Average Error: 31.4 → 12.6
Time: 1.9m
Precision: 64
Internal Precision: 128
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\left(\frac{\frac{2}{t}}{\sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}} \cdot \frac{\frac{1}{\sin k}}{\sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}\right) \cdot \left(\frac{\frac{\frac{\ell}{t}}{\tan k}}{\sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}} \cdot \frac{\frac{\ell}{t}}{\sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}\right)\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.4

    \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

  2. Initial simplification24.5

    \[\leadsto \frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \tan k}}{\frac{k}{t} \cdot \frac{k}{t} + 2}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt24.5

    \[\leadsto \frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \tan k}}{\color{blue}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2} \cdot \sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}\]

  5. Applied times-frac18.3

    \[\leadsto \frac{\color{blue}{\frac{\frac{2}{t}}{\sin k} \cdot \frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2} \cdot \sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}\]

  6. Applied times-frac16.5

    \[\leadsto \color{blue}{\frac{\frac{\frac{2}{t}}{\sin k}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}\]
  7. Using strategy rm

  8. Applied add-sqr-sqrt16.4

    \[\leadsto \frac{\frac{\frac{2}{t}}{\sin k}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\color{blue}{\sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}} \cdot \sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}}\]

  9. Applied *-un-lft-identity16.4

    \[\leadsto \frac{\frac{\frac{2}{t}}{\sin k}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\color{blue}{1 \cdot \tan k}}}{\sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}} \cdot \sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}\]

  10. Applied times-frac13.6

    \[\leadsto \frac{\frac{\frac{2}{t}}{\sin k}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}} \cdot \frac{\color{blue}{\frac{\frac{\ell}{t}}{1} \cdot \frac{\frac{\ell}{t}}{\tan k}}}{\sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}} \cdot \sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}\]

  11. Applied times-frac12.7

    \[\leadsto \frac{\frac{\frac{2}{t}}{\sin k}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}} \cdot \color{blue}{\left(\frac{\frac{\frac{\ell}{t}}{1}}{\sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}} \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{\sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}\right)}\]

  12. Simplified12.7

    \[\leadsto \frac{\frac{\frac{2}{t}}{\sin k}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}} \cdot \left(\color{blue}{\frac{\frac{\ell}{t}}{\sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}} \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{\sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}\right)\]
  13. Using strategy rm

  14. Applied add-sqr-sqrt12.7

    \[\leadsto \frac{\frac{\frac{2}{t}}{\sin k}}{\sqrt{\color{blue}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2} \cdot \sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}} \cdot \left(\frac{\frac{\ell}{t}}{\sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}} \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{\sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}\right)\]

  15. Applied sqrt-prod12.7

    \[\leadsto \frac{\frac{\frac{2}{t}}{\sin k}}{\color{blue}{\sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}} \cdot \sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}} \cdot \left(\frac{\frac{\ell}{t}}{\sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}} \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{\sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}\right)\]

  16. Applied div-inv12.7

    \[\leadsto \frac{\color{blue}{\frac{2}{t} \cdot \frac{1}{\sin k}}}{\sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}} \cdot \sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}} \cdot \left(\frac{\frac{\ell}{t}}{\sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}} \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{\sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}\right)\]

  17. Applied times-frac12.6

    \[\leadsto \color{blue}{\left(\frac{\frac{2}{t}}{\sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}} \cdot \frac{\frac{1}{\sin k}}{\sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}\right)} \cdot \left(\frac{\frac{\ell}{t}}{\sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}} \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{\sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}\right)\]

  18. Final simplification12.6

    \[\leadsto \left(\frac{\frac{2}{t}}{\sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}} \cdot \frac{\frac{1}{\sin k}}{\sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}\right) \cdot \left(\frac{\frac{\frac{\ell}{t}}{\tan k}}{\sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}} \cdot \frac{\frac{\ell}{t}}{\sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}\right)\]

Reproduce

herbie shell --seed 2018362 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  (/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))))

Details

Time bar (total: 1.4m)Debug log

start718.0ms

Algorithm
intervals

setup216.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 21.3b

localize38.0ms

Local error

Found 4 expressions with local error:

13.7b
(/ (/ (* (/ 2 t) (* (/ l t) (/ l t))) (* (sin k) (tan k))) (+ (* (/ k t) (/ k t)) 2))
10.8b
(/ (* (/ 2 t) (* (/ l t) (/ l t))) (* (sin k) (tan k)))
0.8b
(* (/ 2 t) (* (/ l t) (/ l t)))
0.2b
(* (/ l t) (/ l t))

rewrite42.0ms

Algorithm
rewrite-expression-head
Counts
4 → 138
Calls

4 calls. Slowest were:

17.0ms
(/ (/ (* (/ 2 t) (* (/ l t) (/ l t))) (* (sin k) (tan k))) (+ (* (/ k t) (/ k t)) 2))
13.0ms
(/ (* (/ 2 t) (* (/ l t) (/ l t))) (* (sin k) (tan k)))
7.0ms
(* (/ 2 t) (* (/ l t) (/ l t)))

series562.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

289.0ms
(/ (* (/ 2 t) (* (/ l t) (/ l t))) (* (sin k) (tan k)))
214.0ms
(/ (/ (* (/ 2 t) (* (/ l t) (/ l t))) (* (sin k) (tan k))) (+ (* (/ k t) (/ k t)) 2))
37.0ms
(* (/ 2 t) (* (/ l t) (/ l t)))
22.0ms
(* (/ l t) (/ l t))

simplify24.1s

Counts
129 → 150
Calls

129 calls. Slowest were:

1.2s
(/ (/ (* (* (* (/ 2 t) (/ 2 t)) (/ 2 t)) (* (* (* (/ l t) (/ l t)) (/ l t)) (* (* (/ l t) (/ l t)) (/ l t)))) (* (* (* (sin k) (tan k)) (* (sin k) (tan k))) (* (sin k) (tan k)))) (* (* (+ (* (/ k t) (/ k t)) 2) (+ (* (/ k t) (/ k t)) 2)) (+ (* (/ k t) (/ k t)) 2)))
936.0ms
(* (* (* (/ l t) (/ l t)) (/ l t)) (* (* (/ l t) (/ l t)) (/ l t)))
755.0ms
(/ (/ (* (* (* (/ 2 t) (* (/ l t) (/ l t))) (* (/ 2 t) (* (/ l t) (/ l t)))) (* (/ 2 t) (* (/ l t) (/ l t)))) (* (* (* (sin k) (tan k)) (* (sin k) (tan k))) (* (sin k) (tan k)))) (* (* (+ (* (/ k t) (/ k t)) 2) (+ (* (/ k t) (/ k t)) 2)) (+ (* (/ k t) (/ k t)) 2)))

prune2.3s

Pruning

15 alts after pruning (15 fresh and 0 done)

Merged error: 10.7b

localize18.0ms

Local error

Found 4 expressions with local error:

11.3b
(/ (/ (/ 2 t) (sin k)) (sqrt (+ (* (/ k t) (/ k t)) 2)))
9.6b
(/ (/ (* (/ l t) (/ l t)) (tan k)) (sqrt (+ (* (/ k t) (/ k t)) 2)))
7.1b
(sqrt (+ (* (/ k t) (/ k t)) 2))
7.1b
(sqrt (+ (* (/ k t) (/ k t)) 2))

rewrite73.0ms

Algorithm
rewrite-expression-head
Counts
4 → 244
Calls

4 calls. Slowest were:

24.0ms
(/ (/ (* (/ l t) (/ l t)) (tan k)) (sqrt (+ (* (/ k t) (/ k t)) 2)))
20.0ms
(/ (/ (/ 2 t) (sin k)) (sqrt (+ (* (/ k t) (/ k t)) 2)))
13.0ms
(sqrt (+ (* (/ k t) (/ k t)) 2))

series381.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

188.0ms
(/ (/ (* (/ l t) (/ l t)) (tan k)) (sqrt (+ (* (/ k t) (/ k t)) 2)))
117.0ms
(/ (/ (/ 2 t) (sin k)) (sqrt (+ (* (/ k t) (/ k t)) 2)))
39.0ms
(sqrt (+ (* (/ k t) (/ k t)) 2))
37.0ms
(sqrt (+ (* (/ k t) (/ k t)) 2))

simplify22.3s

Counts
362 → 256
Calls

362 calls. Slowest were:

999.0ms
(sqrt (+ (pow (* (/ k t) (/ k t)) 3) (pow 2 3)))
904.0ms
(sqrt (+ (pow (* (/ k t) (/ k t)) 3) (pow 2 3)))
655.0ms
(/ (* (* (/ (/ 2 t) (sin k)) (/ (/ 2 t) (sin k))) (/ (/ 2 t) (sin k))) (* (* (sqrt (+ (* (/ k t) (/ k t)) 2)) (sqrt (+ (* (/ k t) (/ k t)) 2))) (sqrt (+ (* (/ k t) (/ k t)) 2))))

prune5.1s

Pruning

13 alts after pruning (13 fresh and 0 done)

Merged error: 9.9b

localize35.0ms

Local error

Found 4 expressions with local error:

11.3b
(/ (/ (/ 2 t) (sin k)) (sqrt (+ (* (/ k t) (/ k t)) 2)))
10.6b
(sqrt (sqrt (+ (* (/ k t) (/ k t)) 2)))
10.6b
(sqrt (sqrt (+ (* (/ k t) (/ k t)) 2)))
7.1b
(sqrt (+ (* (/ k t) (/ k t)) 2))

rewrite26.0ms

Algorithm
rewrite-expression-head
Counts
4 → 185
Calls

4 calls. Slowest were:

10.0ms
(/ (/ (/ 2 t) (sin k)) (sqrt (+ (* (/ k t) (/ k t)) 2)))
4.0ms
(sqrt (sqrt (+ (* (/ k t) (/ k t)) 2)))
4.0ms
(sqrt (sqrt (+ (* (/ k t) (/ k t)) 2)))

series455.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

164.0ms
(sqrt (sqrt (+ (* (/ k t) (/ k t)) 2)))
135.0ms
(/ (/ (/ 2 t) (sin k)) (sqrt (+ (* (/ k t) (/ k t)) 2)))
118.0ms
(sqrt (sqrt (+ (* (/ k t) (/ k t)) 2)))
38.0ms
(sqrt (+ (* (/ k t) (/ k t)) 2))

simplify15.0s

Counts
258 → 197
Calls

258 calls. Slowest were:

1.2s
(sqrt (sqrt (+ (pow (* (/ k t) (/ k t)) 3) (pow 2 3))))
889.0ms
(sqrt (+ (pow (* (/ k t) (/ k t)) 3) (pow 2 3)))
885.0ms
(sqrt (sqrt (+ (pow (* (/ k t) (/ k t)) 3) (pow 2 3))))

prune4.0s

Pruning

12 alts after pruning (11 fresh and 1 done)

Merged error: 9.9b

localize17.0ms

Local error

Found 4 expressions with local error:

10.6b
(sqrt (sqrt (+ (* (/ k t) (/ k t)) 2)))
10.6b
(sqrt (sqrt (+ (* (/ k t) (/ k t)) 2)))
10.6b
(sqrt (sqrt (+ (* (/ k t) (/ k t)) 2)))
10.6b
(sqrt (sqrt (+ (* (/ k t) (/ k t)) 2)))

rewrite23.0ms

Algorithm
rewrite-expression-head
Counts
4 → 72
Calls

4 calls. Slowest were:

8.0ms
(sqrt (sqrt (+ (* (/ k t) (/ k t)) 2)))
5.0ms
(sqrt (sqrt (+ (* (/ k t) (/ k t)) 2)))
4.0ms
(sqrt (sqrt (+ (* (/ k t) (/ k t)) 2)))

series555.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

151.0ms
(sqrt (sqrt (+ (* (/ k t) (/ k t)) 2)))
142.0ms
(sqrt (sqrt (+ (* (/ k t) (/ k t)) 2)))
136.0ms
(sqrt (sqrt (+ (* (/ k t) (/ k t)) 2)))
126.0ms
(sqrt (sqrt (+ (* (/ k t) (/ k t)) 2)))

simplify7.4s

Counts
52 → 84
Calls

52 calls. Slowest were:

922.0ms
(sqrt (sqrt (+ (pow (* (/ k t) (/ k t)) 3) (pow 2 3))))
904.0ms
(sqrt (sqrt (+ (pow (* (/ k t) (/ k t)) 3) (pow 2 3))))
843.0ms
(sqrt (sqrt (+ (pow (* (/ k t) (/ k t)) 3) (pow 2 3))))

prune1.9s

Pruning

12 alts after pruning (10 fresh and 2 done)

Merged error: 9.9b

regimes389.0ms

Accuracy

0% (1.5b remaining)

Error of 12.6b against oracle of 11.1b and baseline of 12.6b

bsearch5.0ms