Average Error: 29.2 → 7.3
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 29.2

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

  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 
(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

start53.0ms

Algorithm
intervals

setup239.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 7.8b

localize26.0ms

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 → 37
Calls

2 calls. Slowest were:

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

series33.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

18.0ms
(* (/ b a) (/ b a))
15.0ms
(sqrt (fabs (- 1 (* (/ b a) (/ b a)))))

simplify1.4s

Counts
25 → 43
Calls

25 calls. Slowest were:

1.0s
(* (* (* (/ b a) (/ b a)) (/ b a)) (* (* (/ b a) (/ b a)) (/ b a)))
60.0ms
(sqrt (fabs (- 1 (* (/ b a) (/ b a)))))
58.0ms
(sqrt (* (cbrt (fabs (- 1 (* (/ b a) (/ b a))))) (cbrt (fabs (- 1 (* (/ b a) (/ b a)))))))

prune500.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 7.7b

localize12.0ms

Local error

Found 3 expressions with local error:

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

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
3 → 37
Calls

3 calls. Slowest were:

4.0ms
(/ (* (/ b a) b) a)
2.0ms
(* (/ b a) b)
1.0ms
(sqrt (fabs (- 1 (/ (* (/ b a) b) a))))

series51.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

21.0ms
(/ (* (/ b a) b) a)
16.0ms
(* (/ b a) b)
13.0ms
(sqrt (fabs (- 1 (/ (* (/ b a) b) a))))

simplify413.0ms

Counts
18 → 46
Calls

18 calls. Slowest were:

70.0ms
(sqrt (* (cbrt (fabs (- 1 (/ (* (/ b a) b) a)))) (cbrt (fabs (- 1 (/ (* (/ b a) b) a))))))
64.0ms
(sqrt (sqrt (fabs (- 1 (/ (* (/ b a) b) a)))))
63.0ms
(sqrt (fabs (- 1 (/ (pow b 2) (pow a 2)))))

prune453.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 7.7b

localize9.0ms

Local error

Found 4 expressions with local error:

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

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
4 → 46
Calls

4 calls. Slowest were:

8.0ms
(cbrt (* (* (sqrt (fabs (- 1 (/ (* (/ b a) b) a)))) (sqrt (fabs (- 1 (/ (* (/ b a) b) a))))) (sqrt (fabs (- 1 (/ (* (/ b a) b) a))))))
1.0ms
(sqrt (fabs (- 1 (/ (* (/ b a) b) a))))
1.0ms
(sqrt (fabs (- 1 (/ (* (/ b a) b) a))))

series82.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

22.0ms
(sqrt (fabs (- 1 (/ (* (/ b a) b) a))))
21.0ms
(sqrt (fabs (- 1 (/ (* (/ b a) b) a))))
21.0ms
(sqrt (fabs (- 1 (/ (* (/ b a) b) a))))
18.0ms
(cbrt (* (* (sqrt (fabs (- 1 (/ (* (/ b a) b) a)))) (sqrt (fabs (- 1 (/ (* (/ b a) b) a))))) (sqrt (fabs (- 1 (/ (* (/ b a) b) a))))))

simplify1.0s

Counts
26 → 58
Calls

26 calls. Slowest were:

67.0ms
(sqrt (fabs (- 1 (/ (pow b 2) (pow a 2)))))
65.0ms
(cbrt (* (sqrt (fabs (- 1 (/ (* (/ b a) b) a)))) (sqrt (fabs (- 1 (/ (* (/ b a) b) a))))))
63.0ms
(sqrt (fabs (- 1 (/ (pow b 2) (pow a 2)))))

prune827.0ms

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 7.7b

regimes51.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms