Average Error: 29.8 → 7.6
Time: 7.8s
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 29.8

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

  2. Simplified7.6

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

  3. Final simplification7.6

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

Reproduce

herbie shell --seed 2019008 +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: 7.3s)Debug log

sample55.0ms

Algorithm
intervals

simplify290.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

290.0ms
(sqrt (fabs (/ (- (* a a) (* b b)) (* a a))))

prune18.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 6.4b

localize26.0ms

Local error

Found 2 expressions with local error:

6.4b
(sqrt (fabs (- 1 (* (/ b a) (/ b a)))))
0.4b
(* (/ b a) (/ b a))

rewrite8.0ms

Algorithm
rewrite-expression-head
Counts
2 → 41
Calls

2 calls. Slowest were:

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

series50.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

27.0ms
(sqrt (fabs (- 1 (* (/ b a) (/ b a)))))
22.0ms
(* (/ b a) (/ b a))

simplify1.3s

Counts
25 → 47
Calls

25 calls. Slowest were:

869.0ms
(* (* (* (/ b a) (/ b a)) (/ b a)) (* (* (/ b a) (/ b a)) (/ b a)))
66.0ms
(sqrt (* (cbrt (fabs (- 1 (* (/ b a) (/ b a))))) (cbrt (fabs (- 1 (* (/ b a) (/ b a)))))))
65.0ms
(sqrt (fabs (- 1 (/ (pow b 2) (pow a 2)))))

prune435.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 6.4b

localize21.0ms

Local error

Found 3 expressions with local error:

7.9b
(/ (* (/ b a) b) a)
6.4b
(sqrt (fabs (- 1 (/ (* (/ b a) b) a))))
0.2b
(* (/ b a) b)

rewrite16.0ms

Algorithm
rewrite-expression-head
Counts
3 → 43
Calls

3 calls. Slowest were:

9.0ms
(/ (* (/ b a) b) a)
4.0ms
(* (/ b a) b)
2.0ms
(sqrt (fabs (- 1 (/ (* (/ b a) b) a))))

series64.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

26.0ms
(/ (* (/ b a) b) a)
21.0ms
(sqrt (fabs (- 1 (/ (* (/ b a) b) a))))
17.0ms
(* (/ b a) b)

simplify323.0ms

Counts
18 → 52
Calls

18 calls. Slowest were:

64.0ms
(sqrt (fabs (- 1 (/ (pow b 2) (pow a 2)))))
52.0ms
(sqrt (fabs (- 1 (/ (pow b 2) (pow a 2)))))
49.0ms
(sqrt (sqrt (fabs (- 1 (/ (* (/ b a) b) a)))))

prune598.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 6.4b

localize16.0ms

Local error

Found 4 expressions with local error:

7.9b
(/ (* (/ b a) b) a)
6.7b
(cbrt (fabs (- 1 (/ (* (/ b a) b) a))))
6.7b
(cbrt (fabs (- 1 (* (/ b a) (/ b a)))))
0.4b
(* (/ b a) (/ b a))

rewrite9.0ms

Algorithm
rewrite-expression-head
Counts
4 → 67
Calls

4 calls. Slowest were:

4.0ms
(/ (* (/ b a) b) a)
3.0ms
(* (/ b a) (/ b a))
1.0ms
(cbrt (fabs (- 1 (/ (* (/ b a) b) a))))

series165.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

75.0ms
(cbrt (fabs (- 1 (/ (* (/ b a) b) a))))
59.0ms
(cbrt (fabs (- 1 (* (/ b a) (/ b a)))))
17.0ms
(* (/ b a) (/ b a))
14.0ms
(/ (* (/ b a) b) a)

simplify1.3s

Counts
35 → 79
Calls

35 calls. Slowest were:

793.0ms
(* (* (* (/ b a) (/ b a)) (/ b a)) (* (* (/ b a) (/ b a)) (/ b a)))
62.0ms
(cbrt (sqrt (fabs (- 1 (/ (* (/ b a) b) a)))))
61.0ms
(pow (fabs (- 1 (/ (pow b 2) (pow a 2)))) 1/3)

prune1.2s

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 6.4b

regimes75.0ms

Accuracy

0% (0.0b remaining)

Error of 7.6b against oracle of 7.6b and baseline of 7.6b

bsearch2.0ms

end0.0ms

sample1.4s

Algorithm
intervals