Average Error: 29.6 → 7.7
Time: 13.6s
Precision: 64
Internal Precision: 128
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\frac{b + a}{a} \cdot \frac{a - b}{a}\right|}\]

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 29.6

    \[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
  2. Using strategy rm

  3. Applied difference-of-squares29.6

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

  4. Applied times-frac7.7

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

  5. Final simplification7.7

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

Reproduce

herbie shell --seed 2019002 +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: 13.1s)Debug log

sample54.0ms

Algorithm
intervals

simplify265.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

264.0ms
(sqrt (fabs (/ (- (* a a) (* b b)) (* a a))))

prune13.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 7.3b

localize23.0ms

Local error

Found 3 expressions with local error:

7.3b
(sqrt (fabs (- 1 (* (/ b a) (/ b a)))))
0.4b
(* (/ b a) (/ b a))
0.0b
(- 1 (* (/ b a) (/ b a)))

rewrite11.0ms

Algorithm
rewrite-expression-head
Counts
3 → 54
Calls

3 calls. Slowest were:

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

series51.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

24.0ms
(- 1 (* (/ b a) (/ b a)))
14.0ms
(* (/ b a) (/ b a))
14.0ms
(sqrt (fabs (- 1 (* (/ b a) (/ b a)))))

simplify1.4s

Counts
28 → 63
Calls

28 calls. Slowest were:

935.0ms
(* (* (* (/ b a) (/ b a)) (/ b a)) (* (* (/ b a) (/ b a)) (/ b a)))
67.0ms
(sqrt (fabs (- 1 (/ (pow b 2) (pow a 2)))))
65.0ms
(sqrt (fabs (- 1 (/ (pow b 2) (pow a 2)))))

prune631.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 7.3b

localize14.0ms

Local error

Found 4 expressions with local error:

8.5b
(* (* (/ b a) b) (/ 1 a))
7.3b
(sqrt (fabs (- 1 (* (* (/ b a) b) (/ 1 a)))))
0.2b
(* (/ b a) b)
0.0b
(- 1 (* (* (/ b a) b) (/ 1 a)))

rewrite18.0ms

Algorithm
rewrite-expression-head
Counts
4 → 64
Calls

4 calls. Slowest were:

9.0ms
(* (* (/ b a) b) (/ 1 a))
4.0ms
(* (/ b a) b)
2.0ms
(- 1 (* (* (/ b a) b) (/ 1 a)))

series71.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

23.0ms
(* (* (/ b a) b) (/ 1 a))
20.0ms
(sqrt (fabs (- 1 (* (* (/ b a) b) (/ 1 a)))))
18.0ms
(- 1 (* (* (/ b a) b) (/ 1 a)))
10.0ms
(* (/ b a) b)

simplify3.1s

Counts
30 → 76
Calls

30 calls. Slowest were:

1.3s
(* (* (* (* (/ b a) b) (* (/ b a) b)) (* (/ b a) b)) (* (* (/ 1 a) (/ 1 a)) (/ 1 a)))
559.0ms
(sqrt (sqrt (fabs (- 1 (* (* (/ b a) b) (/ 1 a))))))
317.0ms
(sqrt (* (cbrt (fabs (- 1 (* (* (/ b a) b) (/ 1 a))))) (cbrt (fabs (- 1 (* (* (/ b a) b) (/ 1 a)))))))

prune728.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 7.3b

localize19.0ms

Local error

Found 4 expressions with local error:

12.2b
(cbrt (* (* (sqrt (fabs (- 1 (* (/ b a) (/ b a))))) (sqrt (fabs (- 1 (* (/ b a) (/ b a)))))) (sqrt (fabs (- 1 (* (/ b a) (/ b a)))))))
7.3b
(sqrt (fabs (- 1 (* (/ b a) (/ b a)))))
7.3b
(sqrt (fabs (- 1 (* (/ b a) (/ b a)))))
7.3b
(sqrt (fabs (- 1 (* (/ b a) (/ b a)))))

rewrite21.0ms

Algorithm
rewrite-expression-head
Counts
4 → 54
Calls

4 calls. Slowest were:

16.0ms
(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)))))

series76.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

26.0ms
(sqrt (fabs (- 1 (* (/ b a) (/ b a)))))
22.0ms
(cbrt (* (* (sqrt (fabs (- 1 (* (/ b a) (/ b a))))) (sqrt (fabs (- 1 (* (/ b a) (/ b a)))))) (sqrt (fabs (- 1 (* (/ b a) (/ b a)))))))
14.0ms
(sqrt (fabs (- 1 (* (/ b a) (/ b a)))))
13.0ms
(sqrt (fabs (- 1 (* (/ b a) (/ b a)))))

simplify1.2s

Counts
26 → 66
Calls

26 calls. Slowest were:

66.0ms
(sqrt (fabs (- 1 (/ (pow b 2) (pow a 2)))))
66.0ms
(sqrt (sqrt (fabs (- 1 (* (/ b a) (/ b a))))))
66.0ms
(cbrt (* (sqrt (fabs (- 1 (* (/ b a) (/ b a))))) (sqrt (fabs (- 1 (* (/ b a) (/ b a)))))))

prune800.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 7.3b

localize28.0ms

Local error

Found 3 expressions with local error:

20.8b
(/ (- (* a a) (* b b)) (* a a))
7.3b
(sqrt (fabs (/ (- (* a a) (* b b)) (* a a))))
0.0b
(- (* a a) (* b b))

rewrite15.0ms

Algorithm
rewrite-expression-head
Counts
3 → 57
Calls

3 calls. Slowest were:

11.0ms
(/ (- (* a a) (* b b)) (* a a))
2.0ms
(- (* a a) (* b b))
1.0ms
(sqrt (fabs (/ (- (* a a) (* b b)) (* a a))))

series67.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

26.0ms
(/ (- (* a a) (* b b)) (* a a))
23.0ms
(sqrt (fabs (/ (- (* a a) (* b b)) (* a a))))
18.0ms
(- (* a a) (* b b))

simplify2.5s

Counts
34 → 66
Calls

34 calls. Slowest were:

261.0ms
(sqrt (* (cbrt (fabs (/ (- (* a a) (* b b)) (* a a)))) (cbrt (fabs (/ (- (* a a) (* b b)) (* a a))))))
255.0ms
(- (* (* a a) (* a a)) (* (* b b) (* b b)))
226.0ms
(sqrt (fabs (/ (- (* a a) (* b b)) (* a a))))

prune609.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 7.3b

regimes65.0ms

Accuracy

0% (0.0b remaining)

Error of 7.7b against oracle of 7.7b and baseline of 7.7b

bsearch1.0ms

end0.0ms

sample1.2s

Algorithm
intervals