Average Error: 29.2 → 7.6
Time: 7.5s
Precision: 64
Internal Precision: 128
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|1 - \frac{\frac{b}{a} \cdot b}{a}\right|}\]

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. Simplified7.6

    \[\leadsto \color{blue}{\sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}}\]
  3. Using strategy rm

  4. Applied associate-*r/7.6

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

  5. Final simplification7.6

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

Reproduce

herbie shell --seed 2019004 
(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: 7.2s)Debug log

sample43.0ms

Algorithm
intervals

simplify216.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

216.0ms
(sqrt (fabs (/ (- (* a a) (* b b)) (* a a))))

prune11.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 8.2b

localize23.0ms

Local error

Found 3 expressions with local error:

8.3b
(sqrt (fabs (- 1 (* (/ b a) (/ b a)))))
0.4b
(* (/ b a) (/ b a))
0.0b
(- 1 (* (/ b a) (/ b a)))

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
3 → 48
Calls

3 calls. Slowest were:

6.0ms
(* (/ b a) (/ b a))
2.0ms
(- 1 (* (/ b a) (/ b a)))
1.0ms
(sqrt (fabs (- 1 (* (/ b a) (/ b a)))))

series54.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

21.0ms
(- 1 (* (/ b a) (/ b a)))
20.0ms
(* (/ b a) (/ b a))
14.0ms
(sqrt (fabs (- 1 (* (/ b a) (/ b a)))))

simplify1.4s

Counts
28 → 57
Calls

28 calls. Slowest were:

895.0ms
(* (* (* (/ b a) (/ b a)) (/ b a)) (* (* (/ b a) (/ b a)) (/ b a)))
63.0ms
(sqrt (fabs (- 1 (/ (pow b 2) (pow a 2)))))
62.0ms
(sqrt (fabs (- 1 (/ (pow b 2) (pow a 2)))))

prune576.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 8.2b

localize25.0ms

Local error

Found 4 expressions with local error:

8.3b
(sqrt (fabs (- 1 (/ (* (/ b a) b) a))))
6.9b
(/ (* (/ b a) b) a)
0.2b
(* (/ b a) b)
0.0b
(- 1 (/ (* (/ b a) b) a))

rewrite13.0ms

Algorithm
rewrite-expression-head
Counts
4 → 48
Calls

4 calls. Slowest were:

9.0ms
(/ (* (/ b a) b) a)
2.0ms
(* (/ b a) b)
1.0ms
(sqrt (fabs (- 1 (/ (* (/ b a) b) a))))

series58.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

18.0ms
(- 1 (/ (* (/ b a) b) a))
16.0ms
(sqrt (fabs (- 1 (/ (* (/ b a) b) a))))
14.0ms
(/ (* (/ b a) b) a)
10.0ms
(* (/ b a) b)

simplify390.0ms

Counts
21 → 60
Calls

21 calls. Slowest were:

60.0ms
(sqrt (fabs (- 1 (/ (* (/ b a) b) a))))
55.0ms
(sqrt (sqrt (fabs (- 1 (/ (* (/ b a) b) a)))))
42.0ms
(sqrt (fabs (- 1 (/ (pow b 2) (pow a 2)))))

prune575.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 8.2b

localize21.0ms

Local error

Found 4 expressions with local error:

8.3b
(sqrt (fabs (- 1 (* b (* (/ 1 a) (/ b a))))))
7.4b
(* b (* (/ 1 a) (/ b a)))
0.2b
(* (/ 1 a) (/ b a))
0.0b
(- 1 (* b (* (/ 1 a) (/ b a))))

rewrite8.0ms

Algorithm
rewrite-expression-head
Counts
4 → 57
Calls

4 calls. Slowest were:

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

series78.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

32.0ms
(- 1 (* b (* (/ 1 a) (/ b a))))
18.0ms
(* b (* (/ 1 a) (/ b a)))
17.0ms
(* (/ 1 a) (/ b a))
12.0ms
(sqrt (fabs (- 1 (* b (* (/ 1 a) (/ b a))))))

simplify1.7s

Counts
30 → 69
Calls

30 calls. Slowest were:

530.0ms
(sqrt (fabs (- 1 (* b (* (/ 1 a) (/ b a))))))
360.0ms
(* (* (* (/ 1 a) (/ 1 a)) (/ 1 a)) (* (* (/ b a) (/ b a)) (/ b a)))
294.0ms
(sqrt (* (cbrt (fabs (- 1 (* b (* (/ 1 a) (/ b a)))))) (cbrt (fabs (- 1 (* b (* (/ 1 a) (/ b a))))))))

prune599.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 8.2b

regimes47.0ms

Accuracy

0% (0.0b remaining)

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

bsearch2.0ms

end0.0ms

sample1.3s

Algorithm
intervals