# SICP 1.3.3 Procedures as general methods

Published: 2020-06-29T21:32:06.000Z

I'll attach solution from previous section to this post as it was the only exercise there.

## Exercise 1.34 Runtime error with types

``````(define (f g)
(g 2))
(f f) -> (f 2) -> (2 2)``````

2 is obviously not a function, so scheme gives error when you try to apply it as function.

## Exercise 1.35 Use fixed point procedure for finding golden ratio

So, let's divide both sides of equation Φ2 = Φ + 1. We get Φ = 1 + 1/Φ - and this is exactly the formula to find fixed point for.

``````(define tolerance 0.00001)
(define (fixed-point f first-guess)
(define (close-enough? v1 v2)
(< (abs (- v1 v2)) tolerance))
(define (try guess)
(let ((next (f guess)))
(if (close-enough? guess next)
next
(try next))))
(try first-guess))
(fixed-point (lambda (x) (+ 1 (/ 1 x))) 1.0)``````

## Exercise 1.36 Printing sequence of approximations

``````(define tolerance 0.00001)
(define (fixed-point f first-guess)
(define (close-enough? v1 v2)
(< (abs (- v1 v2)) tolerance))
(define (try guess)
(let ((next (f guess)))
(display "Approximation: ") (display next)(newline)
(if (close-enough? guess next)
next
(try next))))
(try first-guess))
(fixed-point (lambda (x) (/ (log 1000) (log x))) 2.0)``````

## Exercise 1.37 Golden ratio from continued fraction

``````(define (cont-frac n d k)
(define (iter i)
(/ (n i) (if (= i k)
(d i)
(+ (d i) (iter (+ i 1)))
))
)
(iter 1)
)

(define (golden k)
(/ 1 (cont-frac (lambda (i) 1.0)
(lambda (i) 1.0)
k))
)

(golden 12)
;Value: 1.6180555555555558
``````

Iterative continuous fraction:

``````(define (cont-frac n d k)
(define (iter i result)
(if (= i 0)
result
(iter
(- i 1)
(/ (n i) (+ (d i) result))
)
)
)
(iter k 0)
)``````

## Exercise 1.38 Euler number

``````(define (e k)
(+ 2 (cont-frac
(lambda (i) 1.0)
(lambda (i)
(if (= (remainder i 3) 2)
(* (+ 1 (floor (/ i 3))) 2)
1.0
)
)
k
))
)

(e 15)
;Value: 2.718281828470584``````

## Exercise 1.39 Continued fraction tangent

``````(define (tan x k)
(let ((n_sqr (- 0 (* x x))))
(/ x (+ 1 (cont-frac
(lambda (i) n_sqr)
(lambda (i) (+ 1 (* i 2)))
k
)))
)
)

(tan (/ pi 4) 100)
;Value: .9999999732051038

(tan (/ pi 2) 1000)
;Value: 37320539.58514773

(tan 0 100)
;Value: 0``````

Approximately so.