# Monthly Archives: July 2009

# Anthropomorphic shower

Video behind the link.

Now for horrific, and yet contemporary ‘piss your pants’ comedy, apply the same formula to Religion, Politics, Economics etc.

# Generating π in Haskell

Haskell beats CL quite comfortably using the same algorithm :

module Main( main ) where import System( getArgs ) arccot :: Integer -> Integer -> Integer arccot x unity = arccot' x unity 0 start 1 1 where start = unity `div` x arccot' x unity sum xpower n sign | xpower `div` n == 0 = sum | otherwise = arccot' x unity (sum + sign*term) (xpower `div` (x*x)) (n+2) (-sign) where term = xpower `div` n machin_pi :: Integer -> Integer machin_pi digits = pi' `div` (10 ^ 10) where unity = 10 ^ (digits+10) pi' = 4 * (4 * arccot 5 unity - arccot 239 unity) main :: IO () main = do args <- getArgs putStrLn (show (machin_pi (read (head args) :: Integer)))

The first 10000 digits.

> time ./machin_pi 10000 31415926535897932384626433832795028841971693993751058209749445923078164062862089 ... real 0m0.381s user 0m0.290s sys 0m0.061s

Saintly Man? - That Mitchell & Webb Look - BBC Two

# Generating π in CL (faster)

Thanks to metacircular for pointing out that (floor (/ x y)) can be written as (floor x y) while avoiding

the intermediate rational.

(defun machin-pi (digits) "Calculates PI digits using fixed point arithmetic and Machin's formula with double recursion" (labels ((arccot-minus (xsq n xpower) (let ((term (floor xpower n))) (if (= term 0) 0 (- (arccot-plus xsq (+ n 2) (floor xpower xsq)) term)))) (arccot-plus (xsq n xpower) (let ((term (floor (/ xpower n)))) (if (= term 0) 0 (+ (arccot-minus xsq (+ n 2) (floor xpower xsq)) term)))) (arccot (x unity) (let ((xpower (floor (/ unity x)))) (arccot-plus (* x x) 1 xpower)))) (let* ((unity (expt 10 (+ digits 10))) (thispi (* 4 (- (* 4 (arccot 5 unity)) (arccot 239 unity))))) (floor thispi (expt 10 10)))))

The first 10000 digits again.

* (time (machin-pi 10000)) Evaluation took: 0.662 seconds of real time 0.634038 seconds of total run time (0.495454 user, 0.138584 system) [ Run times consist of 0.233 seconds GC time, and 0.402 seconds non-GC time. ] 95.77% CPU 1,491,387,858 processor cycles 109,530,592 bytes consed 31415926535897932384626433832795028841971693993751058209749445923078164062862089 ...

Algorithmic optimizations would take us much further. For example the Gauss–Legendre or Salamin–Brent formula.

Then there is the fastest known(at the turn of the millenium), Chudnovsky’s formula :

# Generating π in CL

Update 2009-07-23 : Faster version in CL and a Haskell version.

——————————————————————————–

A trivial approximation using the Leibniz formula.

(defun leibniz-pi() (labels ((local-pi(sum n) (if (< n (expt 10 6)) (local-pi (+ sum (/ (expt -1 n) (+ (* 2 n) 1))) (+ n 1)) sum))) (* 4 (local-pi 0f0 0f0))))

And here’s a longer version(but faster and more precise) using Machin’s formula with fixed point arithmetic to x digits.

(defun machin-pi (digits) "Calculates PI digits using fixed point arithmetic and Machin's formula with double recursion" (labels ((arccot-minus (xsq n xpower) (let ((term (floor (/ xpower n)))) (if (= term 0) 0 (- (arccot-plus xsq (+ n 2) (floor (/ xpower xsq))) term)))) (arccot-plus (xsq n xpower) (let ((term (floor (/ xpower n)))) (if (= term 0) 0 (+ (arccot-minus xsq (+ n 2) (floor (/ xpower xsq))) term)))) (arccot (x unity) (let ((xpower (floor (/ unity x)))) (arccot-plus (* x x) 1 xpower)))) (let* ((unity (expt 10 (+ digits 10))) (thispi (* 4 (- (* 4 (arccot 5 unity)) (arccot 239 unity))))) (floor (/ thispi (expt 10 10))))))

The first 10000 digits.

* (time (pidigits 10000)) Evaluation took: 2.496 seconds of real time 1.149998 seconds of total run time (0.974647 user, 0.175351 system) [ Run times consist of 0.325 seconds GC time, and 0.825 seconds non-GC time. ] 46.07% CPU 5,626,335,988 processor cycles 217,893,792 bytes consed 31415926535897932384626433832795028841971693993751058209749445923078164062862089 ...

# A fast MAPCONCAT implementation in Common Lisp

Here’s an implementation of Emacs Lisp’s MAPCONCAT function for Common Lisp.

(defun mapconcat (fun list sep) (when list (let ((~sep (with-output-to-string (*standard-output*) (map nil (lambda (ch) (princ (if (char= #~ ch) "~~" ch))) sep)))) (format nil (format nil "~~A~~{~A~~A~~}" ~sep) (funcall fun (first list)) (mapcar fun (rest list))))))

timed :

* (time (mapconcat 'identity *mylist* "-")) Evaluation took: 2.805 seconds of real time 2.746358 seconds of total run time (2.736834 user, 0.009524 system) [ Run times consist of 0.004 seconds GC time, and 2.743 seconds non-GC time. ] 97.90% CPU 6,324,642,149 processor cycles 17,734,520 bytes consed "0-1-2-3-4-5-6-7-8-9-10- ... "

And here’s an optimized version.

(setf *mylist* (let ((l (list 0))) (dotimes (i 10000 i) (nconc l (list (write-to-string i)))) (cdr l))) (defun mapconcat(func lst sep) (let ((vs (make-array 0 :element-type 'character :fill-pointer 0 :adjustable t))) (dotimes (i (length lst) i) (let ((str (funcall func (nth i lst)))) (dotimes (j (length str) j) (vector-push-extend (char str j) vs)) (dotimes (k (length sep) k) (vector-push-extend (char sep k) vs)))) vs))

timed :

* (time (mapconcat 'identity *mylist* "-")) Evaluation took: 0.133 seconds of real time 0.098758 seconds of total run time (0.098390 user, 0.000368 system) 74.44% CPU 299,046,898 processor cycles 517,800 bytes consed "0-1-2-3-4-5-6-7-8-9-10- ... "

As someone pointed out the following would be much faster using FORMAT’s powerful directives and turning off *pretty-print* :

(defun mapconcat (function list elem) (let (*print-pretty*) (format nil (format nil "~~{~~a~~^~a~~}" elem) (mapcar function list))))

timed :

* (time (mapconcat 'identity *mylist* "-")) Evaluation took: 0.006 seconds of real time 0.005033 seconds of total run time (0.005001 user, 0.000032 system) 83.33% CPU 11,430,579 processor cycles 539,200 bytes consed "0-1-2-3-4-5-6-7-8-9-10- ... "

However, the FORMAT version does not demonstrate a fast CL implementation.

To get the low-level implementation to match the performance of the FORMAT implementation we simply make a few tweaks.

(We replace DOTIMES/NTH for the outer loop with MAPCAR as NTH is slow).

(defun mapconcat(func lst sep) (declare (type (cons (simple-array character (*))) lst)) (declare (type (simple-array character (*)) sep)) (let ((vs (make-array 0 :element-type 'character :fill-pointer 0 :adjustable t)) (lsep (length sep))) (mapcar #'(lambda (str) (let ((nstr (funcall func str))) (declare (type (simple-array character (*)) nstr)) (dotimes (j (length nstr) j) (declare (type fixnum j)) (vector-push-extend (char nstr j) vs)) (dotimes (k lsep k) (declare (type fixnum k)) (vector-push-extend (char sep k) vs)))) lst) vs))

timed :

* (time (mapconcat 'identity *mylist* "-")) Evaluation took: 0.006 seconds of real time 0.005435 seconds of total run time (0.005261 user, 0.000174 system) 83.33% CPU 11,845,515 processor cycles 605,792 bytes consed "0-1-2-3-4-5-6-7-8-9-10- ... "

(all versions timed on SBCL 1.0.29, OS X 10.5.7, 2.26 GHz core 2 duo macbook)