Average Error: 29.7 → 7.3
Time: 9.2s
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 29.7

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

  2. Simplified7.3

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

  3. Final simplification7.3

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

Reproduce

herbie shell --seed 2019005 +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: 8.8s)Debug log

sample54.0ms

Algorithm
intervals

simplify209.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

209.0ms
(sqrt (fabs (/ (- (* a a) (* b b)) (* a a))))

prune12.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 5.7b

localize30.0ms

Local error

Found 2 expressions with local error:

5.7b
(sqrt (fabs (- 1 (* (/ b a) (/ b a)))))
0.4b
(* (/ b a) (/ b a))

rewrite8.0ms

Algorithm
rewrite-expression-head
Counts
2 → 41
Calls

2 calls. Slowest were:

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

series42.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

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

simplify1.1s

Counts
25 → 47
Calls

25 calls. Slowest were:

791.0ms
(* (* (* (/ b a) (/ b a)) (/ b a)) (* (* (/ b a) (/ b a)) (/ b a)))
57.0ms
(sqrt (fabs (- 1 (* (/ b a) (/ b a)))))
51.0ms
(sqrt (* (cbrt (fabs (- 1 (* (/ b a) (/ b a))))) (cbrt (fabs (- 1 (* (/ b a) (/ b a)))))))

prune498.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 5.7b

localize16.0ms

Local error

Found 4 expressions with local error:

5.7b
(sqrt (fabs (- 1 (* (sqrt (/ b a)) (* (sqrt (/ b a)) (/ b a))))))
0.3b
(* (sqrt (/ b a)) (/ b a))
0.3b
(* (sqrt (/ b a)) (* (sqrt (/ b a)) (/ b a)))
0.2b
(sqrt (/ b a))

rewrite16.0ms

Algorithm
rewrite-expression-head
Counts
4 → 94
Calls

4 calls. Slowest were:

10.0ms
(* (sqrt (/ b a)) (* (sqrt (/ b a)) (/ b a)))
4.0ms
(* (sqrt (/ b a)) (/ b a))
1.0ms
(sqrt (fabs (- 1 (* (sqrt (/ b a)) (* (sqrt (/ b a)) (/ b a))))))

series157.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

85.0ms
(* (sqrt (/ b a)) (/ b a))
30.0ms
(sqrt (/ b a))
24.0ms
(* (sqrt (/ b a)) (* (sqrt (/ b a)) (/ b a)))
17.0ms
(sqrt (fabs (- 1 (* (sqrt (/ b a)) (* (sqrt (/ b a)) (/ b a))))))

simplify2.0s

Counts
69 → 106
Calls

69 calls. Slowest were:

343.0ms
(* (* (* (sqrt (/ b a)) (sqrt (/ b a))) (sqrt (/ b a))) (* (* (* (sqrt (/ b a)) (sqrt (/ b a))) (sqrt (/ b a))) (* (* (/ b a) (/ b a)) (/ b a))))
324.0ms
(* (sqrt (/ b a)) (* (sqrt (/ b a)) (/ b a)))
215.0ms
(* (* (* (sqrt (/ b a)) (sqrt (/ b a))) (sqrt (/ b a))) (* (* (* (sqrt (/ b a)) (/ b a)) (* (sqrt (/ b a)) (/ b a))) (* (sqrt (/ b a)) (/ b a))))

prune956.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 5.7b

localize8.0ms

Local error

Found 4 expressions with local error:

5.7b
(sqrt (fabs (- 1 (* (/ b a) (/ b a)))))
2.5b
(expm1 (log1p (sqrt (fabs (- 1 (* (/ b a) (/ b a)))))))
0.4b
(* (/ b a) (/ b a))
0.0b
(log1p (sqrt (fabs (- 1 (* (/ b a) (/ b a))))))

rewrite5.0ms

Algorithm
rewrite-expression-head
Counts
4 → 63
Calls

4 calls. Slowest were:

3.0ms
(* (/ b a) (/ b a))
1.0ms
(sqrt (fabs (- 1 (* (/ b a) (/ b a)))))
0.0ms
(log1p (sqrt (fabs (- 1 (* (/ b a) (/ b a))))))

series144.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

77.0ms
(expm1 (log1p (sqrt (fabs (- 1 (* (/ b a) (/ b a)))))))
42.0ms
(log1p (sqrt (fabs (- 1 (* (/ b a) (/ b a))))))
13.0ms
(* (/ b a) (/ b a))
12.0ms
(sqrt (fabs (- 1 (* (/ b a) (/ b a)))))

simplify1.4s

Counts
33 → 75
Calls

33 calls. Slowest were:

783.0ms
(* (* (* (/ b a) (/ b a)) (/ b a)) (* (* (/ b a) (/ b a)) (/ b a)))
47.0ms
(sqrt (* (cbrt (fabs (- 1 (* (/ b a) (/ b a))))) (cbrt (fabs (- 1 (* (/ b a) (/ b a)))))))
46.0ms
(log (+ 1 (sqrt (fabs (- 1 (/ (pow b 2) (pow a 2)))))))

prune871.0ms

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 5.7b

regimes50.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms

end0.0ms

sample1.3s

Algorithm
intervals