How to find the minimum positive number that added to 1.0 gives something larger?

Question

While translating some Fortran to Scheme/Racket I have come across the function:

; EPSILON(X)      The  least  positive  number  that added
;                 to 1 returns a number that is greater than 1

How do I find the number in Scheme?

dyoo · Accepted Answer · 2012-07-10T16:48:45.773

6

#lang racket/base

;; http://en.wikipedia.org/wiki/Machine_epsilon
;; approximates the machine epsilon

(require racket/flonum)

(define (compute-machine-epsilon)
  (let loop ([n 1.0])
    (define next-n (fl/ n 2.0))
    (if (fl= 1.0 (fl+ 1.0 next-n))
        n
        (loop next-n))))

edited Jul 10 '12 at 16:48

answered Jul 09 '12 at 19:12

dyoo

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Correct, the only thing to look out for is that this assumes we have a base 2 number. There may be cases in other programming languages that it is a BCD. – Thorsten S. Jul 11 '12 at 14:48

score 5 · Answer 2 · answered Jul 09 '12 at 18:25

Assuming you're using IEEE-754 floating-point (which may not be the case in Scheme, I don't know), then the machine epsilon is well known: for double-precision arithmetic, it's 1.11e-16.

For other platforms or floating-point implementations, Wikipedia shows the formula to compute it as (in Haskell):

main = print . last . map (subtract 1) . takeWhile (/= 1) . map (+ 1) . iterate (/2) $ 1

score 4 · Answer 3 · answered Jul 09 '12 at 21:29

4

This is not a new answer -- it just bothers me that Danny's code makes it look like it's hard to do this kind of thing... it could be simplified to

(let loop ([n 1.0])
  (if (= 1 (+ 1 (/ n 2)))
    n
    (loop (/ n 2))))

answered Jul 09 '12 at 21:29

Eli Barzilay

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Maybe I should talk Neil into adding machine-epsilon to racket/math ? – soegaard Jul 09 '12 at 22:02
IIUC, there are a whole bunch of these useful things, right? He might have some of them in the plot library already. – Eli Barzilay Jul 09 '12 at 22:13
Could be. The Science Collection has machine-epsilon already. – soegaard Jul 09 '12 at 22:30

score 1 · Answer 4 · answered Jul 13 '18 at 19:15

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The existing answers are needlessly complicated, and might lead to non-portable code.

The correct answer is, as documented in Racket's math/flonum package is to use the epsilon.0 value, which will be correct for the current machine, and which won't require you to compute it by hand:

 (require math/flonum)  
 epsilon.0

answered Jul 13 '18 at 19:15

John Doucette

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Good point. The math/flonum library didn’t exist back in ‘12 though. I guess I succeeded in talking Neil into adding it :-) – soegaard Jul 13 '18 at 19:54

How to find the minimum positive number that added to 1.0 gives something larger?

4 Answers4