# SICP 1.3.1 Procedures as arguments

Published: 2020-06-27T17:32:06.000Z

Solving more exercises, trying not to skip to be able to learn next chapters better.

## Exercise 1.29 Simpson rule integration

``````(define (cube x) (* x x x))
(define (integral f a b dx)
(define (add-dx x) (+ x dx))
(* (sum f (+ a (/ dx 2.0)) add-dx b)
dx))

(define (sum term a next b)
(if (> a b)
0
(+ (term a)
(sum term (next a) next b))))

(integral cube 0 1 0.01)
; .24998750000000042
(integral cube 0 1 0.001)
;.249999875000001

(define (simpson-rule-integral f a b n)
(define (inc x) (+ x 1))
(define (term i)
(cond
((= i 0) (f a))
((= i n) (f b))
((even? i) (* 2 (f (+ a (* (/ (- b a) n) i)))))
(else (* 4 (f (+ a (* (/ (- b a) n) i)))))
)
)
(* (sum term 0 inc n)
(/ (- b a) n 3))
)

(simpson-rule-integral cube 0 1 10)
;Value: 1/4
``````

Seems to work very precise. Gives rational numbers, wow. Even when try with different functions:

``````(define (identity x) x)
(simpson-rule-integral identity 0 1 10)
;Value: 1/2``````

## Exercise 1.30 Iterative summation

``````(define (sum term a next b)
(define (iter a result)
(if (> a b)
result
(iter (next a) (+ (term a) result))))
(iter a 0))``````

## Exercise 1.31 Product

``````(define (product term a next b)
(define (iter a result)
(if (> a b)
result
(iter (next a) (* (term a) result))))
(iter a 1))``````

Factorial:

``````(define (inc i) (+ i 1))
(define (identity x) x)
(define (factorial n) (product identity 1 inc n))``````

Approximating Pi:

``````(define (pi precision)
(define (term i)
(define k (* i 2))
(/ (* k (+ k 2)) (square (+ k 1)))
)
(* 4.0 (product term 1 inc precision))
)
(pi 100)
;Value: 3.1493784731686008``````

Recursive product:

``````(define (product term a next b)
(if (> a b)
1
(* (term a) (product term (next a) next b))
)
)``````

## Exercise 1.32: Accumulate

``````(define (accumulate combiner null-value term a next b)
(if (> a b)
null-value
(combiner
(term a)
(accumulate combiner null-value term (next a) next b)
)
)
)

(define (sum term a next b)
(accumulate + 0 term a next b)
)

(define (product term a next b)
(accumulate * 1 term a next b)
)``````

Iterative:

``````(define (accumulate combiner null-value term a next b)
(define (iter a result)
(if (> a b)
result
(iter (next a) (combiner (term a) result))
)
)
(iter a null-value)
)``````

## Exercise 1.33 Filtered accumulate

``````(define (filtered-accumulate combiner null-value term a next b filter)
(define (iter a result)
(if (> a b)
result
(if (filter a)
(iter (next a) result)
(iter (next a) (combiner (term a) result))
)
)
)
(iter a null-value)
)

(define (sum-of-squares-of-primes-in-interval a b)
(filtered-accumulate + 0 sqr a inc b prime?)
)

(define (product-of-relatively-primes-to n)
(define (!rel-prime x)
(not (= (gcd x n) 1))
)
(filtered-accumulate * 1 identity 1 inc n !rel-prime)
)``````