Average Error: 29.7 → 7.3
Time: 6.9s
Precision: 64
Internal Precision: 128
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}\]

Error

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 29.7

    \[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]

  2. Simplified7.3

    \[\leadsto \color{blue}{\sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}}\]

  3. Final simplification7.3

    \[\leadsto \sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}\]

Reproduce

herbie shell --seed 2019005 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))

Details

Time bar (total: 6.6s)Debug log

sample52.0ms

Algorithm
intervals

simplify188.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

188.0ms
(sqrt (fabs (/ (- (* a a) (* b b)) (* a a))))

prune9.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 5.7b

localize13.0ms

Local error

Found 2 expressions with local error:

5.7b
(sqrt (fabs (- 1 (* (/ b a) (/ b a)))))
0.4b
(* (/ b a) (/ b a))

rewrite4.0ms

Algorithm
rewrite-expression-head
Counts
2 → 37
Calls

2 calls. Slowest were:

3.0ms
(* (/ b a) (/ b a))
1.0ms
(sqrt (fabs (- 1 (* (/ b a) (/ b a)))))

series50.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

34.0ms
(* (/ b a) (/ b a))
16.0ms
(sqrt (fabs (- 1 (* (/ b a) (/ b a)))))

simplify1.6s

Counts
25 → 43
Calls

25 calls. Slowest were:

1.3s
(* (* (* (/ b a) (/ b a)) (/ b a)) (* (* (/ b a) (/ b a)) (/ b a)))
59.0ms
(sqrt (fabs (- 1 (* (/ b a) (/ b a)))))
59.0ms
(sqrt (fabs (- 1 (/ (pow b 2) (pow a 2)))))

prune339.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 5.7b

localize17.0ms

Local error

Found 4 expressions with local error:

5.7b
(sqrt (fabs (- 1 (* (* (/ b a) (sqrt (/ b a))) (sqrt (/ b a))))))
0.3b
(* (/ b a) (sqrt (/ b a)))
0.3b
(* (* (/ b a) (sqrt (/ b a))) (sqrt (/ b a)))
0.2b
(sqrt (/ b a))

rewrite16.0ms

Algorithm
rewrite-expression-head
Counts
4 → 85
Calls

4 calls. Slowest were:

10.0ms
(* (* (/ b a) (sqrt (/ b a))) (sqrt (/ b a)))
4.0ms
(* (/ b a) (sqrt (/ b a)))
1.0ms
(sqrt (fabs (- 1 (* (* (/ b a) (sqrt (/ b a))) (sqrt (/ b a))))))

series144.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

69.0ms
(* (/ b a) (sqrt (/ b a)))
39.0ms
(sqrt (/ b a))
24.0ms
(* (* (/ b a) (sqrt (/ b a))) (sqrt (/ b a)))
12.0ms
(sqrt (fabs (- 1 (* (* (/ b a) (sqrt (/ b a))) (sqrt (/ b a))))))

simplify1.8s

Counts
70 → 97
Calls

70 calls. Slowest were:

494.0ms
(* (* (* (* (/ b a) (/ b a)) (/ b a)) (* (* (sqrt (/ b a)) (sqrt (/ b a))) (sqrt (/ b a)))) (* (* (sqrt (/ b a)) (sqrt (/ b a))) (sqrt (/ b a))))
247.0ms
(* (* (* (* (/ b a) (sqrt (/ b a))) (* (/ b a) (sqrt (/ b a)))) (* (/ b a) (sqrt (/ b a)))) (* (* (sqrt (/ b a)) (sqrt (/ b a))) (sqrt (/ b a))))
169.0ms
(* (* (* (/ b a) (/ b a)) (/ b a)) (* (* (sqrt (/ b a)) (sqrt (/ b a))) (sqrt (/ b a))))

prune1.0s

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 5.7b

regimes61.0ms

Accuracy

0% (0.0b remaining)

Error of 7.3b against oracle of 7.3b and baseline of 7.3b

bsearch1.0ms

end0.0ms

sample1.2s

Algorithm
intervals