Average Error: 29.7 → 7.5
Time: 6.2s
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. Simplified7.5

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

  3. Final simplification7.5

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

Reproduce

herbie shell --seed 2019010 +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: 5.8s)Debug log

sample61.0ms

Algorithm
intervals

simplify232.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
232.0ms
(sqrt (fabs (/ (- (* a a) (* b b)) (* a a))))

prune10.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 9.2b

localize20.0ms

Local error

Found 3 expressions with local error:

9.2b
(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
Rules
8×pow1
7×add-sqr-sqrt
6×add-cube-cbrt
6×*-un-lft-identity
5×add-exp-log
5×add-cbrt-cube
4×associate-*l*
4×associate-*r*
3×add-log-exp
3×log1p-expm1-u
3×sqrt-prod
3×expm1-log1p-u
2×div-inv
1×flip--
1×cbrt-unprod
1×pow-prod-up
1×*-commutative
1×associate-*r/
1×prod-exp
1×associate-*l/
1×pow-prod-down
1×pow1/2
1×flip3--
1×pow-plus
1×sub-neg
1×rem-sqrt-square
1×frac-times
1×pow2
Counts
3 → 54
Calls
3 calls:
Slowest
7.0ms
(* (/ b a) (/ b a))
2.0ms
(- 1 (* (/ b a) (/ b a)))
2.0ms
(sqrt (fabs (- 1 (* (/ b a) (/ b a)))))

series47.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
18.0ms
(- 1 (* (/ b a) (/ b a)))
16.0ms
(* (/ b a) (/ b a))
13.0ms
(sqrt (fabs (- 1 (* (/ b a) (/ b a)))))

simplify1.4s

Counts
28 → 63
Calls
28 calls:
Slowest
903.0ms
(* (* (* (/ b a) (/ b a)) (/ b a)) (* (* (/ b a) (/ b a)) (/ b a)))
59.0ms
(sqrt (* (cbrt (fabs (- 1 (* (/ b a) (/ b a))))) (cbrt (fabs (- 1 (* (/ b a) (/ b a)))))))
53.0ms
(sqrt (fabs (- 1 (/ (pow b 2) (pow a 2)))))
53.0ms
(sqrt (fabs (- 1 (/ (pow b 2) (pow a 2)))))
48.0ms
(sqrt (fabs (- 1 (* (/ b a) (/ b a)))))

prune655.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 9.2b

localize20.0ms

Local error

Found 4 expressions with local error:

13.9b
(cbrt (* (* (sqrt (fabs (- 1 (* (/ b a) (/ b a))))) (sqrt (fabs (- 1 (* (/ b a) (/ b a)))))) (sqrt (fabs (- 1 (* (/ b a) (/ b a)))))))
9.2b
(sqrt (fabs (- 1 (* (/ b a) (/ b a)))))
9.2b
(sqrt (fabs (- 1 (* (/ b a) (/ b a)))))
9.2b
(sqrt (fabs (- 1 (* (/ b a) (/ b a)))))

rewrite23.0ms

Algorithm
rewrite-expression-head
Rules
10×add-sqr-sqrt
9×sqrt-prod
7×add-cube-cbrt
7×*-un-lft-identity
4×add-log-exp
4×log1p-expm1-u
4×add-exp-log
4×add-cbrt-cube
4×pow1
4×expm1-log1p-u
3×pow1/2
3×rem-sqrt-square
1×rem-cbrt-cube
1×pow3
1×pow1/3
1×cbrt-prod
Counts
4 → 54
Calls
4 calls:
Slowest
18.0ms
(cbrt (* (* (sqrt (fabs (- 1 (* (/ b a) (/ b a))))) (sqrt (fabs (- 1 (* (/ b a) (/ b a)))))) (sqrt (fabs (- 1 (* (/ b a) (/ b a)))))))
2.0ms
(sqrt (fabs (- 1 (* (/ b a) (/ b a)))))
2.0ms
(sqrt (fabs (- 1 (* (/ b a) (/ b a)))))
2.0ms
(sqrt (fabs (- 1 (* (/ b a) (/ b a)))))

series86.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
26.0ms
(cbrt (* (* (sqrt (fabs (- 1 (* (/ b a) (/ b a))))) (sqrt (fabs (- 1 (* (/ b a) (/ b a)))))) (sqrt (fabs (- 1 (* (/ b a) (/ b a)))))))
20.0ms
(sqrt (fabs (- 1 (* (/ b a) (/ b a)))))
20.0ms
(sqrt (fabs (- 1 (* (/ b a) (/ b a)))))
19.0ms
(sqrt (fabs (- 1 (* (/ b a) (/ b a)))))

simplify1.1s

Counts
26 → 66
Calls
26 calls:
Slowest
64.0ms
(sqrt (fabs (- 1 (/ (pow b 2) (pow a 2)))))
64.0ms
(sqrt (fabs (- 1 (/ (pow b 2) (pow a 2)))))
63.0ms
(sqrt (fabs (- 1 (/ (pow b 2) (pow a 2)))))
63.0ms
(sqrt (sqrt (fabs (- 1 (* (/ b a) (/ b a))))))
62.0ms
(sqrt (fabs (- 1 (/ (pow b 2) (pow a 2)))))

prune774.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 9.2b

regimes97.0ms

Accuracy

0% (0.0b remaining)

Error of 7.5b against oracle of 7.5b and baseline of 7.5b

bsearch2.0ms

end0.0ms

sample1.3s

Algorithm
intervals