tnumbers.lisp - clic - Clic is an command line interactive client for gopher written in Common LISP
(HTM) git clone git://bitreich.org/clic/ git://hg6vgqziawt5s4dj.onion/clic/
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tnumbers.lisp (9688B)
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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 repeat n
65 ;; KLUDGE: get numeric contagion right for the first element too
66 for i = (+ (- (+ start step) step)) then (+ i step)
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 repeat n
84 ;; KLUDGE: get numeric contagion right for the first element too
85 for i = (+ start (- step step)) then (+ i step)
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 (declaim (inline median))
108 (defun median (sample)
109 "Returns median of SAMPLE. SAMPLE must be a sequence of real numbers."
110 (let* ((vector (sort (copy-sequence 'vector sample) #'<))
111 (length (length vector))
112 (middle (truncate length 2)))
113 (if (oddp length)
114 (aref vector middle)
115 (/ (+ (aref vector middle) (aref vector (1- middle))) 2))))
116
117 (declaim (inline variance))
118 (defun variance (sample &key (biased t))
119 "Variance of SAMPLE. Returns the biased variance if BIASED is true (the default),
120 and the unbiased estimator of variance if BIASED is false. SAMPLE must be a
121 sequence of numbers."
122 (let ((mean (mean sample)))
123 (/ (reduce (lambda (a b)
124 (+ a (expt (- b mean) 2)))
125 sample
126 :initial-value 0)
127 (- (length sample) (if biased 0 1)))))
128
129 (declaim (inline standard-deviation))
130 (defun standard-deviation (sample &key (biased t))
131 "Standard deviation of SAMPLE. Returns the biased standard deviation if
132 BIASED is true (the default), and the square root of the unbiased estimator
133 for variance if BIASED is false (which is not the same as the unbiased
134 estimator for standard deviation). SAMPLE must be a sequence of numbers."
135 (sqrt (variance sample :biased biased)))
136
137 (define-modify-macro maxf (&rest numbers) max
138 "Modify-macro for MAX. Sets place designated by the first argument to the
139 maximum of its original value and NUMBERS.")
140
141 (define-modify-macro minf (&rest numbers) min
142 "Modify-macro for MIN. Sets place designated by the first argument to the
143 minimum of its original value and NUMBERS.")
144
145 ;;;; Factorial
146
147 ;;; KLUDGE: This is really dependant on the numbers in question: for
148 ;;; small numbers this is larger, and vice versa. Ideally instead of a
149 ;;; constant we would have RANGE-FAST-TO-MULTIPLY-DIRECTLY-P.
150 (defconstant +factorial-bisection-range-limit+ 8)
151
152 ;;; KLUDGE: This is really platform dependant: ideally we would use
153 ;;; (load-time-value (find-good-direct-multiplication-limit)) instead.
154 (defconstant +factorial-direct-multiplication-limit+ 13)
155
156 (defun %multiply-range (i j)
157 ;; We use a a bit of cleverness here:
158 ;;
159 ;; 1. For large factorials we bisect in order to avoid expensive bignum
160 ;; multiplications: 1 x 2 x 3 x ... runs into bignums pretty soon,
161 ;; and once it does that all further multiplications will be with bignums.
162 ;;
163 ;; By instead doing the multiplication in a tree like
164 ;; ((1 x 2) x (3 x 4)) x ((5 x 6) x (7 x 8))
165 ;; we manage to get less bignums.
166 ;;
167 ;; 2. Division isn't exactly free either, however, so we don't bisect
168 ;; all the way down, but multiply ranges of integers close to each
169 ;; other directly.
170 ;;
171 ;; For even better results it should be possible to use prime
172 ;; factorization magic, but Nikodemus ran out of steam.
173 ;;
174 ;; KLUDGE: We support factorials of bignums, but it seems quite
175 ;; unlikely anyone would ever be able to use them on a modern lisp,
176 ;; since the resulting numbers are unlikely to fit in memory... but
177 ;; it would be extremely unelegant to define FACTORIAL only on
178 ;; fixnums, _and_ on lisps with 16 bit fixnums this can actually be
179 ;; needed.
180 (labels ((bisect (j k)
181 (declare (type (integer 1 #.most-positive-fixnum) j k))
182 (if (< (- k j) +factorial-bisection-range-limit+)
183 (multiply-range j k)
184 (let ((middle (+ j (truncate (- k j) 2))))
185 (* (bisect j middle)
186 (bisect (+ middle 1) k)))))
187 (bisect-big (j k)
188 (declare (type (integer 1) j k))
189 (if (= j k)
190 j
191 (let ((middle (+ j (truncate (- k j) 2))))
192 (* (if (<= middle most-positive-fixnum)
193 (bisect j middle)
194 (bisect-big j middle))
195 (bisect-big (+ middle 1) k)))))
196 (multiply-range (j k)
197 (declare (type (integer 1 #.most-positive-fixnum) j k))
198 (do ((f k (* f m))
199 (m (1- k) (1- m)))
200 ((< m j) f)
201 (declare (type (integer 0 (#.most-positive-fixnum)) m)
202 (type unsigned-byte f)))))
203 (if (and (typep i 'fixnum) (typep j 'fixnum))
204 (bisect i j)
205 (bisect-big i j))))
206
207 (declaim (inline factorial))
208 (defun %factorial (n)
209 (if (< n 2)
210 1
211 (%multiply-range 1 n)))
212
213 (defun factorial (n)
214 "Factorial of non-negative integer N."
215 (check-type n (integer 0))
216 (%factorial n))
217
218 ;;;; Combinatorics
219
220 (defun binomial-coefficient (n k)
221 "Binomial coefficient of N and K, also expressed as N choose K. This is the
222 number of K element combinations given N choises. N must be equal to or
223 greater then K."
224 (check-type n (integer 0))
225 (check-type k (integer 0))
226 (assert (>= n k))
227 (if (or (zerop k) (= n k))
228 1
229 (let ((n-k (- n k)))
230 ;; Swaps K and N-K if K < N-K because the algorithm
231 ;; below is faster for bigger K and smaller N-K
232 (when (< k n-k)
233 (rotatef k n-k))
234 (if (= 1 n-k)
235 n
236 ;; General case, avoid computing the 1x...xK twice:
237 ;;
238 ;; N! 1x...xN (K+1)x...xN
239 ;; -------- = ---------------- = ------------, N>1
240 ;; K!(N-K)! 1x...xK x (N-K)! (N-K)!
241 (/ (%multiply-range (+ k 1) n)
242 (%factorial n-k))))))
243
244 (defun subfactorial (n)
245 "Subfactorial of the non-negative integer N."
246 (check-type n (integer 0))
247 (if (zerop n)
248 1
249 (do ((x 1 (1+ x))
250 (a 0 (* x (+ a b)))
251 (b 1 a))
252 ((= n x) a))))
253
254 (defun count-permutations (n &optional (k n))
255 "Number of K element permutations for a sequence of N objects.
256 K defaults to N"
257 (check-type n (integer 0))
258 (check-type k (integer 0))
259 (assert (>= n k))
260 (%multiply-range (1+ (- n k)) n))