Add solutions from 1-1 to 1-8

ex-1_1-8.scm

@@ -0,0 +1,133 @@
+(display "ex-1.1 - solutions in comments\n")
+; 10 -> 10
+; (+ 5 3 4) -> 12
+; (- 9 1) -> 8
+; (/ 6 2) -> 3
+; (+ (* 2 4) (- 4 6)) -> 6
+; (define a 3) -> a
+; (define b (+ a 1)) -> b
+; (+ a b (* a b)) -> 19
+; (= a b) -> #f
+; (if (and (> b a) (< b (* a b))) b a) ; -> b -> 4
+; (cond ((= a 4) 6)
+;       ((= b 4) (+ 6 7 a))
+;       (else 25)) -> 126
+; (+ 2 (if (> b a) b a)) -> 6
+; (* (cond ((> a b) a)
+;          ((< a b) b)
+;          (else -1))
+;    (+ a 1)) ->
+
+(display "ex-1.2 - ")
+(display (/ (+ 5 4 (- 2 (- 3 (+ 6 (/ 4 5)))))
+            (* 3 (- 6 2) (- 2 7))))
+(newline)
+
+(display "ex-1.3 - ")
+
+(define (sum-squares a b) (+ (* a a) (* b b)))
+
+(define (sum-squares-max a b c)
+  (cond ((and (>= a c) (>= b c)) (sum-squares a b))
+        ((and (>= a b) (>= c b)) (sum-squares a c))
+        (else (sum-squares b c))))
+
+(display (sum-squares-max 2 -6 1))
+(newline)
+
+(display "ex-1.4 - Operator becomes + or - depending on the value of b\n")
+; (define (a-plus-abs-b a b)
+;    ((if (> b 0) + -) a b))
+
+(display "ex-1.5 - Only normal-order terminates\n")
+;(define (p) (p))
+;(define (test x y)
+;  (if (= x 0) 0 y))
+;(display (if (= 0 0) 0 (p)))
+; Will not terminate:
+;(display (test 0 (p)))
+
+(display "\n; Square roots via Newton's Method") (newline)
+(define (sqrt-iter guess x)
+  (if (good-enough? guess x)
+      guess
+      (sqrt-iter (improve guess x)
+                 x)))
+
+(define (improve guess x)
+  (average guess (/ x guess)))
+
+(define (average a b) (/ (+ a b) 2))
+
+(define (good-enough? guess x)
+  (< (abs (- (square guess) x)) 0.0001))
+
+(define (square a) (* a a))
+
+(define (sqrt x) (sqrt-iter 1.0 x))
+
+(display (sqrt 9))
+(newline)
+(display (sqrt (+ 100 37)))
+(newline)
+(display (sqrt (+ (sqrt 2) (sqrt 3))))
+(newline)
+(display (square (sqrt 1000)))
+(newline)
+
+(display "\nex-1.6 - see comment\n")
+
+(define (new-if predicate then-clause else-clause)
+  (cond (predicate then-clause)
+        (else else-clause)))
+
+;(define (sqrt-iter guess x)
+;  (new-if (good-enough? guess x)
+;          guess
+;          (sqrt-iter (improve guess x)
+;                     x)))
+
+;(display (sqrt 9))
+
+; sqrt-iter doesn't terminate because new-if is evaluated in applicative
+; order which results in an endless recursion of sqrt-iter -> new-if.
+
+(display "\nex-1.7 - see comments for explanation\nwrong behavior:\n")
+(display (sqrt 9)) (newline)
+; Very small numbers don't work because the delta between the initial
+; guess and the expected solution is in a smaller dimension than the
+; value used in good-enough?
+(display (sqrt 0.0000001)) (newline)
+; For very large numbers, good-enough? will never return true because the
+; representation of floating point numbers is not accurate enough for their
+; difference to ever fall below the tolerance value.
+;(display (sqrt 9732402478147293489)) (newline)
+(display "better:\n")
+
+(define (good-enough2? guess new-guess)
+  (< (/ (abs (- new-guess guess)) new-guess) 0.00000000001))
+
+(define (sqrt-iter-precise guess x)
+  (if (good-enough2? guess (improve guess x))
+      guess
+      (sqrt-iter-precise (improve guess x)
+                 x)))
+
+(display (sqrt 9)) (newline)
+(display (sqrt 0.00000001)) (newline)
+
+; (display (sqrt 9732402478147293489))
+; works with racket and newer MIT Scheme versions
+
+(display "\nex-1.8 - cube-root") (newline)
+(define (improve-cubic y x) (/ (+ (/ x (* y y)) (* 2 y)) 3))
+
+(define (cbrt-iter guess x)
+  (if (good-enough2? guess (improve-cubic guess x))
+      guess
+      (cbrt-iter (improve-cubic guess x) x)))
+
+(define (cbrt x) (cbrt-iter 1.0 x))
+
+(display (cbrt 27)) (newline)
+(display (cbrt 0.001)) (newline)