Average Error: 1.7 → 1.5
Time: 41.8s
Precision: 64
Internal Precision: 128
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt[3]{\sqrt{\left(\left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) \cdot \frac{\ell \cdot 2}{Om}\right) \cdot \frac{\ell \cdot 2}{Om} + 1}} \cdot \left(\sqrt[3]{\sqrt{\left(\left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) \cdot \frac{\ell \cdot 2}{Om}\right) \cdot \frac{\ell \cdot 2}{Om} + 1}} \cdot \sqrt[3]{\sqrt{\left(\left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) \cdot \frac{\ell \cdot 2}{Om}\right) \cdot \frac{\ell \cdot 2}{Om} + 1}}\right)}\right)}\]

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 1.7

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

  3. Applied unpow21.7

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

  4. Applied associate-*l*1.5

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

  6. Applied add-cube-cbrt1.5

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

  7. Final simplification1.5

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

Reproduce

herbie shell --seed 2019002 
(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: 41.1s)Debug log

sample100.0ms

Algorithm
intervals

simplify334.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

333.0ms
(sqrt (* (/ 1 2) (+ 1 (/ 1 (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2)))))))))

prune22.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 1.8b

localize59.0ms

Local error

Found 4 expressions with local error:

10.7b
(sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2)))))
3.5b
(* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2)))
0.3b
(pow (sin ky) 2)
0.2b
(/ 1 (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))

rewrite20.0ms

Algorithm
rewrite-expression-head
Counts
4 → 79
Calls

4 calls. Slowest were:

13.0ms
(* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2)))
2.0ms
(sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2)))))
1.0ms
(/ 1 (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))

series779.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

320.0ms
(/ 1 (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))
264.0ms
(sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2)))))
178.0ms
(* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2)))
16.0ms
(pow (sin ky) 2)

simplify6.5s

Counts
53 → 91
Calls

53 calls. Slowest were:

634.0ms
(* (pow (/ (* 2 l) Om) 2) (pow (sin ky) 2))
622.0ms
(* (pow (sin ky) 2) (pow (/ (* 2 l) Om) 2))
620.0ms
(sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2)))))

prune1.8s

Pruning

4 alts after pruning (4 fresh and 0 done)

Merged error: 0.8b

localize42.0ms

Local error

Found 4 expressions with local error:

10.7b
(sqrt (+ 1 (* (/ (* 2 l) Om) (* (/ (* 2 l) Om) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))
2.2b
(* (/ (* 2 l) Om) (+ (pow (sin kx) 2) (pow (sin ky) 2)))
0.7b
(* (/ (* 2 l) Om) (* (/ (* 2 l) Om) (+ (pow (sin kx) 2) (pow (sin ky) 2))))
0.3b
(pow (sin ky) 2)

rewrite40.0ms

Algorithm
rewrite-expression-head
Counts
4 → 89
Calls

4 calls. Slowest were:

24.0ms
(* (/ (* 2 l) Om) (* (/ (* 2 l) Om) (+ (pow (sin kx) 2) (pow (sin ky) 2))))
12.0ms
(* (/ (* 2 l) Om) (+ (pow (sin kx) 2) (pow (sin ky) 2)))
1.0ms
(sqrt (+ 1 (* (/ (* 2 l) Om) (* (/ (* 2 l) Om) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))

series627.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

260.0ms
(sqrt (+ 1 (* (/ (* 2 l) Om) (* (/ (* 2 l) Om) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))
175.0ms
(* (/ (* 2 l) Om) (+ (pow (sin kx) 2) (pow (sin ky) 2)))
168.0ms
(* (/ (* 2 l) Om) (* (/ (* 2 l) Om) (+ (pow (sin kx) 2) (pow (sin ky) 2))))
23.0ms
(pow (sin ky) 2)

simplify9.9s

Counts
72 → 101
Calls

72 calls. Slowest were:

540.0ms
(* (* (* (/ (* 2 l) Om) (/ (* 2 l) Om)) (/ (* 2 l) Om)) (* (* (* (/ (* 2 l) Om) (+ (pow (sin kx) 2) (pow (sin ky) 2))) (* (/ (* 2 l) Om) (+ (pow (sin kx) 2) (pow (sin ky) 2)))) (* (/ (* 2 l) Om) (+ (pow (sin kx) 2) (pow (sin ky) 2)))))
528.0ms
(* (* 2 l) (* (* 2 l) (- (* (pow (sin kx) 2) (pow (sin kx) 2)) (* (pow (sin ky) 2) (pow (sin ky) 2)))))
501.0ms
(* (* 2 l) (* (* 2 l) (+ (pow (sin kx) 2) (pow (sin ky) 2))))

prune1.9s

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.8b

localize27.0ms

Local error

Found 4 expressions with local error:

10.7b
(sqrt (+ 1 (* (/ (* 2 l) Om) (* (/ (* 2 l) Om) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))
10.7b
(sqrt (+ 1 (* (/ (* 2 l) Om) (* (/ (* 2 l) Om) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))
10.7b
(sqrt (+ 1 (* (/ (* 2 l) Om) (* (/ (* 2 l) Om) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))
6.2b
(cbrt (sqrt (+ 1 (* (/ (* 2 l) Om) (* (/ (* 2 l) Om) (+ (pow (sin kx) 2) (pow (sin ky) 2)))))))

rewrite5.0ms

Algorithm
rewrite-expression-head
Counts
4 → 58
Calls

4 calls. Slowest were:

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

series945.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

329.0ms
(cbrt (sqrt (+ 1 (* (/ (* 2 l) Om) (* (/ (* 2 l) Om) (+ (pow (sin kx) 2) (pow (sin ky) 2)))))))
222.0ms
(sqrt (+ 1 (* (/ (* 2 l) Om) (* (/ (* 2 l) Om) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))
200.0ms
(sqrt (+ 1 (* (/ (* 2 l) Om) (* (/ (* 2 l) Om) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))
194.0ms
(sqrt (+ 1 (* (/ (* 2 l) Om) (* (/ (* 2 l) Om) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))

simplify3.9s

Counts
38 → 70
Calls

38 calls. Slowest were:

324.0ms
(exp (* 1/6 (- (+ (* 2 (log (/ 1 Om))) (log (+ (* 4 (pow (sin ky) 2)) (* 4 (pow (sin kx) 2))))) (* 2 (log (/ 1 l))))))
294.0ms
(cbrt (sqrt (- (* 1 1) (* (* (/ (* 2 l) Om) (* (/ (* 2 l) Om) (+ (pow (sin kx) 2) (pow (sin ky) 2)))) (* (/ (* 2 l) Om) (* (/ (* 2 l) Om) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))))
285.0ms
(sqrt (+ (pow 1 3) (pow (* (/ (* 2 l) Om) (* (/ (* 2 l) Om) (+ (pow (sin kx) 2) (pow (sin ky) 2)))) 3)))

prune1.6s

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.8b

localize33.0ms

Local error

Found 4 expressions with local error:

10.7b
(sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2)))))
10.7b
(sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2)))))
10.7b
(sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2)))))
6.0b
(cbrt (/ 1 (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2)))))))

rewrite3.0ms

Algorithm
rewrite-expression-head
Counts
4 → 57
Calls

4 calls. Slowest were:

1.0ms
(cbrt (/ 1 (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)))))

series1.0s

Counts
4 → 12
Calls

4 calls. Slowest were:

351.0ms
(cbrt (/ 1 (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2)))))))
238.0ms
(sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2)))))
236.0ms
(sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2)))))
218.0ms
(sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2)))))

simplify4.4s

Counts
37 → 69
Calls

37 calls. Slowest were:

660.0ms
(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)))))))
262.0ms
(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)))))))
249.0ms
(sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2)))))

prune1.8s

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0.8b

regimes1.2s

Accuracy

0% (0.8b remaining)

Error of 1.5b against oracle of 0.7b and baseline of 1.5b

bsearch4.0ms

end0.0ms

sample3.9s

Algorithm
intervals