado: 3.144
99%  encontrado: 3.176
3.1421183999999878

  

Q.: This is just one of the points I found most critical for the   performance. But it is also worth following the instructions of   response.

I was able to optimize, running with python 3.5, the execution time in + - 1.2 secs. What I did to measure the time of excution was:

from math import sqrt
from random import random
import time

start_t = time.time()

#TODO O TEU CÓDIGO AQUI, fora os imports

print(time.time() - start_t)

Since your code takes between 8.0 secs and 8.3 secs, with the code that I was able to optimize now takes between 6.9 and 7.2, it follows the code:

from math import sqrt
from random import random
import time

start_t = time.time()

piavg = 0
samples = 10000
scale = 1000
progress = 0
debug_rate = 5
debug_clock = 0

def pis (scale):
    pi = 0
    points = ((random(),random()) for _ in range(scale)) # points e um gerador em vez de ser uma lista de listas
    for p1, p2 in points: # unpacking points, assim escusamos de estar sempre a aceder aos indices
        if sqrt(p1 ** 2 + p2 ** 2) <= 1:
            pi += 1
    return pi / scale * 4 # tudo numa so expressao

for _ in range(samples): # por convencao se nao se usa o i podemos deixar assim com _ , nao sei se otimiza a performance mas e boa pratica
    cpi = pis(scale)
    progress += 1/samples*100
    if debug_rate > 0 and debug_clock == 0:
        print("{}%  encontrado: {}".format(int(progress), cpi)) # escusas de tranformar em string de formatares assim, menos duas funcoes que executas aqui
        debug_clock = debug_rate
    piavg += cpi
    debug_clock -= 1
piavg /= samples

print (piavg)

print(time.time() - start_t)

What I've done I've commented on to understand, I'm still studying ways to optimize but this is already a good start.

Thank you, with your help I have managed to achieve this result:

from random import random

scale = int(input("quantas amostras ? "))
repeat = int(input("quantas repetiçoes ? "))
pi = 0

def pic(sc):
    p = 0
    points = ([random(),random()] for i in range(sc))
    def dist (p1, p2, dist):
        d = p1 ** 2 + p2 **2
        return d - dist

    for p1, p2 in points:
        if dist(p1,p2,1) < 0:
            p += 1
    p = p * 4 / sc
    return p

print("calculando...")

for _ in range(repeat):
    pi += pic(scale)
pi /= repeat
print (pi)

This algorithm is considerably faster than my previous ones !!