Average Error: 29.4 → 7.4
Time: 5.1s
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 29.4

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

  2. Initial simplification7.4

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

  3. Final simplification7.4

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

Reproduce

herbie shell --seed 2018360 +o rules:numerics
(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: 3.0s)Debug log

start72.0ms

Algorithm
intervals

setup244.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 8.9b

localize19.0ms

Local error

Found 2 expressions with local error:

9.0b
(sqrt (fabs (- 1 (* (/ b a) (/ b a)))))
0.3b
(* (/ b a) (/ b a))

rewrite8.0ms

Algorithm
rewrite-expression-head
Counts
2 → 41
Calls

2 calls. Slowest were:

7.0ms
(* (/ b a) (/ b a))
1.0ms
(sqrt (fabs (- 1 (* (/ b a) (/ b a)))))

series27.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

14.0ms
(* (/ b a) (/ b a))
12.0ms
(sqrt (fabs (- 1 (* (/ b a) (/ b a)))))

simplify1.3s

Counts
25 → 47
Calls

25 calls. Slowest were:

923.0ms
(* (* (* (/ b a) (/ b a)) (/ b a)) (* (* (/ b a) (/ b a)) (/ b a)))
61.0ms
(sqrt (fabs (- 1 (/ (pow b 2) (pow a 2)))))
53.0ms
(sqrt (fabs (- 1 (/ (pow b 2) (pow a 2)))))

prune429.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 8.9b

localize16.0ms

Local error

Found 3 expressions with local error:

9.0b
(sqrt (fabs (- 1 (/ (* (/ b a) b) a))))
8.9b
(/ (* (/ b a) b) a)
0.2b
(* (/ b a) b)

rewrite16.0ms

Algorithm
rewrite-expression-head
Counts
3 → 43
Calls

3 calls. Slowest were:

10.0ms
(/ (* (/ b a) b) a)
4.0ms
(* (/ b a) b)
2.0ms
(sqrt (fabs (- 1 (/ (* (/ b a) b) a))))

series41.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

14.0ms
(sqrt (fabs (- 1 (/ (* (/ b a) b) a))))
14.0ms
(/ (* (/ b a) b) a)
13.0ms
(* (/ b a) b)

simplify316.0ms

Counts
18 → 52
Calls

18 calls. Slowest were:

65.0ms
(sqrt (fabs (- 1 (/ (pow b 2) (pow a 2)))))
54.0ms
(sqrt (fabs (- 1 (/ (* (/ b a) b) a))))
48.0ms
(sqrt (sqrt (fabs (- 1 (/ (* (/ b a) b) a)))))

prune482.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 8.9b

regimes72.0ms

Accuracy

0% (0.0b remaining)

Error of 7.4b against oracle of 7.4b and baseline of 7.4b

bsearch1.0ms