Problem 2
Translate the following expression into prefix form $$ \frac{5+4+\left(2-\left(3-\left(6+\frac{4}{5}\right)\right)\right)}{3(6-2)(2-7)} $$
Problem 3
Define a procedure that takes three numbers as arguments and returns the sum of the squares of the two larger numbers.
Problem 4
Observe that our model of evaluation allows for combinations whose operators are compound expressions. Use this observation to describe the behavior of the following procedure: (define (a-plus-abs-b a b) \(((\) if \((>b 0)+-) a\) b \())\)
Problem 5
Ben Bitdiddle has invented a test to determine whether the interpreter he is faced with is using applicative-order evaluation or normal-order evaluation. He defines the following two procedures: (define \((\mathrm{p})(\mathrm{p}))\) (define (test \(\mathrm{x} \mathrm{y}\) ) (if \((=\mathrm{x} 0)\) 0 \(\mathrm{y})\) ) Then he evaluates the expression (test \(0(\mathrm{p}))\) What behavior will Ben observe with an interpreter that uses applicative-order evaluation? What behavior will he observe with an interpreter that uses normalorder evaluation? Explain your answer. (Assume that the evaluation rule for the special form if is the same whether the interpreter is using normal or applicative order: The predicate expression is evaluated first, and the result determines whether to evaluate the consequent or the alternative expression.)
Problem 6
Alyssa P. Hacker doesn't see why if needs to be provided as a special form. "Why can't I just define it as an ordinary procedure in terms of cond?" she asks. Alyssa's friend Eva Lu Ator claims this can indeed be done, and she defines a new version of if: (define (new-if predicate then-clause else-clause) (cond (predicate then-clause) (else else-clause))) Eva demonstrates the program for Alyssa: (new-if (=2 3) 0 5) 5 (new-if (= 1 1) 0 5) 0 Delighted, Alyssa uses new-if to rewrite the s (define (sqrt-iter guess \(x\) ) (new-if (good-enough? guess \(x\) ) $$ \begin{array}{l}\text { guess } \\ \text { (sqrt-iter (improve guess } x \text { ) } \\ x) \text { ) }\end{array} $$ What happens when Alyssa attempts to use this to compute square roots? Explain.
Problem 7
The good-enough? test used in computing square roots will not be very effective for finding the square roots of very small numbers. Also, in real computers, arithmetic operations are almost always performed with limited precision. This makes our test inadequate for very large numbers. Explain these statements, with examples showing how the test fails for small and large numbers. An alternative strategy for implementing good-enough? is to watch how guess changes from one iteration to the next and to stop when the change is a very small fraction of the guess. Design a square-root procedure that uses this kind of end test. Does this work better for small and large numbers? \({ }^{24}\) Readers who are worried about the efficiency issues involved in using procedure calls to implement iteration should note the remarks on "tail recursion" in section 1.2.1.
Problem 8
Newton's method for cube roots is based on the fact that if \(y\) is an approximation to the cube root of \(x\), then a better approximation is given by the value \(\frac{x / y^{2}+2 y}{3}\) Use this formula to implement a cube-root procedure analogous to the squareroot procedure. (In section \(1.3 .4\) we will see how to implement Newton's method in general as an abstraction of these square-root and cube- root procedures.)
Problem 11
A function \(f\) is defined by the rule that \(f(n)=n\) if \(n<3\) and \(f(n)=\) \(f(n-1)+2 f(n-2)+3 f(n-3)\) if \(n \geq 3\). Write a procedure that computes \(f\) by means of a recursive process. Write a procedure that computes \(f\) by means of an iterative process.
Problem 15
The sine of an angle (specified in radians) can be computed by making use of the approximation \(\sin x \approx x\) if \(x\) is sufficiently small, and the trigonometric identity \(\sin x=3 \sin \frac{x}{3}-4 \sin ^{3} \frac{x}{3}\) to reduce the size of the argument of \(\sin\). (For purposes of this exercise an angle is considered "sufficiently small" if its magnitude is not greater than \(0.1\) radians.) These ideas are incorporated in the following procedures: (define (cube \(x\) ) \((* x x x)\) ) (define (p \(x\) ) \((-(* 3 x)(* 4(\) cube \(x)))\) ) (define (sine angle) (if (not (> (abs angle) 0.1)) angle (p (sine (/ angle 3.0))))) a. How many times is the procedure \(\mathrm{p}\) applied when (sine 12.15) is evaluated? b. What is the order of growth in space and number of steps (as a function of \(a\) ) used by the process generated by the sine procedure when (sine a) is evaluated?
Problem 16
Design a procedure that evolves an iterative exponentiation process that uses successive squaring and uses a logarithmic number of steps, as does \(f\) ast- expt. (Hint: Using the observation that \(\left(b^{n / 2}\right)^{2}=\left(b^{2}\right)^{n / 2}\), keep, along with the exponent \(n\) and the base \(b\), an additional state variable \(a\), and define the state transformation in such a way that the product \(a b^{n}\) is unchanged from state to state. At the beginning of the process \(a\) is taken to be 1 , and the answer is given by the value of \(a\) at the end of the process. In general, the technique of defining an invariant quantity that remains unchanged from state to state is a powerful way to think about the design of iterative algorithms.)
