Average Error: 28.9 → 7.4
Time: 5.5s
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 28.9

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

  2. Simplified7.4

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

  3. Final simplification7.4

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

Reproduce

herbie shell --seed 2019007 +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: 5.1s)Debug log

sample34.0ms

Algorithm
intervals

simplify183.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

183.0ms
(sqrt (fabs (/ (- (* a a) (* b b)) (* a a))))

prune10.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 7.0b

localize28.0ms

Local error

Found 3 expressions with local error:

7.1b
(sqrt (fabs (- 1 (* (/ b a) (/ b a)))))
0.4b
(* (/ b a) (/ b a))
0.0b
(- 1 (* (/ b a) (/ b a)))

rewrite12.0ms

Algorithm
rewrite-expression-head
Counts
3 → 54
Calls

3 calls. Slowest were:

7.0ms
(* (/ b a) (/ b a))
2.0ms
(- 1 (* (/ b a) (/ b a)))
2.0ms
(sqrt (fabs (- 1 (* (/ b a) (/ b a)))))

series75.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

30.0ms
(- 1 (* (/ b a) (/ b a)))
23.0ms
(* (/ b a) (/ b a))
22.0ms
(sqrt (fabs (- 1 (* (/ b a) (/ b a)))))

simplify1.3s

Counts
28 → 63
Calls

28 calls. Slowest were:

792.0ms
(* (* (* (/ b a) (/ b a)) (/ b a)) (* (* (/ b a) (/ b a)) (/ b a)))
65.0ms
(sqrt (* (cbrt (fabs (- 1 (* (/ b a) (/ b a))))) (cbrt (fabs (- 1 (* (/ b a) (/ b a)))))))
60.0ms
(sqrt (fabs (- 1 (/ (pow b 2) (pow a 2)))))

prune553.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 7.0b

localize13.0ms

Local error

Found 4 expressions with local error:

27.1b
(cbrt (pow (/ b a) 6))
7.1b
(sqrt (fabs (- 1 (cbrt (pow (/ b a) 6)))))
0.4b
(pow (/ b a) 6)
0.0b
(- 1 (cbrt (pow (/ b a) 6)))

rewrite5.0ms

Algorithm
rewrite-expression-head
Counts
4 → 60
Calls

4 calls. Slowest were:

2.0ms
(- 1 (cbrt (pow (/ b a) 6)))
1.0ms
(pow (/ b a) 6)
1.0ms
(cbrt (pow (/ b a) 6))

series111.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

39.0ms
(pow (/ b a) 6)
28.0ms
(- 1 (cbrt (pow (/ b a) 6)))
23.0ms
(cbrt (pow (/ b a) 6))
20.0ms
(sqrt (fabs (- 1 (cbrt (pow (/ b a) 6)))))

simplify306.0ms

Counts
29 → 72
Calls

29 calls. Slowest were:

52.0ms
(sqrt (fabs (- 1 (/ (pow b 2) (pow a 2)))))
46.0ms
(sqrt (fabs (- 1 (/ (pow b 2) (pow a 2)))))
41.0ms
(- 1 (/ (pow b 2) (pow a 2)))

prune729.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 7.0b

regimes86.0ms

Accuracy

0% (0.0b remaining)

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

bsearch2.0ms

end0.0ms

sample1.7s

Algorithm
intervals