Average Error: 1.7 → 0.8
Time: 43.8s
Precision: 64
Internal Precision: 128
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;{\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \le 2.7530406583153254 \cdot 10^{+294}:\\ \;\;\;\;\sqrt{\left(1 + \frac{\frac{1}{\left(\sqrt[3]{\sqrt[3]{\sqrt{\left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) \cdot {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} + 1}}} \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{\left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) \cdot {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} + 1}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{\left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) \cdot {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} + 1}}}\right)\right) \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{\left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) \cdot {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} + 1}}} \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{\left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) \cdot {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} + 1}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{\left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) \cdot {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} + 1}}}\right)\right)}}{\sqrt[3]{\sqrt[3]{\sqrt{\left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) \cdot {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} + 1}}} \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{\left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) \cdot {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} + 1}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{\left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) \cdot {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} + 1}}}\right)}\right) \cdot \frac{1}{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{2} \cdot \left(\frac{1}{\sqrt{1 + \left(\left(\ell \cdot \ell\right) \cdot 4\right) \cdot (\left(\frac{kx}{Om}\right) \cdot \left(\frac{kx}{Om}\right) + \left(\frac{ky}{Om} \cdot \frac{ky}{Om}\right))_*}} + 1\right)}\\ \end{array}\]

Error

Bits error versus l

Bits error versus Om

Bits error versus kx

Bits error versus ky

Derivation

  1. Split input into 2 regimes

  2. if (pow (/ (* 2 l) Om) 2) < 2.7530406583153254e+294

    1. Initial program 0.8

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

    3. Applied add-cube-cbrt0.8

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

    4. Applied associate-/r*0.8

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

    6. Applied add-cube-cbrt0.8

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

    8. Applied add-cube-cbrt0.8

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

    10. Applied add-cube-cbrt0.8

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

    if 2.7530406583153254e+294 < (pow (/ (* 2 l) Om) 2)

    1. Initial program 3.8

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

    2. Taylor expanded around 0 24.6

      \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + \color{blue}{\left(4 \cdot \frac{{kx}^{2} \cdot {\ell}^{2}}{{Om}^{2}} + 4 \cdot \frac{{ky}^{2} \cdot {\ell}^{2}}{{Om}^{2}}\right)}}}\right)}\]

    3. Simplified0.9

      \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + \color{blue}{\left(\left(\ell \cdot \ell\right) \cdot 4\right) \cdot (\left(\frac{kx}{Om}\right) \cdot \left(\frac{kx}{Om}\right) + \left(\frac{ky}{Om} \cdot \frac{ky}{Om}\right))_*}}}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \le 2.7530406583153254 \cdot 10^{+294}:\\ \;\;\;\;\sqrt{\left(1 + \frac{\frac{1}{\left(\sqrt[3]{\sqrt[3]{\sqrt{\left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) \cdot {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} + 1}}} \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{\left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) \cdot {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} + 1}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{\left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) \cdot {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} + 1}}}\right)\right) \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{\left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) \cdot {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} + 1}}} \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{\left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) \cdot {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} + 1}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{\left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) \cdot {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} + 1}}}\right)\right)}}{\sqrt[3]{\sqrt[3]{\sqrt{\left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) \cdot {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} + 1}}} \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{\left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) \cdot {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} + 1}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{\left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) \cdot {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} + 1}}}\right)}\right) \cdot \frac{1}{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{2} \cdot \left(\frac{1}{\sqrt{1 + \left(\left(\ell \cdot \ell\right) \cdot 4\right) \cdot (\left(\frac{kx}{Om}\right) \cdot \left(\frac{kx}{Om}\right) + \left(\frac{ky}{Om} \cdot \frac{ky}{Om}\right))_*}} + 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019008 +o rules:numerics
(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: 42.9s)Debug log

sample118.0ms

Algorithm
intervals

simplify277.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

277.0ms
(sqrt (* (/ 1 2) (+ 1 (/ 1 (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2)))))))))

prune13.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 1.9b

localize61.0ms

Local error

Found 4 expressions with local error:

8.0b
(sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2)))))
2.5b
(* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2)))
0.8b
(/ 1 (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))
0.3b
(pow (sin ky) 2)

rewrite38.0ms

Algorithm
rewrite-expression-head
Counts
4 → 87
Calls

4 calls. Slowest were:

26.0ms
(* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2)))
4.0ms
(sqrt (+ 1 (* (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))))))

series844.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

315.0ms
(sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2)))))
314.0ms
(/ 1 (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))
189.0ms
(* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2)))
25.0ms
(pow (sin ky) 2)

simplify6.8s

Counts
53 → 99
Calls

53 calls. Slowest were:

581.0ms
(sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2)))))
525.0ms
(* (pow (sin ky) 2) (pow (/ (* 2 l) Om) 2))
510.0ms
(* (pow (/ (* 2 l) Om) 2) (pow (sin ky) 2))

prune2.1s

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.7b

localize32.0ms

Local error

Found 4 expressions with local error:

10.5b
(cbrt (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))
10.5b
(cbrt (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))
10.5b
(cbrt (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))
8.0b
(sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2)))))

rewrite8.0ms

Algorithm
rewrite-expression-head
Counts
4 → 70
Calls

4 calls. Slowest were:

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

series1.3s

Counts
4 → 12
Calls

4 calls. Slowest were:

455.0ms
(cbrt (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))
328.0ms
(cbrt (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))
321.0ms
(cbrt (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))
228.0ms
(sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2)))))

simplify5.5s

Counts
42 → 82
Calls

42 calls. Slowest were:

293.0ms
(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))))))))
284.0ms
(sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2)))))
284.0ms
(cbrt (* (cbrt (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2)))))) (cbrt (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))))

prune2.1s

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.7b

localize45.0ms

Local error

Found 4 expressions with local error:

10.5b
(cbrt (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))
10.5b
(cbrt (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))
10.5b
(cbrt (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))
10.5b
(cbrt (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))

rewrite4.0ms

Algorithm
rewrite-expression-head
Counts
4 → 72
Calls

4 calls. Slowest were:

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

series1.4s

Counts
4 → 12
Calls

4 calls. Slowest were:

377.0ms
(cbrt (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))
369.0ms
(cbrt (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))
358.0ms
(cbrt (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))
335.0ms
(cbrt (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))

simplify5.8s

Counts
44 → 84
Calls

44 calls. Slowest were:

306.0ms
(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))))))))
287.0ms
(cbrt (sqrt (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2)))))))
280.0ms
(cbrt (sqrt (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2)))))))

prune2.3s

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.7b

localize41.0ms

Local error

Found 4 expressions with local error:

10.5b
(cbrt (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))
10.5b
(cbrt (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))
10.5b
(cbrt (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))
10.5b
(cbrt (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))

rewrite5.0ms

Algorithm
rewrite-expression-head
Counts
4 → 72
Calls

4 calls. Slowest were:

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

series1.2s

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))))))
310.0ms
(cbrt (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))
306.0ms
(cbrt (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))
286.0ms
(cbrt (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))

simplify5.4s

Counts
44 → 84
Calls

44 calls. Slowest were:

650.0ms
(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))))))))
273.0ms
(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))))))))
262.0ms
(cbrt (sqrt (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2)))))))

prune2.3s

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.7b

regimes1.4s

Accuracy

90.2% (0.1b remaining)

Error of 0.8b against oracle of 0.8b and baseline of 1.7b

bsearch13.0ms

end0.0ms

sample3.8s

Algorithm
intervals