Average Error: 29.3 → 7.4
Time: 17.5s
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 29.3

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

  2. Simplified7.4

    \[\leadsto \color{blue}{\sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}}\]
  3. Using strategy rm

  4. Applied associate-*r/7.4

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

  5. Final simplification7.4

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

Reproduce

herbie shell --seed 2019022 +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: 17.1s)Debug log

sample33.0ms

Algorithm
intervals
Results
18.0ms256×body80valid
7.0ms256×pre80true

simplify250.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
250.0ms
(sqrt (fabs (/ (- (* a a) (* b b)) (* a a))))

prune24.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 9.9b

localize24.0ms

Local error

Found 3 expressions with local error:

10.0b
(sqrt (fabs (- 1 (* (/ b a) (/ b a)))))
0.4b
(* (/ b a) (/ b a))
0.0b
(- 1 (* (/ b a) (/ b a)))

rewrite10.0ms

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
6.0ms
(* (/ b a) (/ b a))
2.0ms
(- 1 (* (/ b a) (/ b a)))
1.0ms
(sqrt (fabs (- 1 (* (/ b a) (/ b a)))))

series65.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
26.0ms
(- 1 (* (/ b a) (/ b a)))
20.0ms
(* (/ b a) (/ b a))
19.0ms
(sqrt (fabs (- 1 (* (/ b a) (/ b a)))))

simplify1.4s

Counts
28 → 63
Calls
28 calls:
Slowest
891.0ms
(* (* (* (/ b a) (/ b a)) (/ b a)) (* (* (/ b a) (/ b a)) (/ b a)))
62.0ms
(sqrt (fabs (- 1 (/ (pow b 2) (pow a 2)))))
61.0ms
(sqrt (* (cbrt (fabs (- 1 (* (/ b a) (/ b a))))) (cbrt (fabs (- 1 (* (/ b a) (/ b a)))))))
54.0ms
(sqrt (fabs (- 1 (/ (pow b 2) (pow a 2)))))
53.0ms
(sqrt (sqrt (fabs (- 1 (* (/ b a) (/ b a))))))

prune692.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 9.9b

localize12.0ms

Local error

Found 4 expressions with local error:

10.0b
(sqrt (fabs (- 1 (/ (* (/ b a) b) a))))
6.9b
(/ (* (/ b a) b) a)
0.2b
(* (/ b a) b)
0.0b
(- 1 (/ (* (/ b a) b) a))

rewrite9.0ms

Algorithm
rewrite-expression-head
Rules
7×add-sqr-sqrt
6×add-cube-cbrt
6×*-un-lft-identity
4×add-log-exp
4×log1p-expm1-u
4×add-exp-log
4×associate-*l*
4×add-cbrt-cube
4×pow1
4×expm1-log1p-u
3×sqrt-prod
2×div-inv
2×associate-*l/
1×associate-/l/
1×associate-/l*
1×flip--
1×*-commutative
1×pow1/2
1×flip3--
1×frac-2neg
1×sub-neg
1×clear-num
1×rem-sqrt-square
Counts
4 → 56
Calls
4 calls:
Slowest
4.0ms
(/ (* (/ b a) b) a)
2.0ms
(* (/ b a) b)
1.0ms
(- 1 (/ (* (/ b a) b) a))
1.0ms
(sqrt (fabs (- 1 (/ (* (/ b a) b) a))))

series69.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
22.0ms
(/ (* (/ b a) b) a)
19.0ms
(sqrt (fabs (- 1 (/ (* (/ b a) b) a))))
17.0ms
(- 1 (/ (* (/ b a) b) a))
10.0ms
(* (/ b a) b)

simplify435.0ms

Counts
21 → 68
Calls
21 calls:
Slowest
67.0ms
(sqrt (* (cbrt (fabs (- 1 (/ (* (/ b a) b) a)))) (cbrt (fabs (- 1 (/ (* (/ b a) b) a))))))
63.0ms
(sqrt (fabs (- 1 (/ (pow b 2) (pow a 2)))))
51.0ms
(sqrt (fabs (- 1 (/ (pow b 2) (pow a 2)))))
46.0ms
(sqrt (sqrt (fabs (- 1 (/ (* (/ b a) b) a)))))
41.0ms
(sqrt (fabs (- 1 (/ (* (/ b a) b) a))))

prune708.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 9.9b

localize38.0ms

Local error

Found 4 expressions with local error:

18.9b
(/ (- (* 1 1) (* (* (/ b a) (/ b a)) (* (/ b a) (/ b a)))) (+ 1 (* (/ b a) (/ b a))))
10.0b
(sqrt (fabs (/ (- (* 1 1) (* (* (/ b a) (/ b a)) (* (/ b a) (/ b a)))) (+ 1 (* (/ b a) (/ b a))))))
0.4b
(* (/ b a) (/ b a))
0.4b
(* (/ b a) (/ b a))

rewrite24.0ms

Algorithm
rewrite-expression-head
Rules
19×add-sqr-sqrt
18×add-cube-cbrt
18×*-un-lft-identity
14×pow1
12×times-frac
10×add-exp-log
10×add-cbrt-cube
8×associate-*l*
8×associate-*r*
5×div-inv
4×difference-of-squares
4×add-log-exp
4×associate-/l*
4×log1p-expm1-u
4×expm1-log1p-u
3×sqrt-prod
3×associate-/r*
2×associate-/l/
2×cbrt-unprod
2×pow-prod-up
2×*-commutative
2×associate-/r/
2×associate-*r/
2×prod-exp
2×associate-*l/
2×pow-prod-down
2×pow-plus
2×frac-times
2×pow2
1×flip--
1×flip-+
1×div-exp
1×div-sub
1×pow1/2
1×flip3--
1×flip3-+
1×frac-2neg
1×clear-num
1×rem-sqrt-square
1×cbrt-undiv
Counts
4 → 106
Calls
4 calls:
Slowest
15.0ms
(/ (- (* 1 1) (* (* (/ b a) (/ b a)) (* (/ b a) (/ b a)))) (+ 1 (* (/ b a) (/ b a))))
3.0ms
(* (/ b a) (/ b a))
3.0ms
(* (/ b a) (/ b a))
1.0ms
(sqrt (fabs (/ (- (* 1 1) (* (* (/ b a) (/ b a)) (* (/ b a) (/ b a)))) (+ 1 (* (/ b a) (/ b a))))))

series121.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
53.0ms
(/ (- (* 1 1) (* (* (/ b a) (/ b a)) (* (/ b a) (/ b a)))) (+ 1 (* (/ b a) (/ b a))))
33.0ms
(sqrt (fabs (/ (- (* 1 1) (* (* (/ b a) (/ b a)) (* (/ b a) (/ b a)))) (+ 1 (* (/ b a) (/ b a))))))
23.0ms
(* (/ b a) (/ b a))
13.0ms
(* (/ b a) (/ b a))

simplify10.6s

Counts
83 → 118
Calls
83 calls:
Slowest
855.0ms
(* (* (* (/ b a) (/ b a)) (/ b a)) (* (* (/ b a) (/ b a)) (/ b a)))
834.0ms
(* (* (* (/ b a) (/ b a)) (/ b a)) (* (* (/ b a) (/ b a)) (/ b a)))
688.0ms
(/ (sqrt (- (* 1 1) (* (* (/ b a) (/ b a)) (* (/ b a) (/ b a))))) (* (cbrt (+ 1 (* (/ b a) (/ b a)))) (cbrt (+ 1 (* (/ b a) (/ b a))))))
602.0ms
(- (pow (* 1 1) 3) (pow (* (* (/ b a) (/ b a)) (* (/ b a) (/ b a))) 3))
455.0ms
(/ (* (cbrt (- (* 1 1) (* (* (/ b a) (/ b a)) (* (/ b a) (/ b a))))) (cbrt (- (* 1 1) (* (* (/ b a) (/ b a)) (* (/ b a) (/ b a)))))) (* (cbrt (+ 1 (* (/ b a) (/ b a)))) (cbrt (+ 1 (* (/ b a) (/ b a))))))

prune1.2s

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 9.9b

regimes98.0ms

Accuracy

0% (0.0b remaining)

Error of 7.4b against oracle of 7.4b and baseline of 7.4b

bsearch2.0ms

end0.0ms

sample1.2s

Algorithm
intervals
Results
834.0ms8000×body80valid
272.0ms8000×pre80true