Average Error: 1.7 → 1.7
Time: 24.7s
Precision: 64
Internal Precision: 128
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\left(\frac{\frac{1}{\sqrt{\sqrt[3]{\left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} + 1} \cdot \sqrt[3]{\left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} + 1}}}}{\sqrt{\sqrt[3]{\left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} + 1}}} + 1\right) \cdot \frac{1}{2}}\]

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 add-cube-cbrt1.7

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

  4. Applied sqrt-prod1.7

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

  5. Applied associate-/r*1.7

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

  6. Final simplification1.7

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

Reproduce

herbie shell --seed 2019016 +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: 23.7s)Debug log

sample168.0ms

Algorithm
intervals
Results
256×(pre true 80)
256×(body real 80)

simplify716.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
715.0ms
(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: 1.3b

localize98.0ms

Local error

Found 4 expressions with local error:

8.7b
(sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2)))))
2.3b
(* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2)))
0.4b
(/ 1 (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))
0.3b
(pow (sin kx) 2)

rewrite20.0ms

Algorithm
rewrite-expression-head
Rules
15×*-un-lft-identity
12×add-sqr-sqrt
11×add-cube-cbrt
11×add-exp-log
9×unpow-prod-down
8×associate-*l*
6×sqrt-prod
6×add-cbrt-cube
6×associate-/r*
6×pow1
5×associate-*r*
4×add-log-exp
4×log1p-expm1-u
4×sqrt-div
4×expm1-log1p-u
3×flip-+
3×prod-exp
3×flip3-+
2×unpow2
2×distribute-lft-out
2×pow-flip
2×div-inv
2×associate-/r/
2×associate-*r/
2×pow-exp
2×pow1/2
2×pow-to-exp
1×inv-pow
1×cbrt-unprod
1×*-commutative
1×distribute-lft-in
1×frac-2neg
1×pow-pow
1×clear-num
1×rec-exp
1×rem-sqrt-square
1×distribute-rgt-in
Counts
4 → 87
Calls
4 calls:
Slowest
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))))))
1.0ms
(pow (sin kx) 2)

series740.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
323.0ms
(/ 1 (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))
237.0ms
(sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2)))))
161.0ms
(* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2)))
17.0ms
(pow (sin kx) 2)

simplify6.7s

Counts
53 → 99
Calls
53 calls:
Slowest
634.0ms
(* (pow (sin ky) 2) (pow (/ (* 2 l) Om) 2))
547.0ms
(* (pow (/ (* 2 l) Om) 2) (pow (sin ky) 2))
419.0ms
(sqrt (- 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2)))))
309.0ms
(sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2)))))
287.0ms
(sqrt (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))

prune2.1s

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.6b

localize45.0ms

Local error

Found 4 expressions with local error:

13.1b
(cbrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2)))))
13.1b
(cbrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2)))))
13.1b
(cbrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2)))))
7.3b
(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)))))))

rewrite11.0ms

Algorithm
rewrite-expression-head
Rules
18×cbrt-div
9×flip-+
9×flip3-+
9×cbrt-prod
8×sqrt-div
7×add-cube-cbrt
7×*-un-lft-identity
7×add-sqr-sqrt
4×add-log-exp
4×log1p-expm1-u
4×add-exp-log
4×add-cbrt-cube
4×pow1
4×frac-times
4×expm1-log1p-u
3×pow1/3
2×associate-*r/
2×associate-*l/
1×sqrt-prod
1×pow1/2
1×rem-sqrt-square
Counts
4 → 65
Calls
4 calls:
Slowest
6.0ms
(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)))))))
1.0ms
(cbrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2)))))
1.0ms
(cbrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2)))))
1.0ms
(cbrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2)))))

series1.4s

Counts
4 → 12
Calls
4 calls:
Slowest
395.0ms
(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)))))))
352.0ms
(cbrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2)))))
338.0ms
(cbrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2)))))
336.0ms
(cbrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2)))))

simplify4.8s

Counts
37 → 77
Calls
37 calls:
Slowest
315.0ms
(cbrt (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))
269.0ms
(sqrt (cbrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))
265.0ms
(cbrt (* (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)))))))
235.0ms
(cbrt (* (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)))))))
218.0ms
(cbrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2)))))

prune2.1s

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0.6b

localize10.0ms

Local error

Found 0 expressions with local error:

rewrite0.0ms

Algorithm
rewrite-expression-head
Rules
Counts
0 → 0
Calls
0 calls:
Slowest

series0.0ms

Counts
0 → 0
Calls
0 calls:
Slowest

simplify0.0ms

Counts
0 → 0
Calls
0 calls:
Slowest

prune1.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0.6b

regimes751.0ms

Accuracy

0% (0.9b remaining)

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

bsearch4.0ms

end0.0ms

sample4.0s

Algorithm
intervals
Results
8000×(pre true 80)
8000×(body real 80)