numbers.lisp - clic - Clic is an command line interactive client for gopher written in Common LISP
(HTM) git clone git://bitreich.org/clic/ git://enlrupgkhuxnvlhsf6lc3fziv5h2hhfrinws65d7roiv6bfj7d652fid.onion/clic/
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numbers.lisp (11218B)
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1 (in-package :alexandria)
2
3 (declaim (inline clamp))
4 (defun clamp (number min max)
5 "Clamps the NUMBER into [min, max] range. Returns MIN if NUMBER is lesser then
6 MIN and MAX if NUMBER is greater then MAX, otherwise returns NUMBER."
7 (if (< number min)
8 min
9 (if (> number max)
10 max
11 number)))
12
13 (defun gaussian-random (&optional min max)
14 "Returns two gaussian random double floats as the primary and secondary value,
15 optionally constrained by MIN and MAX. Gaussian random numbers form a standard
16 normal distribution around 0.0d0.
17
18 Sufficiently positive MIN or negative MAX will cause the algorithm used to
19 take a very long time. If MIN is positive it should be close to zero, and
20 similarly if MAX is negative it should be close to zero."
21 (macrolet
22 ((valid (x)
23 `(<= (or min ,x) ,x (or max ,x)) ))
24 (labels
25 ((gauss ()
26 (loop
27 for x1 = (- (random 2.0d0) 1.0d0)
28 for x2 = (- (random 2.0d0) 1.0d0)
29 for w = (+ (expt x1 2) (expt x2 2))
30 when (< w 1.0d0)
31 do (let ((v (sqrt (/ (* -2.0d0 (log w)) w))))
32 (return (values (* x1 v) (* x2 v))))))
33 (guard (x)
34 (unless (valid x)
35 (tagbody
36 :retry
37 (multiple-value-bind (x1 x2) (gauss)
38 (when (valid x1)
39 (setf x x1)
40 (go :done))
41 (when (valid x2)
42 (setf x x2)
43 (go :done))
44 (go :retry))
45 :done))
46 x))
47 (multiple-value-bind
48 (g1 g2) (gauss)
49 (values (guard g1) (guard g2))))))
50
51 (declaim (inline iota))
52 (defun iota (n &key (start 0) (step 1))
53 "Return a list of n numbers, starting from START (with numeric contagion
54 from STEP applied), each consequtive number being the sum of the previous one
55 and STEP. START defaults to 0 and STEP to 1.
56
57 Examples:
58
59 (iota 4) => (0 1 2 3)
60 (iota 3 :start 1 :step 1.0) => (1.0 2.0 3.0)
61 (iota 3 :start -1 :step -1/2) => (-1 -3/2 -2)
62 "
63 (declare (type (integer 0) n) (number start step))
64 (loop ;; KLUDGE: get numeric contagion right for the first element too
65 for i = (+ (- (+ start step) step)) then (+ i step)
66 repeat n
67 collect i))
68
69 (declaim (inline map-iota))
70 (defun map-iota (function n &key (start 0) (step 1))
71 "Calls FUNCTION with N numbers, starting from START (with numeric contagion
72 from STEP applied), each consequtive number being the sum of the previous one
73 and STEP. START defaults to 0 and STEP to 1. Returns N.
74
75 Examples:
76
77 (map-iota #'print 3 :start 1 :step 1.0) => 3
78 ;;; 1.0
79 ;;; 2.0
80 ;;; 3.0
81 "
82 (declare (type (integer 0) n) (number start step))
83 (loop ;; KLUDGE: get numeric contagion right for the first element too
84 for i = (+ start (- step step)) then (+ i step)
85 repeat n
86 do (funcall function i))
87 n)
88
89 (declaim (inline lerp))
90 (defun lerp (v a b)
91 "Returns the result of linear interpolation between A and B, using the
92 interpolation coefficient V."
93 ;; The correct version is numerically stable, at the expense of an
94 ;; extra multiply. See (lerp 0.1 4 25) with (+ a (* v (- b a))). The
95 ;; unstable version can often be converted to a fast instruction on
96 ;; a lot of machines, though this is machine/implementation
97 ;; specific. As alexandria is more about correct code, than
98 ;; efficiency, and we're only talking about a single extra multiply,
99 ;; many would prefer the stable version
100 (+ (* (- 1.0 v) a) (* v b)))
101
102 (declaim (inline mean))
103 (defun mean (sample)
104 "Returns the mean of SAMPLE. SAMPLE must be a sequence of numbers."
105 (/ (reduce #'+ sample) (length sample)))
106
107 (defun median (sample)
108 "Returns median of SAMPLE. SAMPLE must be a sequence of real numbers."
109 ;; Implements and uses the quick-select algorithm to find the median
110 ;; https://en.wikipedia.org/wiki/Quickselect
111
112 (labels ((randint-in-range (start-int end-int)
113 "Returns a random integer in the specified range, inclusive"
114 (+ start-int (random (1+ (- end-int start-int)))))
115 (partition (vec start-i end-i)
116 "Implements the partition function, which performs a partial
117 sort of vec around the (randomly) chosen pivot.
118 Returns the index where the pivot element would be located
119 in a correctly-sorted array"
120 (if (= start-i end-i)
121 start-i
122 (let ((pivot-i (randint-in-range start-i end-i)))
123 (rotatef (aref vec start-i) (aref vec pivot-i))
124 (let ((swap-i end-i))
125 (loop for i from swap-i downto (1+ start-i) do
126 (when (>= (aref vec i) (aref vec start-i))
127 (rotatef (aref vec i) (aref vec swap-i))
128 (decf swap-i)))
129 (rotatef (aref vec swap-i) (aref vec start-i))
130 swap-i)))))
131
132 (let* ((vector (copy-sequence 'vector sample))
133 (len (length vector))
134 (mid-i (ash len -1))
135 (i 0)
136 (j (1- len)))
137
138 (loop for correct-pos = (partition vector i j)
139 while (/= correct-pos mid-i) do
140 (if (< correct-pos mid-i)
141 (setf i (1+ correct-pos))
142 (setf j (1- correct-pos))))
143
144 (if (oddp len)
145 (aref vector mid-i)
146 (* 1/2
147 (+ (aref vector mid-i)
148 (reduce #'max (make-array
149 mid-i
150 :displaced-to vector))))))))
151
152 (declaim (inline variance))
153 (defun variance (sample &key (biased t))
154 "Variance of SAMPLE. Returns the biased variance if BIASED is true (the default),
155 and the unbiased estimator of variance if BIASED is false. SAMPLE must be a
156 sequence of numbers."
157 (let ((mean (mean sample)))
158 (/ (reduce (lambda (a b)
159 (+ a (expt (- b mean) 2)))
160 sample
161 :initial-value 0)
162 (- (length sample) (if biased 0 1)))))
163
164 (declaim (inline standard-deviation))
165 (defun standard-deviation (sample &key (biased t))
166 "Standard deviation of SAMPLE. Returns the biased standard deviation if
167 BIASED is true (the default), and the square root of the unbiased estimator
168 for variance if BIASED is false (which is not the same as the unbiased
169 estimator for standard deviation). SAMPLE must be a sequence of numbers."
170 (sqrt (variance sample :biased biased)))
171
172 (define-modify-macro maxf (&rest numbers) max
173 "Modify-macro for MAX. Sets place designated by the first argument to the
174 maximum of its original value and NUMBERS.")
175
176 (define-modify-macro minf (&rest numbers) min
177 "Modify-macro for MIN. Sets place designated by the first argument to the
178 minimum of its original value and NUMBERS.")
179
180 ;;;; Factorial
181
182 ;;; KLUDGE: This is really dependant on the numbers in question: for
183 ;;; small numbers this is larger, and vice versa. Ideally instead of a
184 ;;; constant we would have RANGE-FAST-TO-MULTIPLY-DIRECTLY-P.
185 (defconstant +factorial-bisection-range-limit+ 8)
186
187 ;;; KLUDGE: This is really platform dependant: ideally we would use
188 ;;; (load-time-value (find-good-direct-multiplication-limit)) instead.
189 (defconstant +factorial-direct-multiplication-limit+ 13)
190
191 (defun %multiply-range (i j)
192 ;; We use a a bit of cleverness here:
193 ;;
194 ;; 1. For large factorials we bisect in order to avoid expensive bignum
195 ;; multiplications: 1 x 2 x 3 x ... runs into bignums pretty soon,
196 ;; and once it does that all further multiplications will be with bignums.
197 ;;
198 ;; By instead doing the multiplication in a tree like
199 ;; ((1 x 2) x (3 x 4)) x ((5 x 6) x (7 x 8))
200 ;; we manage to get less bignums.
201 ;;
202 ;; 2. Division isn't exactly free either, however, so we don't bisect
203 ;; all the way down, but multiply ranges of integers close to each
204 ;; other directly.
205 ;;
206 ;; For even better results it should be possible to use prime
207 ;; factorization magic, but Nikodemus ran out of steam.
208 ;;
209 ;; KLUDGE: We support factorials of bignums, but it seems quite
210 ;; unlikely anyone would ever be able to use them on a modern lisp,
211 ;; since the resulting numbers are unlikely to fit in memory... but
212 ;; it would be extremely unelegant to define FACTORIAL only on
213 ;; fixnums, _and_ on lisps with 16 bit fixnums this can actually be
214 ;; needed.
215 (labels ((bisect (j k)
216 (declare (type (integer 1 #.most-positive-fixnum) j k))
217 (if (< (- k j) +factorial-bisection-range-limit+)
218 (multiply-range j k)
219 (let ((middle (+ j (truncate (- k j) 2))))
220 (* (bisect j middle)
221 (bisect (+ middle 1) k)))))
222 (bisect-big (j k)
223 (declare (type (integer 1) j k))
224 (if (= j k)
225 j
226 (let ((middle (+ j (truncate (- k j) 2))))
227 (* (if (<= middle most-positive-fixnum)
228 (bisect j middle)
229 (bisect-big j middle))
230 (bisect-big (+ middle 1) k)))))
231 (multiply-range (j k)
232 (declare (type (integer 1 #.most-positive-fixnum) j k))
233 (do ((f k (* f m))
234 (m (1- k) (1- m)))
235 ((< m j) f)
236 (declare (type (integer 0 (#.most-positive-fixnum)) m)
237 (type unsigned-byte f)))))
238 (if (and (typep i 'fixnum) (typep j 'fixnum))
239 (bisect i j)
240 (bisect-big i j))))
241
242 (declaim (inline factorial))
243 (defun %factorial (n)
244 (if (< n 2)
245 1
246 (%multiply-range 1 n)))
247
248 (defun factorial (n)
249 "Factorial of non-negative integer N."
250 (check-type n (integer 0))
251 (%factorial n))
252
253 ;;;; Combinatorics
254
255 (defun binomial-coefficient (n k)
256 "Binomial coefficient of N and K, also expressed as N choose K. This is the
257 number of K element combinations given N choises. N must be equal to or
258 greater then K."
259 (check-type n (integer 0))
260 (check-type k (integer 0))
261 (assert (>= n k))
262 (if (or (zerop k) (= n k))
263 1
264 (let ((n-k (- n k)))
265 ;; Swaps K and N-K if K < N-K because the algorithm
266 ;; below is faster for bigger K and smaller N-K
267 (when (< k n-k)
268 (rotatef k n-k))
269 (if (= 1 n-k)
270 n
271 ;; General case, avoid computing the 1x...xK twice:
272 ;;
273 ;; N! 1x...xN (K+1)x...xN
274 ;; -------- = ---------------- = ------------, N>1
275 ;; K!(N-K)! 1x...xK x (N-K)! (N-K)!
276 (/ (%multiply-range (+ k 1) n)
277 (%factorial n-k))))))
278
279 (defun subfactorial (n)
280 "Subfactorial of the non-negative integer N."
281 (check-type n (integer 0))
282 (if (zerop n)
283 1
284 (do ((x 1 (1+ x))
285 (a 0 (* x (+ a b)))
286 (b 1 a))
287 ((= n x) a))))
288
289 (defun count-permutations (n &optional (k n))
290 "Number of K element permutations for a sequence of N objects.
291 K defaults to N"
292 (check-type n (integer 0))
293 (check-type k (integer 0))
294 (assert (>= n k))
295 (%multiply-range (1+ (- n k)) n))