From fde4478f5900c2210cf662eacba93e9d797c2b17 Mon Sep 17 00:00:00 2001
From: Cosmin Stejerean <cosmin@offbytwo.com>
Date: Sun, 12 Feb 2012 23:38:34 +1100
Subject: [PATCH] added exercises 1.15, 1.16, 1.17, 1.18

---
 exercise-1-15.ss | 19 +++++++++++++++++++
 exercise-1-16.ss | 21 +++++++++++++++++++++
 exercise-1-17.ss | 19 +++++++++++++++++++
 exercise-1-18.ss | 18 ++++++++++++++++++
 4 files changed, 77 insertions(+)
 create mode 100644 exercise-1-15.ss
 create mode 100644 exercise-1-16.ss
 create mode 100644 exercise-1-17.ss
 create mode 100644 exercise-1-18.ss

diff --git a/exercise-1-15.ss b/exercise-1-15.ss
new file mode 100644
index 0000000..a032b35
--- /dev/null
+++ b/exercise-1-15.ss
@@ -0,0 +1,19 @@
+(define (cube x) (* x x x))
+
+(define (p x) 
+  (pretty-print 1)
+  (- (* 3 x) (* 4 (cube x))))
+
+(define (sine angle)
+  (if (not (> (abs angle) 0.1))
+      angle
+      (p (sine (/ angle 3.0)))))
+
+
+;; each time we divide the angle by 3 until the angle is <= 0.1
+;; 
+;; 12.15 => 4.05 => 1.35 => 0.45 => 0.15 => 0.05
+;;
+;; therefore we call p 5 times to compute sine of 12.15
+
+;; both the space and complexity of this operation is log(n) (base 3)
diff --git a/exercise-1-16.ss b/exercise-1-16.ss
new file mode 100644
index 0000000..aeb8ae9
--- /dev/null
+++ b/exercise-1-16.ss
@@ -0,0 +1,21 @@
+(define (square x) (* x x))
+
+(define (even? n) (= (remainder n 2) 0))
+
+;; linear recursive algorithm
+
+(define (fast-expt b n)
+  (cond ((= n 0) 1)
+        ((even? n) (square (fast-expt b (/ n 2))))
+        (else (* b (fast-expt b (- n 1))))))
+
+
+;; iterative algorithm
+
+(define (expt b n)
+  (fast-expt-iter b n 1))
+
+(define (fast-expt-iter b n a)
+  (cond ((= n 0) a)
+        ((even? n) (fast-expt-iter (square b) (/ n 2) a))
+        (else (fast-expt-iter b (- n 1) (* b a)))))
diff --git a/exercise-1-17.ss b/exercise-1-17.ss
new file mode 100644
index 0000000..8f1a51d
--- /dev/null
+++ b/exercise-1-17.ss
@@ -0,0 +1,19 @@
+(define (mul a b)
+  (if (= b 0)
+      0
+      (+ a (mul a (- b 1)))))
+
+;; pretend the following are built-in
+
+(define (double a) (* a 2))
+
+(define (halve a) (/ a 2))
+
+(define (even? n) (= (remainder n 2) 0))
+
+;; fast-multiplication -- log(n) steps
+
+(define (fast-mul a b)
+  (cond ((= b 0) 0)
+        ((even? b) (fast-mul (double a) (halve b)))
+        (else (+ a (fast-mul a (- b 1))))))
diff --git a/exercise-1-18.ss b/exercise-1-18.ss
new file mode 100644
index 0000000..23ef03d
--- /dev/null
+++ b/exercise-1-18.ss
@@ -0,0 +1,18 @@
+;; pretend built-ins
+
+(define (double a) (* a 2))
+
+(define (halve a) (/ a 2))
+
+(define (even? n) (= (remainder n 2) 0))
+
+
+;; iterative fast-mul
+
+(define (mul a b)
+  (fast-mul-iter a b 0))
+
+(define (fast-mul-iter a b acc)
+  (cond ((= b 0) acc)
+        ((even? b) (fast-mul-iter (double a) (halve b) acc))
+        (else (fast-mul-iter a (- b 1) (+ acc a)))))