some solutions for MIT Press's "structure and interpretation of computer programs", I guess
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(define (search f neg-point pos-point)
(let ((midpoint (average neg-point pos-point)))
(if (close-enough? neg-point pos-point)
(let ((test-value (f midpoint)))
(cond ((positive? test-value)
(search f neg-point midpoint))
((negative? test-value)
(search f midpoint pos-point))
(else midpoint))))))
(define (average x y)
(/ (+ x y) 2))
(define (close-enough? x y)
(< (abs (- x y)) 0.001))
(define (half-interval-method f a b)
(let ((a-value (f a))
(b-value (f b)))
(cond ((and (negative? a-value) (positive? b-value))
(search f a b))
((and (negative? b-value) (positive? a-value))
(search f b a))
(error "Values are not of opposite sign" a b)))))
(define tolerance 0.00001)
(define (fixed-point f first-guess)
(define (close-enough? v1 v2)
(< (abs (- v1 v2))
(define (try guess)
(let ((next (f guess)))
(if (close-enough? guess next)
(try next))))
(try first-guess))
(define (nice-sqrt x)
(fixed-point (lambda (y) (average y (/ x y)))