# SICP - Solution: Exercise 1.22

##### October 15, 2018

## Exercise 1.22 #

Most Lisp implementations include a primitive called

`runtime`

that returns an integer that specifies the amount of time the system has been running (measured, for example, in microseconds). The following`timed-prime-test`

procedure, when called with an integer $n$, prints $n$ and checks to see if $n$ is prime. If $n$ is prime, the procedure prints three asterisks followed by the amount of time used in performing the test.`(define (timed-prime-test n) (newline) (display n) (start-prime-test n (runtime))) (define (start-prime-test n start-time) (if (prime? n) (report-prime (- (runtime) start-time)))) (define (report-prime elapsed-time) (display " *** ") (display elapsed-time))`

Using this procedure, write a procedure

`search-for-primes`

that checks the primality of consecutive odd integers in a specified range. Use your procedure to find the three smallest primes larger than 1000; larger than 10,000; larger than 100,000; larger than 1,000,000. Note the time needed to test each prime. Since the testing algorithm has order of growth of ${\mathrm\Theta(\sqrt n)}$, you should expect that testing for primes around 10,000 should take about $\sqrt {10}$ times as long as testing for primes around 1000. Do your timing data bear this out? How well do the data for 100,000 and 1,000,000 support the ${\mathrm\Theta(\sqrt n)}$ prediction? Is your result compatible with the notion that programs on your machine run in time proportional to the number of steps required for the computation?

## Solution #

Since `runtime`

is not part of DrRacket, you can use
current-inexact-milliseconds. The code, including the `search-for-primes`

function is:

```
#lang racket/base
(define (runtime) (current-inexact-milliseconds)) ; adapting to DrRacket
; --- Prime computation ---
(define (square x) (* x x))
(define (smallest-divisor n)
(find-divisor n 2))
(define (find-divisor n test-divisor)
(cond ((> (square test-divisor) n)
n)
((divides? test-divisor n)
test-divisor)
(else (find-divisor
n
(+ test-divisor 1)))))
(define (divides? a b)
(= (remainder b a) 0))
(define (prime? n)
(= n (smallest-divisor n)))
; --- Timing ---
(define (timed-prime-test n)
(newline)
(display n)
(start-prime-test n (runtime)))
(define (start-prime-test n start-time)
(if (prime? n)
(report-prime (- (runtime) start-time))
"nothing"))
(define (report-prime elapsed-time)
(display " *** ")
(display elapsed-time))
(define (search-for-primes start-range end-range)
(if (even? start-range)
(search-for-primes (+ 1 start-range) end-range)
(cond ((> start-range end-range)
(newline) (display "done"))
(else (timed-prime-test start-range)
(search-for-primes (+ 2 start-range) end-range)))))
(search-for-primes 10000000000 10000000090)
```

The funcion `current-inexact-milliseconds`

provide data in the microseconds scale, but our current hardware are pretty fast and this is not precise enough. I extended the code for computing the averate time of 1000 runs.

### DrRacket has high variability #

The initial test where done with DrRacket:

log(prime) | prime | time (µs) |
---|---|---|

3 | 1,009 | 2.8010 |

3 | 1,013 | 2.4858 |

3 | 1,019 | 2.3898 |

4 | 10,007 | 7.9331 |

4 | 10,009 | 7.9948 |

4 | 10,037 | 7.8310 |

5 | 100,003 | 44.663 |

5 | 100,019 | 26.879 |

5 | 100,043 | 24.563 |

6 | 1,000,003 | 81.736 |

6 | 1,000,033 | 78.949 |

6 | 1,000,037 | 79.261 |

7 | 10,000,019 | 284.77 |

7 | 10,000,079 | 254.29 |

7 | 10,000,103 | 302.63 |

8 | 100,000,007 | 847.27 |

8 | 100,000,037 | 854.13 |

8 | 100,000,039 | 887.25 |

9 | 1,000,000,007 | 2612.4 |

9 | 1,000,000,009 | 2616.5 |

9 | 1,000,000,021 | 2625.0 |

10 | 10,000,000,019 | 8438.9 |

10 | 10,000,000,033 | 8096.1 |

10 | 10,000,000,061 | 8052.0 |

By averaging the run of the 3 first prime numbers and taking successive ratio, you get:

log(prime) | average (µs) | ratio |
---|---|---|

3 | 2,5589 | |

4 | 7,9196 | 3,09493067 |

5 | 28,702 | 3,624145833 |

6 | 79,982 | 2,786651772 |

7 | 280,56 | 3,507844209 |

8 | 862,88 | 3,075520274 |

9 | 2618,0 | 3,03400981 |

10 | 8195,7 | 3,130518518 |

The ratio is pretty close to $\sqrt {10} = 3.162$, although this is only average of 3 run.

### Chicken Scheme, with compiled code, is less variable #

I wanted to compare to another implementation and managed to get the program compiled with Chicken Scheme. The results are somewhat more consistent:

log(prime) | prime | time (µs) |
---|---|---|

3 | 1,009 | 4.0 |

3 | 1,013 | 3.0 |

3 | 1,019 | 4.0 |

4 | 10,007 | 11.0 |

4 | 10,009 | 12.0 |

4 | 10,037 | 11.0 |

5 | 100,003 | 35.0 |

5 | 100,019 | 34.0 |

5 | 100,043 | 36.0 |

6 | 1,000,003 | 111.0 |

6 | 1,000,033 | 108.0 |

6 | 1,000,037 | 114.0 |

7 | 10,000,019 | 341.0 |

7 | 10,000,079 | 353.0 |

7 | 10,000,103 | 342.0 |

8 | 100,000,007 | 1105.0 |

8 | 100,000,037 | 1073.0 |

8 | 100,000,039 | 1080.0 |

9 | 1,000,000,007 | 3439.0 |

9 | 1,000,000,009 | 3471.0 |

9 | 1,000,000,021 | 3494.0 |

10 | 10,000,000,019 | 11129.0 |

10 | 10,000,000,033 | 11193.0 |

10 | 10,000,000,061 | 11224.0 |

The table can be summarized as above:

log(prime) | average (µs) | ratio |
---|---|---|

3 | 3,6 | |

4 | 11 | 3,090 |

5 | 35 | 3,088 |

6 | 111 | 3,171 |

7 | 345 | 3,111 |

8 | 1086 | 3,144 |

9 | 3468 | 3,193 |

10 | 11182 | 3,224 |

The ratio here is also close to $\sqrt {10} = 3.162$, although this is only average of 3 run.

The variation in speed might be due to interaction with garbage collection. This article about micro-benchmarks could be a clue.