1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
|
#lang racket/base
(require racket/math racket/flonum racket/contract racket/match unstable/latent-contract/defthing
"math.rkt"
"contract.rkt")
(provide (all-defined-out))
(struct invertible-function (f g) #:transparent)
(defproc (invertible-compose [f1 invertible-function?] [f2 invertible-function?]
) invertible-function?
(match-let ([(invertible-function f1 g1) f1]
[(invertible-function f2 g2) f2])
(invertible-function (compose f1 f2) (compose g2 g1))))
(defproc (invertible-inverse [h invertible-function?]) invertible-function? #:document-body
(match-define (invertible-function f g) h)
(invertible-function g f))
(defcontract axis-transform/c (real? real? invertible-function? . -> . invertible-function?))
(defthing id-transform axis-transform/c
(λ (x-min x-max old-function) old-function))
(defthing id-function invertible-function? (invertible-function (λ (x) x) (λ (x) x)))
(defproc (apply-axis-transform [t axis-transform/c] [x-min real?] [x-max real?]
) invertible-function? #:document-body
(t x-min x-max id-function))
;; Turns any total, surjective, monotone real function and its inverse into an axis transform
(defproc (make-axis-transform [fun invertible-function?]) axis-transform/c
(match-define (invertible-function f g) fun)
(λ (x-min x-max old-function)
(define fx-min (f x-min))
(define fx-scale (/ (- x-max x-min) (- (f x-max) fx-min)))
(define (new-f x) (+ x-min (* (- (f x) fx-min) fx-scale)))
(define (new-g y) (g (+ fx-min (/ (- y x-min) fx-scale))))
(invertible-compose (invertible-function new-f new-g) old-function)))
;; ===================================================================================================
;; Axis transform combinators
(defproc (axis-transform-compose [t1 axis-transform/c] [t2 axis-transform/c]) axis-transform/c
(λ (x-min x-max old-function)
(t1 x-min x-max (t2 x-min x-max old-function))))
(defproc (axis-transform-append [t1 axis-transform/c] [t2 axis-transform/c] [mid real?]
) axis-transform/c
(λ (x-min x-max old-function)
(match-define (invertible-function old-f old-g) old-function)
(let ([mid (old-f mid)])
(cond [(mid . >= . x-max) (t1 x-min x-max old-function)]
[(mid . <= . x-min) (t2 x-min x-max old-function)]
[else
(match-define (invertible-function f1 g1) (t1 x-min mid old-function))
(match-define (invertible-function f2 g2) (t2 mid x-max old-function))
((make-axis-transform
(invertible-function
(λ (x) (cond [((old-f x) . < . mid) (f1 x)]
[else (f2 x)]))
(λ (x) (cond [(x . < . mid) (g1 x)]
[else (g2 x)]))))
x-min x-max id-function)]))))
(defproc (axis-transform-bound [t axis-transform/c] [a real?] [b real?]
) axis-transform/c #:document-body
(axis-transform-append
(axis-transform-append id-transform t a) id-transform b))
;; ===================================================================================================
;; Specific axis transforms
(define ((flnewton-invert f f-diff f-inv-guess n) y)
(let ([y (exact->inexact y)])
(let loop ([x (f-inv-guess y)] [n n])
(let/ec return
(when (zero? n) (return x))
(define dx (fl/ (fl- y (f x)) (f-diff x)))
(when (zero? dx) (return x))
(loop (fl- x dx) (sub1 n))))))
(define (sine-diag d)
(let ([d (exact->inexact d)])
(λ (x) (let ([x (exact->inexact x)])
(fl+ x (fl* (fl/ 1.0 (fl* 4.0 d)) (flsin (fl* d x))))))))
(define (sine-diag-diff d)
(let ([d (exact->inexact d)])
(λ (x) (let ([x (exact->inexact x)])
(fl- (fl/ (flcos (fl* d x)) 4.0) 1.0)))))
(define (sine-diag-inv d)
(flnewton-invert (sine-diag d) (sine-diag-diff d) values 10))
(define cbrt
(let ([e (exact->inexact 1/3)])
(λ (x) (let ([x (exact->inexact x)])
(fl* (sgn x) (expt (flabs x) e))))))
(define (cube x)
(let ([x (exact->inexact x)])
(fl* x (fl* x x))))
(define (real-log x)
(fllog (exact->inexact x)))
(define (real-exp x)
(flexp (exact->inexact x)))
(defthing log-transform axis-transform/c
(λ (x-min x-max old-function)
(when ((exact->inexact x-min) . <= . 0)
(raise-type-error 'log-transform "positive real" 0 x-min x-max))
((make-axis-transform (invertible-function real-log real-exp)) x-min x-max old-function)))
(defthing cbrt-transform axis-transform/c
(λ (x-min x-max old-function)
((make-axis-transform (invertible-function cbrt cube)) x-min x-max old-function)))
(defproc (hand-drawn-transform [freq (>/c 0)]) axis-transform/c
(λ (x-min x-max old-function)
(define d (/ freq (- x-max x-min)))
((make-axis-transform (invertible-function (sine-diag d) (sine-diag-inv d)))
x-min x-max old-function)))
;; ===================================================================================================
(define (stretch a b s)
(define d (- b a))
(define ds (* d s))
(λ (x)
(cond [(x . < . a) x]
[(x . > . b) (+ (- x d) ds)]
[else (+ a (* (- x a) s))])))
(defproc (stretch-transform [a real?] [b real?] [scale (>/c 0)]) axis-transform/c
(when (a . > . b) (error 'stretch-transform "expected a <= b; given ~e and ~e" a b))
(λ (x-min x-max old-function)
(match-define (invertible-function old-f old-g) old-function)
(let ([a (old-f a)]
[b (old-f b)])
(define f (stretch a b scale))
(define g (stretch (f a) (f b) (/ 1 scale)))
((make-axis-transform (invertible-function f g)) x-min x-max old-function))))
(defproc (collapse-transform [a real?] [b real?]) axis-transform/c
(when (a . > . b) (error 'stretch-transform "expected a <= b; given ~e and ~e" a b))
(λ (x-min x-max old-function)
(match-define (invertible-function old-f old-g) old-function)
(let ([a (old-f a)]
[b (old-f b)])
(define 1/2size (* 1/2 (- b a)))
(define center (* 1/2 (+ a b)))
(define (f x) (cond [(x . < . a) (+ x 1/2size)]
[(x . > . b) (- x 1/2size)]
[else center]))
(define (g x) (cond [(x . < . center) (- x 1/2size)]
[(x . > . center) (+ x 1/2size)]
[else center]))
((make-axis-transform (invertible-function f g)) x-min x-max old-function))))
|
