Average Error: 29.2 → 7.6
Time: 7.8s
Precision: 64
Internal Precision: 128
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|1 - \frac{\frac{b}{a} \cdot b}{a}\right|}\]

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-*l/7.6

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

  5. Final simplification7.6

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

Reproduce

herbie shell --seed 2019004 +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: 7.5s)Debug log

sample53.0ms

Algorithm
intervals

simplify260.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

259.0ms
(sqrt (fabs (/ (- (* a a) (* b b)) (* a a))))

prune16.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 8.2b

localize13.0ms

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)))

rewrite6.0ms

Algorithm
rewrite-expression-head
Counts
3 → 54
Calls

3 calls. Slowest were:

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

series72.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

33.0ms
(- 1 (* (/ b a) (/ b a)))
23.0ms
(* (/ b a) (/ b a))
16.0ms
(sqrt (fabs (- 1 (* (/ b a) (/ b a)))))

simplify1.5s

Counts
28 → 63
Calls

28 calls. Slowest were:

765.0ms
(* (* (* (/ b a) (/ b a)) (/ b a)) (* (* (/ b a) (/ b a)) (/ b a)))
390.0ms
(sqrt (fabs (- 1 (* (/ b a) (/ b a)))))
59.0ms
(sqrt (* (cbrt (fabs (- 1 (* (/ b a) (/ b a))))) (cbrt (fabs (- 1 (* (/ b a) (/ b a)))))))

prune575.0ms

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 (/ b a)) a))))
6.9b
(/ (* b (/ b a)) a)
0.2b
(* b (/ b a))
0.0b
(- 1 (/ (* b (/ b a)) a))

rewrite13.0ms

Algorithm
rewrite-expression-head
Counts
4 → 56
Calls

4 calls. Slowest were:

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

series94.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

33.0ms
(- 1 (/ (* b (/ b a)) a))
23.0ms
(/ (* b (/ b a)) a)
22.0ms
(sqrt (fabs (- 1 (/ (* b (/ b a)) a))))
16.0ms
(* b (/ b a))

simplify445.0ms

Counts
21 → 68
Calls

21 calls. Slowest were:

63.0ms
(sqrt (* (cbrt (fabs (- 1 (/ (* b (/ b a)) a)))) (cbrt (fabs (- 1 (/ (* b (/ b a)) a))))))
62.0ms
(sqrt (fabs (- 1 (/ (pow b 2) (pow a 2)))))
60.0ms
(sqrt (fabs (- 1 (/ (pow b 2) (pow a 2)))))

prune672.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 8.2b

localize26.0ms

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))))

rewrite15.0ms

Algorithm
rewrite-expression-head
Counts
4 → 65
Calls

4 calls. Slowest were:

6.0ms
(* (/ 1 a) (/ b a))
5.0ms
(* b (* (/ 1 a) (/ b a)))
2.0ms
(sqrt (fabs (- 1 (* b (* (/ 1 a) (/ b a))))))

series67.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

25.0ms
(- 1 (* b (* (/ 1 a) (/ b a))))
15.0ms
(* (/ 1 a) (/ b a))
13.0ms
(* b (* (/ 1 a) (/ b a)))
13.0ms
(sqrt (fabs (- 1 (* b (* (/ 1 a) (/ b a))))))

simplify1.5s

Counts
30 → 77
Calls

30 calls. Slowest were:

335.0ms
(* (* (* (/ 1 a) (/ 1 a)) (/ 1 a)) (* (* (/ b a) (/ b a)) (/ b a)))
314.0ms
(sqrt (fabs (- 1 (* b (* (/ 1 a) (/ b a))))))
268.0ms
(sqrt (* (cbrt (fabs (- 1 (* b (* (/ 1 a) (/ b a)))))) (cbrt (fabs (- 1 (* b (* (/ 1 a) (/ b a))))))))

prune761.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 8.2b

regimes64.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms

end0.0ms

sample1.3s

Algorithm
intervals