# SICP 1.3.4 Procedures as returned values

*Published: 2020-07-02T21:43:40.000Z*

Today I have learned that John Carmack was also challenged by SICP:

100 pages into Structure and Interpretation of Computer Programs. I'm proud of myself for not skipping the Church Numerals exercise. Twitter

I still need to get so far.

## Exercise 1.40

```
(define (cubic a b c)
(lambda (x) (+
(* x x x)
(* a x x)
(* b x)
c
))
)
```

## Exercise 1.41

```
(define (inc x) (+ x 1))
(define (double f)
(lambda (x) (f (f x)))
)
```

## Exercise 1.42

```
(define (sqr x) (* x x))
(define (compose f g)
(lambda (x) (f (g x)))
)
((compose sqr inc) 6)
```

## Exercise 1.43

```
(define (repeated f n)
(if (= n 1)
f
(if (even? n)
(repeated (double f) (/ n 2))
(compose f (repeated f (- n 1)))
)
)
)
```

## Exercise 1.44

```
(define dx 0.000001)
(define (smooth f)
(lambda (x) (/ (+
(f x)
(f (+ x dx))
(f (- x dx))
) 3.0))
)
(repeated smooth 10)
```

# Exercise 1.45

```
(define tolerance 0.00001)
(define (fixed-point f first-guess)
(define (close-enough? v1 v2)
(< (abs (- v1 v2)) tolerance))
(define (try guess limit)
(display guess) (newline); to debug trace
(let ((next (f guess)))
(if (= limit 0)
(error "Reached iteration limit")
(if (close-enough? guess next)
next
(try next (- limit 1)))
)))
(try first-guess 1000)
)
(define (average-damp f)
(lambda (y) (/ (+ (f y) y) 2))
)
(define (npow x n)
(if (= n 1) x (* x (npow x (- n 1)) ))
)
(define (n-root x n)
(fixed-point
((repeated average-damp (/ n 2)) (lambda (y) (/ x (npow y (- n 1)))))
1
)
)
```

I saw on the internet solutions better than n / 2, but I would like to move on.

# Exercise 1.46

```
(define (iterative-improve good-enough next)
(define (iter guess)
(if (good-enough guess)
guess
(iter (next guess))
)
)
iter
)
(define (~= a b)
(< (abs (- a b)) 0.00001)
)
(define (sqrt x)
((iterative-improve
(lambda (y) (~= (* y y) x))
(lambda (y) (/ (+ y (/ x y)) 2))
) 1.0)
)
(define (fixed-point f first-guess)
((iterative-improve
(lambda (y) (~= (f y) y)); not very optimal because we computing f twice
f
) first-guess)
)
```

And I could move to the chapter about data structures.