Lets first take an example of two intervals, and , and compute
and using the procedures
par2 given in exercise.
1 2 3 4 5 6 7 8 > (define I1 (make-center-percent 25 1.5)) > (define I2 (make-center-percent 75 0.75)) > (define rs1 (par1 I1 I2)) > (define rs2 (par2 I1 I2)) > (display-interval rs1) [18.15998452012384,19.35544164037855] > (display-interval rs2) [18.50370662460568,18.995897832817338]
Clearly both results in different answers.
Lets see if the formulas are actually equivalent when applied to intervals.
By algebra, we can simplify it as:
This will be true for intervals, iff:
- for any interval and . If we look closely this actually means that .
- , for any interval .
Lets see using Alyssa’s “interval arithmetic” that above is correct or not:
1 2 3 4 5 6 7 8 9 > (define I1 (make-center-percent 25 1.5)) > (display-interval I1) [24.625,25.375] > (display-interval div) [0.9704433497536946,1.030456852791878] > (define reverse (div-interval (make-interval 1.0 1.0) I1)) > (define reverse_reverse (div-interval (make-interval 1.0 1.0) reverse)) > (display-interval reverse_reverse) [24.625000000000004,25.375]
As we can see by interval arithmetic that . Thus the algebraic equivalent is not same as interval equivalent. However , for any interval holds true which means this farmulae alegebraic equivalent is same as arithmetic equivalent.