Average Error: 1.6 → 1.6
Time: 33.0s
Precision: 64
Internal Precision: 128
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{\frac{1}{\sqrt[3]{\sqrt{\left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} + 1}} \cdot \sqrt[3]{\sqrt{\left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} + 1}}}}{\sqrt[3]{\sqrt{\left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} + 1}}}\right)}\]

Error

Bits error versus l

Bits error versus Om

Bits error versus kx

Bits error versus ky

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 1.6

    \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
  2. Using strategy rm

  3. Applied add-cube-cbrt1.6

    \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\color{blue}{\left(\sqrt[3]{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}} \cdot \sqrt[3]{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right) \cdot \sqrt[3]{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}}}\right)}\]

  4. Applied associate-/r*1.6

    \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \color{blue}{\frac{\frac{1}{\sqrt[3]{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}} \cdot \sqrt[3]{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}}}{\sqrt[3]{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}}}\right)}\]

  5. Final simplification1.6

    \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{\frac{1}{\sqrt[3]{\sqrt{\left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} + 1}} \cdot \sqrt[3]{\sqrt{\left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} + 1}}}}{\sqrt[3]{\sqrt{\left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} + 1}}}\right)}\]

Reproduce

herbie shell --seed 2019005 
(FPCore (l Om kx ky)
  :name "Toniolo and Linder, Equation (3a)"
  (sqrt (* (/ 1 2) (+ 1 (/ 1 (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))))))

Details

Time bar (total: 32.5s)Debug log

sample180.0ms

Algorithm
intervals

simplify364.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

363.0ms
(sqrt (* (/ 1 2) (+ 1 (/ 1 (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2)))))))))

prune26.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 1.4b

localize70.0ms

Local error

Found 4 expressions with local error:

9.9b
(sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2)))))
3.1b
(* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2)))
0.5b
(/ 1 (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))
0.3b
(pow (sin ky) 2)

rewrite29.0ms

Algorithm
rewrite-expression-head
Counts
4 → 79
Calls

4 calls. Slowest were:

16.0ms
(* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2)))
3.0ms
(/ 1 (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))
2.0ms
(pow (sin ky) 2)

series808.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

365.0ms
(/ 1 (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))
230.0ms
(sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2)))))
186.0ms
(* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2)))
27.0ms
(pow (sin ky) 2)

simplify6.2s

Counts
53 → 91
Calls

53 calls. Slowest were:

547.0ms
(* (pow (/ (* 2 l) Om) 2) (pow (sin ky) 2))
510.0ms
(* (pow (sin ky) 2) (pow (/ (* 2 l) Om) 2))
330.0ms
(sqrt (- 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2)))))

prune1.9s

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.4b

localize35.0ms

Local error

Found 4 expressions with local error:

9.9b
(sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2)))))
9.9b
(sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2)))))
9.9b
(sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2)))))
7.1b
(cbrt (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
4 → 58
Calls

4 calls. Slowest were:

2.0ms
(cbrt (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))
1.0ms
(sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2)))))
1.0ms
(sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2)))))

series1.1s

Counts
4 → 12
Calls

4 calls. Slowest were:

330.0ms
(cbrt (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))
276.0ms
(sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2)))))
241.0ms
(sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2)))))
240.0ms
(sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2)))))

simplify4.6s

Counts
38 → 70
Calls

38 calls. Slowest were:

297.0ms
(sqrt (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))
294.0ms
(cbrt (sqrt (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2)))))))
267.0ms
(cbrt (sqrt (* (cbrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2))))) (cbrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))))

prune2.0s

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0.4b

localize45.0ms

Local error

Found 4 expressions with local error:

15.0b
(* (* (* l l) 4) (+ (* (/ ky Om) (/ ky Om)) (* (/ kx Om) (/ kx Om))))
12.6b
(sqrt (+ 1 (* (* (* l l) 4) (+ (* (/ ky Om) (/ ky Om)) (* (/ kx Om) (/ kx Om))))))
0.8b
(/ 1 (sqrt (+ 1 (* (* (* l l) 4) (+ (* (/ ky Om) (/ ky Om)) (* (/ kx Om) (/ kx Om)))))))
0.2b
(* (/ kx Om) (/ kx Om))

rewrite40.0ms

Algorithm
rewrite-expression-head
Counts
4 → 89
Calls

4 calls. Slowest were:

27.0ms
(* (* (* l l) 4) (+ (* (/ ky Om) (/ ky Om)) (* (/ kx Om) (/ kx Om))))
7.0ms
(* (/ kx Om) (/ kx Om))
3.0ms
(/ 1 (sqrt (+ 1 (* (* (* l l) 4) (+ (* (/ ky Om) (/ ky Om)) (* (/ kx Om) (/ kx Om)))))))

series410.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

137.0ms
(* (* (* l l) 4) (+ (* (/ ky Om) (/ ky Om)) (* (/ kx Om) (/ kx Om))))
131.0ms
(/ 1 (sqrt (+ 1 (* (* (* l l) 4) (+ (* (/ ky Om) (/ ky Om)) (* (/ kx Om) (/ kx Om)))))))
117.0ms
(sqrt (+ 1 (* (* (* l l) 4) (+ (* (/ ky Om) (/ ky Om)) (* (/ kx Om) (/ kx Om))))))
24.0ms
(* (/ kx Om) (/ kx Om))

simplify8.2s

Counts
63 → 101
Calls

63 calls. Slowest were:

955.0ms
(* (* (* (/ kx Om) (/ kx Om)) (/ kx Om)) (* (* (/ kx Om) (/ kx Om)) (/ kx Om)))
474.0ms
(* (* (/ kx Om) (/ kx Om)) (* (* l l) 4))
460.0ms
(sqrt (- (* 1 1) (* (* (* (* l l) 4) (+ (* (/ ky Om) (/ ky Om)) (* (/ kx Om) (/ kx Om)))) (* (* (* l l) 4) (+ (* (/ ky Om) (/ ky Om)) (* (/ kx Om) (/ kx Om)))))))

prune1.6s

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0.4b

regimes1.1s

Accuracy

0% (0.9b remaining)

Error of 1.6b against oracle of 0.8b and baseline of 1.6b

bsearch4.0ms

end0.0ms

sample3.8s

Algorithm
intervals