Average Error: 29.8 → 7.6
Time: 7.2s
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 
(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: 6.7s)Debug log

sample63.0ms

Algorithm
intervals

simplify181.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

181.0ms
(sqrt (fabs (/ (- (* a a) (* b b)) (* a a))))

prune19.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 6.4b

localize30.0ms

Local error

Found 2 expressions with local error:

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

rewrite9.0ms

Algorithm
rewrite-expression-head
Counts
2 → 37
Calls

2 calls. Slowest were:

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

series28.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

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

simplify1.1s

Counts
25 → 43
Calls

25 calls. Slowest were:

788.0ms
(* (* (* (/ b a) (/ b a)) (/ b a)) (* (* (/ b a) (/ b a)) (/ b a)))
58.0ms
(sqrt (* (cbrt (fabs (- 1 (* (/ b a) (/ b a))))) (cbrt (fabs (- 1 (* (/ b a) (/ b a)))))))
57.0ms
(sqrt (fabs (- 1 (/ (pow b 2) (pow a 2)))))

prune368.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 6.4b

localize25.0ms

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 → 37
Calls

3 calls. Slowest were:

9.0ms
(/ (* (/ b a) b) a)
4.0ms
(* (/ b a) b)
1.0ms
(sqrt (fabs (- 1 (/ (* (/ b a) b) a))))

series41.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

19.0ms
(/ (* (/ b a) b) a)
12.0ms
(sqrt (fabs (- 1 (/ (* (/ b a) b) a))))
10.0ms
(* (/ b a) b)

simplify303.0ms

Counts
18 → 46
Calls

18 calls. Slowest were:

61.0ms
(sqrt (fabs (- 1 (/ (pow b 2) (pow a 2)))))
49.0ms
(sqrt (* (cbrt (fabs (- 1 (/ (* (/ b a) b) a)))) (cbrt (fabs (- 1 (/ (* (/ b a) b) a))))))
43.0ms
(sqrt (fabs (- 1 (/ (pow b 2) (pow a 2)))))

prune487.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 6.4b

localize25.0ms

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))

rewrite17.0ms

Algorithm
rewrite-expression-head
Counts
4 → 59
Calls

4 calls. Slowest were:

9.0ms
(/ (* (/ b a) b) a)
5.0ms
(* (/ b a) (/ b a))
1.0ms
(cbrt (fabs (- 1 (/ (* (/ b a) b) a))))

series219.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

87.0ms
(cbrt (fabs (- 1 (* (/ b a) (/ b a)))))
85.0ms
(cbrt (fabs (- 1 (/ (* (/ b a) b) a))))
24.0ms
(* (/ b a) (/ b a))
23.0ms
(/ (* (/ b a) b) a)

simplify1.4s

Counts
35 → 71
Calls

35 calls. Slowest were:

845.0ms
(* (* (* (/ b a) (/ b a)) (/ b a)) (* (* (/ b a) (/ b a)) (/ b a)))
63.0ms
(cbrt (* (cbrt (fabs (- 1 (/ (* (/ b a) b) a)))) (cbrt (fabs (- 1 (/ (* (/ b a) b) a))))))
62.0ms
(pow (fabs (- 1 (/ (pow b 2) (pow a 2)))) 1/3)

prune917.0ms

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 6.4b

regimes92.0ms

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