Average Error: 29.2 → 7.3
Time: 7.6s
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 29.2

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

  2. Initial simplification7.3

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

  3. Final simplification7.3

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

Reproduce

herbie shell --seed 2018365 +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: 5.2s)Debug log

start57.0ms

Algorithm
intervals

setup233.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 7.8b

localize13.0ms

Local error

Found 2 expressions with local error:

7.8b
(sqrt (fabs (- 1 (* (/ b a) (/ b a)))))
0.4b
(* (/ b a) (/ b a))

rewrite4.0ms

Algorithm
rewrite-expression-head
Counts
2 → 41
Calls

2 calls. Slowest were:

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

series39.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

23.0ms
(* (/ b a) (/ b a))
16.0ms
(sqrt (fabs (- 1 (* (/ b a) (/ b a)))))

simplify1.3s

Counts
25 → 47
Calls

25 calls. Slowest were:

815.0ms
(* (* (* (/ b a) (/ b a)) (/ b a)) (* (* (/ b a) (/ b a)) (/ b a)))
71.0ms
(sqrt (fabs (- 1 (/ (pow b 2) (pow a 2)))))
67.0ms
(sqrt (* (cbrt (fabs (- 1 (* (/ b a) (/ b a))))) (cbrt (fabs (- 1 (* (/ b a) (/ b a)))))))

prune545.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 7.7b

localize25.0ms

Local error

Found 3 expressions with local error:

7.8b
(sqrt (fabs (- 1 (/ (* b (/ b a)) a))))
7.3b
(/ (* b (/ b a)) a)
0.2b
(* b (/ b a))

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
3 → 43
Calls

3 calls. Slowest were:

6.0ms
(/ (* b (/ b a)) a)
2.0ms
(sqrt (fabs (- 1 (/ (* b (/ b a)) a))))
2.0ms
(* b (/ b a))

series60.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

22.0ms
(/ (* b (/ b a)) a)
22.0ms
(sqrt (fabs (- 1 (/ (* b (/ b a)) a))))
15.0ms
(* b (/ b a))

simplify343.0ms

Counts
18 → 52
Calls

18 calls. Slowest were:

68.0ms
(sqrt (* (cbrt (fabs (- 1 (/ (* b (/ b a)) a)))) (cbrt (fabs (- 1 (/ (* b (/ b a)) a))))))
64.0ms
(sqrt (fabs (- 1 (/ (pow b 2) (pow a 2)))))
46.0ms
(sqrt (fabs (- 1 (/ (pow b 2) (pow a 2)))))

prune558.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 7.7b

localize19.0ms

Local error

Found 4 expressions with local error:

13.2b
(cbrt (* (* (sqrt (fabs (- 1 (/ (* b (/ b a)) a)))) (sqrt (fabs (- 1 (/ (* b (/ b a)) a))))) (sqrt (fabs (- 1 (/ (* b (/ b a)) a))))))
7.8b
(sqrt (fabs (- 1 (/ (* b (/ b a)) a))))
7.8b
(sqrt (fabs (- 1 (/ (* b (/ b a)) a))))
7.8b
(sqrt (fabs (- 1 (/ (* b (/ b a)) a))))

rewrite22.0ms

Algorithm
rewrite-expression-head
Counts
4 → 54
Calls

4 calls. Slowest were:

17.0ms
(cbrt (* (* (sqrt (fabs (- 1 (/ (* b (/ b a)) a)))) (sqrt (fabs (- 1 (/ (* b (/ b a)) a))))) (sqrt (fabs (- 1 (/ (* b (/ b a)) a))))))
2.0ms
(sqrt (fabs (- 1 (/ (* b (/ b a)) a))))
1.0ms
(sqrt (fabs (- 1 (/ (* b (/ b a)) a))))

series63.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

19.0ms
(sqrt (fabs (- 1 (/ (* b (/ b a)) a))))
17.0ms
(cbrt (* (* (sqrt (fabs (- 1 (/ (* b (/ b a)) a)))) (sqrt (fabs (- 1 (/ (* b (/ b a)) a))))) (sqrt (fabs (- 1 (/ (* b (/ b a)) a))))))
14.0ms
(sqrt (fabs (- 1 (/ (* b (/ b a)) a))))
12.0ms
(sqrt (fabs (- 1 (/ (* b (/ b a)) a))))

simplify924.0ms

Counts
26 → 66
Calls

26 calls. Slowest were:

57.0ms
(sqrt (fabs (- 1 (/ (pow b 2) (pow a 2)))))
55.0ms
(sqrt (fabs (- 1 (/ (pow b 2) (pow a 2)))))
52.0ms
(cbrt (* (sqrt (fabs (- 1 (/ (* b (/ b a)) a)))) (sqrt (fabs (- 1 (/ (* b (/ b a)) a))))))

prune895.0ms

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 7.7b

regimes88.0ms

Accuracy

0% (0.0b remaining)

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

bsearch2.0ms