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--- a/chap1.rkt
+++ /dev/null
@@ -1,1452 +0,0 @@
-#lang sicp
-
-;; Chapter 1
-;; Building Abstractions with Procedures
-
-;; 1.1
-;; The Elements of Programming
-
-#| 1.1 |#
-
-#| 10 |#
-#| 10 |#
-
-#| (+ 5 3 4) |#
-#| 12 |#
-
-#| (- 9 1) |#
-#| 8 |#
-
-#| (/ 6 2) |#
-#| 3 |#
-
-#| (+ (* 2 4) (- 4 6)) |#
-#| 6 |#
-
-#| (define a 3) |#
-#| () |#
-
-#| (define b (+ a 1)) |#
-#| () |#
-
-#| (+ a b (* a b)) |#
-#| 19 |#
-
-#| (= a b) |#
-#| #f |#
-
-#| (if |#
-#|   (and (> b a) (< b (* a b))) |#
-#|   b |#
-#|   a) |#
-#| 4 |#
-
-#| (cond |#
-#|   ((= a 4) 6) |#
-#|   ((= b 4) (+ 6 7 a)) |#
-#|   (else 25)) |#
-#| 16 |#
-
-#| (+ 2 (if (> b a) b a)) |#
-#| 6 |#
-
-#| (* |#
-#|   (cond |#
-#|     ((> a b) a) |#
-#|     ((< a b) b) |#
-#|     (else -1)) |#
-#|   (+ a 1)) |#
-#| 16 |#
-
-#| 1.2 |#
-
-(/
-  (+ 5 4 (- 2 (- 3 (+ 6 (/ 4 5)))))
-  (* 3 (- 6 2) (- 2 7)))
-
-
-(#%provide square)
-(define (square x) (* x x))
-(define (sum-of-squares x y) (+ (square x) (square y)))
-
-#| 1.3 |#
-
-(#%provide sos-two-larger)
-(define (sos-two-larger a b c)
-  (if (> a b)
-    (sum-of-squares a (if (> b c) b c))
-    (sum-of-squares b (if (> a c) a c))))
-
-#| 1.4 |#
-
-(#%provide a-plus-abs-b)
-(define (a-plus-abs-b a b)
-  ((if (> b 0) + -) a b))
-
-#| 1.5 |#
-
-(#%provide p)
-(#%provide test)
-(define (p) (p))
-(define (test x y)
-  (if (= x 0)
-    0
-    y))
-
-#| Applicative order: this will loop forever |#
-#| Normal order: this will terminate after one call to test |#
-
-#| (test 0 (p)) |#
-
-
-(define (average x y)
-  (/ (+ x y) 2))
-
-(#%provide sqrt-)
-(define (sqrt- x)
-  (define (good-enough? guess)
-    (< (abs (- (square guess) x)) 0.001))
-  (define (improve guess)
-    (average guess (/ x guess)))
-  (define (sqrt-iter guess)
-    (if (good-enough? guess)
-      guess
-      (sqrt-iter (improve guess))))
-  (sqrt-iter 1.0))
-
-#| 1.6 |#
-
-(define (new-if pred then-clause else-clause)
-  (cond
-    (pred then-clause)
-    (else else-clause)))
-
-(#%provide sqrt-new)
-(define (sqrt-new x)
-  (define (good-enough? guess)
-    (< (abs (- (square guess) x)) 0.001))
-  (define (improve guess)
-    (average guess (/ x guess)))
-  (define (sqrt-iter guess)
-    (new-if (good-enough? guess)
-      guess
-      (sqrt-iter (improve guess))))
-  (sqrt-iter 1.0))
-
-#| 1.7 |#
-
-(#%provide sqrt-delt)
-(define (sqrt-delt x)
-  (define (good-enough? last-guess guess)
-    (< (/ (abs (- last-guess guess)) x) 0.000000000001))
-  (define (improve guess)
-    (average guess (/ x guess)))
-  (define (sqrt-iter last-guess guess)
-    (if (good-enough? last-guess guess)
-      guess
-      (sqrt-iter guess (improve guess))))
-  (sqrt-iter 1.0 (improve 1.0)))
-
-(square (sqrt-delt 479800023432748679))
-(square (sqrt-delt 0.00000024353))
-
-#| 1.8 |#
-
-(#%provide cube)
-(define (cube x) (* x x x))
-
-(#%provide cbrt)
-(define (cbrt x)
-  (define (good-enough? guess)
-    (< (abs (- (cube guess) x)) 0.001))
-  (define (improve y)
-    (/ (+ (/ x (square y)) (* 2 y)) 3))
-  (define (cbrt-iter guess)
-    (if (good-enough? guess)
-      guess
-      (cbrt-iter (improve guess))))
-  (cbrt-iter 1.0))
-
-#| 1.9 |#
-
-;; recursive process:
-#| (define (+ a b) |#
-#|   (if (= a 0) |#
-#|     b |#
-#|     (inc (+ (dec a) b)))) |#
-
-;; iterative process:
-#| (define (+ a b) |#
-#|   (if (= a 0) |#
-#|     b |#
-#|     (+ (dec a) (inc b)))) |#
-
-#| 1.10 |#
-
-(#%provide A)
-(define (A x y)
-  (cond
-    ((= y 0) 0)
-    ((= x 0) (* 2 y))
-    ((= y 1) 2)
-    (else
-      (A
-        (- x 1)
-        (A x (- y 1))))))
-
-#| (A 1 10) |#
-#| (A 0 (A 1 9)) |#
-#| (A 0 (A 0 (A 1 8))) |#
-#| (A 0 (A 0 (A 0 (A 1 7)))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 1 6))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 1 (A 1 5)))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 4))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 3)))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 2))))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 1)))))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 2))))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 2)))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 4))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 4))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 8))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 8))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 16)))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 16)))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 32))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 32))))) |#
-#| (A 0 (A 0 (A 0 (A 0 64)))) |#
-#| (A 0 (A 0 (A 0 (* 2 64)))) |#
-#| (A 0 (A 0 (A 0 128))) |#
-#| (A 0 (A 0 (* 2 128))) |#
-#| (A 0 (A 0 256)) |#
-#| (A 0 (* 2 256)) |#
-#| (A 0 512) |#
-#| (* 2 512) |#
-#| 1024 |#
-
-#| (A 2 4) |#
-#| (A 1 (A 2 3)) |#
-#| (A 1 (A 1 (A 2 2))) |#
-#| (A 1 (A 1 (A 1 (A 2 1)))) |#
-#| (A 1 (A 1 (A 1 2))) |#
-#| (A 1 (A 1 (A 0 (A 1 1)))) |#
-#| (A 1 (A 1 (A 0 2))) |#
-#| (A 1 (A 1 (* 2 2))) |#
-#| (A 1 (A 1 4)) |#
-#| (A 1 (A 0 (A 1 3))) |#
-#| (A 1 (A 0 (A 0 (A 1 2)))) |#
-#| (A 1 (A 0 (A 0 (A 0 (A 1 1))))) |#
-#| (A 1 (A 0 (A 0 (A 0 2)))) |#
-#| (A 1 (A 0 (A 0 (* 2 2)))) |#
-#| (A 1 (A 0 (A 0 4))) |#
-#| (A 1 (A 0 (* 2 4))) |#
-#| (A 1 (A 0 8)) |#
-#| (A 1 (* 2 8)) |#
-#| (A 1 16) |#
-#| (A 0 (A 1 15)) |#
-#| (A 0 (A 0 (A 1 14))) |#
-#| (A 0 (A 0 (A 0 (A 1 13)))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 1 12))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 11)))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 10))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 9)))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 8))))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 7)))))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 6))))))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 (A 1 5)))))))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 4))))))))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 3)))))))))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 2))))))))))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 1)))))))))))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 2))))))))))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 2)))))))))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 4))))))))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 4))))))))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 8))))))))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 8))))))))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 16)))))))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 16)))))))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 32))))))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 32))))))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 64)))))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 64)))))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 128))))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 128))))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 256)))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 256)))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 512))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 512))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 1024)))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 1024)))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 2048))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (* 2 2048))))) |#
-#| (A 0 (A 0 (A 0 (A 0 4096)))) |#
-#| (A 0 (A 0 (A 0 (* 2 4096)))) |#
-#| (A 0 (A 0 (A 0 8182))) |#
-#| (A 0 (A 0 (* 2 8182))) |#
-#| (A 0 (A 0 16364)) |#
-#| (A 0 (* 2 16364)) |#
-#| (A 0 32768) |#
-#| (* 2 32768) |#
-#| 65536 |#
-
-
-#| (A 3 3) |#
-#| (A 2 (A 3 2)) |#
-#| (A 2 (A 2 (A 3 1))) |#
-#| (A 2 (A 2 2)) |#
-#| (A 2 (A 1 (A 2 1))) |#
-#| (A 2 (A 1 2)) |#
-#| (A 2 (A 0 (A 1 1))) |#
-#| (A 2 (A 0 2)) |#
-#| (A 2 (* 2 2)) |#
-#| (A 2 4) |#
-#| (A 1 (A 2 3)) |#
-#| (A 1 (A 1 (A 2 2))) |#
-#| (A 1 (A 1 (A 1 (A 2 1)))) |#
-#| (A 1 (A 1 (A 1 2))) |#
-#| (A 1 (A 1 (A 0 (A 1 1)))) |#
-#| (A 1 (A 1 (A 0 2))) |#
-#| (A 1 (A 1 (* 2 2))) |#
-#| (A 1 (A 1 4)) |#
-#| (A 1 (A 0 (A 1 3))) |#
-#| (A 1 (A 0 (A 0 (A 1 2)))) |#
-#| (A 1 (A 0 (A 0 (A 0 (A 1 1))))) |#
-#| (A 1 (A 0 (A 0 (A 0 2)))) |#
-#| (A 1 (A 0 (A 0 (* 2 2)))) |#
-#| (A 1 (A 0 (A 0 4))) |#
-#| (A 1 (A 0 (* 2 4))) |#
-#| (A 1 (A 0 8)) |#
-#| (A 1 (* 2 8)) |#
-#| (A 1 16) |#
-#| (A 0 (A 1 15)) |#
-#| (A 0 (A 0 (A 1 14))) |#
-#| (A 0 (A 0 (A 0 (A 1 13)))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 1 12))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 11)))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 10))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 9)))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 8))))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 7)))))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 6))))))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 (A 1 5)))))))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 4))))))))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 3)))))))))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 2))))))))))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 1)))))))))))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 2))))))))))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 2)))))))))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 4))))))))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 4))))))))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 8))))))))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 8))))))))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 16)))))))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 16)))))))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 32))))))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 32))))))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 64)))))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 64)))))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 128))))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 128))))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 256)))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 256)))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 512))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 512))))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 1024)))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 1024)))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (A 0 2048))))) |#
-#| (A 0 (A 0 (A 0 (A 0 (* 2 2048))))) |#
-#| (A 0 (A 0 (A 0 (A 0 4096)))) |#
-#| (A 0 (A 0 (A 0 (* 2 4096)))) |#
-#| (A 0 (A 0 (A 0 8182))) |#
-#| (A 0 (A 0 (* 2 8182))) |#
-#| (A 0 (A 0 16364)) |#
-#| (A 0 (* 2 16364)) |#
-#| (A 0 32768) |#
-#| (* 2 32768) |#
-#| 65536 |#
-
-;; f(n) = 2n
-(define (f n) (A 0 n))
-
-;; g(n) = 2^n
-(define (g n) (A 1 n))
-
-;; h(n) = 2^(2^... n times ...(2^2)...)
-(define (h n) (A 2 n))
-
-;; k(n) = 5n^2
-(define (k n) (* 5 n n))
-
-(#%provide count-change)
-(define (count-change amount)
-  (cc amount 5))
-
-(define (cc amount kinds-of-coins)
-  (cond
-    ((= amount 0) 1)
-    ((or (< amount 0) (= kinds-of-coins 0)) 0)
-    (else
-      (+
-        (cc amount (- kinds-of-coins 1))
-        (cc
-          (- amount (first-denomination kinds-of-coins))
-          kinds-of-coins)))))
-
-(define (first-denomination kinds-of-coins)
-  (cond
-    ((= kinds-of-coins 1) 1)
-    ((= kinds-of-coins 2) 5)
-    ((= kinds-of-coins 3) 10)
-    ((= kinds-of-coins 4) 25)
-    ((= kinds-of-coins 5) 50)))
-
-#| 1.11 |#
-
-(#%provide f-rec)
-(define (f-rec n)
-  (cond
-    ((< n 3) n)
-    (else
-      (+
-        (f-rec (- n 1))
-        (* 2 (f-rec (- n 2)))
-        (* 3 (f-rec (- n 3)))))))
-
-(#%provide f-iter)
-(define (f-iter n)
-  (f-iter- 2 1 0 n))
-
-(define (f-iter- a b c i)
-  (if (= i 0)
-    c
-    (f-iter-
-      (+ a (* 2 b) (* 3 c))
-      a
-      b
-      (- i 1))))
-
-#| 1.12 |#
-
-(#%provide pascal)
-(define (pascal row elem)
-  (cond
-    ((> elem row) 0)
-    ((= elem 0) 1)
-    (else
-      (+
-        (pascal (- row 1) (- elem 1))
-        (pascal (- row 1) elem)))))
-
-#| 1.13 |#
-
-;; phi = (1 + sqrt(5)) / 2
-;; psi = (1 - sqrt(5)) / 2
-
-;; note that:
-;; phi^2 = phi + 1, and
-;; psi^2 = psi + 1
-
-;; Proof by induction that Fib(n) = (phi^n - psi^n) / sqrt(5)
-
-;; Base cases:
-;; Fib(0) = 0 = (phi^0 - psi^0) / sqrt(5)
-;; Fib(1) = 1 = = (((1 + sqrt(5)) - (1 - sqrt(5))) / 2) / sqrt(5)
-;;        = (phi^1 - psi^1) / sqrt(5)
-
-;; Inductive case:
-
-;; Suppose
-;; Fib(n-2) = (phi^(n-2) - psi^(n-2)) / sqrt(5), and
-;; Fib(n-1) = (phi^(n-1) - psi^(n-1)) / sqrt(5), and
-
-;; Then
-;; Fib(n) = Fib(n-1) + Fib(n - 2)
-;;        = (phi^(n-1) - psi^(n-1) + phi^(n-2) - psi^(n-2)) / sqrt(5)
-;;        = (phi^(n-1) + phi^(n-2) - psi(^n-1) - psi^(n-2)) / sqrt(5)
-;;        = ((phi + 1)phi^(n-2) - (psi + 1)psi^(n-2)) / sqrt(5)
-;;        = ((phi^2)phi^(n-2) - (psi^2)psi^(n-2)) / sqrt(5)
-;;        = (phi^n - psi^n) / sqrt(5)
-
-;; For all n, psi^n / sqrt(5) < 1/2. So Fib(n) is the closest integer to phi^n / sqrt(5)
-
-
-#| 1.14 |#
-
-#| The process generated by (count-change 11): |#
-
-#| - cc 11 5 |#
-#| | - cc -39 5 |#
-#| | - cc 11 4 |#
-#|   | - cc -14 4 |#
-#|   | - cc 11 3 |#
-#|     | - cc 11 2 |#
-#|     | | - cc 11 1 |#
-#|     | | | - cc 11 0 |#
-#|     | | | - cc 10 1 |#
-#|     | |   | - cc 10 0 |#
-#|     | |   | - cc 9 1 |#
-#|     | |     | - cc 9 0 |#
-#|     | |     | - cc 8 1 |#
-#|     | |       | - cc 8 0 |#
-#|     | |       | - cc 7 1 |#
-#|     | |         | - cc 7 0 |#
-#|     | |         | - cc 6 1 |#
-#|     | |           | - cc 6 0 |#
-#|     | |           | - cc 5 1 |#
-#|     | |             | - cc 5 0 |#
-#|     | |             | - cc 4 1 |#
-#|     | |               | - cc 4 0 |#
-#|     | |               | - cc 3 1 |#
-#|     | |                 | - cc 3 0 |#
-#|     | |                 | - cc 2 1 |#
-#|     | |                   | - cc 2 0 |#
-#|     | |                   | - cc 1 1 |#
-#|     | |                     | - cc 1 0 |#
-#|     | |                     | - cc 0 1 |#
-#|     | | - cc 6 2 |#
-#|     |   | - cc 6 1 |#
-#|     |   | | - cc 6 0 |#
-#|     |   | | - cc 5 1 |#
-#|     |   |   | - cc 5 0 |#
-#|     |   |   | - cc 4 1 |#
-#|     |   |     | - cc 4 0 |#
-#|     |   |     | - cc 3 1 |#
-#|     |   |       | - cc 3 0 |#
-#|     |   |       | - cc 2 1 |#
-#|     |   |         | - cc 2 0 |#
-#|     |   |           - cc 1 1 |#
-#|     |   |           | - cc 1 0 |#
-#|     |   |           | - cc 0 1 |#
-#|     |   | - cc 1 2 |#
-#|     |     | - cc 1 1 |#
-#|     |     | | - cc 1 0 |#
-#|     |     | | - cc 0 1 |#
-#|     |     | - cc -4 2 |#
-#|     | - cc 1 3 |#
-#|       | - cc 1 2 |#
-#|       | | - cc 1 1 |#
-#|       | | | - cc 1 0 |#
-#|       | | | - cc 0 1 |#
-#|       | | - cc -4 2 |#
-#|       | - cc -9 3 |#
-
-#| count-change is Theta(n) in space (max depth of tree) |#
-#| count-change is Theta(e^n) in time (number of nodes in tree) |#
-
-#| 1.15 |#
-
-(#%provide sine)
-(define (sine ang)
-  (define (p x) (- (* 3 x) (* 4 (cube x))))
-  (if (not (> (abs ang) 0.1))
-    ang
-    (p (sine (/ ang 3.0)))))
-
-#| sine 12.15 |#
-#| p (sine 4.05) |#
-#| p (p (sine 1.35)) |#
-#| p (p (p (sine 0.45))) |#
-#| p (p (p (p (sine 0.15)))) |#
-#| p (p (p (p (p (sine 0.05))))) |#
-#| p (p (p (p (p 0.05)))) |#
-
-#| p is called 5 times |#
-
-#| The process generated by sine is recursive |#
-#| Order of growth in space: Theta(log_3(n)) |#
-#| Order of growth in time: Theta(log_3(n)) |#
-
-(#%provide expt-rec)
-(define (expt-rec b n)
-  (if (= n 0)
-    1
-    (* b (expt-rec b (- n 1)))))
-
-(#%provide expt-iter)
-(define (expt-iter b n)
-  (expt-iter- b n 1))
-
-(define (expt-iter- b i prod)
-  (if (= i 0)
-    prod
-    (expt-iter-
-      b
-      (- i 1)
-      (* b prod))))
-
-(define (even-? k) (= (remainder k 2) 0))
-
-(#%provide fast-expt-rec)
-(define (fast-expt-rec b n)
-  (cond
-    ((= n 0) 1)
-    ((even-? n) (square (fast-expt-rec b (/ n 2))))
-    (else (* b (fast-expt-rec b (- n 1))))))
-
-#| 1.16 |#
-
-(#%provide fast-expt-iter)
-(define (fast-expt-iter b n)
-  (fast-expt-iter- 1 b n))
-
-(define (fast-expt-iter- a b n)
-  (cond
-    ((= n 0) a)
-    ((even-? n) (fast-expt-iter- a (square b) (/ n 2)))
-    (else (fast-expt-iter- (* a b) b (- n 1)))))
-
-(#%provide mult)
-(define (mult a b)
-  (if (= b 0)
-    0
-    (+ a (mult a (- b 1)))))
-
-#| 1.17 |#
-
-(define (double x) (+ x x))
-(define (halve x) (/ x 2))
-
-(#%provide fast-times-rec)
-(define (fast-times-rec a b)
-  (cond
-    ((= b 0) 0)
-    ((even-? b) (double (fast-times-rec a (halve b))))
-    (else (+ a (fast-times-rec a (- b 1))))))
-
-#| 1.18 |#
-
-(#%provide fast-times-iter)
-(define (fast-times-iter a b)
-  (fast-times-iter- 0 a b))
-
-(define (fast-times-iter- res a b)
-  (cond
-    ((= b 0) res)
-    ((even-? b) (fast-times-iter- res (double a) (halve b)))
-    (else (fast-times-iter- (+ res a) a (- b 1)))))
-
-#| 1.19 |#
-
-;; T
-;; a <- a + b
-;; b <- a
-
-;; Tpq
-;; a <- bq + aq + ap
-;; b <- bp + aq
-
-;; Tpq^2
-;; a <- (bp + aq)q + (bq + aq + ap)q + (bq + aq + ap)p
-;; b <- (bp + aq)p + (bq + aq + ap)q
-
-;; a <- b(2pq + qq) + a(2pq + qq) + a(pp + qq)
-;; b <- b(pp + qq) + a(2qp + qq)
-
-;; p' = pp + qq
-;; q' = 2pq + qq
-
-(#%provide fib)
-(define (fib n)
-  (fib-iter 1 0 0 1 n))
-
-(define (fib-iter a b p q i)
-  (cond
-    ((= i 0) b)
-    ((even-? i)
-     (fib-iter
-       a
-       b
-       (+ (square p) (square q))
-       (+ (* 2 p q) (square q))
-       (/ i 2)))
-    (else
-      (fib-iter
-        (+ (* b q) (* a q) (* a p))
-        (+ (* b p) (* a q))
-        p
-        q
-        (- i 1)))))
-
-(#%provide fib-slow)
-(define (fib-slow n)
-  (cond
-    ((= n 0) 0)
-    ((= n 1) 1)
-    (else (+ (fib (- n 1)) (fib (- n 2))))))
-
-#| 1.20 |#
-
-(#%provide gcd-)
-(define (gcd- a b)
-  (if (= b 0)
-    a
-    (gcd- b (remainder a b))))
-
-;; Process generated by normal order evaluation
-
-#| (gcd- 206 40) |#
-#| (if (= 40 0) 206 (gcd- 40 (rem 206 40))) |#
-#| (gcd- 40 (rem 206 40)) |#
-#| (if (= (rem 206 40) 0) 40 (gcd- b (rem 40 (rem 206 40)))) ; one call |#
-#| (if (= 6 0) 40 (gcd- (rem 206 40) (rem 40 (rem 206 40)))) |#
-#| (gcd- (rem 206 40) (rem 40 (rem 206 40))) |#
-#| (if (= (rem 40 (rem 206 40)) 0) ... (gcd- (rem 40 (rem 206 40)) (rem (rem 206 40) (rem 40 (rem 206 40))))) ; one call |#
-#| (if (= (rem 40 6) 0) ... (gcd- (rem 40 (rem 206 40)) (rem (rem 206 40) (rem 40 (rem 206 40))))) ; one call |#
-#| (if (= 4 0) ... (gcd- (rem 40 (rem 206 40)) (rem (rem 206 40) (rem 40 (rem 206 40))))) |#
-#| (gcd- (rem 40 (rem 206 40)) (rem (rem 206 40) (rem 40 (rem 206 40)))) |#
-#| (if (= (rem (rem 206 40) (rem 40 (rem 206 40))) 0) ... (gcd- (rem (rem 206 40) (rem 40 (rem 206 40))) ...)) ; one call |#
-#| (if (= (rem 6 (rem 40 (rem 206 40))) 0) ... (gcd- (rem (rem 206 40) (rem 40 (rem 206 40))) ...)) ; one call |#
-#| (if (= (rem 6 (rem 40 6)) 0) ... (gcd- (rem (rem 206 40) (rem 40 (rem 206 40))) ...)) ; one call |#
-#| (if (= (rem 6 4) 0) ... (gcd- (rem (rem 206 40) (rem 40 (rem 206 40))) ...)) ; one call |#
-#| (if (= 2 0) ... (gcd- (rem (rem 206 40) (rem 40 (rem 206 40))) ...)) |#
-#| (gcd- (rem (rem 206 40) (rem 40 (rem 206 40))) (rem (rem 40 (rem 206 40)) ...)) |#
-#| (if (= (rem (rem 40 (rem 206 40)) (rem (rem 206 40) (rem 40 (rem 206 40)))) 0) (rem (rem 206 40) ...) ...) ; one call |#
-#| (if (= (rem (rem 40 6) (rem (rem 206 40) (rem 40 (rem 206 40)))) 0) (rem (rem 206 40) ...) ...) ; one call |#
-#| (if (= (rem 4 (rem (rem 206 40) (rem 40 (rem 206 40)))) 0) (rem (rem 206 40) (rem 40 (rem 206 40))) ...) ; one call |#
-#| (if (= (rem 4 (rem 6 (rem 40 (rem 206 40)))) 0) (rem (rem 206 40) (rem 40 (rem 206 40))) ...) ; one call |#
-#| (if (= (rem 4 (rem 6 (rem 40 6))) 0) (rem (rem 206 40) (rem 40 (rem 206 40))) ...) ; one call |#
-#| (if (= (rem 4 (rem 6 4)) 0) (rem (rem 206 40) (rem 40 (rem 206 40))) ...) ; one call |#
-#| (if (= (rem 4 2) 0) (rem (rem 206 40) (rem 40 (rem 206 40))) ...) ; one call |#
-#| (if (= 0 0) (rem (rem 206 40) (rem 40 (rem 206 40))) ...) |#
-#| (rem (rem 206 40) (rem 40 (rem 206 40))) ; one call |#
-#| (rem 6 (rem 40 (rem 206 40))) ; one call |#
-#| (rem 6 (rem 40 6)) ; one call |#
-#| (rem 6 4) ; one call |#
-#| 2 |#
-
-;; 18 calls to remainder are performed
-
-;; Process generated by applicative order evaluation
-
-#| (gcd- 206 40) |#
-#| (if (= 40 0) 206 (gcd- 40 (remainder 206 40))) ; one call |#
-#| (gcd- 40 6) |#
-#| (gcd- 40 6) |#
-#| (if (= 6 0) 40 (gcd- 6 (remainder 40 6))) |#
-#| (gcd- 6 (remainder 40 6)) ; one call |#
-#| (gcd- 6 4) |#
-#| (if (= 4 0) 6 (gcd- 4 (remainder 6 4))) |#
-#| (gcd- 4 (remainder 6 4)) ; one call |#
-#| (gcd- 4 2) |#
-#| (if (= 2 0) 4 (gcd- 2 (remainder 4 2))) |#
-#| (gcd- 2 (remainder 4 2)) ; one call |#
-#| (gcd- 2 0) |#
-#| (if (= 0 0) 2 (gcd- 0 (remainder 2 0))) |#
-#| 2 |#
-
-;; 4 calls to remainder are performed
-
-(#%provide smallest-divisor)
-(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 (next test-divisor)))))
-
-(define (divides? a b)
-  (= (remainder b a) 0))
-
-(#%provide prime?)
-(define (prime? n)
-  (= n (smallest-divisor n)))
-
-(define (expmod base expo m)
-  (cond
-    ((= expo 0) 1)
-    ((even-? expo)
-     (remainder (square (expmod base (/ expo 2) m)) m))
-    (else
-      (remainder
-        (* base (expmod base (- expo 1) m))
-        m))))
-
-(define (fermat-test n)
-  (define (try-it a)
-    (= (expmod a n n) a))
-  (try-it (+ 1 (random (- n 1)))))
-
-(define (fast-prime? n times)
-  (cond
-    ((= times 0) true)
-    ((fermat-test n) (fast-prime? n (- times 1)))
-    (else false)))
-
-#| 1.21 |#
-
-#| (smallest-divisor 199) |#
-#| (smallest-divisor 1999) |#
-#| (smallest-divisor 19999) |#
-
-(#%provide timed-prime-test)
-(define (timed-prime-test n)
-  (newline)
-  (display n)
-  (start-prime-test n (runtime)))
-
-(define (start-prime-test n start-time)
-  (if (fast-prime? n 100)
-    (report-prime (- (runtime) start-time))))
-
-(define (report-prime elapsed-time)
-(display " *** ")
-  (display elapsed-time))
-
-#| 1.22 |#
-
-(#%provide search-for-primes)
-(define (search-for-primes a b)
-  (cond
-    ((> a b) (newline))
-    (else
-      (timed-prime-test a)
-      (search-for-primes (+ a 1) b))))
-
-;; Smallest primes over 1000:
-;; 1009 3us
-;; 1013 3
-;; 1019 3
-
-;; Smallest primes over 10000:
-;; 10007 9us
-;; 10019 8
-;; 10037 9
-
-;; Smallest primes over 100000:
-;; 100003 22us
-;; 100019 21
-;; 100043 20
-
-;; Smallest primes over 1000000:
-;; 1000003 65us
-;; 1000033 65
-;; 1000037 65
-
-;; 9 / 3 = 3
-;; 21 / 9 = 2.33
-;; 65 / 21 = 3.095
-
-;; sqrt(10) = 3.16
-
-#| 1.23 |#
-
-(define (next n)
-  (if (= n 2) 3 (+ n 2)))
-
-;; Smallest primes over 1000:
-;; 1009 2us
-;; 1013 2
-;; 1019 2
-
-;; Smallest primes over 10000:
-;; 10007 5us
-;; 10019 4
-;; 10037 4
-
-;; Smallest primes over 100000:
-;; 100003 11us
-;; 100019 12
-;; 100043 11
-
-;; Smallest primes over 1000000:
-;; 1000003 34us
-;; 1000033 34
-;; 1000037 34
-
-;; 3 / 2 = 1.4
-;; 9 / 4 = 2.25
-;; 21 / 11 = 1.9
-;; 65 / 34 = 1.9
-
-#| 1.24 |#
-
-;; Using fast-prime?
-
-;; Smallest primes over 1000:
-;; 1009 .28ms
-;; 1013 .29
-;; 1019 .31
-
-;; Smallest primes over 10000:
-;; 10007 .37ms
-;; 10019 .36
-;; 10037 .40
-
-;; Smallest primes over 100000:
-;; 100003 .42ms
-;; 100019 .44
-;; 100043 .46
-
-;; Smallest primes over 1000000:
-;; 1000003 .50ms
-;; 1000033 .49
-;; 1000037 .51
-
-;; Scaling up by a factor of ten adds a constant amount of time
-
-#| 1.25 |#
-
-;; This version of expmod requires too much space to
-;; represent the intermediate result when using very
-;; large exponents
-
-(define (expmod-bad base expo m)
-  (remainder (fast-expt-iter base expo) m))
-
-#| 1.26 |#
-
-;; This version of expmod makes two recursive calls of
-;; size n / 2, making it Theta(n) in time rather than
-;; Theta(log(n))
-
-(define (expmod-slow base expo m)
-  (cond
-    ((= expo 0) 1)
-    ((even-? expo)
-     (remainder
-       (*
-         (remainder (expmod-slow base (/ expo 2) m)
-         (remainder (expmod-slow base (/ expo 2) m))))
-       m))
-    (else
-      (remainder
-        (*
-          base
-          (expmod-slow base (- expo 1) m))
-        m))))
-
-#| 1.27 |#
-
-(#%provide fermat-test-exhaustive)
-(define (fermat-test-exhaustive n)
-  (define (iter i res)
-    (define (try-it a) (= (expmod a n n) a))
-    (if (= i n)
-      res
-      (iter (+ i 1) (and res (try-it i)))))
-  (iter 1 true))
-
-#| (fermat-test-exhaustive 561) |#
-#| (fermat-test-exhaustive 1105) |#
-#| (fermat-test-exhaustive 1729) |#
-#| (fermat-test-exhaustive 2465) |#
-#| (fermat-test-exhaustive 2821) |#
-#| (fermat-test-exhaustive 6601) |#
-
-#| 1.28 |#
-
-(define (expmod-sig base expo m)
-  (define (square-sig x)
-    (define squared (remainder (square x) m))
-    (if
-      (and
-        (= squared 1)
-        (not (= x 1))
-        (not (= x expo)))
-      0
-      squared))
-  (cond
-    ((= expo 0) 1)
-    ((even-? expo)
-     (square-sig (expmod base (/ expo 2) m)))
-    (else
-      (remainder
-        (* base (expmod base (- expo 1) m))
-        m))))
-
-(define (miller-rabin-test n)
-  (define (try-it a)
-    (= (expmod a (- n 1) n) 1))
-  (try-it (+ 1 (random (- n 1)))))
-
-(#%provide mr-fast-prime?)
-(define (mr-fast-prime? n times)
-  (cond
-    ((= times 0) true)
-    ((miller-rabin-test n) (mr-fast-prime? n (- times 1)))
-    (else false)))
-
-(#%provide sum-integers-)
-(define (sum-integers- a b)
-  (if (> a b)
-    0
-    (+ a (sum-integers- (+ a 1) b))))
-
-(#%provide sum-cubes-)
-(define (sum-cubes- a b)
-  (if (> a b)
-    0
-    (+ (cube a) (sum-cubes- (+ a 1) b))))
-
-(#%provide pi-sum-)
-(define (pi-sum- a b)
-  (if (> a b)
-    0
-    (+ (/ 1.0 (* a (+ a 2))) (pi-sum- (+ a 4) b))))
-
-(#%provide sum)
-(define (sum term a next b)
-  (if (> a b)
-    0
-    (+
-      (term a)
-      (sum term (next a) next b))))
-
-(define (inc n) (+ n 1))
-
-(#%provide sum-cubes)
-(define (sum-cubes a b) (sum cube a inc b))
-
-(define (id x) x)
-
-(#%provide sum-integers)
-(define (sum-integers a b) (sum id a inc b))
-
-(#%provide pi-sum)
-(define (pi-sum a b)
-  (define (pi-term x) (/ 1.0 (* x (+ x 2))))
-  (define (pi-next x) (+ x 4))
-  (sum pi-term a pi-next b))
-
-(#%provide integral)
-(define (integral f a b dx)
-  (define (add-dx x) (+ x dx))
-  (* (sum f (+ a (/ dx 2.0)) add-dx b) dx))
-
-#| 1.29 |#
-
-(#%provide simpson)
-(define (simpson f a b n)
-  (define h (/ (- b a) n))
-  (define (single-term k)
-    (f (+ a (* k h))))
-  (define (simpson-term k)
-    (+
-      (single-term (- k 1))
-      (* 4.0 (single-term k))
-      (single-term (+ k 1))))
-  (define (simpson-next k) (+ k 2))
-  (* (/ h 3.0) (sum simpson-term 1 simpson-next n)))
-
-#| 1.30 |#
-
-(#%provide sum-iter)
-(define (sum-iter term a next b)
-  (define (iter a result)
-    (if (> a b)
-      result
-      (iter (next a) (+ (term a) result))))
-  (iter a 0))
-
-#| 1.31 |#
-
-(#%provide product)
-(define (product term a next b)
-  (if (> a b)
-    1
-    (*
-      (term a)
-      (product term (next a) next b))))
-
-(#%provide factorial)
-(define (factorial n)
-  (product id 1 inc n))
-
-(#%provide pi-prod)
-(define (pi-prod n)
-  (define (pi-term x) (/ (* (- x 1) (+ x 1)) (square x)))
-  (define (pi-next x) (+ x 2))
-  (* 4.0 (product pi-term 3 pi-next n)))
-
-(#%provide product-iter)
-(define (product-iter term a next b)
-  (define (iter a result)
-    (if (> a b)
-      result
-      (iter (next a) (* (term a) result))))
-  (iter a 1))
-
-#| 1.32 |#
-
-(#%provide accumulate)
-(define (accumulate combiner null-value term a next b)
-  (if (> a b)
-    null-value
-    (combiner
-      (term a)
-      (accumulate combiner null-value term (next a) next b))))
-
-(#%provide prod-acc)
-(define (prod-acc term a next b)
-  (accumulate * 1 term a next b))
-
-(#%provide sum-acc)
-(define (sum-acc term a next b)
-  (accumulate + 0 term a next b))
-
-(#%provide acc-iter)
-(define (acc-iter combiner null-value term a next b)
-  (define (iter a result)
-    (if (> a b)
-      result
-      (iter (next a) (combiner (term a) result))))
-  (iter a null-value))
-
-#| 1.33 |#
-
-(#%provide filtered-accumulate)
-(define (filtered-accumulate combiner null-value pred term a next b)
-  (if (> a b)
-    null-value
-    (if (pred a)
-      (combiner
-        (term a)
-        (filtered-accumulate combiner null-value pred term (next a) next b))
-      (filtered-accumulate combiner null-value pred term (next a) next b))))
-
-(#%provide sum-prime-square)
-(define (sum-prime-square a b)
-  (filtered-accumulate + 0 prime? square a inc b))
-
-(#%provide prod-coprime)
-(define (prod-coprime n)
-  (define (pred i) (= (gcd- i n) 1))
-  (filtered-accumulate * 1 pred id 1 inc n))
-
-(#%provide pi-sum-lam)
-(define (pi-sum-lam a b)
-  (sum
-    (lambda (x) (/ 1.0 (* x (+ x 2))))
-    a
-    (lambda (x) (+ x 4))
-    b))
-
-(#%provide integral-lam)
-(define (integral-lam f a b dx)
-  (*
-    (sum
-      f
-      (+ a (/ dx 2.0))
-      (lambda (x) (+ x dx))
-      b)
-    dx))
-
-(#%provide f-help)
-(define (f-help x y)
-  (define (f-helper a b)
-    (+
-      (* x (square a))
-      (* y b)
-      (* a b)))
-  (f-helper
-    (+ 1 (* x y))
-    (- 1 y)))
-
-(#%provide f-lam)
-(define (f-lam x y)
-  ((lambda (a b)
-     (+
-       (* x (square a))
-       (* y b)
-       (* a b)))
-   (+ 1 (* x y))
-   (- 1 y)))
-
-(#%provide f-let)
-(define (f-let x y)
-  (let
-    ((a (+ 1 (* x y)))
-     (b (- 1 y)))
-    (+
-     (* x (square a))
-     (* y b)
-     (* a b))))
-
-(#%provide f-def)
-(define (f-def x y)
-  (define a (+ 1 (* x y)))
-  (define b (- 1 y))
-  (+
-    (* x (square a))
-    (* y b)
-    (* a b)))
-
-#| 1.34 |#
-
-;; (define (f g) (g 2))
-
-;; (f square) 4
-
-;; (f (lambda (z) (* z (+ z 1)))) 6
-
-;; (f f) (f 2) (2 2) error
-
-(#%provide search)
-(define (search f neg-point pos-point)
-  (let ((midpoint (average neg-point pos-point)))
-    (if (close-enough? neg-point pos-point)
-      midpoint
-      (let ((test-value (f midpoint)))
-        (cond
-          ((positive? test-value)
-           (search f neg-point midpoint))
-          ((negative? test-value)
-           (search f midpoint pos-point))
-          (else midpoint))))))
-
-(define (close-enough? x y)
-  (< (abs (- x y)) 0.001))
-
-(#%provide half-interval-method)
-(define (half-interval-method f a b)
-  (let
-    ((a-value (f a))
-     (b-value (f b)))
-    (cond
-      ((and (negative? a-value) (positive? b-value))
-       (search f a b))
-      ((and (negative? b-value) (positive? a-value))
-       (search f b a))
-      (else
-        (error "Values are not of opposite sign" a b)))))
-
-(define tolerance 0.0001)
-
-(#%provide fixed-point)
-(define (fixed-point f first-guess)
-  (define (close-enough? v1 v2)
-    (< (abs (- v1 v2)) tolerance))
-  (define (try guess)
-    (display guess)
-    (newline)
-    (let ((next (f guess)))
-      (if (close-enough? guess next)
-        next
-        (try next))))
-  (try first-guess))
-
-(#%provide sqrt-fix)
-(define (sqrt-fix x)
-  (fixed-point
-    (lambda (y) (average y (/ x y)))
-    1.0))
-
-#| 1.35 |#
-
-(#%provide golden)
-(define (golden)
-  (fixed-point
-    (lambda (x) (+ 1.0 (/ 1.0 x)))
-    1.0))
-
-#| 1.36 |#
-
-(#%provide x-to-the-x)
-(define (x-to-the-x)
-  (fixed-point
-    (lambda (x) (/ (log 1000.0) (log x)))
-    2.0))
-
-#| 1.37 |#
-
-(#%provide cont-frac)
-(define (cont-frac n d k)
-  (define (iter res i)
-    (if (= i 0)
-      res
-      (iter (/ (n i) (+ (d i) res)) (- i 1))))
-  (iter 0 k))
-
-;; (cont-frac (lambda (i) 1.0) (lambda (i) 1.0) 11)
-;; 0.6180555555555556
-
-;; Accurate to 4 places after 11 iterations
-
-(#%provide cont-frac-rec)
-(define (cont-frac-rec n d k)
-  (define (rec i)
-    (if (> i k)
-      0
-      (/ (n i) (+ (d i) (rec (+ i 1))))))
-  (rec 1))
-
-#| 1.38 |#
-
-(#%provide e-approx)
-(define (e-approx k)
-  (define (n i) 1.0)
-  (define (d i)
-    (if (divides? 3 (+ i 1))
-      (* 2 (/ (+ i 1) 3))
-      1.0))
-  (+ (cont-frac n d k) 2))
-
-#| 1.39 |#
-
-(#%provide tan-cf)
-(define (tan-cf x k)
-  (define (rec prod sum)
-    (let ((stop (+ 1 (* 2 (- k 1)))))
-      (if (> sum stop)
-        0
-        (/ prod (- sum (rec (* prod x) (+ sum 2)))))))
-  (rec x 1.0))
-
-(define (average-damp f)
-  (lambda (x) (average x (f x))))
-
-((average-damp square) 10)
-
-(#%provide sqrt-avg-damp)
-(define (sqrt-avg-damp x)
-  (fixed-point
-    (average-damp (lambda (y) (/ x y)))
-    1.0))
-
-(#%provide cbrt-avg-damp)
-(define (cbrt-avg-damp x)
-  (fixed-point
-    (average-damp (lambda (y) (/ x (square y))))
-    1.0))
-
-(define (deriv g)
-  (lambda (x)
-    (/
-      (- (g (+ x dx)) (g x))
-      dx)))
-
-(define dx 0.00001)
-
-((deriv cube) 5)
-
-(define (newton-transform g)
-  (lambda (x)
-    (- x (/ (g x) ((deriv g) x)))))
-
-(#%provide newtons-method)
-(define (newtons-method g guess)
-  (fixed-point (newton-transform g) guess))
-
-(#%provide sqrt-newt)
-(define (sqrt-newt x)
-  (newtons-method
-    (lambda (y) (- (square y) x))
-    1.0))
-
-(define (fixed-point-of-transform g transform guess)
-  (fixed-point (transform g) guess))
-
-(#%provide sqrt-ad-trans)
-(define (sqrt-ad-trans x)
-  (fixed-point-of-transform
-    (lambda (y) (/ x y))
-    average-damp
-    1.0))
-
-(#%provide sqrt-newt-trans)
-(define (sqrt-newt-trans x)
-  (fixed-point-of-transform
-    (lambda (y) (- (square y) x))
-    newton-transform
-    1.0))
-
-#| 1.40 |#
-
-(#%provide cubic)
-(define (cubic a b c)
-  (lambda (x)
-    (+
-      (cube x)
-      (* a (square x))
-      (* b x)
-      c)))
-
-;; (newtons-method (cubic 0 0 -8.0) 4.0)
-
-#| 1.41 |#
-
-(#%provide twice)
-(define (twice f)
-  (lambda (x) (f (f x))))
-
-;; (twice inc 1)
-;; 3
-
-;; (((twice (twice twice)) inc) 5)
-;; 21
-
-#| 1.42 |#
-
-(#%provide compose-)
-(define (compose- f g)
-  (lambda (x) (f (g x))))
-
-;; ((compose- square inc) 6)
-;; 49
-
-#| 1.43 |#
-
-(#%provide repeated)
-(define (repeated f n)
-  (if (= n 0)
-    (lambda (x) x)
-    (compose- (repeated f (- n 1)) f)))
-
-;; ((repeated square 2) 5)
-;; 625
-
-#| 1.44 |#
-
-(#%provide smooth)
-(define (smooth f)
-  (lambda (x)
-    (/
-      (+
-        (f (- x dx))
-        (f x)
-        (f (+ x dx)))
-      3.0)))
-
-(#%provide n-smooth)
-(define (n-smooth f n)
-  (repeated smooth n))
-
-#| 1.45 |#
-
-(#%provide flog2)
-(define (flog2 n) (floor (/ (log n) (log 2))))
-
-(#%provide nth-root)
-(define (nth-root n x)
-  (fixed-point
-    ((repeated average-damp (flog2 n))
-     (lambda (y) (/ x (fast-expt-iter y (- n 1)))))
-    1.0))
-
-#| 1.46 |#
-
-(#%provide iterative-improve)
-(define (iterative-improve good-enough? improve)
-  (lambda (guess)
-    (define (iter x)
-      (if (good-enough? x)
-        x
-        (iter (improve x))))
-    (iter guess)))
-
-(#%provide sqrt-it-imp)
-(define (sqrt-it-imp x)
-  ((iterative-improve
-     (lambda (guess) (< (abs (- (square guess) x)) 0.001))
-     (lambda (guess) (average guess (/ x guess))))
-   1.0))
-
-(#%provide fixed-point-it-imp)
-(define (fixed-point-it-imp f first-guess)
-  ((iterative-improve
-     (lambda (guess) (< (abs (- guess (f guess))) tolerance))
-     f)
-   first-guess))